introduction to the fourier transform & pseudo-differential operators

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BENT E PETEISEI

Introduction to the Fourier transform & pseudo-differential operators

Main EditorsA. Jeffrey, University of Newcastle-upon-TyneR. G. Douglas, State University of New York at Stony Brook

Editorial BoardF. F. Bonsall, University of EdinburghH. Brezis, Universite de ParisG. Fichera, Universita di RomaR. P. Gilbert, University of DelawareK. Kirchgassner, Universitat StuttgartR. E. Meyer, University of Wisconsin-Madison J. Nitsche, Universitat FreiburgL. E. Payne, Cornell UniversityG. F. Roach, University of StrathclydeI. N. Stewart, University of WarwickS. J. Taylor, University of Liverpool

Introduction to the Fourier Transform & Pseudo -differential Operators

Bent E . PetersenOregon State University

j tPitman A dvanced Publishing Program

Boston • London • Melbourne

PITMAN PUBLISHING LIM ITED 128 Long Acre, London W C2E 9AN

PITMAN PUBLISHING INC1020 Plain Street, Marshfield, Massachusetts 02050

Associated Companies Pitman Publishing Pty Ltd, Melbourne Pitman Publishing New Zealand Ltd, Wellington Copp Clark Pitman, Toronto

First published 1983

© Bent E. Petersen 1983

AMS Subject Classifications: 42B10, 53S05, 47G05

British Library Cataloguing in Publication Data

Petersen, Bent E.Introduction to the Fourier transform & pseudodifferential operators.— (Monographs and studies in mathematics; 19)1. Fourier transforms I. Title II. Series 515.T23 QA403.5

ISBN 0-273-08600-6

Library of Congress Cataloging in Publication Data

Petersen, Bent E.Introduction to the Fourier transform and pseudodifferential operators.(Monographs and studies in mathematics; 19)Bibliography; p.Includes index.1. Pseudo-differential operators; 2. Fourier transformation.I. Title II. Series

QA 329.7.P47 1983 515.T242 83-8083

ISBN 0-273-08600-6

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording and/or otherwise, without the prior written permission of the publishers. This book may not be lent, resold, hired out or otherwise disposed of by way of trade in any form of binding or cover other than that in which it is published, without the prior consent of the publishers.

Printed in Northern Ireland at The Universities Press (Belfast) Ltd.

Contents

Preface ix Acknowledgment x

Chapter 1 Theory of distributions 1

1. Introduction to Chapter 1 12. Convolution of LF functions 43. Regularization 6

4. Complex Borel measures 8

5. Partitions of unity 126. Integration by parts 137. Distributions 158. Restriction and support 219. Differentiation of distributions 23

10. Fundamental solutions of the Laplacian 2511. The Newtonian potential in [R” 2812. Leibniz’ formula. Classical derivatives 30

13. Distributions with point support 3114. Weak derivatives and integration by parts 3715. Distribution valued holomorphic functions 3816. Boundary values of holomorphic functions 4317. Operations on distributions 4618. Convolution of distributions 4719. Oscillatory integrals 58

VI CONTENTS

Chapter 2 The Fourier transform 65

1. Introduction to Chapter 2 652. Fourier transform. O theory 673. V inversion theory 714. Fourier transform. theory 785. Temperate distributions and 6^ 826. Convolution of temperate distributions. The space 0^ 877. The Fourier transform on Sf'. The exchange formulae 918. The Fourier transform on The Paley-Wiener

theorem 959. Operators defined by the Fourier transform 106

10. Homogeneous distributions 11911. Periodic distributions and Fourier series 12212. Laplace transform 12813. The wave front set of a distribution 145

Chapter 3 Pseudo-differential operators 161

1. Introduction to Chapter 3 1612. Pseudo-differential operators 1663. Smoothing operators and properly supported operators 1784. Operators of the form a'(X, - iD , X) 1855. Transpose and composition 1926. Classical pseudo-differential operators 1987. Invariance of pseudo-differential operators 2048. The pseudo-local property 208

9. Characteristics. The regularity theorem 212

Chapter 4 Hflbert space methods 225

1. Introduction to Chapter 4 2252. Sobolev spaces 228

CONTENTS Vll

3. Operators on Sobolev spaces 2374. continuity of pseudo-differential operators 2475. Local Sobolev spaces 2556. The wave front set 2597. Subellipticity and local existence 2658. Appendix. The Seidenberg-Tarski theorem 293

Chapter 5 Garding’s inequality 305

1. Introduction to Chapter 5 3052. The spaces 3063. The Dirichlet problem 3134. Discussion of Garding’s inequality 3225. Generalized Dirichlet forms 3256. The Friedrichs’ symmetrization 3297. Propagation of singularities 335

Bibliography 346

Index 353

Preface

This book is an introduction to the Fourier transform and to the theory of pseudo-differential operators. As a text it is intended to be used at the second year graduate level. However, care has been taken to keep the text reasonably accessible. Thus large parts of it may profitably be used as a supplement for a first year course in functional analysis.

Chapter 1 presents Schwartz’ theory of distributions. It is possible to cover much ground without invoking the theory of topological vector spaces. One then misses, however, the chance to illustrate some of the basic theorems of functional analysis. Moreover, eventually the functional analysis becomes well-nigh indispensable. Therefore in Section 7, prior to considering distributions, we give a very brief introduction to the theory of locally convex spaces. The reader familiar with Banach spaces should have little difficulty making the transition to the more general setting. In Chapter 2 we continue the study of distributions and develop the theory of the Fourier transform on the space of temperate distributions.

The remainder of the book is given over to pseudo-differential operators. In Chapter 3 we construct an operational calculus for the (non-commuting) operators of multiplication by the coordinate functions and differentiation. The Fourier transform is the major tool here. The resulting operators are the pseudo-differential operators. Chapter 4 concerns the continuity of pseudo-differential operators on Sobolev spaces. In Chapter 5 we first discuss the Dirichlet problem in an setting and Garding’s inequality. We then give a proof of the sharp Garding inequality. This result leads to a theorem on propagation of singularities of solutions of pseudo-differential equations. This result together with the results of Chapter 4 then leads, for example, to a local existence theorem for operators with real principal symbol and simple real characteristics.

Each chapter begins with an introduction which gives a detailed description of the contents of the chapter and, in some cases, some historical remarks. Additional historical remarks and some exercises are scattered throughout the text.

IX

Acknowledgment

This text is prepared from lecture notes for courses the author has given at Oregon State University. The students in these courses gave me many useful comments for which I am very grateful. I also wish to express my gratitude to Oregon State University for providing me with a sabbatical leave to complete a major portion of the manuscript. Finally I wish to thank my wife Marguerite for her patient support, her good-natured tolerance of my late night typing, and for many other things.

Corvallis, Oregon Bent E. Petersen

Til min kone Marguerite.

‘Jeg fik det gjort.’

XI

Chapter 1

Theory of distributions

§1. Introduction to Chapter 1

This chapter is an introduction to Schwartz’ theory of distributions. In Sections 2 through 4 we review convolution of functions and measures. In Sections 5 and 6 we introduce two basic tools; partitions of unity and integration by parts. In Sections 7 through 9 we study distributions and their basic properties. Distributions occur quite naturally even in applied problems. For example let be a unit vector in Euclidean 3-space IR , let t> 0 and consider the electrical dipole consisting of a charge Q at the point to) and a charge - Q at -t(o. We define the moment of the dipole to be p = 2tQ. Associated to the dipole we have an electrical potential given by

= cQ(|x + to)\~ ~

where x g IR , = c is a constant depending on units andthe surrounding medium. If we examine the potential ij/tix) for \x\ very large compared to the separation 2t of the charges we do not expect to be able to detect the charge Q or the separation 2t. Hence we express the potential in terms of the moment p and then consider what happens if we hold p fixed and let t tend to 0. A computation yields

^t(x) = -2pc{(x), x) \x — jx + (jx + tcol + lx - t(o\)~

where (o), x) is the inner product of co and x. Thus if ij/Q = limt^o have

= -pc<6), x) jxl"®

It would be aesthetically satisfying if we could regard iI/q, which after all was obtained in a perfectly reasonable manner, as the potential of some generalized charge distribution. We will now do so, at least formally. If we introduce the Dirac delta ‘function’ 6, which is really a measure, and

1

THEORY OF DISTRIBUTIONS

which we think of as a unit charge at the origin, then our original dipole corresponds to the charge distribution

^ (6(x — toy) — 6(x + to)))

Here we have followed the convention of thinking of 6 as a ‘generalized function’ and have used functional notation. If now we let t —>0 we obtain, at least formally, the infinitesimal dipole with moment p, given by the expression

-p Z

where Dy = d/dXj. While 8 may easily be interpreted as a measure, Dy6 is more difficult to interpret. The theory of distributions enables us to make sense of expressions such as Dj8, and indeed of the above argument. This theory is a powerful tool for handling singularities not only in idealized physical problems but also in the study of linear partial differential equations.

Distributions not only generalize the notions of functions and measures but they have the nice property of having derivatives of all orders. As practice in working with distributional derivatives we obtain the classical formulas for fundamental solutions of the Laplacian in Section 10. For the same reason we briefly consider the Newtonian potential in Section 11. In Sections 12 and 13 we obtain some additional properties of distributions and we show how to obtain Borel’s theorem on Taylor series of C “ functions from the characterization of distributions with point support. In Section 14 we return to integration by parts, this time for distributional derivatives, and indicate how to obtain Heisenberg’s uncertainty principle. In Section 15 we consider distributions which depend holomorphically on a parameter. We carry out some explicit analytic continuations and develop the connection with Cauchy principal values and Hadamard finite parts of divergent integrals. These computations give us a number of non-trivial examples of distributions. In addition we obtain once again the fundamental solutions of the Laplacian. Section 16 is just an example; we indicate how distributions may arise as boundary values of holomorphic functions. In Section 17 we show how various operations on functions may be extended to distributions by duality. In Section 18 we extend the notion of convolution to distributions satisfying a support condition. We also give a discussion of fundamental solutions of partial differential operators and systems of partial differential operators with constant coefficients. This discussion is not very deep and is intended only to illustrate the properties of the convolution product. We also illustrate how the notion of convolution of distributions may be used to make some sense of the example treated formally above.

Finally, in Section 19 we give an example of an important class of

1. INTRODUCTION TO CHAPTER 1

distributions, the oscillatory integrals. These distributions occur in optics, acoustics, and other areas. Consider, for example, the homogeneous wave equation in

d U— 7 —An = 0

where we have written the coordinate functions in as (x, t) withX = (x i , . . . , x„) and where A = Y,d^ldx^ is the Laplacian in IR”. If we set

w(x, 0 = 1 d^

and differentiate under the integral sign we see that we have a solution of the wave equation. The difficulty here is that in general one needs to consider this integral for functions a which are not integrable. Under suitable hypotheses, however, the fact that the integrand oscillates rapidly for large \\ will cause enough cancellation that we are able to make sense of the integral provided we interpret w as a distribution.

Above we introduced the notation Dj =dldXj where x^,. . . , x are the coordinate functions in IR”. In addition to this notation we will use the following notations. Letters from the beginning of the Greek alphabet will usually indicate multi-indexes. We say a is a multi-index or n multi-index if o: = ( « ! , . . . , a„) where each a, is a non-negative integer. We set D “ = D “> • • • where D f = Dy • • • Dy {h times) and we set x" = x“‘ • • • x“". If X is a point in IR” then jx] = (Y whereas if a isa multi-index then These notations are somewhat ambiguous

( f7l\jx "

where the sum is over lo:l<m and where the coefficients are defined by

( YYl\j = m!/(m — jal)! a ! where a ! = • 0!n

If ft is an open subset of R ” we denote by L^(ft, loc) the space of complex valued functions in ft which are integrable on each compact subset of ft. If /g L^ft, loc) the support of /, denoted by supp /, is defined to be the complement in ft of the largest open subset of ft on which f

vanishes almost everywhere. It is easy to see that this definition makes sense. Additional notation will be introduced as it is needed.

Throughout the chapter we review facts from functional analysis, not necessarily in the order in which one would prove them. While it is possible to develop quite a bit of distribution theory without much functional analysis, such an approach would pass up the chance to illustrate some of the main results of functional analysis in a concrete setting. In addition we will also need a fair amount of functional analysis in later chapters. With a few exceptions, for example, the Hahn-Banach

theorem and basic facts about Hilbert space, most of the functional analysis that we eventually use is reviewed in this chapter.

§2. Convolution of functions

Two important techniques in partial differential equations are convolution and integration by parts. We will begin by considering convolution in the setting. Recall Minkowski’s inequality; if a , b e F then l|a + h||p ll< llp+ll llp- More generally, if a ^ e F and Zmlkmllp<°° then the series Xm converges in F and

t J| ^Il|a„,llp." m II p m

That is, if • • ■) thenp\ ^ / \ ^/p

Z Z«mk/ \ *

^ Z ll la m k n

If we replace these sums by integrals we obtain the following lemma which obviously is true in more general measure spaces.

Lemma 2.1 Minkowski inequality. Let LI be an open subset o f R ” and /et 1 < p < 00. Suppose <p is a measurable complex valued function on Ll^Ll. Then

(L 1 “1 (1That is, whenever the integral on the right is finite then the integral on the left exists and the inequality holds.

Proof. First suppose p = 1 and JnJn l<p( , y)l dx dy <oo. Then by Fu- bini’s theorem y) dy exists for almost all x and

j <p(x, y) dyj dx 1 and

C = | Q y)!” dx dy<oo

Then Jnl<pU y)!'’ dx<oofor almost all y. If (l/p) + (l/q) = 1 and geL^(n) then by Holder’s inequality we have

lg(x)<p(x, y)l dxsllgm l<p(x, y)!'’ dx^

2. CONVOLUTION OF FUNCTIONS

and therefore

f f lg(x)<p(x, y)l dx dy < C llglL (1)

By Fubini’s theorem it follows that Jn |g(x)<p(x, y)| dy <oo for almost all x. Since geL^ iO ) is arbitrary Jn l<p(x, y)j dy <<» for almost all x whence In y) dy exists for almost all x. Moreover, Fubini’s theorem and (1) imply IJn g(x)fo<p(x, y)dy d x]< C Hgl|<,. Since q<oo the dual of L f is U . Hence there exists h eL F {(l) , l|hllp:sC, such that

f g (x)f <p(x, y) dy dx = I g(x)h(x)dx. g € L ‘'(ft).Jn Jn Jn

It follows that h{x) = id <p{x, y)dy for almost all x and

(1II »<>•>’>as required.

\ i/P:) =lNlp^<

P \ 1/Pdx

Theorem 2.2 Young's inequality. Let l’ gllr^ll/lUlgll.

The function f * g is called the convolution of f and g. Note that f * g = g * f -

Proof. First consider the case r = In this case (l/p) + (l/q) = l and hence by Holder’s inequality I|/(x-y)g(y)|dy<H/|lp HgHq. Thus / *g (x ) exists for each x and is bounded. Next consider the case l< r< o o . Note p < r and q < r. Let s = p(l —(1/q)) = l-(p/r) so 0 < s < 1. Let t = (r/q) so l< f< o o , and let q' be defined by (1/q) + (1/q') = 1, so K q '< o o . Now let

= I|/(x - y)g(y)l dy = I|/(x - y)P”'‘ lg(y)l |/(x - y)!* dy

Then by Holder’s inequality

h (x )< {| l/ (x -y )r-''-| g (y )l- d y }’'"!! \f\ H,,

If s = 0 then q = 1 and if s / 0 then sq' = p. In either case taking the qth

power above we obtain

h ixT ^11/lirj l/( - l8(y)l" dy

Then by Lemma 2.1

ll/tlfe = llh1l.^|l/P (I( j| / (x -y )r -^ > ‘-lg(y)l‘' dy)' dx)

(| l/ (x -y )r-^ '‘'Mg(y)r d x )’"dy

= ll/llrllgll^Wl=s^^,

1/t

But qt = r and (1 —s)r = p. Taking qth roots we obtain

ii iii ii/irpiigiui/iir=ii/iip iigii.In particular h{x) is finite for almost all x and therefore J/ (x -y )g (y ) dy exists for almost all x. Moreover, 1/* g (x)l< J |/(x - y)l lg(y)| dy = h(x) implies / * g g L*'([R ) and \\f * gH ÛhlU

Corollary 2.3. With respect to convolution, L ((R”) is a commutative Banach algebra and if l< p < o o then LF(W^) is an 0{W^)-module.

Proof. If f , g e L (1R”) and h e UiW^) then (/ * g) * h = / * (g * h) by Fubini’s theorem.

Exercise 2.4. I f (l/p) + (l/q) = 1 and fe L F (W ) and gsLîU^) then f g is a uniformly continuous bounded function on IR”. Hint: if l< r< o o translation is continuous in L‘(IR”).

§3. Regularization

Let Cl be an open subset o f W and let 0 < m <oo. We denote by the space of complex valued functions on ft which are m times continuously differentiable in ft. We denote by C^(ft) the subspace of C'^(ft) consisting of functions which have compact support in ft. A mollifier is a function p g C “(1R”) such that pÔ, the support of p is contained in {x g [R” I lxl< 1} and jp (x )d x ^ l . If p is a mollifier and 8 > 0 we set pe(x) = e'^'pCe'^x). Then the support of p is contained in {x g IR"" \ |x|<s} and Jpg(x)dx = l. As an example of a mollifier we have p(x) = cu{\x\ — l) where c >0 is a suitable constant and w(t) = e * if t <0 and w(t) = 0 if t> 0 .

3. REG ULARIZATIO N

Theorem 3.1. I f p is a mollifier and f e L'CR”, loc) then the convolution

/ * Ps (jc) = I f ix - y)p (y) dy

exists for each x e IR" and each s > 0. Moreover, f * p ^ s C “(IR''), supp f * p is contained in the closed e-neighbourhood o f supp f, and if K is a compact set o f points o f continuity o f f then f* p ^ ~ ^ f uniformly on K as e ^ 0 .

Proof. The first part is obvious, as is the statement concerning the support. If we write / * Pe(x) = J Pe(x-y)/(y) dy then differentiability follows from standard facts concerning differentiation under the integral sign. For the last part recall that a function is uniformly continuous on any compact set of points of continuity. Thus if -rj > 0 there exists 8 > 0 such that if x s K , z€lR" and jx —z l< 8 then 1/ ( x ) - / ( z)1< t). Now

fix) - / * pe (x) = [ i f ix)-f ix-y))pîy)dynyl e

Hence if 0 < e < 6 then

l / (x ) - / *p^ (x ) l< [ l / W - / ( x - y ) l p ^ ( y ) d y < T } L (y)dy = -n J|vlse J

for each x e K.

Corollary 3.2, I f f l is an open subset o f R ” and K is a compact subset o f i l then there exists q> g C“(fi) such that 0 < 0, and let / be the characteristic function of the closed 6-neighbourhood of K. Let 0 < 8 < 6 . Then <p = / * pe € C"(IR'') has support in the closed 26 -neighbourhood of K and so has compact support in fl. Clearly

and cp = 1 on the (6 —s)-neighbourhood of K.

Exercise 3.3. I f f is a function on R ” and define the translationTyf by Tyf{x) = f { x - y ) . Use Lem m a 2.1 to show if gGL^(R”) and p is a mollifier then Hg-g * p llp ^supiîê llg-T„g|lp. Thus if l< p < c o then g * Pe—* g in LFiW') as e ^ O , by continuity o f translation in LFiU’').

A more general result is given in Theorem 3:3.3.

Theorem 3.4. I f l< p < o o and if ft is an open subset o f then Cc(ft) is dense in L^(ft).

Proof. The monotone convergence theorem, for example, implies that the space of functions in L^(ft) with compact support in ft is dense in

8 THEORY OF DISTRIBUTIONS

LF{D). Let let t) > 0 and let p be a mollifier. Choose geW {C l)with compact support K such that \\f- gl|p < rj/2 and let 26 be the distance from K to the boundary of fl. If 0 < e < 6 then g * C “(H), where wehave extended g by 0 outside of fl. By Exercise 3.3 if s is sufficiently small then Hg *p^ ~ gUp < tj/2. Thus ||/- g * p X < h-

Exercise 3.5. It is a standard fact from measure theory that 0^(11) is dense in LF(Cl) if l< p < o o . Use this fact and Theorem 3.1 rather than Exercise 3.3 to deduce Theorem 3.4.

Theorem 3.6. I f f l is an open subset o f R”, if feL^(Ti, \oc) and if J f{x)(p{x) dx = 0 for each <p g (fl) then f = 0 almost everywhere in fl.

Proof. Let co be a relatively compact open subset of f l and let 26 be the distance from co to the boundary of fl. Let K be the closed 6- neighbourhood of co so iC is a compact subset of fl. If ip is the characteristic function of K then i/rf g L^(R”). Now let p be a mollifier. If x e o ) and 0 < e < 6 then

(# ) * Pe (^) = I - y)pe(y) dy

= j/(^ - y)Pe(y) dy = j/(y)Pe(x - y) dy = 0

since x eco and s < 6 imply x — y e K if ly|^e and, therefore, imply Pe( “ y) is in C “(fl) as a function of y. By Exercise 3.3 (ipf) * p ^ in

as 8 -^ 0 . Thus f = 0 almost everywhere in o>. Since f l is a countable union of open sets such as (o the conclusion of the theorem follows.

§4. Complex Borel measures

In this section we review some facts concerning Borel measures. Those proofs which are omitted may be found in many texts; for example, Rudin [1] and Berberian [1]. Let f l be an open subset of R ” and let ^ be the O’-ring of Borel sets in fl. That is, ^ is the o--ring generated by the compact subsets of fl. A (finite) complex Borel measure is a countably additive map p, We denote the vector space of all complex Borelmeasures on f l by SS(fl). If p, G^(fl) we define a (positive finite) measure IP'1 by 1p 1(A) = supX!k-i IM'(A|c)| where A e 0 l and the supremum is taken over all finite disjoint families (A^) of Borel subsets of A. The measure jp l is called the variation of p,. The total variation l|p,H of p, is defined by Hp,|l = lp,l (fl). The total variation is a norm. Relative to this norm ^(fl) is a Banach space.

4. COMPLEX BOREL MEASURES

If /€ L (||ll|) we can make sense out of the integral J /djut and we have |J/djULl<Jl/ldl|ULl. Indeed, for each jULG^(fl) there is a Borel function h such that \h\ = 1 and such that J / dfx = J/h d 1|ll1 for each feO{\^\). This fact may be used to extend Fubini’s theorem to measures in Note if V is any positive Borel measure (not necessarily finite) and if g e V (v ) then ix(A) = ^y^gdv, A e 01 defines fi g If / is a Borel function then fGO(\fi\) if and only if fgGL^{v) and in this case J/dp, = J/g di . Moreover, 1p 1 CA) = Ja Igl A g 01, and the function h for fju satisfies ^g=igi-

By means of stereographic projection we may identify the one-point compactification of (R” with the unit n-sphere S ”. We denote the boundary of n considered as a subset of S'" by doofl. We denote by Co(fl) the space of continuous complex valued functions on ft which when extended by 0 in the complement of ft are continuous functions on S". Such functions are said to vanish at 3ooft. Note a continuous function / on ft is in Co(ft) if and only if for each 8 > 0 the set {x e ft | |/(x)l > e } is compact. If we let BC(ft) be the space of bounded complex valued continuous functions on ft and provide BC(ft) with the supremum norm then BC(ft) is a Banach space and Co(ft) is a closed subspace of BC(ft). Note also that if pGS9(ft) then BC(ft) is contained in L (lp|).

E xercise 4.1. // g€ Co(ft) then g is uniformly continuous on ft.

T h eorem 4.2. R iesz-M arkoff. // p g ^(ft) then / ^ J / dp is a continuous linear functional on Co(ft). This correspondence is an isometric isomorphism o f the Banach space ^ (ft) onto the dual space o f the Banach space Co(ft), where we provide the dual space o f Co(ft) with the usual dual norm.

T h eorem 4.3. R iesz—M arkoff. I f 6 is a linear functional on Q (ft) such that (0 ,/ )> 0 whenever f G Q (ft) is real and non-negative, then there exists a unique (positive) Borel measure v on f l such that (6, f ) = f dv for each fG Q (ft).

The measure v provided by Theorem 4.3 is finite if and only if 6 is continuous relative to the supremum norm.

We have a natural inclusion of L^(ft) in ^(ft) given by associating to f G L^(ft) the complex Borel measure p defined by p(A ) = Ja f M dx. This inclusion is isometric so we may view L^(ft) as a closed subspace of ^(ft). By the Radon-Nikodym theorem L^(ft) is the subspace of SS(ft) consisting of complex Borel measures which are absolutely continuous with respect to Lebesgue measure in ft. If f and g are in L (IR”) then for each h in

Co(IR”) we have

I h {x )f * g(x) dx = Ih(x)|/(x - y)g(y) dy dx

= { { h { x + y)/(x)g(y) dx dy.

This equation shows how we may extend the convolution product to If fjb,vG and h e Co(IR”) then h(x + y) is a bounded continu

ous function of (x, y)e[R^'' and

j j j h ( x + y ) d p , ( x ) d v ( y ) j < |lh|U Hfx|l ||H1

Thus by Theorem 4.2 there exists a unique A e^(lR'') such that

I h(x) d A ( x ) = 11 h ( x + y ) d jL L (x ) d i / ( y ) , h g C q( [ R ' ' ) .

We call A the convolution of jl, and v and write A = ill * v. Clearly —IIM'II ‘ Ikll- The convolution product of measures when restricted to

L (IR”) obviously yields the convolution product we originally defined on L'([R").

Exercise 4.4. ^([R”) is a commutative B anach algebra with respect to the convolution product. Moreover, SS([R'') has an identity, usually denoted by 8 and called the Dirac measure (or Dirac "function') and defined by 6(A) = 1 if A contains the origin and 6(A) = 0 otherwise.

Theorem 4.5. L (IR”) is a closed ideal in the B anach algebra ^(IR”). That is, if feL\U*^) and fxGSS((R”) then f p ieV(W "). Moreover, / /x(x) = J /(x - y) djLt(y) for almost all x.

Proof. We may assume that / is Borel measurable. Let g(x) = J l/(^~y)| d IM'I (y)- Then g (x )> 0 and by Fubini’s theorem

I g(x) dx = j jl/(x - y)l dx d 1 .1 (y) = |1/1|, H|l

It follows that g(x) is finite for almost all x and therefore h(x) = —y)dfi(y) exists for almost all x. Moreover, heL'ClR") and llHlljS

ll/lli 11m.11. Again by Fubini’s theorem if <p € Co(IR") then

I h(x)<p(x) dx = I j<p(x)/(x - y) dM(y) dx

= 1 1 <P(x + y)f(x) dx dM(y)

4. COMPLEX BOREL MEASURES 11

Thus the measure h dx is the convolution of the measures f dx and ju, i.e.,h = / * jUL.

If / is a measurable function on (R"" we say that f is bounded at infinity if there exists a compact set K and a constant M such that l/(x)|<M for almost all x in the complement of K. For locally integrable functions bounded at infinity we have a useful extension of Theorem 4 .5 .

Theorem 4.6. I f /g L (1R”, loc) and f is bounded at infinity and if |UL G then the integral f * jUL(x) = i f { x - y) dfx(y) exists for almost all xand defines a function f loc).

Proof. Let B = { x g 1R” | |x|<jR} and assume |/(x)|<M for almost all x^ B . Then for any compact set K and any yelR” we have Jk 1/U - y)l dx < Jb 1/(x)1 dx + M \K\ = Ck where \K\ is the Lebesgue measure of K. Now if g(x) = J l/(x-y)| d I/ulI (y) then g (x )> 0 and by Fubini’s theorem Jk g M dx = Ir. Jk I/U ~ y)| dx d 1|ul1 (y) < Ck lljutH. Thus / * |ljl(x) exists for almost all x. Moreover, ]/ * tx(x)\ < g(x) implies / * jui is locally integrable.

Lemma 4.7. I f (p g Co(R”) and p is a mollifier then 0 and suppose lcp(x)|<r] if |xl>jR. Then | R + s. Thus <p * vanishes at infinity.

Exercise 4.8. Use Exercise 4.1 to complete the proof o f Lem m a 4.7.

Exercise 4.9. Let p be a mollifier and let Then p, * p g

L^(R") and by Theorem 4.5 and Exercise 3.3 p * Pe —> p if and only if P g L^(R”). Use Lem m a 4.7 to show that for any we have

Pe in the weafc* topology if we regard ^ (R ”) as the dual space o f Co(R").

A Banach space E is reflexive if and only if each strongly closed convex subset of the dual space E' is weak* closed. Since Exercise 4.9 implies that L’ (R'') is not weak* closed in S9(R'') it follows that Cq(R”) is not reflexive. Actually it is obvious that Co(n) is not reflexive since there is a natural inclusion of BC{Cl) in the dual space of S8(fl).

The proof of the next theorem illustrates an important technique of approximation known as ‘cutting and regularizing’. We will encounter this technique several times in the sequel in more elaborate settings.

Theorem 4.10. I f f l is an open subset o f IR" then C *(n) is dense in Coin).

Proof. Let <p6 Co(ft) and let tj> 0. Then K = {x e f t | l<p(x)l > t]/2} is compact. Choose 6 e C“(H) so 0 < 0 < 1 and 6 = 1 on K. Then Ocp e

Cc(lR"') and 6(p — (p\[o <'r\l2. Now let p be a mollifier. Then iS<p)^Pe^6 0 is small enough then iO(p)'^p^e C ”(fl) and WiOcp) * p^-e(p\\„<r}l2.

Corollary 4 .11 . I f pi e SS(fl) and i/ J <p d/x = 0 for each cp e C~(fi) thenjUL = 0.

Corollary 4.12. I f v is a positive Borel measure on f l and if <p dv = 0 for each <p g C “(ft) then v = 0.

Proof. If (1) is an open set in ft and is compact in ft then v\^G^i(o) and so =0 by the previous corollary.

§5. Partitions of unity

Partitions of unity are a useful tool for passing from local results to global results. Recall that an indexed family (AOiei of subsets of a topological space X is locally finite if each x € X has a neighbourhood U such that U n A i is non-empty for only finitely many i e I.

Exercise 5.1. I f (Ai)igj is locally finite then Uiei Aj = Uief Aj and for each compact set K the intersection K H A i is non-empty for at most finitely many i e I.

Theorem 5.2. Let ft be an open subset o f IR” and let (ftj)jej 1 tmy open cover o f ft. Then for each j e J we can choose if/j g C°°(ft) such that

supp i/fy c ft., and such that the family (supp i/ry)yej is locally finite and = 1 for each x in ft.

Proof. Choose an open refinement with refinement maph : I —> J such that (Oi ^fth(i) and is compact for each i e I. Since ft is paracompact by refining once more if necessary we may assume (coi)iej is locally finite. By the shrinking lemma for T4 spaces (see Exercise V, Chapter 5 of Kelley [1]) we may choose open sets o)-, i e l (same index set) such that co-^ for each i e I and such that o)[ is an open cover of_ft. By Corollary 3.2 there exists Oi e Ccio)t) with 0 < < 1 and = 1 on <o[.Let 6 =Ziei By local finiteness of the family of supports 6 e C~(ft) and obviously 1 < 0. Now let i/y = (1/0) Zh(o=j 0i-

6. INTEGRATION BY PARTS 13

Exercise 5.3. Use Exercise 5.1 to verify that the i/f, found above satisfy the conclusion o f Theorem 5.2.

Exercise 5.4. Let Cl be an open subset o f and let g be a locally bounded function in H. Show there exists if/ g C*(fl) such that |g(x)l < ipix) for each x in ft.

Exercise 5.5. Let ft be an open subset o f R ”. I f hsLF{Cl,\oc) there exists (p G C“(ft) such that he~* g LF{Cl).

§6. Integration by parts

In this section we state a version of the divergence theorem and derive a few consequences. A good reference for this material is Loomis and Sternberg [1]. The results presented here form the basis for extending the notion of differentiation to distributions in Section 8. Let ft be an open subset of An open subset 1/ of ft is called a regular domain in ft if there exists a real valued function p g C~(ft) such that U = {xG ft I p (x )> 0}, at/ = {xG ft I p(x) = 0}, and dp^O on dU, where dU is the topological boundary of U relative to ft.

Theorem 6.1. Divergence theorem. Let Y be a continuously differentiable vector field on ft and assume U Hsupp Y is compact, where U and supp Y are computed relative to ft. Then

f (div Y ) dx = - f (Y • p) Idpl"' dSJu dU

where dS is the surface area measure on d U, that is, dS is the Riemannian density corresponding to the Riemannian metric that dU inherits from R".

If X is a continuously differentiable vector field on ft and f e C^ft) and we take Y = fX, since div (fX) = X • / + / div X, we obtain

[ /d ivX d x = - [ ( X - / ) d x - f (X -p)ld p| -'/ d S Ju Ju Jau

if L/nsupp / Hsupp X is compact. This formula is just the integration by parts formula. For example, if g g C^ft) and we take X = gD we obtain

[ /jDfcgdx = - [ gDfc/dx-[ /gDfcP ldp|“* dSJ u J u J d U

if L7 n supp/n supp g is compact.

Exercise 6.2. I f e > 0 and h e C '(U ’') then

[ Dkh dx = f-'lx|<e J s - ' '

and if h e CcCIR") then

a)fch(ea)) do)

I Dkhdx = —s'' M a)fcft(£(t)) d<w* xl>e Js"-'

where S'" is the unit sphere in W" and dco is the usual surface area measure on

Lemma 6.3. Let ft be an open subset o f W" and K a compact subset o f Cl. Then there exists a regular domain U in [R” such that U is compact, K ^ U and U^Cl.

Proof. Choose 6 e C “(H) such that 0 < 0 < 1 and 6 = 1 on K. By Sard’s theorem the critical values of 6 form a set of measure 0 in [0,1]. Hence there exists t, 0 < t < 1, such that t is not a critical value of 0. If we set p(x) = 0 ( x ) - t then peC^'ClR”) and if L/ = {x elR’" | p (x )> 0} then K ^ U , U is compact and U ^ su pp 0 ^Cl. Moreover, if p(x) = 0 then d p (x)^ 0 since t is not a critical value of 6. Clearly p = 0 on d U = U ^ U . For the converse, suppose p(xo) = 0. Since dp(xo) ^ 0 we have Dfcp(xo) ^ 0 for some fc. Thus each neighbourhood of Xq contains points x with p (x )< 0 , whence Xq g U -- U = dU.

Theorem 6.4. I f ft is an open subset o f W" and i f f and g are in C^(ft) and supp f D supp g is a compact subset o f ft then

[ /Dfcgdx = - [ gDfc/dx JQ. Jo

Proof. Let K = supp f n supp g and choose U as in Lemma 6.3. Then

[ (/Ofcg + g W ) dx = [ (/Dfcg + gDfJ) dx

= - f /gD,pldpl-MS = 0JdU

because fg = 0 on dU.

Exercise 6.5. Use a partition o f unity argument rather than Lem m a 6.3 to prove Theorem 6.4. This argument avoids the use o f Sard's theorem.

Discussions of Sard’s theorem may be found for example in Sternberg[1] and Milnor [1].

7. DISTRIBUTIONS 15

Corollary 6.6. Let LI be an open subset o f IR” and let P = Ziai<m where a e I f f , g e C ”(X1) and supp /Hsupp g is a compact subseto f n then fa fPg dx = Jn gP'f dx where

P 'f= I ( - l ) '“'D“( a J )lo£l<m

The partial differential operator P' is called the formal transpose of P.

§7. Distributions

It is quite feasible to discuss distributions without introducing the theory of locally convex topological vector spaces. To do so, however, obscures the fact that spaces of distributions provide many, possibly most, of the important examples in functional analysis. Moreover, in later chapters we will need to use quite a few theorems from functional analysis. Therefore we will use functional analysis concepts and results from the beginning. The books of Treves [1], Yosida [1], and Edwards [1] are for the most part compatible with our terminology and notation.

In principle if one is familiar with Banach spaces then the transition to locally convex spaces presents no conceptual difficulties. None the less, the reader who is unfamiliar with the theory of topological vector spaces may find the ‘review’ below somewhat heavy going. In this case it may be preferable to proceed directly to the definition of distributions following Theorem 7.4, then to Examples 7.5, 7.6, and 7.7, Corollary 7.11 and Example 7.12. If one does not bother too much concerning some details then Chapters 1 and 2 and much of the remainder of the book are quite comprehensible to the reader without a background in topological vector spaces. The student of analysis should, however, master parts of this theory eventually. In the last 30 or so years it has become an indispensable tool.

A brief but very intelligible outline of the theory of locally convex spaces is given in the survey article of Dieudonne [1].

Let E be a complex vector space. A seminorm on E is a non-negative real valued function p on E such that p satisfies the triangle inequality p(x + y) p(x) + p(y) and such that p(ax) = \a\ p{x) for each x and y in E and each complex number a. If S is a family of seminorms on E, the weakest topology on E such that each p in S is continuous, is called the locally convex topology induced by S. The space E provided with this topology is called a locally convex topological vector space. Finite intersections of convex sets of the form { x e E \ p { x ) < s } for p e S and e > 0 provide a base of neighbourhoods of the origin in E. If the intersection of the sets {x e E | p(x) = 0} is {0} then the topology of E is Hausdorff. If E is Hausdorff then it is metrizable if and only if the topology may be induced


by a countable collection of seminorms. If the locally convex topological vector space E is metrizable then it admits a translation invariant metric and, moreover, is complete with respect to one translation invariant metric which induces the topology of E if and only if it is complete with respect to any translation invariant metric which induces the topology. A metrizable complete locally convex topological vector space is called a Frechet space. If Cl is an open subset of IR” and if for each compact subset K of ft we define the seminorm on C(ft) by Pk (/) = supK |/| it is easy to see that the topology induced by the seminorms p may in fact be defined by just a countable number of the p . With this topology C(ft) is a Frechet space. The topology on C(ft) is just the familiar compact-open topology. If m > 0 is an integer we define for each compact subset K of ft a seminorm pK,m t>y PK,m(/) = niaX|o,i m sup^ |D“/|. These seminorms induce the structure of a Frechet space on C ”(ft). The same seminorms with m = 0 , 1 , 2 , . . . induce the structure of a Frechet space on C “(ft). A subset jB of a locally convex topological vector space E is said to be bounded if each continuous seminorm p on E is bounded on B. A locally convex topological vector space in which each lower semicontinuous seminorm is continuous is said to be barrelled. Every Frechet space is barrelled. A barrelled locally convex topological vector space in which each bounded set is relatively compact is called a Montel space. From the mean value theorem and the theorem of Arzela-Ascoli one sees that C ”(ft) is a Montel space. If ft is an open subset of C ' and A (ft) is the space of holomorphic functions in ft then A (ft) is a closed subspace of C(ft) and so is a Frechet space relative to the compact-open topology. It is an important theorem of Montel that A (ft) is a Montel space. Note that a Banach space is a Montel space if and only if it is finite dimensional.

Let E be a locally convex Hausdorff topological vector space and let E ' be the dual space of E, that is, the space of continuous linear functionals on E. Since several different locally convex topologies on E may give rise to the same dual space E' it is important to identify those features of E which depend only on E ' and not on the topology. In this connection we have Mackey’s theorem that a subset B of E is bounded if and only if each f in E ' is bounded on B. We also have Mazur’s theorem that if A is a convex subset of E then the closure of A is the intersection of those half-spaces { x e E \ f ( x ) + f { x ) > k } , f e E ' , \ g R, which contain A. There are a number of topologies on E and E' in addition to the original topology on E which are important. We will use only a few of these. The weak* topology on E ' is defined by the seminorms Px(f) = \fM\, x e E . The strong topology on E ' is defined by the seminorms Psif) = supxeB l/( )l where B runs over the bounded sets in E. The weak topology on E is defined by the seminorms Pf(x) = l/(x)l, f e E \ The strong topology on E is defined by the seminorms Pb M = sup^ e 1/( )1 where B runs over the weak* bounded sets in E'. Note that the dual of E when E is

7. DISTRIBUTIONS 17

provided with its weak topology is still E' so weak bounded and bounded sets coincide and the weak closure of a convex set is just its closure. If we provide E' with the weak* topology its dual is a vector space canonically isomorphic with E via the evaluation map. Moreover, the original topology on E is induced by the seminorms P oM = supyr G where G runs over the equicontinuous subsets of E', If we provide E' with the strong topology its dual space E" is called the bidual of E. There is a canonical inclusion of E in E'\ again via the evaluation map. If E = E" we say that E is semirefiexive. Since E" is the dual of E' with the strong topology we also have the notion of the strong topology on E", defined in terms of the strongly bounded subsets of E '. If E is semireflexive and the strong topology on E" coincides with the original topology on E, we say that E is reflexive. E is barrelled if and only if the strong topology on E coincides with the original topology. E is reflexive if and only if E is semireflexive and barrelled. If E is a Montel space then E is reflexive and E' is also a Montel space relative to the strong topology. If E is a Banach space the strong topologies on E' and on E " are just the usual norm topologies.

Let E be a vector space and let E^ be a strictly increasing sequence of subspaces of E with union E. Assume each E^ is a Frechet space, E^ is a closed subspace of E^+i, and the topology of E^ coincides with the topology induced from E^+i. If we provide E with the strongest locally convex topology such that each inclusion map E^ ^ E is continuous then we say E is the strict inductive limit of the sequence E^, and we call E an LF-space. A seminorm p on E is continuous relative to this inductive limit topology if and only if its restriction to each E^ is continuous. An LF-space is a Hausdorff locally convex topological vector space and is barrelled, but is not metrizable. It is complete. (We have not defined the notion of completeness for non-metrizable spaces.) The topology induced by E on each E^ is the original topology on E^. Finally E is a Montel space if each of the E^ is a Montel space. An important theorem of Dieudonne-Schwartz asserts that a subset B of the LF-space E is bounded if and only if B is contained in E^ for some m and B is bounded in the Frechet space E^. Since a convergent sequence is bounded it follows that the set of convergent sequences in E is just the union of the sets of convergent sequences in the various E^s. Even though the LF-space E is not metrizable we none the less have the following pleasant result.

Exercise 7.1. Let E be an LF-space, let F be a locally convex topological vector space, and let T : E - ^ F be a linear map. Then the following statements are equivalent:

(1) T is continuous.(2) The restriction o f T to each E^ is continuous, where (E^) is the

sequence o f Frechet spaces defining the topology o f E.

(3) T is bounded; that is, T maps bounded sets to bounded sets.(4) T maps bounded sequences to bounded sequences.(5) T maps sequences converging to 0 to bounded sequences.(6) T maps sequences converging to 0 to sequences converging to 0;

that is, T is sequentially continuous.

Show first that (1) and (3) through (6) are equivalent when E is a Frechet space and then show (1) and (2) are equivalent when E is an LF-space. Then the rest follows from the remarks preceding the exercise.

Let n be an open subset of IR”. We saw above that C°°(n) is a Frechet-Montel space. For each compact subset K of Cl let C k be the subspace of C “(fl) consisting of functions with support in K. Then C k is a closed subspace of and so is a Frechet-Montel space. We provideC~(fl) with the strongest locally convex topology such that the inclusion map C k C^iH) is continuous for each compact subset K of ft. Since it suffices to consider a sequence of compact sets with union ft we see that C~(ft) is a Montel LF-space. In particular it is reflexive and non- metrizable, and each C k is a closed subspace. A seminorm p on C “(ft) is continuous if and only if its restriction to C k is continuous for each compact subset K of ft.

Exercise 7.2. Suppose we have an indexed family o f functions f g C ”(ft) and an indexed fam ily o f multi-indexes a (j) , j e J . Assume that the family (supp f)j^ j is locally finite. Then

p(<p) = Z supje J

is a continuous seminorm on C~(ft). Moreover, the set o f all seminorms o f this form determines the topology o f C “(ft).

The bounded sets in C “(ft) and in C “(ft) are easily determined. Indeed we have the following results.

Theorem 7.3. A subset B o f C'^Cft) is bounded if and only if for each compact subset K o f Cl and each integer m ^ 0 there exists a constant such that supjc |D“<pl<CK,m for each <peB and each ja l^ m .

Proof. Obvious.

Theorem 7.4. A subset B o f C“(ft) is bounded if and only if there exists a compact set K in ft, and for each integer m ^ 0 there exists a constant such that (i) supp <p^K for each cp e B and (ii) jD “(pl<C^ for each cpe B and each lal<m.

7. DISTRIBUTIONS 19

Proof. This theorem is an immediate corollary of the Dieudonne- Schwartz theorem discussed above. We can also give a direct proof. Indeed suppose (i) is false. Then there exists a sequence (xy)js.i of points in ft with no accumulation points in ft and functions <Pj in B such that (Pj(xy)^O, for each j. Then we can choose functions g C“(ft) with disjoint supports such that sup j >/, for each /. Now p defined by p(<p) = Zsupl/j<pl is a continuous seminorm on C“(ft) and p(<py)>j, for each j. Thus B is not bounded.

A distribution on ft is a continuous linear functional on C“(ft). We denote the space of distributions on ft by S>'(ft). A linear functional / on C~(ft) is a distribution if and only if for each compact subset 1C of ft there exists a constant Ck and an integer > 0 such that

\if, <P)1 ^ Q c, max sup lD“<pl

for each <p g C k- Other equivalent definitions may be obtained from Exercise 7.1.

Example 7,5 We have a canonical inclusion L^(ft, loc)^2i'(ft) defined by (/, <p) = Jn/(^)<p(^) /GL^(ft, loc) and <pGC“(ft). This correspondence is one-to-one by Theorem 3.6.

Example 7,6. We have a canonical inclusion ^ (ft)^S)'(ft) defined by (jLL, <p) = J (p(x)djUL(x), jULG (ft) and <pGC“(ft). This correspondence is one-to-one by Corollary 4.11.

Example 7.7. If |ul is a positive Borel measure on ft (not necessarily finite) then (p,, <p) = f <p(x) dp,(x), <pGCc(ft), defines a distribution on ft. This distribution in turn determines p by Corollary 4.12.

In each of the above cases we identify the function or measure with the distribution that it determines. We think of distributions as generalized functions or measures. If w is a distribution on ft we will sometimes write J u(x)(p(x) dx for (m, <p). This notation is purely formal but is sometimes convenient especially when parameters are involved. Thus in physics texts and texts on ordinary differential equations one sometimes sees for example f 6(x)cp(x) dx = <p(0).

Exercise 7.8. I f feO)'(Cl) then there exists a positive Borel measure p on ft such that (/, <p) = J <p(x) dp(x), <pGC~(ft), if and only if (/, 0 for each real valued <p g C “(ft).

Here are some hints and comments concerning Exercise 7.8. To do the exercise first show (f, for each real valued non-negative i/re C“(ft).

Give Q (fl) the inductive topology induced by the inclusions Ck ^ Q(H) where K runs over the compact subsets of ft. Here Ck is the space of continuous functions with support in K. We provide Ck with the compact-open topology. Then Q (ft) is an LF-space. Second, show / is continuous on Cc(ft) for the topology induced by Q (ft). Third, show C “(ft) is dense in Q (ft) so f extends by continuity to a continuous linear functional on Q (ft). Fourth, show this extension has the positivity property which allows an application of Theorem 4.3. The dual space of Cc(ft) is called the space of R adon "'measures' on ft, though Radon integrals is probably a better term. The Radon ‘measures’ may be identified with linear combinations with complex coefficients of positive Borel measures. We cannot really interpret them as complex measures because of the difficulty with infinities. Still for each Radon ‘measure’ |ll, if K is a compact subset of ft we can make sense of the restriction ix\k of jLL to K. Then /llIk; is a complex Borel measure.

One important way of producing distributions is by taking limits. To justify the method we summarize some more functional analysis.

T h eorem 7.9. B an ach -S te in h au s . Let E and F be locally convex topological vector spaces and assume that E is barrelled. Let H be a set o f continuous linear maps from E into F. Then the following statements are equivalent:

(i) H is pointwise bounded, that is, for each x in E the set {h(x) | h g H } is a bounded subset o f F.

(ii) H is uniformly bounded, that is, for each bounded set B in E the set { h ( x ) \ h e H ,x e B } is a bounded subset o f F.

(iii) H is equicontinuous, that is, for each neighbourhood V o f 0 in F theintersection neighbourhood o f 0 in E.

C orollary 7.10. I f E and F are locally convex Hausdorff topological vector spaces and E is barrelled, and if (h^) is a sequence o f continuous linear maps o f E into F and for each x in E the limit h(x) = \im h^(x) exists, then h is a continuous linear map o f E into F.

Proof. A pointwise convergent sequence is pointwise bounded and so is equicontinuous by Theorem 7.9. Thus h is continuous.

C orollary 7.11. I f (/ ) is a sequence o f distributions on ft and if the limit {f, <p) = lim^ </ , <p) exists for each e exists for each <p€C“(ft) and hence

8. RESTRICTION AND SUPPORT 21

determines a distribution f which is denoted by P.v. (1/x) and called the principal value of 1/x.

E xercise 7.13. Show that the limit in Example 7.12 exists and show that

(p.-V. “ ’ <p) = 1 1 <P S

We will discuss some more principal values in Section 15. We will see that principal values are just a special case of a more general notion, the finite part of Hadamard. This topic is also discussed for example in Edwards [1].

§8. Restriction and support

Suppose are open subsets of W". Then the inclusion map^ C “(n 2) is continuous. The transpose of this inclusion map

maps Q}'(Cl2) into and is called the restriction map. If f e ^ ' { f l 2) theimage of f in SJ'(fli) is denoted by Note if / is a locally integrable function on CI2 then in view of Theorem 3.6 its restriction to fli as a distribution coincides with its restriction as a function.

L em m a 8.1. Locally equal distributions are equal. That is, if Cl is an open subset o f R ”, (ftj)jej is an open cover o f Cl, f, g eS)'(fl) and /In, = g|n, for each j € J, then f= g .

Proof. Choose a partition of unity e 0 ^ 1,supp (supp a locally finite family, and X i/fy = 1. If <#> e Cc(n)then <f>=Z j<f> where this sum is finite and e C c(d j). Thus (/, <f>) =Z (f, = Z (g, = <g, <>>•

L em m a 8.2. I f H is an open subset o f {Clj)j^j is an open cover o f Cl, feQ}\Clj) and i//i 1000, ="/j|ninn, for each i , j e J , then there exists a unique feQ }'(d) such that /n, =/ for each j e J .

Proof. Uniqueness follows by Lemma 8.1. For existence, we choose a partition of unity as in the proof of Lemma 8.1 and define {f,(j>} = 'Lifp ^j^)- Exercise 8.3 / is continuous. It remains to show /n. =/. If 4> eC c(d j) then <t>=Y k<f> is a finite sum and g Thus(fk, 4>k<l>} = (fi, and therefore (/, <f>> = I </k, 4>k<f>) = Z ifj, k4>) = iU <#>)■

Exercise 8.3. Let be open subsets o f andassume supp Then multiplication by il/ is a continuous linear mapo f C^(fl) into

If n is an open subset of (R"" then the assignment of Q)'{U) to each open subset U of fl, together with the restriction maps defined above, defines a presheaf of abelian groups Q)' on fl. Lemmas 8.1 and 8.2 then imply that S)' ‘is’ the presheaf of sections of the sheaf of germs of the presheaf Si'; somewhat loosely, S ' is a sheaf. See, for example, Godement [1], Bredon [1] or Hormander [1] for the theory of sheaves. For any sheaf of abelian groups one has the notion of the support of a section. In our present context this notion takes the following form. If / eS '(ft) the support of /, denoted supp /, is defined to be the complement in ft of the largest open set U ^ ft such that f\u = 0. This definition makes sense by Lemma 8.1. We note if / is a locally integrable function on ft then the support of f as a function coincides with the support of / regarded as a distribution.

Lemma 8.4. I f ft is an open subset o f [R” then the inclusion map o f C “(ft) into C “(ft) is continuous and has dense image. Thus we have a natural inclusion o f the dual space o f C “(ft) in Si'(ft).

Proof. The continuity is obvious. Choose compact sets (Ky)y>i in ft with Kj contained in the interior of Kj+i and with the union of the K, equal to ft. Then choose Oj e C~(ft) with 0 < < 1 and 6j = 1 on Kj. If < >GC“(ft) then dj<t>GCc(Li) and in C “(ft). Indeed, given anycompact subset K of ft there exists jo such that if j > /q then 6j<f> = <f> in 3. neighbourhood of K. Thus maxi«i<^supjc — = 0 for j^/o-

We denote the dual space of C Cft) by ^'(ft). Since the transpose of a continuous linear map is continuous for any reasonable dual topologies, the inclusion of ^'(ft) in S '(ft) is continuous for example for the weak* topologies and for the strong topologies. We can give a direct description of ^'(ft) as a subspace of Si'(ft) as follows.

Theorem 8.5. I f ft is an open subset o/ IR” then €'(Ll) consists o f the distributions on ft which are compactly supported. 1/ /€Si'(ft), supp / is compact, i/f€C~(ft), and i{/ = l in a neighbourhood o f supp/, then <t>- if, is the unique extension o f f to a continuous linear functional on C “(ft).

Proof. If f e €'(Ci) there exists a compact set K in ft, an integer m > 0, and a constant C such that !(/, <^))|<CmaXl^^<^supK |D“<f)|, for </>€ C°°(ft). Thus if </)ECc(ft) and K IT supp <#> = 0 we have {f,<f>) = 0, whence supp/^K Conversely, assume /GS)'(ft) and supp / is compact.

9. DIFFERENTIATION OF DISTRIBUTIONS 23

Choose i/f as in the statement of the theorem. If <t>GCc(0) then has support disjoint from supp f and therefore (/,</>) = (/, for <t> e Cc (ft). Thus by Exercise 8.6 {f, (f>) = {f, <f) e C ”(fl) defines a continuous extension of / to C“(fl).

E xercise 8.6. I f ft is an open subset o f and ij/eC ciO ) then multiplication by if/ is a continuous linear map o f C “(fl) into

Since we have 0< k< oo. Thus itmakes sense to say that a distribution is of class on an open set. Either by a partition of unity argument or by Theorem 3.6 it follows that distributions which are locally of class are of class C . Thus it makes sense to define the singular support of a distribution f on fl, denoted sing, supp f, as the complement in fl of the largest open set in on which f is of class O ”. In Chapter 2 we will study the wave front set of a distribution. This notion is a refinement of the notion of singular support.

E xercise 8.7. Let n = l. Show sing, supp P.v. (1/x) = {0}.

§9. Differentiation of distributions

Let ft be an open subset of IR”, let a e C~(ft), and let P = X|ai- so P is a linear partial differential operator of order m.

E xercise 9.1. P is a continuous linear map o f C~(ft) into itself and also o f C “(ft) into itself.

In view of Exercise 9.1 the formal transpose P' of P is a continuous linear map of C “(ft) into itself. We may therefore consider the transpose (P')* :S>'(ft) ^ S)'(ft) which will be continuous say for the weak* topology or the strong topology on S>'(ft). By Corollary 6.6 the following diagram commutes for any integer k > 0.

^C^(ft)

(P T > ^ '( f t )1

S)'(ft)-

Thus (P')* is an extension of P and we will denote it simply by P. In summary, P:S>'(ft)-^S)'(ft) is defined by (P/, <f>) = (/, P'< >), /eS)'(ft), < )e Cc(ft). This definition agrees with the classical action of P on C ” (ft). As a special case we have (D “/, </>) = ( - ! ) ’“’{/, for / e S '(ft) andCc(ft)- The derivatives D""/ defined in this way, say for fe L \ L l, loc), are called the distribution derivatives of /, or weak derivatives if they happen

to be locally integrable functions. The distribution derivatives do not necessarily coincide with the classical derivatives when the latter exist. From now on the notation D “/ will always be used to mean distribution derivatives. Special notation will be used for classical derivatives if necessary. As pointed out above, if f e and jal < m then D “/ in thedistribution sense coincides with D “/ in the classical sense. It is important to identify other situations in which the two notions agree, but first the following exercise gives some simple non-classical derivatives.

Exercise 9.2. Let n = l. Show loglxleL\[R\loc) and D log 1x1 = P.v. (1/x). Let Y be Heaviside's function, Y(x) = l i/ x > 0 and Y(x) = 0 if x < 0 . Then Y is locally integrable and D Y = 8.

Multiplication by a function ij/ e C “(ft) is a differential operator of order zero and therefore we have already defined multiplication by i/f on

Explicitly we have 4>) = {f,^4>) for fe & {Q ) and The reciprocal of a polynomial in general is not locally integrable. None the less, it can be interpreted as a distribution. The following exercise gives a simple example of this fact.

Exercise 9.3. Let n = l. Show x(P.v. (1/x)) = 1 .

The following two theorems describe some situations in which the distribution and classical derivatives coincide. We will not give the proofs here since in Section 12 we will prove a simpler result which will suffice for the sequel. The proofs of these theorems may be found in Schwartz [1] Chapter 2, §5.

Theorem 9.4. I f feL^(W^,\oc) is absolutely continuous on almost every line parallel to the Xy axis then the classical derivative [Dy/] exists almost everywhere in (R”. I f [Dy/]E L ([R”, loc) then Dy/ = [Dy/].

Theorem 9.5. I//g L^(IR^1oc) and D jfeL^(W ,\oc), then after m odification on a set o f measure 0, f is absolutely continuous on every line parallel to the Xy axis and the classical derivative [Dy/] exists almost everywhere and Dy/ = [Dy/].

The following theorem is a consequence of Theorem 8.21 in Rudin [1]. In view of Exercise 9.2 it is important to note that we assume the classical derivative exists everywhere in fl.

Theorem 9.6. I f f is a continuous function on f l and the classical derivative [Dy/] exists everywhere in f l and [Dy/]e L^(fl, loc) then [Dy/] =w *

Proof. If then [Dy(/</))] = [Dy/]<;)+/Dy<#> exists everywherein R" and is in L (IR”). Since f<l> has compact support it follows from the theorem of Rudin referred to above that f_oo [Dy(/<#>)] dx = 0 . Then by Fubini’s theorem Jn [ ^ (/<#>)] = 0, that is, Jn dx = fDi<f> dx.

If we replace the theorem of Rudin used above by the simplest version of the fundamental theorem of calculus we obtain Theorem 9.6 under the stronger hypothesis that [D jf]eC(Cl). We will prove a converse of this latter result in Section 12.

Exercise 9.7. I f and 4>eO^{D) then (D “/, <>) =

Remark 9.8. In this section we have used A* to denote the algebraic transpose of a linear operator A. The notation A' is reserved for the formal transpose. In the sequel we will also use A* to denote the formal adjoint and to denote Hilbert space adjoints (where the domain is part of the definition). Hopefully the context will keep the meaning clear and no confusion will result. The alternative is a multiplicity of unfamiliar and unpleasant notations.

10. FUNDAMENTAL SOLUTIONS OF THE LAPLACIAN 25

§10. Fundamental solutions of the Laplacian

If P g 0 [z i, . . . , z„] is a polynomial of degree m, say P(z) = Xiai<m a e 0 , we set P(D) = Xiai<m so P(D) is a linear partial differentialoperator of order m with constant coefficients. A fundamental solution for P(D) is a distribution E on R ” such that P {D )E = 8. In this section we will compute some more interesting distribution derivatives than we have seen so far and also obtain fundamental solutions for Laplace’s operator A = If </>€C~(R") then for each co in the unit sphere

we have

</>(0) = - [ <l>{to}) dt = - ^ [ o}j{Di<l)){ta)) dt.Jo Qt j = i Jo

It follows if h is any infinitely differentiable function on andJs”-> h{o}) do> = 1 then

<#>(0) = - Z f f (OjhMm<t>)ito}) dt dojj = l Js"-’ Jo

= - Z f f it<Oi)r’'h(—]{Dj<l>){t(a)t’'-'<itd(o , = i Js - ' Jo \ t J

= “ Z [ lxl“')D,<;»(x) dx.J = 1 -fo'’

If we set K,(x) = x,-jxj "Ii(x lx l ') then r - ( 0)) and Ki ishomogeneous of degree 1 — n. In particular K, is locally integrable on all of R" and so we may regard K, as a distribution on R". Our computations above then show

l D j K j = d.i = i

(1)

Another derivation of this result is given in the comments following Theorem 15.8. Note it is simply a generalization of the second part of Exercise 9.2. Next we note jxp”” e L ((R”, loc) and if <f>eCc(U^) then

<D,.|xr",<f,) = -<lxr",D,<f.>

= - 11^ 1 "" <ix

= —lim I D.< )(x) dx

= lim] (2 —n)Xj |x|“” <^)(x)dx+8 <#>(sa>)d6>['1' Mx|>e JS"-' J

= |(2 —n)xy lx|“” < )(x) dx

since Xj Ixj"' is locally integrable on W . The integration by parts in the fourth step follows by Exercise 6.2. We have now shown

D j x r " = ( 2 - n ) x j x r . (2)

This formula is of course trivial classically, but only on (R” - (0 ) . The point here is that when interpreted in the distribution sense the formula also holds at the origin. If in the definition of Kj above we take h to be a constant, necessarily r(n/2) 7r“” , we obtain from (1) and (2)

(3)

if ni=2. Thus if n ^ 2 we have found a locally integrable fundamental solution for the Laplacian. It is somewhat remarkable that if in (3) we replace by lx| “'" — 1 and formally let n 2 we obtain a fundamental solution for the Laplacian also in dimension 2. We will now, however, obtain it directly. The difficulty in the argument above is that in dimension 2 formula (2) does not give us a primitive for Xj \x\~ . We consider therefore the function logjxj which for is locally integrable in IR”.

10. FUNDAMENTAL SOLUTIONS OF THE LAPLACIAN 27

Integrating by parts once again we have

(Dj log 1x1, <t>) = -<log 1x1, D,<f)>

= -| log lxl Dj<f>(x) dx

= —lim log 1x1 D,<f)(x) dxJ l x l > s ^

= lim I Xj \x\~' (f>{x) dxM x | > e

+ 8”“ I log |exl <l>{ex) dxj.

In Exercise 9.2 we have already seen what happens here when n = 1. In the present case we are interested in n > 2 . In this case x, \x\~ is locally integrable in [R" and the second term in the last expression above tends to 0 with 8. Thus we obtain

Dj log jxl = Xj \x\~ if n > 2

Now let n = 2. If in the definition of Kj above we take h to be constant necessarily 1/2(7t), and then combine (1) and (4) we obtain

(4)

if n = 2 . (5)

Thus also in dimension two we have now found a locally integrable fundamental solution for the Laplacian.

In the complex plane 0 = [R“ with coordinate functions x and y define the following differential operators.

a i / a .a \ , a i/ a .a \- - 1 — and — = - ---- \-i —: dyJ dz 2 \dx dyJdz 2 \dx dy/ dz 2\dx dyj

Note if f=u-\-iv and we take the real and imaginary parts of the equation dfidz = 0 we obtain the Cauchy-Riemann equations. For this reason a/az is called the Cauchy-Riem ann operator. Its null space, even in the distribution sense, consists of holomorphic functions (see Example 3:9.10).

E xercise 10.1. Note 1/z is locally integrable in 0. Show that (a/az) log |z| = 5(l/z). Show (a/az)(l/z) = 7t6. Show formula (5) follows.

The exercise may be deduced from the results above or may be done directly by using polar coordinates and integrating by parts.

The derivation of formula (1) is due to Sobolev. This formula is the basis for much more general decompositions of 8 obtained by an ingenious algebraic argument in Smith [1].

§11. The Newtonian potential in IR"

Theorem 11.1. I f |x€g8(IR"), feV(U '^, loc), f is bounded at infinity, and P e 0 [ z i , . . -, z„] then for each <l> e C“(R") we have

<P(D)(/ * ^), <>) = j {P(D)f, d^(y)

= I {P(D )n, T_ <#)>/(x) dx

Proof. Recall that f * L ’(IR", loc) by Theorem 4.6. We have

<P(D)(/ * /x), <t>) = (f^ IX, P(-D)<t>)

= |/ * fx(x)(P(-D)<#))(x) dx

= j| / (x - y)(P(-D)){x) d|x(y) dx

= j j/ U - y){P{-D)<(>)ix) dx d/x(y)

= 1 1 f(x)iP{-D )4>){x + y) dx d|x(y)

= |</,P(-D)T_,<#>)d|x(y)

If instead of proceeding to the last integral we first interchange the order of integration we have

= I (itt, Pi-D)T.„<f>)f{x) dx

The applications of the Fubini-Tonelli theorem in the above argument are justified since the estimates in the proof of Theorem 4.6 imply the absolute convergence of the iterated integrals.

Corollary 11.2. I f fiG^([R”), f e L \ W , loc), f is bounded at infinity, and P e 0 [ z i , . . . , z ] we have:

(A) I f P {D )f G then P (D )(f * |ul) = P {D )f * jm.(B) I f P{D)iJi e then P (D )(f * |ll) = / * P{D)^.

11. THE NEWTONIAN POTENTIAL IN 29

Proof. Let </>€C"(IR''). To prove (A) let v = P{D)f. Then

<P(D)(/ * ill), <#>)=!{v, T_y<#>) djUL(y)

= + y) di^U) dfx(y) = <v * jui, <#)).

To prove (B) let v = P(D)fjt. Then

<P(D)(/ * |ul), <;>) = I<v, T-^<l>)f{x) dx

= [ [ <l>(x + y) dv{y)f(x) dx = </ * v, <f)).

An important special case of the corollary occurs when P (D )f = 8. In this case we have P(D)(/ * fi) = [x. This property illustrates a general principle: when the appropriate convolution product is defined we may use convolution by a fundamental solution to solve inhomogeneous partial differential equations. Consider the case n > 3. For each jut e^([R") we define the Newtonian potential of /uc, denoted by Pot (/x), by Pot (jui) = 1x1““” * jUL G L ([R”, loc). By Corollary 11.2 and formula (3) in Section 10 we have

2(2-n)7T” ^A Pot (|ul) = — — fx for each pu g ^(IR”).r(n/2)

If jUL > 0 one can show that Pot (jut) is lower-semicontinuous, and in fact is superharmonic. See Donoghue [1]. Applications of the Newtonian potential may be found, for example in Torre and Longo [1].

Exercise 11.3. I f n > 3 and p. is the uniform distribution o f unit mass over the unit sphere = {x g IR” | |xl = 1} then

_ , s f l if 1 1 < 1P o > W W = j ^

Exercise 11.4. J/ n > 3 and fx is the uniform distribution o f unit mass over the unit ball B ” = { x g (R” | lx|<l} then

Pot (fi,)(x) = I ^(n-^(2—n)\x\ ) if 1x 1 < 1 if 1x 1 > 1 .

Exercise 11.5. I f fx g ^(IR”) show Pot (|lx) is on the complement o fS U p p jUL.

The proofs of these three results are given in Donoghue [1]. In Exercise 11.5 it is actually the case that Pot (ill) is real analytic on the

complement of supp jut. This fact follows from the regularity properties of elliptic operators and from the fact that A Pot (/m) = 0 on the complement of supp fjL. Exercises 11.3 and 11.4 are actually results which are essentially due to Isaac Newton. In his book [1] Bronowski states that he is convinced that it was the difficulty in resolving these two problems, i.e. in computing the gravitational attraction of an extended body such as a ball, that caused Newton a long delay in publishing his theory of gravitation.

§12. Leibniz' formula. Classical derivatives

Let P g . . . , n] be a polynomial of degree m. Note that the differential operator P(D) determines the polynomial P by e-< ’'">P(D)e< ’''> = P(^). For convenience set = D|P(^).

L em m a 12.1. Leibniz' form ula. I f fe9 )'{ft) and then

P (D )(# ) = X (l/a!)(D “./r)P<“>(I»/a.

In particular for any multi-index (3 we have

Vo:/

where a < |3 means a, j = 1 , 2 , . . . ,n and = |3 !/(|3 — a)! a !

Proof. If c l> eC 7 m then {D ^ m ,4> ) = -{f,iifD^) = {f,m iif)<f>- = -/),<;>). It follows that D ,(# ) = (D,-i/.)/+iA(D^/)

and therefore by induction we have P(D)(il/f) = Y,(D'^il/)Q^(D)f for certain polynomials e . . . , Now if we take il/(x) = and take for f the function /(x) = we obtain the relation P(^ + t)) = S^^Qa('n)- Differentiating with respect to | and then setting | = 0 we obtain = a \ QcXv)-

T h eorem 12.2. I f u and f are continuous functions on and DjU = f in the distribution sense then DjU = f in the classical sense.

Proof. If i( /eC c{0 ) then iJ/ueC{H) and by Lemma 12.1 Dj(il/u) = {Djil/)u + i/rf which is also continuous. Hence by Corollary 3.2 it suffices to consider the case when u and / are compactly supported distributions on U'". Choose a mollifier p. By Theorem 3.1 the convolutions pe * u and Pe * / are in C“(U”) and converge to u and f, respectively, uniformly on compact sets, as s [ 0 . Taking classical derivatives and keeping in mind

13. DISTRIBUTIONS WITH POINT SUPPORT 31

that f is the distribution derivative of u we have

D,(pe * u)(x) = I u{y)D^p^(x - y) dy = -|u(y)Dy,p^(x - y) dy

= j/(y)Pe (x - y) dy = pe * fix ).

Thus Pe * M—» M and Dy(pg * w)—> / uniformly on compacta. It follows that DjU exists in the classical sense and DjU = f.

Corollary 12.3. I f and D “u e C (n ) for each a with l a l <mthen u G

There is no really satisfactory way of defining the product of any two distributions, yet in many special cases products do exist. In such cases it is useful to have a Leibniz formula. We will see another example of a Leibniz formula in Section 14.

Exercise 12.4. Let f,gGQ}'(d) and assume sing. supp/H sing, supp g = 0 . Show that the product fg m akes sense and that Leibniz' formula P(D )(fg) = Y,cc m akes sense and is valid.

§13. Distributions w ith point support

As usual, let H be an open subset of IR". If feQ }'{d) and m^O is an integer we say that f has order <m if for each compact set K in Cl there is a constant Qc such that \{f, </>)! < C k maX|o,|< n sup^ 1D “</>1 for each </> g C^. We say f has finite order if / has order <m for some m. If feQ}'{Ct) and (o is a relatively compact open subset of o) then has finite order, and if ip G C“(fl) then ipf has finite order. In particular each distribution with compact support has finite order.

Exercise 13.1. Define an appropriate LF-space structure on C^(fl) so that the space o f distributions in f l with order <m may be identified with the dual space o f C^(fl).

Exercise 13.2. Use translates o f derivatives o f 8 to construct a distribution which does not have finite order.

Exercise 13.3. I f fe € ' ( f l ) , and <l> = 0 in a neighbourhoodo f supp f then (f, <t>) = 0. Give an example o f fe€ '{Q ) and <f> g C ”((1) such that </) = 0 on supp f but (/, (p) ^ 0.

Theorem 13.4. I f there exists an integer m >0 such that if<f> € and = 0 on supp f for la| < m then {f, ) = 0.

Proof. Since fe€ '{ft) there exists a compact set L, an integer m > 0 , and a constant C such that !</, i/r)] < C maX|o,i 0 let be the closed e-neighbourhood of K. Choose a mollifier p and let Xe = Pe * Xie where Xie is the characteristic function of K2e- Then = 1 on K , supp Xe ^ ^3e ind if 3s <dist (K, dfl) then Since x'e = in a neighbourhood of supp/ we have( f,^ ) = (f,Xe^), ipeC^iO ) and therefore !</, <#>)l<Cmaxio,i^ sup |D“(xe<#))l. Assume now D “</>=0 on K for lal<m. If x e K 2e choose y e K with l x - y l < 3 e . Since D “<#>(y) = 0 for \a\^m, by Taylor’s formula with remainder for g(0 = D “</)(y+ r(x —y)) we have

It follows that

1(m + 1 — la|)! o<t\^C max sup 1D <#)1K„. I3l m + 1 K-,,

for all small e > 0 , if lal<m. Also D “XeU) = i “ ey)(F>“p) (y)dy implies |D“;\:'(x)|<Cs“’“' for 8 > 0 . Thus by Leibniz’ formula

3<0£Hence we have \{f, 4>)\^Cs if 8 > 0 is small, i.e. {/ <f>)- 0.

If a € f l let be defined by {8a, <f>) = <l>{a).

Corollary 13.5. I f fe^ '{Q ) and if supp/^ {a } then / = Ziai<m for some integer m and constants c , € 0.

Proof. Let m be the integer provided by Theorem 13.4. If <> 6 C~(fl) then by Taylor’s theorem (l>(x) = Y\cc\^mi l<^ -)f^^<t>M{x-a)' -\-il/{x) where and D^ij/(a) = 0 for \f3\:^m. Thus (/, «A) = 0 and soif, <#>) = Ziai<m i^/oLl)D'^(l>{a){f ( x - a ) “). It follows f has the desired form with c« = ( ( - l ) '“'/of!)</, ( x - a ) “).

We will give an application of Corollary 13.5 as an exercise, but first we recall the closed range theorem. Let E and F be locally convex Hausdorff topological vector spaces. If M is a subspace of E we let M- = { feE'\{ f ,u) = 0 for each u s M } and if M is a subspace of E' we let

= { ueE\{ f , u) = 0 for each f e M}. If T : E F is a continuous linear map then ker T = (im T *)“ and im T * is weak* dense in (ker T)- . Also ker T * = (im T)^ and im T is weak dense in (ker T*)-^.

Theorem 13.6. Banach-Rota-Dieudonne-Schwartz. Let E andF be locally convex Hausdorjf topological vector spaces and let T :E - ^ F be a continuous linear map. Consider the following properties:

(1)(2)(3)(4)(5)(6)(7)(8)

T : E - ^ F is relatively open.T : E - ^ F is relatively open for the weak topologies. im T is closed.im T is weakly closed, i.e. im T = (ker im T * is weak* closed, i.e. im T * = (ker T)- .T * : F '—> JE' is relatively open for the weak* topologies. im T * is strongly closed.T * : F ' E' is relatively open for the strong topologies.

Then we have:

(A) Statements (1) through (6) are equivalent for Frechet spaces.(B) Statements (1) through (8) are equivalent for B anach spaces.(C) Statements (1) through (8) are equivalent for Frechet-Schwartz

spaces.(D) Statements (1) through (7) are equivalent if E is a reflexive Frechet

space and F is a Frechet space.

Schwartz spaces are discussed in Schwartz [2]. C~(fl) is a Schwartz space and and A (fl) (see Section 7) are Frechet-Schwartz spaces.

Exercise 13.7. Let F = 0[[x^,. . . , x„]] be the space o f form al power series topologized by the seminorms E =rnaxi^l<^ lUo l, m =0 , 1 , 2 , 3 , . . . . Show that F is a Frechet space. I f f e F and P e 0 [ x i , . . . , x ], say / = cind P = Zb^x^ define (f, P) = Zoil a^b .Show that this pairing identifies 0 [ x i , . . . , x^] with F'. Let T : Fbe defined by T(< >) = X (l/o^0I^“<#>(0)x“. Show that T * :F '^ ^ '(1 R ”) is given by T" {P) = P {-D )8 . Now use Corollary 13.5 to obtain BoreVs theorem that each formal power series is the Taylor series o f some C°° function, i.e. T is onto.

Exercise 13.8. Find all solutions feSlJ'iU) o f the equation x'^f = 0, where m^O is an integer. Find all solutions gs3)\R ) o f the equation xg = l.

We will now consider distributions supported by a closed submanifold. We begin with the simplest situation. Let k < n . We may regard IR as a subspace of IR” by identifying the point (x i , . . . , x ) in with the point (x i , . . . , Xfc, 0 , . . . , 0) in IR”. We will then write the coordinate functions in IR" as Xi , . . . , Xk, y i , . . . , y„_k. The restriction map C“(IR"") C “(IR ) isclearly continuous and onto. Its transpose therefore gives us a natural

inclusion

If /e2)'(!F5*‘) and <f>eC”(!R") then formally we have

</, <f>) = |/(x)<f)(x,0)dx

This notation is a little misleading. When we view / as a distribution on R ” it becomes possible to take derivatives of f in directions normal to Again formally we have

{D % 4>) = ( - l ) '“l|/(x)(D“<f>)(x, 0) dx.

A distribution of the form D “/, is called a multiple layer o f type|q:1 + 1 carried by R . The double distributions in potential theory (see Kellogg [1]) are multiple layers of type two carried usually by hypersurfaces.

We will now characterize the distributions on R ” supported by R*". We first consider the case of distributions with compact support. Let with supp/clR*" and suppose f has order <m. If <^>€C“(R”) we define i/feC“(R”) by the equation

y)= Z ^ D “ (x ,0)y“ + « (x, y).

Clearly 0) = 0 if ||3|<m. It follows by Theorem 13.4 that(/, i/f) = 0. If we define e ^'(R'") by

( - 1) a !

lo-l

</., 0) = (/, y“0>, 0 € C”(R' )

we obtain

f = 1 D % .lo tl< m

Thus / is a sum of multiple layers.We may remove the restriction that f have compact support by using a

partition of unity. Let <{>j e C“(R”) be chosen such that their supports form a locally finite family and such that X </>/ = 1 in a neighbourhood of R^ If /g S j'(R”) and supp/c|R* then

J J lotl^mj

where and supp ^ supp <#>y. Thus we see that / is a locallyfinite sum of multiple layers.

We consider now the general case of distributions supported by a

submanifold. Let M be a closed submanifold of IR”. Then M inherits a Riemannian structure from and hence has a distinguished positive measure fx, the Riemannian density or volume element of M. If K is a compact subset of M and K ^ U where U is a chart on M we will say that K is a small compact set. We can topologize C k by considering derivatives relative to the local coordinates in U. This topology is independent of the choice of local coordinates in U. This fact follows from the chain rule and the compactness of K. Now that we have the topology of uniform convergence of functions and their derivatives on small compact sets we may introduce distributions on M exactly as we did on R ”. We topologize C“(M) by giving it the strongest locally convex topology such that the inclusion maps ^ C~(M) are continuous for each small compact set K in M. The space of distributions on M is thendefined to be the dual space of C~(M). The space C~(M) may of course be topologized by the topology of uniform convergence of functions and their derivatives on small compact sets and its dual space €'{M) is then, as before, the space of compactly supported distributions on M. We note if K is any compact subset of M we may provide with the induced topology from C“(M). The inclusion map C k “ C "(M ) will then be continuous for any compact subset K of M, small or not. If f e V ( p i , loc) then f defines a distribution by

(/,<;>)=[ fd|Li, <t>eC7(Ml

The restriction map C“(R”) C c ( M ) is continuous. To see that it is onto, by a partition of unity argument it suffices to show that we can extend functions with small compact support; but that is clear. Again, the transpose of the restriction map gives us a natural inclusion

When the Riemannian density jut of M is regarded as a distribution on R"" by means of this inclusion map then it is frequently denoted by and is called the Dirac layer of M. By arguing in local coordinates one may now show, if /g S)'(IR”) and supp/^M then / is a locally finite sum of transversal derivatives of distributions on M, i.e., of multiple layers.

A few comments on computing the Riemannian density are in order. Suppose M has dimension m and codimension p = n — m. If Xq is a point in M then since M is locally a graph, after relabelling the coordinates in R"" if necessary, there is an open neighbourhood ft of Xq in R ” and coordinates Vi , . . . , y in ft such that

[ yj = Xj l < j < m

Y m + j

and M n n is given by the equations y„+, = 0 , We note theinverse transformation of coordinates is given by

{ i = yj l^ / < m

The functions Vi, . . . , restricted to M yield a system of coordinates in M n fl. The corresponding frame of the tangent bundle of M is given by

y/ k yj ^ k k = l yj ^^m+k

The components gij of the metric tensor of M relative to y i , . . . , ym ire then given by

___ ^ -5 1 _L Vyi yj k = \ dyi dyj

since the coordinates are orthonormal. If we introduce thep X m Jacobi matrix = (d<l>JdXi) we have

(g,-) = /'/ + /

where J' is the transpose of the matrix J and where I is the m x m identity matrix. Since x, = y for 1 < / < m we may regard Xi , . . . , as coordinates on M Pifi. The Riemannian density fx in M n n is then given by

djUL = Idet (/7 + J)r^ dxi • • • dx^.

This expression is quite complicated unless m = l or m = n — l , the two cases considered in calculus. The following exercise gives explicitly the computation of the determinant above.

Exercise 13.9. Let be the k x k identity matrix. I f A is an n x n matrix then det (A +I„) is equal to 1 plus the sum o f the determinants o f all the principal minors o f A. That is, det (A +/„) = Xq=o tr where tr denotes the trace and A"*A is the qth exterior power o f A. I f B is a p x m matrix then det (B 'B + 1^) is equal to 1 plus the sum o f the squares o f the determinants o f all the minors o f B. In particular d et(B 'B + I^) = det (B B ' + Jp).

A sometimes more convenient way to compute the density p. is given by the following exercise.

Exercise 13.10. Pappus' theorem. Let M be a closed submanifold o f o f codimension p. I f s > 0 let be the e-neighbourhood o f M. I f

14. WEAK DERIVATIVES AND INTEGRATION 37

<t> € C”(IR") then

(8m, <#>) = lim <fM dx

§14. Weak derivatives and integration by parts

Lemma 14.1. I f feL\ R ^ )' and D jfeL\U ^) then f Djf{x) dx = 0.

Proof. Choose <f> e CcClR""), 0<< ) < 1 such that <;> = 1 in a neighbourhood of the origin. If is an integer let <f>j (x) = cl>ik~^x). ThenIJ Dy/(x)<#>fc(x) dx\< k~ ll/lli sup \Dj<t>\0 as fc oo. Now apply the dominated convergence theorem.

Now Lemma 14.1 together with Leibniz’ formula Dj(fg) = (Dj/)g + /(D^g) immediately gives an integration by parts result. The problem is though that so far we have Leibniz’ formula only if one of the factors is smooth. We handle this problem by regularizing.

Theorem 14.2. Suppose l< p < o o , p~ + q~ = l and f e L ^ ( W ) , D j fe g e L "(R ”), and D ,g€L "(R ”). Then D,.(/g) = (D^/)g + /(D,-g) and

J /(Dyg) dx = - J (Dy/)g dx.

Proof. Choose a mollifier p and let /k=/*Pi/k. Then is in C “(R”) nL^([R”) and /k-^/ in U ( W ) by Exercise 3.3. Moreover, as in the proof of Theorem 12.2 D/k = (Djf) * pi/k and therefore Dy/k € L ([R”) and Dy/k->Dy/ in LFiW ) as fc^oo. Since we may applyLeibniz’ formula to obtain L>y(/kg) - (Dy/k)g+/k(Dyg). This implies that ^(/kg) is in L\ W ) and converges to (Dy/)g+/(Dyg) in L^R”) and therefore weak* in ^ '(R”). But f k g ^ f g in implies that Dy(/kg)—>

weak* in Thus Dy(/g) = (Dy/)g+/(Dyg) which impliesDj{fg)eL^{W^) and implies the last conclusion by Lemma 14.1.

Exercise 14.3. Prove Heisenberg's uncertainty principle, that if g, Dyg, and Xjg are in L^(R") then

^ (1 IgPd^) ^ jx flg p d x jlD jg l^ d x .

As a hint for the exercise apply Leibniz’ formula to XyDy(gg) and to Dy(xygg) to conclude that XyDy(gg) is in This proof of the uncertainty principle, by integration by parts, is due to W. Pauli. See Weyl [1] p. 77 and p. 393 and also Petersen [1]. There are other proofs as well, for example, one based on the Fourier transform.

§15. Distribution valued holomorphic functions

Let n be an open subset of R ” and let U be an open subset of 0. Let z-^ f^ be a function defined on U with values in We say that isholomorphic in U if for each <#> g C ”(fl) we have z ^ </>) is aholomorphic function in U in the usual sense. We note that since

<>) = (— that is holomorphic in U. Alsolim^^o —/z), <#>) exists for each <l>eCc(^) and each z e U andhence by Corollary 7.11 defines a distribution (d/dz)/ which is also holomorphic in 17. Actually Corollary 7.11 deals with sequences of distributions but we can restrict our attention to w ->0 through a sequence to conclude that (d/dz)/ is a distribution. Inductively we define

by = and = Then is holomorphic inU. Now if Zo e U and g C “(ft) if we expand (/z, <>) in a convergent power series we conclude that =YZ=o iiz - Zo)^/kl)fzJ where the series converges weak* in for jz-Z ol< dist (zq, 6U).

Suppose now that is holomorphic in the punctured disc {z g 0 | 0 < |z —Zo|<r}. Then we have the Laurent expansion

ooZ (g k , - Zo)'"k=—co

where

<gfc, <>) = (2-n-i) ’ [ {f , }{z - Zo) ' dz.•'Iz—Znl = r/2•\z-Zo\ = r/2

If we write the integral as a limit of Riemann-Stieltjes sums then again Corollary 7.11 implies gkE^'(fl) and therefore =Xfc=-oo(2:-Zo)^gk for 0 < l z - Zo l < r , where the series converges weak* in The distribution g_i is called the residue of at Zq. In case gk = 0 for fc < —1 we say that /z has at worst a simple pole at Zq. We note if Zq is a simple pole then g_i ^lim^^zj, (z-Zo)/z weak* in S>'(fl). Moreover, in this case, f^ - (z-Zo)“^g-i has a removable singularity at Zq in the sense that it is holomorphic throughout jz — ZqI < r if we define it to be go at Zq. It should be clear now that many of the ideas and results from the theory of holomorphic functions carries over with no essential change to the case of distribution yalued holomorphic functions.

We will now give an example of analytic continuation in the case of a distribution valued holomorphic function. This is an important concept and has been used to solve the division problem and in particular to show that each constant coefficient linear partial differential operator has a temperate fundamental solution (see Atiyah [1]).

Let h be an infinitely differentiable function on the unit sphere and for each complex number z define h ,{x) = \x\ h{x Then h^e

15. HOLOMORPHIC FUNCTIONS 39

C°°((R” —(0)) for all z and h^e \oc) if R e z > —n where R ez denotes the real part of z. It is easy to see that is aholomorphic function in {z 6 0 | Re z > —n}. We will now explicitly continue this function. Note if m > 0 is any integer and if Re z > - n then for each <t> G C"(R") we have

[fi^(x)</>(x)dx= [ h^(x)|<#>(x)- ^ -^ D “</)(0)x“|dx

+ hz(x)<l>(x) dxJ | x l > l

+ X —,D^<l>{0) , I I f a>“h(o))do>.z + n + la| Js -i

The first integral on the right is a holomorphic function of z for Re z > - n —m —1. The second integral is an entire function of z. Thus we have proved:

Theorem 15.1. I f <>) = J hz(x)</>(x) dx for R e z > —n then extends to a meromorphic valued function in the complex plane withat worst simple poles. The poles occur at z = —n — m, m = 0 , 1 , 2 , ___ Theresidue at z = —n — m is = Ziai=m where

=- a ! L do>.

In particular we see that the pole at z = —n — m is a removable singularity if and only if h is orthogonal on to each homogeneous polynomial of degree m. By the Weierstrass theorem if h is orthogonal to each homogeneous polynomial then h = 0. Thus if h is not identically zero there is always at least one pole. Consider now - (z + n + m)~^R^ which has a removable singularity at z = —n — m. It follows that P.f. fi-n-m = liniz^-n-m ~ (z + n + m)“ jR ) defines a distributionP.f. on IR”. This distribution is the ‘partie finie’ or finite part ofHadamard. This is not the way it is usually introduced. See Edwards [1] p. 306 for the usual approach. Explicitly we have

<P.f. <#>) = [ fi_„_^(x)|<^(x)- X -^ D “<^(0)x“|dx

+ fi_„_ (x)<#>(x) dx• xl>l

- z — D ‘‘<t>{0) — V f f (0^h{a>) d<o.U\CmOL' m - a Jc«-1

Now let 0 < e < 1 and replace the first integral above by an integral over

lx|<8 and an integral over 8 < lx | < l. In the second of these two integrals integrate out the part containing the sum and combine the remainder with the second integral above. The integration of the part containing the sum yields a term cancelling the third term above and some additional terms involving powers of s and log e. Thus we obtain

(P.f.h_„_,<^>>= f f i_ „ _ (x )U (x ) - S ^ D “<#.(0)x“|dx

h_„_„(x)</)(x)dx4Ax\>e1 gkl-m r

- z — D “<#)(0)-----p-: (o^h(o})d(ol^K^a! m-\a\ Js -i

y D “</)(0) log 8 [ co“h(co) do>.a| = m«I JS"-la|

If we let 8 i 0 then the first integral here converges to 0 and therefore we obtain

(P.f. <>) = lim{last three terms above}.eiOThus the finite part is found by subtracting from Jix|>e ^-n-m M<l){x) dx terms involving log s and negative powers of 8, i.e. we subtract the divergent part, which is required to have a special form, and then let 8 i 0. This is the approach taken in Edwards [1] and is the general method for obtaining finite parts in various situations. Of course, one needs to specify what sorts of divergent parts are allowed in order to obtain a well defined finite part. In the special case we have considered, analytic continuation happens to pick out the usual finite part of Hadamard. In other situations one can also sometimes use analytic continuation to pick out various finite parts.

If limej,oJixi>e cxists foT each <l>eCc{U'^) we say thatP.f. ^-n-m exists in the principal value sense and we write P.v.From the computations above we see that P.v. exists if and only ifJsn-i a)“h(co)da) = 0 for each la|<m. Thus we obtain

Lemma 15.2. The following statements are equivalent.

(1) P.v. h_n-m exists.(2) h is orthogonal on to each polynomial o f degree <m.(3) Hy. has removable singularities at z = —n — k, fc = 0 , 1 , . . . , m.

The simplest example where the singularity at z = —n is not removable is obtained by taking h = 1. In this case we have

(P.f. 1x1"", 4>) = Hm ( 1 ^ </>(0) log s + 1 Ix l- <f>{x) dx).

15. HOLOMORPHIC FUNCTIONS 41

In this case the singularity at z = —n —(2fc + l) is removable since the integral over of a homogeneous polynomial of odd degree is 0. Thus, for example, = P.f. h_„_i (with h = l) is given explicitly by

(

9 _n/2 r \

Since Xy4>(x) vanishes at the origin this last equation implies

Xj P.f. =P.V. (Xj 1x1“”“ )

Other relations may be obtained in a similar manner or by analytic continuation.

E xercise 15,3. The classical derivatives o f loc) are notlocally integrable. Show if <l>e (IR") then

{Dj lxl “”, </)) = lim ( ( 1 - n ) f x. 4>(x)dx— f a>,4>(6a)) dct> Jlxl>e Js"- f

and therefore Dj 1x1 “” = (1 — n) P.v. (xy |x|“”“ ). Deduce the sam e result by considering the analytic continuation o f Dj \x\ .

We now give a simple example of how analytic continuation may be used to compute fundamental solutions. If Re z > 2 then we have A lx]"" = z(n + z - 2) 1x1 “ . Therefore if is the analytic continuation of jxj then AH = z(n + z — 2 )H^_2 ‘ If we let z —> 2 - n since H 2~n is just the locally integrable function jxp"' the left side converges to A |xp“''. The right side on the other hand converges to (2 — n) times the residue of at the pole at z = —n. Thus we obtain

9 _n/2

which we obtained by a different method in Section 10. Note the convergence referred to above is weak* in

E xercise 15.4.

Xy P.f. Ixl""" = Xj 1x 1“'".

E xercise 15.5. // n = 1 then

| ;( i f j io g w ) = p .f . i x r ' .

E xercise 15.6.

<P.l = lim ( i i * ( 0 ) log e - * < 0 ) e - )

+ [ |x| " ^<^(x)dx|.Jlx|>8 J

E xercise 15.7. Use analytic continuation to show9 _n/2

The following result may be regarded as a generalization of the last part of Exercise 9.2. Not only is it useful, but it also gives a nice example of the distinction between classical and distributional derivatives.

T heorem 15.8. I f K g C “(IR” —(0)) is homogeneous o f degree 1 - n and [DjK] is the classical derivative o f K then P.v. [DjK] exists and we have

DjK = P.v. [D,K] + ( I (OjKiw) dw^8

Proof. We drop the notational distinction between distributional and classical derivatives as there is no risk of confusion. Since DjK is a function homogeneous of degree —n we have

[ DjK{x) dx = [ [ DjK{r(o)r^-^ day dr

= —logs DjK((o)dco,Jsn-X

On the other hand by a modification of Exercise 6.2 and since K is homogeneous of degree 1 —n we have

f DjK(x) dx = [ o jjK M do) - f o)jK{eo)) d<o = 0.J e < | x ! < l J s " - ' J s " - '

Thus P.v. DjK exists. Now

(P.v. DjK, <#>) = lim f DjK(x)<t>(x) dx

= Hm Kix)Dj4>(x) dx - 1 <OjK((o)<t>{e<o) do^

= ~{K, Di<t>)- <f>(0) [ «,R(o)) do)Jcn-l

16. BOUNDARY VALUES 43

where we have used the second part of Exercise 6.2 with the function h occurring there equal to K<t> outside a small neighbourhood of the origin.

Note as a special case we obtain Exercise 15.3. Note also that we may use Theorem 15.8 to recover the first example discussed in Section 12. Indeed if h 6 C“(S"“ ), Js -i h(co) do; = 1 and we set Kj{x) = Xf lxl“” h(x |x|~ ) then in (R” - ( 0) we have

X DjKj(x) = X d^l"” - nx |x|~') + X 1 1"" 1x 1“ )

= |xl“” X XjDjh(x lxl~ ) = 0

since h(x 1x 1“ ) is homogeneous of degree zero and Y XjDj is the radial vector field. Also we have

Yj (OjKjioy) d(o = Y, (x)^h{o)) dct) = 1,JgM-. J n-X

Thus X = 8 by Theorem 15.8.

§16. Boundary values of holomorphic functions

This section consists of an example and a few remarks, not a general theory. The main purpose of this section is simply to illustrate yet another situation in which distributions occur quite naturally. In the upper half of the complex plane consider the function 1/z. We will evaluate the boundary value limy4o l/(x + iy) in the sense of distribution theory. That is, we will evaluate

lim fyio J x + iy for € Cr(IR).

In the upper half-plane we take the branch of log z defined by log z = log |zl + i arg z where 0 < arg z < tt. Then by the Cauchy-Riemann equations

: ^ l o g z = ^ l o g z = l/z, dx dz

and therefore

1 dlim — = — lim log z weak" in S)'(IR). yio z dx yio

Now if <l>eCc{U) then for y > 0 we have

|log (x + iy)<t>{x) dx = 5 J log (x + y )< >(x) dx + i |arg (x + iy)<l>{x) dx

and therefore

lim log {x + iy) = log 1x 1 + iVYy i o

where Y (x )= Y {—x) and Y is Heaviside’s function. Differentiating we obtain

1 T. 1 . Ol i m ---------- = P . V . -------- I7TO.y i o x + iy X

This formula is due to Dirac.By using a suitable branch of the logarithm in the lower half-plane we

obtain1- 1 1 chm-------= P.V. — + iTTO.y i o x — iy X

Taking the difference of these two expressions we obtain

lim p y = 8 weak* inyio

where Py(x) = (l/7r)y/(x^+y^) is the Poisson kernel of the upper halfplane. This fact is a special case of a more general fact, namely:

E xercise 16.1, Let p g V ( W ) and suppose p > Q a n d Jp(x)dx = l . Let p^{x) = e~'^p{e~^x). I f <l>eCciW) show that Pe*</>-^<f> in C “(R”) as 8 i 0 and therefore lime o Pe =

If f e€'{U) and Py is the Poisson kernel as above we define the Poisson integral of f by

Pf(x, y ) = f * P y ( x ) , y > 0.

If cf>e C “(1R) then

(Pf(-, y), <;>) = </ *Py,< )) = </,<#) * p y )^ </,<#>)

since Py = py. Thus P/(*, y) -» / weak* in Q)'(R) as y i 0. In Chapter 2 we will encounter the Poisson integral again, though we will use slightly different notation.

Now notice if z is a complex number not in supp/ then (t-z)~^ is a smooth function of t in a neighbourhood of supp f and therefore

ZTTl

makes sense. Formally

1-z

16. BOUNDARY VALUES 45

and therefore we call F the Cauchy integral of /. It is not difficult to check that F is holomorphic in the complement of supp f and so is holomorphic in the upper and lower half-planes. A simple computation shows if y > 0 then

F ix + iy) - F ix - iy) = Pfix, y).

Therefore we have

lim (F(- + iy )-F (-yiO ■ *y)) = f weak* in

that is, the distribution / is the ‘jump’ of the holomorphic function F across the real axis. Actually we need not restrict ourselves to distributions with compact supports. Any distribution on IR may be realized as the ‘jump’ of a function holomorphic in 0~ [R . These ‘jumps’ are the so- called hyperfunctions, at least in the one-dimensional case. We may give a precise definition as follows. Let f l be an open subset of U and let U be an open subset of 0 such that Cl is relatively closed in U. If A (U ) is the space of holomorphic functions in U let

p(ft) = A (U ~Il)/A (U )

where ~ denotes set difference. If F e A (L / ~ fl) its equivalence class [F] in |3(fi) may be thought of as the ‘jump’ of F on fl, i.e., the difference of the upper and lower boundary values. Of course, these boundary values may fail to exist in any of the usual senses. The space does not depend on the choice of U and is called the space of hyperfunctions in ft. There is a natural proper inclusion Si'(ft)^/3(ft). Indeed one can show that [F] corresponds to a distribution if and only if |F(x±iy)| locally grows at worst as a power of y“ as y i 0. In this case the corresponding distribution f is given by

</, < )) = lim f (F(x + iy) - F{x - iy))<t>(x) dx,yio J

In higher dimensions hyperfunctions are introduced as certain relative cohomology classes for cohomology with values in the sheaf of germs of holomorphic functions. See Komatsu [1].

E xercise 16.2. Show if <t>eCc{U) and y > 0 then

- 1 arg (x + iy)Dix) dx = | y(x^+y^)“V (^ ) dx.

Use this result to obtain Exercise 16.1 in the special case where p is the Poisson kernel o f the upper half-plane.

§17. Operations on distributions

We have already seen how the notion of differentiation may be extended to distributions by duality. Many other operations may also be extended to distributions in the same manner. Suppose fti is an open subset of IR” and Cl2 is an open subset of IR"" and suppose we have a linear map L : C“(fii) ^ C “(fl2)- A formal transpose for L is a continuous linear map L ': C"(H2) -» C “(fti) such that

[ dy = [ dx

for each <#> g C c(Hi) and Cc(^^2)* If L! exists it is unique by Theorem 3.6. If L' exists we may consider its transpose in the sense of functional analysis

It is easy to see that (L')* is an extension of L relative to the natural inclusions Therefore we will denote (L')* simply by L.Explicitly, the action of L on distributions is given by

(Lf, = L » , i l fe C :ia 2 )

It may happen that we have L : C “(f^i) ^ C~(fl2) with formal transpose L ': Cci^i ) C “(fti). In this case a procedure similar to the above yields an extension L Other situations may also arise. Thebasic idea, however, is that in appropriate situations a second transpose will yield an extension. One just has to pick the appropriate spaces in each case.

E xam ple 17.1. Translation. Translation admits a formal transpose, namely t_ . Thus

<t j , <;>)=</, T_a\ <> € cr(iR")

In view of Theorem 3.6 r j has the usual meaning if /g L (IR”, loc).

E xam ple 17.2. H om othety . If A > 0 define (Y^<f>)(x) = cf>(\x). Then Yx has a formal transpose, namely A“”Yi/x. Thus

<Yx/, <>) = X - ( f , Yy <t>), feS )'(R n , <l> € C : m .

Again by Theorem 3.6 Yxf has the usual meaning if /g L ([R”, loc). A distribution /g S)'(IR”) is said to be homogeneous of degree z g 0 if

Yxf = A Y for each A > 0.

18. CONVOLUTION OF DISTRIBUTIONS 47

E xercise 17.3. I f is homogeneous o f degree z then D* f ishomogeneous o f degree z — \a\. The Dirac measure 8 is homogeneous o f degree —n.

E xercise 17.4. Let hGC~(S”“ ) and let HzeQ}'{W^) be the analytic continuation o f \x\ h{x |xl“ ) as in Section 15. Show that the following statements are equivalent.

(A) Hz has a removable singularity at z = —n — m.(B) P.f. h-n-m is homogeneous, necessarily o f degree —n — m.(C) h is orthogonal on S ”“ to each homogeneous polynomial o f degree

m.

Show also that is homogeneous o f degree z if zi^ —n — k, k =0 , 1 , 2 , -----The first part o f the exercise may be done by showing

Y;,(P.f. h _ ._ ) = A— P.f. h _ _ + A— log AR^

where is the residue at the pole z = —n — m.

E xam ple 17.5. Integration. Consider the special case of ‘integration over the fibre’ given by L :C c ( L l x R ^ ) - ^ Cc(Li) where {L<t>)(x) = J* y) dy. If then J i/r(x)(L<#))(x) dx =/]* y) dx dy =

(L'il/)(x, y)<t>{x, y) dx dy where L'il/(x, y) = i/ (x). Then L ': ^C^iDxR^) is a formal transpose. Thus we obtain an extension of L to

E xam ple 17.6. Linear transform ation. Let A be a non-singular n x n matrix and let {L<f))(x) = <l>{Ax) so L : C~(R”) C “(IR”). Then L has a formal transpose V given by (L'<f>)(x) = |det A\~ </>(A~ x). Hence L extends to L:^'(R'^)->^'{R'^) given by

(Lf, 4>) = Idet A|- {f, L-^4>), f e ^ \ R - ) , <f> G C:(R^).

Example 17.2 is just a special case of this one. If A = - I we denote L f byand read it as "f check". For the general non-singular A we will denote

L f by A"\/. We write A"^ rather than A so that A .(B .f) = (AB).f.

§18. Convolution of distributions

In general to define a convolution / * g it is necessary to place growth or support conditions on / and g. In the case of functions and measures we have already considered convolutions under growth conditions in Sections 2 and 4. In this section we will consider the convolution of distributions under fairly general support conditions. In Chapter 2 we will consider the convolution of distributions under some growth conditions.

Since we will study several convolution products, it is necessary to verify that they agree in common domains of definition. In the present section Lemma 18.16 shows that the convolution of distributions agrees with the convolution defined for functions in Section 2. The case of Borel measures is left for the reader to consider.

If we denote by the set of all <>g C "(1R") such thatsupp / nsupp (/) is compact. Then is a linear subspace of C “([R”).

L em m a 18.1. I f f then f has a unique extension to a linearfunctional on C J such that if i/g C c(IR”) cind ij/ = l in a neighbourhood o f supp / n supp (f> then {f,<t>) = (f,

Proof. Suppose and «/f' = l in a neighbourhood ofsupp/nsupp 4>. Then (i/f- /')<> = 0 in a neighbourhood of supp/. Thus (/, = if, and therefore / is well-defined on C “. Since we maychoose a to work for any finite number of (f) linearity is obvious.

In this section we will always consider / to be extended to C “.

E xercise 18.2. J//€S>'([R”) show:

(A) <D7 ,< >) = ( - ! ) '“'</, <l>eCJ(B) {df,<t>} = <f,e<t>}, 0 € C ”(R")(C) I f <t>€ C°°(IR”) and supp /fl supp <]> = 0 then 4> e C J and {f, <f>) =

0.(D) =

Let Aj, j = 1 , . . . , m, be closed subsets of IR". We will say that the m-tuple ( A , , . . . , A^) satisfies condition (C) if for each compact subset Kof F the set (Ai X • • ■ X A „ ) n K ^ is compact, where K\A _

E xercise 18.3. Let A and B be closed subsets o f W and let K be a compact subset o f R ”. Then K f is a closed subset o f R^” and the following statements are equivalent:

(A) ( A x B ) n K 2 is compact in R "'(B) A n ( K - B ) is compact in R"(C) B n { K - A) is compact in R ”

Here K - A = { k - a \ k e K , a e A } . From now on we will drop the subscript m in and simply write K^. Note if Aj are closed subsets of R"" and L and Lj are compact subsets of R"" and ( A i , . . . , A^) satisfies condition (C) then so do ( A i , . . . , A^, L) and (Ai + L i , . . . , A^ +L^).

L em m a 18.4 I f (A, B ) satisfies condition (C) then A + B is closed.

Proof. Here A -^ B = {a + b \ a e A, b e B } . Assume g A, b ^ e B and au,-^bj^-^c. Let K be a compact set containing a + bk, fc ^ l. Then ai = + bf ) - bj e A n (K - B ) and therefore has a convergent subsequence Uk- ^a e A. Then b i = (ak~\-bi) — a i ^ c — a implies c — a e B . Thus c = a {c — a) G A B.

A cone is a subset T of R" such that A F ^ r for each A > 0 . A cone F is convex if and only if F + F ^ F . If F is a cone the dual cone is defined by F" = {^ g [R” 1 <1, x) > 0 , X g F}. F" is closed and convex and F" " is the smallest closed convex cone which contains F. A closed convex cone F is said to be salient if it contains no one-dimensional subspace of R". The closed convex cone F is salient if and only if its dual cone F" has non-empty interior F q. If F is a closed convex salient cone and | g F q then 6 (0 = inf x) 1 X g F, |x| = 1} is equal to the distance from | to the boundary of F^ and we have F ^ {x g R ” | x ) > 6 ( 0 jxl}.

E xercise 18.5. Verify the above statements concerning convex cones and show that if T is a closed convex salient cone then the m-tuple ( F , . . . , F) satisfies condition (C), and if H is a closed half-space with interior normal rj, H = {x g R"" | (r\, x )> 0 }, then the m-tuple (H, F , . . . , F) satisfies condition (C) i/ t g Fo.

E xercise 18.8. Let A, B and C be closed subsets o f R". Then (A, B, C) satisfies condition (C) if and only if (A, B ), (B, C), (A + B , C) and (A, B + C) all satisfy condition (C).

E xercise 18.7. Let A^, . . . , A^ be closed subsets o f R ” with all but at most one o f them compact. Then ( A i , . . . , A^) satisfies condition (C).

If are distributions on R ” we will say that they satisfycondition (C) provided that the m-tuple (supp/i,. . . , supp/^) satisfies condition (C).

T h eorem 18.8. Let f, ge3)'{W^) satisfy condition (C). I/<#)g C c(R") define G^(x) = (g, Then G^g C“ for each <>g C “(R”) and thereexists a unique distribution / * gGSJ'(D^”) such that

Proof. Let A=supp/ and B=suppg. First note that if <#>g C “(R”) then a —> T-a<l> is a continuous map of R ” into C “(R”) and therefore G is a continuous function on R ”. If Cy = ( 0 , . . . , 0,1, 0 , . . . , 0), with the 1 in

the jth place, then r\ G ^ (x + t6j ) -G^(x) ) = <g, and~ T‘-x<l>) 'r~xDj<l>, (t 0), in Cc(IR'') for each x. This fact is a

consequence of the mean value theorem and the definition of the topology on C “(1R”). It follows that DjG^ = Go.<f, and therefore DjG^ is continuous. By induction we conclude that G € C “(IR'') and D"^G = Gd- . We see easily that supp G > ^ (supp <#>) - B. Now for any compact set K we have A C[(K — B) is compact by hypothesis. Thus G^ e C “. To show that / * g exists we must check that the linear functional <t> — (/, G^) is continuous on for each compact set K. Fix now a compact set K and choose 6 e C “(R”) such that 0 = 1 in a neighbourhood of the compact set A n ( K - B ) . If </> G C k then supp G ^ ^ K — B and by the definition of the extension of / to C ” we have

</, G^) = if, 6G^) for each <t> g

Let L = supp0 so supp (0G ^)^L. Then it suffices to show that ^G^ : C k — CZ is continuous. Now G^(x) = (g, and if x g L thensupp (r-x<f>) ^ K — L for each <#>g Ck. Since supp (t_x<>) stays in a fixed compact set K — L we have constants C and m such that

\G^M\ = Kg, t^A)\ ^ G mSiX sup \r. D <t>\< C max sup \D<t>\131 I3l m

for each x e L and each <l> g C k- Then since D" G = Go“<f> we have

suplD“G^ l < C max suplD < >|L l3l:Sm+lotl

for each <> g We may now use Leibniz’ formula to estimatesu p1D “(0G^)1-

Note if either / or g has compact support then the hypotheses of the theorem are satisfied and so / * g is defined. Note also for any distribution f we have / * 5 = 6 */ = /.

T h eorem 18.9. I f f ,ge^'{W^) satisfy condition (C) then supp/ + supp g is closed and supp (/ * g) ^ supp / + supp g.

Proof. Let A = supp/ and B=suppg. Then A + B is closed by Lemma 18.4. Let <>g C “((R”), K = supp<l> and suppose ( A - \ - B ) n K = 0 . Then A H ( K - B) = 0 and supp G ^ ^ K - B . Thus </ * g, c )) = </, G^) = 0. Since A + B is closed it follows that supp (/ * g) ^ A + B.

T heorem 18.10. I f f , gGS>'(U^”) satisfy condition (C) then D°"(f^ g) = (D “/) * g = / ❖ (D “g). In particular D°"f = * / = / * D “6 for any distribution f.

Proof. For each <^>€C”(IR") we have <D“g, = (-1 )'“'<g,T_xD“<#>)=(-l)'“'Go.<,(x) = ( - iy “'D“G*(x). Thus { f * D<‘g,<f>) =

D “G<,) = <D“/,G*) = <D 7 *&</>>. We also have <D“(/=f=g), <{>) = ( - ! ) '“'</ * g, D “< .) = ( - ! ) '“'</, Gd»*) = ( - ! ) '“'</, D “G*> = {D % G^)= <D 7 *&<#►).

Note Theorem 18.10 is not true without some hypotheses on / and g. For example, if n = 1 then D Y * 1 = 6 * 1 = 1 whereas Y * D 1 = Y * 0 = 0, where Y is Heaviside’s function.

E xercise 18.11. then and {f,T-^{J/} =(Tx/> I f f g satisfy condition (C) so do and g'', and i f gY =r * g " -

T h eorem 18.12. I f f, ge 'CM" ) satisfy condition (C) then T^(/*g) = (tJ ) * g = / * (r^g) for each x e U .

Proof. lf(f>e Cr([R”) then {r^(f * g), <>) = </ * g, = </, NowG _ 4,(y) = <g, T_ -y<#>) = G^(x + y) = (T_^G^)(y). Thus t_^G g and if, t- xG^) = <tJ , G^>, which implies t (/ * g) = (tJ ) * g. Also G _^ (y) = (r^g, T_y<#>) implies that t (/ * g) = / * r g.

T h eorem 18.13. ///g S j'([R”), and if f, il/ satisfy condition(C) then / * i/f G C“(1R' ). Moreover, C “ for each x g R'' and

Proof. Let A = supp / and B = supp if/. Then supp ^ {x} — B and A n ( { x } - B ) is compact by condition (C). Thus Let <o be anopen subset of R ” such that a> is compact and choose 0 g C c(R”) with 6 = 1 in a neighbourhood of An(a> —B). If c^GC“(a>) then ( f * = if, OG^) where G^(x) = (il/, We note (0G^)(x) =J 0(x)il/{y)<l>{x-\-y)dy=^ 0(x)i/r(y-x)<;>(y) dy. If s > 0 set

= 0(x)il/{s7r- xk Yett)

where the sum is over ttgZ” where Z is the ring of integers. Note that the sum is finite because <j> has compact support. Ie(x) is just a Riemann sum for the integral which gives (0G^)(x) and we have g C " (R”), supp Jg c supp 6 and —> D" {6 G^) uniformly as 8 0. Thus

le —> OG4, in C “(IR") as £ —> 0. It follows that

</*(/>,<#>) = </, 0G^>

= Mm if, I^)eiO

= lim £" X (f’ 0T ti>'‘)4>{eTv)eiO

= I </> Ty>P'")<t>iy) dy

If y E o) then supp TyiJ/'' a> — B and so since 0 = 1 in a neighbourhood of A n ( o ) - B ) we have (/, = {/, Tyi/r'') for yEo). As in Theorem 18.8where we showed that is C * we can show that y ^ {/, Oryil/'') is in C"(IR"). It follows that (/, Tyil/"') is a C ” function of y e co and therefore isin C“([R”) since o)

*ywas arbitrary. We have shown ( f * i/r, <>) =

J (/, Tyi/fVc Cy) dy for e C “(co). Hence / * i/f is a function in co and / * «/ (y) = (/, for y E o), and so for y e (R” since co was arbitrary.

Exercise 18.14. Let T : C^([R") ^ C 'ClR") be a linear map. Then T is continuous and commutes with translations if and only if there exists /e such that T{(l>) = f <l> for each <f> e C”(IR"")-

Suppose now f, ge3)\W^) satisfy condition (C) and <l> e C cCR"") then by Theorem 18.13 g ''* </>(x) = (g, and therefore (/ *g , <f>) =(f, * ^)- We will now show that we can drop the condition that <#> havecompact support.

Theorem 18.15. L etf, g e S j'CIR'') and i/f e C “((R'') satisfy condition (C)for triples. Then e C “*g, (/* g) **/' = /* (g * <A)-

g'' * i/f'' E C “, i f * g, = </, g'' * i/rv and

Proof. Let A = supp/, B = su p p g and C = suppi/f. Then (A + B )(T (—C) = (A + B )n ( {0 }—C) is compact since (A + B , C) satisfies condition (C). Thus By Theorem 18.13 is C~. SinceA n ( - B - C) = A n ({0} - (B + C)) is compact by condition (C) for (A, B + C) we have g' i/ ''e C " Choose 0 e C “(R”) such that:

(a) 0 = 1 in a neighbourhood of (A + B )(T (-C )(b) T_x0 = 1 in a neighbourhood of B n ( —{x }—C) for x. in a neigh

bourhood of A.

To see that we can choose 6 with these properties let V be an open neighbourhood of the origin such that V is compact and V = — V. Choose

0 such that 0 = 1 on (A + B )n (V + V - C ) + V which we can do since this set is compact by condition (C) for (A + B , C). Then (a) is satisfied. Since V = - V we have (A + B + V) n ( V - C) c (A + B ) n ( V + V -C )+ V . Thus if XG A + V then = 1 on (A + V - { x} + B ) n ( V - { x } —C). Now U = A + V —{x} is an open neighbourhood of the origin and therefore {U + B ) r \ { V - { x } - C ) is an open neighbourhood of B n ( - { x } - C ) and

= 1 on this neighbourhood for each x g A + V. Thus (b) follows. Now by (a) we have (/ * g, i/r'') = (/ * g, = {f, g'' * (Oil/'')) where the second equality follows from the remarks preceding the theorem since 6 il/''e Cr(R"). By (b) and Theorem 18.13 g'' (6 il/'')ix) = (g, (t- J ) ( tA Y ) = (gATx^Y) = g''* for X in a neighbourhood of A. Then (/, g ''*(Oij/'')) = </, g'' * il/''). Finally, /, g and T ij/ satisfy the hypotheses of the theorem for any x g IR and therefore by what we have already proven ( f * g> = (/, * (TjcA' )) = (f, ^Ag * Then by Theorem 18.13 (f ^g)*l/f(x)=/*(g*l/f)(x).

Lemma 18.16. 1 //, gGL\[R",loc) satisfy condition (C) t h e n f ^ g e L (IR”, loc) and f * g(x) = J /(y)g(x - y) dy. In particular f g = g^ f.

Proof. Let co be a relatively compact open subset of IR”. Let A = supp = supp g and choose 6 g C ”(R'') so 0 = 1 on a neighbourhood of A n (o) — B). Then choose x such that x = 1 in a neighbourhoodof o)-supp 0. If (f>eCA<^) then ( f g, <l>) = (f, OG^) where G ^Cy) (gj T-y<l>) = J g( )</>(x + y) dy. Thus we have:

</ * g, <#>) - II/(y)0(y)g(x)<^>(x + y) dy dx

= ||/(y)0(y)g(x -y)d>(x) dy dx

" " I I fiy )0{y)xix - y)g(x - y)<i)(x) dy dx

It follows that / * g - (Of) :5c (^g) in (0. Since Of and xg are both integrable we have / g g L ((R , loc). Finally, if x g o> then f * g(x) =; / ( y ) 0 ( y ) g ( x - y ) d y = J / ( y ) g ( x - y ) d y since 0(y) = l if y e A n ( { x } - B ) and XGO).

Theorem 18.17. f//, g g 2i'(R") satisfy condition (C) then / * g = g * /.

Proof. L et <f>, »/reC~(IR"). Then we have:

{{f * g) * <t>) * iff = (f * g) * ( *«/')) (Theorem 18.15)

= / * ((g (Theorem 18 .15)

= / * ( '/ '* ( g * 4 > ) ) (Lem m a 18 .16)

= (/*«/ ')* (g * <#») (Theorem 18.15)

= (g * < ) * (/ * ^) (Lem m a 18.16)

= g * (<f> * (/ *»/')) (Theorem 18.15)

= g * i ( f * ^ ) * 4>) (Lem m a 18 .16)

= g *(/*(«/<* <#>)) (Theorem 18 .15)

= (g */ )* (« / '* <f>) (Theorem 18 .15)

= (g */)*(</>* lA) (Lem m a 18 .16)

= ((g */)*</>)*«/' (Theorem 18 .15)

In particular ((f * g) * <#>) * t/»(0) = ((g * f ) * <f>)* ilfiO) and therefore by Theorem 18.13 {( f * g )* <f>, 4'' ) = {(g *f)*<t>, for each ip € C“(IR"). Thus (/*g)*<f> = (g*/)*<f> for each <f> e C ”(IR”). Evaluating at 0 again we obtain f * g = g * f.

Theorem 18.18. I f f , g, he3)'{U") satisfy condition (C) for triples then

/ * (g * h) = (/ * g) * h

Proof. If 4>e Cr(IR") then:

( f * ( g * h), <f>) = (f, (g’' * h'') * <f>)

= </,§''* (h'' * <t>)}= ( f * g, h''* <t>)= ( ( f * g ) * h , }

where we have used Theorem 18.15 four times.

Note some hypothesis is necessary for associativity as the following example, with n = 1 , shows.

( Y * D S ) * 1 = 8 * 1 = 1

y * ( D 8 * l ) = Y * 0 = 0

Here Y is Heaviside’s function.If A is a closed subset of IR" we denote by 2)^ the space of distributions

on R" with supports in A.

Corollary 18.19. I f T is a closed convex salient cone in R” then is an associative and commutative convolution algebra over the ring o f constant coefficient linear partial differential operators. Moreover, if H is a closed half-space with interior normal in Tq then is a Q)r-module. The locally integrable functions with supports in T form a subring o f S)f, C~ is an ideal in and is a submodule o f

If M = 1, r = [0, oo) then V {R , loc) H S f is a classical convolution ring. A theorem of Titchmarsh states that it has no zero divisors. The field of quotients of this ring is the subject of the operational calculus of Heaviside and Mikusinski (see Yosida [1]). More generally a theorem of Lions (see Lions and Magenes [1]) states that if T is a closed convex salient cone then Sip has no zero divisors. Note without some support condition we do have zero divisors as the example, for n = l , D 8 * 1 = 0, shows.

Let p be a mollifier. Then p'' is also a mollifier. If <>g C “(R”) then Pe * </) G Cc (R”) and by Theorem 3.1 we have D “(pg * <f>) = p * D “<#> ^ D “<> uniformly on compact sets as 8 —> 0. For 8 < 1 the support of p * <#> stays in a fixed compact set. Thus p * in Cc(R”). If /g S)'(R'') itfollows that

i f * Pe, <l>) = (f, i l <t>)

Thus f * Pe~^f weak* in S5'(U ”). If jE is a Montel space then the weak* and strong topologies induce the same topology on any weak* bounded set. Therefore we also have /* Pe strongly in S)'(IR”). It follows that C “(R”) is sequentially strongly dense in SJ'(IfS”).

Exercise 18.20. Modify the preceding argument, i.e., cut and regularize, to show that for any open subset f l o/ R ” we have Cc(fl) is strongly sequentially dense in 2)'(fl).

Exercise 18.21. Let P(D) be a constant coefficient linear partial differential operator. Show that {<(> g C”(R”) 1 P(D)'(R”) and suppose for each <^>GC*(R”) we have supp (/ * <>) ^ supp cp. Show that supp ( f * g ) ^ supp g for each g e ^'(R”). Conclude there exists a constant coefficient linear partial differential operator P{D) such that f = P {D )8 .

We will now consider a few results which illustrate some applications of convolution products. Let P(D) be a constant coefficient linear partial differential operator. By a result of Malgrange and Ehrenpreis (see Hormander [2]) the operator P(D) admits a fundamental solution E ; that

is, there exists a distribution E such that P {D )E = 8 . Note we can also write P (D ) 8 E = 8 so that we may view E as an inverse for P (D )6.

Theorem 18.23. Let P(D) be a constant coefficient linear partial differential operator with fundamental solution E.

(1) I f fe& iW ") and E, f satisfy condition (C) then u = E * f is a solution o f the differential equation P(D )u =/.

(2) I f ueQ)'{W^) and E, u satisfy condition (C) and P {D )u = f thenu = E ^ f .

Proof. For (1), by Theorem 18.10 P(D )(E */) = (P(D )E) */ = 6 */ = f. For (2), if P(D )u = f then u = 8 u = P (D )E u = E ^ P(D )u = E ^ f again by Theorem 18.10, since the appropriate support conditions are satisfied.

Corollary 18.24. I f P{D ) admits a fundamental solution E with support in a closed convex salient cone T then P(D) is an isomorphism o f 9)y on to itself and an isomorphism o f Cp onto itself. Moreover, if H is a closed half-space with interior normal q in Fq then P(D ) is an isomorphism o f on to itself and an isomorphism o f C h onto itself.

We turn now to systems. Consider first the case of square systems, say P(D) = (PijiD)) is a pXp matrix of constant coefficient linear partial differential operators. Let M(|) = detP(|) and let S(^) be the matrix of cofactors of P(|), so S(^)P(|) = P(^)S(|) = M(^)Ip where Ip is the pXp identity matrix. If M(^) is not identically zero there is a fundamental solution K for M (D). If we let E = S(D)(KIp) then E is a p x p matrix of distributions and P (D )E = Sip, Thus we are justified in calling E a fundamental solution of P(D). As in the scalar case we can use a fundamental solution to solve differential equations. We note the following useful result.

Lemma 18.25. I f M(§) = detP(^), where P(^) is a pXp matrix o f polynomials, then P(D) admits a fundamental solution E with support in the closed convex salient cone F if and only if M (D) admits a fundamental solution K with support in F.

Proof. We use the notation above. Obviously if K has support in F so does E = S(D){KIp), Conversely, assume E = (Ey) is a fundamental solution for P(D ) and supp Ejy ^ F, 1 < i, j < p. Since P (D )E = Sip we have P{D ){8 Ip) * E = dip. Since Sip is a commutative and associative convolution algebra it makes sense to take determinants of matrices with elements in S>p. Thus det (P(D )(6Ip)) * det E = 8 , i.e. M (D ) 8 * det E = 8 , Thus if we take K = det E then KeQ)^ and M {D )K = 8 ,

If F(D ) (scalar or square matrix) admits a fundamental solution with support in a closed convex salient cone T we say that P(D) is hyperbolic with respect to each direction in Tq. See Garding [3] or H5rmander [2] for an algebraic characterization of hyperbolic operators.

Consider now the case where P(|) is a p x q matrix of rank p. If J = 0‘i. • • • ,/p) where l^ / i < j2, • • • let F'i^) be the p x p matrixwhose fcth column is the j^th column of P(^) and let Qj(|) be the determinant of P^(^). That P(^) has rank p just means that at least one of the Qj(^) is not identically zero. It follows that there exist distributions Fj such that X Qj (F>)Fj = 8 ; for example, we could simply take one of the Fj to be a fundamental solution for the corresponding Q j(D ) and take the remaining Fj to be zero, though for various problems it may turn out that there are better choices with various desirable properties. Let Q f{^) be the (i, fc)-cofactor of P^(^). Then if / = ( j i , . . . , jp) we have

Z Pi/ ,(l)Q f(l) = Q/(l) if k = ih = l

0 if k i^ i

If we set Ejk = Sh=i Z Q j ^(D)Fj , 1 < / < q, 1 < k < p, where the inner sum is over / with jh = j, then we obtain a q X p matrix E of distributions and a computation shows that P (D )E = Sip. We may of course use E to solve equations of the form P(D)u =/, but a more important property of E is that we may use it to solve the transpose system P(DYv = g when g satisfies appropriate compatibility conditions. (See for example Lancaster and Petersen [1].)

As a simple example we take P(D) = (D i , . . . , D^). Then a fundamental solution is an n X 1 matrix E satisfying

l D ,E , = 8

Consider now the transpose system P(DYv = g, i.e.

DjV = gj.

Obviously this system has no solutions unless g satisfies the compatibility conditions

DjSk = Dkgj

In this case if E, g satisfy condition (C) and we set u = E * g = Z * g, we have

DfcU = Z P/ * Dkgj = Z Pj * A-gfc = Z A'Pi * gk = gfcThere are other notions of fundamental solutions in the case of systems

in addition to those we have considered above; see for instance Ehren- preis [1]. Convolution products and the notion of fundamental solution when combined with various other ideas and techniques provide useful

tools for the study of partial differential equations. Even our few computations above provide considerable motivation for introducing convolution products whenever possible. We will encounter in Chapter 2 more situations in which we have convolution products with good properties.

Remark 18.26. Now that we have the notion of convolution of distributions available we may extend the concept of the Newtonian potential to distributions with compact support. If n > 3 and wedefine

Pot(u) = |xp-” *M

By Theorem 18.23 we have as before APot (m) = (2(2—n)7r"^ /r(n/2))M. Let In the introduction to Chapter 1 we considered the dipoleYfOjDjS. Using theorem 18.10 and the formula for the derivatives of

given in Section 10 (formula 2) we obtain

Pot (X <0/Dj8) = X ‘O/I)/= (2 —nXa>, x) |xl“”

Up to a constant this potential is the function i/tq considered in the introduction to this chapter.

§19. Oscillatory integrals

In this section we consider as an example a class of distributions, the oscillatory integrals, defined by divergent integrals. Oscillatory integrals are important since they may be used to construct parametrices, roughly approximate inverses, for partial differential operators. See for example Hormander [3], Duistermaat and Hormander [1], and Melrose [2], to mention just a few examples. We will be concerned with integrals of the form

A (x) = |e‘*<"®'a(x,0)d 8

where <f>, the phase function, is a suitable real valued function such that the integrand oscillates rapidly for large \0\. We can then expect, under suitable hypotheses, there will be enough cancellation that we can make sense of the integral. The function a, called the amplitude function, will be allowed to have polynomial growth in 6 . It would be too restrictive to require the integral to define a function A. We will therefore interpret it in the distribution sense. Thus we will actually be concerned with integrals of the type

<A a(x, 0)m(x) dx dd (1)

19. OSCILLATORY INTEGRALS 59

To be specific, let O, be an open subset of IR". A real valued function C ~ (flx(R ^ -(0)) is called a phase function if <f> is homogeneous of

degree 1 in the phase variables 6 = (6 i , . . . , 6 j ) and has no critical points in —(0), i.e. d<l>ix,6)i^0 if 0 ^ 0 . The amplitude a will be afunction in the space S^s(n,IR^) which we shall encounter again in Chapter 3. If m ,p , 8 are real numbers, 0 < p < l , 0 < 6 < 1 , we define

to be the set of functions a G C ”(flx[R^) such that for each compact subset K of Cl, each N multi-index a, and each n multi-index jS we have

PK.o<.3(a) = sup sup (1 + \DeD^a(x, 0)] < oox g K 6

We topologize IR' ) by means of the seminorms Pk.c,,0-

E xercise 19.1. Show that 'S a Frechet space.

It is convenient to introduce also the notations

s “s (a iR ^ )= u s^ 8 (a R ^ )m

s - 6( a = n s - s ( a = s - “( a

L em m a 19.2. I f ^ is a phase function there exists a first order differential operator

L = X Oj^e: + Z + c

such that a,-€ S® (a 15^), k> c € S “’( a a,(x, 0) = O, bk{x,d) = 0 if|01<1 and such that L 'e“*’ = e'**’, where L' is the formal transpose o f L.

Proof. Let be such that (/r(0) = l if l6l < l . Define

a, = id - )i\de<t>\' + \e\-

and let L' = -'ZajDe.-'ZbkD^^ + ilf. Note a, (respectively, b ) is homogeneous of degree 0 (respectively, degree —1) in 6 for large |0l. Thus L has the required form with c = ip+'^ +X

Suppose now </> is a phase function and a 6 S^g(a[R^)- If m + N < 0

and u € C ”( a the integral

(A, w) = I I e ‘* ‘'‘ ®’a(x, $)u(x) dx d 6

is absolutely convergent and defines A e C(fl). Indeed

A (x) = d0

Now let L be a differential operator with the properties in Lemma 19.2. Since L'e'* = e‘" we have for any integer fc > 0

e^L^{au)-e^^L^^\au) = L\t^^)L^(au)-&^L^^\au)= - I De,(a,e‘'^ L "(au ))-Z D ^(b ,e‘*L'^(aM))

By Exercise 6.2 and Theorem 6.4 we have

II 'L^(au) dx dS-f* lel<R

e^^L^^\au) dx dO

_ j^N-i r f ^ jRco)a,(x, R<o) day dxJn

The term containing the integral over is bounded byfor some constant C, where 0 < f :^ l is given by t = min (p, 1 —6). As jR —>00 the first integral converges absolutely if m + N -fc t< 0 and the second integral converges absolutely if m + N -(fc + l ) t< 0 . Thus if m - \ - N - k t - t < 0 we define the distribution A eQ fiS l) by

<A, u )= lim f f ^ ^ L ^ {au)dxde J Jlei<R

= dx d0

and the definition is independent of fc. In particular if m + iV - f < 0 we may take fc = 0. Then we see the distribution A is defined as a principal value at infinity in the phase variables. Summarizing we have:

T h eorem 19.3. I f a e is a phase function there existsa distribution A e QfiO) such that

<A, u) = 1 1 (aw) dx de, u e C”(D) (2)for any integer fc> 0 with m + N —fct<0 where t = min(p, 1 — 6). In particular A has order <fc for any such fc.

Even for small values of fc the operator is very complicated to compute in general. Therefore we will give another formula for A. First note maps continuously into IR^). Inserting (1 +

integrand in (2) and observing that for

m -\ -N -k t< 0 that + is integrable on we see that 1<A, w)l<Cpk,o,o(^*"(^w)) where K = supp w. From the continuity of V" it follows that a-^ {A ,u ):S '^ ^ s{il,R ^ )^ ^ is continuous. In order to exploit this continuity we need some information on the topology of R^), Wehave:

L em m a 19.4. I f m '> m and B is a bounded subset o f S^s(n, R^) then the topology o f pointwise convergence, the topology o f C “(flx[R^) and the topology o f R^) all coincide on B.

Proof. Indeed all three topologies are Hausdorff and the first two are weaker than the third. Thus it suffices to show that B is relatively compact in R^). Let at be a sequence in B. Since the sequenceis bounded in C~(flx[R^) it has a convergent subsequence a . Then a^-^ a in C “(flx[R^). For any compact set K and multi-indexes a, |3 we have

\D-^DWix, e)\^ Ck,„,3(1 +for fc > l, x e K , 0 €lR^. Since D eD fa^ — DJD®a pointwise we see

\D‘SD ta(x,e)\ ^ C K ,.M + \ e\ r

so a€S^s(fi»R^)- Finally

f>l«l+Sl3l

( l + le lr" ''"" '“ '-®'^'lDSDS(a(x, 0 ) - a i ( x , e ))1^2C K .„ .3 (l+ 1 1)’" '" ’'is small for large \6 \ since m '> m. It is also small for small || if k is large, since al,-^ a in C ~(ftxR^). Thus a i ^ a in S^5(fl,R ^).

C orollary 19.5. I f m '> m ,ilfG C^CR" ), ip(0) = 1 and a e S^s(n, R^) then il/(e6 )a{x, 6 ) a(x, 6 ) in R^) as 8 -» 0.

Proof. Indeed il/is6 )a(x, 0 )—> a(x, 0) pointwise and il/{s6 )a(x, 6 ) lies in a bounded subset of S^s(ft,R^) if 0< e < l .

T h eorem 19.6. Let <(> be a phase function and let a e R^). LetA be the distribution defined in Theorem 19.3. I f i/f g Cc(R^) o,nd i/f(0) = 1 then

{A, u) = lim J J 6 )u{x) dx dS

for each u g C “(f )-

Proof. If wGCr(ft) define I(a ) = <A, w), a e S ls i^ lM ^ l Then I : Sp s(fl, R^) —> 0 and the restriction of I to R^) is continuous foreach’ m. If aG S^5(n ,R ^ ) then il,{se)a(x, 0 ) ^ a(x, 6 ) in forany m '> m and therefore I(il/(eS)a{x, 6 )) 1(a).

We will write

A = I do [Osc.]

to indicate the distribution A defined by Theorem 19.6.

Theorem 19.7. I f A=Je*'âdO [Osc.] and de<#>(x, 0 )^ 0 for 0 ^ 0 then A e

Proof. Define a differential operator M = X UyDe. + c as follows. Let ilf(e) = l if \e\<l and set aj = i(l-ili)\de<l>\~ De <f>, Now

define the formal transpose M '= —X ,De + i/f. Then Uy g andc = +Z JêAi IR^). Also Since il/(e6 )a(x, 6 ) hascompact support in d we may integrate by parts as before, but with no boundary terms, to obtain

(A, u) = lini ||e‘"^^ ' ^M (i/f(80)a(x, 6 ))u(x) dx dO

since contains no derivatives with respect to the Xy. Now if m '> m then i/r(e0)a(x, 0) ^ a(x, 0) in S^s(n, DR ) and maps IR )continuously into IR^). Thus A = i dS [Osc.] for anyinteger k > 0 . In particular if m —kp4-N < 0 we have

A(x) = I 6 ) dd

where the integral converges absolutely. We may differentiate under the integral sign as often as we please by taking k sufficiently large. Thus A G c “(n).

Corollary 19.8. // A = J e'â dS [Osc.] and U = {x g fl j de</>(x, 0) ^ 0 for each 0 g IR — (0)} then U is open and sing, supp A ^ U.

Proof. That U is open follows from the homogeneity of and the compactness of The rest follows by Theorem 19.7.

Example 19.9. Let fjeC°°(£l) be real valued and let <f>(x, 0) =

M = {x € f t l/j(x) = 0, j = 1 ,

Then

= Z ld/j(x)P + X \fj{x)\ .

Thus if d/i(x),. . . , d/ (x) are linearly independent for each x g M then <f)

is a phase function and, moreover, M is a submanifold of codimension N, Since lde<#>(x, 6 )\ = X we see that if A = J e‘" a d0 [Osc.] thensing, supp A ^ M.

Note that a priori it is not clear that the distribution A defined in Theorem 19.3 does not depend on the choice of the operator L. Theorem 19.6, however, shows that A is independent of L. Corollary 19.8 may be improved by giving lower bounds for sing, supp A. Such bounds involve the growth of the amplitude function a. The corollary may also be improved by estimating the wave front set of A, WF(A), rather than sing, supp A. The wave front set is a refinement of the notion of singular support and will be considered in Chapter 2. Finally we note that there are classes of oscillatory integrals other than the one we have considered. See, for example, Melrose [1]. The homogeneity of the phase function can be replaced by growth properties. See, for example, Hormander [5].

E xercise 19.10.J e ‘'®' a(0)d 0, a e S

i2 - \ l - i l / ) 0j\0 \- , etc.

M ake sense o f oscillatory integrals o f the form That is find an operator L=X^/L>y + c, a ,€

so L'e‘’®''' = e‘'®' For example, set aj =

In the exercise S'^iR^) denotes the space R^).

E xam ple 19.11. Let aGS'^(R’ ) and consider the distribution A on R""-' defined by

A(x, 0 = 1 d| [Osc.]

If M 6 CrdR"""'), «/'€C“(!R") and ip(0) = l then

<A, u) = lim 1 1 0 dx df d|

If □ = D? - is the d’Alembertian or wave operator integrating by partsin the inner integral we see {A, Du) = 0, that is DA = 0.

Chapter 2

The Fourier transform


About 1750 Daniel Bernoulli was convinced that an ‘arbitrary’ function could be represented by a Fourier series. While Fourier series were used around this time by Euler, d’Alembert, Lagrange and others, the general question of what functions could be represented by Fourier series was not settled. The majority opinion was far too conservative and definitely opposed to Bernoulli and it was not until Fourier’s 1811 paper on propagation of heat that reasonably convincing arguments were given that a fairly arbitrary function could be represented by a Fourier series. Undoubtedly Fourier’s proofs could be, and were, criticized for a lack of rigour. The results he obtained by his methods were impressive, however, and encouraged research in the right direction. Thus, Dirichlet in 1829 after being influenced by Fourier’s work gave some fairly general sufficient conditions for a function to have a Fourier series representation. From the 1850s research on Fourier series was carried out by Riemann, Cantor, and many others. This work finally culminated in Lebesgue’s introduction of his integral and its applications to Fourier series, and in Hilbert’s expansion of functions in series of eigenfunctions of compact self-adjoint operators in the early 1900s.

The Fourier transform was introduced by Fourier in his 1811 paper by a limiting argument based on Fourier series. Even today the same argument is used in many textbooks to serve as motivation. The Fourier transform was further developed by Cauchy and Poisson. Over the years a number of special ‘singular’ Fourier transforms have been introduced. The culmination of this work is Schwartz’ very beautiful theory of temperate distributions and the corresponding Fourier transform. It is Schwartz’ theory that we take up in this chapter.

We begin in Section 2 by considering the Fourier transform of finite measures and integrable functions. In Section 3 we obtain some inversion

65

66 THE FOURIER TRANSFORM

formulae associated with the names of Gauss, Weierstrass, Abel, and Poisson and in Section 4 we extend our results to square integrable functions. The main result here is the Parseval-Plancherel theorem that the Fourier transform is an isometric isomorphism of In Sections5 and 6 we introduce temperate distributions (Sf') and the spaces of multipliers {0^) and convolutors (([ c) on At this point the preparation is complete and in Section 7 we introduce the Fourier transform of temperate distributions by duality. The Fourier transform is then an isomorphism of Sf' and carries convolutors to multipliers, and conversely (exchange formulae).

Distributions with compact support are temperate and their Fourier transforms extend to entire holomorphic functions of exponential type. In Section 8 we study this situation and prove the Paley-Wiener theorem which characterizes distributions with support in a given compact convex set in terms of their Fourier transforms, and conversely. Some a priori lower bounds for general polynomials then enable us to obtain a special case of an important convolution theorem of Titchmarsh-Lions. We also sketch an application of the Paley-Wiener theorem to prove an inequality of Bernstein, important in the theory of linear filters. We also give, without proof, a Paley-Wiener theorem for the singular support and sketch some simple applications such as a special case of the convolution theorem of Malgrange-John-H5rmander, and a very primitive propagation of singularities result for constant coefficient partial differential operators (as an exercise). Section 9 is given over to the study of some translation invariant operators defined by the Fourier transform. Basically we construct a functional calculus for the operators of differentiation Dy. The resulting operators include some singular convolution operators and their consideration leads to some quite explicit computations of Fourier transforms of distributions arising from homogeneous harmonic polynomials.

In Section 10 we discuss some simpler properties of homogeneous distributions and their Fourier transforms. We only scratch the surface. In Section 11 periodic distributions are discussed and we show how the Fourier transform theory developed so far leads to Fourier series expansions of periodic distributions. As an application we obtain the Poisson summation formula and sketch how one derives the sampling theorem.

In Section 12 we develop some of Schwartz’ theory of the Laplace transform. We give a Paley-Wiener theorem which characterizes the Laplace transform of a distribution and another one which characterizes the Laplace transform of a temperate distribution with support in a closed convex salient cone. Results in this vein are important in quantum field theory. As an application we give necessary and sufficient conditions for a linear partial differential operator with constant coefficients to have a temperate fundamental solution with support in a closed convex salient

2. FOURIER TRANSFORM. O THEORY 67

cone. The wave operator is the canonical example. This result relates to stability theory for ordinary linear differential operators.

Section 13 is an application of the Fourier transform. We use it to introduce Hormander’s wave front set (or singular spectrum) of a distribution. The wave front set may be regarded as a refinement of the notion of the singular support. In addition to locating the singularities it also gives the frequencies in the spectral decomposition that give rise to them. One may loosely think of it as giving the singular directions. The study of the wave front set makes use of an old idea (see Remark 13.7), the principle of stationary phase, for estimating integrals of Fourier type. Using this idea we give an upper bound for the wave front set of an oscillatory integral. The notion of the wave front set will be very useful to us in Chapter 3 in connection with pseudo-differential operators.

By the end of this lengthy chapter we will have travelled a long way down the road that Fourier set analysis on in the early 1800s. We will of course have omitted many topics. An appreciation of much of the earlier work in this area may be gained by consulting the excellent work of Morris Kline [1].

§2. Fourier transform. theory

Recall L^(R”) is an ideal in the Banach algebra ^(R"^) of complex Borel measures on R ”. If fx e ^ (R ”) we define the Fourier transform /i or of jLL by the integral

A(l) = I e ‘- i < x , 0 d|LL(x), I g O

T heorem 2.1. R iem a n n -L eb esg u e . I fbounded uniformly continuous function on R” L^(R'^) then f vanishes at infinity.

|LtGSS(R”) then and suplfx|<||jLi

jUL is aII- I f f e

Proof. For the inequality we have |/i(^)l:<Jd |fx| = l|jUL||. If t] g R ” then

lA(l + il ) -A (l)| ^ 2 | jsin ^^^^^)jdlfi.l(x).

By the dominated convergence theorem the integral goes to 0 as 17 ^ 0 through a sequence. Since the integral does not depend on we have uniforai^continuity. Suppose now e C^(R”). Integration by parts shows that Dj< >( ) = i j< ( ). Since Dj<t> e the first part of the proof showsthat 111 < (|) is bounded. Thus <t> vanishes at infinity. Let /g L^R"') and let 8 > 0. By Theorem 1:3.4 there exists <#> g C~(IR”) such that |1 /- < s.Then l/(|)-<^(|)|^||/-</>lli<s. It follows that |/(|)|<28 for large 1||.

Theorem 2.2. I f fx, then f dv = J P djut and (jUL*r) = (iv.

Proof. The first part follows by Fubini’s theorem. For the second part let (f> e C“(R''), <f) = l i n a neighbourhood of the origin. If k > 0is an integer then

I d(jUL * v){x) = 1 1 + y)/fc)e“' ''' ’ djUL(x) dv(y)

by the definition of ill * v. If we let fc oo and use the dominated convergence theorem we obtain

| e v)(x)= I J e di'(y).

Example 2.3.

6 = 1.

Example 2.4. Let n > 1 and let /lx be the uniform distribution of unit mass over the unit ball with centre at the origin. Then

where is the Bessel function of the first kind of order v. A good reference for Bessel functions is Watson [1]. This encyclopaedic book also has many interesting historical comments. To see that the formula holds we note

. - ‘““-a *

nTin/2 )I e " ‘'«"(!- 2\(n-l)/2

( n - l ) r ( ( n - l ) / 2)

The result now follows by Watson [1], p. 48.

■t y dt

Example 2.5. Let n ^ 2 and let v be the uniform distribution of unit mass over the unit sphere with centre at the origin. Then

f>(|) = r(n/2)J(„_2V2(1||)

Indeed let K be the characteristic function of the ball {x g R'" |1x 1< s}. Then

2n/2f 2”'= s'* I dx = s"/x(s^)nT(n/2)

2. FOURIER TRANSFORM. V- THEORY 69

where )x is the measure of Example 2.4. We also have

^s(l) = J J dr

9—n/2 rs

If we differentiate these expressions with respect to s and then set s = 1 we obtain

v(l) = -| -(s"A (s| ))U in ds

= 2<"-"'«T((n + 1)/2) J;./2(l^l)}

= 2<-’-« « r((n + 1)/2) /(„-2)/2(lll)by Watson [1], p. 45, equation (3).

E xam ple 2.6. Let n > 2 . Suppose /g L ([R”) is radial, say f(x ) = F(lxl). Then

f ( i ) = (27t)’'« /(„_2)/2(/- |ll)F(r)r"« dr.

Indeed, we have

/( ) = |F(lxl)e-‘< -«>dx

F(r)r""'e-"<“’«> dw dr[q Js "->9_ m/2 i*oo

■ r W 2 ) l T O r"-'i> (r{)d r

where i/ is the measure of Example 2.5.

E xam ple 2.7. If f{x ) = e”*""’ then

f ^ 2V-^>^^r((n + l)/2)T\ ) Q_|_ j |2yn+l)/2

This formula is obvious for n = 1. For n > 2 it follows by Example 2.6 and Watson [1], p. 386, equation (6). An easy direct computation is also available. See the remarks following Example 2.8.

E xam ple 2.8. If /(x) = e~' ''' then /(^) = Indeed, we have

/ (!)= [ e - ‘<«-">-'"' dx = n i ( 4 )J k = l

where

I( s )= f Jj*

'df

I f Q-it+is/2) 'u

■ 1= e-^dt

— ^V2--s2/4— 7T C

Exercise 2.9. Justify the change o f contour in the integral which occurred in the argument in Example 2.8.

Example 2.7 may be derived from Example 2.8 without using Bessel functions, by using instead the relation

f dt

which holds for b > 0 . See Stein and Weiss [1], p. 7.With regard to the examples above it is worthwhile to point out that

the Bessel functions of half-integral order may be expressed in terms of elementary functions. For example, we have

sin , and

See Watson [1],

cos t

Exercise 2.10. Let n = 1. I f x is the characteristic function o f the interval [a ,b ] then x(s) = sin ((&-a)s/2). I f f{t) = (l-\ t\y,that is /(t) = l - l t l if I t l^ l and f(t) = 0 if |f|>l, then f{s) = (2/s)^sin (s/2).

Theorem 2.11. I f feL ^ iW ') and XifeL^iU") then D f = -i{Xjf)\

Proof. Let l e R " and let e, = ( 0 ,0 , . . . , 1 ,. . . , 0 ) .(T ’ in /th place). Then

i (/(e+fe,) -/(^)) = I y (e-‘« "-^ -> -e-‘«->)/(x) dx.

The integrand here is bounded by 2t“' |/(x) sin (x,t/2)|<|x/(x)|. If we let t —> 0 through a sequence and apply the dominated convergence theorem we see that D,/ exists everywhere (classically) and D jf = - i ^ f By

3. INVERSION THEORY 71

Theorem 2.1 / and Djf are continuous and therefore by the remarks following Theorem 1:9.6 Djf is the distribution derivative of /.

Corollary 2.12. I f m > 0 is an integer, feL^ iW ') and then feC ^ iU ^ ) and Dy=i~^^Kx^fy for |a|<m.

Proof. Indeed if |al<m then implies x^/eL^lR''). ByTheorem 2.11 and induction D"/= and then by Theorem 2.1

is continuous, |al<m. Now we may either use Theorem 1:12.2, or the fact that in the proof of Theorem 2.11 we actually computed classical derivatives, to conclude that feO ^iW ^).

Theorem 2.13. I f f e L \ R ^ ) and D jfe L\R^) then D^(^) =

Proof. By Leibniz’ formula Dj(e“* ^ a n d therefore Dj(e‘ ‘ ’ y)6 L^(R”). Since e“'^’ /6 L ((R”) also it follows by Lemma 1:14.1 that J Dy(e“‘ ’'’ /(x)) dx = 0.

Corollary 2.14. I f D J e L \ R ^ ) for \a\^m then D 7(^ ) =In particular l/(|)l<c(l + |||)“'” for some constant c > 0 .

§3. L** inversion theory

Let e > 0 . We define the Gauss—Weierstrass kernel

W (e s) =

and the Poisson kernel

r((n + l)/2) eP (e £ ) = (27r)-"^(e-''^ ')=- A n + l ) / 2

We will use these functions to obtain inversion formulas for the Fourier transform.

Lemma 3.1. I f ^eL^(R"") and then for each /g L^(R”) andeach e > 0 we have

I - y)/(y) dy

where <f> (x) = In particular

(2 '^ )- 1 d = I P(x - y, s)f{y ) dy

(2^ )-" I d = j W(x - y, s)f(y ) dy;

Proof. This result is immediate from the first part of Theorem 2.2 and from the computation

(e‘<-> Y,<D) (y) = j e - ‘<«’’-*>(8f ) d| = <f>Hy-x).

Lemma 3.2. I f e > 0 then J W(x, e) dx = 1 and J P(x, e) dx = 1 .

Proof. For the Gauss-Weierstrass kernel we have

J W(x, e) dx = | dx = J d x = 1 .

For the Poisson kernel we have

J P(x, g) dx =r((n + l)/2)f 6 dx

. (. 0 / 2 J

r((n + l)/2) f dx(n+l)/2 J(l + lxP)<"^i>'2

2r((n + l)/2) r r " - ' dr'n-*'^n/2)

2r((n + l)/2)ir '^r(n/2)

i•7r/2sin”- 0 d0

= 1

where in the last integral we made the substitution r = tan B.

The following theorem generalizes Exercise 1 :3.3 .

Theorem 3.3. Let leL^{W'), Jd>dx = l . I f f e L^iW') then f*4> ^ -^ f in L ”(IR”) as g -> 0 . In particular the G auss- Weierstrass integral W J(x) = J W (x -y , g)/(y) dy and the Poisson integral P J ix ) = J P(x - y, e)/(y) dy converge to f in L ”(IR") as e ^ 0.

Proof. /*^(x)-/(x) = J [/(x -gy)-/(x )]< f.(y)dy implies

ll/*<^s-/llp = ( I | |[ /(^ -e y )- /W ]< f) (y )d y dx^

= | lk y /- / l lp l<f>(y)l dy

3. O INVERSION THEORY 73

where we have used Lemma 1:2.1. The integrand is bounded by 2 ll/llp and by continuity of translation in l 0. By the dominated convergence theorem it follows that the integral converges to 0 as e —> 0.

E xercise 3.4. The dominated convergence theorem applies to sequences. Give the additional argument needed to complete the proof above. The argument required is quite simple, but see the introduction in Buck [1] to see why one should give it at least once.'

C orollary 3.5. Let ldx = 0. I f f eL^(1R' ) then f * <f) ^ 0 in L^([R”) a se -^ O .

Corollary 3.5 is a corollary of the proof, not the statement, of Theorem 3.3. From Lemma 3.1 and Theorem 3.3 we have:

C orollary 3.6. I f feL^(W^) we have the Gauss-Weierstrass inversion formula

fix ) = L ' -lim (27t)-" f d|e iO J

and the Abel-Poisson inversion formula

d|.fix ) = V - lim (2i7) " [eiO J

C orollary 3.7. I f feL^iU*^) and if feL^iW ^) then

/(x) = (2i r ) - " j e ‘<-«>/(|)d| (1)

for almost all xelR''. After changing f on a null set, f is continuous and vanishes at infinity and we have equality in (1) for each x g R ”.

Proof. Each convergent sequence in L*’ has a subsequence which converges almost everywhere to the limit of the original sequence. This fact is part of the proof of the Riesz-Fischer theorem on completeness of L^. See for example Munroe [1], Theorem 42.3. Hence we have SkiO with

fix ) = lim (27t)‘ " f d^k —>oo J

for almost all x. Since /g L^(R”) the dominated convergence theorem yields (1). In particular / = g almost everywhere where g = (27t)“"/ is in L iU" ) whence gGCoCR"").

Theorem 3.8.image.

^ Cq{W ) is one-to-one and has dense

Proof. ^ is one-to-one by Corollary 3.7. Let g g C“(R''). By Theorem 2.13 and Theorem 2.1 g(^) is a bounded continuous function for anyinteger m > 0 . Thus gGL^([R”) and therefore by Corollary 3.7 g(x) = (27t)"" f e' ' ’ ^gd) d . That is if we set / = (27r)”''g'' then /g L ([R”) and f = g . In particular C~(IR”) ^ im ^ and so im ^ is dense in Co([R' ) by Theorem 1:4.10.

If ft is an open subset of R ” we denote by BC(ft) the space of bounded continuous functions in ft. We provide this space with the supremum norm, so that it becomes a Banach space.

Corollary 3.9. ^ :^ ( R ”)-^ B C (R ”) is one-to-one.

Proof. Indeed by Theorem 3.8 the transpose iSSCR"") ofthe Fourier transform Co(R”) is one-to-one. By Theorem2.2 =

Since BC(W") has the induced topology from L~(R”) we see that neither ^ :L ^ (R ”) ^ Co(R") nor ^ ^ BC (R ”) has closed image,since if either did so would the other by the proof above and by Theorem 1:13.6. But if the first one had closed image then im ^ * = (ker ^ ) ‘‘‘ = (0)- = L “(R'") which cannot be since im ^ * ^ B C (R ”). Thus in Theorem 3.8 and in Corollary 3.9 we do not have onto.

We will now obtain some pointwise inversion formulae. Since we will need a pointwise version of Theorem 3.3 we will pause to consider the philosophy of this theorem. If </>g L^(R") and J<^>(x)dx = l we may regard f <t>e as a weighted average of /. We may then expect that /*</>£ will converge to f in some sense as e 0 provided that <f> is small at infinity, since then the contributions from far out will tend to 0 as s ^ 0. That </>g L \ R ”) implies in some sense <f> is on the average small at infinity (not pointwise of course) and hence we obtain convergence. To obtain pointwise convergence we need more stringent conditions on (f>; namely <{> should be uniformly sufficiently small at infinity. There are various ways of formulating suitable conditions. We have chosen one which is easy to apply, but is not too restrictive to be useful. It should be pointed out that the smallness of <f> required at infinity can be relaxed if we require <t> to behave in such a way so as to obtain enough cancellation to still have /*</>£ converge to / as e ^ 0. Such results are quite delicate. We will see an example in Section 4 where <f> is not integrable.

We will use the pointwise inversion results to obtain the theory of the Fourier transform in Section 4. It is possible to base the theory on

3. O INVERSION THEORY 75

weaker inversion results, but the theorems we prove are also of independent interest.

L em m a 3.10. Let <f> be a measurable function on IR” such that there exist constants C > 0 and \ > n such that l<f>(x)l< C(1 + lx|)“ for almost all X. 1 / 1 0 we have

lim f fix-y)< l> ^ (y)dy= 0 .

Proof. Note <t> e L' (1R'') for any r > 1. Define q by + q~ = 1. Then by Holder’s inequality ^

[ \f(x - y)<#>e (y)1 dy < 11/llp ( [ l<#>(y)1‘' dy]\J|y|>8/e '

if q < 00. Since <#> € the lemma follows for q <oo. In the case q = oo the bound above is replaced by H/Hi ess. sup|y|s,5/e |<>(y)| and again the lemma follows.

Let / be a locally integrable function on IR’". A point x g IR" is called a Lebesgue point of / if

l i m r - 4 | / (x-y )-/ (x )ld y = 0.

For any locally integrable function f almost every point x in IR” is a Lebesgue point. Moreover, any point of continuity of / is a Lebesgue point of /. If n = 1 and we take f to be the characteristic function of the rationals then / has no points of continuity, but each irrational number is a Lebesgue point.

Using the concept of Lebesgue point we can now prove a pointwise version of Theorem 3.3.

T h eorem 3.11. Let <f> be a measurable function on IR” such that there exist constants O O and A >n such that l<#)(x)l<C(l + lx|)“^/or almost all X. Assume moreover that J <t>{x) dx = l . I f 1 e( ) Lebesgue point x o f f

Proof. Let x be a Lebesgue point of /. If 6 > 0, since J </>(x) dx = 1 we have

l/*<f>^(x)-/(x)l<I(8)+ I |/(x-y)<#>e(y)ldy

+

•iylas

l/(x)| [ |<#>(y)| dy.- yl>8/e

By Lemma 3.10 the second term converges to 0 as e 40. Also since (f) e the last term converges to 0 as 810. It remains to estimate thefirst term

J(5 )= f l/(x-y)-/U)ll</>e(y)lciy. (2)

Clearly it will suffice to prove there exists a constant A such that if t) > 0 then for some 6 > 0 we have 1(d) < A t] for all e > 0. Since x is a Lebesgue point we may choose 8 > 0 such that

' ■" f l/ (^ -y )-/ W ld y < -n

for 0 < r < 8. If we let

^(0 = I \f(x — to)) — f(x)\d(x)Jon-l

(3)

(4)

then h(t) exists for almost all f in [0, 8] by Fubini’s theorem and (3). By (3) we have

H(r) = j r - ‘h(t)dt<'r|r"e'’ (5)

for 0 < r < 8/e. The function H defined by (5) is absolutely continuous and we have H'(r) = 8> ”“^h(er). Now we have

I(S)= [ |/(x-sy)-/(x)ll<;>(y)ldyJlyl:<8/e

< c [ l/ (x-£y)-/ (x)l(l + ly|)-^dyJ|yl:S8/erS/e

= c\ r " - ‘ h(£r)(l + r)-'^dr

JrS/e

I H '(r)(l + r)-^dr.0

Since H(0) = 0 if we integrate by parts we have

J* 8 / e

H (r)(l + r)-^-M r.0

In view of (5) the first term here is bounded by

C8-"T|8'*(l + 8/£r^ = CT,(l + e/S)-"(l + 8/£)"-'^<CT] for all £ > 0.

3. INVERSION THEORY 77

For the second term we use (5) to obtain the boundr8/e roo

C£“”aJ Tir"g'*(l + r)-'^-Mr<CAT}J r"(l + r)-^-Mr

where the integral converges since A > n. Thus we obtain 1(5) < Atj with

A = c ( i + a| r"(l + r)-'^-'dry

Corollary 3.12. I f e C(IR"), (0) = (2ir)-'* and if D “^»€ L '(R ") for la l< n + l then for each feL ^ iU ") we have

/(x) = l im fe ‘<-«>/(|)«I>(£|)d|eiO J

for each Lebesgue point x o f f.

Proof. Let <f) = <!>''. By Corollary 2.14 we have |<^>(x)l<C(l + lJcl)“”~ In particular is integrable and so since is continuous, by Corollary 3.7 we have <1>(0) = (27t)“” J <f>{x) dx. It follows that J dx = 1 . Now by Lemma 3.1 we have J d | = /*< >g(x). Therefore the corollary follows by Theorem 3.11.

Corollary 3.13, I f f e L \ R ^ ) then

( 2 r r ) 7 ( x ) = l i m f e ‘ < - « > / ( | ) e - ^ l « 'd ^e W J

and

( 2 t7) 7 ( x ) = l i m f e ‘ < - « > / ( | ) e - ^ '« l ’ d |elO J

for each Lebesgue point x o f f.

Proof. The second part follows immediately from Corollary 3.12. For the first part since D “e“’ ' is not integrable at the origin if \a\>n-^l we have to do a little work. If we set <f> = (e“'^*)'' then by Example 2.7 we have = + + Now the rest of the argument in Corollary 3.12 goes through.

Exercise 3.14. Let n = 1. 1/ (t) = (1 — \t\y the hypothesis o f Corollary 3.12 is not satisfied. However, <l>(s) = (s/2)“ sin (s/2) and therefore the proof o f Corollary 3.12 goes through. Conclude i f f e L ^ { R ) and ifM(t, r) = (27t)" ds then /(t) = limb oo b~ Jo r) dr, both in the sense o fL^-convergence and pointwise at each Lebesgue point t o f f . This formula is the first arithmetic means inversion formula or the Cesaro inversion formula.

We have already seen that ^ B C {W ) is one-to-one and isnot onto, indeed does not even have closed range. There is no effective description of the range of but an important theorem of Bochner gives a result in this direction. For details see Donoghue [1]. A continuous function / on R"" is said to be of positive type if for any integer N and any jc i,. . . , Xn in R ” and any Zi , . . . , in C" we have

N

u = l

It follows that f ( - x ) = f(x) and l/(x)l</(0) for each x g R ”. Moreover, by a theorem of Schur the product of two functions of positive type is of positive type. Thus the functions of positive type form a closed convex cone P in BC(W^) and P P ^ P . If we denote by PM the convex cone of positive measures in ^(R"") then Bochner’s theorem asserts ^ (P M ) = P. A corollary of the proof is that /g BC (R ”) is of positive type if and only if Jj* f ix - y)<f>ix)(f>iy) dx dy > 0 for each cf> g (R”).

Exercise 3.15. Let n = l . Let 0 < a < l and let b be an odd integer with a b > 1 + 377/2. Weierstrass showed that the continuous function

oof i t ) = X cos (a 'TTt)

k=0is nowhere differentiable. Show that this function is o f positive type by exhibiting a finite positive Borel measure fx such that ( l = f .

Exercise 3.16. Prove the easy half o f BochnePs theorem: if p ie is positive, then fx is o f positive type.

§4. Fourier transform. theory

The following lemma is the crucial result in developing the theory of the Fourier transform.

Lemma 4.1. Suppose h g L^R"'), 0 is a Lebesgue point o f h and h > 0. Then h eL \ R ^ ) and h(0) = (277)"” J h(^) d^

Proof. Since 0 is a Lebesgue point of h by Corollary 3.13 we have

h ( 0) = l i m ( 2 t7) - " f h ( ^ ) e - ^ ' ^ ' d ^eiO J

Since h > 0 the monotone convergence theorem implies h{0) = (2t7)“” j h(|)dg. Since h(0) is finite it follows that heL^{U^).

4. FOURIER TRANSFORM. THEORY 79

L em m a(27rr^M 2.

4.2. I f feL 'iW ')n L H W ') then f e L ^ U " ) and \\f\\2 =

Proof. Let g = f^ so g = f. Now let h = f * g . By Exercise 1:2.4 h is continuous and by Corollary 1:2.3 h e L^K "). Moreover, by Theorem 2.2 li=/g = l/p. By Lemma 4.1 it now follows that l i € L ’(IR") and (2-n-)"h(0) = J li(g) dg = II/II2. On the other hand we also have h(0) = J /(y)g(O- y) dy = J f(y)f{y) dy.

T h eorem 4.3 P arseva l-P lan cherel. The restriction o f the Fourier transform ^ to nL^(R'") extends uniquely to a bounded linearoperator on L^((R”). I f we denote the extension also by ^ then isan isometry o f L ‘ {W) onto.L^W"). Explicitly, for f , g e L \ W ) we have the following relations.

(A) (/|g)=(2^r(/ lg)(B) J/ gd x= J/ gd x (c ) f = { 2 T T r f

Proof. By Lemma 4.2 (277)“'* ^ extends uniquely to an isometry of L^CIR") into itself and (A) holds by polarization. Moreover, (B) holds by continuity, since in view of Theorem 2.2 it holds on the dense subspace L ’(IR") nL^(IR"). Now let E be the image of Then E is closed since ( 2 is an isometry. Suppose geE-^. By (B) we have for each f e L^(IR") that 0 = J /g dx = J /g dx. It follows that g = 0 and therefore by(A) g = 0. Thus E = L^(IR"). If /e L^dR") and g € L'flR") DL^dR") then by(B) J/g dx =f/|dx =J/g''dx. Thus ^ * g = g'' for each g€L (IR'’)nL^(IR'*) and so by continuity for each g€L^(IR"). Since (217)“"'^^ is an isometry, (277)“" = 1 and therefore (C) now follows.(Here * denotes the Hilbert space adjoint.)

E xercise 4.4. Let T = (277)“"^^ so T is an isometry o f L^(IR") onto itself. Show that T* = 1. Let

Pi=z{1- - i T - T ^ + iT^)E2 = K 1 -T + T ^ -T ® )P3 = K l + iT -T ^ -iT ^ )

P4 = 3(1 + T+T^+T^).Show that each P is an orthogonal projection, L^(R") is the orthogonal sum o f the images o f the P , and TPk = i*‘Pk- Show that each P is non-trivial.

Note if /€L^(IR") then /(x)e"®''‘'eL'(IR'')nL^(IR") and by the dominated convergence theorem J |/(x)P ( 1— dx —» 0 as 8 —> 0. Thus

fiO = L^-lim [ e - ‘<*«>/(x)e"®' ' dxeiO J

Similarly we see

f(| ) = - lim f e-'<*«>/(x) dx4x1s r

Using by now familiar techniques we proceed to obtain pointwise convergence results.

T h eorem 4.5. Suppose g L^((R”) and <t>=^e L\IR”) and j < )(x) dx = 1. Then for each fsL^(W^) we have

/(I) = (27t)~" L ^ -lim f dx,e iO J

I f moreover there exist constants O O and X.>n such that |< >(x)l< C (l + lxl)“ for almost all x e W , then

/( ) = (27t)”” lim [ e "‘ *’ /(x)(8x) dxe 4,0 J

for almost all indeed, at each Lebesgue point o f f.

Proof. ( = <i) = (277^^'' and therefore

(27r)-"e-‘< «>(ex) = | e-'<"-«-'’'> (T j) dr,

= 8-"|e-*<-«><#>(e-H|-r,))dT,

= ir,cf>i)\x)

Thus

(2-77)"" j e"‘< - V(x)(£x) dx = I f(x)(Tf<)>iy(x) dx

= I f(v)rf<l>i(v) dr)

= j f (v)<f>Ai-v )dv

where we have used part (B) of Theorem 4.3. Since feL^ifR"^) the theorem now follows by Theorem 3.3 and Theorem 3.11.

4. FOURIER TRANSFORM. THEORY 81

C orollary 4.6. I f f e L\U ") then

f(| ) = lim [ dx (Gauss-Weierstrass)eiO J

= lim [ e “* ’‘- /(x)e~®''‘'dx (Abel-Poisson)eiO j

almost everywhere and also in L^-norm.

E xercise 4.7. I f feL^iW") and geL^{R^) then { f^ g f = fg.

We would like to have the result of the exercise for f e L ^ i W ) as well, but in this case /*g is in general not even in L^(R”) + L (1R'"). We will extend the Fourier transform to a larger space than L^(IR”) and we will see that the result of Exercise 4.7 holds in considerable generality. For the present the best we can say in case f e L \ W ) is as follows:

T h eorem 4.8. I f f g e . L H W ' ) then f *g (x ) = i d^

Proof. A computation yields = ( t _ for f eL^iW') andtherefore also for /eL^(R") by approximation. Thus

(217)-" I e*<-«>/(|)g(|) d^ = (2rr)-" j (t_ J^)'(^)g(|)

= (2rr)-"jT _J(y )| (y )d y

/(^ +y)g(-y) dy= 1 -

= | / U -y )g (y )d y

where we have used Theorem 4.3(B) and (C).

C orollary 4.9. I f f g e L^(IR") then f * g = (2ir)"(/g)\

Proof. Replace f and g in Theorem 4.8 by / and g, respectively and then use Theorem 4.3(C).

R em ark 4.10. Let x 1*® the characteristic function of the unit ball in R" and let ip = x. If f eL^(U") and g > 0 then by Corollary 4.9

'V(x) dx = I e " ‘<^-^y(x)x(ex) dx

= (2-7r)~7*«/>e(|).

- i< x .€ > /

By the comments preceding Theorem 4.5 we therefore have

/=(27rr"L^-lim /*i/r,.40

By the Parseval-Plancherel theorem the Fourier transform maps L (1R”) onto itself. Thus if we replace / by g and recall Example 2.4 we obtain

g(x) = L2-lim (2iT sr"'^ [ g(x-y)|y|“" /„/2(e“Myl)dye i O J

for each geL^(IR"). In particular if n = 1 we have

g(s) = L ^ -lim — [ g ( s - e t ) ^ ^ d te4.0 7T J t

= L^-limg*<#>^(s)e 4 0

for each gGL^(IR), where <;>(?) = (l/7r)(sin t)/t. This result is not implied by Theorem 3.3 since Note, however, that we do havej" (l/7r)((sin t)/t) df = 1 as an improper integral, so this part of the hypotheses of Theorem 3.3 is satisfied, at least in a generalized sense. It is typical of the Parseval-Plancherel theorem that it leads to such delicate results. It is a very powerful tool when it is at all applicable.

§5. Tem perate distributions and

We define Sf to be the space of all functions <f> e such that foreach pair of integers m, k > 0

= max sup (1 + 1x1 )'' |D“<f>(x)l <<».lotl m X

If we topologize Sf by the seminorms 1*1 then 5 becomes a Frechet- Montel space.

E xercise 5.1. We have continuous inclusions

Cr(IR”) c ^ c C “(IR") and

L em m a 5.2. Cc(IR”) is a dense subspace o f Sf and therefore the restriction map is one-to-one.

Proof. Let (l>eSf. Choose O e C c i W ) such that 6 {x) = l for l x l <l . Define 6h{x) = 6 {h~^x). Then OheCciW) and by Leibniz’ formula

sup(l + lxp)'' lD“(0h<#)-<#))l<C X sup l(l + lxp)'‘D “" <f>(x)l Uls:h

5. TEMPERATE DISTRIBUTIONS AND 83

since 6h — l and D 'Sh (yi=0) vanish for \x\<h. Since (1 + 1x1 )'"' \D <t>(x)\ is bounded we see + lD < >(x)l —>0 as \x\^00. Thus 6h<f> — in 6 as fi —> 00.

The image of Sf' in we will identify with Sf' when it is convenientto do so. Occasionally some caution is needed. The image of Sf' we call the space of temperate distributions.

If then / is temperate if and only if there exist integersm, fc> 0 and a constant O O such that

\{f, <#>)! ^ C for each <f> g Cr(R”).

We may extend various operations on functions to temperate distributions as we did in the case of distributions. Suppose we have a linear map L :S f-^ S f and L has a formal transpose that is, a continuouslinear map L ':S f-^ S f such that

I (L<#))i/f = I <f>(EV) for ®ach <l),il/eSf.

Then we may extend L to Sf' by taking the extension to be (L')*. Note we have an obvious inclusion and the usual commutative diagram

Note if L' happens to map C~(R”) continuously into itself then L extends to as before. If this extension is restricted to Sf' it agrees with theextension obtained above.

Exercise 5.3. I f P e 0 [ ^ i , . . . , In] ^hen P{D)fGSf' and

The point here is that the equation is true for <>g Cc(IR”) but we wish to show that it is true for <f>GSI. It suffices to show that P{—D ) maps Sf continuously into Sf.

Exercise 5.4. I f feS f' , P e 0 [ x i , . . . , then PfeSI"' and

{Pf,<t>} = {f,P<f>), <t>GSf.

It suffices to show that multiplication by P maps Sf continuously into

Exercise 5.5. Let A be a non-singular nX n matrix over the reals. I f <l>eSf we define A.<l> by A.<l>(x) = Then A. i S f ^ S f has a formaltranspose and hence extends to 5 '. I f fe S f' then

(See Exam ple 1:17.6.) I f A > 0 and we take A = A I we obtain the homothety or dilation Y . {See Example 1:17.2.) I f we take A = —I we obtain f .

Exercise 5.6. I f feL^(W ^) then 4> (/, <#>) = J dx is continuous onCc(iR'') for the topology induced by Sf and therefore f is temperate. The extension o f f to a continuous linear functional on ^ is given by the integral; that is, if feL^{W ^) then

</,<#>)=}/<#> dx, <j)GSf.

Caution 5.7. Exercise 5.6 is trivial, but it is necessary to pay attention to such things to avoid going astray. Suppose f eSI"' r\L^{W",\oc). Then

J /</> dx is continuous on (IR”) for the topology induced by Sf, and therefore f extends uniquely to a continuous linear functional on Sf. The extension of f to Sf, however, is not necessarily given by an integral.

Indeed, we can give an example as follows. Let n = l . By Exercise 1:5.4 we can find a function i/f € C“([R) which grows at infinity as rapidly as we please. Let 0 g C“((R) be a primitive of il/ and let g(x) = sin 0(x). Then g is bounded and therefore geSf'. By Exercise 5.3 it follows that f = D geSf'. Since g e C “([R) we have f is the classical derivative of g. Thus /(x) = i/f(x) cos 0(x). Admittedly f oscillates very rapidly near infinity, but none the less it has the same order of growth as ij/. Thus contains smooth functions of arbitrary large growth. Since feS I',

<t>-^if,<f>}=^ cos 6{x) <l>{x) dx, <t> G c r w ,

is continuous for the topology induced on C“([R) by Sf. By suitably choosing i/r we can arrange that this integral diverges for some in Sf. Thus the extension of / to 5 is not given by an ordinary integral. An explicit example may be given as follows. Suppose and <f>{x) = e~'"for x > 0 . Let i/f(x) = 2e and 6{x) = e ''. Then

I|i/ (x)cos 0(x) <l>{x)\ dx > I 2e'' jcos e "" 1 dx

Icos u\ du

5. TEMPERATE DISTRIBUTIONS AND €m 85

In the last integral we may bound the integrand below by (2u) on intervals of the form [fc r —tt/4, fc7r + 7r/4]. Thus this integral diverges.

Remark 5.8. The comments above imply if /(x) = e cosCe'') then \f\ is not temperate. Thus we see that there exist /€5^'nL^(IR,loc) with 1/1 Sf'. Note that if f(x ) = e cos (e'') and g(x) = sin (e"") then geS f' and \f\ is the classical derivative of (sgn f)geS f'. The distribution derivative of (sgn/)g is of course in 6 ', but it differs from 1/| by a distribution supported by the zeros of /.

The inclusion of Sf in C"(R”) is continuous and has dense image, since Cc (U”) is dense in It follows that each distribution with compactsupport is temperate. The following exercises give some more examples of temperate distributions.

Exercise 5.9. and if ix e ^ iW ) then

<|LL, </)) = j </>(x) djLt(x), <f>eSf.

Exercise 5.10. I f v is a signed (real) Borel measure and if for some integer k we have J + d \v\ (x)<oo then veS f' and

(v,<t))=^ <l)(x) di/(x), <l>GSe.

Exercise 5.11. I//€L\[R”, loc) and if for some integer k the function ( l + lxp)~*" \f(x)\ is essentially bounded then fe S f' and

As a special case of Exercise 5.10 we note if jui a Tf S where a ^ e ^ and the sum is over k in the integer lattice and if there is aninteger m such that Z + then yueSf' and

Exercise 5.12. I f then multiplication by ij/ maps SI continuously into and maps SI' into continuously for the weafc*topologies or strong topologies.

We denote by the space of all functions such thatil/<l)eSI for each 4>g SI. Recall now the closed graph theorem; if T : E - ^ F is a linear map with closed graph in E 0 F where E and F are say Frechet spaces, then T is continuous.

L em m a 5.13. I f then multiplication by ij/ maps Sf into Sfcontinuously, and maps Sf' into Sf' continuously say for the weak^ topology or the strong topology.

Proof. Suppose is a sequence in 5 and <t>j^<f> in Sf and 0in 5 . Since pointwise and —> 0 pointwise we must have0 = Thus multiplication by if/ has closed graph and so is continuous. The last part follows since multiplication by ip on Sf' is the transpose of multiplication by ip on Sf'.

E xercise 5.14. I f g€S>'(IR”) and (pgeSf for each <peSf then ge^M-

We will need a simple inequality frequently in the sequel. First note that (|x|-l)^>0 implies 2|x|<l + lxp and therefore

( l + | x n < ( l + l x l ) " < 2 ( l + l x n .

We also have 2<x, y )< 2 [xj ly|<ly| (1 + jxp) and therefore ( l + |x + yp)<1 + lxp + |yp + ly| + ly| \x\ . If we add lyl + jy] Ixp + jx] ]yp to the right side we obtain ( l + |x+ yp)<(H-|y|)^(H-lxl^). Now let t be a real number. Replace y by y - x . If t < 0 raise both sides to the —t power and if t> 0 , first interchange x and y and then raise both sides to the t power. We obtain then

< (1 + lx - y 1) '" < 2"'«(1 + lx - y-lyl

for any real number t and any x, yelR”. We will call this inequality Peetre's inequality.

T h eorem 5.15. Let ipe Then ipeOj^ if and only if for eachmulti-index a there is an integer m and a constant Q, such that

l D X x ) l < C J l + lxl"r«

Proof. First suppose ip satisfies the inequalities. Then by Leibniz’ formula it is clear that multiplication by ip maps Sf into 5 . Thus iPg Om- Conversely, suppose iPe Om- Since multiplication by tp maps Sf continuously into Sf, given any multi-index a there is a constant C and integers m, fc > 0 such that

sup lD“(«/»< )(y)l< Cmaxsup (1 + lyl )’” \D <t>(y)\y l3Mfc y

for each <peSf. Choose <f>G5 so<#> = l i n a neighbourhood of the origin.

6. CONVOLUTION OF TEMPERATE DISTRIBUTIONS 87

Then

lD“«;»(x)l = 1D“(«K</>)( )I ^ c max sup (1 + lyl )"* \D <f>{y - x)]y

< C m axsu p (l + lx + y P r \D^iy)\I3 l^ k y

< C(1 + |x|) "’ max sup (1 + \y\^r \D <t>(y)\I3 l^ k y

<c'(i+|xi^r.

Corollary 5.16. o,nd if if/g € m then

(il/, < )) = I il/{x)<t>{x) dx for each g 5 .

The corollary is obvious and we state it only in view of Caution 5.7. We will no longer belabour this point.

§6. Convolution of temperate distributions. The space C'c

In this section we will define a convolution product for distributions satisfying suitable growth conditions rather than support conditions. Before we proceed to the definition we need to observe that translation is continuous in 5 .

Lem m a 6.1. I f x e W then ^ 9 " is continuous. I f <l>e 9 then the map R” — 5 : X Tx is continuous.

Proof. For the first part we have

sup(l + lyf)'' lD“(T,,< )(y)l = sup(l + lx + yp)'' lD“< >(y)|y y

< (1 + 1x1)2'= sup(l + lyl2)'= lD“</>(y)l.y

For the second part we use the mean value theorem. If x', x"e!R" then

sup (1 + ly 12)*= lD“[T .<f> - T «< ](y)ly

= sup (1 + ly p)'= lD“<;>(y - x') - D “<#>(y - x")ly

ssup(l + lyl2)'y,y'

|;D,.D“<^(y')(x"-x;.)|

(where ly - y'l —1 * 1 I* l

<n^''2 lx "-x 'l max sup ( l + (lyl + lx'l + lx"l)2)'' lD,D“< (y)lj y

<n^'2 ( l + lx'l + lx"l)2'= maxsup ( l + lyl2)'= lD,■D“< '(y)lJ y

As a corollary we note if x e R " and f&Sf' then and for each<t>€Sf, not just in C^(R"), we have <f>) = (f,

If f e i f ' and we define the convolution /*«/> by

By Lemma 6.1 is a continuous function.

Theorem 6.2. I f fe S f' and il/eSf then C“(IR") and there exists an integer m, and for each multi-index a a constant such that

lD“(/ * .A )w i^ c „ (i+ ix p r .

In particular f^ il/e€i^ . Moreover, for each multi-index a we have

(/ * i/f) = (D “/) * .

Proof. It follows directly from the definition that =If (f>GSf then by the mean value theorem

-< # > )- Dj<l}){y) = Dj<l)iy + ste-) - Dj<l>(y)

where 0 < s < 1 depends on t and y, and where = ( 0 , . . . , 1 , . . . , 0) with the 1 in the jth place. By Lemma 6.1 it follows that

r\r-te,<l>-<f>)-> D (f> in Sf.

Replacing 4> by we have

r\ f^ il/(x - tej)-f^ il/{x)) (f, Dj{r-^il^Y)

as t -> 0 . Thus -Dy(/*i/f) exists classically and we have

DYf^ il ) = (DY) iP^It follows that Djif'- il ) is continuous, and therefore by an inductive argument g C~(1R”) and the last part of the theorem holds. Since fe S f' we have a constant C and integers m, k > 0 such that

|</, <f>>! < C max sup (1 + ly|")'" lD^<f.(y)lI3 l^ k y

for each 4>eSf. Thus we have

< C m ax su p (l + |yp)"' “ y)lI3 l^ k y

< C ( l + lxl) '” max sup(l + |yp)"* lD V (y)lI3 l < k + la l y

< 0 1 + 1x1)"-.

Lem m a 6.3. I f ij/eSf then ^ is a continuous linear map o f Sf into ^ and if feS f' then <#>) = (/, for each <t>eSf.

6. CONVOLUTION OF TEM PERATE DISTRIBUTIONS 89

Proof. We have

( l + lxp)"* (1 + lxp)'" l«|f(y-x)D“<#)(y)|dy

<2'"'^ I (l + lx - y p r |»/»(y-x)l

x ( l + ly p r lD“<#.(y)ldy

^ C s u p d + lyR '” lD“</>(y)l.y

Therefore ^ is continuous. To show that </>) = (/,for each <l>eSf it suffices to consider <f> e C c(W ) since by the first part of the proof both sides are continuous functions o f <l>eSf. If 4>e C“([R”) then

«/f''*<#,(x) = I il/{y - x)<l>iy) dy

= lim e” Yj — ek)6 >10

where Z"" denotes the integer lattice in R ”. Thus we have

= s" X < (efc)Tefc«/r''.e i O ^

Moreover, the convergence here is actually in the topology of 5 . Thus we have

if, <f>) = lim £" X </. TskV}<t>{£k)slO ^

= I if, rA'')4>(x) dx

= i f **!>,<{>)■

Exercise 6.4. In the proof o f Lem m a 6.3 we used the fact that a certain "sequence' ofR iem ann sums converges in the topology o f Sf. Prove this fact.

Suppose now that f,geSf'. Motivated by Lemma 6.3 we attempt to define /*g by {/*g, < )) = (/, g' *<#>). Unfortunately while g' *c/>G 5*M in general it is not in Sf and therefore the definition does not make sense. If, however, we put appropriate conditions on g all will be well. We therefore define the space O'c to be the set of all geS f' such that </) —> g * <#) is a continuous linear map of Sf into Sf. By Lemma 6.3 we have Sf . We note that g' *< > = (g*< >' / and therefore if g e ^ c th^ng" eO c. We will be able to obtain a concrete description of O'c once we have extended the Fourier transform to Sf' in Section 7.

If /e5^' and gÔ'c then <(> —> {f, is a continuous linear functionalon 5 which we denote by /*g. Thus

</*g, §''*<#>>, sse.

Lemma 6.5. I f geO'c and a is a multi-index then D^^geO'c and for each fe S f' we have D “(/*g) = (D “/)*g = / *(D “g).

Proof. First note <^>-^(D“g)*< > = D “(g*<f)) is a continuous linear map of 5 into 5 . Thus D^^geOc- Now by Theorem 6.2 we have

<D“(/*g), </>)=</^g,(-Dm =(f,g''*i-D r)= </,(D“g)"*<f>>= </H=D“g,4>).

We also have by the same theorem

if, g ''*i-D r< i,}= {f, ( - D )“(g =*=<#>))= (D%g^*<t>)= <(D7)*g,<#>>.

Exercise 6.6. I f feS f' , g e & c < nd x e R " then r^geC'c and

Tx (/ * g) = (tJ ) * g = / * (X<g).

Exercise 6.7. I f f e l f ' and geC 'c then ( / * g / = /^*g'^.

Lemma 6.8. I f f e l l" and i f / e l f then ipfeO'c.

Proof. We have a constant C and integers m, fc & 0 such that

|</, <f))lsCm axsup(l + lyp)"' lD <#>(y)|, <j>eSf.I3 l^ k y

To show liffÔ'c we must show that ^(</(f)*</> maps I f continuouslyinto If. Let iV > 0 be an integer and a a multi-index. Then

( l + lx| T lD“((#)*<#>)Wl = ( l + |x|T \{f, •lnA{D“<t>y)}\

s C ( l + |xp)' m axsup(l + |yp)"' |D^(tK(I>“<f»)'')(y)|I3 l< k y

s C m a x X ( l + lxp)'^sup(l + lyn"'lD'^(/f(y)l |D“' ®“"'<;>(x-y)|

rsC m ax X ( l + lxp)'^sup(l + lx -y p ) '” lDî/^(x-y)| lD“ ' “X y ) l

< C ( l + lxp)'^'""' max sup(l + ly p r lT > X x -y )l ljD <f>(y)l13l< k+ la l y

7. THE FOURIER TRANSFORM ON <f' 91

where we have used C to indicate several different constants, as we shall continue to do. Since il/eSf -we have for each

- y)l < C(1 + lx - y C ( 1 + lxp)-^-"'(l + ly

Thus

( l + |xlT lD“((#)*<f>)(x)l<C max sup(l + lylT"^'"|r>«<f.(y)l131<k+lotl y

which shows that is in Sf and depends continuously on <{>.

C orollary 6.9. We have

Proof. If we can choose il/eC c iW ) such that i/ = l in aneighbourhood of supp/. Then f= ilr f is in € c by Lemma 6.8.

If fe S f' and geO'c we define g*/ to be /*g. This definition makes sense as our next result shows the convolution of distributions in O'c is commutative.

T h eorem 6.10. I f f e S f ' and g, h e O c then g^heO 'c, g^ h = h^ g and / *(g *h ) = (/*g)*h .

Proof. If <l>£Sf then (g*h)*<^(x) = (g 5 h, Tx<#>'') = (g, h'' (Tx</>'')) = (g, Txi(h^<l>Y)) = g^{h^(t))(x). Thus (g*h)^ regarded as a map of Sf into Sf is the composition of the maps h* and g* and therefore is continuous. Thus g^'heO'c- Since we have (g*h)*<^> = g*(h*</)) the same argument as in Theorem 1:18.17 will show that g *h = h*g. The same argument as in Theorem 1:18.18 will show / *(g *h ) = (/*g)*h.

C orollary 6.11. € c is a commutative associative convolution algebra and Sf' is an O'c^module. Moreover, '([R”) is a subalgebra o f 0*0

E xercise 6.12. Show that all the convolution products defined so far agree on common domains. Note the last statement o f Corollary 6.11 will then follow from Theorem 1:18.9.

§7. The Fourier transform on Sf"\ The exchange formulae

In this section we extend the Fourier transform to Sf'. The resulting theory encompasses the classical theory for L^(W") ( l< p < 2 ) ,generalized Fourier transforms of functions of polynomial growth, assorted singular Fourier transforms, and even Fourier series. We begin by considering the Fourier transform on Sf. Since Sf" if <t>eSf thenhas already been defined.

T h eorem 7.1. The Fourier transform is a topological isomorphism o f S’ onto S’ and if we have the following relations.

(A)(B)(C)(D)(E)(F)(G)(H)(I)(J)(K)

(P(D)<^>)"(|) = P(i|)<^(|) for each polynomial P. P{D )${^) = {P(—i ■)<t>) (^) for each polynomial P.

(Ty<t>) (|) = e for each y G[R^for each rj e

(Yx< ))"(|) = for each A > 0.

= (27t)“”< * i/lf J dx = J <l>4f dx.

Proof. (A) follows by Theorem 2.13, (B) by Corollary 2.12, (C) by Theorem 4.3, (D) and (E) are obvious, (F), (G), and (H) are simple computations, (I) follows by Theorem 2.2, (J) by Corollary 4.9, and (K) follows by Theorem 2.2. If <t>GSf then by Corollary 2.12 However, (1 + 1 | )"T>“< (|) is the Fourier transform of (1 — A)” (-ix)“</)(x) and therefore by Theorem 2.1

sup(l + l|| )"‘ |D“<^(|)1< j l(l-A)'"(-ixr</>(x)|dx

< C s u p (l + lxl )” l(l-A )" '(x “<;)(x))l.X

It follows that < € 5 and ^ is continuous. By property (C) above ^ maps Sf onto Sf. By the open mapping theorem ^ is a topological isomorphism.

If fe S f' we define the Fourier transform /g 5 ' of / by

(l<l>) = {f,4>\

By Theorem 2.2 this definition agrees with the previous one on ^ (R ”) and by Theorem 4.3(B) it agrees with the previous definition on L^(R”). Clearly the Fourier transform is an isomorphism of 5 ' onto Sf' and indeed is a topological isomorphism say for the weak* topology, or for the strong topology. Moreover, all the properties listed in Theorem 7.1, appropriately interpreted, continue to hold for the Fourier transform on 5 '. This fact is obvious for properties (A) through (H), where in connection with property (E) we note if / is a distribution its conjugate f is defined by </, <t>) = (/, $). Property (K) is just the definition of the Fourier transform on Sf'. Properties (I) and (J) are called the exchange formulae. Our next result gives the extension of these formulae to 5 '.

7. THE FOURIER TRANSFORM ON if 93

Theorem 7.2. Exchange formulae. (A): I f geC 'c then geCM and for each f e S f we have

(B): I f fhen >(fG€c and for each f & i f we have

W y = (2-ir)~''f*ip.

In particular the Fourier transform maps <?c one-to-one onto 0 ^ cind maps Cm one-to-one onto C'c-

Proof. Let f e S f . For (A) let g € C c and Then for eachwe have

d>) = <g, liKf) = <g,= {2'n)~"{g, 4 f*$ } = {27r)~''(g*4i'', (f>)

= (2-n-r"<(g *«?-''), <#))

Thus »/rg = (2'jr)“"(g*i^'')". Since g€€5'c we have g*^ '^ sif, and therefore iffg eif. By Exercise 5.14 we have geCM- By the computations above we have (i/>g) =g'^*tj/ and therefore

<(/* g f , = =

=</,(< >g )=</,< >g>= (gf, <t>)

For (B) let ^ & € m and <j>eif. Then and for each OeSF we have

, e ) = {ii/4>, 6 }= {iif, $6}= <t/>, (< * eY)=(4>,d>* 6}= (4> *<!>•', 0).

Thus = eS f. In particular the map is given by thecomposition (multiplication by (‘check’) and therefore is continuous. Thus tj/ 6 C'c- Now from we have =(2'n-)-"(^''*<^)\ Thus

<(#) , <t>) = <#, <#►) = if, 4>)= (27r)-"</, («^''*<#.)> = (2rr)-"</,= (2'ir)~''(f*4f, <f>).

The exchange formulae above are proved by completely formal arguments starting from the definition of Cm as the space of multipliers on i f and C'c as the space of convolutors on i f . This approach is a departure from the more conventional one of first introducing Cm and C'c concretely. In Theorem 5.15 we obtained a concrete description of Cm and

we may now, in view of Theorem 7.2, use the Fourier transform to obtain a concrete description of €'c.

E xercise 7.3. The distribution g is in €'c if o,nd only if for each integer N > 0 there exists a finite family o f functions h^,\a\^m{N), such that g =Zioti m(N) such that for each |a|<m(N) the function(1 + lx|^)^h«(x) vanishes at infinity.

E xercise 7.4. I f feLîW ^) then f is a temperate distribution. Show if l < p < 2 then f is actually a function, and indeed is in L (IR”) + L “(IR”). Use the Riesz'" interpolation theorem to deduce if l< p < 2 , p“ + q~ = l , andfeL'îW ^) then/€L^([R”) and ll/IL-’*

We will now compute one example of the Fourier transform on in this case on Some more examples will be given in Sections 9 and 11. We will need the following lemma which is frequently useful in other situations as well.

L em m a 7.5. Let P be a harmonic homogeneous polynomial o f degree m and let g e C“(0, oo). Then

P{D)g{\x\^) = 2^P{x)g^^\\x\^) for x 6 lR "-(0 ) .

Proof. If m = 0 then P is constant and the formula is obvious. Assume m > 0 and assume inductively that we have proved the formula for homogeneous harmonic polynomials of degree m - 1 . Let P = D^P so Pk is a harmonic homogeneous polynomial of degree m —1. Since P is homogeneous of degree m by Euler’s relation we have mP(x) = X and therefore mP(D) = 'Zl^kPk(I^)- Since P is harmonic we have 0 = AP = X J kf k C )- Applying the inductive hypothesis we have

P(D)g(|xp) = m-'2fcDkP,(D)g(lxR= m-'2fcDfc(2"’-'Pfc(x)g<"’-^XlxR)= m -’2k2'"-'Pfc(x)Dfcg^"’-'X|xH (by (*))= m -’2,2"'P,(x)x„g<"'X|xP)

= 2"’P (x)g<"’>(lxl^).

E xam ple 7.6. Let P be a harmonic homogeneous polynomial of degree m and let z be a complex number with Re z > 0 and zÔ. If /(x) = P(x)e"^'^'" then f e O ^ and and we have

f(|) =

where = z“” (z ^ )~” and z ^ is determined so as to be positive on(0, oo). Indeed by Lemma 7.5 and Theorem 7.1(B) it suffices to compute

8. THE FOURIER TRANSFORM ON %' 95

the Fourier transform of gz(x) = e” ''‘'’ . From Example 2.8 we have gt(l) = if t > 0. If <f> € then <&, <f>) = (gz, < ) is holomorphicin Re z > 0 and continuous in Re z a 0. On the other hand

</>) is holomorphic in Re z > 0 and continuous in Re z ^ 0, z^O. By uniqueness of analytic continuation we obtain the desired formula.

Note it follows that is in € q and in and its Fourier transform is(ttO'n/2_-il€l2/4

§8. The Fourier transform on The Paley-Wiener theorem

Since distributions with compact support are temperate we may consider their Fourier transforms. In this section we give an important and very useful characterization of distributions with compact support in terms of their Fourier transforms.

T heorem 8.1. I f gG 'ClR"") then g e C“(1R”). Indeed, g is the restriction to R" o f a holomorphic function, also denoted by g, on 0 ” and we have

g(z) = <g, for z € 0 " .

Proof. Let G(z) = (g, First note that z —»■ is a continu-ous mapping of 0"" into and therefore G is continuous. If z g 0 ”we write z = + iTj. The usual difference quotient for converges to the derivative in the topology of C “(R”) and therefore dGId^j exists and

(z) = -i{X jg,e

(A similar argument occurs in the proof of Theorem 6.2.) Since Xjge we see that dG/d^j is continuous. Similarly dGIdrij is continuous

and

| ^ (z ) = <x,g,e-‘<->). dVj

Thus, G e C '(0 '*) and, moreover, G satisfies the Cauchy-Riemann equations

aG 1 / a . a ^ “ I" “ X — )G = 0.dZj- 2 \d j drjj /

It follows that G is holomorphic. If <> g C”(R"") then

< = lim s'' Xe iO ks2"

in the topology of C”(!R"). Therefore we have

<g, </>> = <g, < >= lim X <g,e-‘<*"->)<f>(8fc)8'*

E J,0 keZ"

= |G(i)<#>(i)de

It follows that G restricted to IR” is equal to g.

Let A be a non-empty subset of R ”. The support function Ha of A is defined by

Ha (^) = sup<|, x).x e A

Clearly Ha is lower semicontinuous with values in ( - 00, 00], Ha is convex, Ha is homogeneous of degree 1, and Ha (0) = 0. If we set

cch(A ) = {x e R " |<4x)< H a (|) for each

then cch(A ) is the closed convex hull of A, i.e., the smallest closed convex set which contains A, and we have H wa) = Ha-

Exercise 8.2

(A)(B)(C)

(D)(E)

(F)

(G)(H) (I)

/ / A = {x e R "| lx l< R } then Ha (|) = R ||.1/ A = {x € R" I maxi,sj,£„ \xj\< R} then H a (I) = jR S |§1.I f p > l , p“' + q“’ = l and A = {x € R " | (X < R } thenHa (|) = R (I1§1"V ^I f A = {x} then H^i^) = < , x).1/ A and B are non-empty closed convex sets then Ha+b = Ha +H b .I f A is a non-empty set then Ha CD + c 111 for e > 0 is the support function o f the closed e-neighbourhood o f the closed convex hull o f A.I f A are non-empty thenI f A, B are non-empty and Ha ^ H b then A ccch (B ).I f A = { x g W |Z|Xj 1< jR} then Ha (|) = B maxi<j<n ll/l-

Exercise 8.3. I f A is a non-empty subset o f R'" then the following statements are equivalent

(A) Ha is finite valued.(B) Ha (|)^ R 111 for some R > 0 .(C) Ha is continuous.(D) The closure o f A is com pact(E) The closed convex hull o f A is com pact

8. THE FOURIER TRANSFORM ON ^ 97

Exercise 8.4. I f H is a lower semicontinuous function on H is convex and homogeneous o f degree 1, and H(0) = 0, then H is the support function o f some non-empty subset o f IR”. In particular H is the support function o f {x g R'" | for each

Lemma 8.5. Let <j) g C~(R”) and let H be the support function o f supp Then for each integer N > 0 there is a constant such that

|< (z)1<Cn(1 + |z1) where z = $-\-i7}.

Proof. |(iz)“< (z)l = lD“< )(z)l

II'= e

a<'n x> \D^<l>{x)\dx

< e H(ti)I \D‘‘4>(x)\ dx

Lemma 8.6. Suppose F is a holomorphic function on 0 ”, H is the support function o f a compact convex set K, and for each integer N > 0 there is a constant Cj such that

lF(z)l<CN (l + l2 ir^ e” <’''

for each z = + i't] in 0 ”. Then there exists a unique <f> e C“(R”) such that 4> = F. Moreover, supp (f)^K.

Proof. On R ” we have \F(^)\ is bounded by any power of (1 + | |)~ and therefore is certainly in Sf'. Thus F = $ for a unique <j)eSf'. Then (j) = (27t)“'"F'. Since any polynomial times F is integrable on R ” it follows by Corollary 2.12 that </>g C“(R”). Explicitly we have

</>(x) = (2i7)-'’ | e‘<"-«>F(^)de

Fix t] g R '" - (0) and t > 0 and change the contour of integration to R"" + itri. Then we have

<t>(x) = (2-7T)-" I e‘<’‘-«>-'< -”>F( + itv) de

T h e integrand is bounded by C n ( 1 + 1 |) and if N ^ n + 1then ( l + lll)” ' is integrable. Thus

for each t > 0 and each t i€ I R " - ( 0 ) . N ow if x ^ K then there exists

7] g IR” — (0) such that H{r]) — (x, t) ) < 0. Letting f ^oo we see that < >(x) = 0 .

Exercise 8.7. Justify the change o f contour o f integration in the proof o f Lem m a 8.6.

Theorem 8.8. Paley-Wiener. Let ond let H be the support function o f supp g. Then there is a constant C and an integer N such that

lg (2 )l< C (l + lzire»<’”

for each z = | + it] in 0 ”. Conversely, suppose F is an holomorphic function on (p , H is the support function o f a compact convex set K and

lF (z )l< C (l + |zire” ^^

for some constants C and N and for each z = + irj in (p . Then there exists a unique gG^\W^) such that g = F. Moreover, supp g ^ K .

Proof. Let K be the convex hull of supp g. Choose u e C^(U) such that u{t) = l if r < 1 and u(t) = 0 if t> 2 . Fix z = + ir] e 0 ” and let

< >(x) = u((T^,x)-H(r^))e-^<-^>.

Since (t), x) - H ( t] ) < 0 for x in K we see that <#>(x) = on a neighbourhood of K. Thus g(z) = {g, <l>) by Theorem 8.1. Since gG^'(IR”) there are constants N and C' and a compact set L such that

\{g, ij/)\ < C' max sup for each if/ g C CIR" ).la j< N L

It follows that

ll(z)| < C max sup |D"< |.lotl<N

By Leibniz’ formula

a !!>“</>= Z jT)3e-i<z.>jr)T„_

0+y=aP'-y'-Since u and its derivatives are bounded on IR the term D'^u is bounded by CCI + ItjI)' '. The term is bounded by C jz^j Since 1/31 +l7 l< N we have

lD“(f>(x)l<C'(l + Uire<’ ’">

for la l< N , where C' is independent of z. By the choice of u if x Gsupp <l> then (rj, x) —H (t] ) < 2 and therefore we have

\D <l>{x)\ < C "e\ l + IzD e'^ ’ ^

It follows that

lg(z)l < C 'C V d + lz i r e " '’’\

For the converse, 1F(^)1^C(1 + ||1)' implies there exists a unique g e i f such that FIr- = g. Choose a mollifier (see Section 1:3). Let ge = g*Pe- Then g^eOM and we have g = gPe- It follows that g extends to a holomorphic function on and ^ (z ) = p^(z)g(z) for each z in (S". By Lemma 8.5 for each integer fc > 0 we have

lp ,(z )l< Q ,,( l + lz|)-'‘e^'"'since has its support in the ball of radius s centred at the origin. Thus

1&(Z)| ^ C Q .d l +By Lemma 8.6 we have e C~(IR' ) and supp is contained in the closede-neighbourhood of K. Since ge g weak* in (see commentspreceding Exercise 1:18.20) we have supp g ^ K .

Exercise 8.9. The convolution algebra has no zero divisors.Show by example that the convolution algebra € c ^oes have zero divisors.

We will now give an application of the Paley-Wiener theorem, but first we will need to obtain some inequalities for polynomials. We denote by 0 (m ,n ) the complex vector space of polynomials of degree <m in Z i,. . . , z . Then 0(m , n) is a finite dimensional vector space with dimension (m + n)!/m!n!. If z € (? ” and r > 0 we define the norm on 0(m , n) by

llJ lUr = sup lP(z + w)l, P e 0{m , n).lwl<r

We will denote \\-\\z,i simply by \\'\\.

Lemma 8.10. For each P e ^ (m , n), Pi=0, we have

l< K r - - IIPIU,

where fx is the degree o f P and K is a constant depending only on m, n and the top-order coefficients o f P.

Proof. Since any two norms on 0*(m, n) are equivalent there is a constant O O which depends only on m and n such that

C-’ I lhJ<llQllo<C I \ K \

k l ^ m la l< m

if Q(w) = Xi<j,|sm Therefore if Q has degree p then

C - ' I Ibal^llQllo.|a|=(i

Now let 2 G 0'”, r > 0 , P e 0 { m , n), F(w) = X|ai£m and suppose P hasdegree p,. Define Q by 0(w ) = P(z + rw). Then

Q(w) = X + (lower order in w)lotl = fA

Thus

C -V - Z |aJ ||0|lo = llP|L-

Lemma 8.11. I f f l is any open subset o f then there exists a finite set A e f t and polynomials Q a e 0 (m , n), a e A, such that

P (z )= X P(a)Q<.(z)aeA

for each P e 0 ( m , n) and each z e (p .

Proof. The image of the evaluation map ft 0{m , n)' generates the dual space 0(m , n)' of 0(m , n) since a non-trivial polynomial cannot vanish identically on ft. Now choose A e ft such that the image of A is a basis for 0(m , n)' and let Q , a e A , be the dual basis of 0(m , n).

Lemma 8.12. There exists a constant C > 0 depending only on m and n such that if P e 0(m , n), P(z) = Z then

X la J < C ( l + r-" ') ( l + l2 ir ilP lUjal m

for each z g 0 ” and r > 0 .

Proof. In the proof of Lemma 8.10 we saw that Ziai^mkal^^CHPHo where C depends only on m and n. Let z g 0 ” and let r > 0 . Then P(w) = P(z + rr“Hw —z)) and if we apply Lemma 8.11 with ft = {x g IR” I |xl<l} and consider P(w) as a polynomial in r~^(w-z) we obtain

P(w )= Z P iz + ra)Qa{r~^(\^-z)).aeA

Thus lP (w )l< C '(l + r“' |w —zl)"* |lPllz,r which implies

llPlloSC'(l + r - ‘ + r -M 2 ir llPlUr < C '( l + r - ‘r ( l + lz i r llPlLr.

Finally ( l + r - ' r < 2 " '( l + r-'”).

Lemma 8.13. I / 0 = {x g R" 10<lxl < 1} then there exists a finite set ©

in Cl and a constant O O , both depending only on m and n, such that

llPll ,, < C max inf \P{z + rrd)!0 e © 1t 1 = 1

Te(T

for each P e 0 { m , n), and r > 0 .

Proof. If Q{w) = P(z-^rw) then ||Qllo = IWUr and therefore it suffices to show

llPllo^Cmax inf |P(t0)|" 0 e © l T | = l ' '

for each P in 0{m , n). Choose a finite set A c f l and polynomials Q , a e A , as in Lemma 8.11. Since P(z) = YP(o,)Q a(z) we have

llPlIo^C' maxlP(a)l.aeA

Fix a e A and let /(t) = P ira) so / is a polynomial in one variable. Let T i,. . . , be the roots of f repeated according to multiplicity so

/('r) = c n ( '> '-T ,) .J=1

Since jut ^ m if we subdivide the interval [1/2 — l/4m, 1 + l/4m] into m + 1 equal subintervals of length l/2m at least one of these subintervals contains none of the numbers |Ty| in its interior. If s is the midpoint of such a subinterval we have for some integer fc, 0 fc m, that

s = l/2 + fc/2m

and

\s - Ixy11 > l/4m for each j = 1 , . . . , jui.

Note

|/(i)l = lcinii-T,Nlc|Il(i+hl).J = 1 i = l

Consider now t on the circle 1t | = s. We have for such t that jr —Tyj

ls-|Tyll>l/4m and therefore if lTy|<l + l /4m then

|t —Tyl l/4m1 + jTyl 2+l/4m

= (8m +1)

On the other hand, since (f —s)/(l + t) is an increasing function of t and since s < l , for l ryl>H-l /4m we have ,

T -T ; iTyj —s l + l/4m —s 1 + lryl 1 + lryl 2+l/4m

>(8m + l)"\

Thus o n the c i r c l e It ) = s w e have

I/(t)1 £: 1/(1)1 (8m +1)-*^ & 1/(1)1 (8m + 1 )-" .

It follows that

1/(1)1 < (8m +1)"* max inf 1/(t)1s \t \ = s

where s = 1/2 + fc/2m, k = 0 , 1 , . . . , m. Therefore if we set

& = {sa I a G A, s = 1/2+ fc/2m, fc = 0 , 1 , . . . , m}

then

l|Pllo<C'(8m + i r max inf |P(t0)10 e © 1t | = 1

and the lemma follows.

Theorem 8.14. I f 0 ^ P g ^ [z i, . . . , z ] has degree m there exists a constant K and a finite set 0 = {x g (R"' | 0< 1x 1<1} such that

1 < Kr~^ max inf lP(z + rrOM0 e © 1t |=1 '

for each zG jS" and r > 0 . Here 0 depends only on m and n, and K depends only on m, n and the top-order coefficients o f P

Proof. The theorem follows by Lemma 8.10 and Lemma 8.13.

We may also combine Lemmas 8.12 and 8.13 to obtain a similar estimate. If we take r = l in Theorem 8.14 it says roughly that a polynomial P on the average is large. Indeed there are a finite number of circles centred at the origin such that for any z g if we translate the circles to be centred at z, then \P\ is bounded below by a positive constant, independent of z, on at least one of the circles. This type of estimate is very useful in problems involving division by P since if the quotient is known to be holomorphic, then the maximum principle allows us to obtain an estimate of the quotient at z when we have an estimate on a circle about z. Our next theorem, a special case of the convolution theorem of Titchmarsh and Lions, gives a simple example of this argument and is also the promised application of the Paley-Wiener theorem. We denote by ch (A) the convex hull of the set A.

Exercise 8.15. I f A is a compact subset o f then ch (A) is compact.

Theorem 8.16. Let O ^ P g 0 [ z i, . . . , z^], u,fe<^'(W) and suppose P(D )u = f. Then ch (supp u) = ch (supp /).

8. THE FOURIER TRANSFORM ON V 103

Proof. If / = 0 then w = 0 by Theorem 1:18.23(2). Assume // 0. If we take the Fourier transform we have P{iz)u(z) = f{z ) for each z € 0 ”. If H is the support function of supp / then by the Paley-Wiener theorem

l/(z)l < C(1 + \z\re^^-\ z = + IT] G r .

Let z e (p". By Theorem 8.14 there is 0 € 0 such that

l^ K \ P {iz + iT$)\

for 1t1 = 1. Now if T = s + it then

l/(z + t 6)\ < C(1 + Iz + (recall 6 is real)

and therefore

I u(z + t 6)\ < K C il + Iz +

Now H is convex and homogeneous of degree 1 and therefore subadditive. Thus H{7] + t6)^H(r\)-\-c where c =m ax{H (±0) | 0 €©}. Also we have (1 + 1z + t01)^<2^(H-1z1) . It follows that

lw(z + T0)|<2'^e"jR:C(l + lz|) e" ^\ for H = l.

By the maximum principle we have the same upper bound for |w(z)l. By the Paley-Wiener theorem we then have supp m^ ch(supp/). Since we trivially have supp / supp u the theorem follows.

Note if P{D)u =/ we can in general not say anything about the support of M if u does not have compact support, though there are some other cases in which we can estimate the support of u.

The theorem of Titchmarsh-Lions referred to above says if w, ve^'{R'^) then ch (supp u*w) = ch (supp u) + ch (supp m). If we write P {D )u = f as / = P (D )6 * m we see that Theorem 8.16 is indeed a special case.

Example 8.17. Another application of the Paley-Wiener theorem is to prove an inequality of Bernstein: if /g L°°(1R) and supp/c[—1,1] then l|/'||«,<||/m where IHloo is the supremum norm. Bernstein’s inequality is important in obtaining a lower bound for the response time or resolution of a linear filter with finite bandwidth. A very pretty proof of the inequality due to Carding (see Donoghue [1]) goes as follows. Since / is a distribution with compact support / is the restriction to R of an entire function. Assume first that / is a measure. Since the integer N which occurs in the Paley-Wiener theorem may be taken to be the order of / we have l/(z)l<Ce’ ’, z = x + iy. Let be the square in 0 centred at the origin with sides of length Skir parallel to the axes. Let

= 2 ^ 1/(z) ■dz.

z cos z

A simple estimate shows 4 ^ 0 as fc -» oo. By the residue theorem

T v4/(mir/2) . ,4 = / (0) - 2 . ---- 5—2— sm (mir/2)m 7T

where the sum is over odd integers m with \m\<8k. Thus we have

f ( 0 ) = — I (-1 )'fjrmr - TT/2)-fi-rmr + rr/2)

(2m -1 )2 ( 1)

Since the Fourier transform of sin x is 5(8_i - S j ) we see that (1) applies to sin X and yields

1=A I (2m -1 )2 -

Combining (1) and (2) we have

(2)

8If (0)1 Z 1

(2m - 1)^'

By translating f we obtain ||/'m: ll/l|oo. The case when / is not a measure is handled by regularization.

Exercise 8.18. Fill in the details in the argument in Example 8.17, Show i/ / eL “([R”) and supp/^j^elR” | |^l<jR} then llld/|||oo: jR H/IU. I f in addition / >0 then 1| jd/j Hoc ^ (i /2) l|/|loo.

A number of interesting Paley-Wiener theorems which arose in the study of mathematical aspects of the problem of reconstructing cross- sections from X-ray data may be found in the survey article of Smith, Solmon and Wagner [1].

Exercise 8.19. I f P e 0 [ z i , . . . , z„] and fe^ '{W ) then the differential equation P {D )u = f has a solution ue^'iW^) if and only if f{z)/P {iz) is locally bounded in (p .

Exercise 8.20. 1/P e 0 [ z i , . . . , z„] has degree ^ 1 , E e ^ ' iW ) and P {D )E = 8 then E does not have compact support.

Exercise 8.21. I f P ,Q e 0 [z „ z J , E e S)'(R") and P {D )E = 8then Q {D )E has compact support if and only if there exists a polynomial R such that Q = PR.

Exercise 8.22. I f u e^ \ R ), supp u ^ [ -a , a] and Ui , . . . , are distinct points in [—a, a] there exists a unique v e^ '(R ) with supp v ^ [ —a, a]

8. THE FOURIER TRANSFORM ON V 105

and unique complex numbers b i , . . . ,bj^ such that

D^V = M + X bjTa.8.

I f N is the order o f u and N —M + k + 2 < 0 then v e Cc(R).

As a hint for Exercise 8.22 note that if we take the Fourier transform, differentiate and evaluate at z = 0 we obtain a system of linear equations for the coefficients bj.

There is also a Paley-Wiener theorem for the singular support. Here, and also in Section 12, we adopt the convention that a Paley-Wiener theorem is any theorem which gives information about u in terms of the growth of u in complex directions.

T heorem 8.23. Let and let K be a non-empty compactconvex subset o f with support function H. Then sing, supp u ^ K if and only if there is a constant N and a sequence o f constants such that

lM(z)l<C^(l + lz ire» '^ 'if lr)l<m lo g (l + lzl), m = 1, 2 , . . . , where z = ^ + ir|. In order that u e C”([R”) it is necessary and sujficient that there exist constants Cm,k such that

iu (2 )i< c„ ,k (i+ iz ir ''

if lT7| < m log(l + l2l), m ,k = 1 ,2 , -----

We will not give the proof here. It may be found in Hormander [2]. Note that if we imitate the proof of Theorem 8.16 we obtain the following result as a corollary of Theorem 8.23.

E xercise 8.24. I f u e^ '{W ), . . . , z„] and Pi^0 then

ch (sing, supp P{D )u) = ch (sing, supp u).

Exercise 8.24 is a special case of the convolution theorem of Malgrange-John-Hormander. This theorem states that if u, g6^'(F^”) and one of them is supported by a finite set of points then

ch (sing, supp g * m) = ch (sing, supp g) + ch (sing, supp u).

We obtain the conclusion of the exercise by taking g = P(D)S. As an application of Exercise 8.24 we obtain a very primitive result on propagation of singularities.

E xercise 8.25. Let P g 0 [ z i , . . . , z„], u,/6S>'([R”) and P{D )u =/. I f ft is an open subset o f IR” and K = ft H sing, supp w and L = ft n sing, supp/, then, if K is not contained in ch(L), then K is not compact.

Indeed if K is compact choose </> e C“(H) with <#> = 1 in a neighbourhood of K and show P(D){<l>u) = <l>fif/ where il/eC ci^ )-

Exercise 8.26. I f O ^ P e 0 [z i , . . . , z,], E e P (D )E = 8, Cl is an open subset o f and K = ft (T sing, supp E is compact then K = {0} or K is empty. In particular sing, supp E = {0} or else sing, supp E is unbounded and contains no compact pieces at a positive distance from the rest.

In Section 12 we will see some more Paley-Wiener theorems in connection with the Laplace transform. In the typical situation in Section 12 the Fourier transform u does not extend to an entire holomorphic function but occurs instead as a boundary value of a function holomorphic in a tube. Aside from a trivial interchange of the real and imaginary parts of the variables it is the emphasis on the Fourier transform as a boundary value rather than a restriction of a holomorphic function that distinguishes the theory surrounding the Laplace transform from the Fourier transform theory of this section. Thus the difference between the two theories is one of philosophy rather than essence.

§9. Operators defined by the Fourier transform

In this section we will study some ‘constant coefficient’ operators defined by the Fourier transform. In Chapter 3 we will be concerned with certain classes of operators similarly defined, but not necessarily with constant coefficients. We first note if P is a polynomial then (P(-iD)w ) = Pu. This formula suggests that for any aeS f' we should define the operator

by

a(-iD ):0 'c -^ S f'

(a(—iD)w)" = ua, ueO'c

By Lemma 6.8 we have

a ( - iD ) :S f^ O j^ .

From the exchange formulae we see that

a (—iD)u = ^ ~ ^ (a)*u .

In particular a {—iD) commutes with translations and it is therefore that we say that a (—iD) has ‘constant coefficients’. We note that if a e O ^ then a (- iD ) maps Sf continuously into Sf and Sf' continuously into 6 '. If a e S f then a (- iD ) maps Sf' into Om-

An important special case occurs if a is a bounded measurable function

9. OPERATORS 107

on R". In this case by the Parseval-Plancherel theorem a { - iD ) :S f extends to a bounded linear operator on L (1R”) and

lla(-iD)ll = ess. sup la(^)l.

As a simple example consider the bounded function a(^) = wherec g R ”. In this case we obtain the translation operator

E xam ple 9.1. The functions (1 + z e 0 , are in and give riseto a group of isomorphisms

E xam ple 9.2. Let b g and suppose Re b < C for some constant C Then eO ^ and hence maps Sf' continuously into 5 '. Takingb(|) = - s where Re s > 0 we obtain a semigroup of operators

(R e s> 0 ).

Since W(x, s), where W is the Gauss-Weierstrass kernel, we have

^u(x) = I W(x - y, s)w(y) dy, u eS f.

E xam ple 9.3. Let a g S^(U^) c if e 6 , i/r g C~(1R”) and (^(0) = 1 then by the dominated convergence theorem

= (277)-"! a(^)<^(-|)d|

= (2t7)-" lim fe i O J

= (277)--' lim 1 1 e‘<" «>a(a«/'(£|)</>(x) d| dx.

In view of Theorem 1:19.6 we have

S^-\a) = (2t7)'" I d| [Osc.].

Thus in this case a (- iD ) is convolution with an oscillatory integral. By Corollary 1:19.8 we have sing, supp ^ “^(a) ^ {0 } for each aGS""(IR”). Moreover, we have such amusing formulae as

a = (277)-"|e‘<-«M| [Osc.].

The result in Example 9.3 concerning the singular support may also be obtained directly without appealing to Corollary 1:19.8.

Lemma 9.4. I f a g S' CIR”) then sing, supp d ^{0}.

Proof. Given any multi-index a choose the integer k such that m + laj —2fc + n < 0 . Then A^(x'^a)GL^(W') and therefore |^p^D"^dG C(IR”). Thus is continuous on IR” —(0).

Corollary 9.5. I f fe S f' is homogeneous and sing, supp / c {0} then sing. supp/c{0}.

Proof. Assume / is homogeneous of degree z. Then /|iRn-(o) is a C°° function homogeneous of degree z. Choose </> g C”(IR"') with <#> = 1 in a neighbourhood of the origin. Then <l>fG€'(W ) implies <#>/€C~(IR”) by Theorem 8.1. The corollary now follows by Lemma 9.4 since {l-<f>)fG

Note now that all the arguments of Section 1:15 are valid with C“(IR”) and replaced by 5 and 5 ', respectively. Indeed all the distributionsconsidered there are temperate. Note also if is an 5^'-holomorphic function so is since (f , <f>) = {f , 4>)- We will use the results of Section 1:15 without further comment.

Let b G C°°([R"-(0 )) be homogeneous of degree - n and suppose

b{o)) do) = 0J g n -

so P.v. b exists and is a temperate distribution homogeneous of degree — n. Then the operator

<#)-^(P.v. b) <t> :€'c-^Sf'

is called a singular convolution operator. Let a be the restriction to IR '-CO) of (P.v. b) . Then a e C“(1R” -(0 )) and a is homogeneous of degree 0, and therefore is locally integrable on R ”. If we regard a as a distribution on R"" then a -(P .v . b) is supported by {0} and therefore by Corollary 1:13.5

a -(P .v . i>) = X c„ D “5

for certain constants But the left side is a distribution homogeneous of degree 0 and D “6 is homogeneous of degree -n -\ a\ . Thus Co,=0 for each a and we obtain

a {- iD )u = (P.v. b) * u, ug €'c .

9. OPERATORS 109

We note since a is bounded the singular convolution operator a {—iD) is a bounded linear operator on L^(R”). Indeed the norm is sup la(<r>)l. Finallywe note that a has integral 0 on S ” \ Indeed, let (f>{x) = e

Then

f a(o))d(o\ r"-'e-^M r = (a,</>)Jgn-l Jo

= <P.v. b, 4>)

= lim b{x)4>(x)dx Jlx|>e

= lim b((s)) do) eiO J^n-l

-1x12 SO

r7r"/^rn-ie’ ""/4dr

= 0

by hypothesis. Thus Js--* a{o)) d(o = 0 .Consider now any a g C“(R” — (0)) homogeneous of degree 0. Such a

function we will call a symbol. (This term will be used in a more general sense in Chapter 3.) We have

a {- iD )u = ( 2 7 r ) u g O ‘

and d eS f' is homogeneous of degree —n. Let fe = d|Rn_(o) so b e C^CR '-CO)) by Corollary 9.5 and b is homogeneous of degree -n . The finite part P.f. b is then a temperate distribution and for each <) g 5 we have

<P.f. b, (f)) = lim ( f b(x)</)(x) dx + <f>(0) log e f b(a>) da>).Js«-. /

If A > 0 then for the homothety we have

<Y, P.f. b,<l>) = X-^(P.f. b,Y,^,d>)

and a simple computation shows that

Yx P.f. b = A"" P.f. 6 + A-" log A^l b(a>)

Since d - P.f. b is supported by the origin we have

d = P.f. fe + X c„ D “5

for certain constants Since d is homogeneous of degree —n and D “5 is homogeneous of degree —n — \a\ we must have Jsn-i i>(<o) do> = 0 and

no THE FOURIER TRANSFORM

c« = 0 if a 0. It follows that the principal value P.v. b exists and

d = P.v. b + c8.

It remains to compute the constant c. Let </>(x) = Then

<d, <l>) = {a, 4>)

= I dx

= f a{w )d(o [ r”- 'e “' '/ drJgn-l Jo

= [ a{(o) dco [Js"-' Jo

= j* a(o)) d(o.

Since </> is radial we have (P.v. b, <j>) = 0. Thus

c = I a{o)) do).

Recalling now that a (-iD )u = (27r)“''d''*u we see that we have proved the following theorem.

T heorem 9.6. I f a e C~((R” — (0)) is homogeneous o f degree 0 and b is the restriction o f to IR” —(0) then 6 g C~((R” — (0)) is homogeneouso f degree - n and fs" ' b{(o) do) = 0, so P.v. b exists. Moreover,

a {—iD)u = CM + (P.v. b)^u, u e Ocy

where

c = 2 - V""/^r(n/2) f a((o) do)Js"-'

is the average value o f a on The operator a {—iD) is a bounded linear operator on L\R^) with norm sup la(ci))|. Finally a —» a {—iD) gives a one- to-one correspondence between singular convolution operators and symbols a with average value 0 on

The Fourier transforms of homogeneous functions may be computed by invoking the theory of spherical harmonics. The basic facts are that there exist homogeneous harmonic polynomials Qm,k, k = l , m = 0,1, • • •, where = (2m + n -2 )(n + m -3)!/(m ! (n -2 ) !) = 0(m”~ ) and

k is homogeneous of degree m, such that (Qm,k) is an orthonormal basis of L^(S"“ ). Moreover there exist constants depending only ona and n such that lD“Om,k( )l — lo t l- l+ (n / 2 ) 1 |m -lotl . Another important

9. OPERATORS 111

fact is that if Q is a harmonic polynomial homogeneous of degree m then Q is orthogonal in L^(S”“ ) to each homogeneous polynomial of degree <m.

If a € C“(1R” -(0 )) is homogeneous of degree z and a=Y^am,kQm.k in L ((R”) then for each r and

a(x) = Z

converges in C“((R” -(0 )) . (See Seeley [1] or Stein and Weiss [1].)

Exercise 9.7. 7/ Re z > —n use the dominated convergence theorem or FubinVs theorem and the estimates above to show that the series for a converges in 5 '.

In view of Exercise 9.7 in principle we can compute a and therefore the operator a (- iD ) once we have computed (Qm,kM • We nowcarry out this computation. Let Q be a harmonic polynomial homogeneous of degree m. Then Qz(x) = Q(x \x\~ ) = Q(x) is locallyintegrable if Re z > - n and continues analytically to an 5^'-meromorphicfunction with simple poles at z = —n — m — 2k, k = 0 , l , ----- Now(Q(x) 1x1"'“"')" = i”'Q(D)(|x|^“' )" and our problem therefore is to compute (1x1''“"')". Let hz be the analytic continuation of 1x1 so h is 5^'-meromorphic with simple poles at z = —n — 2k, fc = 0 , 1 , . . . . If —n < Re z < 0 then is locally integrable, homogeneous of degree z, radial, and C“ on R " - (0 ) . It follows that h is homogeneous of degree —n — z, radial (i.e., invariant under the orthogonal group; see Exercise 5.5), and C“ on R " - (0 ) . Therefore if is the restriction of to (R "-(0 ) then

gj(x) = T(z) 1x1""“^for a certain constant t ( z ) . Since R e z < 0 the function is locally integrable on IR" and so defines a temperate distribution, also denoted by gz- Since gz — K is homogeneous of degree - n — z and supported by {0}, and since R e (—n —z ) > —n a by now familiar argument shows that gz = hz- Thus

(IxH = t ( z ) 1x1“"“'" if - n < R e z < 0 .

To compute r(z) let < (x) = e“*"'’'' so < (x) = Then

277sfr

r(n/2)2n+z^n f<x>

77

r(n/2)

^„+^_ie-r2/4<jr

j- l+ (n + z )/2 g -<

= 2"-"V„ r((n + z)/2)

r(n/2)

A similar computation shows

<(1x1 )', <t>) = T(z)<lxl“''“^ <l>) = Tr"^^T(z) ^r(n/2 )It follows that

(lxl"r = 2'’-""ir"'2T((m + z)/2)

F ( - z/2)

at least for — n < R e z < 0 . By uniqueness of analytic continuation thisequality holds for all z with z i = - n - 2 k , k = 0 , l , ___ We note that theright side has singularities at z = 2fc, fc = 0 , 1 , . . . but they are removable since T(—z/2) has simple poles at these points. We now have

(0 (x ) Ixp-”*) = rQiD){\x\^-"'y

= „/z.^r ( ( n - m + zV2)T((m -z)/2)

If Re z < —n then is on IR”. If we apply Lemma 7.5 we have

Q(£>) Ixl-"-"-" = 2 ’ Q{x)(^ ~ ~ ~ ~ 1^

( m - n - z \ ,X • • • y----- ^------- m + ljlx | -'" - ''-"

= 2 -Q (x ) I P .+ » ^ - n - z ) / 2 ) , r((2 —m - n - z ) / 2 )

If we recall

r(m + u ) r ( l - m - M ) = L ^ L E ^sin ('TTz)’

we see

T((n - m + z)/2)T((2+ m - n - z)/2) = r ( - m + E ± | ± f j

I nt + n + z\i ' * ' " — — )

X r( 1 + m —

r r (- l)"sin (irCm + n + z)/2)

and

9. OPERATORS 113

In view of the uniqueness of analytic continuation we have proved:

T h eorem 9.8. I f Q is a harmonic polynomial homogeneous o f degree m then

(Q ix) Ix r - ) ^ =r((m - z)/2)

if —n — m — 2k, fc = 0 , 1 , . . . . In particular if m > 0 wc obtain

r(m/2)(P.v. Q{x) 1x1“"-" ’) = r " ’i7n/2 _ Q ( i ) i r '" -r((m + n)/2)

E xam ple 9.9. The Riesz operators Rj on R" are defined by

say for <l>GSf. From Theorem 9.8 we see

From this formula it is immediately clear that the Rj are bounded linear operators on L^(R”) and that we have the following properties.

(a) Kl| = l(b) R * = -R i(c) l R f = - I(d) ZK/lll= = ll/llK/eL^(R").(e) Rj commutes with translations and homotheties.(f) The Rj have nice properties relative to rotations.

The exact meaning of (f) may be worked out as an exercise or may be found in Stein [1].

E xercise 9.10. I f j ^ k then RjR^ is a singular convolution operator. On the other hand R f = —(l/n )I +A j where Aj is a singular convolution operator.

E xam ple 9.11. If n = 1 there is only one Riesz operator. It is called the Hilbert transform on R and is usually denoted by H. We have

H<l>(x) = \im— f ^ ^ d t>10 77 J| _,1 3 X - t

say for More conveniently, on the Fourier transform side we have

(H<#>) (s) = -i(sgn s)<^(s).

where sgn s is +1 if s > 0 and - 1 if s < 0 . The following properties are now obvious.

(a) H is an isometric isomorphism of L\U ) onto L (IR).(b) H * = - H(c) H^ = - I(d) H commutes with translations and homotheties.(e) H anticommutes with reflections.

Theorem 9.8 may be used to compute the residue of Q(x) at thepoles and also to compute the Fourier transforms of the corresponding finite parts. We will now carry out these computations. First note if h is a meromorphic function with a simple pole at z = a then we have

h{ z ) = +foo + 0(|z-a|)z — a

where h_i is of course the residue of h at z = a and where we will call ho the jinite part of h at z = a. If g is holomorphic at a we have

(gh)_i= g(a)h_iigh)o = g{a)ho-\- g '(a)h-i.

For any integer fc > 0 we have

7T

sin ( t t z ) z + fc

and therefore

77

+ 0(|z + fcl)

r(z) =F ( l - z ) sin ( t t z )

(-1 )"fc!

ijfik + 1) + 0(lz + fcl)

where i/f(z) = r'(z)/F(z) is the digamma function. Now let: — m _ n / 2 ^ n + z I 7T Z

g(z) = ; Q (l) 1 1—m—n—zr((m -z)/ 2)

and let

h(z) = F((m + n + z)/2).

In view of Theorem 9.8 the residue of gh at z = - n - m - 2 k is the Fourier transform of the residue of 0 (x ) at z = - n - m - 2 k .

9. OPERATORS 115

Moreover, the finite part of gh at z = —n — m — 2k is the Fourier transform of (P.f. 0 (x ) 1x1“" “^"'“ '"). By the computations above we have

h -i =2 { - l f

k\

(-1 )''ho = - ^ 4 > ( k + l).

Therefore

(gh)_i = g (-n - m -2fc)h_i

fc! T(m-H fc-1-n/2)

If we now take the inverse Fourier transform we see that the residue of Q(x)lxl"-"* at z = - n - m - 2 f c is

Q i e

:! r^m + kQ(D)A^8.

For the finite part we have

(g^)o = g(~^ — m — 2fc)ho + g '(-n - m — 2 k )h -i

:! r^m + fc+^^• (w -lo g l^ l)Q (^ )

where

C n ,m .k = log 2 + + 1) + + k +

Taking the inverse Fourier transform we obtain/__1 \ m n/2 2 l “ »n—2 k

P.f. Q(x) = ----------- — Q{D)AHcn.^,k8-A)(-l)m^"/22'"'”“2fc

fc !r(m + f c + 0

where A(^) = logl^|. To compute A we take m = 0 and fc = 0. Then wesee

A =c„.o,oS-2;jj^P-f-|xl ",

where

c„.o,o = log 2 +^»A(1)


and

«/'(2) = -T ~ 2 1 o g 2

'/'(I) = - yHere y = linip^ (If=i (1//)“ log p) = 0.57721566 • • Mascheroni constant. If p is an integer we have

is the Euler-

p-1 i«/'(p) = - y + Z " (p — 2)

j=i ]

>lfipH) = - 2 \ o g 2 - y + 2 (P ^ D -

Taking Q = 1 in our computations above we have

P.f. 1x1-"- ' = -t i r ( ) c + f )

A‘‘(c„,o,fc6-A).

If n is even and n + 2 k = 2 p and we multiply by Ixp” we obtain

nl'lrs\-2k1 = ------- -------T A^A.

: ! r ( c + 2 )

If we now take the Fourier transform we haven / 2 o l - 2 k

(2 ir rs =-^^— -------- ( - i r A ' ’( ( - l ) ' ‘ Ixp" log|x|).fc!F (-i)

Thus

A p / = X\7r"/"2 '’- 'f c ! (p - l ) !/

if n is even and n-\-2k= 2p. Fundamental solutions of powers of the Laplacian in other cases may be computed as well.

E xercise 9.12. I f

Q{x)F =

2("^")/2r((m + n + z)/2)

where Q is a harmonic homogeneous polynomial o f degree m then is an Sf'-entire function and Compute AF . ComputeF-n-m-2k-

9. OPERATORS 117

Exercise 9.13. Let n = l and let Y be Heaviside's function. Show

= - 2 7 - 21oglsl

(logltl) =-27776 - 77 P.f. 7^I5l

-177 sgn s(p .v . i) — i,

(sgn ty = - 2 i P.v. 0 ^

Y = Tj-S-iP.v.

Exercise 9.14. Let n = l and let Y be Heaviside's function. Investigate the analytic continuation o f t^'Yit) and its Fourier transform.

Exercise 9.14 may be worked out from Theorem 9.8 since FY{t) = 2 W ‘ 1 + 2 f and 1 and t are harmonic homogeneous polynomials.It may also be done more directly; see, for example, Gel’fand and Shilov[1].

Remark 9.15. In Theorem 9.8 if we take 2 = -n /2 we obtain

(Q{x) = r^ (2'T rr'^ Q ie

if Q is a harmonic homogeneous polynomial of degree m. Thus from the point of view of expansion in series of spherical harmonics the Fourier transform of a function homogeneous of degree -n/2 is particularly simple to compute. In particular if n = 2 we obtain a rather curious result.

In IR with polar coordinates r, 0, apart from a normalizing factor, an orthonormal basis of L^(S^) is given by the restrictions to of the homogeneous harmonic polynomials

Om,iM = Re (xi -1- 1X2)” = r' cos (mO) (m > 0)Qm.2(x) = Im (xi -h 1X2)"' = r"" sin (mS) (m > 1).

If M G C“([R^-(0)) is homogeneous of degree - 1 then00

M(x) = aor~ -h X (a^r~ cos (m0) + b r~ sin (mO))m = l

where the series converges in 5 '. Thus for the Fourier transform we have00

u(x) = 2-7raor“’ + ^ 2'jn'-'”(a^r“' cos (mO) + b„,r~ sin (md)).

If u is even then = 0 for odd m, and if m = 2k then

cos (m6) = (-1 )'' cos (2k0) = cos m(^0

sin (m$) = (-1)'" sin (2k0) = sin

Thus apart from a factor of 27t the Fourier transform is a rotation through an angle 7t/2. Explicitly, if u is an even function homogeneous of degree - 1 on then

X2) = 2 t t u ( - X 2 , Xi)This formula is a special case of a more general result for even functions homogeneous of degree 1 - n on R ” -(0 ) . See Semyanistyi [1], equation (7).

E xam ple 9.16. Identify 0 and as usual and let z = x + iy. Then 1/z = z lzl“ and z is a harmonic polynomial homogeneous of degree 1. Then by Theorem 9.8 (1/z) = -2 ir i{\ lz ). If {dldz) = \{{dldx)-\-i{dldy)) is the Cauchy-Riemann operator we have therefore

dz z ) 2 \ z J

Thus

dz \7rz/

R em ark 9.17. We have so far considered operators which on the Fourier transform side are multiplication operators. Another class of operators, perhaps even more natural, is the class of multiplication operators. These, of course, do not have constant coefficients, that is, do not commute with translations. We let X, denote the operator of multiplication by the jth coordinate function. (We will not be overly consistent in the use of this notation.) More generally if b €5^' we define the operator

b(X ):O j^ ^ se'

just to be multiplication by the distribution b. If a, b e 5 ' we then have the composition

b (X )a {- iD ):S f-^ S f'

If a ,b e € M then b (X )a(—iD) maps 5 continuously into Sf and Sf' continuously into Sf'. In this case for any we have

b{X)a{-iD){x) = (217)-" I e‘<-«>i)(x)a(a<^(D

10. HOMOGENEOUS DISTRIBUTIONS 119

This formula suggests that for any reasonable function p we should define the operator p{X, —iD) by

p(X, -iD)4>ix) = (277)-"I e'<=‘-«>p(x, |)< (|) d|

= (277)-’' ! j e ‘<"- «>p(x, ^)4>{y) dy d^

These expressions are purely formal at present but we will encounter them in a rigorous setting in Chapter 3. Note for suitable functions a, b, c one may also consider operators of the form a (-iD , X ), b (X ,-iD , X ), c{—iD, X , —iD) and so forth. Usually such generalizations lead to no significant extension of the class of operators under consideration, but do have the advantage of permitting a more refined symbolic calculus.

§10. Homogeneous distributions

Let U be an open cone in [R” and let fe 2 ' {U ) . Recall f is homogeneous of degree z if Y J = A Y for each A > 0 (see Example 1:17.2). The homogeneous distributions are characterized by a relation due to Euler.

T heorem 10.1. Euler. I f U is an open cone in I f is homogeneous o f degree z if and only if

XjDjf.1 = 1

r and fe ^ ' iU ) then

Proof. Let </>g C~(17). If A > 0 then lime_^o(lM)(Yx-bs</>-Yx< )) = A“ in Cc(U ). If f satisfies Euler’s relation let cr(A) =</,Yx< >). Then a ( l ) = </, <#>) and o-'(A) = A- I:(/,x,D,-Yx<^>) = —A (n + z)o-(A) where we have used Leibniz’ formula. It follows that o-(A) = A-"-^(/, <#>) whence (Y J , <>) = A""</, Y„ <l>) = A (/, <f>). Conversely, assume that / is homogeneous of degree z. Then (/, Yx< >) =Thus - (n + z)(/, <t>) = or'(l) = Z </, xPj<l)) = -n { f , (XjDjf, </>) by Leibniz’ formula. Then Euler’s relation follows.

L em m a 10.2. I f H is an open interval in then h is a constant.

and h e Of (Cl) and Dh = 0

Proof. Choose x (D) with J dx = 1. Let <f> 6 C“(D) and let c =i<t>dx. Let il/(s)=itoo(<f>(t) — cx(t))dt. If [a, fc] is the smallest closed interval in IR which contains supp (<f> — cx) then [a, b]^ C l since ft is an interval. Moreover, by the choice of c we have supp ij/ ^ [a, b]. Thus

ij/ e C“(fl) and Di/f = ( f) -c x ^ Now {h,<t>) = {h , D il/) + c (h , x ) = c (h , x ) = J ( h , x )< l> M d x . Thus h is the constant ( h , x ) -

E xercise 10.3. I f H is an open subset o f R, heO)'{0) and Dh e C^(H) then h e

C orollary 10.4. I f hG2)'(R — {0)) is homogeneous o f degree z and Y is Heaviside's function then h{t) = at"'Y(t)-\-b(—tY Y (—t) for some constants a and b.

Proof. By Euler’s relation tDh = zh and therefore D (t“'"h) = 0.

A distribution / on R" or on R ’' — (0) is said to be radial if it is invariant under the orthogonal group: that is, if R •/ = / for each Re0{W ^) (see Exercise 5.5 and Example 1:17.6).

T heorem 10.5. I f f is a radial distribution on R ” —(0) and f is homogeneous o f degree z then f is a function and for some constant c we have f(x) = c

Proof. Define T : 0 ( 0 , cx>)) O R " - (0)) by (Tw)(x) = u{\x\). ThenT is a continuous linear map and therefore g = f ° T e ^ ' { ( 0 , ^ ) ) . If ug C“((0, oo)) since T commutes with homotheties we have

(g, Y xm) = </, Yi,Tu) = Tu) = A“"“Yg, «)-

It follows that g is homogeneous of degree n + z —1 and therefore by Corollary 10.4 g{t) = Cor^^~\ If <#> g C r(R "- (0 )) is radial <l>(x) = u(\x\) then

( f <#>) = <g, m) = Co j u{t) dt = Cl j |xl <f>{x) dx.

Define h g S)'(R” - (0)) by

<h, il/) = {f, i/^)-Ci I |x| il/(x) dx.

Then h is homogeneous of degree z, h is radial, and if </> is radial then (h, <t>) = 0. Let djR be the normalized Haar measure on the compact group 0(n). If i/fGCr(R"-(0)) let

<#)(x)= f i/f(jR-'x)djR. Jo(n)

Then <f> g C“(R" — (0)) is radial. Therefore if we approximate the integral

10. HOMOGENEOUS DISTRIBUTIONS 121

by Riemann sums we have

0 = (h ,< l> )= [ ( h , ijj) dR = <h, i/f),-'O (n )

where we have used the fact that h is radial. It follows that h = 0.

We will now show that each homogeneous distribution on DR” is temperate. First we need to construct a special partition of unity on IR”. We will use this partition of unity again in Section 11.

L em m a 10.6. For each e > 0 there exists x C“([R”) such that 0 < x: — 1, supp X has diameter less than (1 + and t x, v g Z”, is a partition o funity on IR”.

Proof. Let K be the cube with vertices (fci, fc2, . . . , K ) where each kj is 0 or 1. Choose i/fGC”(lR”) with such that ip = l on aneighbourhood of K and supp if/ is contained in the open e j l - neighbourhood of K. Then the diameter of supp i/f is less than (1 + e)u^ . Let ^ = X V e Z”. We may suppose 8 < 1. Then 1 < <

2”“'" is the number of fc-dimensional

r ) and = ^ for each i/e Z”.

since

faces in the n-cube. Moreover, ''Pe C“ Now let X = ^/^*

T h eorem 10.7. IffeQ)'{W^) is homogeneous th en feS f'.

Proof. Suppose f is homogeneous of degree z. If A > 0 let = {x e [R” I lx|<A}. There exist constants Co and m such that

|</, cf))|<ComaxsuplD“c )l, if </>g C“(1R”), supp</>cBi.|a|<m

Now if supp<f>sBx then Yx<f> has support in B i and therefore </>)!:£ Co maxi„|sm '“'sup 1D“<#>1. Thus if a = R e z we have

i</,<f))l<A"'"“'"'"Comax|„|^^suplD“<f>|, if (j>eC7{U''), supp<^cB^. According to Lemma 10.6 for some tj> 0 there is a partition of unity t x, v er fl' ' , where suppx has diameter less than 1, and Oesupp X- If </> e C^(R") then has support in Bxfor A = 1 v| -I-1 we have

Kf, (t, x )« >)1 (1»'I + l) " ' '“''"’Ci max sup lD“<f>l.

But if X e supp (t„x )<#> then lv|-l<|xl<lv|-l-l and therefore][)tt+a+m<2^(l4-|x|)^(l + |vD* if t = n -ha + m -N . Thus

l(f, 4>}\^(L (1 + k l)‘)2'^Ci max sup (1 + lx|) \D°‘<t>{x)\

where the sum is over vgtjZ”. Replacing \v\ by Xk/I and recalling that

the number of multi-indexes of length m is ^ which (for fixed

n) is 0(m''~^) by Stirling’s formula, we see that the series converges if f + n —1 < — 1; that is, if N > m + a + 2n. It follows that / is continuous for the topology induced by Sf on C~([R”).

Exercise 10,8. I f f is a radial distribution on R ” and f is homogeneous o f degree —n then f = c8 for some constant c.

Note Exercise 10.8 implies that A 1x1 “” = c8 for some constant c. The constant c is easily computed and is not 0 if n 2. Thus once again we obtain a fundamental solution for the Laplacian.

Additional information concerning homogeneous distributions may be found in Carding [2].

§11. Periodic distributions and Fourier series

Let and define G ={u elR” | = Then G is a closedsubgroup of R ”, called the group o f periods of /. We say that f is periodic if G spans R' . In this case we have

R- = V + G o and G = ( G n V ) + Go

where Go and V are subspaces of R ”, G fi V is the free abelian group generated by some basis of V, and the sums are direct. This result on the structure of vector groups may be found for example in Hochschild [1]. We define

G * = {^ € R ” I (I, v) e 27tZ for each v g G}.

Then G * is a closed subgroup of R ” and indeed is the free abelian group on some basis of the annihilator (or orthogonal space, depending on one’s point of view) G q. We call G * the dual group of G. Let Ui, . . . , be a basis for V which generates G Pi V and let . . . , be a basis for G q. Then

r m n I 'j

G = I 2] kjVj + X h' i fc, eZ , f, elR [.M = 1 i = m + l • J

Let Ml,. . . , be the basis of R" dual to Oj , . . . , v„. Then Ui, . . . , u„, is a basis of Go and we see easily

G* = |2'7t I fc,M,.| fc, €z|.

11. PERIODIC DISTRIBUTIONS AND FOURIER SERIES 123

Let L be the free abelian group generated by Ui, . . . , so L is a discrete subgroup of G. Let K be the convex hull of the points X where each fcy is 0 or 1 and j = 1 , . . . , n. We will call L a fundamental group o f periods and K a fundamental parallelepiped. As in Lemma 10.6 there exists a function x C“(IR”) with support in a small neighbourhood of K such that 0 < X — 1 and u € L, is a partition of unity on Since r j = f for each u e L we have

/= I T„(xf).ueL

The sum is locally finite and in particular converges weak* in

Exercise 11.1. I f ipeSf then (v g L) converges in

Theorem 11.2. IffeQ}'(W^) is periodic th en feS f'.

Proof. Let il/eSI. Since xf^^'i^"") Exercise 11.1 implies that

Z = Z M x f ) . «/>-

By the Banach-Steinhaus theorem there exists geS f' such that

<g, =

for each il/eSf. In particular, if i/feC“([R'") the sum is finite and by the remarks above is (/, ij/}. Thus the restriction of g to C“((R'') is /, that is, f is temperate.

Since the periodic distribution f is temperate we may consider its Fourier transform. The relation r j —f = 0 for each v e G implies (e-i<u, >_ 1)^^ 0 for each v g G. Thus the support of f is contained in G*. If u g G * the restriction of / to a neighbourhood of u must be a linear combination of derivatives of r^S by Corollary 1:13.5. Actually no derivatives occur, and indeed we have:

Theorem 11.3. Fourier series representation. I f f is periodic with group o f periods G then

/= Z "uTuSueG *

where a ^ ^ 0 and where the sum converges weak* in Sf'. In particular

/ = (2 i7 )-" Zu sG *

where the sum converges weak* in Sf'.

Proof. Let ueG"^. By the remarks above f = in a neighbourhood of u. If </)6C“([R”), G *nsupp = { m} and </> = 1 in a neighbourhood of u then

0 = <(e-‘<“->-l)/,<#,)= X K i i v r

for each v e G . But any polynomial which vanishes on G vanishes identically. Hence ho, = 0 if a^^O, and so /= in a neighbourhood of u. If we choose il/eCciU"^) such that i/f=l in a neighbourhood of the origin and such that G * Plsupp = {0}, then and ifthen X (u g G *), converges in Sf. It follows that f = 'Zi'^u^)f wherethe sum converges weak* in Sf'. The last part now follows by the continuity of the inverse Fourier transform.

The last part of Theorem 11.3 gives a decomposition of f into a series of plane waves. The plane wave has amplitude a and (radian)frequency lu| in the direction of u. Alternatively ifM = (wi , . . . ,u„) then Uj is the frequency in the direction of the jth coordinate axis. For this reason one calls G * the group o f frequencies of /. The frequencies which actually occur in the decomposition are just the points in supp/. Therefore the support of / is often called the spectrum of /.

Example 11.4. Let n = l, a>>0 and let /(t) = sin (cot). Then f is periodic and the group of periods of f is the cyclic group generated by 27t/co. The dual group is the cyclic group generated by co. Since / = (i72)(t_^6 - t 6) the spectrum of f consists of the two points {-co} and {co}.

In the proof of Theorem 11.3 we made use of the result of the following exercise.

Exercise 11.5. Let G be a subgroup o f W and let P be a polynomial on I f P vanishes on G then P vanishes on the span o f G.

Remark 11.6. Let fGQ}'{W^). We asserted above that the group of periods G of / is closed. This fact follows from the continuity of translation in C“(1R") (mean value theorem). For functions the situation is less simple. If / is the function on IR defined by /(f) = 0 if t is rational and /(f) = 1 if f is irrational, then the group of periods of the function / in the usual pointwise sense, is Q, the rational numbers. Viewed as a distribution / is the constant 1 and the group of periods is of course R.

Theorem 11.7. Let G be a closed subgroup o f R ” which spans R ”. Then the trigonometric series

(2t7)-" X o„e'ugG*

i<u,-> (a G 0 )

converges in SF' to a distribution f if and only if there exist constants C and N such that

10itl:<C(l + lMl) for each ueG"^.

In case the series converges, then f is periodic and G is a subgroup o f the group o f periods o f f.

Proof. The last part is obvious since G is a subgroup of the group of periods of for each u e G *. Suppose first that the series converges. By continuity of the Fourier transform we have

f ^ uTu 'ugG*

Since / is a temperate distribution there exist constants C and m such that

\(f, il/)\ s C naax sup (1 + jxp)'” lD“i/>(x)ll<xl:<m X

for each il/eSf. Choose <t> g (IR”) such that <#> = 1 in a neighbourhood of the origin and G * Pisupp </> ={0}. If m g G* then {f,T ) = a and therefore

la„l< C niaxsup (l + |xp)'” \D°‘<j>{x-u)\la| m X

< C max sup (1 + jx + wp)"" lD"*<f>(x)ljo ilrS m X

< c'(i+ iw iy


2m

by Peetre’s inequality. Conversely, assume we have the polynomial growth condition satisfied by the a . If <>g 5 then the series u G G *, converges. Therefore, by the Banach-Steinhaus theorem the series X converges weak* to a temperate distribution. Taking the inverse Fourier transform we obtain the desired conclusion.

Consider now a periodic distribution fe S f' with group of periods G, so

/ = (27t) - Z o„e‘<“-> and /= X a„T„8.ugG* ueG*

If h e O c and v e G then T^(f^h) = {T j)^ h = f^ h . Thus /*fi is periodic and G is a subgroup of its group of periods. Since

( f * h y = h f= X h(u)a^T^8ueG *

we obtain

f * h = (27r) ” X h{u)a c' ' ''\ugG*

In particular, if we can choose h e Cc such that h(i<) = a„ for each u € G * we obtain / = g * h where

g = (2rr)-'’ I e‘<“-> and g = I t „ 8 .ueG* ueG*

If G * spans [R” then g is periodic. Thus we are led to investigate periodic distributions with periodic Fourier transform.

Theorem 11.8. I f / = then f is periodic with group o fperiods 27tZ”. Moreover, f is periodic with groups o f periods (27tZ")* = Z” and f = 'Zuer' I f h e is periodic with group o f periods lirlT' and h is periodic with group o f periods Z” then h = c f for some constant c.

Proof. Clearly / = I]'T’u5, u g 27tZ”, defines a periodic distribution / with groups of periods 27tZ”. Thus, / = Z w € Z”. If u e IT- thenclearly 1)/ = 0 and therefore r j = f It follows that f is periodicand is a subgroup of its group of periods. Then we must have a = ao for each u e IT'. For the homothety we have

A”Y ,(T ,.6) = T.a.

Taking A =27t we obtain

/= ao Z 'T'uS = ao(2ir)"Y2,, Z ' 2-.ru8ueZ” ueZ”

= ao(27T)''Y2

Since f ' = / we have

/ = (27T)-/=ao(Y2^/r

= ao(2'77) "Y i/(2„)/= Uq/-

It follows that ao= 1. To see that ao= 1 it suffices to show that Uq is a positive real number. If <f> € 5 then

0 Z <l>iu) = {f,<t>) = { f , $ } = Z < (2'7TU).ueZ" ueZ"

If we let < >(x) = e“'''' then 4>( ) = and therefore we have ao>0.For the last part the periodicity of h implies h = Y u e Z'"^Then theperiodicity of h implies = bo for each u e Z". Hence, h = bof.

Corollary 11.9. Poisson summation formula. I f <l>eSf then X<^(27rii) = X u eT " , and Z <#>(277 ) = (27t)“” X u g Z"", where the series are absolutely convergent.

For example, if t> 0 the Poisson summation formula gives the result

X e-''“' = (7r/t)"' X eu e Z ”

n/2 V g-'n-2lul2/f u e Z "

R em ark 11.10. Let / = S t2„u , ueZ ", so / = Z t„8, u eZ ". Let U be the cube { x s R " | lx,l<ir, ; = 1 , . . . , n} and suppose h e G ^ and H has compact support contained in U. Choose <f> 6 C^(U) such that <f> = 1 in a neighbourhood of supp h. Let = Since f*^ i = T.'r2m,h we have ft = <#)(/*/i). Thus

h = (2ir)-'’h''

= (2v)~’'ip*(hf).

But h f = Y, h(w)T„S then implies

h = (217)“" Xu e Z "

where the series converges in In other words we may reconstruct the function h at all points merely by sampling it at the points of the integer lattice This remarkable fact is known as the sampling theorem; any function which is band-limited, i.e. whose Fourier transform has compact support, may be reconstructed by sampling it at the points of a suitably spaced lattice. For a discussion in the one-dimensional case see for example Brigham [1].

If / is a locally integrable periodic function then the Poisson formula enables us to obtain the usual formulae for the Fourier coefficients of /.

T h eorem 11.11. I f /€L'(IR”, loc) is periodic with group o f periods 27tZ” and

/ = (2Tr)-'* X o„e‘<“->u e Z "

where the series converges weak* in 5 ' then

a^= { dxJk

where K is the fundamental parallelepiped [0, 2 ttY.

Proof. If <^€C“(1R") then

(f,4>)= \ f(x )^ ix )d x = I X <P(x + 27ru)f{x) dxj Jk u e Z "

since the sum is finite. Since 4>(x + 2 ttu) = {2ttu) the Poisson

summation formula now gives

J k u g Z "

The right side here depends continuously on <l> for the topology of Sf as does the left side. Thus the equality holds for all in 5 and in particular for </) G Cc(R'"). Since </, = 4>) and / = E we also have

ueZ"

If u g Z” choose <>g Cc(1R”) so < >(u) = 1 and <#>(m) = 0 if mgZ” and Then we obtain the desired formula for a„.

E xam ple 11.12. Let f be the function on IR which is periodic with period 2 7 and satisfies

« '> - { o if'0<X <77 77 < X < 2t7.

Then by Theorem 11.11 we have

^ ( , ) - 2 £ sin[(2 tH-l),]77fc=o 2fc + l

where the series converges weak* in Sf'. Since differentiation is continuous on Sf' we may differentiate term by term and we obtain

00 oo

Z ( - l ) " T fc ,S = - Z cos[(2fc + l)t]k = —oo 77 =Q

where both series converge weak* in 5 '.

Fourier series of periodic distributions together with many interesting calculations and applications are treated in Schwartz [3]. For a different point of view see Lighthill [1]. For a more applications orientated point of view and many illuminating diagrams see Brigham [1].

§12. Laplace transform

If has compact support the Laplace transform of f is theentire function F defined by

F ( z )= ! ■

.-<2.X>fix ) d x = f ( r i - i i ) .

12. LAPLACE TRANSFORM 129

This definition extends immediately to distributions with compact support. If we define the Laplace transform F of / by

F(z) = </,e-< ->).

If I, T) elR" we see

By the Paley-Wiener theorem an entire function F is the Laplace transform of a distribution f with compact support if and only if there exist constants C, N and R such that

|F(z)l<C(l + |zl) e' * ’ where z = + iTj.

Consider now any distribution We formally define the Laplace transform F of / by

To make sense of this definition (when possible) we see that we should focus attention on the set

This approach is the one taken by Schwartz [1]; see also Streater and Wightman [1]. A more restrictive situation is considered by Hayakawa [1] and Gel’fand and Shilov [2]. Let be a convex function whichincreases sufficiently rapidly as Ixj^oo. Then for each we maydefine a function in 5 by Now if / is a distributionsuch that e' f is temperate we define the Laplace transform F of / by

F(z) = (c^/,e-^-<"">)

In this case one obtains theorems of the Paley-Wiener type where the exponent R lll above is replaced by </>*(— ) where is the function conjugate to c) in the sense of Young.

Let ^1, . . . , g R ” and define the function a byN

a (x ,z ) XJ = 1

If $ lies in the convex hull of |i , . . . , say | = X h — 0. Z then

e = nJ = 1

implies

mine ^’""^^maxej j

Therefore

0<|a(x, | + iT])l<l

for I in the convex hull of . . . , If we consider derivatives of a with respect to x we obtain linear combinations of products of functions of the type a(x, z) but with z in some of the factors replaced by the and with coefficients which are polynomials in rj and the It follows that we have

|D:a(x,| + iT ,)l< Q (l + h l)'“i

for I in the convex hull of , 1 .

L em m a 12,1. Let U be the interior o f the convex hull o f andlet a be the function defined by

a{x, z) X = ■<2,X>

I f U is non-empty and geS f' then a(-, z)g is an Sf'-holomorphic function o f z in the tube U + iW .

Proof. We note we may differentiate a relative to and rj as well as X, and we obtain

\D-DfD-a(x, I + iT,)l < C„.3,^(l + 1t,1)'“ '(1 + 1x1)' '"'"' (*)

for in the convex hull of l i , . . . , In- Now let <t>eSf. Then a(x, | + it])< )(x) is C " in the variables x, |, r\. Moreover, for each /3, y we have DfD^;a(*, | + and the inequality (*) implies that the usualdifference quotients for these derivatives converge to the derivatives in the topology of 5 (by mean value theorem). Thus (g, a(*, z)<f>) is C~ in U +ilR”. It is holomorphic in U + iR ” since the Cauchy-Riemann equations are obviously satisfied.

L em m a 12.2, I f feO )'(W ) then the set H/ = {|eIR” | isconvex.

Proof. Let r), and let | = tT j+(l —t)cr. Define thefunction a by

:,x>a(x)(e + e = e

As noted above a eO ^ . Indeed a and all its derivatives are bounded. Since

e-<€,->y:=ae-< ’*>/+ae-< ">/

we see

Lemma 12.3. Let feO fiR "). Let w, and assume t + ( l —1)«> is in H/ for 0 < f ^ 1. Then w e S f if and only if

tio

exists for each <f)eSf. Moreover, the limit is necessarily </>).

Proof. Define the function a by

Then

lD E 'D ?a a x )l< Q ,,( l + lx|)'

for It follows if <l>e^ then t ^ a(t, •)<#> is a continuous, indeedinfinitely differentiable, function from the interval [0 ,1 ] to 5 . Let g =

and h = e “ " ' /. Then geSf'. If <^eSf then

d>) = {g , a { t , -)4 > )+ { h , a i t , •)<► ).

If <0 e H/ then h e ^ f’ and the right side is continuous on [0 ,1], Thus the limit as f|0 exists. Conversely, if the limit exists then by the Banach- Steinhaus theorem there is a distribution h e ^ ' such that

lim<e-<‘«"< -‘>‘“->/,</.) = <h,«f.)tio

for each <f>eSf. But if <f>e C“(1R”) then the limit on the left side is clearly Thus, e-<" ’->/=hG^'.

Exercise 12.4. I f feQ)'{R*^) and A is the convex hull o f a finite subset o f H/ then z —> e“^ ^ i s a continuous map o/ A + iU' into 6 ' with the wcafc* topology. I f B is any subset o f H/ and e“^ ^ i s weak* bounded in Sf’ for ^ e B then z - weak* topology.

e '7 is a continuous map o fB + ilR" into 5 ' with the

The first part of the exercise says e~ ''' f has non-tangential boundary values at those parts of the boundary of H/ + ilR” that have real part in Bf. The second part gives a condition for continuity up to the boundary. The simplest way to do the second part of the exercise is to note that on the weak* bounded subsets of Sf' the weak* topologies of Sf' and Si'ClR"") coincide. In the exercise one may replace ‘continuous’ by ‘infinitely differentiable’.

Lem m a 12.5. Let and let U be the interior o f Hf. Letd{x) = (1 + I f K is a compact subset o f U there exists £ > 0 such thate0-<2, i/ z = + IT) and E K. Indeed there exists a continuous

seminorm p on Sf and an integer N such that if <l>eSf then

< )1 < p(<t>)(l + hl)^ for z = ^ + i'n eK ^ iU \

Moreover, is an Sf'-holomorphic function in KQ-\-iR* where K q isthe interior o f K.

Proof. We choose e > 0 and a finite set A = { ^ i , . . . , In) in U such that the closed s-neighbourhood of K is contained in the convex hull of A. Let a be the function associated to A by Lemma 12.1 and let

b(x ,z) = a(x,z)e^^^^^

If z = I + IT) then

\b{x, z)l<e®e*'^' la(x, z)|

= e sup lelories

-<<T,X>a(x, z)l

se® sup |a(x, z + cr)|.l o - |< e

Since + cr lies in the convex hull of A we have

lb(x,z)l<e" if ^ g K

If we differentiate b relative to | and rj the effect is simply to multiply b by a polynomial in x. If we differentiate b relative to x we obtain sums of terms of the form D “(e®®)Dxa. The factors D^a are linear combinations of products of functions of the type a(x, z), but with z in some, but not all, of the factors replaced by some of the and with coefficients which are polynomials in t) and the The factors D “(e ®) are bounded by constants times e®®. Incorporating these exponential bounds with factors of the form a(x, z) as above, we obtain

|D :D fD ;b(x,^ + iT,)l<C„.3,^(l + lT,l)'“'(l + |x|y®i ' ' for ^ e K .

As in the proof of Lemma 12.1 it follows if g g 5 ', g 5 then <g, b(-, z)<t>) is holomorphic in K o+ iW . Moreover, from the estimates on b and from the continuity of g we see there exists a continuous seminorm p on 5 and an integer N such that

Kg, b{-, z)<#))l<p(<f.)(l + lT}ir

for IG K and Since

X b(-, z}e~^^ryJ = 1

and e 3/ the conclusion follows.

Theorem 12.6. Let and let U be the interior o f 3/. I f ^ e Uthen is in Oq. Moreover,

F ( i + iv) = (e-< ^ y f(v )

defines a holomorphic function F in the tube U-\-iR* and for each compact subset K o f U there exist constants C and N such that.

lF(^ + iT])l<C(l + l'nl) for each ^ eK .

Proof. If then Since it follows thate~^ '' feO'c. Thus (e“^’ /) and therefore the definition of F(^ + ir])makes sense. Let K be a compact subset of U and choose e > 0 as in Lemnia 12.3. Note if g e ^ ' and then (<f>gy and (<f>g) (0) =(27r)”"g*<^(0) = (27r)“”(g, = <f>). Thus if we have

F(^ + iT]) = (e-<^"->/r(0)= (e-"®e"®-<"’ >/) (0)

It follows that F is holomorphic in K q+ iIR” and

|F(^ + ^T])l^p(e-")(l + lT]|rfor | g K where p is the seminorm given by Lemma 12.5.

We call the holomorphic function F the Laplace transform of the distribution f. We now prove the converse of Theorem 12.6.

Theorem 12.7. Let U be an open convex subset o f IR"' and let F be a holomorphic function in the tube 17+iR”. Assume for each compact subset K o f U there are constants C and N such that

lF(| + iT,)l<C(l + b i r forieK.Then there exists a unique distribution /e2i'(IR") such that U ^ S f and

F(|+iT,) = (e-«->/)"(T,).

Proof. The estimate on F certainly implies that there exists such that gg = F(4 + i •) for each ^ eU . If <f>e C”(IR") then

<gg,e<«-><#.) = (27r)-"<gg,(e<«-V)''')

= (2ir)-’’ | F (i+ ir ,)< b (H -v )d v .

The integrand here is a holomorphic function of l + ir] g l7+i(R” and if K

is a compact subset of U then the integrand is rapidly decreasing in t), uniformly for | e K Therefore we may change the contour of integration. It follows that the integral is independent of $ for | in the convex set U. But then / = e * ' gg is independent of | for $ e U . Since we haveU £ a f. Moreover, = gj = F (|+ i ■).

Remark 12.8. With regard to the proof of Theorem 12.7 note that for the holomorphic function F, if e > 0, one has the Cauchy inequalities

lD ?F(2) l < a ! e“'“' sup lF(z + w)l.lw ,l<e

These inequalities may be used to obtain estimates of the form

+ + for

It then follows that g^eOc and for each we have (g , <#>) is smoothin and

It follows that < e ^ ^ <l>) is independent of e 1/ if <#> g C“([R”). Thus we obtain an alternate proof of Theorem 12.7 and in addition we see that the fact that e“^ ■ /g 0'c for | g t/ (see Theorem 12.6) is a consequence of the Cauchy inequalities.

We turn now to a consideration of convolution products. The Laplace transform enables us to introduce quite general convolution products via multiplication of holomorphic functions. Let /, gGS)'(IR”) and suppose that f and g satisfy the support condition, condition (C), of Section 1:18. Let A = supp f , B = supp g and let K be a compact subset of IR' . If g Ck define = so G ^ e C k - b - By condition (C) the setA n ( K - B ) is compact and therefore we may choose 0 g Cc(IR'') such that 0 = 1 in a neighbourhood of A n ( K - B ) . Then

</*g, <#>) = </, <#>€Ck .

If IGIR” then <g, = e~ ''' G' (x) where G'^(x) =It follows that (/*g, = 6G'^) and therefore

for each ^ g [R”.

-<e*>* g,

Theorem 12.9. Let /, gGS>'(II ”) CLi d assume that the interior U o f nHg is non-empty. Then there is a unique distribution f^ g such that

* g) = Y) * (*)

for each ^ e U where the convolution on the right side is the convolution o f distributions in Oq-

Proof. If F is the Laplace transform of f and G is the Laplace transform of g then by Theorems 12.6 and 12.7 there is a unique distribution /*g with Laplace transform FG. Then (e“^ ‘ (/*g)) = (e-<€ -> ) (e“^’' g) for each ^ eU .

Remark 12.10. If / and g satisfy condition (C) then by Exercise 6.12 the convolution on the right side of (*) agrees with the convolution defined in Section 1:18. Thus by the remarks preceding the theorem the definition of /* g agrees with the definition of /* g given in Section 1:18.

We will now prove a Paley-Wiener type theorem characterizing when r is a closed convex salient cone (Theorem 12.15). Together with Theorem 12.9 it will imply the perhaps unexpected fact that is a convolution algebra. Another application of this theorem is to obtain some results for partial differential operators which are related to the Routh-Hurwitz stability theory for ordinary differential operators. We will discuss these facts and give a few examples. First we need some preliminary results.

Lemma 12.11. Let m g C“((0, 1]) and suppose for some constants and r we have

\u^^\t)\^c^r^

for k > 0 and 0 < f < 1. Then there exist constants C and N depending only on r such that

\u^^\t)\^C max Cj

for k > 0 and 0 < t < l .

Proof. We may of course increase r since that only weakens the hypothesis. Thus we may assume r > 0 and r is not an integer. Then

•It

implies

iu < '^ > ( o is a ( | i - ^ | + i )

where is the maximum of and C^+i. If l - r > 0 we are done. If

1 — r < 0 we have

Now the lemma follows by induction.

Lemma 12.12. Let / g^'([R”), let U be the interior o f H/ and let F be the Laplace transform o f f. Assume t^ oeU for 0 < r< 1 and for some constants C, r and N we have

\F(t 0 + iv )\^ C r ’’( l + 1t)D^

for and T h en feS f' .

Proof. Let . If <#> g 5 then

<a, <#>)=!F(t^o+ d i l

is a C°° function on (0,1]. If we use the Cauchy-Riemann equations and integrate by parts we have

A<g(, </>) = Z f ^ o P i^ F { t io + iv ) < t > iv ) dT|j=i

= - i Z j io iD .r ,F it $ o + iv ) < t > iv ) dT?

= ' Z I F ( t i o + h)iojF> i< (> (v) dt?

It follows that

0^(g„ = I f F(tio + iT,)^gD“< .(1 ) dr,.lal = k ^

Therefore by hypothesis

where is a continuous seminorm on 5 . By Lemma 12.11 we then have

l(gt.<f>)\^p(<f>), 0 < t < l

where p is a continuous seminorm on 6 . Replacing <#> by (27r)"”< '' we have

l<e-*< o’->/, < / > ) ! < 0 < t < l ,

where q is a continuous seminorm on 6f. Since weak* in

2 i'(IR") as ( —> 0 we obtain

for each <I)g C7{U") Thus

Remark 12.13. With regard to the proof of Lemma 12.12 note that if K is a compact subset of U such that tK ^ U for 0 < t < 1 then we may choose the continuous seminorm q independent of K, Indeed the depend polynomially on ind p is a constant multiple of the maximum of a finite number of the p . Since C“(IR”) is dense in 5 a simple argument shows that if </> e 6 then

lim <t>) = if, )t i o

uniformly for ^ e K Of course since that is, Oe H/, the uniformityalso follows from Exercise 12.4, but here we see it explicitly rather than by abstract functional analysis.

Lemma 12.14. Let /e S)'([R”), let U be the interior o f H/ and let F be the Laplace transform o f f. Let assume ^Q-\-têU for t >0 .Assume there are constants C, N such that

IHio + + iv )\^ C(1 + + ir\\)

for r\eW and t> 0 . Then

s u p p f^ H = { x e W |<4x)>0}.

Proof. Let K be a compact convex set disjoint from H and let (f) E C k’ Then

= (2-n -r" | F{ô+ti+iy\)ît$-'r\)<iir\.

By Lemma 8.6

for each fc > 0, where h is the support function of K. It follows that

for t > 0 . Since K is compact convex and disjoint from H we have ^(^)<0. Letting t-ôo we see that K H su pp f = 0 .

We now obtain our Paley-Wiener type theorem characterizing 5 f. Recall if r is a closed convex salient cone we denote its dual cone by

and we denote the interior of the dual cone by Fq . If | we let d{^) be the distance from to the boundary aF" of F" . As pointed out in Exercise 1:18.5 we have

5(1) = inf {<1, x) I X € r , lx| = 1}.

Theorem 12.15. I f T is a closed convex salient cone and F is a holomorphic function in Fo + ilR" then F is the Laplace transform o f a distribution f e S f r if tind only if there exist constants C, m and N such that

lF(z)| s C(1 + 5(^)-'”-'")(l + Izl)^ (*)

for each z = | + it] g Fq + iW . Moreover, in this case if then

lim \ F{t^-\-i7})(t>(r])dr\={f,(f>)tio J

uniformly for | in any compact subset o f Fq .

Proof. Suppose (*) holds. By Theorem 12.7 there is a unique f e such that Fq ^ H/ and F is the Laplace transform of /. Note if t > 0

and êFo then d(t$) = t8{^). Thus Lemma 12.12 implies feS f' . The last part of the theorem now follows by Remark 12.13. Finally if and tÔ then 8(ô-^t^)^8(ô)>0 and therefore Lemma 12.14 implies that supp (Fo)" = F. Conversely, assume that f e ^ r - For some integers

N > 0 we havem.

|(f, <f))l < C max sup (1 -|a l< N

\x\r \D^4,(x)\, <f,eSf.

We will prove that (*) holds with this choice of m and N. Choose p €C "(R ") such that p(t) = 0 if t < - 2 and p(t) = l if t > - l . Then p«^, •))e“^-' is in C'm for any |eIR" and if then p«|, •» = 1 on aneighbourhood of T. It follows that is temperate if Thus

and the Laplace transform F of / is holomorphic in Fq + iIR". Choose 6 e C"(!R") such that 0(x) = 1 for j x j s 1. Then and itsLaplace transform Fi is given by Fi(| + it]) = (0/)'(Tj-i|) and therefore by Theorem 8.8

lFi(| + iT,)l C(1 + II + IT]

where h is the support function of supp $f and where we have used that $f has order <N . Clearly h (-| )< 0 if leF"^ and so

|Fi(|+iT])l<C(l + ll + iT ]r , l e F " , t)€ R ".

It remains to estimate the Laplace transform F 2 of (1 - 0)f. Let u e C”(IR) be chosen so that u(t) = 0 if t < l and u(t) = l if t>6/5. If |€Fo define

tlf (x) = u(4<|, x) 8(|)-'(1 + lx R - ‘'2).

Then .^^ eC“(IR") and <g, x )> 6(| ) lxj/4 if If

L/ = {x €|R'’ |lxl>3/4 and <|, x )> 5(| ) |xl/2}

then U is an open neighbourhood of T n s u p p (l-0 ) . If x e U then

4<|, JC) ^ 2 1x1 (1 + IxR-^/">6/5.

It follows that = 1 on a neighbourhood of supp (1 - $ )f and therefore if z = | + iT] and we set = then

F^Cz) = (e-«->(i - e)f)\'n) = {Ai - o)f)\ o).

If we show that <t>z^ then by the argument in the proof of Theorem 12.6 we have

F2(z) = </,(1-0)<#>,).

Since the derivatives of 1 — 0 2ire all bounded we will obtain the desired estimate for F 2 provided we show that for any integers fc, M > 0 we have

niax sup ( l + lx|)'‘ lD“<#> (x)l:<Cfc,M(l + 8 ( i r ' ‘"'^)(l + lzl)'^,

| €T o, z = | + iT}.

If |a|^M then by Leibniz’ formula

lD“<# .,(x)l< C l lD^e-«^’’'-’=>| • |D>j(x)l

where the sum is taken over lj3l + lyl:^M. On the support of we have (4 1 1/4 and therefore

on the support of We may choose a common bound for the first M derivatives of u on the real line and therefore if l7 l< M then

lD > ^ (x )l< C (l + l^l6(|)-^y^'< c ( i+ i i iH i+ s ( ^ r '^ ') .

Thusmax sup (l + lxl)*' lD“<f> (x)llal<M X

< C(1 + lz|)' max (1 + 8(g)"’’) sup (1 + |xl)'‘e"® *''’‘' "'.p<lVf X

Now clearly

e-®<«)ixl/4(l + |x|)'= < 1 + (41 )“ 8(1)”'

and if 0 < p < M

(1 + 8(|)-'=)(l + 8(1)-'’) < 4(1 + 8(a-''-®4).

These estimates complete the proof.

Exercise 12.16. I f ueL^{U ) is real show there exist unique functions u,/g L (IR) such that

(a) V is real(b) supp/c(-oo,0](c) u-\-iv=f

Show that V is the Hilbert transform o f u (see Example 9.11). Let F be the Laplace transform o f f , so F is a holomorphic function in the left half-plane. Show that if x < 0 then

I lF(x + iy)p dy = 2 tt| e

Let F(x-\-iy)= Lf(x, y)-hiV(x, y). Show f(t) = (l/7r)Y (—t)u(—t) and therefore

U(x

Conclude

:,y) = - - f7T J

xu(s)x + ( y - s y

:d5, X < 0 .

f(y ) = L^ — lim F(x + iy).xTO

Finally show if F is any function holomorphic in the left half-plane and there is a constant M such that

lF(x + iy)p dy < M, x < 0 (+)

then F is the Laplace transform o f a function /g L^(R) and supp/c(-00, 0].

The exercise gives a precise meaning to the informal statement that the Hilbert transform of the boundary values of the real part of a holomorphic function is the boundary value of the imaginary part. The space of holomorphic functions satisfying (+) is called the Hardy space of the left half-plane and is denoted by H^ (see Hoffman [1]). Caution: the same notation is used for Sobolev spaces (see Chapter 4).

Lemma 12.17. Let fGQ)'(R'^), let , z^] and let P (D )f= g.Then H/ ^ Hg. I f the interior o f H/ is non-empty, F is the Laplace transform o f f and G is the Laplace transform o f g then G (z) = P(z)F(z).

Proof. If I G E/ then

e-< ’->D,/ = D,-(e-< ’->/) + §e-<^’‘>/

implies Taking the Fourier transform then implies that theLaplace transform of Djf is ZyF(z).

E xercise 12.18. Let P(A) = XlT=o «k ^ ^ 0 and /ef P(D) bethe corresponding ordinary differential operator. Show that P(D) has a unique fundamental solution E with support in [0, oo). Note that E is analytic in the complement o f the origin and if m > 2 then EeC^~^(^). Show if M is the maximum o f the real parts o f the roots o f P(A) then s = O’ + IT G He if O’> M . Show

(A) Unit^^E(t) = 0 if and only if M < 0 .(B) E{t) is bounded if and only i/ M < 0 and the pure imaginary roots

o f P(A) are simple.(C) E is temperate if and only i/ M < 0 .

The exercise may be done by solving P(D )g = 0 with appropriate initial conditions and then setting E = Y g where Y is Heaviside’s function. Sufficiency is clear but necessity requires a little argument. The results of the exercise also hold for systems of linear first order ordinary differential equations with constant coefficients, where the above criteria are phrased in terms of the real parts of the eigenvalues of the coefficient matrix. The only difference is that in (B) the correct condition is that the pure imaginary eigenvalues should be simple roots of the minimal polynomial.

Case (A) in the exercise is stability in the sense of Routh-Hurwitz. According to Hurwitz (see Marden [1]) there exist certain (explicit) polynomials g Z[Xq, . . . , X^] such that if the are real, andao>0 then (A) holds if and only if Qk(ao, . . . , o^)>0 , fc = 1 , . . . , m. Case (B) has been studied by Viola [1].

A result for partial differential operators related to case (A) has been obtained by Kannai [1]. Kannai shows if P g 0 [ z i , . . . , z„] and T is a closed convex cone then P(D) has a fundamental solution E eO 'c with support in T if and only if P(z) has no zeros in T^-\-iW. We will give a generalization of case (C) to partial differential operators. We will consider only salient cones. The general case has been settled by Enqvist [2].

T h eorem 12.19. I f T is a closed convex salient cone and P g 0 [ z i , . . . , z„] then P(D) has a fundamental solution in If only if P(z) ^ 0 for each z g Tq + iR”.

Proof. Let E e S f r h e 3. fundamental solution for P(D). The Laplace transform F of E is holomorphic in ro + iR"" and P(z)F(z) = l. Conversely, assume P(z) has no zeros in To+iR^ Then F(z) = l/P(z) is holomorphic in TJ + iR''. By Theorem 8.14 there is a constant C and a finite set © in {x g R'' | 0 < lx | < l} such that

1 < Cr~" max inf \P(z + rr6)\

for each z g 0 ” and r > 0. Fix z g Tq + iR” and let r be the distance from z

to the boundary of F J + ilR". Choose do€& , depending on z, so

l < C r ' " inf |P(z + irr0o)l-1t 1 = i

By the maximum principle

lF(z)l < sup |F(z + rr0o)l ^ Cr“"'.k l = i

If z = I + IT] then r = 6(^) and therefore

|F(z)l<C6(^)—

By Theorem 12.15 F is the Laplace transform of a distribution in

If P(D) admits a fundamental solution E in where F is a closed convex cone we call F a propagation cone or a causal cone for P(D) and we call E a causal fundamental solution. The dual cone F^ is called a cocausal cone and its interior Fq, if non-empty, is called a hyperbolicity cone. The elements of the hyperbolicity cones are called hyperbolic vectors for P(D). They may be characterized algebraically. See Garding [3] and Hormander [2].

The reason for the above terminology is as follows. If P(D ) admits a fundamental solution E with support in the closed convex cone F and

then P (D )u = f has a solution given by u = E ^ f.Since supp u ^ F + supp/ the solution u may be thought of as a disturbance caused by f and propagating along the directions in F. If tj g F^ we may think of H t= {xeW " |(t], x) = f} as the space of some observer at time t. Then Ht Hsupp u increases with t. If tj g Fq the speed of propagation is observed to be finite in all directions.

The canonical example is the wave operator or d'Alembertian which we will now consider.

E xam ple 12.20. Let P(D) = c~^Dn-T4 l l D f be the wave operator in IR". Here c > 0 is a constant. If z = + ir] and P(z) = 0 then

andc~^L-nn = v'}

where |' = ( l i , . . . , Assume l^„l>c Then 1t}„|>c Itj'I and therefore = ■r}')l^c^lg'll'rj'l<l4T]„l which is a contradiction.Therefore, if

F = {x g (R'' I

then P(z) has no zeros in F J + iU' where

n = {| € R "| 4 > c l| 'l} .

It follows that P(D ) has fundamental solutions E+sSf'r and E -eS f'-r. These are called the forward and backward causal fundamental solutions, respectively. Note if e 5 then

M v)(E , , 4>) = lint f :tio J j - dT|

uniformly for in any compact subset of Fq. It follows that E+ is homogeneous of degree 2 —n.

Recall if A G GL(n, R), <#> € C”(R”) we define

(A.<#>)(y) = <l>{A~^y), y e R ”.

If we define A .mg '(IR' ) by

(A m , <f>) = Idet A j {u, A-^<^>), <f> e O R " ) .

Clearly supp (A m ) = A(supp u). If A ' is the transpose of A then we have

P(A'D )u = A ,(P(D )(A -\u)),

We will say that P(D) is invariant under A if P(A'^) = P(|), ^g R ”.

Exercise 12.21. Let T be a closed convex salient cone and assume P(D) has a fundamental solution p G ^ f. I f A eG L (n ,U ) , det A = ± l , P(D ) is invariant under A and there exists t) g Fo such that (tj, x )> 0 for each X in A(F) then A .E = E.

Exercise 12.22. Let P(D) be the wave operator and let E+ be the forward causal fundamental solution. Use the previous exercise to deduce that E+ is invariant under the restricted Lorentz group. Using also the homogeneity o f E+ deduce that supp E+ is either T or dT where T is the forward causal cone.

The restricted Lorentz group is the connected component of the identity in the Lorentz group and consists of the orthochroneous Lorentz matrices of determinant 1. Both situations in the exercise actually occur. Indeed, by Huygens’ principle, if n = 2 or n is odd then supp P+ = F whereas if n is even and ni=2 then supp E+ = dT. See Treves [2].

Exercise 12.23. In R the forward causal fundamental solution o f the wave operator c~ ^ D l~ D l is the bounded function E+ defined by

£+(Xi, X2)J c l 2

loif CX2>l l| otherwise.

Exercise 12.24. In R the forward causal fundamental solution o f the wave operator c“^ D | -D ? -D i is the locally integrable function E+ defined

by

£+(Xi, X2, X2)=<— if CX3>|x'l

0 otherwise.

Exercise 12.25. In the forward causal fundamental solution o f the wave operator c~ ^D l~D i — D l~ D l is the measure E+ defined by

(E^, <!>} = —477 Jo

<l>(r(o,c V)rdct>dr,

where is the unit sphere in IR = {x € | X4 = 0}. One customarily writes

E^ = \x'\~^ 6(0x4- Ix'l)477

since r d(o 6 x4 = r c~ dco dr on dT = {(rco, c~V) | co g S , r > 0}.

Lemma 12.26. Let C be a convex open cone and let U be a convex open subset o f C. Then U + C = U if and only if tU ^ U for t > l andc = u>0 tu.

Proof. If 6 U then | = + (1 — s)^ and for some s with 0 < s < 1 wehave s ^ e U since U is open. Thus L7c U + C Assume now U + C = U. If t > l then t U c U + ( f - l ) U c U + C = U + C + C = L / + C = U . Let ^ g C and choose any r j e U . Since C is open if s > 0 is sufficiently small then sct)_= 1 + 8( — 7] )G C Thus (j jeC. If t = 8( l + e)“ then = t(i7 + 6))G t(L7+ C) = tU. Conversely, assume tU U for t > 1 and C is the union of tU for t > 0 . It suffices to show U + C ^ U. Let 7]g U and ^g C Choose r > 0 so t ^ e U and let s = ( l + 0 “\ Then 0 < s < l and 1 —s = fs. Then sr(T] + ) = ( l - s)T] + s t ^ G ( l - s ) U + s U c U and so t)+ ^ g (sO“'U c U.

Theorem 12.27. Let P g 0 [z i , . . . , z„] and let T be a closed convex salient cone. I f there exists a convex open set U in Fq such that tU ^ U for t > l and Tq is the union o f tU for f > 0 , and if P (z )^ 0 for zg U + zR ” then P{D) has a fundamental solution E e ^ p . Moreover, U

Proof. By Lemma 12.26 U-\-Tq = U. Thus if ( o e U then P(co + z) ^ 0 for each z g Fo+ zR ”. By Theorem 12.19 it follows that P(o> + D) has a fundamental solution E'^GSfp. By Leibniz’ formula (Lemma 1:12.1) we have P(D)(e<"’'->E0 = (l/« \)o) e "'' P^"\D)E' = e<‘"">P(co + D )EL = 8.It follows that E = is a fundamental solution for P(D) in ^p. ButP(D ) has at most one fundamental solution in ^p since Q)p is a convolution algebra (see also Exercise 12.21). Therefore E is independent of co. Since e“^ " = E' we have (o g He-

The converse of Theorem 12.27 is also true but the proof requires the Seidenberg-Tarski theorem and in addition some facts about hyperbolic polynomials or Bochner’s theorem on tubes. For the present we content ourselves with a statement of Bochner’s theorem and we show how it allows us to dispense with the convexity hypothesis on U. In Remark 4:8.18 we will use this observation to give a proof of the converse of Theorem 12.27.

T h eorem 12.28. B och n er's th eo rem on tu bes. Let V be a connected open set in R ” and let U be the convex hull o f V. I f F is a holomorphic function on V-\- iR” then F has a holomorphic continuation to the convex tube U+iR"'.

For the proof of Theorem 12.28 see for example Hormander [1].

C orollary 12.29. Let P g 0 [ z i , . . . , z„] and let T be a closed convex salient cone. Suppose for each compact subset K o f Tq there is a constant tK>0 such that P(t^ + ir j)^ 0 if and t>tf^. Then there is a convex open subset U o f F J such that tU ^ U for t > l , Fq is the union o f tU for t > 0, and P(z) for z e U ^ iR^

Proof. The hypotheses imply there is a connected open set V in Fq such that tV c V if t > 1, F J is the union of tV for f > 0 , and P (z )^ 0 if Z6 V + iR ”. Then 1/P is holomorphic in V + f R ” and therefore by Bochner’s theorem 1/P is holomorphic in U + iR” if U is the convex hull of V.

E xercise 12.30. I f P(D) has a fundamental solution in Oq then it has no other fundamental solutions in \ I f F is a closed convex salient cone and P(D) has a fundamental solution in then for any coeFo the operator P(a> + D) has a fundamental solution in (O r-

The first part of the exercise follows from the fact that Sf' is an O'c convolution module. Give a direct argument using the Fourier transform instead.

13. THE WAVE FRONT SET OF A DISTRIBUTION 145

§13. The wave front set of a distribution

The wave front set of a distribution is a refinement of the notion of the singular support. It may be thought of as giving at each point the directions in which a distribution is singular. The wave front set was introduced by Hormander [3, 6, 7] and is related to an analogous concept introduced for hyperfunctions by Sato [1].

Lemma 13.1. I f then ueC^ClR") if and only if for eachinteger N s 0 there is a constant such that

Proof. Indeed the inequality implies is integrable for each aand therefore is continuous. Conversely, if v is smooth then is the Fourier transform of and so | “t3(|)|<|lD“u||Li.

Let ft be an open subset of !R”. If ueS)'(ft) then <t>uG '(R' ) for each </)GC“(ft). Moreover, by Theorem 8.1 we have

$u(^) = (u,

Corollary 13.2. Let u G^'(ft) let U be an open subset o f ft. Then u\uG C^(U) if and only if for each <f> e C^{U) and each integer N > 0 there is a constant such that

l<u, e- < '-><#>)l < C ,, (1 +111)-^ IE R".

Recall the singular support of me Si'(ft) is the complement of the largest open subset of ft on which u is infinitely differentiable. Motivated by Corollary 13.2 we introduce a refinement of the notion of singular support as follows. The wave front set, WF ( m ) , of u is defined to be the complement in f tx (R ” -(0 )) of the set of all points (xq, o) in f tx ([R” —(0)) such that for some neighbourhood U of Xq and some conic neighbourhood V of we have for each <>e Cc(1/) and each integer N > 0 a constant such that

l<M,e- < ’•></>)l<C ,, (l + l |)- for each e V.

We denote by tt the projection of ftx((R" —(0)) onto ft. A subset T of ftx([R” —(0)) is called a conic set or a cone if its intersection with each fibre of 77 is a cone in the usual sense: that is, if (x, e T implies (x, t ) e T for each t > 0 . Note that complements, unions and intersections of conic sets are conic sets.

Lemma 13.3. Let ft be an open subset o f R” and let MES)'(ft).

(a) WF (m) is a closed conic set in f tx (R ” -(0 )) .(b) I f W is an open subset o f ft then WF ( m |w^) = WF ( m ) n7r“^(W).(c) sing, supp M = 77 (WF (m)).(d) I f il/e C“(ft) then WF (i/ fu ) c WF ( m ) .

(e) I f il/e C“(ft) and W = {x e ft | i/r(x) i= 0} then

WF ( i i / u ) n 7 T - \ w ) = WF ( m ) n T T - \ w ) .

Proof, (a) The complement of WF ( m ) is the union of the open conic sets U x V provided by the definition, (b) Obvious, (c) If Xo^sing. supp u

then by Corollary 13.2 (xq, W F (m) for each (0). Conversely,suppose 7t(WF (u)). By compactness of there is a neighbourhood U of Xq such that for each <(> g C^iU) and each integer N > 0 there is a constant with 1<#>m(^)1<Ctv,^(1 + 1 1)" . Thus by Corollary 13.2 Xo sing, supp M. (d) Obvious, (e) By (b) it suffices to show that WF (i/ ulw) = WF (wlw) and therefore we may assume il/{x) ^ 0 for each x. Then (e) follows from (d) since u ^ (l/i/f)(i/fw).

Let P = Y e C“(n), jaj < m. Then clearly supp Pu ^ supp u foreach u^Q)'{Q). This property is called the local property and characterizes the differential operators in the class of linear operators; see Peetre [1,2]. There are, however, non-linear operators which have the local property but are not differential operators. We also clearly have sing, supp Pu ^ sing, supp u. This property is called the pseudo-local property. In Chapter 3 we will see that pseudo-differential operators have the pseudo-local property, and indeed have the following stronger version of the pseudolocal property.

Theorem 13.4. Strong pseudo-local property. I f P =a G C°°(n), jal < m, then WF (Pm) c WF (m) for each u g

Proof. In view of Lemma 13.3(d) it suffices to show W F(D ,m) c W F (m). If 4>GCr(n) then <f>D,M = Dy(< >M)-(D,-<#>)M. If (xq, W F (m) there is a neighbourhood U of Xq in H and a conic neighbourhood V of go in IR” - ( 0 ) such that if c )g Cc(U) then l<#>M(g)l< CN< (l + lg|)“^ for each N > 0 and each gGV. Then l(A■(< >M)) (g)l<|g,•fC ,, (l + lgl)-^ and l(( -c^>)M)7g)l < Thus (xo, go) WF (D,m).

In the definition of WF (m) we have to estimate $u for an inconveniently large number of functions 4>. The following theorem shows that we can get away with fewer estimates.

Theorem 13.5. I f uGO)'(Ci) then W F(m) is the complement in f ix (R” -(0 )) o f the set o f all points (xo, g o )e flx (R ” -(0 )) such that there exists (f) G C“(ft) with <t>(xo) ^ 0 and there exists a conic neighbourhood V o f go in IR” - ( 0 ) such that for each integer N > 0 there is a constant such that

1<U, ^ Ck (1 + 111)-"' for each | € V.


Proof. Let W 'F(u) be the set defined by the hypotheses of the theorem. If a point is not in WF (w) then it is certainly not in W'F (w). Thus W'F (u) c WF (m). For the converse suppose (x©, lo) W'F (m). Then there exists <f> € C”(fl) with <f>(xo) i= 0 and a conic neighbourhood V of |o in IR^-CO) such that |^(|)|s Cn(1 + ||1)~' for each integer N > 0 and

each I e V. To show (xq, I o) WF (w) it suffices by Lemma 13.3(e) to show (^ o > l o ) WF since < ^ » ( x o ) ^ 0 . Let j/re C ^ C I R " ) . Then

i/><#>u(|) = (2 ir)~ '‘ | dr).

Let

jTi( I )= f <^(|-T|)^(T})dT|Jv

l2(^ )= f 'A (| -'n )^ ('» j)d ’r}.4?"~V

We will estimate these integrals. Since i/rGC“(lR”) by Lemma 13.1 for each integer N > 0 we have a constant such that

|iA(|)1^An(1 + 1H) for each | eR ".

Thus

f ( l + l| -T 7 l)-" '( l + |7,l)-'^-'‘ d7,.Jv

Since l + l^ N l + ||-'nl + |'r}l<(H-l^-T}l)(l + lT|l) we have

lIi(|)l<CN,fcAN(l + ll|)-^| ( l + h l)- '' dr,.

If we choose fc > n + 1 the integral converges and so for each N 0 we have a constant CJv such that

lli(^)l ^ +111)-^ for each 6

We now estimate l 2(^). First note since by the Paley-Wienertheorem there are constants B and M such that

l<#)u(^)|<B(l + l l)' for each ^ g IR”.

Thus

f ( l + l ^ - t j | r ^ ^ - " '“ '‘( l + l T , r dr,

<A2M+N+fcB(l + l|l)' [ (l+l^-T ,l)-^-"'-'=dT,

where we have used —rjl)(H-|T]|). Now let V' be an opencone in V such that ^ osY ' and such that the closure of V 'n S ”“ is compact in V. Let 6 > 0 be the distance from V' H to the boundary

of V. If e V ' and -n € R" ~ V then HI"' € V' O S "“' and 17 e R" ~V. Thus 1| - tj1>S 111. If 1 6 V ' it follows that

lI2( l ) l^ A 2M + ^ ,, ,B ( l+ l l lr ( l+8 iiir^^-^'l (i+|^-iî)-fc dT,

< A Js ,(l + |||)-'^|(l + l'rilr" dr,.

Again if we choose fc > n + 1 the integral converges and so for each N > 0 we have a constant Cl such that

ll2(l)l ^ c;^(l +111)'^ for each | € V'.

If = Cn+ Cn we have

for each |€ V', each if/e CîW) and each N > 0 . Thus (xq, I o)WF(</>w).

The coordinates X i,. . . , in f l induce the so-called canonical coordinates X i,. , . , Xn, , in in the cotangent bundle If p e f l andCO G T^(fl) one defines Xy(co) = Xy(p) and |y(co) = (co, D,): that is

= Z lf ( " ) dx,(p).

The canonical coordinates allow us to identify T *(n) with ftx(R ” and hence to consider WF (u) as a subset of T*(fl). We will now show that WF (m) regarded as a subset of is independent of the choice ofcoordinates in fl. We will prove this fact by giving an invariant description of WF (m). Note the previous two descriptions are not invariant since they involve linear functions in the coordinates. (See Section 5:7 for additional discussion of canonical coordinates.)

Theorem 13.6. Invariance of W F(u). Let f l be an open subset o f and let u e S)'(fl). I f Xoe£l (o e T*^(fl) and co7 0 then (o WF (u) if and

only if for each real-valued i/co€C°°(fl) with dipoixo) = o) there exists a neighbourhood W o f i/cq in C“(fl) (real-valued) and there exists <f> e C“(fl) with <l>(xo)i=0 such that for each integer N >0 and each bounded set B ^ W we have

(u, e”‘*‘ </)) = 0(t“^) as t-^ ^ , uniformly for if/g B.

Proof. Identify T*(fl) and f lx R " by means of canonical coordinates so ct) = (xq, io). Assume first the hypotheses of the theorem hold. Let ô(x) = d o , x) so di/fo(xo) = io. The map [R” C "(fl) : •) is continuous. Thus there is a neighbourhood V of io such that i e V implies <1, •)g W. Moreover, we may choose V compact so •) lies in a compact

and therefore bounded subset of W. Then by hypothesis

(m, ■<)) = 0(t~^) as uniformly for e V.

It follows that (xo, ^o)^WF (u). Conversely, assume (xq, WF (m). Let i/fo e C“(ft) and assume di/ro( o) = o- There is a neighbourhood U' of Xq such that for each integer M > 0 and for each <f>GC^{U') we have

<f>u(t ) = as t ^ o o

uniformly for 1 — ol — for some r > 0 . Let L7 be a relatively compact open neighbourhood of Xq in U' such that

\dil/o(x) - ol ^ r/4 for X 6 U

and let

W = {il/e C“(n)l \dil/{x) - d<AoU)l ^ r/4 for x € U}.

Then W is a neighbourhood of ij/o and

ldi/f(x)- ol — r/2 for x g U and ipeW .

It follows that

— 5 l l~ lo l

for «/f s W, X € t/, V where V is the ball with centre radius r. IfV then l|-^ol ( l + lll)”' is bounded below, say by 2s > 0 . Thus we

have shown

for x e U , ilfsW , ^ iV . (1)ld « /r (x )-^ l> s (l + |ll)

Let <J>eC^(U) and let B be a bounded subset of W. Choose d eC ^ iU ) such that 0 = 1 in a neighbourhood of supp <f>. Then

(u, e - ‘“ </)) = (<f»u), ^ - ‘ (e-“' 0)>

= (217)-" I (e - '" "0 ) '( -| )^ (l) d i

= (2ir)-"t"| (e -‘"^ 0 y (-t i)^ (t i) d i

= (2Tr)-'‘t " | | dx d^

In general this iterated integral is not absolutely convergent and we must be careful to integrate in the order indicated. We have shown

<M, = (2rr)-"r | I(t, |)<f>u(t|) d |

where

I(t, ^) = I e -“’<* 0(x) dx

and

p{x) = {^,x)->l/(,x).

We will now estimate I(t, |) by integration by parts and we will see that it is rapidly decreasing as t oo, provided V. The technique used is called the method of stationary phase and in the present case depends on the fact that p has no stationary points on supp 0 for V. The method is the same as the technique used to treat oscillatory integrals in Section 1:19. See Remark 13.7 for a discussion of the method of stationary phase.

Let L be the differential operator

SO

Since supp O ^ U by (1) we have ld p l> 8(l + lll) on supp 6 ii and ij/eW . Thus L makes sense and we may integrate by parts to obtain

I) = I e'«’(L')'‘0 dx

for t > 0, V and if/eW , where L ' is the formal transpose of L. We have

L '0 = -X a ,D ,0 + h0where

Oj = ldpl“ DjP

and

b = ldpl“" d^p • dp-|dp|“ Ap

where d p is the Hessian matrix of p. By an inductive argument

i L T e = \dpr^Q ^(p,e)

where Qk(p, 0) is a polynomial in dp, 6 and their derivatives with degree 3k in the derivatives of p. Since dp = —dik, ^ € jB, B is bounded and supp 6 is compact we have

lQfc(p,0)l<A^(l + |^ir

for t/reB, |€(R" and x e R ". By (1) it therefore follows that

l a r e i ^ A ^ d + i i i r "

for ifteB , V and x elR". Since supp {L')’ d is contained in the compact set supp 6 we have

for i\f&B and V. (2)

(3)

(4)

(5)

l i (U ) l< A fc r H i+ l l ir ' '

Of course, we also have

U (f,l)l^ | le(x )ld x = B i

for all t, and il/.Since c )U is a distribution with compact support we have

+ for and

By the choice of V and since 1/ c t/' we have

for t > l , V and fc>0.

If D is the volume of the ball of radius r we obtain

1<U, e - ‘‘V ) l ^ (2ir)-’'f" f I7(t, |)^(t^)l d^Jv

+ ( 2 i r ) - " r f \I{t,$)^{t$)\d$

<(2ir)-''BiCfcD t''-''

If we choose k large enough so that the integral converges and so that n -\ -M -k < - N we have (w, e“‘*' <f>) = 0(t“^) as uniformly for i/fGB.

R em ark 13.7. In the course of the proof of Theorem 13.6 we had to estimate an integral of the form J 0(x) dx. We used the fact that p has no stationary points to show that the integral is 0(f~^) for each N as t 00. The argument is similar to Hormander [8], Lemma 3.4 where the Fourier transform of is estimated for large A. Arguments of this type have a long history and are referred to as the method of stationary phase. The intuitive idea is as follows: consider the integral

I ds

where g is real and smooth. If we replace g by the first two terms of its Taylor series near Sq, g(so) + g'(so)(s - So) we see that near Sq the exponential may be regarded as periodic with frequency tg'{so)l2'ir. If

g'(so) ^ 0 and if h is sufficiently smooth at Sq then as f oo we expect to obtain a lot of cancellation in the integral. In other words, as t -> oo the major contribution to the integral will come from near the stationary points of g. This fact is referred to as the principle o f stationary phase and is usually regarded as being due to Kelvin (1887). According to Watson[1], Section 8.2, the principle also occurs in a footnote of Stokes (1856), in a paper of Riemann (1876 posthumous) and even in the abundant work of Cauchy (1815). See also Erdelyi [1], Section 2.9, for another discussion of the history of the principle of stationary phase.

R em ark 13.8. In discussing the invariance of WF (u) we should be a little careful concerning the invariance of distributions on fl. It is clear that the space C“(n) and its topology are invariant under changes of coordinates. The dual space &(£l) is therefore also an invariant object. Indeed, the only difficulty that arises is in the identification of locally integrable functions with distributions. Here we take the point of view of Section 1:13. As a submanifold of the manifold ft has a natural Riemannian structure. If / e L ^ ft, loc) then


{f, </>) = ! fix)<t>(x) d)x(x)

where p. is the Riemannian volume element. In any rectangular coordinate system p, is of course just Lebesgue measure relative to the coordinates and the present point of view is consistent with the approach of Chapter 1. It is an important point that it is the existence of a distinguished volume element that allows us to interpret distributions as generalized functions. In the absence of a distinguished volume element we are forced to regard distributions as generalized measures. In this case they are sometimes called twisted distributions. It is possible to introduce generalized functions even in the absence of a distinguished volume element, but they are elements of the dual space of the space of smooth volume elements (densities) with compact supports.

In the case of a distribution with compact support we have a nice characterization of the totality of singular directions. Let

7 T ':f tx | R "-(0 )^ (R "-(0 )

be the projection map.

T h eorem 13.9. Let u g € ' ( C I ) and let To = 7t'(W F(u)). Then Tq is a closed cone in (R'^-(O). I f F is a closed cone in IR” - ( 0 ) and r n T o = 0 then for each integer N > 0 there is a constant such that

1w( )1 Cnx(i+1 I) for each ^ € T. (*)

Conversely, if T is a closed cone in IR” —(0) such that (*) holds for each N > 0 then Tq is disjoint from the interior o f F. In particular Tq is the smallest closed cone in — (0) such that (*) holds for each closed cone T disjoint from F q.

Proof. Let K = suppu, Then W F (M )n (n x S "“') = W F (w )n (K x is compact. Its image under tt' is compact and generates Fq

and therefore Fq is closed. For the last part assume Fi is a closed cone such that (*) holds for each closed cone F disjoint from Fq. If let Fbe a closed cone disjoint from Fi with in the interior of F. Assuming the first part of the theorem Fq is disjoint from the interior of F, that is

Thus Fo^ Fi.Consider now the first part of the theorem. We prove the converse

result first. Let F be a closed cone such that (*) holds for each N > 0 , Let lo be an interior point of F and choose an open cone V with Let <l> E C“(ft) with <> = 1 in a neighbourhood of K so <j>u = u. Then 1< )M(|)1<CN(1 + Ig|)“^ for each ^ e V and each N > 0 . If x e K then <l>{x) = l and therefore by Theorem 13.5 (x, WF (m). ThusI o$ 7t'(W F(u)) = Fo.

For the direct statement, let F be a closed cone with F n F o = 0 . Then ft X F is disjoint from WF (u). Fix x e K . For each | e F n there is an open neighbourhood U' of x and a conic open neighbourhood V of | and a constant C^, depending on <f>, 17' and V, such that CJv(1 + 1t]1)“ for each iV >0, </>e C“([7') and rjE V. A finite number of the neighbourhoods V cover F n S ”“\ If we let U be the intersection of the corresponding U' we obtain an open neighbourhood U o f x such that l$U(T,)l<C"(l + h l)-^ for each N > 0 , cf>eC:(U ) and t /e F . A finite number of the neighbourhoods U cover K. If is a finite partition of unity in a neighbourhood of K with each (f>j having support in one of the neighbourhoods U then u=Y<l>iU implies (*).

We will now obtain an upper bound for the wave front set of an oscillatory integral of the type considered in Section 1:19. Let </>e C“(ftx[R ” -(0 )) be a phase function, let aE S^ s(ft,R ^ ) where 0 = 0 on ftx R ^ -(O ) then WF (A) = 0 . Hence for any phase function <j> we introduce the set

Q = { ( x , 0) E ft XR - (0) I d e< ^ )(x , e) = 0}.Then Q, is a closed conic set in ftx R ^ -(O ). By the same argument as in

the proof of Theorem 13.9 if

= {(x, I) € D X R " - (0) || = d,(x, 01 (x, 0) e Q }

then is a closed conic set in — (0). We will show WF (A) £ A< .Indeed, we can obtain a better result if we also take into account the growth of the amplitude function a. If W is an open conic set in f t X [R - (0) we define S^s( W) to be the set of all functions a e C“( W) such that for each compact set K in ft and each closed cone F in IR — (0) such that K x T ^ W , and each integer N > 0 , and each pair of multi-indexes a, P there exists a constant r,«,3.ic such that

lD?D?a(x, 6)\ < CN,r.„.e.K(l +

for each x e K and each 0 € T with l0|s:l. We set

S -“(W ) = n S^s(W).m

Now if a G S^5(ft, R^) we define the essential support of a, ess. supp a, to be the complement in ftx R ^ —(0) of the largest open conic set W in ftx R ^ -(O ) such that a lw e S ““(W).

Lem m a 13.10. I f a e cind V is a conic neighbourhood o fess. supp a there exists a ' e such that a'(x,6) = 0 ifsupp a' £ V and a — a ' e S ““’(ft, R^),

Proof. Let h e C“(ft xR ^ — (0)) be a function homogeneous of degree 0 chosen so h = 1 in a neighbourhood of ess. supp a and so supp h^ V . Let x ^ C ciR ^ ) be chosen so x:(^) = l if 1 1 < 1- Let

a'(x, 0) = { l-x (0 ) )h {x , e)a{x , 0).

Then a - a' = + (1 — x )(l ~ h)a.

Lem m a 13.11. I f 4> is a phase function, a g S^^Cft,IR^), m + fc< —N and A e*" a dS [Osc.] then A e C^(ft). In particular if a e S~“(ft, IR ) then W F(A ) = 0 .

Proof. Indeed, = 6)d0 converges absolutely anddefines A g C(ft). Moreover, we may differentiate under the integral sign k times.

If <> is a phase function on ftx lR ^ -(O ) and F is a closed conic set in ftx lR ^ -(O ) we define

I) l l = X<t>(x, 0), de<l>(x, e) = 0, (x, e) G F}.

Then is a closed conic set in ft x[R” - (0). If a g S^s(ft, IR ) we define

A< ,a A< ess.suppa*

Theorem 13.12. Let <f> be a phase function on —(0) and leta € [R ) where 0 < p ^ l and 0 ^ 6 < 1 . I f

A = I e' a [Osc.]

then

WF(A)cA^^^.

Proof. Let F be a convenient conic neighbourhood of ess. supp a. In view of Lemmas 13.10 and 13.11 we may assume a{x, 0) = 0 if 1 1 < 1 and that supp a ^ r . By Lemma 1:19.2 we have a differential operator

L = X aP e, + S + c

with aj G IR^), b , c g U^) and aj(x, 0) = bk(x, 0) = 0 if 1 1 < 1,such that L'Ce*" ) = e** where V is the formal transpose of L. Let cr = min (p, 1 — 6) and let u g C~(ft). If m + N —k a < 0 then

(A, = J J 6)u{x)) dx dS

= Z f'“' r I I e)Dû(x) dx d$

where the sum is over \a\^k and l|3l<fc, and a^^(x,6) = 0 if 1 1 < 1 , suppao,3 ^ r , and the double integrals are absolutely convergent. Let s > 0 and set a = + a"^ where the a^ anda"p are in !R^), vanish for 1 1 < 1 , have their supports in T andsatisfy

a'oc ix, 6) = 0 if \de(l>(x, 0)1 < e/2,

and

a"^(x, 0) = 0 if lde< (x, 0)1 > s.

Let

r;p(f) = 1 1 e)Dû(x) dx d0

_ I= e J^q(x)Dû{x) dx

where

Since a'^ {x, 0) = 0 when \decl>(x, 0)1 < s/2 the argument in Theorem 1:19.7

implies e Since = (Ja^Dûy(t^) it follows that I'aeit) =as for any M, uniformly for | in a bounded set. Let

(^0. and assume (xq, lo) A ,a- To show that(xo, lo) WF (A) we must produce a neighbourhood U of Xo, a neighbourhood V of |o> and £ and F as above, such that if

r«3(0 = IIthen for each M and for each u e C~(t/)

IU (t) = 0 (t-^ ) as

uniformly for | g V. Note that a"^ depends on the choice of e and F. We will choose s and F in a manner depending only on <f> and ess. supp a. Note that \i — d <l>(x, 0)| is continuous and homogeneous of degree 1 in (I, 0) g R”'^^-(O). Since (jcq, o) it follows if (xq, o) ^ss. supp a and de(/)(xo, 6q) = 0 then ô^dx<^(xo, Oq), Thus there is a neighbourhood U of Xo and a conic neighbourhood W of do such that d <f>{x, 6) lies outside a conic neighbourhood V of if FT and 6 eW . Since l| -d x < / )( x , 0)1^0 in U X V X W, if we shrink these neighbourhoods, then for some e > 0 we have

\ ^ -d M x ,e)\ ^ 2 ei\ e + \e\) i f x e U , ^ g V, S g W.

On the other hand if de< >(xo, Sq) i= 0 there is a neighbourhood U of Xo and a conic neighbourhood W of Sq such that lde</>(x, 6)\>2s if x g U, Og W and 8 > 0 is sufficiently small. It follows that for each (xq, 6o) ecss. supp a we have a neighbourhood U of Xq, a conic neighbourhood V of q, and a conic neighbourhood W of such that

\de<t>(x, 0)1 +|g-d,<f>(x,0)|

ICl + lOl> 2 e (*)

if x e U , and 6 g W. A finite number of the neighbourhoods U x W cover {(x, 0) g ess. supp a | x = Xq}, say Uj x W,-, j = 1 , . . . , h. If B is their union then ess. supp a ~ B is a closed conic set and hence its projection into fl is a closed set disjoint from {xq}. It follows that there is an open neighbourhood U of Xq with U ^ C m such that U x W ^ 7 7 I/) Hess, supp a, where W’= | jW j and tt is the projection map. Letting V be the intersection of the corresponding V, we have shown that there is a conic neighbourhood F of ess. supp a, an open neighbourhood U of Xo, a conic neighbourhood V of and 8 > 0 such that (*) holds if x g U, I g V and (x, 0) g F. N o w if m g CîU) then

lUt) a"e(x, tO)D u(x) dx d6

where

x i x , e ) = 4>{x, d ) - { $ , x ) .

By the choice of a"p if a'^ {x, t6)D ^u{x)^0 then (x, 0 )6 F , x e U and |do<f>(x, <f>)l<£. Therefore by (*) |d < (x, 0 ) -| l> £ ( l l l + l0|) for ^ eV . In particular if Vq S V is a bounded neighbourhood of |o and Vq is bounded away from the origin then a ’l ^ i x , t e ) D ^ u ( x ) i ^ 0 implies |d;,xUI, 0)1^ c ( l + l0l). If L' = - i Idjc^r^Z t~'L '(e“ ) = e“ and|(r*L)^(a"3(x, t0)D^u(x))l is bounded by C r '' ( l + l0l)-'*(l + l (0 ir - '“''"''®. If m — fccr+ h 8 ^ 0 and t ^ 1 this bound may be replaced byQ m - ( c ^ - M i - 8 ) ( i + |g,|)m-fc<T-h(i-8) u n i f o r j n j y f o r T i j u s l " p ( 0 =

G(r'^) as r ^ 00, for any M, uniformly for | € Vq.

Example 13.13. Let F i , . . . ,F „ e C“(ft) be real valued and assume that Fjix) = 0, j = 1 , . . . , N implies that d F i(x ) ,. . . , dF^Cx) are linearly independent. Let

<f>ix, 0 ) = Z F ; ( x )0,..j=l

Then is a phase function and

A«. = {{x, e\ F t(x ) = 0 , j = l , . . . , N and | = X e,dF;(x)}.

Thus is the conormal bundle of the submanifold

M = {xea\F^{x) = 0, 7 = 1 , . .- ,N } .

If aG S^s(ft,R^) and A = J e ‘ ad0 [Osc.] then W F(A ) is contained in the cone in given by X dFy(x), x g M, (x, 0)Gess. supp a.

Exercise 13,14. I f hG C “(R” -(0 )) is homogeneous o f degree z with Re z > —n and A = then WF (A) = {0} x supp h.

The exercise shows if T is any closed cone with non-empty interior we can produce a distribution with wave front set contained in {0}xr. Actually we can construct a distribution with wave front set equal to a single ray as the following example of Hormander [9] shows. See also Example 4:6.12.

Example 13.15. Le if/eCc{W^) be chosen so Ji/^(x)dx = l . Then 4f{0) = 1 and if <) = then <f) g C~(R") and $ = \4f\ . Thus > 0 and < (0) = 1. Fix t) g R ” - ( 0) and define

u(x)= ^ k <p{kx)e'ikH'T\,x)

Clearly ueQ C R ”). Since the series for u is locally finite on R ” - (0 ) we have that u is C“ on R"" —(0). Hence WF (m) c {0 }x (|R’ —(0)) and therefore by Theorem 13.9 WF (u) is {0}xFo where Fq is the smallest closed cone in R ” — (0) such that u is rapidly decreasing in any closed cone T in W ^-(0) disjoint from Fq. N ow

Thus

u(m^r]) = ^ — k)ri) >

which implies {(0, t) | t > 0 }^ W F(u). Suppose now F is a closed cone disjoint from {tTj | t > 0}. Then 1| - tr] j (1 1 + t)"Ms a homogeneous function of degree 0 in 0eR'"'^^-(0) and does not vanish on FxR + . Therefore there is a constant c > 0 such that \^-t7]\>c(\^\ + t) for | € F and t> 0 . It follows that krjl = fc" 1 —fc^T)l>ck“ (| l +fc^)>cfc“^ ( 2 f c ) =2c Since + for any N, we have

l< (fc sup 1< M1la>l>2c>/iil

< C n (1 + 4 c l^ir'Then

lM (l)l^C K (Z k-^-’') ( l+ 4 c 111)- N

for ^g F.

for ^ g F.

It follows that

WF(w) = {(0,rT])|t>0}.

In Chapter 3 we will obtain some results concerning the wave front sets of solutions of partial differential equations. These results will enable us to compute wave front sets of some distributions in certain simple cases. The resulting examples will in a sense be more elementary than the ones above, but will give additional insight into the nature of the wave front set.

Chapter 3

Pseudo-differential operators


Let Xj denote multiplication by the jth coordinate function and let D, denote differentiation relative to the jth coordinate function in In this chapter we will develop an operational calculus for the operators Xj and Dj. For a function f {x) there is no difficulty in defining f {X) ; it is just multiplication by the function /. Similarly, since the Fourier transform converts differentiation into multiplication by a coordinate function, for a function g(^) we define g(—iD) to be multiplication by g on the Fourier transform side. Thus g{—iD )u=^~^{gu). We have already considered such operators in Section 2:9. As an example we have the Gauss- Weierstrass integral

W,u(x) = dy.

According to Lemma 2:3.1 (see also Example 2:9.2) we have

W,=e*''

where A is the Laplacian.Consider now a function P(x, |). In attempting to define p(X, —iD) we

run into the problem that Dj and Xj do not commute;

If we take the point of view that we should first carry out the differentiations and then the multiplications we are led to the formula

p(X, -iD )u (x ) = (27t)-" j e ‘<«’’=>p(x,

= (27t) - " | I e‘<«--^>p(x, l)u(y) dy d|.

161

162 PSEUDO-DIFFERENTIAL OPERATORS

Alternatively, if we first carry out the multiplications and then the differentiations we are led to the formula

p '(-iD , X)u(x) = (2ir)-"|| e‘<«-"-'’>p(y, |)u(y) dy d|

where we have set p'( , x) = p(x, ^). Since formally the transpose of p(X, —iD) is p'(iD, X ) it is necessary to consider both formulae.

Another approach is to introduce the Fourier transform p(rj, z) of p with respect to all 2n variables,

p(t), z) = j j I ) dx d|.The inverse Fourier transform is then

p(Xy z) dr] dz.

If in this integral we formally replace x by X and | by - iD we are led to the operator Wp defined by

Wp = (27r)-’ l l ' '

2n .i<-n,X>+<z.D>p(v, z) dT| dz.

This formula was given by Weyl [1, p. 274] as the prescription for passing from classical to quantum mechanics. If we set

T(v, z} = i(v ,X ) + (z ,D }

we have

(Wpu)(x) = (2rr)-^"| j p(v, z)(e' <”’ ^u)(x) dr, dz.

Since

[<z, D ), i<tj, X>] = i(z, v )

we have to exercise some care in computing We will use themethod of the integrating factor. First note if we set

<#>(x) = ilzl ^<Tj,x)<z,x>--lzl ^<Z,x)^<T1, z>

then

<z, D)<f>(x) = i(v, x).

It follows that

T(tj, z)u = e~*{z, D)(e^u)

1. INTRODUCTION TO CHAPTER 3 163

and thereforeeT(Ti,z) _ e-<f>e<Z’ >(e^t().

In Section 2:9 we commented that is just the translation operatort_2. We therefore obtain

+ z).

Now a little algebra shows that

<t>{x + z) — (t>{x) = i{T], X + z/2).

Thus

(WpU)(x) = (27t)“ ”| J p(rj, + z) dz dt)

= (27t)“”I p(x + z/2, z)u(x + z) dz

where we have inverted the Fourier transform relative to the first variable so p denotes the partial Fourier transform relative to the second variable. Inserting the definition of this partial Fourier transform we have

{WpU)(x) = (277)“''1 1 + z/2, $)u{x + z) dz d^

= (2ir)-'*||e^<«--^>p(^ , ^)u(y) dy d|.

We will take this last integral as the definition of the operator Wp and we will call this operator the Wey? operator associated to p. Note our calculations above show

i<'n,X>+<z,D> ^ (i/2K'n,z> i<T,,X> <z,D>

The product of exponentials of non-commuting variables is usually studied by means of the Campbell-Hausdorff formula; see Hochschild [1].

We now have three quite reasonably defined operators associated to p(x, ^). Any one of them is given by a formula of the form

Pu(x) = (2it) y>a(x, y, | )u (y )d y d ^ .

We obtain p(X, - iD ) by taking a(x, y, |) = p(x, ^), p'(-iD , X ) by taking a(x, y, I) = p(y, |), and Wp by taking a(x, y, |) = p((x + y)/2, 0 . We will see in Section 5 that if p e and 0 < 5 < p ^ 1 then, modulo smoothing operators, the three classes of operators that arise as above are in fact all the same. Indeed, we may take any a 6 S™s(flxft, R ") in the


definition of P above and still stay in the same class. The resulting operators we call pseudo-differential operators of order m and type (p, 8). If a given operator P is written, modulo smoothing operators, as p(X, — iD) or Wq, then we will call p a symbol of P and q a Weyl symbol of P. While the class of operators we obtain does not depend on the choice we make among the definitions above, the symbol calculus, that is the behaviour of the symbols relative to transposition, composition and change of variables, does depend on the choice. We will work only with the symbol as defined above, but we point out some of the properties of the Weyl symbol as well.

The study of the Hilbert transform and the study of operational calculus for non-commuting operators in quantum mechanics, both in the 1920s, contain some of the basic ingredients of the theory of pseudodifferential operators, as does the earlier work on parametrices of elliptic operators. The main development, however, occurred in the 1950s in the work of Mikhlin, Calderon and Zygmund on singular integral operators. Calderon and Zygmund freed the Hilbert transform from its too intimate connection with complex variables techniques and introduced algebras of singular integral operators based on the salient features of the Hilbert transform. They introduced the commutative (principal) symbol calculus which reflects the calculus of singular integral operators modulo compact operators. Calderon in the late 1950s and early 1960s applied the calculus of singular integral operators to obtain very general existence and uniqueness theorems for partial differential equations. In another direction, in the early 1960s Atiyah and Singer made use of singular integral operators (later replaced by pseudo-differential operators) in the proof of the index theorem. The point here was that the smooth symbols of singular integral operators are more suitable for studying homotopy invariants than are the polynomial symbols of differential operators. These developments stimulated interest in singular integral operators, as did other applications; for example, the treatment of boundary value problems by Friedrichs and Lax. If one considers the algebra generated by singular integral operators and partial differential operators one arrives at the pseudo-differential operators. This theory was developed by Kohn, Nirenberg, Hormander, Seeley, Unterberger and Bokobza in the mid 1960s. The theory has been developed in many directions and has been applied to the study of hypoelliptic operators, boundary value problems, propagation of singularities of solutions of partial differential equations, and to prove local solvability of equations of principal type. In addition th,e theory allows one to consider fractional powers of elliptic operators. The possibility of taking square roots, for example, gives a pretty approach to Garding’s inequality. More general considerations of refined symbol calculi give rise to other Garding type lower bounds.

The contents of this chapter are as follows:In Section 2 we introduce the space of pseudo-differential

operators of order m and type (p, 6) as consisting of operators locally of the form p (X ,—iD) where the symbol p is in IR' ). We alsointroduce the important concept of an asymptotic sum of symbols and prove the standard existence theorem. In Section 3 we show that the pseudo-differential operators of order -o o coincide with smoothing operators, that is, integral operators with smooth kernels. We introduce the important concept of proper support, extend pseudo-differential operators to distributions, and show that the symbol map p p(X, - iD ) induces an isomorphism

if -oo< m '< m . In particular if there is p g R'') suchthat P —p(X, —iD) is a smoothing operator, that is, p is a symbol for P.

In Section 4 we consider operators of the form a'(X, - iD , X) where y) = y, and a E S^5(n x f l , [R”). The main result is that if

8 < p then such operators are pseudo-differential operators and there is an asymptotic formula for the symbol. In particular we obtain the relation between the symbol of P and the Weyl symbol of P. The main result of Section 4 is used in Section 5 to show if 6 < p then the transpose of a pseudo-differential operator is a pseudo-differential operator of the same order and type and to obtain an asymptotic formula for the symbol of the transpose. We also show, again using the main result of Section 4, that the composition of properly supported pseudo-differential operators is a pseudo-differential operator and again we obtain an asymptotic formula for the symbol. Modulo an error of order p — 8 less than the sum of the orders, the symbol of a composition is the product of the symbols.

We specialize our results in Section 6 to the case of classical pseudodifferential operators, that is, operators with symbols p that are asymptotic sums of homogeneous functions. The top order part in such an expansion is called the principal sym bol We outline Hdrmander’s invariant approach to classical pseudo-differential operators and show that the principal symbol may be regarded as an invariantly defined function on —(0), the complement of the zero section in the cotangentbundle of H. We also show that the subprincipal symbol drops out nicely from the Weyl symbol and indicate how one obtains the subprincipal symbol of a composition and of a commutator from the symbol calculus. Finally we show that the subprincipal symbol is invariant under coordinate changes with constant Jacobian.

Having studied the invariance of classical pseudo-differential operators in Section 6 we turn to the problem of invariance in general in Section 7. We show if 1 —p < 6 < p then is invariant under changes of

coordinates and we show that the equivalence class of the symbol in [R”) may be regarded as invariantly defined on

—(0). The basic tool here is the main result of Section 4.In Section 8 we obtain the pseudo-local property of pseudo-differential

operators, namely WF (Pu) c WF (m) H ess. supp P where ess. supp P is the complement of the largest open conic set in which the symbol of P is rapidly decreasing. The proof is a typical stationary phase argument of the type encountered in Chapter 1 in connection with oscillatory integrals.

In Section 9 we introduce the characteristic set Z(P) of P. In the case where P is a classical pseudo-differential operator, Z(P) is just the zero set of the principal symbol. At the non-characteristic points we use the symbol calculus to construct microlocal parametrices. In this way we obtain the regularity theorem, W F (u )c W F(Pu)U Z (P). This theorem leads to a direct invariant definition of the wave front set of a distribution. We also note that the existence of a global parametrix for P implies that P is hypoelliptic. We then give sufficient conditions for the existence of a global parametrix and as a non-elliptic example we show that the heat operator has a parametrix in We finish the section by indicating how the subprincipal symbol intervenes when one tries to obtain a certain canonical form for a pseudo-differential operator with double characteristics.

The results of this chapter are largely standard and form a basic introduction to the calculus of pseudo-differential operators. In order to make this chapter as accessible as possible the proofs are worked out in considerable detail, though hopefully not to the point where the elegance of the subject is concealed. In Chapters 4 and 5 we will consider the action of pseudo-differential operators on Sobolev spaces. The introduction of Hilbert space techniques will lead to much more refined results than those of this chapter.

§2. Pseudo-differential operators

Recall Remark 2:9.17. Given functions a(x) and b{^) we formally define operators a(X ) and b(—iD) by

a{X )u = au (b(—iD)u) =bu.

Then

a{X )b {-iD )u {x ) = {27r) ”a(x)(bu)''

= (27r)-Me^<^->a(x)b(|)u(^) d^

2. PSEUDO-DIFFERENTIAL OPERATORS 167

andm D )a{X )u )\ ^ ) = b (-i)(au )\ ^ )

= e '^ '"^a{x)b{— )u(x)dx.

Therefore for a suitable function p(x, if we set p'(^, x) = p(x, we define operators p(X, —iD) and p\iD, X) formally by the expressions

p(X, -iD )u (x ) = e‘<« ’ >p(x,

(p'iiD, X)u)"(|) = j - | ) u ( x ) dx.

Before proceeding to study these operators we note that any continuous linear map P : C “(fi) is locally of the form p(X, —iD). Indeed letCl be an open subset of R” and let P : C ~ ( f i ) b e a continuous linear map with transpose P*:'^'{C1)-^^'{CI). If u eC ciC l) then

u(x) = (2'ir)-'’| d^

and therefore if 0 g C “(ft) then

0u = (27r)-"|e‘<«->0u(|) d|

where this integral converges as a vector valued integral in C “(ft). It follows that the continuous linear operator P commutes with the integral. We can also see this fact without appealing to the notion of vector valued integrals. If fe^'(C l) then Ofe^'iCl) and therefore by Theorem 2:8.1 we have

if, 6u) = {df, u) - (2it)""<(0/)'', u)

= (2i7)-''|^(-|)u(g)d|

= (27r)-'’|</,e'<«->0)u(|)de

If g s^ '(fl) £ind we set f = P*g we obtain

<g,P(0u)) = <P*g,0u)

= (2irr"|<P*g,e'<«’ >0)u(|)d|

= (2irr''|<g,P(e‘«->0))u(^)de

If X € n and we take g = 6 we obtain

P(eu)ix) = (27r)"”|P(e‘^' '^e){x)u{$) d$.

That is, P commutes with the integral. It follows that

P{0u) = Pe(X, —iD)u

where

We will call the operator Pe(X, - iD ) a localization of P.We will now study operators of the form p (X ,- iD ) where p e

0 < p < l , 0 < 6 < 1 . We will see that the localizations of such operators are of the same form, that is, that p© € S^s(fl, IR”). We will then introduce pseudo-differential operators as those operators which are locally of the form p(X, —iD) where p € IR' ).

E xam ple 2.1. Let f l be an open subset of IR", and letP = Z a « ( - iD r , \a\^m. If u g O H ) then

Pu(x) = X a^{x)(-iD )^uix)

= (27t)-" Z a jx ) | e ‘<«’ >rii(^) d|.

Thus P = p(X, - iD ) where p(x, |) = Z a „ (x )r - Note that p € R").By Leibnitz’ formula. Lemma 1:12.1, if S e C c i ^ ) then

a !Thus

; - k l

Pe (x, I) = Z ^ ^)D°‘e where p‘“ = D|p.a !

E xam ple 2.2. Let ft be an open subset of R", X a compact subset of R" and assume ( f t + X ) D f t = 0 . Choose x ^ C ^ R " ) with su p p ^ sX . Define P : C ”(ft) ^ C “(ft) by

Pu(x) = (2ir)-'* j e‘«-">;e()M(l) deThen Pu = x * “ and x * w has support in ft + X. Hence P = 0. Note P = x i- iD ) and x « S -”(ft, R").

L em m a 2.3. Let ft be an open subset o f R" and let 0 0 there is a continuous

seminorm Hn on C"(H) such that

11 '

, - i< T i,x > p(x, dx

for each v e C “(ft) cind

Proof. Integrating by parts we obtain

(i7])“ j ^)v(x) dx = ^)v{x)) dx.

Let K be a compact subset of ft and assume v e C “(ft) has support in K. By Leibniz’ formula

1D :(p (x, | ) o(x ))1 s Q k (1 + max lD^o(x)|

which completes the proof.

T h eorem 2.4. Let 0 < p ^ l , 0 < 6 < 1 , pGS^s(ft,R '') letp'(i, x) = p(x, Then p (X ,-iD ) and p'{iD ,X) map C “(ft) continuously into C~(ft). Moreover, these operators are formal transposes o f each other.

Proof. If u e C“(ft) then differentiation under the integral sign shows p(X, —iD)u G C “(ft). If X is a compact subset of ft then

D “(p(X, -iD )u )(x) = (27T)-''jD “(e^«->p(x, |))u(^) de

Now

lD“(e'«->p(x, ))| < I I ("^^a^r-^D^pix, I)l<x:S3

< ^ ^ Q -I- |^|y«|-|0l + m-l-Sl3l

< C ( l + l l)"'-"'“' for x g K

Hence

sup |D“(p(X, -iD )u)l ^ C f (1 + 1m(|)| d eK J

The right side of this inequality is a continuous seminorm on Sf and so certainly on C "(n). It follows that p(X, - i D ) : C ”(n) C ”(ft) is continuous. If o € C "(n) let

= j ^ ) u ( x ) dx.

According to Lemma 2.3 with rj = —i

where Hn is a continuous seminorm on C “(ft). We have similar estimates for the derivatives of q . Therefore v ^ is a continuous linear map of C “(fl) into Sf. Since (p'(iD, X)v) = q'i it follows that p\iD, X) maps C “(n) continuously into Sf and so certainly continuously into C~(ft).

Finally if u , v e C “(fl) then

<p(X, -iD )u , v) = (27t) ~ ^ | I d 6x

= (27r)-"|q„(|)u(^)d|

= (27r)~”<u, (p'(iD, X)u)"'')= <u, p V D , X ) v ).

Thus p'{iD, X ) is the formal transpose of p(X, —iD).

C orollary 2.5. I f peS^s(n,lR "), 0 < p < l , 0 < 6 < 1 and p '(£ x ) = p(x, I) then p(X, —iD) and p'{iD, X) extend uniquely to continuous linear maps o f into (weafc* or strong topologies).

R em ark 2.6. The last part of the proof of Theorem 2.4 shows that if u e then

<p(X, -iD )u , v) = (27t)“' ' | I l)w(^)t;(x) dx d^

for each v e C~(fl). Note in general this double integral is not absolutely convergent and therefore we cannot change the order of integration.

L em m a 2.7. I f p e IR"") and 6 e C “(ft) then the localization peo f p is in S^sCft, IR”).

Proof. We have

Pe(x, ^) = e-^<^->p(X, -iD)(e^<^’->0)(x)

= e - ‘<«-">(2'n-)-'| 'n)(e‘<«- >0) (T,) dr,

= (2rr)-'| e‘<"-«->p(x, v ) 0 ( v - e dr,

= (2rr)-" j e ‘<”->p(x, | + r,)0(r,) dr,.

Symbolically we may write pe(x, = p(X, iD)$(x). If we let = L>|p we have

D^D^Peix, i ) = (2'n-)-'| D ?(e‘<”-">p<“\x, | + T])0(t7)) dr,

= (2^)“" +

Since 6 is rapidly decreasing, for each N > 0 we have

|D|DPpe(x,^)l<CNf X ( l + lT,l)'^l-'^'-'"(l + l^-^T,|r"«W-'’'“'dT,.J

Thus

|D^D?p«(x, 1)1 < C|s,|(1 + It7|)'®'- (1 +1^ + dr,

— Jvr(l J (1 + dT|

where we have used Peetre’s inequality

(i+ l^ + T iiy < 2 'n i+ l^ | y (i+ h l) '‘'

(see Section 2.5). Taking N sufficiently large the integral converges and the lemma follows.

The formal Taylor expansion

-kip(X, i - i D ) = Z '— r P “'(2s .

rv 0l\

suggests;-|ocl

Pe(x, e = l ' — r P‘“'(x, $)D -e{x).rv 0l\

Of course this series need not converge. None the less we can make sense of the sum asymptotically as the following lemma shows.

Lemma 2,8. 7 / p g IR”) and OeC^iil) then

P e - Z ^ p ^ “^D“0€S^s"‘’'^(fl,R")lotl<N ^ •

for each integer N^O. In particular if (o is a relatively compact open subset o f Cl such that 6 = 1 on (o then Pe—p e S~° (o), IR' ).

Proof. Let-i«l

q(x, ) = P e(x ,.f)- X ^ ^)D“0(x)lcxl<iV

= {2tt) "[e'<”-^>(p(x, |-t-T})- X A )-n“)0(-n) d-rjJ \ la l< N ^ • /

= (2'n-)""| e‘<” *^h(x, 4 i i)0(t)) dtj.

Then

D|D?q(x,|) = (27T)-" I (^ ) f(iT,)^-^e'<’>’'>DJh<“>(x,eTj)0(T,)dT7.\y y J

By Taylor’s theorem with remainder

lD?h'“>(x,e 11)1 = 1 I ^ f 'D V “"^Xx,| + tT,)T,i'(l-fr-'dr|•h =n 7*-1o 1

< c | (1 11+ |t,1' (1 - dt

^ C \ \ + + |y^pN + l»1^1+PN+p|«l+Sl3l

where we have used Peetre’s inequality. It follows that

lD|D?q(x, 1)1 < C(1 -h III)— pN-pl«l+8lel

X I ( l + |T,|)N+lm|+pN+p|o,|+|0| |^(^)|

where all the estimates hold for x in a compact subset of Cl.

Lemma 2.8 suggests the following notion of the asymptotic sum of symbols py. If my is a sequence of real numbers, lim my = —oo, and Py G 'vve say p is the asymptotic sum of the py, and write

P ~ I P ]provided that

P - I p,es;i(aiR"')j c k

for each k, where m] = maXk>y m . It is easy to see that it suffices to have

p - ZJ<k

for each k > 0 , where /jl oo. Indeed, one just considers the left side with k replaced by k '> k sufficiently large that i^h^ml, for h > k ', and then puts the additional terms on the right side.

With the present notation Lemma 2.8 says if pe R"). 0 < p < l ,and 6 e C”(fl) then

p(X, I - iD)O(x) = p«(x, I ) ~ I — P<“\x, ^)D“0(x).rv IQL •

It will be convenient to consider a more general situation. Let W be an open conic set in f l x R ^ — (0). In Section 2:13 we introduced the space S^5(W). If lim nij = -oo, m- = maXk>y and py € S^^(W) we will write

P ~ Z Pi in W to mean that

p - Z P ie S -i(W )j<k

for each fc> 0 (where the empty sum is 0). If p and the py are in Sp sCn, (R ) we will write

P ~ Z Pi in W

to mean that

p Iw ~ X P j Iw in W.

Note if W = flxlR^ —(0) we recover the first definition of an asymptotic sum since the assumption that p and py are in IR ) takes care ofthe smoothness at flx(O ). Note also if p e IR ) and W is an openconic set in flxIR^ — (0) then p ~ 0 in W if and only if W is disjoint from the essential support of p (see Section 2:13).

L em m a 2.9. Let W be an open conic set in ftxIR^ —(0) and let qGS^s(W)- I f ^ Is a compact subset o f II and T is a closed cone in [R^-(O) such that K x T ^ W then there exists pGS^s(fl, R^) such that p = q in a conic neighbourhood o f K x T in f t x R ^ —(0).

Proof. Since K x ( m S ^ ” ) is compact there is an open set a> in H and an open cone U in R ^-(O ) such that K x T ^ c j x U ^ W . Choose X e C~(R^) so X = 1 in a neighbourhood of the origin, choose h e C~(S^“i) with compact support in UnS^~^ such that h = 1 in a neighbourhood of r n in and choose i/f e C “(co) such that i/f = 1 in aneighbourhood of K. Now let p(x, ) = il-xi^))^M h{^\$\~^)q(x, g).

L em m a 2.10. Assume lim my = —oo and let m =max my. I f pj e Sp‘s(fl,U ^) there exists p g Sl^s(fl,R^) such that P'^'ZPr Moreover if q^ Z P ] then p — qG R^j.

I f W is an open conic set in H x R ^ - ( O ) , py e 5*5(11, R^), Py|w Sp's(W), K is a compact subset o f II, T is a closed cone in R ^-(O ) and

K x T ^ W then there exists p e such that p^'ZPj in a conicneighbourhood o f K x T . Moreover if Pj in conic neighbourhood o f K x T then the essential support o f p — q is disjoint from K x T .

Proof. The last part follows immediately from the first part and Lemma 2.9. For the first part, for the uniqueness we have p ~ q = (p —T,j<kPi)~(q~Zj<kPj) is in for each k, and therefore inSpJCn, R ”). For the existence we set p{x, ) = Y.<f>Wti)pj(x, $) where </)g C “(R^), <) = 0 in a neighbourhood of the origin, </> = ! in a neighbourhood of infinity, that is, l — <j> has compact support, and where the tj are chosen to increase sufficiently rapidly as j oo. This method of summation of an asymptotic series is due to A. Erdelyi and is a modification of an argument of van der Corput. See Erdelyi [1], Section 1.7. We consider now the details. Choose <#>€C°®(R ) such that 0<<^><1 and <#>(!) = 0 if 1 1 <1/2 and </>( ) = l if | 1> 1 . Let be a sequence of compact sets in Cl such that each compact subset of f i is contained in for some fc. Assume for the present that each tj is chosen with tj > 2. We will add more conditions shortly. Note

and if > 1 , xeA^. then

Now t j > 2 implies (D^<#>)(^/ty) = 0 if 1 | | < 1 and therefore for x e A and | g R^ we have

— C'kJ,oc,l3 Xy oc

1 1 \y\

tj 1

(since p ^ l and since it suffices to consider | |> 1), where /lLj, / = — 1, 0,1, 2 , . . . is any sequence decreasing strictly to —oo such that juty > for fc>/. Now

If 77^0 then tj) = 0 unless l/2<l|/fjl<l. Thus 1 whenD X | / t,)^ 0 . Also if <t>Wtj)i=0. It follows ifand |€(R'^ then

Inductively we may choose t, sufficiently large so that if |al-l-l/3l + fe£/ and X € Afc then

lD|D?(<#>(|/f,.)p,.(x, 111*-,-

If A is any compact subset of £i choose fc so A ^ A . If a, |3 are any multi-indexes choose h so lal + ||3l + fc:^fi and iJLh-i<p\a\ — S\P\, Then \D%D i<t>{i/tj)Pj{x, |))1 ^2~' for j > h, since = 0 if li|< 1. It followsthat Y.\D%D^{^{^tj)pjix, $))\ converges uniformly on AxR'^. Hence

p(x, ) = Z Wtj)pj(x, I)

defines p e C “(ft x R'^). From the estimates above if x € A then for any k we have

z \D%Dti<l>Wti)Piix, |))1 < CA.fc(l + l i m -

It follows that p(x,i)-Z i<k4>W ti)Pi(x,^) lies in By theremarks preceding Lemma 2.9 the proof is complete.

Let / be a smooth function on IR. By Taylor’s theorem

f i y ) - f (x )f (x ) = ^ y - x

-i/ "(t)(y -x)

where t lies between x and y. Taking ly-x| = 6 > 0 we obtain

sup l/'l < - sup 1/1 sup If'lI S r L V

where I is an interval and I' is its £-neighbourhood. If |/1<M and |/"1<N on R taking e^ = 4M /N we obtain \f'\:^2(MNy^ , By using Taylor’s theorem twice one can improve the estimate to obtain (2M/V) ^ as the upper bound. By entirely similar methods one may prove the following result.

Exercise 2.11. A rapidly decreasing function with derivatives o f polynomial growth has rapidly decreasing derivatives. That is, if <t> Om and for each integer N > 0 we have |<^>(x)l<Cn(1 + 1x 1)"' then <t>eSf.

Theorem 2.12. I f Pj e S^s({l, R^), limy co m, = -oo, jUL, i -o°, p e C “(n x|R^) and for each compact subset K o f ft, multi-indexes o l, |8 and integer k > 0 , we have constants Qc,a,3 tind v(a, P , K) such that

lD|D?p(x, 1)1 < Ck.o,.3(1 + (1)and

p(x, ) - Z Pi(x, I) ^ CK.fc(l + 111)““ I i<k I

(2)

for x e K and k > 0 then p € IR ) where m = maXj^o Moreover,

P ~ I Pi -

Proof. By Lemma 2.10 we may choose q e IR ) such thatq ~ X Pj. Now (2) implies that p — q is rapidly decreasing in uniformly for X in compacta and (1) implies that the derivatives of p — q are of polynomial growth in uniformly for x in compacta. By the argument needed to establish Exercise 2.11 it follows that p - q G S “”(ft,IR^).

We define to be the linear space of continuous linear mapsP :C ^ (^ l)-> C '"(h ) such that for each 0 g C^({1) if pe(x,^) = e“*< ''>P(e‘< ‘>0)(x) then pe e IR”). The elements of we willcall pseudo-dijferential operators of order m and type (p, 8). In view of Lemma 2.7 we have a linear map

given by p p(X, —iD). Example 2.2 shows that this map is not one-to- one in general. Recall

P (0u)= p^ (X ,-iD )u, w eC r(n).

We may therefore regard operators in as those linear operatorswhich are locally of the form p {X ,—iD) where p g S^g(a IR'')- The following lemma makes this statement explicit.

Lemma 2.13. Let P ; (ft) C“(fl) be a linear map. Then P e ' ''^s(ft) if and only if for each 6 e C"(ft) there exists g S^s(ft? f "") such that

PiOu) = q^ X , -iD )u , u e Cr(ft).

Moreover, in this case pe~q^ e S~“(ft, [R”) for each 0 g C~(ft).

Proof. Indeed if the q® exist then P is certainly continuous and so P(6u) = pe(X, - iD )u where p is defined as above. Choose x e C “(ft) so X = 1 in a neighbourhood of supp 6. Then

Pe(x,|) = e - ‘«->P(e‘«->0x)(x)= e - ‘«-^>q"(X,-iD)(e‘«->x)(x)

= (q '')x U I).

Thus Pe e IR")- By Lemma 2.8

(q'')x(x, I) ~ I D-lq^x, |)D“x(x).rv '

Since we may choose x so x = 1 in a neighbourhood of any compact subset of Cl that we please, we see — q* € S “"(n , R").

Lem m a 2.14. I f peSps(Cl,U") and p (X ,-iD)eT'^s(f^) then pe

Proof. By Lemma 2.8 if is a relatively compact subset of ft, 0 e C "(f t) and 0 = 1 on then p e -p € S ““(<u,R"). But by definition of 'ifp.siCl) we have pe e S^s(ft, R")-

We define

'p M (ft)= U ^ ^ s(ft)m

’p -“(ft)= ri'p ^ 8(ft).m

If P : Cc (n) —> C *(n) is a continuous linear map and we define pe as before then if and only if pe e (R") for each 6 g C“(n).As a simple example of an operator in we have the followingresult.

L em m a 2.15. I/PG ^^s(fl), <t>, if/e and supp </> n supp i/f = 0then R = <t>{X)Pil/iX) g

Proof. Indeed, jR is a continuous linear map of C " i n t o C~(fl) and if 0 G Cc (ft) and we choose x C “(ft) such that x = 1 in a neighbourhood of supp (6ip) then

reix, 0 = e“‘ *-* <f>(x)P(e‘< ' X:0i/r)(x)= e~‘^ -^^4>(x)pJX, -iD){e'^^ '>e>lf)(x)

= <l>ix)(py)e4,ix,

Z — r <f>(x)D|p (x, $ ) D “ {e>{f){x) „ a!

= 0 .

Thus Yq g S “"(ft, [R"').

The property of pseudo-differential operators obtained in Lemma 2.15 is called the pseudo-local property. In Section 8 we will reformulate it in the form that it was obtained for partial differential operators in Section 2:13.

In the next section we will see that T “”(ft) coincides with the space of integral operators with smooth Schwartz kernels on ft x fl. Such operators are in some senses negligible in many situations. As a corollary of Lemma

2.14 note that the map p - p(X, —iD) induces a one-to-one linear map

This map, called the symbol map, is onto. This fact we will also prove in the next section. Thus modulo every pseudo-differential operatoron ft is an operator of the form p(X, - iD ) where p e S “s(ft,lR”).

E xercise 2.16. I f C “(fl) is a continuous linear map andsupp Pu c supp u for each u e C “(fl) then there exist e C^{Cl) with locally finite supports such that P u = Y aJD ' u.

Probably the easiest way to do the exercise is to consider P'(tx6), x g ft, where P' is the transpose of P. Note that Peetre’s theorem [1 ,2 ] is stronger since Peetre does not assume continuity.

§3. Smoothing operators and properly supported operators

Let f t i , f t 2 be open subsets of IR”, IR"", respectively. According to the Schwartz kernel theorem (see Treves [1] or Maurin [1]) if P :C ^ {£ 1 ^ ^ Si'(fti) is a continuous linear map then there exists a unique distribution G G2)'(fti x f t 2)> the Schwartz kernel, such that

{Pu, v) = {G, v ® u )

for MGCc(ft2)> u € C “(fti). Here (v<S)u){x,y) = v{x)u(y). We will not actually use the Schwartz kernel theorem at any time. In those cases where we consider a Schwartz kernel we will have an explicit expression for it. In the case where G is a locally integrable function, P is called an integral operator, and

Pu(x) = I G(x, y)u(y) dy.

Our first result is that pseudo-differential operators of sufficiently low order are in fact integral operators.

L em m a 3.1. I f PG^^s(ft), m + n + fc<0, where f c>0 is an integer, and ft is an open subset o f IR'", then there exists G g C^(ft x ft) such that

Pu{x) = I G(x, y)w(y) dy, u e C “(ft).

Proof. Let i/r, g (ft) be a partition jof unity on ft. Since = Z ^ 0 if we define <f> = then </>, g (ft) and Z <#>/ = 1* Now let Pj(x, |) =e-i<€.x>p(ei<€.> .)( ) p, GS^s(ft,lI^"). If MGCr(ft) then u=I<f)^u is

3. SMOOTHING OPERATORS 179

a finite sum and therefore

Pm = I P(<f.?M) = X Pi(X, -iD M iU ).let

G,(x, y) = (2 ir ) - ''je ‘«- -^>pj(x, |)

Since m + n + k < 0 and 6 < 1 , differentiation under the integral sign at most k times yields absolutely convergent integrals. Thus G, g C ^ (flxfl). Moreover, if t; g C “(fi) then

j G,(x, y)u(y) dy = (2ir)“''||e‘< -*“'’V;(x, |)u(y) d dy

where the double integral is absolutely convergent. Interchanging the order of integration we obtain

(277)-'’! e-<«->p,(x, d = Pj(X, -iD )u(x).

Replacing v with <l>jU we see that the lemma follows with

G(x, y) = X Gj{x, y)<#>j(y).J

An integral operator is called a smoothing operator if its Schwartz kernel is of class C ".

T h eorem 3.2. Let ft be an open subset o f R". I f P : C “(ft) C"(ft) is a continuous linear map then P is a smoothing operator if and only ifPexp-“(ft).

Proof. Necessity follows from Lemma 3.1. For sufficiency assume there is G € C “(ftx ft) such that Pu(x) = J G(x, y)u(y) dy for u e C “(ft). Let 6 € C"(ft) and let

p,(x,|) = e--<«-*>P(e‘«->0)(x)

= I y ) 0(y) dy.

Integrating by parts we have

i-'“'(-i|rD ?D |p,(x, e)

= I r (-iir""(y-x)“e‘«-’->Dr'G(x,y)0(y)dy

= I r e‘« - - > D r i ( y - x r D ? - G ( x , y ) 0(y)]dy.czS(3Therefore pe € S ““(ft, R").


Let X, Y and Z be metric spaces. If C ^ X x Y and B ^ Y x Z we may regard C and B as relations. Generalizing the notion of composition of functions from Z to Y and functions from Y to X we define the composition o f relations

C o B = {(jc, z) I (x, y )€ C and (y, z ) e B for some y e Y}.

Recall a map is proper if the inverse image of each compact set is compact.

Exercise 3.3. I f C and B are closed and one o f the projection maps C ^ X or B —> Z is proper then C ° B is closed.

Let 0 be a space consisting of one point. For any space X we identify 0 X X and X x 0 with X. Then if A ^ X , B ^ Y , C ^ X x Y the compositions A o C c Y and C ° B c X make sense. Explicitly

A o C = {y G YI (x, y) G C for some x g A }

and

C o B = {x G X I (x, y) G C for some y g B }

According to the exercise if A, B and C are closed then A ° C is closed if A is compact or C Y is proper, and C ° B is closed if B is compact or C X is proper.

Exercise 3.4. Let A, B and C be as above. Then A ° C (resp. C ° B ) is compact for each compact A ^ X {resp. B ^ Y ) if and only if the projection C —> X (resp. C ^ Y ) is proper.

Let P : C “(fl2) — be a continuous linear map. We define thesupport of P, suppP, to be the support of the Schwartz kernel of P. Likewise, we may define the singular support of P, sing, supp P. We see that supp P is the complement of the largest open set U in fl iX D 2 such that if (Oj is an open subset of flj, / = 1, 2, and x a>2 ^ L7 then Pm = 0 on (i)i for each u g C “( 02). We say that P is properly supported if the projection maps (supp P) and (supp P) f l2 are proper maps.

Exercise 3.5. I f P : C c(^ 2) 2)'(fli) is a continuous linear map andu G Cc (D2) then

supp (Pm) c (supp P) o (supp m).

Exercise 3.6. Let P : S)'(Di) be a continuous linear map.

Show that P is properly supported if and only if the following two conditions hold,

(1) For each compact subset K o f 0.2 there is a compact subset K' o f i l i such that if u e C “(^ 2) and supp u ^ K then supp P u ^ K '.

(2) For each compact subset L o f f li there is a compact subset L' o f Cl2

such that if u e C ”( 2) tind u = 0 in a neighbourhood o f L then Pu = 0 in a neighbourhood o f L.

E xercise 3.7. Let P : C “( 12) be a continuous linear map, let<l>j (resp, il/j) be a locally finite family in C~(^li) (resp. C “(fl2)) ttnd define Qu='Zj<f>jP{il/jU) for u e C c ( ^ 2)- Then Q : C “(fl2) i s a properly supported continuous linear map.

Note in the exercise that the formula for Qu makes sense for any u 6 C “(fl2) since the sum is locally finite. Moreover, if x G "(fli) then xQu = Z is a finite sum for any u e C“(fl2) ^md clearly defines acontinuous linear map of C“(fl2) into It follows then that Q mapsC “(H2) continuously into Here and elsewhere we use either theweak* or the strong topology on the space of distributions. The same continuity argument works for other topologies as well, as for example in the proof of the following lemma.

Lemma 3.8. Let be an open subset o f IR” and let P: C"(fl) —> C*(n) be a properly supported continuous linear map. Then P extends uniquely to a continuous linear map o f C “(fl) into C “(n).

Proof. Since Cc(Li) is dense in C “(I1) the uniqueness follows. Let A = supp P and let be a locally finite open cover of ft with each o)jrelatively compact in ft. Since P is properly supported the sets o)j ° A and A o o)j are relatively compact open subsets of ft. If w g C “(o)y) then P u e C c iA o (o ). If M €C“(ft) and u = 0 on cOj^A then Pu = 0 on (Oj. Indeed, if t> e Cc(<Oj) then (Pu, v) = 0 since (supp u)x(supp u) is disjoint from A. Choose a partition of unity g Cc(<Oj) and choose Xj G~(ft) with Xj = 1 ir coyoA. If UGCc(ft) then (1 -X j)u = 0 in (Oj ^A implies P ((l —Xj)w) = 0 on (Oj. Therefore

(bjPu = <(>jP(xjU) for each u g C*(ft).

If M G C “(ft) we define

Pu = X <f>iP(Xiu).i

The sum is locally finite and so defines Pu g C “(ft) depending linearly on

u and extending P. If x s C“(D) then

xPu = X x<f>iP(X/u)i

is a finite sum. Therefore xP maps continuously into C “(fl). If K isany compact subset of ft we may choose x = 1 in a neighbourhood of K and therefore conclude that P : C "(ft) ^ C “(ft) is continuous.

If P : Cc (ft) C “(ft) is a properly supported continuous linear map we will extend P to a continuous linear map on C "(ft) without any comment. Note since (supp P)o K is compact for each compact subset K of ft we also have that P maps C "(ft) continuously into Cc(ft).

Lemma 3.9. I f PG^^s(ft) is properly supported and p(x,^) = e-i<ex>p(ei<€,->)( ) p g S^s(^ ,1R") and P = p (X ,- iD ),

Proof. Choose and Xj is in the proof of Lemma 3.8. Then

Pu = X <l>iP(XjU\ u e C~(ft).j

Let p, (x, |) = e“‘<* * P(e'< ' X/)( ) so p,-€ R") and P(%m) =Pj(X,—iD)u for each u € C "(ft). Clearly

p(x, I ) = Z 4>,(x)Pj(x,

and therefore p g S^s(ft,R”). If mg C “(ft) then

p(X, -iD )u (x ) = (27t)"” j X </>,( )Pj(x, I ) m(^) d^

= Z </>j( )Pj(X, -iD )u (x )

where the sums are finite for each x.

Theorem 3.10. J/PG'^^^sCft) there exists a properly supported operator Q g '^^s(^) such that P = Q + P where P G '^ ““(ft). In particular if -oo< m '< m then the mapping p — p(X, — iD) induces an isomorphism o f vector spaces

s^s(ft,

Proof. The first part and Lemma 3.9 imply that S^s(ft, ^ ^ 8(ft)/^^8(ft) is onto and Lemma 2.14 shows that the kernel is

R"). For the first part let be a locally finite open cover of ftwith each a>j relatively compact in ft. Choose a partition of unity <f>j G Cc(<Oj) and then for each j choose Xj Cc(<Oj) with Xj = 1 in an open

neighbourhood of supp <f>,. If w € C "(fl) define

Qm= Z iP(XjU).Then Q : C“(ft) —> C "(fl) is a properly supported continuous linear map. U tj/e Cc (H) then for each u e C “(fl) we have

i//R(u) = i/fP(u)- iffQ(u) = Zj

Since supp Plsupp (1 -X/) = 0 the sum is finite by Lemma 2.15 we see that i{fR e If Oe it follows that iK e IR”) foreach i/f € Cc(fl)- It follows that IR”) and therefore RThen Q = P - R

If Pe'^^siW and p € IR' ) we call p a symbol o f F i f P -p ( X , a n d we call p a principal symbol o f P i f P - p(X, — iD)GT^^3(fl) where m '<m . If poG C “(nxlR ” -(0 )) is homogeneous of degree z, m = Re z, and there exists x C“(IR”) such that x = 1 in a neighbourhood of 0 and (l-x(l))Po(^» is a principal symbol for P we will also call po a principal symbol of P, or if we wish to be explicit, a homogeneous principal sym bol If P admits a homogeneous principal symbol it is unique.

Example 3.11. Let a g C"([R” - (0 )) be homogeneous of degree 0. In Section 2:2 'wt considered the continuous linear map a (—iD ):S f^ € i^ defined by a ( - iD )u = ^ ~ ^ {a )* u. Since {a {—iD)u) = a u we have

a i-iD )u (x ) = (2-77) - " ! d l

Let x e C ” ((R"). Then implies = g for some geC"(IR'*). IfX: = 1 in a neighbourhood of 0 we have b = (1 - e S®(R", IR"). Then

a {-iD )u (x ) = H -iD )u (x ) + 1 g(x - y)u(y) dy

implies a(-iD)eT'^CIR"), b(|) is a symbol for a ( - iD ) and a(|) is a homogeneous principal symbol for a {—iD).

Corollary 3.12. I f then P admits a formal transpose

Proof. Indeed P = q {X ,- iD ) + R where Ru(x) = J G(x, y)u(y) dy, G € C “(f lx f l) . By Theorem 2.4 we have P' = q ’(iD ,X ) + R' where q ' i t x) = q(x, I) and R'u(y) = J G(x, y)u(x) dx.

Corollary 3.13. I f Pe'^^s((l) then P extends uniquely to a continuous

linear map o f into I f P is properly supported then P extendsuniquely to a continuous linear map o f Q)'(d) into

Proof. We use weak* or strong topologies throughout. The first part follows by considering the transpose of P'. For the second part note that supp P' is the image of supp P under the map (x, y) ^ (y, x). Thus if P is properly supported then so is P' and therefore P' maps C“(ft) continuously into Cc(ft). Now transpose.

In Section 5 we will see that if 0 < 6 < p < 1 then the formal transpose of a pseudo-differential operator is also a pseudo-differential operator.

The terminology ‘smoothing operators’ for operators in '4 ““(ft) is justified by the following result.

Lem m a 3.14. I f jRG^~“(ft) and wG^'(ft) then R u e C ’ iH). I f R is properly supported then R u € C “(ft) for each ue& {C l). Moreover, R C “(ft), and in the properly supported case R :S>'(ft)-^ C “(ft),is continuous if we provide the distribution spaces with the strong topology.

Proof. There is G € C “(ft x fl) such that R'v{y) = J G(x, y)u(x) dx for uG C “(ft). It follows if MG^'(ft) (or MGS»'(ft) in the properly supported case) then

(Ru, v) = (u , I G(x, ')v{x) dx^

for V G Cc (ft). Since v has compact support

1G{x, y)u(x) dx = lim e" ^ G (ek, y)u(gfc)

where the convergence takes place in C “(ft) (or in C“(ft) in the properly supported case). Hence we may commute u with the integral to obtain

(Ru,1, u) = I (u. G(x, '))v{x) dx.

Thus Ru = h where h(x) = <w, G(x, •))• K we set g^(x) = D^G(x, •) then go,: ft — C “(ft) is continuous (or : ft —> C~(ft) is continuous in the properly supported case) and D ‘"h(x) = (u, g«(x)). Thus h g C “(ft). If K is a compact subset of ft then ={gocM | is a bounded subset ofC~(ft) (or of C~(ft) in the properly supported case) and

sup lD“hl < Po,(w) = sup 1<M, v)\.K

Since p is a strongly continuous seminorm on ^'(ft) (or on 2)'(ft) in the properly supported case) we obtain the announced continuity.

4. OPERATORS OF THE FORM a'(X, -iD, X) 185

Exercise 3.15. Give a proof o f Lem m a 3.14 based on the formula in R em ark 2.6; namely, (R idu), v) = (2ir)“" JJ |)u(|)u(x) dx d| foru €<g'(n), V, 6 € C "(fl).

§4. Operators of the form a'{X, - iD , X )

Formally we have

ib (-iD )a iX )u ) (|) = | e~‘^^'^^a{y)b{$)u(y) dy

and therefore

c(X )b i-iD )a iX )u {x ) = (2ir)""||e‘< - “'’>c(x)a(y)i)(^)u(y) dy dg.

Thus if a € S^ s(ftxft, R") and a'(x, y) = a(x, y, |) we define

a'iX, - iD , X)u(x) = (217)-"! j y , |)u(y) dy d|

for u 6 C "(fl).

Theorem 4.1. Let 0 < p ^ l , 0 < S < 1 , and leta'{x, 4 y) = a(x, y, and a"{x, y) - a(y, x, ^). Then the operators a'ix, - iD , X) and a''{X, iD, X ) are continuous linear maps o f C “(n) into C “(H). Moreover, these operators are formal transposes o f each other.

Proof. If u € C“(fl) then integration by parts and Leibniz’ formula yield

y, ^)w(y) dy

= I ( ) I y, |)u(y) dyy

= 1 (^ )| e ‘''-^""'J^r^F>riaU y.aw (y)]dy

If K and L are compact subsets of H and 7 < ^ then

lD f-^D “-"la(x , y, ^)w(y)]l ^ Q ,3 .k.l(1 + III)m+8(wi+iei) lD< M(y)lcr ot+3

for x g K and supp u ^ L . Thus for each compact set K ^ O , and each integer N ^O

|d ?| e‘<*’=-'’>a(x, y, |)u(y) dyj<|ulN.K.3( l + |||)m+8|3l+(8-l)N

where is a continuous seminorm on Cc(ft). Since 6 < 1 thisestimate with N large enough shows that in the definition of a'(X ,—iD ,X )u we may differentiate under the integral sign to obtain a'{X, - iD ,X )u e C~(D). Moreover, we have

sup \Dta’{X, - iD , X)ul < 1u1n,k.3 f (1 +K J

which implies the continuity of a'(X ,—iD ,X ). The continuity of a"(X, iD, X) follows by the same argument. Finally if u ,v e (fl) then

(a '(X ,-iD ,X )u , v ) = { 2 7 t )<«.x-y> a(x, y, |)u(x)u(y) dy d| dx.

In general this integral is not absolutely convergent and we have to be careful concerning the order of integration. If we set

h(x, $) = j e‘«--^>a(x, y, |)u(y) dy

then hG C “(flx[R”) and the estimates above show

\h{x, +

for each integer N > 0 and x e supp v. Thus we may interchange the integrations relative to and relative to x. In the resulting triple integral since u and v have compact supports we may interchange the integrations relative to y and relative to x in the inner two integrals. We now have

<a'(X, - iD , X)u, v) = (2'n-)"'’| 1 1 y , ^)u(x)u(y) dx dy d|.

The same argument as above shows we may interchange the integrations relative to | and relative to y. The result of this interchange is (u, a"{X, iD, X)v).

T h eorem 4.2. Let 0 '>a(x, y, |)d| [Osc.].

Proof. Let A be the distribution defined in the statement of the theorem by the oscillatory integral. Admittedly there is considerable abuse of notation here. The definition means that if u ,v e C “(ft) then

(A, v ® u ) = (27t)“”| 1 1 y, |)u(x)u(y) dx dy d| [Osc.].

4. OPERATORS OF THE FORM a'(X, -iD,X) 187

Let be such that </>(0) = 1. By Theorem 2:19.6 we have

(A, v<8)u} = (217)“" I I I V(£|)a(x, y, |)u(x)u(y) dx dy d|.

If we set

h(x, I) = I e‘< - ”'’ a(x, y, |)u(y) dy

then h€C "(flx|R ") and we saw in the proof of Theorem 4.1 that

ih (x ,i)i< C N (i+ iiir"< «-»^

for each integer JV > 0 and x e supp v. Since S < 1 and

■‘<*’*-''>«/>(e )h(x, i)v {x ) d dx<A,u0 M> = (27rr" lim f [e ‘<e i O J J

if we take N large enough we may apply the dominated convergence theorem to obtain

{A, V ® u) = ( I tt) ”1 1 |)u(x)d^dx

= (a '(X ,-iD , X)w, V).

By Theorem 2:13.2 if

A(x, y) = (2i7)-"| e ‘<«- -'’>a(x, y, d [Osc.]

then

WF (A) £ {(x, X, 4 -^ ) I (x, X, 6 ess. supp. a}.

In particular the wave front set of A is contained in the conormal bundle of the diagonal in fl x fl, and therefore A is C~ in the complement of the diagonal in f i x f l .

Corollary 4.3. I f for each x e f l there is a neighbourhood (o o f x in f l such that a(x, y, is rapidly decreasing in | uniformly for {x,y)G<oXo} then a'(X, -iD , X ) e

Proof. As in Exercise 2.11 the derivatives of a are also rapidly decreasing in (oXo), Thus a e S“"(co x o>, DR”) and therefore ess. supp a is disjoint from {(x, x, |) | x 6 fl, | g IR” — (0)}, whence WF (A) = 0 . Now apply Theorem 3.2.

Corollary 4.4. I f C^{H) and supp <f> fl supp i/r = 0 then<#>(X)a'(X, -iD , X)iA(X) G

Proof. Indeed the Schwartz kernel of <t>(X)a’(X, —iD, X)iffiX) is

(2ir)""| y, |) d| [Osc.].

We will now show, under mild hypotheses, that operators of the form a'iX, —iD, X) are pseudo-differential operators and we will obtain an expression for the symbol. These facts will be used in the next section to study the transpose and composition of pseudo-differential operators.

Before proceeding note the converse fact: that modulo smoothing operators, pseudo-differential operators are of the form a'(X, - iD , X). Indeed, if p e [R”) and we set a(x, y, = p(x, |) then

a\ X ,-iD , X) = p (X ,- iD ).

In particular, the Schwartz kernel G of p(X, —iD) is given by

G(x, y) = (2ir)-" j |) d| [Osc.].

This formula yields Lemma 3.1. Also note if p'(|, x) = p(x, |) then

a"(X, iD, X) = p'iiD, X).

Thus Theorem 2.4 is a special case of Theorem 4.1.

Theorem 4.5. I f a € S ^ s ( f lx f l , R") and 0 : s 5 < p ^ l thena '(X ,-iD ,X )€ T '^ s (0 ). I f peS^s(D,IR") is a symbol for a ' (X ,- iD ,X ) then

P(x, I) ~ I ^ D^D^aix, y,„ a!

Proof. If w'e set h<,,(x, ) = D|D“a(x, y, then b^eR"). Thus the hypothesis that 8 < p guarantees that we

actually have an asymptotic series. Let P = a'(X, —iD, X ). If 0 e C “(fi) let pe(x, I) = e“' *"‘^P(e‘^ -' 0)(x) so P {6u)= pe(X, -iD )u . Now

P e (x , I) = (2-jt)""| |e‘<’’"*'‘" ’’ a(x, y, Tj)0(y) dy dtj

= (2ir r "| j e ‘<”-->'>a(x, y, e + t,)0(y) dy dr,.

Let h(x, y, ) = a(x, x-hy, ^)O(x-hy). Then h e since 0has compact support in D. After a change of variables in the integral

4. OPERATORS OF THE FORM a'{X, -iD,X) 189

above we have

p«(x, i ) = (2T r ) " " | y , | + tj) dy dt]

= (2'7r)“"| fi(x, v ,^ + v ) dv

where h denotes the partial Fourier transform given by

fi(x, T}, ) = j y, dy.

If K is a compact subset of ft and x e K then

since h(x, y, |) = 0 if x + y^supp 6. Integration by parts yields

iir,yD%Dth{x, V, I) = (2'n-)-’' j e - ‘<’>-’'>D|D?D>(x, y, dy

which implies

lD|D?h(x, + +

for x e K and for each integer N^O. Since (1 + 1^+17!)'^ ^ r(l+ + |t]|)'’ we obtain

lD|D?Pe(x, e\ < C;c.iV.«.0( l + (1 + l^l)m-6l31-Pl«l-(«-i)N

for x e K . Choosing N sufficiently large, which we can do since 6 < 1, we see that the derivatives of pe(x, |) are of polynomial growth, that is we have verified hypothesis (1) in Theorem 2.12. Now if x e K then by Taylor’s theorem with remainder and the estimates above on the derivatives of h we have

j^( 11, l + i l ) - Z ^ I>|li(x, T}, |)Vla l< k

^ Q .n,k !ill'‘ ( l + |ill) sup (l + | +fT)l)m —8N—pk

for any N > 0 . If m-\-8N — p k > 0 and |^|<21t)| we obtain the bound

If m + 6 iV -p k < 0 we have the bound

c d + h i r ^ .

Since 6 < 1 we see if we take N sufficiently large then for any / > 0 we have

the bound

C d -^ h ir ' for 111^2 h i.

Taking N = k in the estimate we obtain the bound

c h i '^ d + h i r ' ' sup d + ii+ fr ,ir^ < « -‘'>\0<t<l

If 2 h N l l l then ll-t-t-n lsIH -f h ls :lll-h l^ 5 1 1 1 - Since 8 0 we have the estimate

jped, I ) - ZI lo tlck OL . J Ilotl<k ^ •

d+hir'drj-i-cf d+iiir^‘*“‘’''‘dT,^ q {M\

The last estimate holds since the first integral decreases as asrapidly as we please by taking I large and the second integral is over a ball with volume proportional to | |''. Now by the Fourier inversion formula

(-iDy)^h(x, y, I ) = ( 2 - n - ) " " | - r j , |)t}“ drj

and therefore

(2rr)-’| D^H(x, r,, |)t,“ dr, = (~iD yrD ^h(x, y, |)Uo

= (-iD y)“D|(a(x, X -1- y, |)0(x -I- y))|y=o = i- '“'D“D|(a(x, y, | )0(y))U ,.

Thus if k is sufficiently large so that m + (6 -p )fc < 0 then

lp ed ,| )- I ^ D “D |(a(x,y,|)0(y))U , <C|’(l + ll ir " '”"<«-'’^1 la l< k

We have now verified hypothesis (2) of Theorem 2.12. It follows therefore that pe G S^s(n, R'") and therefore P g ' ^ sCII). Moreover, we have the asymptotic expansion

Ped, ^)~^^— rD%D^{a{x, y, |)e(y))|y= .« a !

4. OPERATORS OF THE FORM a'(X, -iD,X) 191

Now if [R”) is a symbol for P, i.e. ifand (p)e is the localization of p then (p)e — pe e R ”). But byLemma 2.8 if co is a relatively compact open subset of f l and if 0 = 1 on co then p - (p)e e R ”). Thus p - p ^ e S~°°(<o, R ”) which implies thestated asymptotic expansion for p.

C orollary 4.6. I f a € S ^ s ( f lx f l ,R ”), and q(x,^) =a(x, X, I) then q g R ”) and

a'iX, - iD , X) - q (X , - iD ) €

E xam ple 4.7. Let p € R") , 0< 5 p ( ^ , l)u (y ) dy d|.

By Theorem 4.5 Wpe^^jCfl) and if r is a symbol for Wp then

,-l«lr(x, €) ~ Z 2-'“'D|D;p(x, 0 .

The correspondence p Wp has many properties which are nicer than the corresponding properties of the correspondence p ^ p ( X , —iD). For example, by Theorem 4.1 the formal transpose of Wp is W where ci(x, i ) = p(x, -| ). As we shall see in the next section the formula for the symbol of the formal transpose of p(X, —iD) is considerably more complicated. Weyl operators are studied, for example, in Hormander [10].

R em ark 4.8. The asymptotic expansion in Theorem 4.5 may be conveniently remembered by writing it formally as

p(x, y,

E xercise 4.9. Let 0 < 6 W induces an isomorphism

S^s(fl, R")/S-~(fl, R") ^

Note if 0 < 6 < p : < l , qG R' ) and p is a symbol for W then we

know that

P(x, ~ I 2->“ lD |D 2q(x, I ) . (*)

If we write this expression formally as

then we are led to expect

that is.,-ki

€ ) ~ I ~ 2-'“'D|D;p (x, a

This asymptotic expansion is correct as may be established by differentiating (*). As a consequence we see that if then qeS ~ °°({l,W )and therefore we have the injectivity of the Weyl symbol map in Exercise 4.9. The first part of the exercise may of course also be based on the asymptotic expansion above, rather than on the successive approximation approach suggested in the exercise.

Exercise 4.10. Asymptotic series o f the type we consider may be commuted with continuous operators. Moreover, they may be commuted with formal series o f continuous operators provided in the formal series at most a finite number o f operators decrease order by less than any fixed amount.

Part of the exercise is to figure out what it means. At any rate the main thrust of it is that formal arguments with formal exponentials as above are in fact perfectly valid.

§5. Transpose and composition

I f 0 < 6 < l , 0 < p < l and P g then by Corollary 3.12 P admits a formal transpose P ': (ft) — C“(ft).

Theorem 5.1. 1/ 0< 6 < p < l and PG^^s(ft) then P'GT^^s(ft)- I f p G S^sCft, IR" ) is a symbol for P and q g S^s(ft, IR”) is a symbol for P' then

q (x ,| ) ~ I ^ (D | D ;p ) ( x ,- | ) .a !

5. TRANSPOSE AND COMPOSITION 193

Formally

q(x, i ) ~ -| ) = ■ p(x, -| ).

Proof. Indeed, P = p {X ,—iD) + R where jR € ^ “~(fl). By Theorem 2.4 we have P' = p'{iD,X)-\-R' where p'( ? ^) = pU, and Ifci(x, y, = p(y, — ) then p'{iD, X) = a'(X, —iD, X). Now apply Theorem 4.5.

Corollary 5.2. I f 0 < 6 < p < l and p € S ^ 5(Il,IR") is a symbol for P e then

Note in particular that Corollary 5.2 implies

p’iiD, X )-p (X , ID) €

This fact may be viewed as a generalization of the commutation rule

D ,X ,-X ,D ,= I .

Remark 5.3. If P e the formal adjoint P* of P is defined by

J (Pu)v “ I uP^V dx, M, U E C “(fl).

Clearly

P*u = Ffi.

If 0 < 6 < p < l then it follows that P*E ^ ^ s(n). If p e is asymbol for P then a simple computation shows that a symbol for P* is given by the asymptotic sum

-l«lI ^ D | D ? p ( x ,| )

a !

that is, formally bye - K D , . D > -

In particular we have

p* - p(x, - I D ) €Exercise 5.4. I f P s 0 < S < p < l , p € R ") is a symbol

for P and P* = P then

Im p € R").

Remark 5.5. Note if O s S < p < l , and as usuala'{x, 4 y) = a(x, y, |) and a"0c, y) = u(y, x, |), and if P = a'(X, - iD , X) then by Theorem 4.1 P * = a" (X ,-iD , X). In particular if q € R")and Wq is the corresponding Weyl operator then

( w ,r = w,.

Hence if q is a Weyl symbol for P and P * = P then

Im q e R").

This fact is a considerable improvement over the result of Exercise 5.4 and shows one of the advantages of the Weyl symbol.

The Weyl symbol has other advantages, but it has disadvantages as well. For example, if € C "(fl) then the differential operator

P= I a J - i O rjal m

has symbol

p (x , i )= Z|ct|m

whereas the Weyl symbol is

9(x,i)= S Z (")(|)"'DV(x)r- .I n f i r m x D / \ Z /lotl m 3

Undoubtedly one can foresee situations even for differential operators where this formula has advantages, but it would certainly be perverse to work exclusively with Weyl symbols, especially in the case of differential operators.

We turn now to composition.

Theorem 5.6. I f 0 < 6 |p(x, ^)D“[«/»(x)q(x, |)].ot ac.

Formally

Proof. First note that P«/r(X)Q: C“(ft) C°°(ft) is a continuous linear map. If M € C "(fl) then

(i/>(X)Qu)"(|) = |e dx

= I Q'(e~‘^’ V)(y)M(y) dy

= |e“‘<*”>q^(y,-^)u(y) dy

where

q^(y,'r,) = e - ‘<’'-^>Q'(e‘<”-V)(y)

is a symbol of Q'«//(X). Now choose <#> € C “(D) so < = 1 in a neighbourhood of supp iff. Then

P(«/»Qm)(x ) = P(<f»«/rOM)(x)

= p ^ {X ,- iD ) i^ u ) ix )

= (2 ir )- ''j e‘«-*>p<t(x, |)(«^Qu)'(|)d|

= (2rr)-" j e ‘« - - ’'>p*(x. |)q^(y, -^)u(y) dy d^

Thus P ^ (X )0 = a'(X, - iD , X ) where

y, ■?) = P<t(x, |)q*(y,

Clearly aeS™ ^'"'(fIxfl,|R'') and so P«/f(X)Q€N[f^^'"'(fl) by Theorem 4.5, and moreover,

j-lotlrix, I ) ~ Z — r l)E>?q^(x, -|)].a !

Now q is in S "“ in the complement of supp ip and, since <t> = l in a neighbourhood of supp ip, we have p — is in S ““ in a neighbourhood of suppi/r. Thus (p -p ,j)D “q„,6 S “”(fl,IR'') and therefore

Kx, ) ~ Z — r -^)]-a !

At this point, rather than rearranging series, it is more convenient to work with the formal exponentials. First note by Leibniz’ formula for suitable functions f and g we have

= (A +-D^)“[f(^)g(q)]|,=,.

Thus we have

r(x, I ) ' (p(x, |)q^(y, -|))|y=*

= e ^)%(y, -■n))l(y,Ti)=u,«)= e~‘<* ‘-®v>(p(x, -•ri))|(y,-n)=(x.«)-

Now since q^(x, i ) is a symbol for Q > ( X ) and ilf(x)q(x, |) is a symbol for i/>(X)Q and </f(X)0 = ( Q X X ) ) ' by Theorem 5.1 we have

i}f(y )q {y ,v )~ e q ,{y, ~v)-Corollary 5.7. J / 0 < 6 < p < l , P s O e and either P

or Q is properly supported then PQ e (D). I f p is a symbol for P, q is a symbol for Q and r is a symbol for PQ then

•-loti

r(x, I) ~ X — p D|p(x, i)D ^q(x, |)

or formally

r{x ,i)~

a i

(p(x, ^)q(y, T7))|(y,.„)=(x.«)-

Proof. Since a properly supported pseudo-diiferential operator maps Cc (H) into itself continuously and maps C~(fl) into itself continuously we see that P Q : C “(ft) C “(I1) is a well-defined continuous linear map. Suppose first that Q is properly supported. Let 6 e C “(fl) and let K = supp 0. Then there exists a compact subset K' of f l such that K ^ K ' and such that u g C“(fl), supp u ^ K implies supp Qu ^ K'. Choose ij/e (f ) such that i/f = 1 in a neighbourhood of K', Then

= e-'< ’">Pi/r(X)Q(e'< ">(9)(x)

implies re e by Theorem 5.6. Thus If P isproperly supported then PQ = (Q 'P')' is in by the argumentabove. It remains to compute the symbol. Let co be a relatively compact open subset of ft and choose 6 e C^(i2) with 0 = 1 in <o. In the case where O is properly supported, choose i{/ e C~(ft) as above. In the case where P is properly supported, choose i/f € C~(ft) such that i l /= l in (o and 6P(u) = 6P{il/u) for each weC^ft ) - In either case we have

e{x)re(x, = e "‘^’ 0(X)Pi/r(X)Q(e*^^’‘ O)(x).

It follows that 0(X)Pi/f(X)Q0(X) = (0re)(X ,-iD ). If

qeU |) = e- < ->Q(e < '->e)(x)

then qe is a symbol for Qd(X). By Theorem 5.6 we therefore havei~M

e{x)re(x, ) ~ X — r D|(0(x)p(x, |))D“ (i/»(x)qe(x, 0 ) . a !

Since 0 and i/f are both 1 in we have r — and q — qe are in S ““(o),IR”) and the claimed expansion of r(x, follows for x e c j .

C orollary 5.8. Under the hypotheses o f Corollary 5.7 if p, q and f are the Weyl symbols o f P, Q and PQ, respectively, then

fix, 1 l))| (y ,-n )= (x ,€ )-

Proof. Indeed by the comments following Exercise 4.9

_g(i/2)<D,.D,>(g-i<D,.D,>(p( |)q(y, -n ) ) l (y ,T ,)= (x ,€ ))

where in the last step we used Leibniz’ formula. Now

p(x,|)~e-«^^x^-^.>p(x,|)

and similarly for q{y,r]) gives

fix, ^ )~ e ‘ ipix, |)q(y, q))|(y,.„)=u,«)

where

T = i(D ,+ D ^ , D , +D ,)-<D ^, D ,) - i ( D „ D J-^ (D ^ , D ,)= ^<D ^ ,D ,)-K D „D ,).

C orollary 5.9. I f 0 < 6 < p < l then the operation o f composition o f properly supported operators induces the structure o f an associative algebra on ^^5(n)/''P~“(ft). Moreover, the map p -^ p { X ,—iD) induces an isomorphism o f algebras

S U ^ , R")/S"“(n, R") ^

if the product o f symbols p and q is defined by

(p • q)(x, | )~ e '‘<°'-°v>(p(x, |)q(y, ii))|(y..p)=(x.?)-

Similarly the Weyl map p —> Wp induces an isomorphism o f algebras if the product o f symbols p and q is defined by

ip#q)ix, >'(p(x, ^)q(y, T }))| (y ,T ,)= (x .€ )-

C orollary 5.10. I f 0 ^ 5 < p < l , and Qe'P^sCfl) havesymbols p and q respectively, and if P or Q is properly supported then

P Q -b (X ,-iD )s^ '^ ,^in-l-m'—(p—6)(D)if

b(x, i ) = p{x, ^)q(x, §)

C orollary 5.11. Under the hypotheses o f Corollary 5.10 if

then R = r(X, —iD) has order m + m' — (p — 6) and

P Q - Q P - R e

Corollary 5.11 states that modulo lower order the symbol of the commutator [P, O] is - i { p ,q } where {•, •} is the Poisson bracket. The Poisson bracket plays an important role in Section 5:7.

R em ark 5.12. Multiplication by is a properly supportedpseudo-differential operator of order 0. Thus if then Pi/f(X)E

A symbol r for Pi/^(X) is given byi~\cc\

r(x, I) ~ Z — - D1p (x, i)D °‘t{/(x) a !

in case if/ has compact support by Lemma 2.8 and by Corollary 5.6 for any i/fEC“(fl). In particular, if we set P " = i~ '^\D'^p){X,—iD) then

P iifiX )- X ^ ( D “./.)p <“’ €la l< k <

A comparison of this formula with Lemma 1:12.1 suggests that Leibniz' formula would be a good name for it.

§6. Classical pseudo-differential operators

Let z be a complex number with real part m. Let Uy e — ( 0 ) )

and assume

ay(x, Ag) = A ”'ay(x, g)

for A > 0. If ij/e C “([R”) and i/f = 1 in a neighbourhood of the origin then (l-i/^(|))ay(x, I) is in R"'). (Recall we drop the p ,d subscripts

when p = l and 8 = 0.) Therefore by Lemma 2.10 there exists pe S"*(n,(R") such that

P(x, I) ~ Z (1 ~ I).J

Moreover, modulo R' ), p is unique and independent of the choiceof i//. Therefore we write

An unravelling of definitions shows that p ~ Z simply means

D|D?(p(x, i ) - Z Oi(x, =

as 1 1 ^ oo uniformly for x in compacta in Cl. As usual it is not necessary to bother about the exponent in the ‘big-o’ term; it suffices to have M'k i We note

6. CLASSICAL PSEUDO-DIFFERENTIAL OPERATORS 199

akix, i ) = lim ^(p(x, A^)- Z ^))\ y<k /

and therefore the formal sum

Z <h(x, 0j

is uniquely determined by p(x, ^). If R g ' ~°°(CI) then P = p (X ,—iD) +JR is called a classical pseudo-differential operator with total symbol

o’(F) = Z «)■The top order part, o-o(P) = Uq, is called the principal symbol. The other terms, aj(P) = Oj, are called lower order symbols. We denote the set of classical pseudo-differential operators of order <m by

Note if PeC^"^{Cl) has symbol p then the Weyl symbol

q~e(^/2)< -D,)p

also has an asymptotic expansion Z Tk(P) in a series of terms homogeneous of degree z - k . We have

lT ,(P ) = e«®<'^ -' ->Z<r,(P)

or, explicitly

Z Tfc(P) = z 2 - '“'D |D ;o-,.(P).j.ct ^ •

The equality of these formal series means the equality of those parts that

are homogeneous of the same degree. For the first two terms we have

To(P) = 0-o(P)

We will call S Tk(P) the total Weyl symbol of P. The function t i(P) is very important. It is called the subprincipal symbol of P.

It will be convenient to set a''(x, = a(x, — ).

E xercise 6.1. 1/ P € C ^"'(D ) then P 'e CT^"'(D). Moreover,

T ( p r = T ( p )

E xercise 6.2. I f at least one o f PeCT^'"(D) and Q e C ' ‘P"'{H) is properly supported then PQ e (ft). Moreover,

o-(PO)= Z ^(D |o-j(P))(D >fc(Q )).

Explicitly for the first two terms we have

(To(PQ) = o-o(P)(To(Q)

cTi(PQ) = o-i(P)oro(0)-l-o-o(P)cri(0)-i Zj = 1

For the total Weyl symbol r we have1-131

t ( P Q ) = Z '-^2-'-'-'^K D tD lTYP )m D % r^(Q )).j,k,a,3 P *

Explicitly, for the subprincipal symbol we have

Ti(PO) = Ti(P)<7o(Q)-l-cro(P)Ti(Q)-^{a-o(P), <ro(Q)}.

Let us briefly turn now to the commutator [P, Q] = PQ — QP. Since we view [P, Q] as an operator of order m + m ' - l we note

O-y ([P, Q]) = 0-y + i(PQ) - 0-y + i(QP)

Ty([P, Q ]) = T y ,i(P Q )-T y ^ l(Q P ).

E xercise 6.3. I f P and Q are classical pseudo-differential operators and at least one o f them is properly supported then

o-o([P, Q]) = -iW o(P \ o-o(Q)}.

Moreover,

T2(PQ) = W P ) , T,(Q )}-^ {ti(P), cto(Q)}+ HP, Q)

whereh{P ,Q ) = h{Q, P).

Therefore for the subprincipal symbol o f the commutator we have

Ti([P, O]) = -iW oiP ), Ti (Q )}- i{ri(P), ao(Q)}.Note in Exercises 6.2 and 6.3 {*, •} is the Poisson bracket defined by

. ^ (d a db da db\

It is a fairly simple matter to find a formula for cti([P, Q]) as well, but it is not as nice as the formula for t i([P, Q]). Indeed, one of the features of the Weyl symbol is that it behaves well relative to the symplectic structure of n x[R ”. This fact already shows up in the formulae above.

R em ark 6.4. Hormander [8] has given a beautiful invariant definition of classical pseudo-differential operators. A continuous linear map P : Cc (H) —> is a classical pseudo-differential operator if and only ifthere is a complex number z such that if u € C”(H) and if K is a compact subset of such that g e K implies g is real and dg^O on supp uthen for each fc > 0

/<k /g e K }

is a bounded subset of C "(n). Here Pj (m, g)e C ”(fl). If we define ajix, i ) = Pjiu, ( , ■})(x) where u = 1 in a neighbourhood of x then Hormander shows that cr(P) = X^j. Moreover, he shows if g G C ”(fl), g real, and dg^ 0 on supp u, then one has

-lalZ Pk(u, g)(x) = X (D?aj)(x, dg(x))D“(ue‘ ’')(x)

a , j « 1

where K (y) = g ( y ) - g ( x ) - ( y - x , dg(x)). As usual equality of these formal sums means equality of those parts that have the same degree of homogeneity in g. Explicitly for the first two terms one has

Po(w, g)(x) = ao(x, dg(x))u(x) (1)

Pi(w, g)(x) = ai(x, dg(x))w(x) - i X (d g (x ))^ (x)i = l H i d X j

9 ^ dgU)) ^ f (x)u(x).2 j,k = i dij dXj dXf,I

Note Hormander’s definition is completely invariant. It works directly on a manifold fl. Thus the left side of (1) is an invariantly defined function. On the right side of (1) we have the principal symbol evaluated at (x, dg(x)) where by dg we mean 3g/3xi,. . . , dgldx^. Identifying dg with this particular n-tuple just amounts to using canonical coordinates on the cotangent bundle. Thus if we compute o-o(P)(x, |) relative to some choice of coordinates in ft and then regard it as a function on T * ( f t ) - (0 ) by viewing (x, as canonical coordinates the resulting function on T*(ft) —(0) is independent of the choice of coordinates in ft. If Xi , . . . , x are coordinates in ft the corresponding canonical coordinates on T*(ft) are defined by

Xy(6>) = Xy(7r(o>)), (o e T*(ft),

where tt : T*(ft) —» ft is the canonical projection. Thus the principal symbol of a classical pseudo-differential operator is an invariantly defined function on T*(ft) —(0). Explicitly, if o> G T j(ft)-(0 ) choose g real such that dg(x) = o> and then choose u € C “(ft) such that w(x)=l and such that dgT O on supp w. Then

<Jo(P)(6>) = lim A" ^e"''^ (x)p(e'^«M)(x).

The invariance of the Py(w, g) may be used to prove a nice transformation result for the subprincipal symbol.

T h eorem 6.5. Let ft be an open subset o f and let P e C^"^(ft). Let Ti(P) be the subprincipal symbol o f P computed relative to some coordinate system in ft. I f we view t i(P) as a homogeneous function on T*(ft) — (0) by means o f the corresponding canonical coordinates, then it depends on the choice o f coordinates in ft, but it is invariant under coordinate transforma- tions with constant Jacobian,

Proof. Let Uy be defined as above and for convenience write a for Uq. If we consider a as a function on T*(ft) — (0) then we have the associated Hamiltonian vector field H. In canonical coordinates

/ da d da d \

In Section 5:7 we give an invariant definition of H. Alternatively one can work out the transformation formulae for canonical coordinates arid then laboriously show that H is invariantly defined. Let wGC“(ft), gEC~(ft) with dg^O on supp m. If we evaluate H at the points dg(x) and then

project it into ft we obtain an invariant vector field Lg in a neighbourhood U of supp u. In coordinates we have

j dXj

Let jUL be a positive density in ft, i.e., fi is a Borel measure and if Xi , , . . are coordinates in ft then djut = h dx where h e C “(ft) and h > 0 . We denote by L( g) the formal transpose of Lg relative to fju. Thus


j(LgU)wdfi = dfi

for v,w g C c(U). Clearly L(^g) is invariantly defined. In coordinates we have

or

= LgW + LgClog h) + Zd^a

wd^a

wT jX Hi Hk dXj dXk

If CO is a relatively compact open subset of ft and w = 1 in co then we obtain

g) = T i ( P ) ( d g ) L g ( l o g h)

in CO. The left side is invariant. Thus if ti and h are computed relative to coordinates Xi , . . . , x and f i and h are the same functions computed relative to coordinates y i , . . . , y„ and we observe that the quotient of h and h is the Jacobian we obtain

fi(P)(dg)=Ti(P)(dg) + - Lg(log/)

where

J = det © 1-R em ark 6.6. The presence of the term iLg(log in the transforma

tion formula for the subprincipal symbol suggests that the so-called 1/2 density bundle should intervene. Indeed it turns out if one views classical pseudo-differential operators as acting on 1/2 densities rather than on functions, then the subprincipal symbol is an invariantly defined function on T*(ft) —(0). See, for example, Jackson [1].

Example 6.7. Since

2-2k

we see that (1 -A)^^ is a classical pseudo-differential operator with principal symbol lll .

Example 6.8. Let a e C “([R" - (0 )) be homogeneous of degree 0. The operator a {—iD) (see Example 3.11 and Theorem 2:9.6) is then a classical pseudo-differential operator with total symbol a(^). Thus the singular convolution operators considered in Section 2:9 are classical pseudo-differential operators of degree 0. As an example we have the Riesz operators Rj on IR”. More particularly, the Hilbert transform on !R

Hu(x) = l im - [siO 7T Jlx-tl>e

u (t)dt x — t

is a classical pseudo-differential operator with symbol —i sgn s. If we take then we obtain a pseudo-differential operator of the form

a(-iD )u {x ) = - djkU{x)-\-\im f bji,{x - y)u(y) dyn eio J|j(._y|>e

where bj e C “([R” - (0)) is homogeneous of degree 0 and has integral 0 on the unit sphere. Indeed, a { - iD ) = Since

a {—iD)A = DjDk

we see that we can estimate D Dj u in terms of Aw for any norm relative to which singular convolution operators are continuous, for example, in the norm. We will investigate some aspects of the continuity of pseudo-differential operators in Chapter 4.

Remark 6.9. The invariance of the functions Pj(w, g) may be used to give an interpretation of the lower order symbols of P eC ^ '^ {Q ) as functions on certain jet bundles (the so-called higher order cotangent bundles). See Jackson [1], Petersen [2] and Shih [1].

§7. Invariance of pseudo-differential operators

In the last section we saw that classical pseudo-differential operators may be defined invariantly directly on a manifold. In the present section we will see that ^^s(ft) is invariant under changes of coordinates if 0 ^ 1 —p < 8 < p < l . Moreover, we will see that modulo

IR"'), the symbol of a pseudo-differential operator may be

7. INVARIANCE 205

interpreted as a function on It follows that one may introducepseudo-differential operators on manifolds. The procedure is as follows. If f l is a manifold and is an open cover of O by coordinate charts,a continuous linear map P : C“(H) —> is said to be in if foreach j the map C “( f i j ) E w l n . e relative to the coordinates infly, is in In addition, since pseudo-differential operators are notlocal in general, one has to bother a little about the nature of the Schwartz kernel away from the diagonal, if we allow disconnected coordinate charts we can simply make the requirement that (nyXfty)y>i is an open cover of f ix ft. If we wish to stay with more conventional charts then it is necessary to assume that whenever U and V are disjoint open subsets of ft then the map C^iU) b u P u \ v e C^{V) is a smoothing operator.

Let us now consider the details. Let ft and ft' be open subsets of R ”. Let </): ft ft' be a diffeomorphism of ft onto ft'. If P : C~(ft') C°°(ft')is a continuous linear map then we define <^>*(P):Cc(ft)-^ C “(ft) by

<(>*(P)(u) = (P(uo,f>-^))ocf>,

It is clear that <(>*(P) is continuous. Moreover, if P is a smoothing operator then it is clear that < >*(P) is also a smoothing operator. Therefore in order to consider the action of <f>* on pseudo-differential operators it suffices to study Q = < >*(p(X, -iD )) where p € S^s(ft', R^). If u g C ~ ( f t )

we have

Q m ( x ) = ( 2 7 r ) - " | I e '

= (2ir)-" j I e‘<”-(v.Mx) z>p(< (x), T})u(<f) *(2))dzd-n

ri)hiy)u{y) dy dt]

where h = ldetJl and J(y) = (a</>,79yk)is,,fcs„- By-Taylor’s theorem with remainder

<Tj, < (x) - <#>(y)) = (B ’ix, y)Tj, x - y)

where B' is the transpose of the matrix B = (h ,) defined for x near y by

bkiix, y) = f ^ (y + - y)) dt.Jq OXj

Note bfc, is a C ” function in a neighbourhood of the diagonal in f i x ft and B (x ,x ) = J(x). Thus we can choose x e C ”(f txf t ) so ; = 1 in a neighbourhood of the diagonal and such that B(x, y) is defined and nonsingular in a neighbourhood of suppx- Then Q = Qq+ R where

Qou(x) = (217)“" j j y)p(<f>(x), Ti)h(y)u(y) dy d-p.

If we make the change of variables | = B'(x, y)rt we obtain

QoU{x) = (217)“" ! I y , |)u(y) dy d|

where

a(x, y, y)p(<l>M, A'(x, y)^)h(y) jdet B(x, y)l"’

where A = B “\ Now a g C “( f t x f t x R ”) and derivatives relative to decrease the order of growth in by p. Derivatives with respect to x or y increase the order by 6 or 1 —p. Thus if 0 < 1 —p :< 6 ( x ,y ,| ) | ,= ,O L !

The lower order terms are quite complicated to compute, but the top order term is just

a{x, X, i ) = p(<t>(x), J'(x)~ '0

where J'(x)~^ is the inverse transpose of the Jacobian matrix of <t>. For the operator R we have

Ru(x) = (2ir)“' ' | I - X ( x , y))p(</>(x), r])h{y)uiy) dy d-rj.

Thus R = 4>*{S) where

Su(x') = (2rr)-"||e‘<’’-"'- '>(l - x'(x', y'))p(x', n)v(y') dy' dt,.

Since l~ x '{x ', y') = vanishes in a neighbourhoodof the diagonal in f l 'x f l ' we have by Corollary 4.3. Therefore jRG''P~~(n). We have proved:

T h eorem 7.1. Assume 0 < l - p < 8 - )) o d> then 4>*(P) e

We will now consider the symbol of < >*(P) and continue with the notation above. In the expression for the symbol of Qo if we carry out the differentiation relative to y and g we obtain

X w„g(x)g'^(D“p)(<f)(x), J'(x )a.3

n

7. INVARIANCE 207

for certain functions The general term in this series comes fromi-M— D }D ^a(x,y,

with l7 l>l^l and lalslYl + ljS!. Hence in (*) we only have terms in which 1 1 ^^\a\. The functions depend only on <f) and therefore we may compute them by considering various special symbols p. First note Wqo = 1. Consider now P = -iD j which has symbol rjj. Clearly </>*(P) has symbol

On the other hand by (*) the symbol is

where Sy = ( 0 , . . . , 1 , . . . , 0). Thus w o = 0 if k l = 1* We have proved:

T h eorem 7.2. Under the hypotheses o f Theorem 7.1 if p is a symbol for P and q is a symbol for < >*(P) then

q(x, ^)-p(<f>(x),J'(x)~^^)+ X ^cc&M(D^p)(<f>(x), J'(x)~^^)2 kl

2l3l:Slalwhere the functions w 3 depend only on 4>. In particular

q(x, i)-p(cf>(x),

Note that l - p < 6 J so that the last statement of the theorem does have some content. Note also that m + 1 — 2p < m — (p — 6).

Consider now P g T'' s(11') where we have coordinates y i , . . . , Vn in fl'. Let p(y, ri) be a symbol for P. If we consider a transformation of coordinates y = <f>(x) and compute the symbol q(x, |) of P relative to the coordinates x^, . . . ,x^ then from the point of view above q is just the symbol of <#>*(P). These two viewpoints, one, regarding <t>*{P) as a different operator and, two, regarding < >*(P) as a different representation of the same operator, are called the alibi and alias interpretations, respectively.

If we view (y, rj) and (x, |) as canonical coordinates on then for(o G we have

f ,M - (» . ^ ) = I ^ (» . = I ^

Thus

y = <f>(x) v = J '(x ) -^ i

Therefore we conclude:

C orollary 7.3. I f 0 < 1 — p < 6 < p < l then is invariant underchanges o f coordinates in Cl, The symbol p(x, o f P€T^^s(fl) may be

regarded as a function on T *(ft) by viewing (x, |) as canonical coordinates, The coset o f p in [R") is then invariant.

E xercise 7.4. I f we consider changes o f variables o f the form y = <f}{x) and t] = A ( x) it is clear that is invariant provided that1 — p < 6. Therefore if Cl is a manifold, E is a real vector bundle on ft and 1 — p < 6 we can define S^^CE) by using the local trivializations o f E. Then i /0 ^ 1 — p: ^6 Sâ(T*(ft))/S^5 '-^"(T*(ft)).

Note we may replace l - 2 p by —( p - 6) if that is preferable.

R em ark 7.5. Let PeT^'^îXl). In Section 3 (following Theorem 3.10) we introduced the notion of a principal symbol of P. If 0 < 6 < p < l a more satisfactory definition may be given as follows. If p€ S^s(ft,R”) and P -p (X ,- iD )e ^ ^ a ”^ - (ft) then the coset of p in S^s(ft, R ”)/S^^^ ‘"^ (ft, IR”) is called the principal symbol of P. If we denote the principal symbol by o-o(P) then by Corollary 5.2

CTo(P')U ) = <ToiP)(x, -^ )

and by Corollary 5.10 if at least one of P or Q is properly supported then

o-o(PO) = o-o(P)o-o(Q).

By Theorem 7 . 2 i f l — p < 6 then

CTo{<l> {P))(x, = oroiPM ix), J'ixrÔ^

These results are exact analogues of the corresponding ones for classical pseudo-differential operators.

§8. The pseudo-local property

Let 0 < 6 < 1 , 0 , i/f g C “(ft) and supp 4> Hsupp il/ = 0 then </>(X)Pi/f(X)G'^““(ft). We called this property the pseudo-local property. We will now reformulate the pseudo-local property in terms of an estimate of the singular support of Pu and then in terms of an estimate for the wave front set of Pu, We have already obtained these results for differential operators in Section 2:13, but their usefulness is in applications to parametrices, i.e., approximate inverses, of differential operators.

8. THE PSEUDO-LOCAL PROPERTY 209

Theorem 8.1. Pseudo-local property, and u g €'(0)then

sing, supp Pu c sing, supp u.

I f P is properly supported the conclusion holds for ueQ)'{H),

Proof. Suppose Xq sing, supp u. Let V be an open neighbourhood of Xq disjoint from sing, supp u. Choose <f> g C ”(ft) with <f> = 0 on V and <#) = 1 on a neighbourhood of sing, supp u. Choose i/r g C “( V) with i/f = 1 in a neighbourhood W of Xq. Then

Pm = (1 — i/r)Pu + il/P<t>u + P(1 - (f>)u.

Now (1 —i/f)Pu = 0 in W. By Lemma 2.15 and Lemma 3.14 i/fP<f>ue C'^Cft). Finally P (1 -</>)m g C “(ft) since (1 -< ))m g Cc(ft). It follows that Pu is on W and therefore Xq sing, supp Pm. Consider now the case where P is properly supported and mg Si'(ft). As in the proof of Lemma 3.8 we can find a partition of unity g C“(ft) and functions Xj C“(ft) such that P v = Y <l>jPiXj' ) for each i; g C “’(ft). This equation also holds for uGS'(ft) since the extension of P to S '(ft) is defined by taking the transpose of the formal transpose P ' : C“(ft) — C“(ft). Now by the first part of the proof sing, supp <f>jP{xjU) ^ sing, supp m.

According to Theorem 3.10 the symbol of a pseudo-differential operator is uniquely determined modulo S “"(ft, R”). Therefore it makes sense to define the essential support, ess. supp P, o f P g ^s(ft) to be the essential support of its symbol. (See Section 2:13.) Note that PG ^"~(ft) if and only if ess. supp P is empty. If 0 < 6 < p < l and at least one of P, QG'?^~5(ft) is properly supported then by Corollary 5.7 we have

ess. supp (PQ) c ess. supp P (T ess. supp Q.

Lemma 8.2. I f P g ‘ '‘ ^(ft) and u g ^'(ft) then WF (Pm) c ess. supp P. I f P is properly supported then the conclusion holds for MGS)'(ft).

Proof. Choose 0 g C c (ft) with 0 = 1 in a neighbourhood of supp m. Then Pu = P{du) = Pe(X, -iD )u . If p is any symbol for P then by Remark 5.12 we have

l~lcc\

Pe(x, I) ~ Z — j- £>|p(x, eD -O (x)cc Oil

Therefore ess. supp pe ^ ess. supp P. Suppose (xq, tiq) ess. supp P. Choose an open cone V in IR "-(0 ) with r^oGV and a relatively compact open set o> in ft with XqGO) such that the closure of o) x V in ft x R " — (0)

is disjoint from ess. supp P. Then P u = f + g where

</, u) = (277)“" f f e‘<’*V eU dx (1)JV Jto

<g, u) = (2t7)"" [. [ e‘ *-’‘V eU ^)w (l)uW dxd| (2)J<o

for V e C“(o>). (See Remark 2.6.) Since pe and its derivatives are rapidly decreasing in a> x V and u is of polynomial growth we may change the order of integration in (1) and deduce that /g C “(o)). Thus WF (Pu\ ) = WF (gU). Let V e C ^ M . We will estimate

(g ,e -‘< '>t;) = (277)-"[ [ e'<«- ’*>pe(x,^)M(|)u(x)dxd|. (3)

Choose an open cone V' in [R"" —(0) such that V ' n S ”" is relatively compact in V and such that 7]q g V'. Let 0 < e =dist ( V ' n S ”“\ 3V). Then clearly

1|-t]1>£|t)1 if T]€V' and

Let

JSO L' = —L and

Since v has compact support integration by parts yields

<g,e-‘<’'->u) = (277)-"f f e‘« “’'-">u(^)L'^(pe(x,|)t;(x))dxdeJr "~V -L

For some M we have lu(^)|:<C(H-lll)'^ and therefore

Kg,e“‘<’'- >t;)l<CN f 11-7,1“^ d iJr "~V

(4)

We split this integral into two parts. If | ^ |> H - 2 | tj1 thenl j —|T]|>i(l + | l). Given N, choose k so M + 6N +(6 —l)fc + m < —n - 1 .Then for | |> 1 + 2 1t]|, rj e V\ | g (R” ~ V we have

(1 + < 2* (1 + | |)M+«N+(S-l)fc+m

Thus if Tj G V' then in (4), with N replaced by N + fc, the part of the integral which is over l j > 1 + 2 jr]! is bounded by for any N > 0,

8. THE PSEUDO-LOCAL PROPERTY 211

where Cn is independent of tj. Next consider < 1 + 2 |t]|. Then H-1 | ^ 2(1 + 1t]1) and since we may assume iTila:! we also have ||-T}l>e j-nls (e/2)(l + Iril). Assuming M + m > 0 we have

(1 + i a M + 8 ( N + k ) + m 1 > __

^ ^ - N - k 2 M - ^ ( 8 + \)(N+k)+m^-^ |^M -t-(8-l)(N + k)+ m

< -N-k2M+(S+l)(N+k)+m 'l |^| -n-l_N

Since in (4) when we integrate over 1|| s 1 -i-2 Itj] we are integrating over a portion of a ball of radius 1-1-2 jril we see that this part of the integral is bounded by for r je V' and l-nl^l. It follows that

<g,e-'< -->u) = (|T,|- )

for each N > 0, for t] e V', 1t]1 > 1, and v e C “(<«)). Thus (xq, tjo) WF (g).

Theorem 8.3. Pseudo-local property. I f P e 0 < 8 < 1, 0 <p ^ l and ue^'(D ) then

WF(Pu) £ WF (m) n ess. supp P.

I f P is properly supported then the conclusion holds for each u e3 f(H ).

Proof. Suppose (xq, tjo) 4 WF (u), t/q ^ 0. Choose an open bone V in R" - (0 ) and a relatively compact set « in H with Xoeo>, r io e V such that

l^ ( i)N C K ^ (i+ i^ ir " '

for each N > 0, <pe C"(co) and | e V. Choose <j> € Cc(<o) so = 1 in a neighbourhood a>' of Xq. If u € € '(fl) or if P is properly supported and we2i'(n) then by Theorem 8.1 P {l — <t>)u is C ” in to' and therefore it suffices to show that (xq, tjo) 4 WF (P(<f>u)). Choose 6 e C ”(fl) with 6 = \ in a neighbourhood of supp <f>. Then P(<f>u) = P(0<f>u) = pe(X,-iD)(<f>u) and therefore for each v e C“(to) we have

<P(tf)M), n) = (217)“'* f f e'<*'‘V e(^ ,^)^(l)u (^)dxd|.Jr" J<o

As in the proof of Lemma 8.2 we write P{6u) = f-^g where f is defined by the integral over o) x V and g is defined by the integral over (o x (R” ^ V). Since pe and its derivatives are of polynomial growth and (j)u is rapidly decreasing in V we may change the order of integration as before to deduce feC°^{o>). We then integrate by parts in (g, andestimate exactly as in the proof of Lemma 8.2 to conclude Uo, 'no)^WF(g).

Remark 8.4. The integration by parts argument in Lemma 8.2 and Theorem 8.3 is a typical stationary phase argument. See Remark 2:13.8.

§9. Characteristics. The regularity theorem

Let (xo, “ (0) 2ind let p be a symbol for P. Wesay that P is (microlocally) elliptic of order m at (xq, |o) if there is a neighbourhood <o of Xq and a conic neighbourhood V of ^md constants C > 0 and jR > 0 such that

1p (x, ^ ) i> c ( i+ i^ ir

for X G 6), G V, and j l > R. By modifying p by an element of R ”)we may assume R = 0. If m' < m it is clear that ellipticity is not disturbed by adding an element of S^s(n, R ”) to p, that is, ellipticity is a property of the principal symbol. The complement of the set of points at which P is elliptic is denoted by Z(P) and is called the characteristic set of P. Z{P) is a closed conic set and by Corollary 7 . 3 i f 0 : ^ l - p ^ 6 < p < l then Z(P) is an invariantly defined subset of —(0). We note if P g then

Z(P) = {(x, G n X R - - (0) I o-o(P)(x, I) = 0}.

In general if a notion in the base fl is transferred to the cotangent bundle one says that it is microlocalized. We will now microlocalize the notion of order. Let and let (xq, We will saythat P has order p, < m at (xq, I o) if there is a conic neighbourhood U of ( 05 o) such'*that if p is any symbol of P then plu^SpsCU). By cutting down p we see that an equivalent definition is that P has order p, at (xq, o) if there exists q e Sp Cfl, R") such that q = p in a conic neighbourhood U of (xq, I o)* We say that P has order p, on an open conic set U if P has order p. at each point of U. We note that ess. supp P is the complement of the largest open conic set on which P has order —oo.

L em m a 9.1. Let PkG'^^sCfl) and let (xq, Let Lf' bean open conic neighbourhood o f (xq, o) assume P has order in U', where - » -o°. Let mk = supj>kmj. Then there exists a conic open neighbourhood U o f (xq, o) there exists P g such that

p - l P ij<khas order in U for each fc>0.

Proof. Let Pk be a symbol for P . Choose e R ”) such that~Pk = 0 in a conic open neighbourhood U of (xq, o)* By Lemma 2.10

there exists pG S^s(n,R ”) such that p~Z^k- Let P = p {X ,—iD), The symbol of P -Z j< k B j is equal to p-Zj<k^Jj in Lf and therefore the conclusion follows.

9. CHARACTERISTICS. THE REGULARITY THEOREM 213

We will indicate the situation described in the lemma by writing

P ~ Z P fc in U,

and we will say that P is an asymptotic sum of the P in U. We may also consider asymptotic limits of sequences.

Corollary 9.2. Let U q, ~(0). Assume thatQ k “ Q k - i order —oo in a conic neighbourhood o f ( x q , | o ) - Thenthere exists OG^^s(n) such that 0 — 0^ has order ^ml, in a conic neighbourhood o f (xq, $q).

Lemma 9.3. I f at least one o f P , Q e ^ p s i ^ ) is properly supported, 0 < 6 < p < l , P has order in an open conic set U and Q has order in U then PQ and QP have order + in U.

Proof. The lemma is an immediate consequence of the formula for the symbol of the composition given by Corollary 5.7.

Lemma 9.4. Let 0 < 6 < p < l , let P€T^^s(fl) and assume (^0. Then there exists a properly supported operator Q g ^ “s ( )and an open conical neighbourhood U o f (xq, q) such that

P Q -1 and Q P - 1

have order - o o in U.

Proof. We are free to modify P by an operator of order —oo. Hence we may assume P is properly supported so P = p {X ,—iD) where p e

[R”). Moreover, we may assume that there is a neighbourhood (o of Xq in Q and a conic neighbourhood V of in (R” - ( 0 ) and a constant c > 0 such that

lp (x ,| )l> c(l + i a '"

if X e « and 1|| < 1 or | € V. Choose (f> e C"(o)) with <f> = 1 in a neighbourhood &)' of Xq. Let V' be a conic neighbourhood of |o in R" — (0) such that V 'n S " “’ is relatively compact in V. Choose h e C ”(V) so h = 1 in a neighbourhood of V ' fl Finally choose iff e C“(IR”) so that if/{ ) = 0 if l l l s : l and «/ (|) = l if 1 1 Let

9o(x, = < t> {x )( i- i f ,im ii\ ^ r )p(x, I)

Then qo€ R") (see Exercise 9.5 below) and

<1o(x, i)p(x, I) = 1

if xe<o', V and |^1>1. Choose Q oe’PpsCft) properly supported with

symbol qo- Then a symbol for QqP is given by the asymptotic sum

I ^ D | q o U | ) D :p ( x ,aa OLl

It follows that OqP - 1 € s(n) has order at most - ( p - 8 ) in U = (o'x V'. Similarly POq- I has order at most - ( p - 6 ) in U. Since Q o P -1 and PQq — 1 are properly supported we may consider powers of them. We define

k k

Qfc= I (l-Q o P y Q o = I O o(l-P Q o)’.j = 0 j = 0

Then Q^e'^p aCD) and

Q ,- Q ,_ i = (l-Q o P )"Q o

has order at most —m - k ( p - 8 ) in U. By Corollary 9.2 there exists such that Q -Q k has order at most - m — k(p — 8) in a conic

neighbourhood U' of (xq, o)- Now

P Q - l = P (Q -Q k ) + (P Q k -l)= P (Q -Q k )-( l-P Q o )" "^

has order at most -(fc + l)(p - 8) in U'. Thus PQ - 1 has order —oo in U'. Similarly

QP - 1 = (Q - Q k )P - (1 - QoP)"^'

implies that Q P -1 has order -o o in U\

Exercise 9.5. I f U is a conic set, p g S^s(L7) and a + 13 / 0 show thatk l+ l3 l

D | D ?(p -')= Ik = l

where

Moreover, gk,a,3 is a sum o f constant multiples o f products o f k factors o f the form D| D f p where the sum o f the a' is a and the sum o f the j3' is jS.

We say that is elliptic if P is elliptic at each point off tx lR " - (0 ) ; that is, if Z(P) is empty.

Corollary 9.6. I f 0 ^ 8 < p : ^ l then P e is elliptic if and only ifthere exists a properly supported operator Q e such that PQ — 1 andQP — 1 are smoothing operators.

Proof. The proof of sufficiency (of ellipticity) is the same as in Lemma 9.4. For necessity note if p g IR”), q g S “s ( f t , a n d p q - lG Sm-(p-s)^Q^^^n) compact set K in ft there are constantsjRk ^ 0 and Ck > 0 such that jp(x, )\> Ck (1 + for x e K and \$\> R^.


Rem ark 9.7. Parametrices and hypoellipticity. Let PG^^6(ft). If QG^^s(ft) is properly supported and P Q - 1 and Q P -1 are smoothing operators then we call Q a (pseudo-differential) parametrix of order m' for P. According to Corollary 9.6 P has a parametrix of order -m if and only if P is elliptic. If we consider parametrices of other orders, however, then the situation changes. Note though that if P is a classical pseudodifferential operator of order m and P has a classical pseudo-differential parametrix Q of order m' then the homogeneity of the principal symbol implies that m' = - m and P is elliptic. Thus if we wish to consider parametrices for non-elliptic partial differential operators, then the class of classical pseudo-differential operators is not large enough.

Parametrices are important in the study of existence and regularity of solutions for (pseudo-) differential equations. Let P g '^^^(ft) be properly supported and suppose that P admits a parametrix QG'^^^s(ft). Let /GSJ'(ft) and consider the equation

Pu = f.

If we have a solution u then

w = 0 / - ( Q P - 1 ) m.

Since (Q P -l)w g C “(ft) we have by Theorem 8.3 WF(w) = W F(Q /)c WF (/). Also by Theorem 8.3 we have WF (/) = WF (Pu) c W F (u). Thus

W F(u) = WF(/).

A properly supported pseudo-differential operator P which satisfies sing, supp Pu = sing, supp u for each uGS)'(ft) is said to be hypoelliptic. Thus a pseudo-differential operator which admits a pseudo-differential parametrix is hypoelliptic. When one is working with systems it is the existence of a left parametrix which implies hypoellipticity. There are differential operators, for example the wave operator, which are not hypoelliptic. Such operators cannot have a pseudo-differential parametrix; indeed, cannot have a pseudo-local parametrix. One is thus encouraged to consider classes of non-pseudo-local operators, for example Fourier integral operators, in order to have a more general notion of parametrix.

With regard to existence, consider again

P u = f .

This equation is satisfied by w = Qv provided that

{PQ — l)v + v =f .

Since PQ — 1 is a smoothing operator we have an integral equation, with kernel, for v. Such equations can be studied by classical methods.

When one is working with systems, it is a right parametrix that intervenes in questions of existence. Again we note that it would be advantageous to consider parametrices which are not pseudo-differential operators.

Let us now consider the possibility of finding a parametrix in the non-elliptic case. Let 0 < 6 0 such that

ip (x ,^ )i> C K (i+ iiir (1)

if x e K and Assume also that we have constants CK,o<.e>0 suchthat

|p(x, 1)1 > Ck.„.3 |D|D?p (x, )| (1 +

for x e K and In Exercise 9.5 we saw if a + j3^ 0 then

1 « H 3 I

D | D ?(p -')= Z p-'=-'gfc.«3.k = l

(2)

(3)

Here gk,a,3 is a sum of constants times products of k factors of the form D fD ^ p where the sum of the a' is a and the sum of the |8' is ^. If we pair each such factor with one of the powers of and estimate these by (2) and then estimate the remaining p~ by (1) we obtain

lD|D?(p(x,|)-')l<Ck...3(l + llir pla|+8l3l (4)

for x g K and l^l>jRj^. Choose such that qo( , ) =p(x, for large \$\, Then by (4) R"). Chooseproperly supported with symbol qo- Then a symbol for QqP —1 is given by the asymptotic sum

Z ^~T i)D^p(x, I).a=70 ^ •

The general term in (5), apart from a constant factor, for large 1| isklZ P '‘gfc.a.oP 'D ?p .

k = l

(5)

(6)

Apart from constant factors (6) is a sum of terms of the form

(p-'D Sp)(p-‘D fp ) • • • (p - 'D fp )

where a ' + • • ’ + = a. By (2) the first factor is dominated by (1 +and of the remaining factors the jth one is domonated by + Ifwe apply the operator to the product above we obtain, apart fromconstant factors, a sum of products of factors of the form

p-^ D fD l'p

where the sum of the |3' is a + j3 and the sum of the y' is a + y. Estimating each factor by (2) we obtain for each term in the sum an upper bound of the form It follows the function in (6) is in5 -iai(p-8)(^, QoP - 1 € Similarly PQo - 1 e

If we now repeat the last part of the argument in the proof of Lemma 9.4 we see that there exists a properly supported operator

such that Q P -1 and PQ —1 are smoothing operators.Summarizing, we have shown i/ p, < m then (1) and (2) imply that P has

a parametrix Q in In particular P is hypoelliptic.We will see in Chapter 4 that if P has a parametrix of order - p where

p = m — r then P satisfies a stronger condition than hypoellipticity; namely, P is subelliptic with a loss o f r derivatives. This notion will be studied in Chapter 4.

Condition (2), which looks rather daunting, turns out to be quite practical to check in many cases. None the less let us consider what happens if we drop condition (2) and assume only the lower bound (1). In(3) the best we can now say is that , 3 € S p , s R " ) . Then by (1) we obtain estimates of the form

\p(x, ^)-'gk.„.p(x, 1)1 < C(1 + )-l«lP+l3|8_

Since /x < m the worst estimate comes from fc = |al + l|3|. Hence we have

It follows that q o ^ R " ) where p' = p - ( m - p ) and 8' = 6 + (m —p). Since p '^ p and 6 '> 6 we may consider P to be a pseudodifferential operator of type (p', 6'). Thus QqP - 1 is a pseudo-differential operator with symbol given by (5) provided that 5 '< p ', that is, provided that m - ^ ( p - 6)< p . Also we note that the general term in (5) is in the space Thus QqP - I has order at most ( m - p ) -(p ' - 6' ) < - K p “ )* Once again we may repeat the last part of the argument in Lemma 9.4 to obtain a parametrix for P.

Summarizing, we have shown if m - 5( p - 6) < p < m then (1) implies that P has a parametrix Q in where p' = p —(m —p) and 8' =6 + (m —p).

The condition on p here is rather stringent. We will, however, use this result later in the proof of Theorem 4:7.36.


Example 9.8. Consider the heat operator in IR"

P(D ) = D „ - X D l1=1

The symbol of P(D ) is given by

P (l)= i€n+ll'P

where = (| i,. . . , Clearly P(D ) is not elliptic. Suppose l|p =^ l + W f ^ 2 . If l f ls | 4 l then implies

e . + III" I l f max [ I l f (1+1||)^ (1 + i i in

If llnl^ ll'l then implies

1^+I l f s 2 IIJ I l f > 111 I l f > I l f (1 + 111)

and

i ^ + i i r ^ i ^ a ^ i i p s K i + i i i f .

Then certainly

| ^ + lir> ^ m a x [l| f (1 + 111), (1 + I l f ] if I II" ^ 2 .

It follows that

|p(|)|>>ax[||'l(l + l | f f ( l + lll)]

for large \\. This inequality implies

1p (^)I^4(i + 1I1)

and

ip(i)i^5ip>“p (i) i( i+ iii) '“""

for large \\. Hence P(D ) admits a parametrix in 2ind inparticular is hypoelliptic.

E xam ple 9.9. The Laplacian A = X is obviously elliptic. It follows that W F (m) = W F(A m) for any weSJ'Cfl), ft an open subset of IR In particular the only solutions of Am = 0 are the usual harmonic functions.

E xam ple 9.10. If ft is an open set in 0 , u e^ '(D ) and {duldZj) = 0 for 7 = 1 , . . . , n then m is a holomorphic function on ft. Indeed 0 = Yid^u/dZi dZj) = l A u implies u is smooth by Example 9.9. But then the Cauchy-Riemann equations imply that u is holomorphic. A much deeper result of Hartogs states if m is a function on ft and u is separately holomorphic then u is holomorphic. Note that if u is assumed to be locally integrable then Hartogs’ result follows from the above. However, in Hartogs’ theorem there are no assumptions on m .

If we construct a microlocal parametrix at each of the elliptic points of an operator we obtain the following pretty regularity theorem.

T heorem 9.11. R egularity th eorem . I f 0 < 6 < p < land then

W F (u) c W F(P m)U Z (P ).

I f P is properly supported then the conclusion holds for any u e^ '{d ) .


Proof. Assume (xq,^q)^ Z (P )\ JW F (P u). By Lemma 9.4 there is an open conic neighbourhood U of (xq, I o) ind a properly supported operator Q such that if jR = Q P - 1 then jR has order —oo in U.Note that R is properly supported if P is properly supported. Now

u = Q(Pu) — Ru

implies

WF (u) c WF (Q(Pm)) U WF (Ru)c (WF (Pm) n ess. supp Q) U (WF (m) H ess. supp R) c WF (Pm) U ess. supp R.

Since U Pi ess. supp R = 0 we have (xq, o)$ WF(m).

In particular if P is, for example, a linear partial differential operator on d then

WF (Pm) c WF (m) c WF (Pm) U Z(P)

for any u e Q ) ' { d ) . This fact may be used to compute, or at least estimate, the wave front set of a distribution when we can find suitable partial differential equations satisfied by the distribution.

Example 9.12. Let ue^'(R^) be the characteristic function of the square {(xi, X2) | |xil<l, lx2l< l } . Then sing, supp u is just the boundary of the square. Clearly D j M is supported by the two vertical sides of the square. In particular DiU vanishes along the upper and lower edges, so at these points the wave front set of u must lie in Z(D i). Similar remarks apply to D2M. Throwing in the worst possible case at the vertices we have WF (m) c A U JB U C where

A = {(xi, X2, 0 , 12) IB = {(x i, X2, li,0)|

C = {(Xi,X2,

W < 1 ,X 2 = ±1,^2 7^0}X i = ± l , | X 2 | < 1 , ^ 1 ^ 0 }

X i = ± l , X 2 = ± l , ^ 7 ^ 0 }.

We will now show that in fact we have WF (u) = A U B U C First note

D 1D 2U = Ta i)6 — T(i + T(_i _i)6 — T(_i 1)6.

Since WF (D 1D 2M) ^ WF (w) we see that C ^W F(w ). Next we consider the edges. If g C “(R ) then

<DiM, <>) = - f (<#>(1, s )-<#>(-1, s)) ds.

In particular if </> has compact support in a small neighbourhood of (1, Sq), where lsol< l, then

<Di M, e ‘^'* <)) = - J e s) ds

= e"'^>h(^2)where h e S f . We may choose so that <#>(so)^0 and h(0) = - J <#>(!, s) ds^ 0. Then e“'^'h(0) is not rapidly decreasing whence (1, So, ^1, 0) is in WF (D im) c WF (m) if f 0. Thus B c WF (m). Similarly A c WF (m). Note that the boundary of the square may be viewed as the union of eight submanifolds of IR ; the edges and the vertices. Then we have shown that WF (m) is the union of the conormal bundles of these eight submanifolds.

C orollary 9.13. I f mgS>'(D) then

W F(u)= n z (P )

where P runs over all properly supported pseudo-differential operators P e C ^ \ n ) such that Pu e C~(n).

Proof. If PueC °°{(l) then WF (u )cZ (P ). Conversely, suppose ( 0. W F(m). Choose h € C “(flx[R” -(0 )) homogeneous of degree 0 with h{xo,^o) = l and h = 0 in a conic neighbourhood U of WF (u). Choose x ^ C c i W ) with ^ = 1 in a neighbourhood of the origin and let p(x, ) = { l -x (^))h {x , ). Let P be a properly supported pseudodifferential operator such that P - p ( x , - iD )€ Then P ecTo(P) = h and P has order -00 in U. Thus WF (Pm) c WF (m) Hess, supp P = 0 which implies Pu g C “(fl). Since <Jo(P)(xo, 0) = 1 we have (xq, o)^ Z (P ).

Since C'^^(fl) has a nice invariant definition which works directly on a manifold (see Section 6) the corollary yields a very pretty invariant definition of WF (m) as a subset of T * ( f l) - (0 ) . This definition may be carried over directly to distributional sections of smooth vector bundles on a manifold.

So far in this section we have considered problems involving division by a symbol with a suitable positive lower bound. We will now consider a simple problem involving division by a symbol which has zeros. We will not actually prove any theorems, but will simply illustrate some of the ideas. We will work exclusively with classical pseudo-differential operators with symbols with real degree of homogeneity.

Let and let p = ao(P). Recall we may regard p as aninvariantly defined function on —(0) by identifying T*(ft) withflx lR ” via canonical coordinates. The zero set of p is just Z(P), the characteristic set of P. We will also write Z(p) and refer to the characteristics of p. A characteristic (xq, o) e Z(p) is said to have multiplicity r if the restriction of p to the fibre T *(fl) vanishes of order r at (xq, o)* This notion is clearly an invariant one. Characteristics of multiplicity 1 are said to be simple; those of multiplicity 2 are said to be double; and so forth. Note in canonical coordinates we have (xq, o) is a characteristic of multiplicity r if and only if D|p(xo, o) = 0 for jal < r and D|p(xo, lo) ^ 0 for some lal = r.


L em m a 9.14. Let qG C“(T*(I1) —(0)) be homogeneous o f degree 1 and have simple characteristics. I f p g C~(T*(fl) — (0)) is homogeneous o f degree m then each point o f Z{q) is a characteristic o fp o f multiplicity > r if and only if there exists a g C “(T*(ft) —(0)) such that p = aq\ Moreover, a is unique and homogeneous o f degree m — r, and (xq, o) is a characteristic o f p o f multiplicity exactly r if and only if a(xo, lo) ^ 0.

Proof. Since d^qÔ on Z{q) we certainly have dqÔ on Z(q). Thus Z(q) is a (2n - l)-dimensional submanifold of —(0), and thereforethe complement of Z(q) is dense in —(0). Since we must define ato be pq“' in the complement of Z(q), the uniqueness and homogeneity of a follow. Thus to construct a it suffices to do so locally in neighbourhoods of points of Z(q). Fix canonical coordinate (x, and a point (xq, o) in Z(q). Since d^q(xo, ô)^0 by relabelling coordinates we may assume {dq/d^n)(ô, o) 0. Then

7]j j, j — 1 , n ~ 1q = q(x,

is a change of coordinates in a neighbourhood of (xq, q), and

d _ d

7 = 1, . . . , M -1

Thus near (xq, o) we have Dj __p = 0 on Z (q) for 0 ^ fe < r . Therefore by Taylor’s theorem with remainder

P(y, = v ', t v n ) ( i - t r ' dt

where tj' = ( r ji , . . . , iin-i)* The integral defines the smooth function a in a neighbourhood of (xq, q)* Conversely if p = a q ’’ it is clear that p vanishes of order on Z(q). If \a\ = r then at points of Z(q) we have D|p = a(D^^q)“' , . . . , (since other terms vanish) and so the last partfollows.

Consider now a simple application. Let Q e be properly supported and have simple characteristics. Let P g and suppose thateach point of Z(Q ) is a characteristic of P of multiplicity at least 2. By Lemma 9.14 if cr(P) = ZPj and (r(Q) = Y,qj then

Po~

where Uq is homogeneous of degree m -2 . Choose A e withcto( A ) = ao and set cr(A) = Z ^r We have

P -A Q ^ e C ^ ^ -\ £ l) .

O ut goal is to express P to still lower order in terms of Q. Since we view P - AQ^ as an operator of order m — 1 we have

o-o(P - AO^) = P, - <ri(AQ2).

Now

o-i(AO^) = aiqo + «oO’i(Q^) “ ' Z (jD€,«o)(L>x,(<?o))

and

cti(Q^) = 2qoqi~ i Xj = i

Thusn

<To{P-AQ^) = Pi + iao Z (Di^<lo)iD^<lo) + b'qo i = i

where b' is homogeneous of degree m -2 . Since Po = o<Io for the subprincipal symbol Ti(P) of P we have

Tl(P) = Pl+Z Z I^^PxiPo i = l

n

= Pi + iao Z iDsflo)(D^flo) + b"qoi = i


where b" is homogeneous of degree m -2 . It follows that

a-o(F-AQ") = Ti(P) + (b'-b")qo.By Lemma 9.14 we see that there exists B e such that

P - AQ^ - B Q e

if and only if

Ti(P) = 0 on Z(Q).

In this case we then have the canonical form

P = A Q ^ ^ B Q ^ C with A , B , C e C ^ ^ ~ H ^ ) ,

Such canonical forms (in the case of characteristics of higher multiplicity) were used by Chazarain [1] to reduce the problem of studying the singularities of solutions of P u = f in case of characteristics of constant multiplicity to the case of systems with ‘scalar’ top order part QI where I is the identity matrix. Such systems behave as scalar operators with simple characteristics and for such operators quite a bit is known about the singularities of solutions. Some results are given in Section 5:7.

Chapter 4

Hflbert space methods


In Section 2 we introduce the Sobolev spaces The space consists of temperate distributions u such that m gL^(IR^Ioc) and

llwlls = (2'n-; |m(I)P d|<+oo.

H" is a Hilbert space and H'' decreases as s increases. In view of the Parseval-Plancherel theorem if m > 0 is an integer then

It is also possible in the case of non-integral s to give a description of W which is independent of the Fourier transform (see Theorem 2.17). The utility of the spaces W arises in part from the fact that they permit precise quantitative regularity assertions concerning solutions of pseudodifferential equations. From our point of view such results arise via parametrices from the continuity of compactly supported pseudodifferential operators of order m as operators from FP to Thereare more general spaces which in some cases give more precise results, but the spaces FT' are the simplest. The connection with ‘classical’ regularity is Sobolev’s lemma (Theorem 2.13); if fc>0 is an integer and s > fc + n/2 then c C^(1R").

In Section 3 we present a potpourri of results concerning continuous linear operators on Sobolev spaces. In particular we show if is compactly supported then P maps FT' continuously into FT'“'” for each s. The continuity of operators of type (p, 8) with 6 > 0 is more difficult to study. In Section 4 we discuss the theorem of Calderon and Vaillancourt which implies for example that if P g '^^5(IR'') is compactly supported and 0 < 6 < p < l then P maps FT' continuously into FT'”"'. We do not prove this theorem, but instead prove the continuity in the case 8 <p. The proof

225

226 HILBERT SPACE METHODS

which we give in this case, which is due to Hormander, makes very elegant use of the symbol calculus developed in Chapter 3. The proof also gives for operators of order 0 an estimate of the norm modulo compact operators, and therefore a criterion for compactness.

It is not very convenient to work exclusively with compactly supported pseudo-differential operators. Therefore in Section 5 we introduce the local Sobolev space loc) consisting of distributions u on i l such that(f)ueH^ for each <t> e Cc(n). We topologize this space by the seminorms u 11<)m11s, E C“(fl). It then becomes a Frechet space. From the results of Section 4 we now have if P E ^ ^ 5(fi) is properly supported and if 0 < 6 < p < l then P maps jFP'(fl, loc) continuously into loc).

In Section 6 we introduce the FT wave front set WFs(w) of a distribution u on II. We give a direct definition in terms of the Fourier transform but show right away that (xq, ^o) WFs(w) if and only if there exists Ml E H""{Cl, loc) such that w = Mi + M2 and (xq, 0) W F(m2). It is easy to see that WFs(m) increases with s and that the closure of the union over all s is W F(m). Moreover, for any ueQ)'(CT) and (xq, ~(0) there existss such that (xq, o)^ WFs(m). This observation frequently permits one to make inductive arguments on s, a technique which is not available when one considers only W F(m). The main result of Section 6 is Theorem 6.5 that if P E '^^ s(Il) is properly supported, 0 < 6 < p < 1 and m e S)'(I^) then

WFs(Pm) c WFs+^(m) Hess, supp P W F ,,^ (m) c W F,(Pm)U Z (P ).

( 1) (2)

Property (1) is a refinement of the pseudo-local property and follows from it in view of action of pseudo-differential operators on Sobolev spaces described above. Property (2) is a refinement of the regularity theorem and follows from (1) by constructing microlocal parametrices at non-characteristic points. In particular if P is elliptic one obtains W F , ( P m ) = W F , , ^ ( m ).

In Section 6 we also prove the invariance of the space H^{Cl, loc) and of its topology under changes of coordinates. We also give an invariant description of WFs(m) and give an example of a distribution m on IR" such that WFs(m) is empty if s < - n l 2 and WFs(m) consists of a single ray over the origin if 5 > 0 .

Section 7 is basically abstract functional analysis. We first discuss various types of subellipticity. If P e is properly supported and Padmits a parametrix Q of order — m + r, then P is strictly subelliptic with a loss of r derivatives in the sense that W F s+^_^(m ) W F s(P m ) for each s and each distribution u on Cl, The canonical example is the heat operator which is strictly subelliptic with a loss of 1 derivative. There is a weaker notion of subellipticity. One says that P is subelliptic with a loss of r derivatives if co an open subset of H, and (Pm)I e jFP'(a>, loc) imply

a still weaker notion is semiglobal subellipticity.

One says that P is semiglobally subelliptic with a loss of r derivatives if M and Pu e loc) imply u e loc). All three of theseproperties are invariant under perturbation by operators of order t if t< m — r. All three properties are equivalent if 0 ^ r < p - 6 and in this case depend only on the principal symbol of P. This case is the one frequently considered in the literature and therefore the term subellipticity is used for any one of the properties above, or for variations on them. It is, however, useful to consider large values of r as well. For example, classical pseudo-differential operators of real principal type are semiglobally subelliptic with a loss of 1 derivative.

Semiglobal subellipticity with a loss of r derivatives is a very weak property when r is large. For example, if P(D) is any constant coefficient partial differential operator of order m then P(D) is semiglobally subelliptic with a loss of m derivatives. Nevertheless there is an even weaker property which it is useful to consider. We say that P satisfies the subelliptic estimate with a loss of r derivatives if for each compact subset K of ft and each 5 there is a constant Ck,s such that

for each with PueH^ and supp u ^ K . In Theorem 7.36 weshow that if P satisfies the subelliptic estimate with a loss of r derivatives and 0 < r < (p -6 )/ 2 then P admits a parametrix in wherep' = p - r and 6' = 6 + r. It follows that P is strictly subelliptic with a loss of r derivatives and is elliptic if r = 0. Moreover if P is a classical pseudodifferential operator, for example a linear partial differential operator, P is subelliptic with a loss of r derivatives in any of the four senses above and r< l/ 2 then P is actually elliptic.

The main point of Section 7 is to show that semiglobal subellipticity of the formal transpose P' gives local existence theorems and semiglobal Fredholm alternatives for P. Under additional hypotheses on P' or P one even obtains global existence theorems of Fredholm alternative type. The additional hypotheses needed to obtain theorems of this type for the space of all distributions is related to a subtle topological fact concerning L F spaces: namely, a closed subspace of an L F space may not have the same continuous linear functionals relative to its subspace topology as it has relative to its natural L F space structure (see Remark 7.30 and Remark 7.31). We indicate the connection between the results of Section 7 and the theory of P(D)-convexity and strong P{D)-convexity for constant coefficient linear partial differential operators P(D).

Criteria for subellipticity have been obtained by Egorov, Hormander and others. We do not discuss this problem. The purpose of Section 7 is simply to lay the functional analysis groundwork for part of Chapter 5 and to discuss the relations among some of the notions subellipticity


which occur in the literature. The terminology used here is not standard. Indeed there seems to be no standard terminology.

In Section 8 we discuss the Seidenberg-Tarski theorem and some of its consequences. The facts that the image under a polynomial map of a semialgebraic set is semialgebraic and that a semialgebraic subset of the plane is bounded by a finite number of pieces of algebraic curves have quite a few consequences for partial differential equations. In particular we use a Seidenberg-Tarski argument to give Hormander’s characterization of subelliptic linear partial differential operators with constant coefficients.

A proof of the Seidenberg-Tarski theorem may be found in Friedman[1]. The required facts concerning the boundary of a semialgebraic subset of the plane may be found in the proof of Lemma 2.1 in the appendix of Hormander [2]. Here too one finds a statement of the required fact concerning algebraic curves, the Puiseux expansion. A proof may be found in the beautiful exposition of Knopp [1] and also in more modern texts such as Hille [1].

§2. Sobolev spaces

Recall that we have a continuous inclusion of 5 in L^(IR”), and therefore by transposition, also a continuous inclusion of L ([R”) in 5 ', if 6 ' is provided with the strong topology. Recall also the pseudodifferential operator A"'= (1 — Thi s operator is a topological isomorphism of 5 onto Sf and of 5 ' onto 5 '. If s is a real number then we define the Sobolev space to be the image of L^(R") under the map A“ We provide H" with the Hilbert space structure which makes A""' an isometric isomorphism of L^(R”) onto H\ Thus if u ,v e H ^ their inner product is given by

(m I v)s = (A^u I A^v)

The inclusion of Sf in may be factored as

^ l H\

and therefore is continuous and has dense image. It follows that iscontinuously and densely contained in H\ Therefore we may view H" as the completion of CcCR"") relative to the inner product

( u I v\ = (2 i r ) - - ' I (1 + d eThe inclusion of H* in 5 ' may be factored as

■9”

2. SOBOLEV SPACES 229

and hence is continuous. Note that H* is just the set of u e 5 ' such that u € L '(R ", loc) and

llull? = (2ir)-’*| (l + |inMu(|)pd| <00.

Finally note that A' is a real operator, that is it commutes with conjugation, and moreover A* is an isometric isomorphism of H* onto

E xam ple 2.1. If s < —nl2 then 8 e Thus if = A~ 8 then Gt e for s < - n l 2 + t . The distribution Gt is called the Bessel potential of order t. It is clear that Gt lies in €'c and that the following properties hold; Go = 5, ( 1 - A)Gt = Gt_2, Gt^Gs = Gt+s, and A~ u = Gt^u for each u in Sf'. In view of Example 2:2.7 we have

G „+ i (x ) = 2 V_o —n _ —(n - l ) /2 . 1r((n + l)/2)

-1x1

A discussion of Bessel kernels and the corresponding Bessel potentials, that is the convolution products Gt*u, may be found in Adams, Aronszajn and Smith [1] and Donoghue [1].

L em m a 2.2. I f s < t then ^ fT' and this inclusion is continuous and has dense image.

Proof. If s < t then (1 + 1 1 )“" + and so and \\u\\ ^

L em m a 2.3. I f ueH ^ then ueH^^^ if and only if DjueH'' for / = 1 , . . . , n. Moreover, in this case

M ^.r = \\ur.+ i\\DM\lJ = 1

Proof. If u e H" then u is locally integrable, Dju{^) = i ju(^) and

d+iipr"' iu(i)i"=(i+i id' \u{0?+ z (i+iipri = i

C orollary 2.4. I f P is a polynomial o f degree m then P(D) maps continuously into

C orollary 2.5. // m > 0 is an integer then consists o f all functionsu e L^{W) such that D “u e L^(W) for each a with la| < m. I f u e then

M llm = Z|al m '

where

n —\ a / oc\

m l

(m —la|)!

E xercise 2.6. Let P = Y,a^D^, a « e C “(ft). Then P : C r ( f t ) C ~ ( f t ) may be regarded as an unbounded linear operator on L^(IR”). Let Q : Cc ^ Cc (n) be the form al adjoint o f P. Then the closed linear operator Pi = Q * is called the maximal extension or maximal operator o f P. Show that the domain o f Pi consists o f the functions ueL'^(W^) such that Pu, computed in the distribution sense, also is in L ((R"‘). I f we consider the maximal operator Oi = P * then the closed operator Pq = (Q i)* is called the minimal extension or minimal operator o f P, Show that Pq is just the closure o f P, where as above we regard P as having domain ( ). denotes the Hilbert space adjoint.)

R em ark 2.7. Let u eL \ R ^ ). If A ueL\ U ^ ) then A^m = (1 -A )m€ L^([R”) implies u g H^. It follows that is the domain of the maximal operator of A as an unbounded linear operator on L\R^). More generally let P g 0 [ zi , . . . , and suppose P(D ) is elliptic of order m. Then P(i^) = where Q and R are polynomials and |Q(^)|^Ciand 1 P ( | ) | ^ C 2 I t follows that

\ p m \ ^ ci

if 1^1>2c2/ci. Thus for some constant c > 0 we have

c-^ lP(i^)|"< (1 + < c(|P(i^)l + 1).

Thus if u g H then P(D)u e if and only if w € In particularis the domain of the maximal operator of P(D ) regarded as an unbounded linear operator on L^R"^). Note from the inequality above we obtain for u e

In the sequel we will obtain an a priori inequality of this type for elliptic operators with variable coefficients. We will use the existence of a parametrix to obtain the inequality. Another approach is to use the ‘Korn trick’. One first shows that if the coefficients are allowed to vary a little then the error made in ‘freezing’ the coefficients at a point can be absorbed in the terms appearing in the inequality. Then one attacks the general case by a partition of unity argument. In this part of the argument one has to commute the operator with multiplication by a function. This operation, however, also leads to an error of lower order which may be absorbed. We will use the a priori inequality to obtain a local existence

theorem for elliptic equations. We will also investigate the subelliptic estimates obtained by replacing m + s by m + s — r.

If ueSf' and <t>eSf we define

(u I <t>) = {u, < >.

The sesquilinear form (• | •) is then the usual pairing between distributions and functions except we have inserted a conjugation.

L em m a 2.8. I f u g H then (w | •) extends uniquely to a continuous conjugate linear functional on I f v e then

<u I u) = (A*m I A”"u)o = (277)"” I u(^)v{^) d|.

Moreover, we have

(A) \{u I v)\ < \\u\l llulU if u g W and v g(B) {u I A*w) = (A*m I w) if ueH ^ and w e

i f u G Wif u g H and v g H “*.

(C) (u\A^û) = \\u\\(D) {u\v} = {v\u)

Proof. The uniqueness follows since Sf is dense in If w g fT' then V {Aû I A~^v)o is a continuous conjugate linear functional on and

(A"u I A-"u)o = (27t)""(A^| aS ; ) l-

= ( 2 7 7 ) - ' ! u (| )i)(| )d e

In case u g 5 we have therefore

(Aû I A~^v)o = (27t)“”{m, v) = {u, v)

by the Fourier inversion formula since v = v' . Thus we have proved the first part. Then (A) follows by the Cauchy-Schwarz inequality since 11A''m1|o = Hulls. (B) follows since the transpose of A* is A* and since A' is a real operator. For (C) note if u g H then A u g and therefore

<u I A^*u) = (A"u I A*m)o = llu|l

Finally (D) is certainly true for u ,v € .9 ‘ and therefore in general.

T h eorem 2.9. The pairing <• | •) identifies H~“ isometrically with the antidual o f I f u e 3)'{R’') then u e W if and only if there is a constant C such that

l<M,<f.)|<Cll<f.lU /or<f>€Cr(R").

Moreover the best value o f C is ||u||s, that is !|u||j < C.

Proof. Let (H*)* be the antidual of that is, the space of continuous conjugate linear functionals on Define G : H~'‘ —> (H“)* by

[G (t;), u]s = {v I u) = I A"m)o = (A"^"u | u\

where [*, L is the canonical pairing on (jFT')*x H* and where the last equality follows since A”"' is an isometry of H^ = L^(IR") onto If is the canonical isometry of the Hilbert space onto its antidual (H^)* then the calculation above shows that

G = J,oA-2^

Thus G is an isometry of H""' onto (f f ') * . Since ( l + |-p)* is a real and even function conjugation is an isometry of Thus if u eH ^ then

Hull, = |1G (m)11 = sup {\{u, <f)>l/ll< ll_, I <f> e Cr(R")}.

Finally if ueQ}'(R'^) and |{m, < >)l<C||< )ll_s for each <t>eCc(U'^) then <l> {u, $ ) = (u\ (f)) extends to an element of (H""')* with norm < C Thusthere exists a unique such that [G(w), <>]s = ( m | <>) for each<t> e Cc ([R”). But then (w, <#>) = (u, 4>) for each 4> g C“(IR”), i.e., w = w.

Corollary 2.10. Let linear map. Then Aextends uniquely to a continuous linear map A :H^ ^ if and only if

\{Au, v)\ < C Hull, llulU, for M, u G C^(R^).

Moreover, the best value o f C is the norm llAjlse o f the linear operator

It is convenient to introduce also the intersection and the union of the Sobolev spaces We let

H~= n and H~^= U

We provide with the weakest topology so that the inclusion map jFT' is continuous for each s (projective topology). We provide

with the strongest locally convex topology so that the inclusion map is continuous for each s (locally convex inductive topology).

Note that

H~ = { m gL^(1R")|D“m gL^([R") for each a }

and that the norms

u 11d “u1|l2,

where a runs over all multi-indices, define the topology of Thus JT° is a metrizable locally convex space. If (Uk)k^o is a Cauchy sequence in H * then -> v in L^(lR") for each a. Since convergence in L ((R”)

implies weak convergence in S)'((R”) we have = D^Uq. Thus UqG H*" and Mfc ^ Vq in H “. Hence is a Frechet space.

We note that 5 is dense in H“. Indeed if w g H " for each integer m > 0 we may choose such that \\u — < 1/m. Then for fc < m we have11 “ mWk < 1/^ 2ind therefore w in for each fc. But then —> uin

With regard to the topology of H “~ we note that a seminorm on H~°° is continuous if and only if its restriction to FT' is continuous for each s.

E xercise 2.11. Hdense in

is the strong dual o f F T and ^ is sequentially

Note some authors denote by and by ^^2.We denote by C^(R”) the space of all functions u g C^([R”) such that

D “m vanishes at infinity for |al<fc. We set

O R " ) = n O R " ) .k>0

We topologize C«(R") by the norm

u max sup lD“ul

and provide C^(R") with the obvious projective topology. Then C^(R") is topologized by the norms

max sup iD^wl\oc\k fc = 0 , l , 2 , . . . .

Note that C^(R") is a Banach space and C“(R") is a Frechet space. Let S" be the unit sphere in R""^\

S "= {(x , 0 | x g R ", tG R , lx l^ + t^ = l } .

The stereographic projection from the north pole N = (0,1),

(x, t) -1 - t

is a diffeomorphism of S " - (N ) onto R" and identifies S" with the one-point compactification of R".

E xercise 2.12. I f /eC«(R") then f and all its derivatives extend to continuous functions on S". Show that the extension o f /, however, is in C“(S") if and only if feSf.

At first it seems curious that if / g C^(R") then f and all its derivatives extend to continuous functions on S", but in general the extension of / is not smooth. A short reflection, however, shows that this fact is not so

strange. The point is that the Cartesian coordinates on R ”, when transplanted to S'" by means of the stereographic projection from the north pole, are rather singular at the north pole N. If we choose good coordinates near the north pole, say by stereographic projection from the south pole S = (0, — 1), then the derivatives of / relative to these new coordinates are linear combinations of the D “/ with coefficients that blow up at the north pole. Hence, if / and its derivatives do not vanish fast enough at infinity then the extension of f will not be smooth.

Theorem 2.13. Sobolev. 1/ fc> 0 is an integer and s > k + n/2 then and the inclusion is continuous.

Proof. Let s = s — k — n/2 so e > 0 . Then (1 + 1*1 )“®“'" is integrable. Therefore if lal<fc then

\ e m \ = i n ( 1 + + \er'^ m )\< ( 1 + 1 u(i)i

is a product of functions and so is integrable. By the Riemann- Lebesgue lemma (Theorem 2:2.1) we have D “m gCoo(R”) and by the Cauchy-Schwarz inequality 1D “m(x)1<11 “m1Il i<C1|m||s where C =

Corollary 2.14. We have = {u e C^(U^) \ D^u e L^U^) for each a } , and Moreover, the inclusion is continuous.

Corollary 2.15. We have ^ H ““. Indeed if ue^'iW ') and u hasorder < k then u g for s > fc + n/2.

Proof. If u has compact support and order <fc then for some constant C we have

|(u, </>)!< C max sup |D“4> I for each < )g C“(R'").lal<fc

By Theorem 2.13 we then have \{u, </>)| < C' l|</)lls for each <l> g C“(R"") and therefore u g H “" by Theorem 2.9.

If m > 0 is an integer then Corollary 2.5 gives a description of H"" which does not depend on the Fourier transform. Theorem 2.9 then allows a description of H "”" which is also independent of the Fourier transform. These descriptions, aside from an appealing concreteness, are important since they permit the introduction of the Sobolev spaces k e Z , in situations where the Fourier transform is not available. We will now obtain a description of for any s, which does not depend on the

Fourier transform. In view of Lemma 2.3 it suffices to consider the case 0 < s < l .

For convenience if ueSf' and u is locally integrable we set

llullf = (2 r r r " | ( l + |^l^riM(^)lM|

where we allow +oo as a value for the integral.

L em m a 2.16. 7/ 0 :S s < l then

( i + i i p r ( l + a)*. If h > 1 and a = l lb then 1 + h" = a - ' ( l + a " ) > a “= (l + a)" = (1 + Thus l + |^p^>( l + ||p) On the other hand if a > 0 and s > 0 then a* is a non-decreasing function of a and therefore l l P ^ l + lll^ implies (1 + l^Pr and so l + l^ p s 2 ( l + l|Pn

T heorem 2.17. I f 0 < s < 1 and u e L^(IR") then u eH ^ if and only if

| u (x ) -u (y )p

In fact, there exists a constant > 0 such that

( 2 ^ ) - 1 II = A. II d . dv

Proof. The Fourier transform of M(x + y )“ M(y) as a function of y is 1). Therefore by the Parseval-Plancherel theorem (Theorem

2:4.3) we have

I lu(x + y) - u(y)p dy = (2rr)-''| |e‘«'^>- Ip |u( )p d$

Then by the Fubini-Tonelli theorem

jI dx dy = II|u(x-1- y)- w(y)P |xl“”"^' dy dx

= (2rr)-"|G(|)lM(|)pd|

where

G (|) = |le' Ip = dx

= 4 | l s i r ( ^ ) 'dx.

If t > 0 then G(t|) = P'‘G(|) and if ReO{n) , i.e., R is an orthogonal matrix, then G (R |) = G (|). Since 4 lx r ”'^'‘sin «|, x)/2) is bounded by llp lx p ”""^" for lxl:< l and by 41x1“"” ' for I x j s l and since 2 — n — 2 s > —n and —n — 2 s < —n we see that 1G(|)1<~ for each f Thus G (|) = l^P'‘/As for some constant A .

If 0 < r < l let Cc'(IR") be the space of compactly supported Holder continuous functions on R" with Holder exponent r, that is, functions w eCc(IR") such that for some C > 0 we have

lu (x )-u (y )l< C Ix -y T fo rx , y eR ".

In the case r = 1 the Holder functions are called Lipschitz functions.

Corollary 2.18. 1/ 0 < s < r < 1 then C‘' *(R") s H \

Proof. If u e Cc''XR") define

lu(x)-M(y)l lx - y l“"~" if I x - y j a l 0 otherwiselu(x)-u(y)| lx - y l“"“"' if l x - y l : < l 0 otherwise

hi(x, y) =

h2(x, y) =

Since wgL ([R”) and s > 0 we have hj g L^(R” xR"") by Exercise 2.19 below. Since u has compact support we have for some N that w(x) = 0 if |x|>N. Then h2(x, y )^ 0 implies jxj+ l,yl < 2 N + 1. Since s < r it follows by Exercise 2.20 below that h2G L^(R” x R ”).

Exercise 2.19. Let h be a measurable function on R'^xR”, let s > 0 and assume h{x ,y) = 0 almost everywhere for |x —y|<8. I f u g L^(R”), t > n l 2 and | (x, y ) l< C lM(x)-w(y)| |x-y|“* almost everywhere then h e L “(R” x R ").

A hint for Exercise 2.19 is to let x = u + w and y = v — w and then to change the integration with respect to w to spherical coordinates.

Exercise 2.20. Let h be a measurable function on R" x R ” and assume that h = 0 almost everywhere in the complement o f a compact set. I f t < n/2 and lh(x, y ) l< C jx —yl“* almost everywhere then hGL^(R” x R ”).

3. OPERATORS ON SOBOLEV SPACES 237

Remark 2.21. If A is a bounded linear operator on L^(R”) and (A*A)^ is compact for some fc> 0 then A is compact. This fact together with Exercises 2.19 and 2.20 above is very useful for establishing compactness of operators since an integral operator with kernel h e L ([R” x[R”) is a Hilbert-Schmidt operator and therefore compact.

We finish this section with a useful inequality.

Theorem 2.22. I f u g H^ then logOJuUs) is a convex function o f s.

Proof. Let a , b > 0 , 0 < f 1 and define g(r) = + (1 - Theng takes its minimum on (0, oo) at r = b/a. Thus g (r )> a ‘b^“‘. Taking a = (1 + and b = { l + and s = fSi + (1 - t)s2 we obtain

(1 + ^ g(r) = tr^-\l + rr* + (1 - O r - (1 +

which implies that for any r > 0 we have

llMll?<tr* 'l|ulg, + ( l - t ) r '11m'

Taking r = llMlgyi|uH?,we obtainIf.

Corollary 2.23. I f S i < s < $2, e > 0 and then

llwlls^e llu|0

where p = {s -S i ) l {s 2 - s ) .

Proof. If r > 0 , s = tsi + {l — t)s2 and u e S f then by the proof of Theorem 2.22

l l u l l ? < ( l - r ) r - l K + t r

< r ulP+r'-||Mie,

Now set r = 8— -i/t

§3. Operators on Sobolev spaces

In this section we collect a number of useful estimates concerning linear operators on Sobolev spaces. Quite a few of the arguments use Young’s inequality (Theorem 1:2.2) and therefore it will be convenient to introduce certain norms. If ueSf' and u is locally integrable we define

= (217)-" j (1 + llirlM(|)l d|

where we allow +oo as a value.

E xercise 3.1. I f s > r + n/2 then for each u g H \ I fm > 0 is an integer then N^(m) < oo implies ueC^{W^) and s u p 1D" m1< N^(u) for I f r < —n then N^(6)<oo.

T h eorem 3.2. Assume b is a measurable function on IR"" xlR ', m and N are real numbers and

m , T|)l ^ Cod + II - 'rj|)-"'(l + h f

I f N > n + \s — m\ then

(Bu)*(|) = (2-7r)“’’| 6(1, t7)u(t|) dij

defines a continuous linear map

and

1|BNCoN,„_,|_n(S).

Proof. By Peetre’s inequality (see comments preceding Theorem 2:5.15)

( 1 = (1 +|t)1 )<"'-" ' (i +

^ (1 +11 - T}|)'"'- '(1 + Ill )<"‘- >' (1 +

Thus

(1 +1|12)<— 1(|)| < (2)-'-Co j (1 + II -x ( l + lr,n^«lM(t,)ldr,.

Since the integral here is a convolution of an and an function the theorem follows by Young’s inequality.

T h eorem 3.3. Let p€ C~(R” x(R”), let m be a real number, and assume for each multi-index a there is a non-negative function g L (IR”) such that

\ D -p {x ,a^ g ^ (x )a+ \ ^ \ r

for X, ^ g IR”. Then p(X, —iD), initially defined on Sf, extends uniquely to a continuous linear map o f into I f u eH ^ and u g 5^ then

<p(X, -iD )u , v) = (2ir)-" j| e‘<’’ ">p(x, t7)u(r})u(x) dx dt].

Proof. Note that the double integral in the statement of the theorem is not absolutely convergent in general. It must be evaluated in the order indicated. Let B = p(X, — iD). U u e S f then

Bu(x) = (2'tt) ”1 rj)u(ri) dr]

converges absolutely. Thus initially we have B :Sf L^{W). Taking the Fourier transform we obtain

B u i O = ( 2 i r ) - ' ' | j t i ) u ( t |) d t j d x .

Since this double integral is absolutely convergent we may interchange the order of integration. Then

B m(|) = (2ir) " I Hi, t})m(t}) dt}

where

H i, il) = I 11) dx.

For any integer iV > 0 we have

b ii, I)) = ( l + I|-T}p)”' j [(l-A )'^ e‘<’ “®-*>]p(x, ■Ti)dx

= (1 +1^ - T7 I - A )'^p(x, T}) dx

(see Lemma 1:14.1) and therefore

1})1 —C'N(l + l|--»]f)“'^(l + |Tj|)'"< c m + II - -nD-^'^d+

Thus the first part follows by Theorem 3.2. For the last part if u e and veS^ then

{Bu, v) = (2tt)~"{Bu, H )

= (2ir)'^"II H i, tj)m(tj)u(-|) dr} d|.

As in the proof of Theorem 3.2 the integrand is bounded by

Therefore the integral is absolutely convergent. Interchanging the order of integration and inserting the definition of b we have

This triple integral in general is not absolutely convergent, but the inner double integral is absolutely convergent. Interchanging the order of integration relative to x and and using the Fourier inversion formula we have

<jBm, u) = (27t)~” JJ t])m(t])u(x) dx dr]

as required.

Corollary 3.4. I f f l is an open subset o f R", <f>, if/e 0 (X)Pi/f(X) maps fp' continuously into

Proof, Indeed, <#>(X)Pi/f(X) = </)(X)p^(X, - iD ) where p e IR”)-

The continuity of pseudo-differential operators of type (p, S) on Sobolev spaces when 6 > 0 (and 6 < p) is more difficult to prove. We will consider this problem in the next section.

Theorem 3.5. I f 4> e l f then multiplication by <{> maps jFP' continuously into H\ Moreover,

llwlls

for each u e

Proof The first part is just a special case of Theorem 3.3 but we want the estimate as well. If ueH ^ and <l)eSf then

^ = (27t) " " m * = (27t) " ”<u,

Since u lies in a suitable space (see Caution 2:5.7) we have

5 m(|) = (27t)"" j < (|- 'n)M(T]) dr].

Then by Peetre’s inequality

(1 + < (27t)-" j (1 +1 - r, 1)'' - T})1x(i+l'nP)^'^lM('n)|d'n-

Now Young’s inequality finishes the proof.

Corollary 3.6. I f <t>GSf, ueH ^ and xeW^ then

- <#>)wll, < 1x1 N|,|+i(<#>) llwll,.

Proof. Let — </>. Then = — implies

|^(^)1<2 Isin « 4 x)/2)\ 1< (|)1<| 1 |xl

T h eorem 3.7. Let Then ueH ^ if and only if there is aconstant C > 0 such that

- u\\s-i C |jc| for each x g IR”.

Proof. First suppose u g H . Then

\{t u - u) (1)1 < 2 jsin «^, x)/2)l |u( )| < jxj (1 + lw( )l

implies

|1t, w- m1Ui < 1x1 Hull,.

Conversely, suppose u € and Ht m — uHs_i < C 1x1. Let e, =( 0 , . . . , 1 , . . . , 0) be the jth vector in the standard basis of R ”. If 0 < f < 1 then

is bounded by l|j| and converges to |§| as t^ O . If R > 0 then

(27t)' "I•'ISISR( 1 + m2 \ s - l f - 2 U-i<eie,.>_ i l 2lplu(|)l^d|

< t 11t„.m- m1Ui < C ^

If we let t 0 through a sequence then by the dominated convergence theorem we obtain

( 2 7 7 ) - " f ( 1 + \ey- i § w ( ^ ) i ^ d ^ < c "

If we let R'foo then the monotone convergence theorem implies that DjUefT~^ and 11D,m||s_ i < C Since we also have mgjFT'-’ it follows by Lemma 2.3 that u e H \

C orollary 3.8. Translation in is continuous, that is, if u e FT' then the map x :R " ^ FT is continuous.

Proof. If yGR" then Ty is an isometry of Thus Ht u —Tyujls = ||Tx_yM - m|1s and therefore it suffices to prove continuity of translation at the origin. If s > 0 choose <f>eSf such that ||<#>- w|ls< e/3. Since < >GjFT'' we have ||t </>-< |1s < C jxj. Thus

IkxM - mIL 1Itx(m - <f>)\l+llTx<> - 4>\\s+11“ - <>1L<28/3 + C1x 1< s for small lx|.

As a special case of Corollary 3.8 we obtain continuity of translation in L^([R”). Alternatively we may use the continuity of translation in L ([R”) and the isometries to conclude continuity of translation in

If </)6C“(1R”) then the commutator A*</>(X)-c/)(X)A* is in We might expect therefore that if (f>eSf then this commutator maps continuously into This fact is Friedrichs’ lemma, which we willnow prove. First though we will need to obtain some preliminary estimates.

Let t be any real number and a, b > 0. By the mean value theorem

l(l + a^)‘ - ( l + b )*l = 2 \v\ jtl \b-a\

for some v between a and b. Since + and + ismonotone for u > 0, the right side above is bounded by

2 M lb - al max [(1 + (1 +

Thus for 1, 17 g [R” we have

< lt| h i max [(1 + (1 + I I -

By Peetre’s inequality we have

(1 + < (1 + It, l) '- 'l( l + II -

and trivially ( l + l|-T,p)‘‘“ ' ^ has the same bound. We therefore obtain if t e U , I, T, elR" then

I d + ( 1 + II - n m ^ lf| ( 1 + |t,1 ) '-» '" ‘ ( 1 + I I -

Theorem 3.9. Friedrichs. If<t>e£f then A'<#>(X)-<#>(X)A' maps W continuously into 7/ r = lt —Ij + js —t + lj + 1 then the norm o f thisoperator is bounded by lt| Nr(<t>).

Proof. If ue77* let u = (A'<f>(X) —<#>(X)A')m. Then v g Om and

tK i)= ( 1 + iip)‘'^ ^ ( i ) - (</> A 'urci)

= (2TT)-" I [d + + I I - T,p)'«]u(|- dr,.

Since (1 + ||R<^-'^'>«<(1 + 1| - t,| T “'^*'"(1 + 1t, 1)' -‘^" we obtain from the inequality preceding the statement of the theorem

(l + ||12)(.-,.t)/2,.(|)l

< ( 2 1 7 ) “ " I t l j ( l + l | - T j p ) " ' ^ l u ( | - t , ) l ( l + l-rjiy 1<^(t j) 1 d r .

The theorem now follows by Young’s inequality.

Exercise 3.10. By applying the mean value theorem to the function h(t) = (t^+l^ + t'np)* show that

for s e R , t] and \ \> 1. Conclude

for uniformly for <f> in a bounded subset o f 5 .

If A is a closed subset of R" we define

H a = { w g H® I supp u c A}.

Then is a closed subspace of and in particular is a Hilbert space.

Lemma 3.11. I f K is a compact subset o f R ” then the Fourier transform is a continuous linear map o f H k into C“(R''). Indeed if u e H^ then

10|u(|)l < C k,„,.(1 + l|u|L.

Proof. C hoose^ g C“(IR' ) with </> = 1 in a neighbourhood of K. If u G H k then u = (f>u = (27t)~"< * u. Then by the Peetre and Cauchy- Schwarz inequalities we have

(l + llPr/^lu(|)l

< 2'^''^(2ir)-" I (1 + I I - 1<^(|- T,)l (1 + h p r 1« (t})1 dr,

Since D “u = (27t)”''D“<^*m = (27t)“”((— * w we now have

(1 + lD“u(|)l < 2' '« llx“<#.ll|,| IImIL.

Theorem 3.12. Rellich. I f K is a compact subset o f and s < t then the inclusion map H k H is a compact operator.

Proof. Let g HJc, Wu Wt fc = 0 , 1 ,2 ,___ It will suffice to showthat the sequence (ujJ has a Cauchy subsequence in Since C"(R'') is a Montel space (theorem of Arzela and Ascoli) Lemma 3.11 implies that stays in a compact subset of C^CR' ). Since C CR"") is metrizable, by passing to a subsequence we may as well assume that w in C“(R”). Let jR > 0 . If veH ^ then let

Ii(t;) = (2 i7 )-"[ ( l + l|p)Mi3(|)|"d|*’|€|s R

< Cr sup lt3(|)ll€l R

and let

i ^ i v ) = (277)-" f ( 1 + m ) \ ^- €1>R

Let s > 0. Note hiuj, - u ) ^ (1 + \W ~ hWf 4C^(1 + R y ~ \Hence, we may choose R > 0 so large that l 2(Wk — < s^/2 for all k andh. We fix such an R. Then the estimate for Iiiu^ — Uh) shows that if k and h are large enough then Ii{ui — Uh)<s^l2 since is a Cauchy sequence in C(R"). Adding the estimates for I i and I 2 we obtain \\uu,-Uh\\s<s for large k and h.

Example 3.13. Here is a variation on Friedrichs’ lemma and also on Lemma 3.11. Let peS'^^oiW) where p > 0 . Thus

Then clearly p{—iD) maps continuously into for each s. Let<t>eSf and consider the commutator [p(-zD), <#>(X)]. If wefT' and v = [p (-iD ), <#>(X)]m then

o(|) = (2-7r)“"I (p(i) - p ( i - v ) ) u ( i - v)<P(v) drj.

By the mean value theorem

P(l) - P (^ - il) = Z D jp(^-r\ + tv)Vi

where 0 < t < l . Thus

1p (I) - p ( | - Tj)| s C ( 1 + l ^ - n i r - ^ d +

and therefore

(1+iiir-'""'’ ii5()i < c j (1+11 - T,i)dud - T7)i (1 +i-nir i<(t,)i dt,.where r = lm -p l + ]s — m + pl + 1. Applying Young’s inequality we obtain l|u||s-m+p—0 Hulls whereas applying the Cauchy-Schwarz inequality we obtain lu(|)l^C Hulls ( l + l l)"’”* '”. Consider now a second commutator. Let and let u = [[p(—iD), <#>(X)], «/r(X)]u. Then

V (e = (277)-^" I |(P (€) - P(| - T1 + U,) - p(| - O)) + p(| - t,))

X u(g — — (o) dt] do).

Applying the mean value theorem three times we obtain

lp (l)-p (^ -T ) + <o)-p(|-w ) + p (| -il)l< c(i+II - Tj +h -

Therefore

(1 + 111)— l£(|)l < c| (1 + II- T,!) 1«(|- Tj)l e ( v ) dr,e

^ ( 17) = I ( l + h - c o i y 1< ^ ( t) - co) 1 ( l + l o ) i r " ^ \{(o)\ do)where

andr = |m — 2p| + |s —m + 2p|+1.

Applying Young’s inequality we have ||u|ls_ +2p — C |1m|1s whereas applying the Cauchy-Schwarz inequality we have |u(^)|<C |1m|1s +Continuing in this fashion we can show if <#>y g 5 and

T i = [ p ( - f D ) , <^>i]

T/+1 = [TJ, </)j+i]

then

ll7;-Mll,_^+yp<Cy Hull,

and

As an application we can now obtain estimates for (p(-iD)w)"(^) = p(l)w(|) as in Lemma 3.11 but without assuming compact support for p{—iD)u.

T heorem 3.14. 1/p g S^o(R”), P > 0 , u g ^'(R"), <t>eC7(U"), = 1 in a neighbourhood o f supp w and (f>pi—iD )ueH ^ then p{—iD )ueH ^ and 1p (^)w(|)1:^C(H-1^1)“\ In particular if p {-iD )u e C^{W) then p(—iD)uGse.

Proof. Indeed, since u e^ 'iW ) we have u g H for some s. Making s smaller if necessary we may assume that s - m + fcp = t for some integer fc > 0 . Set <l>j = <t> for each j in the definition of T) above. For convenience set To = p{—iD). Since TJw = T)<#)m we have

TqU =<t>ToU + TiU TiU =(f>TiU + T2U

Tk_2M = <#)Tk_2M + Tk_iM Tk_iU = <f)Tk_iM+ TfcW

Since <l>ToU e and so ct>(f)ToU e the first equation implies </>TiU g L ikewise from successive equations we obtain <f>TjUeH for 0< 7<fc. Since u eH ^ we have Tj u e Since <l>Tk-iueH^ the lastequation now implies that Tfc_iWGH*. But then since < >Tk_2W6 H* the second last equation implies Tfc_2U € H\ Working backwards we see that TjU G H* for 0 j ^ fc. Since ueH ^ and m — s — fcp = —t we know that |T)^(^)l<C(l + lll)~^ But <#)Tk_iMGH* has compact support and so by Lemma 3.11 \(<t>Tk-iu) C (l + lll)“*. Thus, from the last equation we obtain |(Tk_lM)"(^)l<C(l + l l)“ Working backwards through the equations we obtain 1(TJm) (| )1< C (1 + ||1)”* for 0 < 7<fc. For the last part if p(—iD)u G C“(1R”) then the first part implies that p(^)w(^) is rapidly decreasing. But this function is clearly in and so by Exercise 3:2.11 is in Sf. Thus, p(—iD )u e^ .

E xercise 3.15. Fill in the details in the argument in Example 3.13 concerning the operators T) for 7> 3.

R em ark 3.16. In Corollary 3.4 we obtained the continuity on Sobolev spaces of pseudo-differential operators of type (p, 0). We will consider the case of operators of type (p, 6) in the next section, but first we note that even in this case the methods of the present section do give some continuity results. We will make use of this observation in Remark 4.12. Fix s and m. Let K be a compact subset of IR”. Choose an integer N so that 2 N > n + \s — m\ and let fc<m —8N. Let B be a bounded subset of

R'') such that if p e B then p(x, ) = 0 if x ^ K . ClearlyS .8'

\D tp{x,e\^g^ix)ii+\4\r

if \a\^N. Here g„eQ (IR"). If

H i, il) = I T]) dx

then as in the proof of Theorem 3.3

\Hi, T))i ^ C od + I I - T,i)-"^(i+

where Co is estimated by derivatives of p and hence may be chosen independent of p g B. By Theorem 3.2 it follows that p(X, - iD ) maps continuously into with norm bounded by C i Cq where Ci is aconstant depending only on m, s, N and n. Thus p(X, —iD) is a bounded set of bounded operators of into as p runs over B. Note when8 = 0 we may take k = m ; otherwise fc < m, a deficiency that we will remove in Section 4.

4. CONTINUITY 247

§4. continuity of pseudo-differential operators

Let ft be an open subset of IR”, P g ' ^ sCft), </>, € C~(ft). If 0 < 6 (X)Pil/(X) maps continuously into A simpler proof which makes elegant use of the symbolcalculus was later given in Hormander [3]. It is this proof that we will present below. A more difficult theorem of Calderon and Vaillancourt [1] allows provided that 8 < 1 .

Before proceeding to the result we actually prove we will give a short and rather informal discussion of the reduction of the Calderon- Vaillancourt theorem to the case of operators of order 0 acting on and then a brief indication of how Calderon and Vaillancourt handled this case. We follow the presentation of Unterberger [1], but omit all the hard facts, and simply provide a sketch of the ideas. Let 0:^ 6, p < 1. We define BS^s to be the subspace of C°°(IR” x[R”) consisting of functions p such that for each pair of multi-indexes a, j3 we have a constant 3 such that

|D|D?p(x, 1)1 < C„,3( l + 111)—

for all X, ^ g R ”. We topologize BS^s by the seminorms

Iplit = max max sup (1 + 1||)— lD|D?p(x, |)|.lal<k |3l h

These seminorms also depend on p and 8 but the notation is bad enough without indicating this additional dependence.

Exercise 4.1. BS^s is a Frechet space.

The topology of BS^s is not very convenient because the analogue of Lemma 1:19.4 is false. Therefore we introduce a weaker notion of convergence. A sequence (or a net) in BS^s is said to converge loosely to p if Pk stays in a bounded set in BS^s and if Pk converges to p pointwise. As in Lemma 1:19.4 if pk p loosely then Pk p in C ”(R '"). Note that if p g BS^s, i/r g (R^”), i/ (0, 0) = 1 and p (x, ) =i/f(sx, 8 )p(x, ) then p^-^ p loosely as e 0.

Exercise 4.2. / / p g BS^s then p(X, —iD) maps Sf continuously into Sf. Indeed if 1|*|1 is a continuous seminorm on Sf then there exist non-negative integers k and h and a continuous seminorm IHl' on Sf such that

iip(x,-iD)uii<ipiirh’ iiMii'for p G B S 5 and u eS f.

Suppose now that Pk g BS^g and Pk ' p loosely. By Exercise 4.2 if

u € 5 then Pk(X, —iD)u remains in a bounded set in i f . Since

Pfc(X, -iD )u (x ) = I ) m(I)

the dominated convergence theorem implies that p^(X ,-iD )w -» p(X, —iD)u weak* in Sf'. Since 5 is a Montel space, the weak* topology of Sf' restricted to a bounded subset of Sf agrees with the topology of Sf. Thus Pk(X, —iD)u —» p(X, —iD)u in Sf. This fact is important since as we saw above we may approximate p e BS^s loosely by p € C~

An important property of BS^s is that if 6 < 1, p g BS^s and q e BS'^s then there exists such that p (X ,- iD )q (X ,- iD ) = r (X ,- iD )on Sf. If p and q have compact support then one can write down an integral for r. At this point it is probably best to pass to ‘multiple’ symbols as in Section 3:4, but roughly what one does is to make a delicate integration by parts argument to show that r depends continuously on p and q when p and q (having compact supports) lie in bounded sets. Then the general case follows by continuity.

Suppose now that peBS ^s where 0 < 6 < p < l and 6 < 1 . Then p(X, - iD ) maps H* continuously into One shows this fact bywriting p (X ,- iD ) = A" ""A"“X X , F r o m the discussion above we know there is reBSp^s such that r(X ,—iD) = A^~"^p(X,-iD)A~\ Thus we are reduced to showing that if rsBSp^^ 0 < 6 < p < l and 6 < 1 then r (X ,- iD ) extends to a bounded linear operator on L^([R”). This fact is the theorem of Calderon and Vaillan- court [1]. The theorem moreover gives that if \\r{X, —iD)|l is the norm of r(X, - iD ) as an operator on then

llr(X ,-iD )| l< C l4^ U

where k and h are integers satisfying 2 k > n and h { l — d )> n and the constant C depends only on n and 8. In view of the discussion following Exercise 4.2 above it suffices to prove this estimate for reC^iU'^*^). The proof of the estimate proceeds by an integration by parts argument to write r(X, — iD) as an integral of operators whose norms may be estimated and then using a modification of a lemma of Cotlar to estimate the norm of J d .

We turn now to the argument of Hormander in the case 8 < p , This proof is based on the possibility of forming a squcure root of an operator with positive symbol suitably bounded below.

T h eorem 4.3. Let P €^ ^ s(fl), 0 < 6 < p < l , l< fc< oo, and p e (R”) and assume

(1) P — p(X, —iD) G that is, p is a principal symbol for P.(2) P -P ^ e^ ^ ^ sm —k ( p —(

4. CONTINUITY 249

(3) For each compact set K c Q , there are constants Tk s 0 and Ck > 0 such that

Re p(x, > Ck(1 +1^1)'” if x e K and HI > r .

Then there exists a properly supported operator B e such that

Proof. Modifying P by an element of we may assume P isproperly supported. Modifying p by an element of IR") we mayassume that in (3) rK = 0. Let p(x, i ) = p(x, so that P * - p ( X ,—iD )e

Since fc > l it then follows from (2) that p - p e R"). Thus p = a + ia' where a and a' are real, a € S ”g(ft, R"),

and a(x, |)>CK(1 + H|r for x e K and |eR ". Let bo = a ' . Then D|D?6o is a sum of constant multiples of products of the form

a ’'^(a"’D | 'D f a) • • • (a - ’D|"Df "a)

where a ' + , = a and P ' + , . . . , + /3" = |3. Thus S^8^(fl,R").Choose Bo€T^^8^(fi) properly siyjported with B(,—boiX,-iD)e'^~°°i£t). Then the symbol of B * B q is bobQ=bobo = a modulo R").Since R") we now have

R , = P-BiBoe'¥'^p<^-^\Cl).

If k = 1 we are done. Suppose fc>2. Since R j is properly supported we have R i = r i(X ,- iD ) for some ri € S^g R"). Clearly R * - R i =p * _p g ^m-fc(p-8)^^) Since fc> 2 it follows as before that Im r i€ 5 m-2(p-8)(fi R"). Since bo(x, ^ ^ CJc(l + for x e K , and D^DîrJbo) is a sum of constant multiples of products of the form

( D f D f r i ) h o ' ( h o b o ) , • • •, (bo'D|'"D?'"f)o)

where a ' + a " + , . . . , + a"' = a and |3' + ^ " + , . . . , + /3"' = |3 we see that if bi = ribo' then bi e R") and Im bj € R").Choose B i € properly supported such that B j - bi(X, -iD ) e

Then B j B q and B qB i both have symbol bobi = 2''i modulo ^m-2(p-8) j jjn) Since bo is real we have B q- B * a n d since ImbieSî^-^'^-^XaR") we have B^-Bê^'î^-^^’'‘-^\a). Thus

R 2 — R-i ~ B *B q~ B *B i — B*Bj= (Rj — BiBo~BoBj) + (B] — B*)Bq + (B0 — Bo)Bj — B*Bi

is in Substituting R i = P - B * B o we have

= P - (B * + B f )(Bo+ B i) e >p-g-^< -®>(a.

It is now clear that we may inductively construct By

properly supported such that

\ch ' \ch /l<h / V<Kfor h < k. If fc = oc we take B to be an asymptotic sum Xy

Before proceeding note if jR g and C“(fl) then by Corollary 3.4 <l){X)Ril/(X) is a continuous (indeed compact) linear operator from to H* for any s and t.

Theorem 4.4. Girding's inequality. Let P e 0 < 6 and e

Then for each s g R, each e > 0 and each compact subset L o f Cl there exists a constant C such that

Re (Pm I u) > - s HmH/2- C ||m||

for each u g C*(H) with supp u ^ K .

Proof. Note that Re {Pu | u) = J((P* + P)u | u) for each u g C"(ft) and that the symbol of 5(P* + P) is Re p(x, ^) modulo R”). Hence,we may assume p is real, p ^ 0, and P * = P Then the formally self-adjoint operator P + sA" has principal symbol p(x, ) + 6 (l + ||p)"' which is bounded below by 8(1 + H P ) T h e r e f o r e by Theorem 4.3 there exists a properly supported operator B g T^ s (ft) such that

P + 8 A " '= B *B + R

where j R G ^ “ ~ ( f t ) . If u eC ciC l) it follows that

(Pu I u ) > - e l|Ml|/2+llBulg+(RM I m).

Choose <t> G C~(ft) with <#> = 1 in a neighbourhood of K. If supp u ^ K then \(Ru\u)\ = \(R<t>u\^u)\^\\<l>R(l>u\U,\\ul<C\\uf, since {X)R{X) maps continuously into

We will return to a discussion of Garding’s inequality in Chapter 5. We will now show that the technique used to establish Garding’s inequality may also be used to prove the continuity of pseudo-differential operators on Sobolev space.

A continuous linear map P : C“(ft) S)'(ft) is said to be compactlysupported if its support (see Section 3:3) is a compact subset of f tx f l . Note if P is properly supported and 6 g C”(ft) then 6{X)P and P6{X) are compactly supported.

4. CONTINUITY 251

Suppose is compactly supported. Then we may choose<#>, i/f G C“(ft) such that P = < >(X)P = Pi/r(X) = <#>(X)Pi/f(X). In particular if pU, = < >(x)e“‘^’''^P(e‘^ ‘V )U ) then P = p(X, -iD ). Clearly p g

Thus we may regard P as a pseudo-differential operator on IR” and moreover we see that P admits a symbol vanishing for x outside a compact subset of O.

Suppose now that P g '4 s([R”) is compactly supported. Choose a symbol p(x, I) for P vanishing for x outside a compact set. Then for some constant Mi > 0 we have

Let M > M i. Then the operator —P *P has principal symbolM^Cl + l^P)"* — 1p (x , 1)1 which is bounded below by (M ^ -M f)(l + |^p)'". Therefore by Theorem 4.3 there is a properly supported operator B e ^^s(IR") such that

- P *P = B * B + R

where If ueC^(W ') it follows that

l|Pulg<l|Pulg+l|BMlg = M l|M|l -(Pu I u).

Choose (/feC“(IR") such that P = Ptp{X) and replace u by in the inequality above. Since <Ci HulU (Theorem 3.5) and \{Rilm | »/>u)l< C2ll lPm (comments preceding Theorem 4.4) we obtain

\\Pu\\o C\\u\L

for each m g Cc(1R”). Thus P extends to a continuous linear map of fP ” into Since is continuously included in and P mapscontinuously into itself, the extension of P from C“([R”) to jFP must agree with its action on Si'ClR"").

Theorem 4.5. I f 0 ^ 6 , il/eCciU'^) such that P = (l)P(f) and i/f = 1 in a neighbourhood of supp <t>. Then = < and therefore

P = M " ”'A '“"P

where S = ilfA ~' P(l> and R = — Here for conveniencewe have denoted the operators (X) and i/ (X) simply by <l> and i/f, respectively. Now S g T psCR"") is compactly supported and so maps continuously into by the comments preceding the theorem. Then

<#>A'” ’'S maps H'' continuously into If we write R in the form

then we see by Theorem 3:5.6 that R and that a symbol for Ris given by the asymptotic sum

</>( ) - 1 ^ D^i<f>ix)(l + 1|P)0D;(^(x)(1 + m - ' )a O i l

where r = {m — s)l2. But if a^O then D “i/f vanishes in a neighbourhood of supp . Thus jR It follows that R P 6 ^ ““(IR”) and thereforeij/RP(l) maps continuously into H* for any t.

R em ark 4.6. The proof given for Theorem 4.5 is perhaps overly fussy. It is necessary, however, to keep in mind that A* is not properly supported unless t is a non-negative integer. Thus while it is clear that T = — has order —oo our calculus provides no assertion concerning R = since neither operator is properly supported. Also wecannot apply Theorem 3:5.6 directly to R = <t>A' ~ {l — il/)A ~' since 1 — iJ/ does not have compact support. Finally it is tempting to apply the comments preceding the theorem directly to the operator A ~" P<j>, but unfortunately this operator is not compactly supported.

C orollary 4.7. I / 0 < 6 < p : < l , P € and P is properly supported,then for each compact subset K o f [I the operator P maps continuously into

Proof. Choose 0 e C~(fl) with 0 = 1 in a neighbourhood of K. Then P0{X) is compactly supported and P6(X)u = Pu for any distribution u with supp u ^ K .

C orollary 4.8. I f 0 < 6 < p < l , jRe^^gClR”), m < 0 and R is com pactly supported, then R is a compact operator on L^(IR”).

Proof. Indeed R maps L ((R'") = H^ continuously into for some compact set K. Since m < 0 the inclusion map is compact byRellich’s lemma.

Corollary 4.10 below is a more precise version of Corollary 4.8.

T h eorem 4.9. I f 0 : ^ 8 < p ^ l , P e P is compactly supported, p(x, I) is a principal symbol for P, M is a constant and

lim sup sup 1p (x, )1 < MK

4. CONTINUITY 253

for each compact subset K o f U"' then there exists a compactly supported operator 1 such that

l\Pu-Rul\o^M\\u\\o for each u g L^(R”).

Proof. In view of the continuity of the operators it suffices to consider u g C“([R''). Since there is 0 g C“(W ) such that 0{X)P = P we see that 0(x)p(x, $) is also a principal symbol for P. Hence we may assume \p( , — for all x g IR"" and for 1 1 for some r^O. If M i<M2< M then by Theorem 4.3 there is a properly supported operator BiG'^psClR'") such that

M i - P * P = B ? B i + Siwhere Si g '4 "“(R"). Since P is compactly supported we may choose 6 G Cr([R”) with 0 < 0 < 1 such that Pd(X) = P. Then

M le { X f - P *P = B *B + Swhere B = B i6 {X ) and where S = 0(X )Si^(X) is compactly supported. We note

S m(x ) = I G(x, y )w (y )d y

where G g C“([R” x R "). Now |lPu|g<|lPulg + llBulg = M i ||0M|g-(Sw | w) implies

WPuWl^Ml l|u|g+llSu|lo Hullo for each u g C r ( R ” ) . (*)

Choose il/eCciW ') such that i/r(0) = l , and 0 (0) = Ji/r(x)dx = 1.Then set <> = * i/r''. Since il/(x) dx we have = Notealso that J <f>{x) dx = < (0) = liA(0)l = 1. Let s > 0 and set <l>(x) = s ”''<f)(8“^x) and then define

jR u = P(u*<#>J.

Since u u * is a smoothing operator we have g It is clearthat Bg is compactly supported. Since

(u - u * = u(^)(l - < (e ))

we have

|(u - u *<;)3H^)1<1u( )|

and therefore by Parseval-Plancherel

h-u*4>^\\o^\\u\\o.

Therefore replacing u by u-u*4>^ in (*) we obtain

\\Pu-RM\I^MI\\u\ + \\T M \o \\u \\o

where

T^m(x ) = S ( u - m *<#>J(x ) = I Ge(x, y )u ( y ) d y

where

G^(x, y) = G(x, y ) - | G(x, y+ ez)< >(z) dz

= I (G(x, y) - G(x, y + ez))c^.(z) dz.

Applying the Cauchy-Schwarz inequality and integrating we obtain

If 0 < 6 ^ 1 then Gg has support in a fixed compact set and by the mean value theorem y)| <C 8. Thus

\\Pu - Re«!lo (M i + Ce) llwlg

which completes the proof since M2< M

Corollary 4.10. I f 0 < 6 0. By Theorem 4.9 there exists R e compactlysupported such that 11P - jRH<8. By Corollary 4.8, P is a compact operator on L^(lR''). Since the compact operators form a closed set in the operator norm topology it follows that P is compact.

Remark 4.11. There is also a converse to Theorem 4.9 (and therefore to Corollary 4.10). See Seeley [2], Kohn and Nirenberg [1] and Hormander [12].

Remark 4.12. Let X, Y and Z be Frechet spaces and suppose that we have a continuous inclusion map of Y into Z. Let /: X -» Y be a linear map and suppose that f : X - ^ Z is continuous. Suppose > x in X and fi^k) y in y. Then /(Xfc) /(x) in Z and /(x^) ^ y in Z. Thus y = /(x) and so /: X -> Y has closed graph and therefore is continuous. Let 0 < 6 n + |s - m - 8N\. Let <f>, e C"([R''). By Theorem 4.5 we have a linear map

^P,8V (*)

5. LOCAL SOBOLEV SPACES 255

given by mapping p to <^>(X)p(X, -iD)i/f(X). Here L(jF^ is theBanach space of bounded linear maps from to Suppose nowthat p varies in a bounded set in S^s(l

<^>(X)p(X, - iD )^ (X ) = q(X, - iD ) where

r ) . Then

qU, ) = (27t) -n j< /) ( x ) p ( j C , | + T ) ) l ^ ( T ] ) d i r j .

By the estimates in the proof of Lemma 3:2.7 q varies in a bounded set in Moreover q{x,^) = 0 for x outside supp< >. Therefore by

Remark 3.16 q {X ,—iD) varies in a bounded set in L(H^ ByExercise 1:7.1 it follows that

[R") L(H^ : p q(X, - iD ) (**)is continuous. But then (*) is continuous by the closed graph theorem as above. Explicitly we have shown that if B is a bounded subset of S^8([R”, DR”) and <f>, if/e C”([R”) then for each s there is a constant such that

ll< / > p (X ,-iD )^ u lL _< C Jlu lL

for each u g H" and each p g B.

§5. Local Sobolev spaces

The Sobolev spaces are not very convenient when considering pseudo-differential operators defined just in an open set ft in R ”. Therefore, we introduce the space Hc(ft) of distributions in with compact support in ft and the space LT'Cft, loc) of distributions on ft which are locally in These new spaces are very convenient and may easily be introduced even on manifolds. They are, however, not Hilbert spaces. Considering the wealth of properties peculiar to Hilbert spaces, or at any rate most naturally expressed in a Hilbert space setting, this fact is undoubtedly a disadvantage. On compact manifolds, however, these spaces coincide and are Hilbert(ian) spaces. See Palais [1].

There are certain Hilbert spaces H*(ft) and Ho(ft) analogous to We will encounter these spaces later (for integral s) in connection with the Dirichlet problem. These spaces, however, do not have good invariance properties. In addition the pseudo-differential operators that we have considered do not act on jFT'(ft) spaces in a nice way because we do not control the growth of the symbol p(x, ) as x tends to the boundary of ft or to 00.

Let ft be an open subset of R ” and set

H^(ft) = H^nr( f t ) .

We provide H liil) with the strongest locally convex topology such that the inclusion map is continuous for each compact subset Kof ft. A seminorm H*|l on is continuous if and only if for eachcompact subset K of ft there is a constant such that

for each u e H l

Since the topology of may be defined by considering a sequence ofcompact sets increasing to ft we see that Hl(Q) is an LB-space. In particular it is a complete Hausdorff non-metrizable locally convex space. Each H k is a closed subspace of Hc(ft) and a subset B of Hc(ft) is bounded if and only if B is a bounded subset of H k for some compact subset K of ft.

Exercise 5.1. Let (i/ k)k>o be a partition o f unity in Cc(ft). I f a = (t k)k o is any sequence o f non-negative integers define

I|mIL = Z UkuWsk

for u eH K Q ). Then the (uncountable) fam ily o f seminorms the topology o f Hl(Q).

defines

Lemma 5.2. The inclusion map o f C"(ft) in HJCft) is continuous and has dense image.

Proof. Let K be a compact subset of ft. Then C k ^ H k and H k ~ Hc(ft) are continuous. But the continuity of C k Hc(ft) for each compact subset K of ft implies the continuity of C“(ft) ^ Hc(ft). If w € Hc(ft) choose i/f € C“(ft) such that ij/u = u. Now choose Uj^eSf so u ^ ^ u in Then ij/Ut ~ \\fu = u in If K = supp il/ then i/fWk -^ u in H k and so inH J(ft).

Exercise 5.3. 1/ m > 0 is an integer we have a continuous inclusionC^(ft) ^ H^(ft). I f k > 0 is an integer and s > fc + n/2 we have a continuous inclusion H c(ft)—> Cc(ft).

We set

H^(ft, loc) = { u e S>'(ft) \<l>ueH for each <t> g C~(ft)}.

We give LT'(ft, loc) the weakest topology so that the mapping H^(ft, loc) — LT': u —> </)M is continuous for each <f> e C“(ft). Clearly there exists a sequence <#>k e C“(ft) such that whenever i/f g C“(ft) then there exists fco such that = i/f for fc > fco- Then for fc > fco we have

^ Q \\<t>k \\s

5. LOCAL SOBOLEV SPACES 257

Thus the seminorms u —> \\<f>ku\\s, fc = 0,1, 2 , . . . determine the topology of loc). In particular loc) is metrizable.

Lemma 5.4. jFT'(fl, loc) is a Frechet space.

Proof. Let be a Cauchy sequence in loc). If <;> g C“(H) then<f>Uk in If <#>, € C“(fl) then obviously <I)V = ilw^. Thus thereexists u such that = <f>v for each <f> e Then <f)veH^ and

in implies v GH^{il,\oc) and Uj —>v in loc).

Lemma 5.5. The inclusion map o f C“(fl) in loc) is continuous. Moreover, C“(ft) is dense in loc).

Proof. If <^eC^(fi) then multiplication by </> maps C°°(H) continuously into Cc(ft) which in turn is continuously included in Thus u 1I</>m||s is a continuous seminorm on C“(0). For the last part, choose functions <f>u, e Cc(Ll) such that if i/f € C“(I1) then there exists fco such that fc>fco implies = Let ueH^{Ll,\oc). Since 4>i u g H there exists v^eCciLl) such that \\vk — <l>ku\\s^llk. If il/eCciLl) choose fco as above. Then for fc > fco we have

H'/'CUfc - u)l|, = l|lA(Ufc -<#>IcM)IL Q.-

Thus Ufc —> M in loc).

Exercise 5.6. 1/ m > 0 is an integer we have a continuous inclusionC ^(fl)^ jFT^(ft,loc). 1/ fc>0 is an integer and s > k + nl2 we have a continuous inclusion jFT'Cfl, loc) —> C^(fl).

Exercise 5.7. The strong dual o f is loc). The strong dualo/H^ ( a loc) is H:\CI).

From Exercise 5.7 it follows that the inclusion maps ^'(fl) andloc) —> & {0 ) are continuous even with the strong topologies on the

distribution spaces.

Exercise 5.8. I f s < t then the inclusion map loc) H^(H, loc)is continuous and the inclusion map Hc(H) is compact.

Exercise 5.9. I f (fl) = Hs is given the weakest topology suchthat the inclusion map H“(fl) —> is continuous for each s, then theinclusion map C“(fl) ^ (ft) is an algebraic isomorphism, but is not a homeomorphism.

Note that Exercise 5.9 gives an example of spaces for which the open mapping theorem fails. The simplest way to see the last part of the exercise is to exhibit a continuous linear functional on C“(fl) which is not continuous on Hc(ft). Exercise 5.11 below indicates what linear functionals to consider.

E xercise 5.10. I f H°°{£l,\oc) = C\sî (Âoc) is given the weakest topology such that the inclusion map lo c )^ H ''(fl , loc) is continuousfor each s then loc) = topologically.

E xercise 5.11. We have

= U and 2 )^ 0 )= (a loc)

where 2)p(a is the space o f distributions on H o f finite order. Moreover, if ue^piCl), u has order and s > k + n/2 then ueH~^(Cl, loc).

T h eorem 5.12. 1/ 0 < 6 < p < l and then P mapscontinuously into loc). I f P is properly supported then P mapsH c (a continuously into and P maps HîH, loc) continuously into

loc).

Proof. Let K be a compact subset of H and choose ij/eCciTi) such that i/f = 1 in a neighbourhood of K If </> g C“(ft) then </>(X)Pi/r(X) maps jFP' continuously into PT'"”" by Theorem 4.5. Thus \\<l>PiJ/u\\s-m ^ C \\u\\s for each u eH \ By definition of the topology on PF'“"'(ft, loc) this estimate means that Pi/ (X) maps continuously into jFP'“"'(ft, loc). Since Pi/f(X) = P on H k it follows that P maps continuously intojFP'“”"(ft, loc), by definition of the topology on Hc(ft). Suppose now that P is properly supported. By Corollary 4.7 P maps continuously into PT'"'"; indeed into PT^"" for some compact set K'. Since the inclusion of PTk~”" in is continuous we now have that P maps PT continuously into for each compact subset K of ft. But then P mapsPfc(ft) continuously into by definition of the topology on PTc(ft).Finally for the last part, if (fie Cc(ft) then there exists «/fg C“(ft) such that (fi{X)Pil/{X) = <fi(X)P. Since ifiiX) maps PT'Cft, loc) continuously into where K = supp ifi we see that (fi{X)P maps loc) continuously intopjrs-m each <f)GCc(ft). By the definition of the topology on H"“""(ft, loc) the last part now follows.

The argument given for Theorem 5.12 is not really a proof. It is simply an unravelling of definitions. The theorem, however, is a convenient formulation of the continuity of pseudo-differential operators.

6. THE WAVE FRONT SET 259

Exercise 5.13. I f u g Q}'{H) and K is a compact subset o f H then for some t we have <l>u e for each <f> e Ck- Conclude if DjU e FT' (fl, loc) for / = 1 , . . . , n then u e loc).

Use Leibniz’ formula and induction for this exercise.

§6. The wave front set

In this section we will introduce a refinement of the wave front set of a distribution. This refinement is based on a microlocalization of spaces in the same way that the wave front set was obtained by microlocalizing C”. We will then obtain a refinement of the pseudo-local property (Theorem 3:8.3) of pseudo-differential operators and of the regularity theorem (Theorem 3:9.11). These results are important since they enable us to make quantitative assertions concerning regularity. In conjunction with Sobolev’s theorem they even permit statements concerning classical derivatives. The results, however, are more precise.

Before proceeding we will quickly consider how the localization of the H" spaces leads to a refinement of the notion of singular support. If usQ)'iQ) we define the -singular support of u, FT-sing. supp u, to be the complement in Cl of the largest open subset U o f Cl such that u\u e Fr(U , loc). Note if W is an open neighbourhood of fT'-sing. supp u then there exist Ui g H (Cl, loc) and U2 eQ)'(Cl) such that u = Ui~\-U2 and supp U2 ^ W. It is not difficult to see that sing, supp u is the closure of the union over all s of FT'-sing, supp u.

We proceed now to the definition of the FT'-wave front set. Let Cl be an open subset of If u g Q}'(CI) we define the H^-wave front set of u, denoted by WF,(u), to be the complement in ClxU'^-iO) of the set of points (xq, o) such that there is an open neighbourhood U of Xq in Cl and a conic open neighbourhood V of in (R” - ( 0 ) such that for each <f>eC^(U) we have

f | $ u ( ^ ) p ( l + l| p )’* d | < o o .Jv

If (^oj ^o) W F,(m) one says that u is microlocally in at (xq, |o)» or simply that u is in FT*' at (xq, |o).

The definition of WFs(w) given above has the advantage of being immediately related to the definition of the FT spaces. It is, however, rather cumbersome to work with. Our first job therefore is to obtain a convenient description.

Theorem 6 .1. Let u e Q)'(Cl) and let (xq, lo) x ~ (0). Then (xo, o) WF, (u) if and only if there exists «i e l o c ) and U2^3)'(Cl)

j v

+ 2

such that u = Ui + U2 and (xq, o) WF(m2)- Moreover, we may choose u . e H i m .

Proof. The last statement is trivial. Simply choose x (Lt) with X = 1 in a neighbourhood of Xq and then replace Ui by x^i ^md U2 by (1 —X:)mi + W2- Suppose we have Ui and U2 with the properties indicated. Since (xo, lo) WF(u2) there is an open neighbourhood U of Xq and a conic open neighbourhood V of 0 such that for each N > 0 and each <l> e Cc(U ) we have

for 6 V. But then

Jv Jv

J v

<2ll<^.Ul|l^+2a,^j(l + leP)-^^d^

which is finite if N is large enough. Conversely, assume {xq, WFs(u). Choose an open neighbourhood U of Xq and a conic neighbourhood V of lo such that

f l^ (| )p ( l + l|prd|<oo Jv

for each <I)g C c{U). Choose x ^ C c iU ) such that x = in an open neighbourhood <o of Xq. Define g(|) = l if and g(^) = 0 if V.Then

I| g (a i^ (l)l" (1 +111") = £ 13^(1)1" (1 + 111")-' d^

Therefore there exists Ui g H such that Ui(^) = Then u =M1 + U2 where M2 = (1 — + w and w = xu — Ui. In particular for any <t>eCci<o) we have <\>U2 = w. Note that implies w g H' forsome t and moreover

w(|) = xu(|) - Wi(|) = (1 - g(|))^(|).

If <> € C~(6>) then

<#>2(1 ) = <#>w( ) = (27t) * w(^) = (2t7) ” J —g(ir]))xw(T]) dr].

(The convolution is given by an integral since w eH \ See Caution 2:5.7.)

6. THE IT WAVE FRONT SET 261

Since we have

i ^ ( T 7 ) i < c ( i + h i rfor some M Thus

k M i ) ! ^ c^,,^ f (1+11 - T jir'^d + h i)^ dti

for each N > 0 . Choose an open cone V' with ^nd V 'H S ” ^relatively compact in V. Let 0 < r = dist (V 'n S ''”\0V ). Then

1^-T]l>rl^| if | g V' and

Then ( l + j — <( l + r 1 1)“ for ^ eV ',T ]^ V and N > 0 and (assuming M + f c > 0 ) + + + for all and t].Thus

i^ 2 (i)i ^ c^,^M.fc,<,(l+r i i i r ^ d + i i i r j (1 + it, i)-'= dr,

if | g V'. Taking k > n and choosing N as large as we please we see

for each 4> e C“(co) and ^ g V'. Thus (xo, WF(u2).

Lemma 6.2. Let ueQ)\H), I f K is a compact subset o f fl, T is a closed cone in —(0) and K x T is disjoint from V /F iu) then

for each <f> e and | g T.

Proof. Let XqgK and ogF. Since (xq, WF^Cu) we have u = Ml 4- U2 where u e loc) and (xq, |o) W F(m2). Thus there is an openneighbourhood U of Xq and a conic open neighbourhood V of such that

l^ 2(l)lsc^ ,.< ,d + l^ l)-"'

for each < >GC“(t/), V and N > 0 . By Lemma 3.11 we have

Thus

l ^ ( ^ ) l ^ Q ( i + i i i r (*)

for <l>eCc{U) and V. In view of the compactness of m S " ” ‘ we can find open conic sets V i , . . . , covering T and corresponding open neighbourhoods Uu - ■ . ,U „ of Xq such that (*) holds for <t> e C”(L/,) and

IG Vj, / = 1 , . . . , m. Setting D = f l we now have (*) for <t> e C"{U) and | g T. A finite number of such sets U will cover K and therefore by a (finite) partition of unity argument the lemma follows.

Theorem 6.3. Let u e^ 'ifl) . I f t < s then W Ft(u)c WFs(w)c WF(w). Moreover, WF(u) is the closure o f the union over all s o f WF^{u).

Proof. If t < s then WFt(w)cWF^Cw) is clear by definition. If W F(m) then w = 0 + w is a decomposition of the type occurring in

Theorem 6.1. Thus (xq, WF^Cu). Finally if (xq,^q) is in the complement of the closure of the union of the conic sets WF^iu) then there is a conic neighbourhood W of (xq, o) disjoint from WF^Cw) for each s. Choose an open neighbourhood U of Xq and an open conic neighbourhood of lo such that L / x ( V n S " “ ) is relatively compact in W. Then by Lemma 6.2 we have + for each cjyeCciU ), Vand N > 0 . Thus, (xq, WF(w).

Theorem 6.4. Let 77 — (0) ^ ft be the projection map. I f u eQf(P) then

7t(WFs(u)) = H" - sing, supp u.

Proof. If Xo FT - sing, supp u there is an open neighbourhood U of Xq such that <j) G C“(l/) implies <l>u g H\ By definition of WF^(u) we then have (xo, WFs(m) for each | and therefore Xq 7t(WFs(m)). Conversely, suppose that Xq 7t(WFs(m)). Then (xq, WFs(m) for each By compactness of we can find open cones V i , . . . , covering — (0) and open neighbourhoods U i, o f Xq such that

1<#>m(|)1 (1 + 1|P)''d|<ooJv,

if <#) G Cr(D/). Thus if U = C]Uj then <t>u g for each <j> g C^(U). Thus X() FT - sing, supp u.

As the proof shows. Theorem 6.4 is immediate from the definition of WFs(u). The only reason for separating it out as a formal statement is that we will wish to make use of the corollary that u e H^{H, loc) if and only if WF^(m) is empty.

Theorem 6.5. I f 0 < 6 < p < l , PG^^s(fl) and ue^'iCl) then

W F,(Pm)^ WF,+^(M)ness. supp Pand

W F,,^ (u )^ W F ,(P u )U Z (P ).

I f P is properly supported then the conclusions hold for u g Q)'{H).

6. THE WAVE FRONT SET 263

Proof. Let {xq, |o) ~ (0). If {xq, q) 4 ess. supp P then (xq, I o) 4WF(Pu) by Lemma 3:8.2. Thus (xq, WF,(Pw). If (xq, |o) WF,+Û) then u = Ui + U2 where and (xo,ô)^ W F(m2)- Thus PuêH^'Caioc) and by Theorem 3:8.3 (xo, |o) WF(Pm2). Thus

WFs(Pm). For the second part if (xq, o) then by Lemma 3:9.4 there is Q G ^ “s(fl) properly supported such that R = Q P -1 has order -oo in a conic neighbourhood of (xq, o)- Note if P is properly supported so is R. Now u = QPu + Ru implies by the first part

WFs+^(m) c (WFs(Pm) n ess. supp Q) U (WF(w) H ess. supp R) c WFs(Pm) U ess. supp R.

Since (xq, o) ss. supp R we see that if (xq, |o) 4 WFs(Pw) thenUo, o)^W Fs+^(m).

C orollary 6.6. I f 0 ^ 6 < p < l , P € ^ ^ 5(fl) and P is elliptic then

W F,iPu) = W F ,^ M

for each ue€'{H ). I f P is properly supported the conclusion holds for each ueQ)\Cl),

C orollary 6.7. I f 0 ^ 6 < p ^ l , PG''T^s(fl) and P is elliptic then for u e€ '( fl) we have ueHl'^'^(0) if and only if PuG fP'(fl, loc). I f P is properly supported and ue^'{Ci) then ueH^^^(H,\oc) if and only if P u eH ^ in , loc).

C orollary 6.8. I / 0 : ^ 6 < p : < l , P g P is properly supported andelliptic, h and k are integers with 0 : ^ k < m + h —m/2 and ueQ)'{Il) then Pu G C^(fl) implies u e

T heorem 6.9. The space fP' {Cl, loc) and its topology are invariant.

Proof. By the change of variables formula for an integral it is clear that H^{Cl, loc) = L\C1, loc) is invariant, together with its topology. Now since (ft, loc) is just the set of u€^'{Cl) such that PueH^(Cl, loc) for each properly supported operator PeC^^{Cl) it is clear that ^ ( f t , loc) is invariant. Now let s > 0 . If P g C^^{CI) and <>, i/fg C“(ft) then w— ||i/rP<)Ml|o is clearly a continuous seminorm on (ft, loc). This class of seminorms is invariant and so it suffices to show that it determines the topology. If <f>eCc(Cl), u eH ^ (Cl, loc) then

11wIU.s =11<>m1L=1Ia '< m1|o.

Choose (f>i G Cc (ft) such that <>i = 1 in a neighbourhood of supp <f>. Then

A*</)U = A (l>iU = + [A*, <l>{X)]<l>iU.

Therefore by Friedrichs’ lemma (Theorem 3.9)

llwlU.s ll< >A < >lMllo+Cl|< >lUlU,.

Repeating the argument we obtain inductively

l|u|U.s ^ C t l|<f>,._iA - u|lo + C UM\s-kJ = 1

where </>j e C”(fi) and <t>o = 4>. Choose k so that s - f c < 0 . Sinceand <ll<f)fcM|lo we have the invariance of the topology of

loc) for s > 0 . If K is a compact subset of ft then has the induced topology from H^(ft, loc). Thus is invariant together withits topology for s > 0 . Then the invariance of H “^(ft, loc) follows since it is the strong dual of H^Cft).

Theorem 6.10. then

W F , ( u ) = n Z(P)

where P runs over properly supported operators in such thatPm G L^(ft, loc). In particular WFs(m) is invariant if we regard it as a subset o f T*(ft) by identifying T*(ft) with ftx[R ” via canonical coordinates.

Proof. Let S denote the set of all properly supported PeC ^^((l) such that Pm 6 L^(ft, loc). If P g S then WFq(Pm) is empty and therefore WF,(m) ^ Z (P ). Conversely, suppose (xq, ^o) WFs(m). Choose a e C‘*(ftx[R" —(0)) homogeneous of degree s such that a = 0 in an open conic neighbourhood of WFs(m) and so a(xo, &) = 1. Choose A'^CcCIR”) such that X = 1 neighbourhood of the origin, let p(x, ) =(1 —X:(^))a(x, ) and choose P e C ^ ^ id ) properly supported such that P - p(X, - iD ) e ^ - “(ft). Then

WFq(Pm) c WF,(M)ness.'supp P = 0

implies Pu e L^(ft, loc). Thus P e S . Now a(xo, o) = 1 implies(xo,^o)^Z(P).

Remark 6.11. Let M be a smooth manifold. If L/ is a chart on M then we may define H^{U, loc) by using the coordinates in U. By Theorem 6.9 the topological vector space H^(U, loc) is independent of the coordinates used to define it. If s > 0 we define FT'(M,loc) to be the set of functions m on M such that if 1/ is a chart on M and </> e C“(ft) then d>u € loc) and we take the weakest topology so that m <>m isa continuous linear map of FT'(M, loc) into FT'(17, loc) for each <#>€ C ciU ), for each chart U. If K is a compact subset of M we set H k equal

7. SUBELLIPTICITY AND LOCAL EXISTENCE 265

to the set of ueH ^ (M ,loc) such that supp u ^ K and we give the subspace topology. We then set Hl(M ) equal to the union of the H k and give it the inductive topology. If we are interested in H^(M, loc) for s < 0 then we have to be more careful. The best way to handle it is to go back to the beginning and carefully distinguish distributions (= generalized functions) and twisted distributions (= generalized measures). The careful reader will have noted that this point turns up at the end of the proof of Theorem 6.9 where we rather cavalierly ignored it. Let’s leave it as an exercise to set it up properly.

Example 6.12. Let rieS''~^. Recall in Example 2:13.15 we constructed u e C c (IR”) such that WF(u) = {(0, tT)) 1 f > 0}. Clearly then WFo(m) = 0 and by Theorem 6.3 WF,(w) = {(0, tr ) j t > 0 } for all sufficiently large s. We can construct similar examples by constructing a pseudo-differential operator with essential support equal to Dx(ray). For example let p g C“(1R”) satisfy p(|) = for l l l > l . Note if we set

= then g(^) = 2(1 1-<|,T|». Thus if a > 0 then

Now D “p(^) for 1 1 > 1 is a sum of terms of the form

where |8<a, h is homogeneous of degree —fc and 2k-l-lj8|>|a|. Such terms are bounded by C Thus |D"p(|)|<Co,(l + l l)“'“' , that isP S?/2,o(1R'')- Now obviously p and all its derivatives are rapidly decreasing in the cone | < , tj) < 8 1||} if 0 < e < l . Thus ess. supp p ={(x, tr\) I t> 0} . It follows that if m6^'([R”) then

W F,(p (-iD )w )c{(x , tri) 1 r>0}nWF,(w).

Consider now the case u = S. Since p(tTj) = l we have lp(^)l:^C(l + ||l)“'' implies s < 0 . Thus by Theorem 3.14 p(-iD )S ^ H^(W, loc) if s > 0 . Hence we have

W F,(p (-iD )8) = {{(0, tT]) 11> 0 } if s > 0 0 if s < —nl2

§7. Subellipticity and local existence

Let ft be an open subset of IR". Let P e '^^ s(ft) and assume 0 < 6 < p < 1. Note that the space T^^s(ft) increases if 8 is increased or p is decreased. Therefore, when discussing P we may increase 8 or decrease p should it

be convenient to do so. Of course we must preserve the condition 0 ^ 6 0 then a sufficient condition for P to admit a parametrix is that a symbol p of P satisfy theinequalities

i p U D i ^ Q d + i i i r - ^

for x e K and > r , and

\p(x, )| ^ Ck,.,3 )| (1 +

(1)

(2)for x g K and Here K runs over compact subsets of ft. Notethat if p satisfies these inequalities, a e Sp,s(ft, IR”) and t< m — r then p + a satisfies these inequalities (with different constants).

Note that P is elliptic if and only if p satisfies (1) with r = 0. In this case (2) is trivially satisfied.

The property of admitting a parametrix is stable under perturbation by operators of sufficiently low order. Explicitly we have the following result.

L em m a 7.1. I f 0 < 6 < p < l , admits a parametrix Q eand Te'4^ps(n) where t < m - r then P - T admits a parametrix

in

Proof. It R e then any parametrix for P - T is also a parametrix for P - T + R , and conversely. Hence we may assume that T is properly supported. That Q is a parametrix for P means that Q is properly supported and P i = P Q ~ l and R 2 = Q P ~ i have order —o°. Now define

S k = t ( Q T ' ) ‘Q = I Q iTO y.j=o i=o

Then and Sk+i~Sk has order at most - m + r - { k + l)( ^ - r - t ) . Hence by Corollary 3:9.2 there exists S € properly supported such that S~~Sk has order at most —m + r —(fc + l)(m —r —t) for each fc. Now

(1 - QT)Sk = [1 - (QT)'‘" ']Q = Q[1 -

and

s , (1 - TO) = Q[1 - (TQ)*'"'] = [1 - (OT)'‘"dO.

Since P - T = P ( l - Q T ) + P i T we have

( P - T)Sk - 1 = PQ[l-(TQ)'"'"^]+RiTSk - 1

= P Q (T Q f" ' + R i( l + TSk).

This operator has order at most —(fc + l)(m —r —t) since PQ has order 0 and R i has order -oo. Thus

( P - T ) S - l = ( P - T ) S k - l + ( P - T ) ( S - S f c )

has order at most r — (k + l)(m — r — t), i.e., (P —T)S —1 has order —oo. Similarly S(P —T) —1 has order -oo. Thus S is a parametrix for P - T .

Operators which admit a parametrix satisfy a strong regularity condition. We have the following quantitative version of the hypoelliptic property of Remark 3:9.7.

Lemma 7.2. I f P e ^p 'sC fl), 0 < 6 < p < l , admits a parametrix Q e then

W F ,,^ _ ,(u)^ W F ,(P m) (3)

for each real number s and for each In case P is properlysupported then the conclusion holds for each uG^'iCl).

Proof. Indeed if jR = Q P - 1 then u = QPu — Ru and Ru e C“(fl). Thus WFs+^_^(m) = WFs+^_^(OPm) c WFs(Pm) by Theorem 6.5 since Q has order — m + r. Note by definition Q is properly supported. Therefore R is properly supported if P is properly supported and so the last part follows.

Since WF^CPu) cWF^+^Cw) the best possible result in Lemma 7.2 occurs in the elliptic case (since then r = 0). Therefore, we will say that P is strictly subelliptic with a loss o f r derivatives if (3) holds for each real number s and each ue^'{H). Before proceeding let us note that the last conclusion of Lemma 7.2 is not restricted to operators which admit a parametrix.

Lemma 7.3. Assume Pe'^^s(f^)> 0 < 6 < p < l , is strictly subelliptic with a loss o f r derivatives. I f P is properly supported then

W F,^^_,(u)^W F,(Pu)

for each real number s and each ue^'{Q ).

Proof. Let uGQ)'(il) and suppose (xq, WF^CPw). Choose <t>e C“(fl) such that </> = 1 in a neighbourhood of Xq. Then Pu = Pc )M + P (l —c/))m and P (l — 4>)u is C°° in a neighbourhood of Xq. Thus (xo, WFs(P</)w) and so by hypothesis (xq, WF^+ _X<#)m). Since <#)(xo) ^ 0 we have (xq, &) WF, + m-r (u).

Strict subellipticity is stable under perturbation by operators of sufficiently low order.

L em m a 7.4. I f P g T g 5(H), 0 < 6 < p < l , t < m - r and Pis strictly subelliptic with a loss o f r derivatives then so is P + T.

Proof. Let u g '{£1) and let / = Pw + Tu. Assume (xq, q) WF^C/). For some s' we have (xq? WF +^-^m). If s '> s there is nothing to prove. Assume therefore that s'< s. By Theorem 6.5 we have WFs'+m-r-ti.Tu) ^ WFs'+m-r(u) and therefore (xq, q) ^WFs'+m-r-t(Tu). Since Pu = f - Tu for any s" we have WF,«(Pm) c WF,«(/)UWF3 <Tm). Hence if s" = min (s, s' + m — r - 1) then (xq, 0) WFs«(Pw). By hypothesis therefore (^0? 0)^ WFs»+^_Xm). If s " > s we are done. If s" < s then s" = s' + (m —r - t ) and (xq, 0)^ WFs«+^_^m). Hence we may repeat the argument with s' replaced by s". Since m — r — t > 0 we obtain by induction that (xo, |o)^WF, + m —r (u).

R em ark 7.5. The inductive argument in the proof of Lemma 7.4 is a typical application of Theorem 6.5. The point is that since any distribution u is locally of finite order one has Pis WFs(u) = 0 so there is always a point at which to start the induction. Note that one even has 0^ H osing. supp u = 0 so similar inductions may be carried out using the jFP-singular support.

If we replace the jFP'-wave front set with the FT'-singular support then we arrive at the notion of subellipticity. Explicitly, if 0 < 6 <p < l , then P is subelliptic with a loss o f r derivatives provided that

jjrs+m '^-sing. supp u c - sing, supp (Pu) (4)

for each real number s and each u e^ '(fl). An equivalent formulation is that for each u g €'{£1) and each relatively compact open subset co of we have that (Pu)I gjFP'(o), loc) implies u\ g H ' ' ~''(o),Ioc).

The proofs of Lemma 7.4 and Lemma 7.3 may be modified to give the following result.

E x ercise 7.6. Let P g be subelliptic with a loss o f r derivativesand as usual assume 0 : ^ 6 < p : < l . J / T e §(11) and t< m — r then P + T is subelliptic with a loss o f r derivatives. I f P is properly supported then (4) holds for each u g Q)'{(1).

There are weaker versions of (4). Let P e and 0 < 6 < p < 1. Wewill say that P is semiglobally subelliptic with a loss o f r derivatives if for each uG^'iCl) we have that P u g H ^ ( ( 1 , \ o c ) implies uGHc'^'^~^(£i). We will say that P is globally subelliptic with a loss o f r derivatives if P is properly supported and for each u g ^ ' { £ 1 ) we have that P u g H ^ ( ( 1 , \ o c )

implies u g Even for properly supported operators thesetwo notions are distinct. For example, if P is the wave operator on IR"

then P is semiglobally subelliptic with a loss of 1 derivative, but P is not globally subelliptic since the equation Pu = 0 admits non-smooth solutions

In the literature strict subellipticity, subellipticity and semiglobal subellipticity are not always distinguished. The reason is quite simple. One is usually interested in the case 6 = 0, p = l and 0 ^ r < l . In this case all three notions agree as the following two theorems show.

Theorem 7.7. Let where 0 < 6 < p < l . Assume that P issemiglobally subelliptic with a loss o f r derivatives. I f

0 < r s there is nothing to prove. Assume s'< s. Choose SeC'^^(n) compactly supported such that (xq, lo)^ and such that ess. supp S is disjoint from WFs(Pw) and WFs'+m-rM- Then WFs(SPu) c WFjj (Pm) H ess. supp S = 0 implies SPu e HKCl). Since ess. supp [P, S] cess, supp S and [P, S] has order m —(p —6) we have WFs'_r+p_s([P, S]m) c WFs'+^_Xu) n ess. supp S = 0 which implies [P, S]m g jFP loc). Since PSu = SPu + [P, S]m we have PSu ePr'Xft, loc) where s" = min (s, s ' - r + p —6). Since SuG^'(fl) it follows by hypothesis that Thus we have

W F,.,^_,(m) c WF,«^^_.(Sm) U Z (S) = Z(S)

which implies (xq, WFs»+^_r(u). If s" > s we are done. If s" < s then 5" = s' + (p —6 —r). Since p —6 —r > 0 if we repeat the argument with s' replaced by s" we obtain by induction that (xq, 0)^ WFs+^_ (m).

We can do a little better if we are only interested in subellipticity.

Theorem 7.8. Let ^here 0 < 6 < p < l . Assume that P issemiglobally subelliptic with a loss o f r derivatives. I f

0 < r s there is nothing to prove. Assume s'<s. Let < >GC“(ft) and choose i/f G Cc(Ll) such that i/f = 1 in a neighbourhood of supp </>. Then

= <f) and therefore

P<l>u = Pu - <f>P(l -il/)u + [P, <l>(X)]i{fu. (7)

The first term on the right side of (7) is in The second term is insince The third term is in

jjs+p-r(£|^ loc) because by Lemma 3:2.8 [P, Thus P<l>u e(Ct, loc) where s" = min (s, s' + p - r ) and therefore by hypothesis <f>uG

^s"+m-r(£|) conclusion holds for each <f)GCci^l) we haveu\ g H'"'' ' ~\o),\oc). If s " > s we are done. If s" < s then s" = s' + p - r . Since p —r > 0 we may repeat the argument with s' replaced by s" to obtain by induction that u\^GH^’ '^~''io),\oc).

We will now introduce a certain estimate which is satisfied by semiglobally subelliptic operators. We will see that in certain cases the estimate implies subellipticity. Finally we will come to the motivation for all these considerations. We will show that an operator P with semiglobally subelliptic formal transpose has the local existence property; that is, we can solve P u = f locally. We will even be able to solve semiglobally if f satisfies a finite number of compatibility conditions; that is, we have a Fredholm alternative.

Before introducing the subelliptic estimate we make a convenient convention. If we set

l|ul| = (2 ,7 )-''| (l + | im M (l)Pd |

where we allow +oo as a value for the integral. Either Hm||s <<» for all s, in which case u is smooth, or else there exists Sq such that fors<So and Hulls = +oo for s > S q. From Theorem 2.21 and the monotone convergence theorem if u^O then log (Hulls) is an increasing convex left-continuous function of s g R with values in (—oo, +oo]. It is of course continuous on the interval where it is finite. The function s —> Hulls has the same properties since the exponential is a convex increasing function.

Let PG^^siP') where 0 < 6 0 . We will say that P satisfies the subelliptic estimate with a loss o f r derivatives if for each pair of real numbers s, t, each compact subset K of f l and each <f> g C“(H) such that <#) = 1 on a neighbourhood of K there exists a constant Cs,(,k,4>^0 such that

Cs,,,K,<f,(ll<#>Pul|f + Hull?) (8)for each u g Note that the inequality (8) is trivially satisfied unless<\>Pu g H and t< s + m - r .

If the subelliptic estimate (8) holds for a single pair s, t then it trivially holds for s, t' with f > t If the subelliptic estimate (8) holds for a single pair s, t with t < s + m — r then for t'< t and s > 0 we have

(Corollary 2.22) and therefore choosing s > 0 sufficiently small we obtain

the subelliptic estimate for the pair s, t'. Thus the estimate (8) holds for all t if it holds for some r < s + m — r.

Note if T € p7s"( ) then (f>T maps continuously into H\ Therefore

llc Pwll < 2 U iP + T)wie + 2 \\<t>Tu\\

<2 ii<)(p +t)m11s+ c 1|m11?.If we substitute this inequality into (8), with the same t, then by the remarks above, if t < s + m — r then P + T satisfies the subelliptic estimate. Thus if P satisfies the subelliptic estimate (8) and T has order strictly less than m — r then P + T satisfies the subelliptic estimate (8).

In the case of properly supported operators the subelliptic estimate takes a simpler form. Indeed suppose P is properly supported and satisfies (8). Given the compact set K there is a compact subset K' of Cl such that supp u ^ K implies supp (Pm) c K '. Taking <f> so that </> = 1 in a neighbourhood of K U K ' we see that for each pair of real numbers s, t and each compact subset K of Cl there is a constant Cs,t,K^0 such that

llwlls+m (9)

for each Conversely, suppose P is properly supported andsatisfies (9). Let </> e C“(I1) and suppose <#> = 1 in a neighbourhood <o of K. Choose {{/ e C“(c*>) such that i/f = 1 in a neighbourhood of K. Then

Pu = <l>Pu + (1 - <l>)Pil/u

if mg^k. Since (1 — (l>{X))Pil/(X) e^~°°(Cl) is compactly supported it maps continuously into fP' for any t. Thus

11Pm 11?< 211< ^ p u 1 1 ^ + c , , J | m ||?

for MG^k- Therefore (8) follows from (9).

Theorem 7.9. Let P g ^^ sCO), 0 < 6 0 . I f P is semiglobally subelliptic with a loss o f r derivatives then P satisfies the subelliptic estimate with a loss o f r derivatives.

Proof. Let K be a compact subset of Cl. Let (f> g Cc(Cl) and assume that </> = 1 in a neighbourhood o) of K. Let r < s + m — r and let

E = {uGH^K\<f>PueH^}. (10)

On E we define a norm \\‘\\ by 11m|P = H<)Pm|1s +|lMlf. If u is a Cauchy sequence in E then u^ -^ u in Hk and Pui w in H\ From the continuity of <>P we also have <l>Puj <f>Pu in and therefore(fiPu = w. It follows that E is a Hilbert space. Choose i/fG C“(co) such that

= 1 in a neighbourhood of K If m g E then

Pu = <fiPu^-{l-<t>)Pi\fu. (11)

Since {l-<t>(X ))Pilf(X )e^-^ift) the last term in (11) is in C“(fl). The first term on the right in (11) is in by hypothesis. Thus PueH ^(il,\oc). Since P is semiglobally subelliptic and u has compact support it follows that u e Thus E ^ Suppose ^ w in E and in

Since ^ u in E implies u in H* we must have u = v. Therefore the inclusion E has closed graph and so is continuous by the closed graph theorem. Thus the subelliptic estimate (8) holds for each u e E . Since we have that E =

Pu g H^}. Hence (8) is satisfied for each We will now prove a partial converse to Theorem 7.9. The natural way

to obtain regularity results from inequalities between norms is to make use of Theorem 3.7. Therefore we need to consider the commutator of a pseudo-differential operator with translation.

Lemma 7.10. Let be compactly supported and assume0 < p < l . Then for each real number s there is a constant such that

l|[Q,Ta]ull,<Q lalllulU^

for each and each aelR”.

( 12)

Proof. Since [O, t ] maps continuously into jFP' it suffices tocheck (12) for m g C“(IR”). Since Q is compactly supported there exist c>, i/f G CciUn such that Q = </>(X)Oi/f(X). Thus Q = q(X, - iD ) for some q G q(x, |) = 0 for x outside some compact set. If now weset

b(^, Tj) = I T)) dx

then for each N

m , Tj)i ^ c^, ( i+ II -

(see Theorem 3.3) and

Qm(|) = (2ir)""| b(|, -h)u(t}) dt)

for u eC tiU " ). Now 0 ‘(Tj) = e-'<” ">u(Ti) and (TaOM) ( ) = e“' “^6u(l). Thus

([O, tJ w) '(I) = (27r)-"| (e -‘<’' “>-e-^<«-‘*>)h(|, 'n)Mv) dt}.

Since|e-i<ni.a>_e-i<€.a>| ^ 2 Isin « | - t), a)/2)| < (1 + 1|- tjI) k l

the lemma follows by Theorem 3.2.

T heorem 7.11. Let 0 < p < l , and assume

0 < r < l . (13)

I f P satisfies the subelliptic estimate with a loss o f r derivatives then P is semiglobally subelliptic with a loss o f r derivatives.

Proof. Let u e and assume Pu e loc). Let e > 0 be chosen so that if K is the closed e-neighbourhood of supp u then X is a compact subset of ft. Choose (f) g C“(ft) such that <> = 1 in a neighbourhood cj o f K and then choose ij/ e C“(co) such that i/f = 1 in a neighbourhood of K. For some s' we have If s ' > s there is nothing to prove.Assume s '<s . If lx|<e then r u — u e Thus if s" =min (s' +1 - r, s) we have

IITxM - wlls"+m-l-rS C(1|<#>P(t M - U . ls"+ m -2 . (14)

We will use C to stand for various constants. Since s"—l:^ s ' —r we haveTheorem 3.7

^_2^C|xl.

Now if |xl<8 then

( I > P ( t ^ u — u ) = T x { ( t > P u ) — ( l ) P u - \ - [ Q , T ^ ] u

where Q = c >(X)Pi/ (X) is compactly supported. Since s"< s and 4>Pu g PP Theorem 3.7 implies

llT*(< )PM) - <#>Pull,.._i < C lx|.

Since Lemma 7.10 implies

11[Q, T;,]m1L»_i ^ C \ x\ .

From (14) we now have

lkM-w|L"+m-i-r^Clx| for lx|<£.

Since we also have Theorem 3.7 im-plies If s "> s we are done. If s " < s then s" = s' + l - r . Since1 - r > 0 we may repeat the argument with s' replaced by s" and obtain by induction that

E xercise 7.12. Let P€^^s(ft ), 0 < 6 0 . Let s', t be fixed real numbers with t < s ' + m — r. Assume for each compact set K in ft and each <f> e C“(ft) with > 0 such that

l|uik„_<CK,<,(ll4>PuB+||uB

for each u e Show that if r < p —8 then P satisfies the subellipticestimate with a loss o f r derivatives.

As a hint for the exercise first bound \\u\\ + -r by using the subelliptic estimate (with a loss of 0 derivatives!) for a properly supported elliptic pseudo-differential operator of order s - s'.

Exercise 7.13,1 then

”) is compactly supported and 0 ^ 8 < p ^

11[Q, Ta]w|L =l|(T-aQTa “ Q)w|ls

for each I f q(x,$) is a symbol for Q then r-aQ'^a~Q ^pseudo-differential operator with symbol q(x + a, ) —q(x, |)g S^s(IR”, IR”). I f 0 < l a | < l then \a\~ a, ^) -q {x , ^)) stays in a bounded set inS^8"^([RMR").

We now investigate some of the consequences of semiglobal subellipticity. The subelliptic estimate will play a key role in the proofs, but we do make full use of the semiglobal subellipticity hypothesis as well. In the case 6 = 0 and p = 1 Theorem 7.11 shows that we could base all the arguments solely on the subelliptic estimate if we restrict ourselves to the case r < l . It turns out, however, that one of the major non-elliptic applications is in the case r = 1.

Since we will be interested in local solutions of equations P u = f we should point out that strict subellipticity and subellipticity are local properties; that is, the regularity of u on an open set (o is determined by the regularity of Pu on co. Semiglobal subellipticity, however, is not a local property (unless r<p) . Thus semiglobal subellipticity will not permit us to make any regularity assertions concerning local solutions. As we shall see, however, it does give us existence of local solutions with nice regularity for the formal transpose P'.

Theorem 7.14. I f P g 0 < 6 e C“(ft) and </> = 1 in a neighbourhood (o o f K then

Nk = { ug '(ft) I supp u q K, 4>Pu = 0}

is a finite-dimensional subspace o f C*(ft). Moreover, there is a neighbourhood (Oi o f K such that L = d)i is a compact subset o f o) and Ni = N^-

Proof. Choose ij/e Cc((o) such that i/f = 1 in a neighbourhood of K. If u € ^ k then Pu = <l>Pu-^{l — <l>)Pil/u. Since (1 —<#>(X))Pi/f(X)G'?^““(ft) we see that <f>Pu = 0 implies Pu g C°°(ft). Thus ^ C“(ft). Since <t>P:H*”'” is continuous and ^ C“(ft) ^ we see that Nk is a closed

subspace of for each s. If u e then from the subelliptic estimate we have

l|ul|f<C(ll<#>Pul| _.,+llul|?) = CllMl|ffor any t< s . Thus the identity map (Nk, |Mlt) — (Nk, lH|s) is continuous. Now by Rellich’s lemma (Theorem 3.12) since t < s the identity map

is compact. Thus the identity map (Nk,|H1s)-^ is compact. It follows that Nk is finite-dimensional since

IHls) is a Banach space. For the last part let <o' be any neighbourhood of K with L' = 6)' compact in w. Then ^ Nl and since <#> = 1 in a neighbourhood of L we have that Nl is finite-dimensional. Suppose there is u E Njl with v^Nk . Then there is supp v such that Xq^K. Let <o" be a neighbourhood of K such that and Xq^L" = (o", ThenNl"5 N l '. After a finite number of steps we arrive at the desired conclusion.

Since Nk ^ C“(H) we see that if K has empty interior then Nk = (0).

Corollary 7.15. I f P e 0 < 6 = 1 in a neighbourhood o f K, then there is a neighbourhood (o o f K such that u E supp u ^ d) and 4>Pu = 0 imply u = 0.

Corollary 7.15 shows that we have a local one-to-one property for semiglobally subelliptic operators. An appropriate closed range result will therefore give local surjectivity, i.e., local solvability, for the formal transpose. The next theorem is precisely the closed range type of result needed. (In the case r = 0 it is a closed range result.)

Theorem 7.16. I f P e ^^^ (II), 0 < 6 e C~(fl) and = 1 in a neighbourhood o f K, if Nk is the kernel o f

</>(X)P:Hr

and if S is the orthogonal projection o f constant such that

llu - Su|U„_, < Q ll< >PulL

for each u e H k ' ~' with <t>Pu e

onto Nk then there is a

(15)

Proof. Recall first that Nk is actually contained in Ck and in particular is independent of s. Let t < s + m — r and as in the proof of Theorem 7.9 introduce the Hilbert space E = { u € H k \ <l>PueH^}. Then E is continuously included in and in fact consists of distributions mejyr +m-r <l>PueH\ Moreover, both sides of (15) are continuous

seminorms on E and we wish to prove that (15) holds for each u e E . Assume that (15) is false. Then we can choose a sequence in E such that

1 = llMfc - SUfclU„_, > k l|< PMfc|l, (16)

for fc> 1. Since t<s-\-m — r Rellich’s lemma (Theorem 3.12) implies that there is a subsequence ui such that u — Sul^-^v in H^. Since e it follows that <\)Pul—>(f>Pv in On the other hand by (16) we have4>Pui ^ 0 in Thus, <f>Pv = 0 and so u g N . But then v e Now

and </>F(Mk-Sw^-u) = g fT* implies Ui —Suk — v e E . Since u i — Sul — v ^ O in 2ind <t>Pui^^0 in fP we have u i~ Su'i — v 0 in E and therefore in But v e implies vis orthogonal (in to u — Sul. Thus

1 = llWfc - Sufc|L+„_, < llu - Sufc - ulU 0.

This contradiction shows that (15) must hold for each u e E .

Note in the proof of Theorem 7.16 we used that E is continuously included in This fact is just a restatement of the subellipticestimate. We used also the fact that ^ in order to guarantee that Nk is independent of s. If r > p this statement does not follow from the subelliptic estimate, but uses the semiglobal subellipticity (or at any rate hypoellipticity).

With the aid of some functional analysis we can deduce a very nice closed range theorem from Theorem 7.16. The statement of the theorem, however, is in terms of the weak* topology on and the proofrequires more functional analysis than we have assumed up to the present. Therefore we begin by summarizing some facts from functional analysis. These facts may be found in just about any text on topological vector spaces.

One of the disconcerting features of weak topologies is that they are not metrizable in general. Thus theorems which reduce questions of convergence to questions of bounded convergence or even sequential convergence are particularly useful. Undoubtedly the most famous of such theorems is Banach’s theorem.

Theorem 7.17. Banach. Let E be a Banach space and let B be the unit ball in the dual space E'. Then a convex cone C in E' is weak* closed if and only if B H C is weak" closed.

The unit ball B in E' is actually compact in the weak* topology. Moreover, if E is separable, that is, contains a countable set which spans a dense subspace, then the weak* topology restricted to B is metrizable.

C orollary 7.18. Let E be a separable B anach space. Then a convex cone C in E' is weak* closed if and only if B C\C is sequentially weak* closed.

A subset C of a topological space X is sequentially closed if whenever Xfc is a sequence in C and x in X then x e C . It is clear that closed sets are sequentially closed. In a metrizable space, or more generally a first countable space, the converse is true.

Let E be a topological vector space with dual space E'. Let D be the closed unit disk in the complex plane. A subset A of E' is equicontinuous (see Section 1:7.9) if U =r\f~^{D), f e A , is a neighbourhood of the origin in E. In view of the Minkowski construction of a seminorm associated with a convex balanced absorbing set, we see that A is equicontinuous if and only if there is a continuous seminorm h on E such that |(/, c)|<k(e) for each f e A and each e e E . In case E is a barrelled space, for example a Frechet space, then by the Banach-Steinhaus theorem the equicontinuous subsets, the weak* bounded subsets, and the strongly bounded subsets of E' all coincide.

We can now state an important generalization of Banach’s theorem.

T h eorem 7.19. K rein S m u lian , Let E be a Frechet space and let C be a convex subset o f E'. Then C is weak" closed if and only if A H C is weak* closed for each weak" closed equicontinuous subset A o f E'.

If E is any topological vector space then by a theorem of Banach- Alaoglu-Bourbaki each equicontinuous subset of E ' is relatively weak* compact. If E is a separable locally convex space then each weak* closed equicontinuous subset of E ' is a compact metrizable space in the weak* topology. (See for example Horvath [1], ch. 3, section 11, prop. 4.)

C orollary 7.20. Let E be a separable Frechet space. Then a convex set C in E' is weak" closed if and only if it is sequentially weak" closed.

L em m a 7.21. I f Cl is an open subset o f C^(Cl) is separable.

then the Frechet space

Proof. Let M be a countable dense subset of [R” and let E be the subspace of C"(fl) spanned by the exponentials for ^ eM . Let

If f vanishes on E then /(^) = 0 for each ^ eM . Since / is an entire holomorphic function it follows that / = 0. Thus f = 0. Then by the Hahn-Banach theorem E is dense in C“(fl).

T heorem 7.22. I f P e 0 < 6 e C“(fl) and

</) = 1 in a neighbourhood o f K, then

< l> {X )P :€'K ^ r{a)

has weak" closed range. In particular if P is properly supported then

has weak" closed range for each compact subset K o f ft.

Proof. We prove the last part first. If P is properly supported then for each compact subset K of ft there is a compact subset K' of ft such that su p p u ^ K implies supp {P u )^ K '. Taking </> = 1 in a neighbourhood of K U K ' -WQ see that the last part follows from the first. Now for the first part. In view of Lemma 7.21 and Corollary 7.20 it will suffice to show that <f>{X)P:^K^'S'{H) has sequentially weak* closed range. Let be a sequence in and assume that <f>Pu v weak* in ^'(ft). By Banach- Steinhaus the sequence is equicontinuous and so there exist constants C and M and a compact set L in ft such that

\(<f>Puj , i/f)l < C max supl a l < M L

for each if/e C°°(ft) and so for each iI/g By the Sobolev lemma ift> M + n/ 2 then is continuously included in C ^ (W ). Thus

\(<l>Pu„ eA)l^C'lWc

for each il/eCc{U*'). Thus \\<l)Puk\\-t C'. Therefore by Theorem 7.16 llwk —SUk|l_t+^_r^C" for each fc. By Rellich’s lemma there is a subsequence Mfc such that u'l — S u i ^ u in Then <l>Pul = <l>P(ul — Suj )converges to <f>Pu in and so certainly weak* in ThusV = <t>Pu. Since supp w ^ K we are done.

Exercise 7.23. Prove Theorem 7.22 directly from Theorem 7.19 without resorting to sequences and therefore without using Lem m a 7.21. Also remove the dependence on RellicWs lemma by using a well known fact about the weak topology in a Hilbert space.

Corollary 7.24. Let P€'^^s(ft), 0 :^ 6 < p < l, be properly supported and be semiglobally subelliptic with a loss o f r derivatives. Assume for each compact subset K o f II there is a compact subset K' o f ft such that

(P^^(ft)) Pi — P^K' •

Then P has w eak* closed range.

(17)

Proof. Let be a sequence in ^'(ft) and assume Pu v weak* in ^'(ft)- As in the proof of Theorem 7.22 the sequence Pu is equicontinuous and so supp (Pu^) ^ L for some compact set L, for example,

the compact set L which occurs in the proof of Theorem 7.22. By (17) we may choose such that Pw = Pu . But by Theorem 7.22 Pw vimplies that v is in the image of P.

R em ark 7.25. Note if for each compact set K in ft there is a compact set K" in ft such that we^'(ft) and sing, supp (Pw) c K implies sing, supp u c K" (for example, if P is hypoelliptic or if P is a constant coefficient linear partial differential operator and ft is convex; see Exercise 2:8.24) then (17) is implied by the apparently weaker condition

(P cr(ft))n cK ^ P C K '. (18)

Indeed, let u e and assume supp (Pm) c K. Choose i/r e C~(ft) with (/f=l in a neighbourhood of K" and let Mi = i/ru €^'(ft) and U2 = (1 —i/f)MGCc(ft). Then supp Mi ^ supp i/ and supp (PM2) ^ supp (Pm) U supp(PM i)cL implies supp M2 L ' by (18). Thus supp m c (supp i/f)UL'.

We are now ready.to prove an existence theorem.

T heorem 7.26. Let P e ^^s(ft), 0 6 < p < 1 and let P ' be the formaltranspose o f P. Assume P is properly supported and assume that P' is semiglobally subelliptic with a loss o f r derivatives.

(A) Semiglobal Fredholm alternativeLet K be a compact subset o f ft and let

N K = {^ eC K | P > = 0}.

Then there exists a neighbourhood U o f K such that if feQ }\fl) then there exists u e S»'(ft) such that P u = f in U if and only if (/, ilf) = 0 for each i l / eNK - Moreover, if feH ^ (Cl, loc) we may choose m g jFP‘ ""~'’(ft, loc) and if feC°°{Q ) we may choose MGC°°(ft).

(B) Local existenceIn (A) if K is a point, or more generally a compact set with empty

interior, then the orthogonality condition on f is vacuous.

(C) Global Fredholm alternativeI f for each compact set K in ft there exists a compact set K' in ft such

that

(19)

and if

N = {i/rGC*(ft)|P> = 0}

then for any f e C°°(ft) there exists u e C ”(ft) such that P u = f if and only if (fy = 0 for each iI/g N.

Proof. We prove (C) first. In view of (19) Corollary 7.24 implies that P' has weak* closed range. By Theorem 1:13.6 it followsthat P : C“(fl) has closed range and the range is the annihilatorof Nq = { v e€'(Cl)\P'v = 0}. But Nq = N since P' is semiglobally subelliptic. Part (B) follows trivially from part (A) since if K has empty interior then Nk = (0), We will now prove (A). If Pm =/ in U and i/f e Cc(U ) then </, if/} = (Pu, \if) = <m, P^)- Thus if if/e Nk then {f, if/) = 0. Conversely, assume and {/, il/) = 0 for each i/tgNk. Let a> be arelatively compact neighbourhood of K such that if L = 6) then = Nl (see Theorem 7.14). Then

NK={iAeCr(o>)|P> = 0}.

Choose X C~(o)) such that x = 1 in a neighbourhood U of K. Then for some s if feQ }'(0) and for the given s if loc). Let S

be the orthogonal projection of HZ onto Nl = N k- By Theorem 7.16 we have

1I»A- S»/»1U Q lljP>lU-m+rfor each j/te C“(co). (We can dispense with the function </> which occurs in Theorem 7.16 since P' is properly supported. This argument has occurred several times.) If i/fGC (co) then SiI/g Nk and therefore

(xf, =

Thus

\(xf, A)| llxflls WxfWs llP>ll-s-m+rfor each i/f g C“(o>). Thus if we define a linear functional u on P 'C “(6>) by

(u,P'ilf) = {xf,ilf) (20)

then M is well-defined and

1<M, P'ilf)\^C\\P'llf\U.m.rfor each iI/g C c{<o). By the Hahn-Banach theorem u extends to a continuous linear functional on Thus u extends to an element ofjLjs+m-r gy ^20) Pm = x f in o) and so Pm =/ in U. Finally we come to the case f e C“(fl). Let E be the subspace of C~(fl) consisting of all functions vanishing of infinite order on L and let C~(L) = C*(fl)/E. By Theorem 1:13.4 the dual space of C ”(L) is By Theorem 7.22 P' has weak* closed range. Therefore by Theorem 1 :13.6(A) P :C ^ {Q )^ C°°(L) has closed range and the range is the annihilator of ker P' = Nl = Nk- If / g C“(H) and (/, i/) = 0 for each if/ g Nk then the equivalence class of f in C°°(L) is in the annihilator of Nk- Thus there exists m g C“(fl) so P u —f vanishes of infinite order on L. In particular Pm = / on <o.


Lemma 7.27. Let P satisfy the hypotheses o f Theorem 7.26(C). Let

N = {iiJG C 7 m \ p 'iiJ= o}.

Assume that the induced map

p : ^ '(n)/c“(n) (21)

is surjective. I f feQ}'(Cl) then there exists ueQ)'{Q) such that P u = f if and only if {f, lA) = 0 for each il/eN .

Proof. Suppose {/, i/) = 0 for each ij/eN . Since (21) is surjective there exists t; eSi'(f^) such that P v —f = g e C*(ft). Obviously (g, = 0 for eachiI/g N. Thus there is w e C“(fl) so Pw = g. Now let u = v - w .

In Remark 3:9.7 we introduced the notion of hypoellipticity. Actually there are several notions of hypoellipticity. We say P is strictly hypoelliptic if WF(Pw) = W F(m) for each u e 'S'(Q), hypoelliptic if sing, supp (Pm) = sing, suppu for each ue^'(Q ), and semiglobally hypoelliptic if ue^'iH ), P u € C “(n) implies ueC^iCl). Clearly each type of subellipticity implies the corresponding type of hypoellipticity.

Lemma 7.27 shows that the key to obtain a global Fredholm alternative in Of {££) is the surjectivity of (21). Hdrmander in [6] shows by completely pure abstract functional analysis that if P is properly supported then (21) is surjective if P' is semiglobally hypoelliptic and if for each compact subset K of ft there is a compact subset K' of ft such that if u g ^'(ft) then

sing, supp (P'u) ^ K implies sing, supp v c K\ (22)

Note that the semiglobal hypoellipticity assumption just means that if K is empty then we may take K ' to be empty.

It follows therefore that under the hypotheses of Theorem 7.26(C) with the addition of (22) we obtain a global Fredholm alternative for S)'(ft). When P' is semiglobally hypoelliptic then (19) in Theorem 7.26(C) clearly implies that for each compact subset K of ft there is a compact subset K' of ft such that

(P 'c r (ft))n cK ^ P 'C K '. (23)

Note that Remark 7.25 shows that (23) and (22) together imply (19). Theorems 6.3.3 and 6.3.4 in Duistermaat and Hormander [1] show that for operators of real principal type (22) is equivalent to the surjectivity of (21), and if (22) is assumed then (23) (or (19)) is equivalent to the global Fredholm alternative in S)'(ft), and also in C“(ft).

It is interesting to note that if we are willing to assume some regularity on P then we may dispense with (22) and still obtain the surjectivity of (21). We replace the functional analysis argument of Hormander [6] by a

Cech cohomology argument. Our result will unfortunately not apply to operators of real principal type. However, in view of the importance of the surjectivity of (21) it is worthwhile exploring different ways to get a grasp on it. First we need a well-known lemma which is just part of the assertion that the Cech cohomology groups of ft with coefficients in the sheaf of germs of C~ functions is trivial.

Lemma 7.28. Let be a locally finite open cover o f ft. I fhjf G n o>fc) and hjj + = 0 in coj Pi and hij + hjj + = 0 in(Oi n <Oj n cofc then there exist gj g C"(coy) such that = g k — g , in (Oj H

Proof. Let Xj t>e a partition of unity in C“(ft) with supp Xj ^ < r efine = L xAk- Then gt € C”(wfc) and

Sk S/ ? i(. k Xi jk ^jk‘i i

We now show that we can replace (22) by a hypoellipticity assumption on P and still obtain the surjectivity of (21). At first it may seem strange to apply sheaf theoretic methods to a non-local operator. TThe pseudolocal property of pseudo-differential operators, however, means that in a certain sense they are local modulo C“.

Lemma 7.29. Let PGT^^s(ft), 0 < 6 < p ^ l , and let P' be the formal transpose o f P. Assume P is properly supported and hypoelliptic. Assume P' is semiglobally subelliptic with a loss o f r derivatives. Then the induced map

P :S i '( f t ) -^ ^ '(n )/ C “(ft)

is surjective.

Proof. By Theorem 7.26(B) we may choose a locally finite cover(o)j)y>i of ft by relatively compact open sets such that if /eS)'(ft) then there exist u g 2)'(ft) such that Pwy = / in coy for each /. Since P(wy — w ) = 0 in coyHcOk the hypoellipticity of P implies that hy = (uy - m )! C“(ct>y rict>k). The hypotheses of Lemma 7.28 are trivially satisfied. Thus there exist gy g C“(o>y) such that hy^ = g k — gy in <Oj fi o>k. But then

Wj + gj = i k + gk in coyHfOfc.

Thus there exists u g 9f{H) such that

u = Uj-\- gj in (Oj.

Let fj = P(u — Uj)G^'(H). Since u — Uj is equal to gyGC°°(ct>y) in (Oj the pseudo-local property of P implies that f j \ ^ . g C“(coy). Now

Pu = PUj-\-fj

implies {Pu-f)\^.eC^{<x)j). Thus P u -fe C ^ iH ) .

It now follows that if we assume (19) then under the hypotheses of Lemma 7.29 we have the global Fredholm alternative for S)'(n). See Lemma 7.27.

Remark 7.30. The difficulty in establishing a global Fredholm alternative in the space of all distributions is related to an interesting subtle topological fact concerning subspaces of LF-spaces, in particular ofcr(n).

Let F be a subspace of C“(n) and let tq be the subspace topology on F. Assume for each compact subset K of f l that F H Ck is a closed subspace of Ck- Note that tq and induce the same topology on F Pi C^. Let be the strongest locally convex topology on F such that the inclusion F n Ck ^ F is continuous for each compact subset K of H. Since these inclusions are continuous for tq we have that Note either F iscontained in Ck for some compact set K in which case F is Frechet and

= or else F is not contained in any Ck in which case (F, Ti) is an LF-space. In this second case it may happen that the inclusion isstrict. Indeed, it can even happen that some closed subspace of an LF-space fails to be L F in the subspace topology.

Suppose now that 0 < 6 < p < l , is properly supported.Assume that P' is semiglobally subelliptic with a loss of r derivatives. Let /ESi'(ft) and assume that = 0 for each ij/eC c iil) with P'il/ = 0.Define a linear functional u on F = F 'C ”(H) by

If u is To continuous then by the Hahn-Banach theorem we may extend u to a continuous linear functional on C“(fi). Thus we obtain ue^ '(Q ) with Pu = /. At question, therefore, is the continuity of u.

Let us assume that (23) holds. Then for each compact subset K of H there is a compact subset K' of ft such that

F H C k ^ P 'C k .

Thus

F n C K = (P'CKOnCK. (24)

Let <f>i be a sequence in F Pi Ck such that in Ck- By (24) we maychoose such that P' / k = <>k• By Theorem 7.22 P 'i^ K '^ ^ '(ft)has weak* closed range. Since PVk -^ <#> it therefore follows that there isV e such that P'v = <f>. But P' is semiglobally hypoelliptic and thereforeV e Ck'. Thus <#) G F Pi C^. It follows that F Pi Ck is a closed subspace of Ck and therefore is a Frechet space.

Now let E = {i/f G Ck' I P'lj/e Ck}. Then E is a closed subspace of Ck -

Therefore

P 'i E ^ F D C k (25)is a continuous surjective linear map of Frechet spaces and so is open. Let N be its kernel. If <) g F Pi Ck and we choose i j /e E with P'il/ = <t> then the fact that (25) is open means that the coset + N in E/N depends continuously on </>. Since / induces a continuous linear functional on E/N and

(m, <t>) = {u, P'il/) = {f, l/f)it follows that u is continuous on F fl Ck- By the definition of ti it now follows that M is a Ti continuous linear functional on F. We wanted tq continuity. The Mackey-Arens theorem gives a condition for each ti continuous linear functional to be tq continuous. That may be too strong a condition, however, since we are only interested in the linear functionals that arise as above. Thus the natural approach is to look for conditions on P' or, if we are not interested in all distributions, on /, which guarantee To continuity of u.

The author has occasionally confused tq and ti continuity in various guises in his lectures and has been subsequently consoled by Remark 13.2 in Treves [1]. See Remark 7.31 below for an ‘example’ of a ti continuous linear functional which is not tq continuous.

We now indicate the form the above results take in the case of constant coefficient linear partial differential operators. This material is for illustration only. Therefore we will not give all the proofs. First note if O^^Fg 0 [ z i , . . . , then the formal transpose of P(D) is P (-D ). Since P(—D) is one-to-one on for example by Theorem 2:8.1, the orthogonalitycondition on f above will be vacuous and we will be looking at global existence theorems.

One feature of constant coefficient linear partial differential operators is that they admit fundamental solutions. By choosing a fundamental solution with good local properties one sees that if P(D ) ^ 0 has order m then P(D ) is semiglobally subelliptic with a loss of m derivatives. Indeed, if c Xa — + then P(—iD) is semiglobally subellipticwith a loss of r derivatives. For a more precise statement see Hormander [2], Theorem 3.2.1. Thus semiglobal subellipticity with a loss of many derivatives is a very weak property. On the other hand if aG C “(lR”) vanishes of order la| + l at the origin then aD “6 = 0 and so aD “ is not even semiglobally hypoelliptic.

If O=^Pg 0 [ z i, . . . , z„] then in view of the injectivity of P (-D ) on distributions with compact support condition (19) becomes; for each compact subset K o f Cl there is a compact subset K' of Cl such that

u G '(^ ), supp P(—D)u ^ K implies supp u c K'. (19)'


Condition (23) becomes; for each compact subset K of Cl there is a compact subset K' of f l such that

u G C~(ft), supp P(—D )u ^ K implies supp u c K'. (23)'

The conditions (19)' and (23)' are equivalent. If they hold one says that ft is P{D)-convex. The basic fact here is that ft is P(D)-convex if and only if P (D ): C~(ft) ^ C~(ft) is onto. The ‘only if’ part follows from Theorem 7.26(C) in view of the above remarks.

Note Theorem 2:8.16 implies that if each component of ft is convex then ft is P(D)-convex. Conversely, if ft is P(D)-convex for each first order linear differential operator then each component of ft is convex.

Condition (22) now becomes; for each compact subset K of ft there is a compact subset K' of ft such that

u G ^'(ft), sing, supp P(—D)u ^ K implies sing, supp u c K'.(22)'

By Exercise 2:8.24 this condition is satisfied if each component of ft is convex. One can show that (22)' is equivalent to the surjectivity of the induced map P:&{Cl)-^Qf(Cl)IC^{Cl). Note that according to Lemma7.29 this map is surjective for each ft if P(D ) is hypoelliptic.

We will say that ft is strongly P(D)-convex if conditions (19)' and (22)' both hold. The basic fact here is that ft is strongly P(D)-convex if and only if P(D):S)'(ft)-^S>'(ft) is onto. In particular open sets with convex components are strongly P(D)-convex. If P(D) is hypoelliptic then by the comments above ft is P(D)-convex if and only if ft is strongly P(D)- convex.

One can show that P(D) is elliptic if and only if each ft is P(D)-convex. Moreover, in this case each ft is in fact strongly P(D)-convex.

All the facts concerning P(D)-convexity and strong P(D)-convexity mentioned above may be found in Hormander [2] and Treves [3]. Many of the concepts and proofs have their origin in the work of Malgrange [1].

Remark 7.31. If n = 2 then ft is P(D)-convex if and only if ft is strongly P(D)-convex. (See Hormander [2].) If n > 3 this is no longer the case. (See Treves [3], Remark 6.4.) In particular if n > 3 we may choose a connected open set ft (necessarily not convex) and P(D) (necessarily not hypoelliptic) such that ft is P(D)-convex but not strongly P(D)-convex. Since (23)' holds all the hypotheses of Remark 7.30 are satisfied. Since there exists /g S '(ft) not in the range of P(D) the construction in Remark7.30 provides a ti continuous linear functional which is not tq continuous.

Remark 7.32. Since the formal transpose of an elliptic operator is elliptic and since elliptic operators are strictly subelliptic with a loss of 0 derivatives the results of this section apply to elliptic operators. In

particular we have the semiglobal Fredholm alternative of Theorem 7.26(B), and under hypothesis (19) the global Fredholm alternative on S)'(fl) by Lemma 7.27 and Lemma 7.29.

In [1] A. Plis gives an example of a fourth order real elliptic linear partial differential operator Q with C“ coefficients on [R such that Qi/f = 0 has a non-trivial solution «/rEC“([R ). Thus Q'u = ij/ has no solutions defined on a neighbourhood of supp i/f.

Let Q be a properly supported pseudo-differential operator. We will say that Q has the unique continuation property if ue^'(W > ^ connected open subset of ft, Qu = 0 in co and w = 0 in an open subset of co imply that u = 0 in co. More succinctly, Q has the unique continuation property if ft ~ supp w is a union of components of ft ~ supp Qu.

Assume now that Q has the unique continuation property. Let K be a compact subset of ft, u e^ '({l), and assume supp Q u ^ K . Let K' be the union of K and those bounded components of f t ~ K whose boundaries are contained in K. Here the boundary is computed relative to R ”. Since the complement of K' is a union of components of ft —K it is open. Thus K' is a closed subset of ft. Let U be one of the components of ft~ iC '. If U is unbounded then u = 0 in U by hypothesis, since u has compact support, and so must vanish on an open subset of U. Suppose now that U is bounded. By definition of K' there is XoEdU with Xq^K. If Xoeft then there is an open ball B centred at Xq with B c ft-^ K . Since B H U is not empty we have B U C7 is connected. Since 1/ is a component it follows that B ^ U which contradicts XoEdU. Thus JCo^ft and so XoEdfl. But then there is an open ball B centred at Xq such that w = 0 in B H [/. By hypothesis we have w = 0 in U. We have therefore shown that supp u c K'.

Let H be a closed half-space in R ” such that If a component Uof ft —K meets the complement V of H then either it contains V in which case it is unbounded, or else it has boundary points in V in which case dU is not contained in K. Thus K '^ H which implies that K' is contained in the convex hull of K. In particular K' is bounded.

Let U b e a bounded corryDonent of ft — K such that d U ^ K . Since U is compact there exists Xq e U such that dist (xq, dCt) = dist (U, dCi). Clearly XqESU and so Xq e K. Thus for each x e K' we have dist(x, 5ft)> dist(K, aft). Since K ^ K ' we obtain

dist (jK:,aft) = dist (K ',aft).

We have shown that K' is a closed subset of ft, is bounded, and is bounded away from aft. It follows that K' is a compact subset of ft.

Theorem 7.33. I f P e ^^(ft), 0 < 6 < p < 1, is properly supported and if the formal transpose P' is semiglobally subelliptic and has the unique continuation property then P : C“(ft) ^ C“(ft) is onto.

Proof. Indeed, the unique continuation property implies that for each compact subset K o f i l there is a compact subset K' of f l such that

u G supp P 'u ^ K implies supp u c K'. (26)

This condition is the analogue of (19)' in the case of variable coefficients. It certainly implies (19). If ij/eC ciO ) and P'i/f = 0 then the unique continuation property implies i/r = 0. Now the theorem follows by Theorem 7.26(C).

Regarding Theorem 7.33 note by Lemma 7.27 and Lemma 7.29 if in addition to the other hypotheses P is hypoelliptic then is onto. More generally if (22) holds then P:Q)'(fl)-^3)\H) is onto.

The formal similarity between (26) and (22) should be noted. Results related to (26) often go under the heading of propagation o f zeros whereas results related to (22) go under the heading of propagation o f singularities.

R em ark 7.34. Let Q = Ziai m a^eC*^{H), where C "(fl) denotes the space of analytic functions in Cl. If Q is elliptic and u e then one can show that u is analytic on any open subset of ft where Qu is analytic. (See Hormander [2], Theorem 7.5.1.) In particular Q has the unique continuation property.

It follows if P is an elliptic linear partial differential operator with analytic coefficients in ft then PS>'(ft) = S)'(ft), PC°°(ft) = C“(ft) and PC"(ft) = C "(ft). The last assertion here follows from the first and from analytic regularity.

R em ark 7.35. The previous remark introduces the question of global solvability in the context of analytic functions. Cattabriga and DeGiorgi [2] have shown that if P(D ) is a constant coefficient linear partial differential operator in then P(D)C"(R^) = C"(IR^). If n ^ 3 this result fails for example for the heat operator, see Piccinini [1]. A general study of real analytic solutions has been given by Hbrmander [13].

We finish this section with some results on consequences of the subelliptic estimate. We already know that semiglobal subellipticity with a loss of r derivatives implies strict subellipticity if r < p — 8 (Theorem 7.7) and implies subellipticity if r< p (Theorem 7.8). We will now show that if r<(p-S)l2 then our weakest subellipticity condition, namely the subelliptic estimate with a loss of r derivatives, implies the strongest, namely the existence of a parametrix of order — m + r. As a corollary we obtain if P is a classical pseudo-differential operator and P satisfies the subelliptic estimate with a loss of r derivatives where r < 1/2 then P is elliptic.

T h eorem 7.36. Let PG^2^s(ft), 0 < 6 < p < l , be properly supported and let p be a symbol for P. I f 0 < r< (p —6)/2 then the following

statements are equivalent.

(A)(B)(C)(D)(E)

P satisfies the subelliptic estimate with a loss o f r derivatives.P is semiglobally subelliptic with a loss o f r derivatives.P is subelliptic with a loss o f r derivatives.P is strictly subelliptic with a loss o f r derivatives.For each compact subset K o f Cl there are constants Qc and a^ such that

111" : C k 1p (x , 1)1 for x e K and ||1 > a^.

Proof. Since 0 < r < (p -6 )/ 2 (E) implies that P has a parametrix Q € where p' = p — r and 8' = 8 + r. See Remark 3:9.7. Thus byLemma 7.2 (E) implies (D). Obviously (D) implies (C) and (C) implies (B). By Theorem 7.9 (B) implies (A). It remains to show that (A) implies (E). This implication will also use the hypothesis that r < (p — 8)l2. Assume (A). Let K be a compact subset of Cl. Choose such that<#) = 1 in a neighbourhood of K. Let x g K, 1 6 R" — (0), i/r e C“(R"') and let

w(y) = <#>(y)<A((y -

where 0 = (p + 8)/2. Since u has its support in a fixed compact set we have

l|ulP„_,<CllPu|g+CM?

which certainly implies

lluL_,<CllPullo + Cllu|L (27)

where C depends on t, but does not depend on i/r, or x. We begin by estimating the first term on the right side of (27). Let Q = P< >(X). Since P is properly supported we have that Q is compactly supported. Moreover, if p is a symbol for P then Q = q(X, —iD) where

; - | « l

<i(y, n) ~ Z ~ Ta !

and q(y, v ) — for V outside a compact subset of fl. If we introduce

w(y) = «/'((y-x) 1 1®)

v(y) = w(y)e‘ *’'

then

t;(T)) = w(t} - ^ )

and

Pu = Qv.

Now

Qu(y) = (2-jt) " I T))t5(Tj) dr?

= ( 2 i r ) - ’' j | + T ,)w (t,) d r,.

To estimate this expression we consider the Taylor expansion of q(y, + 17). We have

q(y ,^ + 'n)= Z A 9 ‘“’(y,l)T}“ + i?N (y ,ill)\ a \ < N ^ •

where

(28)

R N (y ,lv )= I ( i - 0^ " V “’(y ,l+ f^ )d fT ,“, (29)|al = N< - Jo

(30)

|al

where From the Fourier inversion formula we have; - k l

Qu(y)= Z — r e ‘ *-” q‘“'(y, |)D“w(y) + gN(y)|a|<N OL-

where g^iy) arises from the remainder terms. Since

D “w(y)= ( D - m y - x) i^n

and -p + 0 = -(p - 8)/2 we have

lq‘“'(y, |)D“w(y)l < C(1 + 1 1)— l^Kp-sw | (D » ((y - x) lll«)l

which implies

llq<“>(s ^)D“w|lo< C(1 + \^\r-M(0-BV2 | |-„e/2 l|£>“,/,llo.

Using this estimate for l^ | al we obtain

111"*'" llOllo^ C (llq(-, |)wllo + llg ,||o)+ C’(lll"’-<p-« /2)

Now

g^,(y) = (2 ir ) - " j e^«^-^>R^,(y, t, ) w ( tj) dr,

vanishes for y outside a fixed compact set. Hence we may estimate the norm of gj by a constant times its supremum norm. From (29) we have

i-R N (y ,i,h )i^ c sup ( i + i i + t T ,r - ' ’ h r .0 :S t< l

(31)

Thus

igN(y)Ncf sup (i+i^+tt,ir-‘-^hi i r«i (iii-%)idT,

s c f sup ( i + i i + t i i r T , i r - " " ' i i r iT,n^(T,)idT,J 0<t<l

If m - p N < 0 then the part of the integral over 1tj1>(1 1 ”®)/2 is bounded by

c f i ^ r h n ^ A M d r ,

which is rapidly decreasing in since 6 < 1 and since ipeSf, If 1tj|< (||| “ )/2 then |^|/2<l^+t 'nl —2 1 |. Thus the part of the integral over|t]|<(|^1 “®)/2 is bounded by

+ j h r \4>i'n)\ dr,

where we have used -p-\r6 = —(p —6)/2. Taking N large enough so that m — N p < 0 and nO < (N — l){p — S) we see from (31) and the above that

llQullo^ C \\q{-, ^)wl|o+<?(l^r-'"-"''"). (32)

To estimate the first term on the right in (32) we use a Taylor expansion again. We have

q(y, I) = + S(x, y, I)

where

n r l

Six, y, I ) = Z + t(y - x), |)(y, - Xj) d(.j = i Jo

Now

lS(x, y, ^)w(y)l < C(1 + |y - x| h ((y - x) 1||*)|

implies

\\S{x, ■, |)wlloSC'(l + |ll)"’- “’-* ''" 111-"®'"

where we have used 8 — 6 = —(p — 8)/2. Since

llq(x, |)wl|o lq(x, |)1 ||wHo = jq(x, |)1 \\ilf\\o

from (32) we now have

111"®'" llOllo^ C lq(x, 1)1 lhllo+C?(lir-‘"-«>'"). (33)

To estimate the remaining two terms in (27) we need an upper and a

lower bound for Hu||s. First we estimate the norm of v. We have

l|u|f = (2 rrr" (1 + |4' + T|py

= (277)--’ l i r « | ( l + ll + lllNH-^ |« (Tj)pdn (34)

If h l< (l^ r*)/ 2 then 1|1/2<1 + ||1%1<21||. Hence the part of the integral which is over |r||<(l|| “®)/2 is bounded above and below by

*'h l< (l€ l'-« )/ 2(35)

To obtain a lower bound we simply discard the rest of the integral and note that the integral in (35) is

(277)” l|i/fllo- [ |i/ (ir])l dii.

The integral appearing here is rapidly decreasing in Thus for ll| ^ 1 we have

Iloilo (36)for any N, where Ci is independent of i/r. Now in (34) consider the part of the integral which is over \r]\ > (l^p~®)/2. If s < 0 we have the upper bound

f liA(il)pdi7

which is rapidly decreasing in C K s > 0 we use 1 + | |*tj|s 1|| ( l + h|) for 1 1 s i to obtain an upper bound of the form

C U P "I(1 + hP)" |i (t})P dT|.

In either case we obtain

(37)

say for Since u = <f>v we have Hm|[, < C Hulls- We also have v =u-\-{l-<t>)v which implies

\\v\\s^\\uim-<t>)v\iSince lx — y 1 is bounded below for y e supp (1 - <l>) we have for \a\ M

llD“(l-</>)ulg<Cmax f l y - x ^ lD^u(y)^dy131<M J

< C max f |y - xp \(D%)({y - x) l^HP dyI31^M J

where we have used Leibniz’ formula and the fact that 0 < 1 . Changing variables we have

11D“ (1 - 4>)v\\l < C max ||pM-«ce-ne|01<M jlzl*' lD V (z)Pd2-

Since 6 > 0 , taking k as large as we please we see that |l(l-<;>)i;||h is rapidly decreasing in for any integer f i> 0 . If h t h e n H-H <||*|| . Thus we have

lk lls^ lk lL + ^ (ir

for any N. Now (27), (38) and (33) imply

-N)} (38)

1 1ne/2 1 . , < c k ( x , i ) i i i A + i i r " i k i i+ e m m -(p -6 )/2 ').

Taking t< m —(p —6)/2, using (36) and (37) and taking i/f with ||i/rllo=l we obtain

i^r = C lq(x, )1+ (|^|m -(p -S )/2 '

uniformly for x e K . Since <> = 1 in a neighbourhood ct> of K we have p — qeS~°°(o),W'). Thus we have the same inequality with q replaced by p. Since r< (p -6 )/ 2 it follows that

i i r -^ < 2 c ip (x , 1)1

for large \$\, uniformly for x e K ,

C orollary 7.37. Let 0 < 6 < p < l , satisfy the subellipticestimate with a loss o f r derivatives where 0 < r < ( p —6)/2. I f p is a principal symbol for P then for each compact subset K o f d there are constants Ck and a^ such that

l^ r-^ < C k 1p (x , g)l for x e K and 1 1 > Uk -

Proof. Modifying p by adding an element of S ”“(fl, [R" ) preserves the conclusion. Hence we may assume p(X, —iD) is properly supported. Since p{X, - i D ) - P has order strictly less than m - r it follows that p(X, - iD ) also satisfies the subelliptic estimate with a loss of r derivatives. Now the conclusion follows from Theorem 7.36.

C orollary 7.38. Let P e C^"”(fl) satisfy the subelliptic estimate with a loss o f r derivatives. I f 0 ^ r < l/ 2 then P is elliptic.

Proof. Let a be the principal symbol of P so a is homogeneous of degree m. Choose pGS” (fl, IR"") so p(x, ) = a(x, for Since

P — Corollary 7.37 implies that 1 1' “' ^ C k lp(x, )|for x e K , > a . Thus a(x, ^ 0 for x g K and ^ g R ” - (0), that is P is elliptic.

E xercise 7.39. I f PeC'^'^iQ) admits a parametrix where 0 6 < p < 1 and 0 ^ r < p — 8 then P is elliptic.

According to Corollary 7.37 the subelliptic estimate with a loss of 0 derivatives is equivalent to ellipticity even for non-classical pseudodifferential operators. The subelliptic estimate with a loss of 0 derivatives is usually called the a priori inequality for elliptic operators.

E xercise 7.40. Give a simple direct proof that the a priori inequality implies ellipticity for classical pseudo-differential operators.

R em ark 7.41. The lower bound for the symbol in Theorem 7.36 is a special case of a much more precise inequality of Hormander [12] and [11]. The estimates in the proof are taken from Hormander [12]. The more precise inequality is obtained by taking Taylor expansions of the p " as well. Corollary 7.38 is contained in Remark 1.1.1 in Hormander[4] as Hormander points out in [11]. Classical pseudo-differential operators subelliptic with a loss of 1/2 derivative have been characterized by Hormander [4]. An extension of this result to l / 2 < r < l has been given by Egorov [1, 2, 3]. A consequence of Egorov’s work is that if fc > 0 is an integer and fc/(k + l ) < r < ( f c + l)/(k + 2) then the set of classical pseudo-differential operators subelliptic with a loss of r derivatives is independent of r. Note Corollary 7.38 is the case k = 0 . A presentation of Egorov’s arguments with some modifications and simplifications is given by Hormander [11]. Subellipticity with a loss of r derivatives when r > l has been studied by Popivanov and Popov [1].

8. APPENDIX. THE SEIDENBERG-TARSKI THEOREM 293

§8. Appendix. The Seidenberg-Tarski theorem

Many of the properties of the constant coefficient differential operator P (- iD ) may be studied by considering the zeros of the polynomial P. One of the important tools for studying the zeros of a polynomial is the theorem of Seidenberg-Tarski. In this section we will discuss the Seidenberg-Tarski theorem and then apply it to obtain Hormander’s characterization of hypoelliptic constant coefficient partial differential operators.

A semialgebraic set in IR’" is a finite union of subsets of IR" each determined by a finite number of simultaneous real polynomial equations and inequalities of the form P(x) = 0, Q (x )> 0 . Clearly finite unions and


finite intersections of semialgebraic sets are semialgebraic. Since the negation of P{x) = 0 is P{x) > 0 or -P (x ) > 0 and the negation of Q(x) > 0 is Q(x) = 0 or —Q (x )> 0 , the complement of a semialgebraic set is semialgebraic.

Let A be the subset of IR"" determined by the finite number of polynomial conditions Pi(x) = 0, and Qy(x)>0, LetR (x ,y ) = 'Z{Piix)y-^'L(yfQj{x)-l)'^. Then JR is a polynomial in n + M variables and the image of the zero set of R under the projection of onto the first n variables is the set A. If we have a finite number of such sets A, say A , fc = 1, 2 , . . . , and form associated polynomials Rîx, y) as above (where some of the jR may be independent of some of the y variables) then jR(x, y) = I l Rk(^, y) is a polynomial whose zero set projects to the union of the A . Thus we see that each semialgebraic set is a projection of the zero set of some polynomial. The converse is also true.

Theorem 8.1. Seidenberg-Tarski. A subset A o f W is sem ialgebraic if and only if it is the projection o f the zero set o f a polynomial in

As an example consider a real polynomial P in n variables and suppose that P is quadratic in the last variable. Setting x' = ( x j , . . . , x„_i) we have

P(x) = a(x')x^ + 2b(x')x„ + c(x').

The projection on the first n — 1 variables of the real zero set of P then consists of the points x' such that {a{x') = b{x') = c{x') = 0) or (a(x') = 0 and b(x')Ô) or (a(x')Ô and b{x')^ — a{x')c{x')>0). Thus the projection is semialgebraic. For a proof of the Seidenberg-Tarski theorem see for example Friedman [1].

Let A be a subset of IR" . A map /: A -> [R"” is said to be a semialgebraic map if its graph is a semialgebraic subset of In view of theSeidenberg-Tarski theorem we see that the domain and the range of a semialgebraic map are semialgebraic sets. Indeed, they are projections of the graph and so are projections of the zero set of a polynomial since the graph is. More generally if B is a semialgebraic subset of IR” then f(B ) is semialgebraic, and if C is a semialgebraic subset of IR' then /“^(C) is a semialgebraic set. If /:[R” ^[R"" is a polynomial map its graph is defined by polynomial equations yj—fj(x) = 0. Thus polynomial maps are semialgebraic maps.

The Seidenberg-Tarski theorem may be used to show that many operations on semialgebraic sets yield semialgebraic sets. Suppose a subset B of IR” is defined by some polynomial conditions. In general the statement defining B may contain several quantifiers. In order to use the Seidenberg-Tarski theorem to prove that B is semialgebraic one first has

to replace all the universal quantifiers in the definition of B by existential quantifiers. As an example let B be the closure of the semialgebraic set A. Then

x g B < ^ V s > 0, 3 y e A , |x —yl<£.

In terms of existential quantifiers we have

x ^ B ^ 3 s > 0 , ~(3yGA, l x - y l < e ) .

Now let C be the set of (x, y, e) with 6 ^ - l x - y l ^ > 0 and y e A. Then C is semialgebraic. Thus D, the set of (x, s) such that (x, y, e) e C for some y is semialgebraic by Seidenberg-Tarski. Then E, the set of (x, s) with (x, s )^ D and e > 0 is semialgebraic. Now the complement of B is just the projection of E on the first n variables. Thus B is semialgebraic.

We conclude that the closure, interior and boundary of a semialgebraic set are semialgebraic.


Exercise 8.2. I f A is a semialgebraic subset o f dist (x. A) then d:W^ is a semialgebraic map.

and d(x) =

Exercise 8.3. The cone generated by a semialgebraic set is semialgebraic.

If A is a semialgebraic subset of the plane R‘ then the boundary of A consists of a finite number of pieces of algebraic curves. This fact, together with the Puiseux expansion of the roots of a polynomial in two variables, yields the following result.

Theorem 8.4. Let A b e a semialgebraic subset o f IR and suppose there exists T o such that if t > T q then (|ul, t ) € A for some pu. Define

| U L ( T ) = S U p { jU L | (jU l, t ) g A } .

Then there is Ti such that either |ul(t) = + oo for t > T i or else i i (t) is an algebraic function o f r for t > T i and

|ul(t) = CT (1 + o (l)) as T^OO,

where c is a real number and a is a rational number.

Theorems 8.1 and 8.4 acting in concert are very powerful. Typically they are used to establish inequalities concerning the growth of polynomials or the behaviour of their roots. Alternatively they are used to improve on existing inequalities. In this alternative case one might for example know a priori that jLt(r)^logT for large t . Then in the notation of Theorem 8.4 one must have a < 0 and therefore |ul( t ) < C for large r.

E xam ple 8.5. Let F e 0 [ z i , . . . , z„], let A be a closed semialgebraic set in [R” and assume that F(^) i= 0 for each e A. Assume for each large T there is g A with 1| = t. Let

B = {(fji, t) G I T = 111, IX = 1F(|)1 for some | g A }.

Then B is the projection on the first two variables of the set of points (/UL, T , D g R""^ such that t > 0 , t^-||P = 0, jLi^-lF(|)l^ = 0 andIG A. Thus B is semialgebraic. Now let

jUL(T) = inf {jUL I (jUL, t) gjB}.

If T is sufficiently large then jutCr) is the infemum of |F(|)| over a non-empty compact set where P(|)t 0. Thus |ui(t) > 0 . By Theorem 8.4 we have jLt(T) = 2 cr ''( l + o(l)) and necessarily c > 0 . Thus

lll if | g A and 1|| is large.

As an example consider F (li , I 2) = + and A=IR^. SinceP(t, 1/t) —> 0 as t 00 we see that in this case we have a < 0 .

Example 8.5 has an interesting consequence.

L em m a 8.6. Let F g 0 [ z i, . . . , z„] be a non-constant polynomial

(A) The real zeros o f P form a bounded set in W if and only if there exists Q eOi^ and jR g C“([R”) such that

F (| )0 (| )-1 = R (|).

(B) P {—iD) has a fundamental solution in O'c if and only if F(|) has no real zeros.

Proof. By Example 8.5 if the real zeros of F form a bounded set then if lIl^ T i. Choose Q g C“([R”) such that F(|)Q (|) = 1 for

large jlj. Then clearly Q g 0‘m- For the converse if F(|) = 0 then jR(|) = - 1 . Since R has compact support the zeros of F form a bounded set. Part (B) follows from the fact that the Fourier transform is an isomorphism of O'c onto Om.

From Lemma 8.6 we see that if the real zeros of F form a bounded set then there is Q g Om such that

P {- iD )Q (-iD ) - 1 G

Thus Q (-iD ) is a parametrix for F (—iD). In general Q (-iD ) is not a pseudo-differential operator. Indeed, Q (-iD ) is a pseudo-differential operator if and only if the distance from points of R" to the complex zeros of F tends to 00 at 00 in R ”. We will prove this fact below and will see that the condition on the complex zeros is necessary and sufficient for F (—iD) to be hypoelliptic.

E xam ple 8.7. Let P € ^ [ z i , .. and let

, z„] be a non-constant polynomial

Zp = { z e 0 - '| P ( z ) = O}.

If |eiR" let d{^) be the distance from | to Zp. Note if + ir)eZ p for some T]€lR’' then d(|)^l-nl. We may view 0" as Then

A = { ( / x, t )|(x = | | - z |", T = l l l ,P ( z ) = 0 }

is the projection on the first two variables of the set of points (fi, t, z) in satisfying = rÔ, t^ -1 | P = 0 , |P(z)| = 0 . Thus A is

sem ialgebraic. Now let

/x (t) = inf {/X I (jtt, t ) € A } .

T hen |x (t ) is the infemum over = t of (d(^))^. If the real zeros of P form a bounded set then d ( | ) > 0 for large 1 1 1 and therefore

|x (t) = 2cV “(1 + o(D)

as T —> 00, w here c > 0 . N ote a > 0 if and only if jx (T )—>°o as t —> oo.Suppose d (|) ^ 00 as 1|| -> oo. If jzj ^ oo, z e Zp, z = | + 1 7 7 then ||1 ^ 00

or 1t)1—> 00. Since d(|)^lT]| the first possibility implies the second. C on versely, suppose z = | + iT jeZp and jzl-ôo implies |t} | -^ ° o. T h e intersection of Zp with the ball of radius d (| ') + l centred at |' is com pact. H ence we can choose z = I + i't) € Z p with l l ' —|P+| t}P = L e t ||'| ^00. If 1|'-|1 is bounded then and so [z l^ o o . B y hypothesisItjI ^ oo and so

W e have shown that the following conditions are equivalent.

(1) d(|) ^ 00 as 111 00, I € R".(2) There exist constants a > 0 and c > 0 such that

d ( | ) > c l l l “ for large 1 1 1 .

(3) If z = 1 +17} € Zp then ItjI —> 00 as Izl —> 00.

B efore we discuss the criterion for hyperbolicity we need a few facts concerning polynom ials.

Lemma 8.8. Let Pe 0 [ z i , . . . , z„]. Then the following conditions are equivalent.

(A ) For each afÔ, D “P (| )/P (| ) 0 as HI ^ 00.(B ) For each 0 € R " we have P ( | + 0 ) / P ( l ) — 1 as l l l ^ ° o .(C) 1 / 1 < / < n then 0 as HI ^ 00.

Proof. N ote each condition implicitly contains the hypothesis that the set of real zeros of P is bounded. Trivially (A ) implies (C ). A ssum e (C).

Let delR" and set g(t) = P(^ + t0). If 0 < t < l then l|+ tel> l|| -l0l and therefore

g '(t) /g (f ) = I ePiPi^+te )iP (i+ te)^oj = l

uniformly for 0 < f < 1, as l j -> oo. Now

where 7 is the total change in the argument of g(t) as t goes from 0 to 1 . Since 7 = arg (g (l))-a rg (g(0)) + 2fc7r for some integer k we have

g(l)g(0)

= exp a'f-)Thus g(l)/ g(0)-> l as and so (B) follows. Assume (B). Let Edenote the space of all polynomials of degree < m where m is the degree of P. Then E is a finite-dimensional vector space. If a eW ' then Q(a) is a linear functional on E. Thus we have an imbedding of R” as a subset (not a subspace!) of the dual space E'. Clearly R ” spans E'. Hence there is a finite set A such that A is a basis of E'. Then

max lQ(a)lasA

is a norm on E. If Q(0) = 1] qoc “ then H a ! lq<| is also a norm on E. Since E is finite dimensional any two norms on E are equivalent. Thus

for each Q e E .X a ! Iq J^ C max lQ(a)lasA

Consider now the polynomial

By Taylor’s theorem

By the inequality above we therefore have

^ 1 D “P (| )1 ^ ^ |P( + a) I

for each | e with P(|) ^ 0. Thus (A) follows.

L em m a 8.9. I f P € ^ [ z i , . . . ,z„] then

D^PiO Ilog 2— — <max nd(|) “’‘o P (0

{*)

where d(^) is the distance from to Zp.

Proof. Let c = n~ log 2. Then

I i lnV=e--l = l.<jt#o o:! k = i

Now

P (l + Z) . V 1 P (l) P (l)

Hence if A is the right side of (*) and \z\<c/A then

| P (l + z) ikl Ulkl < 1.1 P(l) I « 0'

Thus P(| + z) t 0 for each z with lzl<c/A and so d(|)>c/A.

L em m a 8.10. I f P e ^ [ z i , . . . , z„] then

|D“P(|)< 2-"a! loti

I P ( e

where m is the degree o f P.

Proof. Let |zlsd(|) and let g(T) = P(| + Tz), re^C. Then g is a polynomial of degree ix^ m . Let t i, . . . , be the roots of g. By definition of d(|) we have lT,z|>d(|)>:lz| and therefore Thus|t, - 1 | < 2 1 t,1 and therefore

s ( 1 ) L tT IZiTll g(0)lP (l+ z)|P (l)

< 2 "

Since the polydisk {z|lz,l:Sn j = l , . . . , n } is contained in theball of radius d(|), Cauchy’s inequalities imply the desired bounds on the derivatives.

L em m a 8.11. Assume that for some s and some open subset O, o f R"

N = { u e loc) I P (-iD )u = 0}

is a Montel space relative to the topology induced by loc). It follows ifz =^ + ir}eZ p and \ z \ o° then |t j| ^ oo.

Proof. Assume that the conclusion is false. Then there is a sequence Zfc G Zp such that Iz j is unbounded and |T]k| < C Passing to a subsequence if necessary we may even assume that Let

Mfc(x) = ( l + l4pr"e'<^“- \

Then u e N . If (fye C“(IR”) we have

= ( 2 i r ) - | l<?(|-Zfc)P(l + | |P r ( l + l 4 l V de

By Peetre’s inequality

(1 +1^-41)'"''' (1 + \em +141")-' (1 + II - 41)"'''.Thus

I l<^(|-iT,fc)l"(l + lll)""'''d | s(2i7)-' Ilufcllf

 then for each M > 0 we have

1< (I - it7fc)l s C ,(l + II - ivk

Since is bounded we have

(**)

By taking M sufficiently large we see that (*) implies <f>Uk is a bounded sequence in H\ Thus is a bounded sequence in N. Since N is Montel there is a subsequence such that u!,-^ u in N. If <f> e Cc(II^") then

<Ufc, <;►> = (1 + I4l")"'<^(-Zfc).

Thus by (**)

Taking M large enough and noting jz j oo 1t] | < C implies l k| ^ o© we see that U k^O weak* in Thus u = 0, that is, in H* forany <f> e By (*), (**) and the dominated convergence theorem itfollows that

|l<^(|-iT,o)P(l + lll)-^'^'d| = 0.

Thus < (| - iTjo) = 0 which implies <> = 0 for any 4> e C ^ft). Contradiction.

Theorem 8.12. Let P e 0 [ z i , . . . , z„], Zp = {z£jZ'" |P(z) = 0} and if $ eR " let d{^) = dist (|, Zp). Then the following statements are equivalent.

(A) P (-iD ) is hypoelliptic.(B) P(—iD) is strictly subelliptic.(C) 00 os 111 ^ 00.(D) 1/ 1 < 7 < n then DjP{^)/Pi^) 0 as 1|| oo.(E) There exist constants C > 0 and a > 0 such that

lP (| )laC lD “P(|)llll“'“'

for each a, for large ||1.

Remark 8,13. Other conditions equivalent to (C) are listed in Example 8.7 and other conditions equivalent to (D) are listed in Lemma 8.8. If m is the degree of P and in (E) we take \a\ = m such that D “P is a non-zero constant we see that a ^ 1. In the course of the proof we will see that in (B), P (-iD ) is strictly subelliptic with a loss of (1 - a)m derivatives where a is given by (E).

Proof of Theorem 8,12. Assume (A). Let

N = { u e SD'iUn I P {-iD )u = 0}.

Then iV is a closed subspace of C“([R”) and so is a Mon tel space. For any s we also have that iV is a closed subspace of fT'ClR”, loc). Since the identity map (N, C “ topology) (N, local jFP' topology) is continuous the open mapping theorem implies that it is a homeomorphism. Hence by Lemma 8.11 and Example 8.7 (A) implies (C). By Lemma 8.10 (C) implies (D). By Lemma 8.9 and Lemma 8.8 (D) implies (C). Assume (C). By Example 8.7 there are constants c > 0 and a > 0 such that d(^)>c III"" for large jlj. Therefore by Lemma 8.10 (C) implies (E). Trivially (E) implies (D) and (B) implies (A). Assume (E). By Remark 8.13 we have a < l and |P(^)l^c for large \\. From this estimate, the other estimates in (E) and Remark 3:9.7 we see that P {—iD) admits a parametrix in ^ “^(IR”). Thus P(—ID) is strictly subelliptic with a loss of (1 - a)m derivatives.


Exercise 8.14. Let Q be the function given by Lem m a 8.6. Show that Q {—iD) is a pseudo-differential operator if and only if P (—iD) is hypoelliptic.

Let P € ( ? [ z i , . . . , z j have order m, so P(|) = R„(|) + , . . . , + Fo(|) where Pj(|) is homogeneous of degree j. Let ^ g IR” - ( 0) be a single characteristic of P (-iD ), that is Pm(l) = 0 and D,P^(|)/0 for sot j Assume P (^ )^ 0 and let k < m —1 be the largest integer so Pfc(

Then

Q-p(fg)[ i r - ^ Q P ^ ( g ) +> c rP(t ) I I +

for large t. We have proved:

C orollary 8.15. I f P e 0 [ z i , . . . , z^] awd P{—iD) is hypoelliptic then P(—iD) has no simple real characteristics. Thus P (—iD) is elliptic or else all its real characteristics have multiplicity at least two.

R em ark 8.16. If P g j ? [z i , . . . , z„] and P (-iD ) is hypoelliptic then lP(^)|->co as 1 1—>00. Indeed, this fact follows from Theorem 8.12(E) with \a\ = m. It is a very convenient necessary condition.

E xam ple 8.17. Let P(^i, I 2) = 1) + li- The principal symbol ofP (—iD) is $i$ 2- Thus the characteristics all have multiplicity two. We also note P (^)>0 and so by Example 8.5 we have

P (| )> C (1 + 1 1)"

for some O O . However, P (—iD) is not hypoelliptic since P(t, t~ ) 0 as

R em ark 8.18. As another application of the Seidenberg-Tarski theorem we will now prove the converse of Theorem 2:12.27. Let P e 0 [ z i , . . . , z^] and let T be a closed convex salient cone in [R". Assume that P(D) has a fundamental solution Choose <>e C“([R'') suchthat <#>(x) = l if l x l < l and < >(x) = 0 if lxl>2. Then

P(D)(<t>E) = S ^ g

where gE 'ClR"") and supp g c { x e T | 1 <1x1 < 2}. Taking Laplace transform we have

P(z)F(z) = l + G(z)

where F and G are entire holomorphic functions. Note P(z) = 0 implies G(z) = —1. Now by the Paley-Wiener theorem

lG (z) l<C(l + lzl)" e" - >

where z = g +117 and h is the support function of supp g. We have

Ii(-| ) = sup{-<|,x)| l < l x l < 2, x e T}

If I e T" then ( , x ) > 0 and therefore

h (-$ ) = -in f {< , x) 11x1 = 1 , X e T}

= - 8 ( 6

where 5(^) is the distance from to dT" . If P(z) = 0 then

implies

5(^)<c + N lo g ( l + |zl). (*)

Let K ^ S ”” be a compact semialgebraic subset of TJ and let V be the cone generated by K, If 0 < a = d ist(K ,dT' ) then ^ e V implies that 8{ ^) >a 111- We now have if z = + 117, P(z) = 0, V, t = \z \ and fx = 1 1 then

a f x ^ c + N log ( 1 + t ) .

Applying the Seidenberg-Tarski theorem as we have done several times before we conclude Hence for each compact semialgebraic subsetK of To there is a constant such that P(t^ + it]) ^ 0 if | g X and t > If we now apply Corollary 2:12.29 we obtain the converse of Theorem 2:12.27. Note that if T is semialgebraic then 6 is a semialgebraic function and from (*) we then conclude Thus in this case we obtainP(^ + it]) ^ 0 if G and 6(^) > c'. Since S is concave we now obtain the converse of Theorem 2:12.27 without using Corollary 2:12.29 and therefore without using Bochner’s theorem. This observation is useful in connection with systems. See Lancaster and Petersen [1].

Chapter 5

C oding’s inequality


In Section 2 we study the Sobolev space JFT”(X1) consisting of functions u such that D°"u e for ja] < m. These spaces are used in Section 3 as a natural setting for the Dirichlet problem. We solve the Dirichlet problem in an abstract setting under strong hypotheses and also obtain the Dirichlet principle in the positive Hermitian symmetric case. We then sketch the proof of Garding’s famous inequality and indicate how it leads to a Fredholm alternative for the Dirichlet problem in the uniformly strongly elliptic case.

Section 4, which should be regarded as part of this introduction, compares Garding’s inequality to the G^ding inequality of Theorem 4:4.4 and to the sharp Garding inequality of Hormander, Lax-Nirenberg, Friedrichs and others. In Section 5 we describe a notion of generalized Dirichlet forms and in Section 6 we use these and the Friedrichs’ symmetrization to give a proof of the sharp Garding inequality. In Section 7 we follow Hormander to show how the sharp Garding inequality leads to a result on propagation of singularities. We show that if hashomogeneous principal symbol p = a + ib, if P u = f , if y : I ^ T *(ft) is an integral curve segment for the Hamiltonian vector field of a, if b > 0 in a neighbourhood of y{I) and yU) H WF^C/) = 0 and 7 (fo)^ WFs+m-\M then 7 (1) ^W F,+^_ i (m). Moreover, p = 0 and H b - 0 on y(I). As a consequence we see if P has real homogeneous principal symbol and simple real characteristics then P is locally semiglobally subelliptic with a loss of 1 derivative. Since the same conclusion applies to the formal transpose P' the methods of Chapter 4 lead to local existence results for P.

305

306 GARDING’S INEQUALITY

§2. The spaces

Let Cl be an open subset of R". If m^O is an integer then in analogy with Corollary 4:2.5 we define the Sobolev space

H"'(Cl) = { u e L^iCl) I D “m e L^Cl), la| < m}.

If ueH"'iCl) we define Hu|l„ by

lal m 'When is provided with this norm it becomes a Hilbert space. Thecorresponding inner product we denote by (• | Om-

If Ij are indeterminates then

l a | < m + l \ OL /

and

( 1 + ^ , I y y + z z y w .lal<m ' j = l lal<m '

It follows therefore that if then

llM|lm + l=llMllm+ Z im-“ llm- ) = 1

We define to be the closure of C "(fl) in Since andinduce the same topology on Cc(ft) we may also view H o(fl) as a

closed subspace of H "'.Consider now the differential operator — If

u 6 and </> g (H) then

<(l-Aru,</.)= X ( y <f.)lal<m ^

= zjal^m

= (W I )m-

Thus the kernel of (l-A )"" in is the orthogonal complement ofHo(Cl). We define to be the image of under ( l-A )"”.Thus

We make into a Hilbert space by requiring that (1-A )"" be anisometric isomorphism of Ho (H) onto Then (1 —A)"" is a normdecreasing linear map of onto We denote the norm on

2. THE SPACES tT{Q) 307

by |H|_ and the inner product by (• | If k is an integer then the norm on H^(H) will also occasionally be denoted by if there is a risk of confusion with the norm.

Let M be the orthogonal projection of jFiT” onto If v e H ~ ”then M = (1 — g If m = Mu + w then as above (1 — A)'^w = 0 in fl and therefore u|n = (1 — A)"^Mm. Since Mu g we haveand

llu lnll-m.n = 11(1 - A)"’M(1 - A)-'”ul|_ .

In particular

lluy-m.n^llull-m

with equality if and only if (1 - A)"""i; g H q (H). If vGH~*^(d) then V = (1 - A)" u for a unique u g H o(£l). If we regard u as an element of H"" then w = (1 -A )'”m gH “' , w|n = t; and Hwl|_ = l|t)|l_ ,n- Thus the restriction map -^3)'(Cl) maps onto and is norm decreasing.Moreover, we have a canonical isometric extension map E

Since the restriction map L^(W) —> L^(fl) is onto we now have L^(d)^ Since € 7 (0 ) is dense in and (1 - A)^Cr(Xl) £ Ce(n) we

see that C“(ft) is dense in One should note, however, thatand do not induce the same norm on Cc(fi) and the canonicalisometric extension map E does not map C~(fl) toitself.

E xam ple 2.1. Let n = l and let = 1). Let g C c(0>), We willregard cf) as an element of and compute the extension EgH~ .We have E(f> = (1 - A)w where u g HK^I) and (j) = (1 — A)w in H. If x is the characteristic function of ( - 1 ,1 ) we have u = x^ where </> = i/f - i/f". For certain constants A, B we have

= ( ^ “ 1 1 < >(5)6"' dsje' + “ I I ds|e“‘.

Since u is orthogonal to the kernel of 1 —A in we have

r(</>(() + dt = 0

(ip(t) — i//'(t))e ' dt = 0.

These conditions imply

A e“' = - B e = -e e^b - a2 e^ -1

where

- i: </)(s)e ds and b = <l>{s)c ds.=i:It follows that

eb — eTâiA '(-l) = e " -e -2

e“'fe -e a e^-e-^ •

Now (1 - A)m = M- u" = x i^ - V ) - 2i(/'x’ - ihc", X<t> = <t>, X'= S - i - 8 , and x" = S l i - 5 'i . Since 0 8 '= -0 '(c )8 ,+ 0(c)S' for 0 € C “(1R) and c e R we have

, e“’ a - e b ^ e ~ ^ b -ea Ê<f> - 4> + ^2_^-2 -1 + 22 _ -2 1-e —e e —e

If u e there is a unique u e H o if l) such that v = (1 - A)"’u in ft.If w = (1 -A )'"m then weH ~"' and llull_„.n = llu|L =l|wl|_„. Since for any weH ~"' we have l|wyi-m,n —11 11- it follows that

llull-m.n = inf {llwll_„ | w e H w|n = u},

that is, the norm on is the quotient norm for the restriction mapThe following theorem gives another description of the

norm on

Theorem 2.2. Let Cl be an open subset o f R” and let m > 0 be an integer. Then is canonically isometrically isomorphic to the an tidual o f H^SiCl). In particular if v g H~'^{fl) then

llul|_.n = sup {1<U, </>)l/|l<f>lL| <#> € cr(ft)}.Proof. If V there is a unique u eH o (ft) such that u =

If (^GCciCi) then <u, <> = (w | < >) implies l<u, </>)l< C Thus (f) {v, (\>) extends to a continuous conjugate linear functional onH o(fl). The norm of this functional is Hm|| =Hu11_ . Conversely if / is a continuous conjugate linear functional on H o (d ) , then by the Hahn- Banach theorem and duality of H" spaces there exists w e such that {w\u) = f(u ) for each u eH o iC l). (See Lemma 4:2.8.) Thus /(</>), (f>eCc{Cl), coincides with the distribution w\êH~'^{Cl).

Lemma 2.3. Let Cl be an open subset o f R” and let m > 0 be an integer. I f ueH'^{Cl) then there exists w eH '^ such that u — weC°°{Cl).

2. THE SPACES IJr(Q) 309

Proof. Since (1 - A)’ m g (fl) there exists v e such that v\ci =Since (1-A )"" maps onto there exists w g H"" such

that u = (1 — A)""w. Then (1 — A)"” (w — u) = 0 in f l implies w — u e C ”(ft), if m > 0 , since (l-A )"" is elliptic. The case m = 0 is trivial.

C orollary 2.4. I f m > 0 is an integer then C “( f t ) is dense in

Proof. If u g H'^(£1) choose w g IT ” such that u - w eC^(H ). Now choose </>k G C “(R”) such that <>k w in H"'. Then m - w + </>k g

C “(fl) n H ’ (fl) converges to u in

Corollary 2.4 is due to Adams, Aronszajn and Smith [1, p. 28]. A proof has also been given by Meyers and Serrin [1]. Under fairly mild conditions on the boundary of Cl one may obtain genuine extension results. See, for example. Smith [1].

Multiplying by a cut-off function 6(k~^x) where 0(x) = 1 if lx l< l and 6(x) = 0 if |xl>2 and using Leibniz’ formula we obtain the following result.

E xercise 2.5. 7/ m > 0 is an integer then the functions in 77"”(ft) with bounded supports form a dense subspace o f 77"”(ft).

Note by a partition of unity argument (in IR”) each function in 77"”(ft) with bounded support is a linear combination of functions in 77"” (ft) with ‘small’ bounded supports.

We say that ft satisfies the segment condition if for each Xq g dCl there is a neighbourhood U of Xq and a point yo^lR” such that

ftn U + ty o ^ f t for 0 < t < l .

Suppose now u e 77"” has compact support and supp u ^ flO U . Let Ut(x) = u(x - tyo) so that supp li* c ft for 0 < t < 1. If tj > 0 then by Corollary 4:3.8 there is to such that 0 < t < to implies ~ u\\ < t]I2. Choose a mollifier pg. If e > 0 is sufficiently small then p^^u^sCcii^) and Up. * Mto“ “>ollm < V 2, since D “(p * u j = p^* in as8 -^ 0 , for lo:|<m, by Theorem 2:3.3. In view of the proof sketched for Exercise 2.5 and the comments following the exercise we have proved the following result.

T h eorem 2.6. I f ft is an open subset o f I and m > 0 is an integer then 77o(ft) = 77g.

r with the segment property

If t > 0 let = { x g IR” I lx|<t}. We say that ft satisfies the restricted cone condition if for each Xo € ft there is an open neighbourhood U of Xo,

t> 0 , and an open cone F such that

d n [/ + r n j5 t^ d .

If d satisfies the restricted cone condition then one can prove the conclusion of Theorem 2.6 by using a mollifier p with support in F. This argument then does not use continuity of translation in fT ”.

E xercise 2.7. I f dd is locally the graph o f a continuous function then d satisfies the segment condition. I f 8d is locally the graph o f a Lipschitz function then d satisfies the restricted cone condition.

Part of the exercise is to formulate a precise statement of the hypotheses. Note that one must assume d lies on one side of 3d.

If u e H ^ (ID leti(x) if XGd

lO if x^ d .

Then UoGL^iW^). In the proof of Theorem 2.6 we used the segment property to push the support of u g H^ into d . One may also use the segment property m case u e jFT^(d) to push the part of the support of Uq in 3d away from d. In this way one can show that if d has the segment property then C “([R”)|n is dense in jFF”(d). See Agmon [1, Theorem 2.1].

Uoix) =

C orollary 2.8. I f d satisfies the segment condition and u g jF]T”(d) then Uo eH' if and only if u e H q (d).

Proof. Suppose wGHo(d). Choose < )fcGCc(d) with <f>k->u in fT^Cd). Clearly u in FT”. Since a subsequence converges almost everywhere (property of convergence) it follows that v = Uq almost everywhere. Conversely, suppose Uq e IT^. Then UoGHg = H o(d) by Theorem 2.6.

E xercise 2.9. Ifv eH *^ (Q ) and wGHo(d) then

I D “u(x ) m(x ) dx = (—l) ’" ’ f v(x)D^u(x) dx

for |al<m.

Corollary 2.8 and Exercise 2.9 both motivate thinking of H q (d) as the space of MGH""(d) such that D “w = 0 on 3d in some generalized sense for la | < m - l . This point of view is further supported by the following facts.

Let MGH\d). Since UoeL\W^) we have

D Uo = iDjU)o-gj

2. THE SPACES H'"(Q) 311

where g, € H ' and supp g, c an. If <f> e C“(IR") then

<D,Mo, <f>> = - 1 uiDj<t>)dx

and

<(DjU)o, <f>) = f {DjU)<t> dx.•b

Thus by Leibniz’ formula

(gj, <t>) = 1 A •b(<t>u) dx, <f> e Cr(lR").

Suppose now we also have m6 C '(0 ) . Let L be a line parallel to the Xj-axis. Then L f l f t is a union of disjoint open intervals w, and

I dxjJnnL

is a linear combination of the values of <t>u at the endpoints of the (Oj. These endpoints are in the complement of Cl, but are limits along L of points in Cl. Thus they are in dCl. In particular, if w = 0 on afl then by Fubini’s theorem

I Dj(u) dx=0,Jn

that is, gy = 0. We conclude if ueC^(Cl)nH^(Cl) and u = 0 on then DyUo = (Dyw)oe L^(R”) for each j. In particular, If f l satisfies therestricted cone condition then the converse is also true. See Agmon [1, Lemma 9.1]. Thus if Cl satisfies the restricted cone condition and u e C ^ m n H ^ iC l) then u eH ^ {Q ) if and only if D “w = 0 on dCl for |al< m — 1.

In case n is bounded and dCl is smooth we have a natural restriction map H \n)-^W ^\dCl). In this case if then ueH ^{Cl) if andonly if = 0 for jaj < m - 1 . See Lions and Magenes [1, Section 1.3].

E xercise 2.10. Let Cl be an open subset o f W . Show that C“(n) is dense in if and only if u e ft H supp u = 0 implies u = 0. Conclude if Cl then C“(ft) is not dense in for s>n/2.

E xercise 2.11. Let m ^ 1 be an integer. Show if H q (ft) = H"^(ft) then IR" —ft has measure 0. Conclude H q (Cl) = (Cl) if and only if C~(ft) is dense in H* . In particular i/ ft^IR ” then H o (ft)^ H ”'(ft) for m > n l2 .

On we have the seminorms ] *lk» 0 fc < m, defined by

l«| = k \OL/

We note

i« iL i= 1 1 A « 'j=i

and

M I = - Z ^ ( " ) l « E .

We will show that under a width condition on the seminorm | | is a norm on Ho W and is equivalent to the norm \\-\\ on this space.

Let L be a line in R". We say that Cl has L-width at most d if each line parallel to L meets ft in a set of width at most d.

T h eorem 2.12. P oin care inequality. Let ft be an open subset o f R” with L-width at most d for some line L. Let m ^O be an integer. Then

for each ueH"S(Cl) and 0 < k < m . In particular

+ luL

for each u e H q (ft).

Proof. Since is a continuous seminorm on H o(ft) it suffices to prove the inequalities for u e C “(ft). By induction it suffices to prove

jwlo^dluli for ueC ciC l).

If v e Cc (R) has support of length at most d, then for some a, supp v c [a, a + d]. Then u(a) = 0 and

v(t)= f u'(s)ds.•'a

By the Cauchy-Schwarz inequality we have

lu (f)P< | f-al f lu'(s)Pds.

Since v{t) = 0 for t^[a, a + d] we have

|u(t)P<d|lu'(s)p ds.

3. THE DIRICHLET PROBLEM 313

Thus

I lu(r)l^df max lrpsd^|lu'(s)P ds.

By a rotation of coordinates we may assume that the line L is the x„ axis. If x' = (x i , . . . , x„_i) then

|m|o = j j W(x', x„)p dx„ dx'

< lD„u(x', x„)p dx„ dx'

= \D„u\l^d^\u\l

Exercise 2.13. I f ueH^SiH) then 1u1 = ((-A )"'m 1 u). In particular if Cl has L-width at most d for some line L then

((-A)-</> I <f>) > (1 + 1|<;>L for each <f> 6 C:(H ).

Lemma 2.14. I f m and k are integers, k > 0 and m >k-\-nl2 then H'îCl)^C^(Cl).

Proof. Let ueH'îCl). If g C c(.CI) then by Leibniz’ formula 4>ue Now use Theorem 4:2.13.

Note if we choose a suitable function </> in the proof of Lemma 2.14 (by translating and dilating a fixed function) then Theorem 4:2.13 and Theorem 4:3.5 allow us to estimate |D“u(x)l if \a\:^k, ueH'îCi), The estimate obtained in this way is very rough, but it holds for any Cl. When Cl has a suitable cone property then a good estimate may be obtained for |D“m(x)|. This estimate allows one to conclude that if in_ addition Cl is bounded and has the segment property then H'îCl) c C^(Cl). See Agmon [1, Theorem 3.9].

It is possible to introduce spaces fT '(fl) for non-integral s either by modelling the definition on Theorem 4:2.17, or by interpolation (see Lions and Magenes [1]).

§3. The Dirichiet problem

Let Cl be an open subset of IR”. Let P be a differential operator on Cl of order 2m. Let / be a function on Cl and let gk, 0:^ fc m — 1, be functions

on 0ft. Then the Dirichlet problem is the boundary value problem

P u = f on ft

d^u— r= gk on 0ft, 0 < fc < m —1.0n

Here (d/dn) denotes the normal derivative to 0ft. Not only are the normal derivatives somewhat inconvenient to work with, but they necessitate some smoothness conditions on 0ft. However, at points of 0ft, each derivative of u may be written as a linear combination of the normal derivatives of u with coefficients which are polynomials in the tangential derivatives. Thus we may regard the boundary conditions as specifying D “u on 0ft for la l< m —1. We could formulate the boundary conditions as D “u = on 0ft for —1. However, the cannot be specifiedarbitrarily, but must satisfy some compatibility conditions. We circumvent this difficulty by specifying a function g on ft and then requiring that D “u = D “g on 0ft for la l< m —1. Finally since we will wish to consider weak solutions we need to make sense of the condition that D “(w — g) = 0 on 0ft for l a l < m - l . In view of Corollary 2.8, Exercise 2.9 and the remarks following the exercise a reasonable formulation is u — g e H q (ft). In this case it is natural to consider u and g in FT” (ft) and, since P has order 2m, to consider f in H “'”(ft). The Dirichlet problem may now be formulated as follows.

Given f e and g € FT” (ft) find u e FT” (ft) such that P u = f in ftand u -g e H ^ (£ l) .

We still have the difficulty of finding a suitable class of operators. The operator P should map FP”(ft) into FF“"'(ft). This requirement entails inconvenient hypotheses on the coefficients of P, conditions which ensure that multiplication by the coefficient of maps FP”“' ’(ft) into FF“"”(ft). We can circumvent this difficulty by integrating by parts to replace P by a suitable sesquilinear form. If u € FP”(ft), <f> e C*(ft) and the coefficients of P are in C "(ft) we may integrate by parts to obtain

(Pu\lal m •In I3l m

If m > 1 then integrating by parts in different ways will in general associate many such integrals to the operator P. We now dispense with P and focus our attention on the integral. We note that the integral makes sense if the a are bounded measurable functions in ft, so we can allow ourselves quite a bit of generality.

A Dirichlet form B of order m in ft is a function

\<x\mI 3 l < m

x e i l , l . 'n s R " , where the coefficients are bounded measurable functions in fi. Associated with the form B we have the Dirichlet integral

B[u 1 u] = X f dx,jal m "Tl I3l m

Since the are bounded there is a constant Cb > 0 such that

IBCu IvII^Cb IIuL llullmfor u,veH '^(Q ). Thus the Dirichlet integral is a bounded sesquilinear form on

E xam ple 3.1. In IR the Dirichlet forms (gf - gi)(^ i ~ ^2) + 1V2

and (^? + l 2)('ni+ Tli) t)Oth give rise to the Dirichlet integral JAuAudx on H ^xH ^

We can now state the Dirichlet problem in the form that we will consider it.

Dirichlet p rob lem . Given a Dirichlet form B o f order m on f l and data fGH~"^{Cl) and find u g H*^{Q) such that B[u\<f)] =( f I <l>) for each g C~(ft) and such that u - g e H q (ft).

As it stands this problem is much too general. We will first show that if B satisfies a strong lower bound condition then the Dirichlet problem has a unique solution. In the case that B is also Hermitian symmetric and positive we will obtain the classical Dirichlet principle: the solution of the Dirichlet problem is the unique solution of a variational problem. We then proceed to consider more general B satisfying a strong ellipticity condition. In this case Garding’s inequality provides a lower bound for the Dirichlet integral. This lower bound, weaker than the first one we consider, suffices to yield a Fredholm alternative for the Dirichlet problem in the case that ft is bounded.

The solution of the Dirichlet problem depends on an abstract functional analysis result of Lax and Milgram [1].

T h eorem 3.2. Lax-M ilgram . Let H be a Hilbert space and let B be a sesquilinear form on H x H . Assume there is a constant C > 0 such that

1jB [ u , I u]|<C||u|ll|u|l f o r u, v g H.

Then (Au \v) = B[u,\ u], u ,v e H , defines a continuous linear operator A on H and ||A||<C. I f there is a constant 8 > 0 such that

|B[m, I m]|>6 llu|p for u e H

then A is one-to-one and onto and 1|A“ ||<6“ .

Proof. The first part is obvious by the Frechet-Riesz representation theorem for continuous linear functionals on a Hilbert space. For the second part

6 llulp<lB[u, I u]l = l(Au I m)|<11Am1|11u1|

implies HAmH. Thus A is one-to-one and |lA“’||= 6“ on therange of A. In particular the range of A is closed. If v is orthogonal to the range of A then

0 = \(Av I v)\ = \B[v I u]l > 6 IjulP

implies v = 0 . Thus A is onto.

T h eorem 3.3. Let B be a Dirichlet form o f order m on H. Assume that there is a constant 8 > 0 such that

\B[u I m]1 > 5 11m|1 for each u eI f feH ~'^ (Cl) and then there exists a unique ueH "^(0) suchthat B [ m I </)] = (/1 <#)) for each and such that u —gG H o(fl).Moreover,

llu L < (i+ 6-^ C B )llg lL + 6-M l/ U .

Proof. The uniqueness follows by the inequality in the last part of the statement of the theorem. Consider now existence. By Theorem 3.2 we have an isomorphism A of H q (fl) such that (Au | v)^ = jB [m | u] for each u and V in Moreover, HA|1< C b and 1|A“ 1|:<6“\ Define aconjugate linear functional F on H q (ft) by

<F|t;) = </|r)-B[g|t;],

Then

W l^ ll/IU +C al|glL

Therefore by the Frechet-Riesz theorem there is a unique wGH o(ft) such that

(w|u)m = </|u)-B[g|u], veH'SiCl).

Moreover, l|wl|„, = |lfl|. Now let u = g + A “'w. Then u e H "’(fl), u — g e H'Si.Cl) and

B[u I <1>] = B [g I <#.]+B [A “'w I d>] = B[g I <>] + (w I <p)„ =</!<#»)

for each </>€C“(ft). Moreover, l|ulL ^ l l g l L H w | L .

T h eorem 3.4. D irichlet principle. Let B be a Dirichlet form o f order m on ft. Assume that B is Hermitian symmetric and positive on H q (ft);

that is.

B [u\ v] = B [v\ u] for u ,veH "Si^ )

B[u I m] > 6 llwll for u e H q (ft)

where 6 > 0 . I f g e H ”'(ft) then there is a unique function ueH"^{H) such that B[u\<f)] = 0 for each < >GC“(ft) and such that w —geH o(ft). Moreover, u is the unique function in {g} + H o(ft) which minimizes B [u I m].

Proof. Let u be the unique solution of the Dirichlet problem in the first part of the conclusion of the theorem (exists by Theorem 3.3). Let u^{g} + Wo(ft). Then v - u e H o ( I l ) and so 6 ||m — —u | u-w ]. Since C “(ft) is dense in H o(ft) we have B [u \ v -u ] = 0. It follows that B [v — u\v — u] = B [v\ v] — B [v\ u] which implies jB[u | w] is real. But then we also have B [ v - u \ v - u ] = B[v\v]-2B[v\u~\-^-B[u\u] and therefore B[u \ u] = B [v | w]. We now have B [v ~ u \ v — u] = B [v I v] — B[u I u] and therefore

B[u I u]:^ B [v I v] — 8 llw-ull

It follows that B[u j u ] 0 such that

R e B [u I u ]> S

for each u e E . Here Re means real part. If B is strictly coercive over E and Hermitian symmetric then B[* | •] is an inner product on E and the corresponding norm is equivalent to the ET” norm on E. In Theorem 3.4 we have a Hermitian symmetric Dirichlet form strictly coercive over H o(ft). Had it been strictly coercive over jFT”(ft), then Theorem 3.4 would be just a special case of the orthogonal projection theorem for the Hilbert space consisting of provided with the inner productB[-1 •].

E xam ple 3.6. Let ft be an open subset of R'". Assume that there is a line L such that the L-width of ft is at most d. Let

B [u I u] = ^ I DjUDjV dx, u ,v ej=i Jq

Then B is a Dirichlet integral corresponding to the Laplacian —A. In view of the Poincare inequality (Theorem 2.12) we have

B [ m I M ]>(l + d )" ' l|M||? for each ueH o(Q ).


Thus by Theorem 3.4 if geH^{Cl) there is a unique such thatAm = 0 in Cl and u — Moreover, u is the unique function in{g} + Ho(XI) which minimizes the classical Dirichlet integral

[ Z \DiU\ dx.Jn j = i

Moreover, we have

R em ark 3.7. The Dirichlet principle (for the Laplacian) was used as early as 1835 by Green in potential theory and by Riemann in 1851 to obtain the conformal mapping theorem. It was also used in the 1840s by Gauss and Thomson (Lord Kelvin). Riemann bestowed upon it its present name apparently because he learned it from Dirichlet. In these applications there was no difficulty in showing that B[u\u~\ has a suitable lower bound to guarantee uniqueness of the minimizing function. There was, however, a gap concerning the existence of a minimizing function. Riemann, Green and others were undoubtedly aware of the gap but found the principle useful none the less. In 1870 Weierstrass showed that the Dirichlet principle is not universally valid. His critique served to discredit the Dirichlet principle until Hilbert put it on a firm foundation in 1899. In our abstract setting the Dirichlet principle is easy enough to prove. We have, however, assumed strict coercivity and allowed u to assume the boundary values in a generalized sense. The behaviour of u up to the boundary is studied, for example, in Agmon [1].

We turn now to the question of lower bounds or coercive inequalities for Dirichlet integrals. Note that a Dirichlet integral defines a continuous linear operator P by

{Pu 1 u) = B[u \vl UG V G HS^(fl).

Explicitly we have

Pm= Ilal mI3l:sm

If aa3 € C°°(d) then P is a partial differential operator with top-order part given by

\a\ = m iei=m

The principal symbol of P is then

\ot\ = m I3 l = m

For any Dirichlet form

J5(x, 4 t))= Y.jal^ml3l m

We therefore define the principal symbol of B, denoted by B(x, |), by

la l = m I3l = m

Note if < € Cc (fi) then

1 = £ B(x, \(x)\ dx +

Suppose now that the are bounded continuous functions in ft for 1 1 = iPl “ Since

for any <#> 6 C"(ft) it follows that the strict coercive inequality

Rc B [m I u]>Co11m|L ueH oiO .),

implies

ReB(x,^)>Co|^p^ xG ft, (* )

We will say that B is uniformly strongly elliptic in ft if (* ) holds for some constant Cq > 0 . Such a constant Cq is called an ellipticity constant for B.

Let E be a closed subspace of H""(ft) with C“(ft) ^ E . A Dirichlet form B is said to be coercive over E if there exist constants 6 > 0 and A such that

Re B[u I u ]> 8 llwll^-A \\u\ u e E .

Clearly this inequality also implies (*). The basic fact, due to Carding [1], is that the converse is true; that is, any uniformly strongly elliptic Dirichlet form is coercive over H q (ft).

Theorem 3.8. Girding's inequality. Assume B is a uniformly strongly elliptic Dirichlet form in ft with ellipticity constant Cq and assume that the coefficients a Q, are uniformly continuous bounded functions in ft for lal = l|3l = m. Then there exist constants C i>0 and Aq such that

R eB[it I m]>C iCo Ilwllm-Ao Hullo

for each u €H ^ (ft). The constant Ci depends only on m and n and the constant Aq depends only on m, n, Cq, the upper bound o f the coefficients o f B and the modulus o f continuity o f the coefficients a for [a] = ||3l = m.

We will only sketch the proof. A very detailed proof may be found in Agmon [1]. It suffices to prove the inequality for u g (fl). Suppose first that the coefficients are constants for \a\ = \\ = m and that B(^) is the principal symbol of B. Then

B [u I m] = (2 i7)-"| b (^) \a($)\^d^+R{u)

where 1jR(m)|<c l|uH ||ul| _i. It follows that

Re B[u I u]>Co lulm -c l|uL l|ulU_i^Collull^-c IImIU IlMlL.i-mcollMll^ _ i .

Now for any s > 0 we have

iiu L -iiiu iL < | iiu r^ + ^ iiu ip _i.

If we choose e sufficiently small it follows that

R e B [u I u]>icollull^-c'llwll^_i.

By Corollary 4:2.22 if e > 0 then

Taking e sufficiently small we obtain

R e B [u I u]> | co11m11^-c"11u11.

Next one shows that the inequality persists if the top-order coefficients of B are allowed to vary a little. The general result is then obtained by a partition of unity argument. This method of proof, freezing the coefficients, varying them a little, and then using a partition of unity, is known as the K om trick.

We finish this section with a few remarks on the Dirichlet problem in the uniformly strongly elliptic case. Consider a Dirichlet form B which satisfies the coercive inequality of Theorem 3.8. For any complex number A. we define

B x[m I v] = B [u I u] + A(u I u)o.

We note that if ReA>Ao then B is strictly coercive over H'SiO). If is the operator associated to B then

<(P+A)u|<^>) = B;,[u|<#>]for each u e H " ' ia ) and each <f> 6 0^ (0). Thus if g eH " '{a )and ReA>Ao then there exists a unique ueH"'(D,) such that

Pu + \u = f

u - g e H S i f l ) .

Moreover, u depends continuously on / and on g (Theorem 3.3). In case g = 0 we denote the solution u by T J . Then is acontinuous linear map and

(P + A)Tx = l on

Tx(P + A) = l on

Let Po be the operator P considered as an unbounded operator on L^(fl) with domain

dom (Po) = {u e | Pu e

If Re A > Ao then the resolvent jR = is given by the composition

L^(H)-----> HS^(n)-----> L^(fl)

and so is continuous. Thus Pq is a closed operator. If Cl is bounded then by Rellich’s lemma (Theorem 4:3.12) the inclusion Hg ^ L\Cl) is compact. Therefore, in this case the resolvent jR is compact which implies that Po has closed range. Indeed if we set Kx = AR Kx is compact and

PoRx = l- i^ x on L^O)

R kPq = 1 ~ on dom (Po).

Moreover, the kernel of 1 - jKx is contained in dom (Po). Therefore by a theorem of Riesz (see Palais [1, Chapter VII, Theorem 4] or Yosida [1,X.5 Theorem 3]) Pq is an (unbounded) Fredholm operator with index 0. That is, dim ker Pq is finite, Pq has closed range, and if P * is the Hilbert space adjoint of Po then dim ker P * = dim ker Pq.

Setting N = ker P * we obtain the following result.

Theorem 3.9. Let B be a uniformly strongly elliptic Dirichlet form on the bounded open subset Cl o f IR". Assume that the top-order coefficients o f B are uniformly continuous in Cl.

Then there exists a finite-dimensional subspace N o f L^(Il) such that if feL^(Cl) then the Dirichlet problem

B [u I < ] = </| <#>),

u eH 'S id )

4 > e c : im

has a solution u if and only if f is orthogonal to N. The set o f solutions u is a subspace o f H q (Cl) with dimension equal to the dimension o f N ; that is, equal to the number o f independent conditions on f.

Since our main interest in this chapter is Garding’s inequality we will

leave off the Dirichlet problem at this point. One final comment, however: considering the Dirichlet problem for the Laplacian in the context of limits the boundary values, in the case that 3ft is smooth, to liein H^^ (3ft). The restriction is an artefact of the method and is not always desirable. Classical formulations involve continuous functions, U functions, or even measures as boundary values. A nice discussion of the Dirichlet problem for the Laplacian in the unit disc in these cases is given in Hoffman [1, Chapter 3].

§4. Discussion of GSrding's inequality

Let ft be an open subset of IR' . Let m = 2k where k > 0 is an integer. Let

P = I|ot|<m

where eBC^(Q ). Here BC*"(ft) is the space of functions on ft having bounded continuous derivatives through order fc. The principal symbol of P is then

p(x, |) = ( - l ) ' ‘ Z a<,(x)|“.la| = m

Assume that P is uniformly strongly elliptic with ellipticity constant Co>0; that is.

Rep(x, |)>Coll|’”.

Then by Garding’s inequality (Theorem 3.8) we have

Re<Pu I u)>CiCo llMllm/2-c|lM||o

for each Here C i>0. Trivially then we have

Re <Pm I u) > - c llujg, u e

(1)

(2)

(3)

Actually (3) is no weaker than (2). Indeed, if (1) holds for some real number Cq, not necessarily positive, and if e > 0 then P - ( c o “ s)(l-A )'" has principal symbol p(x, ^ ) - ( c o -e ) 1 1"". The real part of this symbol is bounded below by s 1 1"" and therefore (3) implies

Re<Pu I u > > (co -e) \\u\\l,/2 -C e 11m|!o (4)

for each u e and each e > 0 . Trivially for s s 0 and by Corollary 4:2.2 for s < 0 we then obtain

Re <Pu I M> > (co - e) l|wl| /2 ~ Q,. \\u\ (5)

for each each e > 0 and each s ^ m/2. In discussing inequalities such as (5) it will of course suffice to consider the case Cq = 0.

4. DISCUSSION OF GARDING’S INEQUALITY 323

Inequality (5) is the inequality that we called Garding’s inequality in Chapter 4 in the context of pseudo-differential operators. To avoid hypotheses on our symbols at the boundary of f i we must content ourselves with a semiglobal version. Explicitly in Theorem 4:4.4 we showed if 0 < 6 0 , each s < m/2 and each compact subset K o f Cl there is a constant c = c(s, s, K ) such that

Re (Pu I m) > - 8 ||m|m/2- C u (6)for each u e Here m is any real number. Actually we proved (6) only for u € but it then follows for u € by the L^-continuity of pseudo-differential operators given in Theorem 4:4.5; just replace P by xPx where \ C “(fl) is real and ; = 1 in a neighbourhood of K, This form of Glurding’s inequality is given in Kohn and Nirenberg [1, Lemma 6.1].

Kohn and Nirenberg [1, p. 284] point out that, with p = 1, 6 = 0 and s = (m —1)/2, the inequality (6) holds with 8 = 0 in some cases. This sharper result is called the sharp Gdrding inequality and is true in general. Explicitly if P€'4^^s(fl), 0 < 6 0 then for each compact subset K of ft there is a constant Ck such that

Re (Pu 1 u) > - C k llM||fm-p+8)/2 (7)

for each u €Hormander [4, Theorem 1.3.3] first proved a sharp Garding inequality.

Lax and Nirenberg [1, 2] proved a sharp Garding inequality for systems. Friedrichs [1] introduced a clever proof using multiple symbols and a symmetrization. Kumano-Go [1, 2] extended the result to operators of type p, d and Vaillancourt [1] introduced a simplification in the proof. We will prove the sharp Garding inequality (7) by the method of Friedrichs et a l , following the presentation of Taylor [1] and Unterberger [1].

Note P = p {X ,- iD ) + Q where Q If we chooseCc (ft) with ; = 1 in a neighbourhood of K then xQ x maps H* continuously into when 2f = m - p + 6 and (Qu | u) = (xQx^ I whenever u has support in K. Thus

Re (Qu 1 m) > —Ck ' ll(m—p+8)/2

Since Q is arbitrary it is clear that (7) is the best possible result unless we impose conditions on the lower order symbols of P. Alternatively we can focus attention on the operator p(X, —iD) and study conditions on p(x, |) (which is then the total symbol). For example, Fefferman and Phong [1] have shown that if peS'^(ft,[R”) and p > 0 then

Re (p(X, -iD) M I w) > -C k l(m - 2)/2

for u G Horaiander [10] gives a similar result with a weaker lower bound, but relaxes the condition that p > 0.

In another direction if P eC ^'^iC l) has principal symbol p with Re p(x, then the shsirp Garding inequality (7) certainly implies

Re <Pu I M>> - e |lMl|?m-i)/2- c llwllf, u e C k (8)

for some e > 0 . Mellin [1] has given a condition on the lower order symbols of P necessary and sufficient for (8) to hold for each e > 0 (where C now depends on K, s and e).

We note that there is a converse to Garding’s inequality. Suppose we have P g with symbols a, and assume

Re <Pu I u) > - C Hulls, u e Cr . (9)

By Lemma 3:2.8 and the remarks at the beginning of Section 3:6 we have for <l) G C k the asymptotic expansion

j ^ O a

uniformly for ll| = 1, as A —> oo. Thus

:-\cc\ r$)D “4>(x)<f>(x) dx

)

‘ 00 we conclude

<Pe' < ’•><) I e‘ < ’-></)) = A"' f ao(x, \<l>{x)\ dxJft

On the other hand by Exercise 4:3.10 we have

l|e <’•>c>lg = A ll<#>|g + 0(A^-^).

Therefore, if in (9) we set u = and let A

Re ao(x, ^ )> 0

provided that s < m/2. We obtain the lower bound —C if s = m/2.We finish this section with a word on the proofs that will be presented.

The G^ding inequality (6) we proved in Theorem 4:4.4 by constructing a properly supported operator B g such that

K P* + P) + 6 (l-A r^ ^ = B * B + R

where jR e T h e s > 0 was required in order to be able to take a smooth square root of the real part of the principal symbol of the operator on the left side above. The sharp Garding inequality (7) we will prove by constructing an operator Q g such that

1(P* + P) = Q + K

where R and where Q is positive, that is, (Q</> | forG C k- The difficulty is that the symbol calculus does not reflect positiv

ity. For this reason one enlarges the class of symbols (but not the class of operators) to obtain a symbol calculus in which we can identify a class of

5. GENERALIZED DIRICHLET FORMS 325

symbols which give rise to positive operators. The method is entirely analogous to the introduction of Dirichlet forms in Section 3. The new symbols will be mixed symbols of the form a(r],x,^). We will refer to them as generalized Dirichlet forms. Alternatively lower bounds may be investigated by means of the Weyl calculus. See Hormander [10].

§5. Generalized Dirichlet forms

In Section 3 we associated with the Dirichlet form

B(x, 4 T}) = X

the operator P defined by

PM = Z(-l)'® 'D ®(a„3D “u).

Thus P = b(^iD, X, -iD) where 6 (t), x, |) = B{x, —ir]). If we formally compute Pu and then, in the resulting expression, introduce the inverse Fourier transform of the Fourier transform of D “w we obtain

Pu(r\) = (27r) b(Tj, X, ^)u($) d| dx. (1)

We will adopt (1) as the definition of Pu when b belongs to a suitable class of mixed symbols, or generalized Dirichlet forms.

The mixed symbols which will play the role of Dirichlet forms are the functions b g C “([R ”) such that for each compact subset K of and for any multi-indexes a, |3, y there is a constant C = C(K, a, /3, y) such that

-plotl-t-8'|3l^j|^ _j_ j^m"—p 1-y1+6"131lD-D?D|fe(T], X, ^)1<C(1 + I||r (2)for each x g K and t ) g IR"". We denote this space of symbols by and will refer to these symbols as having order (m', m") and type ip,8',8"). We will always assume that 0 < p :^ l and 0 < 6 ', 6 "< 1 . The best value of the constant C in (2) we denote by |blic.a,3.'r We topologize

by the seminorms HK,a,3,T- If Frechet space. If X is acompact subset of DR” we denote by the space of symbols

such that fc(7], x, ) = 0 if x^ K . It is also convenient to introduce the notation S^s for the space S^s(IR”, IR”) and the notation for the symbols p such that p(x, |) = 0 for x^ K .

Assume now beS'^'s^s\K for some compact subset K of R ”. If u e Cc(R'') then the integral in (1) is absolutely convergent. Since

we may integrate by parts in (1) to show |T]yPM(i7)|<C(l + |i7|)'” If we iterate this argument and keep in mind that 6 "< 1 we see that Pu is

rapidly decreasing. Thus Pu e C “(IR'*) and

Pu(x) = (2ir)-"| e ‘<''- >Pu(ri) dri. (3)

If we change the order of integration in (1) and introduce the Fourier transform

= j e - ‘<“- >i)(T), y , i ) d y

we obtain

^ ( t,) = (277)-"! b(v, m ( i ) d l

By (3) and (5) we have

Pu(x) = (277)-^''! I e ‘<”->b(7,, m c e d^ dr,.

(4)

(5)

(6)

For any iV > 0 and M > 0 the integrand in (6) admits a bound of the form

c ( i+h +II - 'oD-^d+as we see by integrating by parts o)"b(ir), co, in (4). If we let t = 1 — 6 " > 0 then by Peetre’s inequality we have a bound of the form

It follows that the integral in (6) is absolutely convergent. We may therefore change the order of integration to obtain

Pu(x) = (2t7) I ) m(I) (7)

where

pix, e = (277)-"! +77, 7,, I) dT,. (8)

Estimates similar to those above show that p e

Theorem 5.1. Let beS^ s' ' K for some compact subset K o/ IR Let m = m' + m", 8 = 8' + 8" and assume 0 ^ 6 < p < 1.1/ p is defined by equation (8) then p € S^s, p(X, - iD ) = b {-iD , X, - iD ) and we have an asymptotic expansion

p ( x , a ~ I ^ D ; D “b(7,,x,|)U=,. (9)a a !

Moreover, if b varies in a bounded set in then p varies in a bounded

5. GENERALIZED DIRICHLET FORMS 327

set in S^6 and the expansion (9) is uniform. In particular p{x, - b{^, x, varies in a bounded set in

Proof. We have already seen that p E C “(R^”) and p {X ,-iD )u = b ( - iD ,X ,- iD )u for m€ C “(R”). Consider now the Taylor expansion

b(^ + r\,ri,$)= X 7J, |)|e=j + i?N('n. !)•x \ < N ^

Here

1 )| < Cn sup \DJb(^ + tv, V, 1)1 In 1

(10)

(11)where the supremum is over l7 l = N and 0 < f < l . If we take the inverse Fourier transform in (10) and recall (8) we obtain

-IdP(x,€)= I — D :P | h (0 ,x ,| )| e = 4 + R N (x ,| )

l a l < N ^

where

R^(x, e = (2rr)-"| I) dr,.

( 12)

(13)

The general term in the asymptotic series in (9) is in Thus byTheorem 3:2.12 it suffices to show that for each compact subset L of R ”, each pair of multi-indexes a, |8 and each integer N > 0 we have constants C'l,n> Cl /3, L), and jUL such that i and such that

lD|D?p(x, 1)1 < Cl.«,3(1 + l|l)''<“-P- > (14)

and

lR^,(x,|)l<C^^,(l + lll)-- (15)

for X € L. First note we have

lT,^D2b(0, V, )1 = ||e-‘<^->D?D3h(d, X, |) dx|

< c (i+lil)"''' *'' '(i+leD- '-oi i+sieisince h = 0 if x^ K . Therefore for any integer M s :0 we have

\D b(e, V, 1 ) 1 ^ Q m ( 1 + 1 I 1 ) ' " ' " ® ' ' ^ ( 1 + l 0 l ) ' " " - ^ l ^ i - « " M ( i + ( 16 )

From (16) and (11) we have

iR N (v ,e \^ c^ .M sup ( i + i i r '" « ’'^(i+ii+tr,r"-o^-«"^(i+iT7i)^-^0 < K 1

(17)

for any integers M, N > 0 . Suppose first that 1^1<21t)1. If the first two

exponents in (17) are positive we have a bound of the form| ^ | )m + (l-p )N -(l-8 )M

If either of the first two exponents is non-positive we discard the corresponding factor and obtain a similar bound. In any case we see that if M is sufficiently large we have a bound of the form

for any v > 0 . Consider now the case 21t)1<1|1. In this case

Therefore if in (17) we take M = N we obtain the bound

From (13) we then have

1 .■Nix, i ) i < c : f (1 + hi)"*' dr) + CA f (1 +11)— dr,.l€l 2hl M\ 2\n\

The first integral here is as rapidly decreasing in | as we please by taking V sufficiently large. The second integral is equal to the integrand times the volume of the ball of radius \\. Thus

1)1 < c ; i ( i + (18)

for X, ^elR". Now (15) certainly follows. Next we prove (14). From (8) we have

D|D?p (x,^) = (277)-'-

= (2ir)-"

D|D?(&(ti, y, |)e‘<«-’*- -’'>) dy dr,

W/

X D p{7], y, dy drj

Since we wish to show that we have polynomial growth in as x varies in a compact set it will suffice to estimate

jI r,^y“D ^b(v, y, |)e-«-^-^-"> dy dr,. (19)

Let bo(v, y, I) = y°‘b(ri, y, ^) so Then (19) may be written as

I j n ^ D p o iv , y, ^)e‘<«-’’- dy dv

= I v ^ D X ( v. V - l 0)le=se‘< -«-"> dr,

= |(tj +a^D3ho(T) + 1, V, 0)le=«e‘<’ -"> drj. (20)

6. THE FRIEDRICHS’ SYMMETRIZATION 329

Let M s 0 be an integer. If lal = M we have

T)“D3bo(h + 4 V, 0) = r ' “'| D^D^boiv + e y, 0)e-‘<’> '’> dy.

Since bo = 0 if y ^ K we obtain

ID X ( v + 1 V, 0)1 ^ Cm (1 +1^ + + l^D-'-oWI+a-M + 1. |)-M

In view of (20) it follows that (19) is bounded by

Cm j dt}.

If we apply Peetre’s inequality we obtain the bound

c ^ (l + l|l)"'+s' -oi W0i| (1 + lT,l)'"'"W®i-<*-«">' dTj.

Since 6 "< 1 we may take M large enough to ensure that the integral converges. The estimate (14) then follows. For the last part assume that b varies in a bounded set B in In this case the estimates (14) and(15) are uniform for b e B . Moreover, the general term in the asymptotic series in (9) will vary in a bounded set in By the proof ofLemma 3:2.10 there is a uniform asymptotic sum q of this series. Note q varies in a bounded set in S^s. The uniformity of the estimate (15) then implies that p(x, | )-q (x , |) is rapidly decreasing in uniformly for x in compacta and uniformly for b e B . The outline of the proof of Theorem 3:2.12 and the uniformity of estimate (14) then implies the same result for the derivatives of p - q . Thus p — q varies in a bounded set in Sp 5 for any t.

(1)

§6. The Friedrichs' symmetrization

Let K be a compact subset of R ” and let p e Let i/r g C“(R^'') be a suitable function and consider the symmetrization

I) = j B)p{x, e)\i>{ , e) de.

If P = h(—iD, X, —iD) then formally for u e C“(R"") we have

(Pm I m) = (27t)“”(Pm I m)

” 1 1 0)u{x)\^ p(x, 6) d0 dx.

In particular if p > 0 then P is a positive operator.

(2)

The symmetrizer we take to be given by

0 ) = ( 1 + - m + (3)

where r = (p + 6)/2, <f> € (IR”), <f> > 0, is an even function, < >( ) = 0 forl l l > l and i d$ = l . For each 0 we have il/(-, 6) e S~°° and so the operator 6) makes sense. Moreover, the formal calculations whichlead to (2) may easily be justified.

L em m a 6.1. I f p eS ^ sK '^here K is a compact subset o f R ” and 0 < 6 < p < l , and if b is given by (1) then beS i^^^K- I f P varies in a bounded set then so does b.

Proof. We have

0) = (1 + I (4)

where = (0 — )(1 + and the sum is over I7 I < |/3| and y '^ 7 . Here^ ^ c - l3 l+ ( i—t)1y —/'I/— o - t131

If e, = (0 ,0, . . . , 1 , . . . . , 0) where 1 is the /th component then

<#>Ei,0,o(l) = 1/(1 + 1 1 )“'

These formulae verify (4) in case j/Sj = 0 or 1. The general case follows by induction. Now by (1) and Schwarz’ inequality we have

\ D Z D ^ ,D % b ( r i , X , 1)1 < ( I \ D Z 4 > (r ,, 6 ) D ^ M x , 0)P

x(j|D |«/.(e0)Pd0y'^. (5)

By (4)

I \D%Hl 0)p d0 £ C„(l + l|p)-i“i— ^ j | e£,2;< >( o)Dr< >(a.)l d0.

Making the change of variables = (0 — + so dco =+ and integrating over lco|<l, since 4>(co) = 0 if |<o|>l, we

obtain

|lD|»;r(e0)pd0<a(i+l P)-- t 1«1 (6)

for suitable constants C„. Consider now the first integral on the right side of (5). If w = (0 - t})(1 + 1t}P)“’' we have by (4)

f 0)D ip(x, e)\ d e < c , ( i + « f ]d ?p (x, de

< C ^ (l + lt i lT "w [ 1D ?p (x, t,

+ (7)

Here we have used that 0 < t < 1 and la>l<l imply

Cl lTjl<lT]+Ct)(l + lT]l )' ^ l<C2 h i

for large hi- Combining (5), (6) and (7) we obtain

lD ^D ? D |b (ri, X, |)l^C<,,3,^(l + lT,l)'”-«'^'-^W(l + l^ ln '“ '

as desired. The constants here involve certain universal constants and the estimates on p. Thus the last part follows.

We have not yet used the fact that <l> is even. This hypothesis leads to some pretty cancellation as the next theorem shows.

T heorem 6.2. Let peS^s,K y^here K is a compact subset o f IR” and 0 ^ 6 < p < l . Let be given by (1). Then

(A) b ( - iD ,X ,- iD )E T ^ -(R ”)

and

(B) b(-iD , X, - iD ) - p(X, - iD ) 6

Proof. By Theorem 5.1 we have b { - iD ,X ,- iD ) = q (X ,- iD ) where q e and

i~\cL\q(x, i ) q c X x , ^ )

a a!where

qa(x,|) = D “D “i)(7},x,|)|^=5. (8)

Since b € S°Io.s,k ^nd t — 8 = (p — S)/2 we have^ o m - ( p - 8 ) l a l / 2 / qx4a r,8,K (9)

Our first job is to improve (9) by showing that actually has type (p, 8).

By (1), (4) and (7) we have

q jx , I) = j 0)Dtp(x, 0)i/>(l 0) d0

= (1 + I 4><.y.Ai) I <a' 'Dl4,{u>)Dtp{x, 0)4>{oi) d0

+ (D{l + \ \ y'^WDZ<l>i<o)<l>i<o) d(0

where the sum is over |7 |<la| and y '^ y and where (o = + Differentiating this last integral and keeping in mind

that T < 1 we see that e Since t — 6 = (p — 8)/2 we obtain

(10)It follows that there is q 'eS^ s such that q '~ Z Then q — q 'eS ““ implies q e S^s and so (A) follows. We can improve further on (10). For laj = 1 we have

(x, I) = I [ £>;p(x, I + 0)(1 + \er'^)

X <o 'Dl<l>{<o)<f>{(o) d(0 (11)

where the sum is over I7 I < 1 and 7 ' < 7 . Note that 7 ' = 0 or 7 ' = 7 in each term. If 7 ' = 7 then we have <l>oc,y,y Spg and hence these terms give rise to an element of If 7 ' = 0 and |7 l = l then <t>cc,y,o S~l andwe need to estimate

.[<1D tpix, i + <o{l + \^mDZ<t>i<o)<t>M day (12)

where la] = |y| = 1. Here we resort to a Taylor expansion

D tp ix ,i + ayil + \ ^ m

= D?p(x, ^) + (1 + t is even

[ DZ<t>i<o)<t>Md(o = 0 ,

Thus if we substitute (13) into (12) the first term on the right of (13) yields no contribution and we obtain

[ ( l + llP)^'" i fu ;D e^p(x,|j^l Jo

+ to (l + \4\y>^)<opZ<l>M<f>M dt day

(14)

which clearly is in In view of (11) we obtain

if lal = l .

From (10) and (14) it follows that q — Hence to prove (B) itwill suffice to prove

n —/7 c:: C» -(P-S) (15)

We have

q o i x , I ) = f p ( x , I + <0(1 + d<t).

By Taylor’s formula

p ( x , I + <v(l + = p ( x , ! ) + ( ! + I l l X D e ,p ( x , |)<Dj)=i

+ ( l + lll")^ I <0-^ -. f ( 1 - t )1t1=2 7 Jo

X D2p (x, I + ro)(H-1 1 )" ) dt.

Recalling J 4>(o)) dco = 1 and noting dco = 0 since 4> is evenwe have

qo(^ ,l)-p(^ , ! ) = ( ! + HP)'' Z [ ( 1 - f )|-/l = 2 7- Jlo)|<l 01 1 = 2 7 i J\o>\

X D2p (x, I + t0)(l + \$\Y )o)' <l>(o)) dt do).

Since m —2p + 2r = m - ( p - 6 ) (15) follows and the proof is complete.

R em ark 6.3. Let p vary in bounded set B in and let b be given by (1). Then b varies in a bounded set in L^t

q(x, I) = (2rr)-"|e‘<” ’‘>h(| + rj, t,, |) dr,

SO q varies in a bounded set in S ”s (see Theorem 5.1) and X !(L’“V«0^a uniformly for p e B . The expressions in the proof of Theorem 6.2 show that q varies in a bounded set in fQj-a, and varies in a bounded set in if |a| = 1, and moreover thatqo~ p varies in a bounded set in for p e B . As in the proof ofTheorem 5.1 there is q'eS ^ s such that (i”'“'/o:!)qa uniformly forp e B . Then q -q 'eS ~ °° uniformly for p e B . It follows that q varies in a bounded set in S^s and p - q = r varies in a bounded set in Inparticular

p(X, - iD ) = b{-iD , X, - iD ) + r(X, - iD )

where r varies in a bounded set in for p e B .

Theorem 6.4. Sharp Girding inequality. Let ft be an open subset 0/ R M / 0 < S 0 for x e f t , |elR" (17)

then for each compact subset K o f Q, there is a constant such that

Re <Pm I u) > - C k |lul|f _p+s)/2 (18)

for each u e

Proof. It suffices to prove (18) for u e C k for each compact subset K of fl . Indeed if K' is a compact subset of f l and K is contained in the interior of K' then is contained in the closure of in and both sides of (18) are continuous on The symbol of (P* + P)/2 isRep(x, ) modulo Thus we may assume that p is realand p(x, | )> 0 . If x :^C c(n), a: —0 and a: = 1 in a neighbourhood of K then {Pu I u) = {Pu | x^) for u e C k* Thus we may replace P by and p(x, ) by x(^)p(^? I), i o., we may assume that p has compact support in X. By Theorem 6.2 there exists B e ^ ^ 5(IR”) such that R = B - P = (B - p(X, - iD )) + (p(X, - iD ) - P ) g and such that (Bu | w) > 0for each ueC^(lR”). Thus

Re (Pu I u) = (Bu I u) — K e(R u | u).

If supp u ^ K then

\{Ru I u)l = \ixRxtt I m)1<Ck l|Mllf„_p+8)/2-

Here Ck is the norm of the operator x^X ■ - H -(m -p +S)/2

We can also prove a uniform version of the sharp Garding inequality.

T h eorem 6.5. Let 0 < 6 0. Then for each compact subset K o f there is a constant Ck such that

Re <p(X, - iD )u I m) > - C k |lul|f„,_p+s)/2

for each u e and each p e B .

Proof. Choose x e C ”(lR") such that x —0 rid x = l in a neighbourhood of K. Then P*{xu) = q {X ,- iD )u where

q(x, I) = (2ir) - j j e - <*n,x-y>p(y, ? + 'n)x(y) dy dr).

7. PROPAGATION OF SINGULARITIES 335

The estimates in the proof of Theorem 3:4.5 show that q varies in a bounded subset of S^s and moreover that

, - l « l

D% Dtixix)pix, D)a !

uniformly for p g B. It follows that

^iP*X + xP) = h (X ,- iD ) where

h(x, = Re x(^)o-(p)(x, i ) + r'(x, i )

and r' varies in a bounded set in But then

^(P*X + xP) = Q + r {X ,- iD )

where O is a positive operator and r varies in a bounded set in Thus

Re (xPu I u) > Re {r(X, —iD)u \ u)

for each u g C “(IR”). If u e C ^ then

l<r(X, - iD )u I m)| = l<r(X, - iD )x u | lluHf„_p+s)/2

for p e B by Remark 4:4.12.

§7. Propagation of singularities

Let ft be an open subset of IR”. If Xi , . . . , are coordinates on ft then we have coordinates Xi , . . . , . . . , on the cotangent bundle T*(ft)defined by

a = 1 1 , ( « ) d X f , a G T *(n).

Such coordinates on T*(ft) are called canonical coordinates. We have already discussed canonical coordinates in Section 3:7. We noted there if y i , . . . , y , 7]i,. . . , is another canonical coordinate system and J is the Jacobi matrix

Vaxfc/

then

= J'vwhere the prime denotes the transpose. Since dy = / dx we see that

<|,dx) = <J'T,,/-' dy) = <r,,dy).

In other words the 1-form

fi, = X dxy

is an invariantly defined 1-form on T *(n ). It is called the canonical l-/orm. The differential

o) = djuo = ^ d ^ j A d X j

is called the canonical 2-form.Consider now vector fields

and

dXj

on We have

<o(X, Y) = i Z m i , XXdx,-, Y)-<dx,, XXdly, Y »

In particular

b,. = <Ydxy) = 2 c o (^ , y )

i.;=<y.ds->=-2<.(A,y),

It follows that if F is any smooth function on T *(fl) then there is a unique vector field Hp on T *(fl) such that

<X,dF) = 2co(X,Hp)

for each vector field X on This invariantly defined vector field iscalled the Hamiltonian vector field of F. From the calculations above we see

H - T (— —__dX jdiJ'

If F and G are smooth functions on T*(ft) then

^ \a§ dXj dXj /

is called the Poisson bracket of F and G and is denoted by

{F, G}.

Note that G is constant on the integral curves of Hp if and only if Hp • G = 0. Since Hp * F = 0 we see in particular that F is constant on the integral curves of Hp,

The first jet bundle may be identified with T*(n)x(R in acanonical way. If u denotes the canonical coordinate on IR then for any coordinates Xi , . . . , on f l we have corresponding canonical coordinates Xi , . . . , x„, M, ^1, . . . , on J\Q ).

E xercise 7.1. I f F is a smooth function on show that the vector field

dXj ^ i u dXj du /

is invariantly defined on /'(ft) and that F is constant on the integral curves ofX p .

If ifieC '(Q ) then /i(j/>):ft—»/'(ft) is defined in canonical coordinates by

h W M = , x„, tfiix), ^ (x), • • ■. ^ (^))-

Then the general first order (non-linear) partial differential equation is F (iiW ) = 0. The vector field Xp plays an important role in the study of this equation. See, for example, Garabedian [1].

Consider the flow <#> :[R x —> T *(fl) defined by a ) = e*a. Theinfinitesimal generator E of this flow is called EuleFs vector field. It can be defined on any conic manifold. A tensor field T on —(0) is saidto be homogeneous o f degree m if «S?e T = mT where is the Lie derivative relative to E, Thus a vector field X is homogeneous of degree m if and only if [F, X ] = mX.

E xercise 7.2. 1/ Xi , . . . , x , ^i , . . . , are canonical coordinates show that

yr d

Show that a vector field X = aj(dldXj)-\-Y ti'jidld j) is homogeneous o f degree m if and only if

Oj(x, A ) = A”"aj (x, and a'(x, A ) = |)

for A >0.

Let P and Q be properly supported pseudo-differential operators on f l so P = p(X, —iD) and Q = q(X, —iD) with say p € S^s(n, IR”) and q e

IR") and 0 < 6 < p < 1. Then PQ has symbol

1 dp dqi ^ d j dXj


where r g IR”). Moreover, r varies in a bounded set if pand q vary in bounded sets in the appropriate spaces. It follows that [P, Q] = P Q -Q P has order m + m ' - ( p - 8 ) and symbol

-i{p , q} + r'

where r' e Moreover, r' varies in a bounded set if pand q vary in bounded sets in the appropriate spaces. The assertions concerning r and r' follow from the estimates in the proof of Theorem 3:4.5.

Let P be a pseudo-differential operator with principal symbol p e S^s(n, R"") so P -p (x , Then P * has principal symbolp .’Thus if A = (P* + P)/2 and B = (P *-P )/ 2 i then P = A + iB, A has principal symbol a = Re p and B has principal symbol fc = Im p. We use this notation below without further comment.

Assume now is properly supported and has homogeneousprincipal symbol p = a + ib. Let be the Hamiltonian vector field of a. The integral curves of on which a = 0 are called the null bicharacteristic strips of A. Classically a curve in the cotangent bundle is called a strip. The reason is that by associating with each cotangent vector its annihilator in the tangent bundle we may regard the strip as a smooth family of tangent hyperplanes. If P u = f we recall (Theorem 4:6.5)

WF,(/) c WF,(/) U Z(P)

where

z ( P ) = { ( x , i ) |x € n , ^ g R "-(0 ), p(x, | ) = 0}.

Note that Z (P )^ Z {A ) and Z(A ) is a union of null bicharacteristic strips of A.

We will now show that if P u = f then the singularities of u which do not correspond to singularities of f propagate in the forward direction along those null bicharacteristic strips of A for which b > 0 in a neighbourhood of the strip. This theorem is due to H5rmander [6].

T h eorem 7.3. Let Pe'^TA ^) properly supported and have homogeneous principal symbol p = a + ib. Let y be an integral curve o f and let I = [to, ti] be contained in the domain o f y. Let ue^\Li), P u = f and assume b > 0 in a conic neighbourhood o f y(I).

I f 7 (/)nW F,(/) = 0 and 7 ( 0)^ WF,+^_i(u) then 7 (1) ^ WF,+^_i(u). Moreover a = b = 0 on y(I) and Hj, = 0 on y(I).

Proof. We prove the last assertion first. Since

7 (1 ) ^ WF,+m-i(i^) ^ W F,+^(m) ^ WF,(/) U Z(P)

and 7 (I)n W F ,(/ )= 0 we have y { I ) ^ Z { P ) ; that is, a = b = 0 on y(I).

(1)

Since b > 0 in a neighbourhood of y{I), b cannot change sign at y{I) on any curve passing through y(I). Since h = 0 on y{I) it follows that db/dXj and db/d j must vanish on y(I); that is, H f ,= 0 on y(I). F o r the first conclusion, if y{I) is not contained in WFs+m-\M then there is t' with

such that y{t')^WFs+m-iM- R eplacing t' by we see that we have to prove:

(If 7 ( J ) n W F , ( / ) = 0 and y { t , ) 4 ^ s ^ m - i M then| 7 (J )n W F ,,_ i(u ) = 0 .

B y replacing P by GP w here is a properly supportedelliptic op erator with real principal symbol we see that we m ay assume that m = 1. B y modifying u and / outside a com pact set we m ay assume that ue€'{£l). A ltering p outside a conic neighbourhood of 7 (1 ) and modifying / by a suitable distribution with w ave front set disjoint from y(I) we m ay assume h > 0 everyw here. Since u e we have ueH^ for som e t and so y{I)r\W Ft(u) = 0 , If r> s we are done. If t< s , making t sm aller if necessary we m ay assume t + 5 < s . If we now show it follows that 7 ( I )n W F t+ i / 2 (w) = 0 then an inductive argum ent yields the theorem . H ence we m ay assume that 7 (1 ) H W F s_ i /2( m) = 0 . If r(x , ^) = 1 in a small conic neighbourhood of 7 (1 ) and r(x , |) is hom ogeneous of degree 0 in | and R is a properly supported op erator with total symbol r then u = Ru-\-(l-R)u w here Ru^Hl~^'^ and ( 1 - jR )m has w ave front set disjoint from a conic neighbourhood of y{I). T hen PRu = f - P ( \ — R)u has the sam e fT '-w ave front set as / in a conic neighbourhood of 7 (1 ). M oreover, WF^ ^Rw) = WF^ ^w) in a conic neighbourhood of 7 (1 ) for any s'. H ence we m ay assume that u e If7 ( J )n W F s (M )^ 0 let t2 = sup t w here and y{t)eW P sM - SinceW F s( m) is closed we have y{t2)eWFs(u). If we show that 7 ((t2 , t j ) disjoint from WF^Cw) implies 7 (t2)^ W Fs(w ), then this contradiction establishes that 7 (I )n W F s(M ) = 0 . W e m ay of course assume that t2 =to. H en ce we are now reduced to proving:

fj/ 7 U )n w F,(/ ) = 0 ,and

yiito, ti])nW F,(u) = 0 everywhere, then 7 (to) WF^(u).

(2)

L e t r be a small closed conic neighbourhood of y{I) with com pact projection R in f i chosen so that r n W F s C f ) ^ 0 . L et

M ^ { q e R ”) | q real and supp q ^ F },

and assume that M is bounded in S i o(ft, IR”)- W e will construct M later. C hoose i/f G (^I) so that i/r = 1 in a neighbourhood of supp u. F o r each q e M let Q = q (X , — iD )i/r(X ). Then Q is com pactly supported and

{ Q u \ q e M } ^ i s bounded in (3)

B y R em ark 4 :4 .1 2 . Since F H WF^if) = 0 we have Qfe If we choose

R as above suitably then / = R / + ( 1 R ) f w here R f^H ^ and Q{1 — R ) g B y the hypotheses in (2) Since 0 ( 1 - R ) = h ( X , - i D )

w here h varies in a bounded set in Si.o(H , IR") for ^ny convenient s ' we see that Q { \ - R ) f lies in a bounded set in H r . Since R f e H l we also have that Q R f varies in a bounded set in H r . Thus

{ Q f \ q e M }S H r is bounded in H r . (4)

Since A * = A we have Im <A Q m | Qw) = 0 . Thus

Im {Qf I Qu) = Im {QPu | Qu)= Im {QAu 1 Q M )+ R e {QBu \ Qu)= Re {BQu I Qw>+Re <[Q, B ] m I 0 “ )

+ Im ([Q , A ] m I Qu) (5)

Since b s O and QueH]p the sharp G ird in g inequality implies

R e < B O M | Q M )s-C ,llQ u | li for each qeM. (6 )

T h e principal symbol of Q * [Q , B ] is -iq{q, b}. This symbol is in Sj%:k and stays in a bounded set in S\%,k- Since it is pure im aginary the sharp

G arding inequality implies

R e <[Q, B ]u 1 Qu) = R e <0*[Q, B ]u 1 u)a - C 2 lN las- . ) / 2 = - 0 2 for qeM. (7 )

(See com m ent following (9 ).) If we set R - 2 iQ * [Q , A ] then R^ has principal symbol - 2q{q, a } = H .(q ^ )= {« ,9 \w h ich has ord er 2 s - 2 and L y s in a bounded set in M oreover, R , = r , (X , - i D ) w here r, has

corresponding properties. Since

Im <[Q, A ] m I Q m) = i R e I “ >

we have from (5 ), (6) and (7)

Im {Qf I Q u > 2 - C l IIQmIIo- C 2 + 5 R e <Rq« I «)•

Now

Im (Qf I Qm):s 1|Q/11o 11Qm|1o^2 11Qw11o+ 5 llQfllo

and by (4)

2C2+11Q/11o^ C .

Thus we obtain

R e< R ,u | u )^ (2 C i + l)llQMlg + C for qeM.

T h erefore , if we set T q = R q - ( 2 C i + l ) Q 'Q then

R e < T „ u | u )^ C for qeM. (8)

N ote that the principal symbol of is {a, q ^ } - ( 2 C i + l)q ^ . W e will now construct the set M L et

M = {q I 0 < 8 < 1}w here

w here h(x,^) is hom ogeneous of degree s in g for large 1||, UqG S%(Cl, [R""), ai e is elliptic on supp fi, and w here A is a realnum ber. W e will choose h, ao, ai and A later. Clearly IR")and q lies in a bounded set in Si,o(Î, R ”). W e will denote simply by Tg and we will denote its principal symbol by Now

{a , ql} = H «(h -f ea?)"^)

= {a , + 2 Ah^{a, ao}e^^'"‘>(l + e a f )“ ^

- 2 e a i h ^ { a , eaf)~^.T h erefore

te ={(^, q l)-(2 C i-\ -l)q l= {{a, h^} + (2A {a, a o } - 2 C i ~ l)h^)e^^“«(l + eî)~^

—2eaih^{a, saf}~ .

C hoose t' < to with t' in the dom ain of 7 . W e m ay assume that Ha is not radial at 7 (f) for any t (since otherw ise there is nothing to p rove; see E xercise 7 .7 ). Then we m ay choose a conic hypersurface S through 7 (t') transversal to at 7 (t') such that each integral curve segm ent of Ha starting in S ends up in a convenient small conic nieghbourhood Ti of 7 (ti), say with Ti ^ T and T 1 IT W Fs(w ) = 0 and such that these segments fill up a conic neighbourhood U of y{I). L e t F q be a conic neighbourhood of 7 (to) with F q^ U and F o lT F i = 0 . C hoose a sm ooth function w > 0 hom ogeneous of degree s such that w (7 (to) ) > 0 and such that supp w c F q. B y integrating along the integral curves of Ha we m ay solve the initial value problem

Hah' = w and h % = 0 .

N ote that W is hom ogeneous by the uniqueness of the solution to the initial value problem , by the hom ogeneity of Ha, by the hom ogeneity of w and by the fact that S is conic. Outside Fq we see that h' is constant along the integral curves of H^. If we multiply h' by a suitable cut-off function with support disjoint from a conic neighbourhood of 7 (ti) we obtain ^ ^ 'î,o,jc hom ogeneous of degree s for large 1 | and in a neighbourhood of 7 ( 1 ). M oreover, h > 0 , supp h ^ F , J 4 h > 0 outside F i and H ^ h > 0 on 7 (1 ) ~ F i . Now we m ay choose aoeS% K such that H ^ ao= 1 in a neighbourhood of supp h and e Sl,o,jc such that H âi = 0 in a neighbourhood

of supp h. Indeed, we may solve these differential equations to produce homogeneous functions in a conic neighbourhood of supp h and then cut them off by multiplying by a suitable homogeneous cut-off function. Thus we may assume that Uq and Ui are homogeneous for large | 1 and also in a neighbourhood of 7 (1). Since is constant on the integral curves of in a conic neighbourhood of supp h we may even arrange that Ui > 0, and therefore ai(x, | )> c lll , in a conic neighbourhood of supp h. Taking A = Cl + 1 we now obtain

te = {{cl, h^}+

Since {a, h^} = 2hHah we see that

({a,

is non-negative outside Ti and is positive on 7 (1) — Ti. Moreover, it is homogeneous of degree 2s for large 1| and in a neighbourhood of y{T). Enlarging Ti slightly (but keeping it disjoint from WFjj(w)) we see that we may choose r, g g S i ,o,k homogeneous for large ||1 and in a neighbourhood of 7 (1) such that r > 0 on 7 (1), supp r c r, supp g ^ Ti and such that

({a, g .

We set

Choose if/ e C “(fl) with i/f = 1 in a neighbourhood of K U supp u and let = r^{X,—iD)il/{X) and G = g^(X,—iD)il/{X), These operators are

compactly supported since r(x, = g(x, ) = 0 if K. Then the operator

GtG, + T,-RtRshas principal symbol

Since Ui is elliptic in a neighbourhood of supp h this symbol lies in for 0 < 8 < 1. Moreover, it stays in a bounded set in S?fo.ic- Thus the sharp Garding inequality implies

Re{{GtG, + T , -R tR J u | u)^-C\\u\\Un- (9)

Actually Theorem 6.5 gives (9) only for ueH ^ but for e > 0 both sides of (9) are continuous on (This observation also applies to the proofof (7).) Now (9) implies

\\RM £ Re <T,u 1 u)+ llG .u i+ C'.

Therefore by (8)

l|ReMlloS|lGeMlg+C for 0 < S < 1 . (10)

Now supp g ^ T I and Ti HWFs(w) = 0 . Choosing an operator of order 0 with symbol 1 in a neighbourhood of Ti as in the proof of (4) we see that Gû G H ]p is bounded in Thus (10) implies

1|K,m11o^ C " for 0< 8 < 1 . (11)

Now Rû G and if jR = r(X, —iD)«/r(X) then by Remark 4:4.12 Rû —> Ru in If <f>€Ce((R^) then <R,u, </>). By (11) it followsthat for each 4> g C"(IR”) we can choose e > 0 such that

\{Ru, <>)l < ||<>||o + \(R u, <>1 < (1 + C") ll<#>l|o.

Thus R u g H k- Since R g C^^(H) is elliptic on y(I) we conclude that 7 (J)n W F 5(M) = 0 . In particular y{to)^W Fs{u) and so (2) is proved.

Remark 7.4. Under the hypotheses of Theorem 7.3 we also have the conclusion if yiI)C\W F(f) = 0 and y(to)GWF(u) then 7 (/)^W F(m). Indeed, if to < t'^ t i and 7 (t')^WF(w) then there is a conic neighbourhood r ' of y(t') disjoint from WF(w) and an open conic neighbourhood Tq of 7 (fo) such that each integral curve segment of starting in Fq and ending in F' is disjoint from WF(/). Then by Theorem 7.3 Fq is disjoint from WFs(u) for each s and therefore disjoint from WF(w) since this set is the closure of the union of the sets WFs(w). But now we have contradicted that 7 (fo)GWF(w).

Corollary 7.5. Let P g Jô( I) properly supported and have homogeneous real principal symbol p. Let y be an integral curve o f Hp and let I = [to, ti] be contained in the domain o f y. I f y (I)n W F s(f) = 0 then either 7(1) HWFs+^_i (m) = 0 or else 7 (1)^ WFs+^_i(u). In the second case p = 0 on y{I). I f 7(1) H WF(/) = 0 then either y{I) C\WF(u) = 0 or else 7 (I)^ W F (u). Again in the second case p = 0 on y(I).

Proof. Since b = 0 we may apply Theorem 7.3 to P and to —P to observe that singularities propagate in both directions along the null bicharacteristics.

Let Xi , . . . , be rectangular Cartesian coordinates in f l and let x, | be the corresponding canonical coordinates on T*(fl). Then

S = {ix,i)eT*(a.)\\i\ = l}

is called the cosphere bundle of ft. The natural projection tt : T*(ft) ^ ft restricts to i f r S ^ f t . If g e C “(S) we define g € C ”(T *(ft)-(0 ))


homogeneous of degree 0 by

g U I) = g U ^

Let X be a vector field on —(0) homogeneous of degree 0. Then[E, X ] = 0 and so the local flows of X commute with homotheties in the fibres, i.e., are homogeneous of degree 1. Thus if y(t) = {x{t),^(t)) is an integral curve of X and A > 0 then 7 i(t) = (x(t), A|(t)) is also an integral curve of X It follows that if the maximal integral curves of X are projected onto the cosphere bundle S then the projections are either disjoint or coincident. These projections are in fact integral curves of a vector field X on S where

X - g = (X-g)\s for g e C -(S ) .

Note if co(t)) is an integral curve of X it is the projection of an integral curve of X of the form (x(f), e * o)(t)). A simple calculation shows that

if

X = \ cij----- '"Xu.- .^ 'dXj ^

Thus we may easily recover the integral curves of X from those of X

Exercise 7.6. Verify the statements concerning X above.

Exercise 7.7. Let Y be a vector field on T * ( f l) - (0 ) homogeneous o f degree r. Show that the ray through (xq, |o)> appropriately parametrized, is an integral curve for Y if and only if Y(xo> o) = cE (xq, q) where 0 and E is Euler's vector field.

Lemma 7.8. Let pG C "(T *(ft) —(0)) be real and homogeneous o f degree m. Let a eC l. I f no null bicharacteristic strip o f p is contained in the fibre T *(n ) then there is a compact neighbourhood K o f a such that no null bicharacteristic strip o f p remains in Tr~^{K). In particular this is the case if for each (0) we have that p(a, ) = 0 implies d^p(a, | )^ 0.

Proof. Let q(x, Then pq is homogeneous of degree 1 andHpq = qHp + pHq. In particular the null bicharacteristic strips of pq coincide with the null bicharacteristic strips of p except for the parametriza- tion. Thus we may assume that p is homogeneous of degree 1. In this case Hp is a vector field homogeneous of degree 0. Let Y be the induced vector field on the cosphere bundle S. If the conclusion of the lemma is


false then there is a decreasing sequence of compact neighbourhoods Kj of a such that H ={^^} and for each j, L, = 7r“ (JfCy) contains a maximal integral curve 7, of Y such that p = 0 on 7, (since p is homogeneous). Note that 7j(t) is defined for all t since is compact (see, for example, Nemytskii and Stepanov [1, Theorem 1.21]). Since the components of jjit) satisfy a system of differential equations and L j is compact, if we differentiate the equations we see that these components and all their derivatives are uniformly bounded on IR. In particular the components are bounded in C ”([R). Thus there is a subsequence which converges, say to 7, in the C“ topology. Obviously 7 is an integral curve of Y, p = 0 on the image of 7 and '77(7 ) = {a}. Thus we have a contradiction. For the last part if 7 (t) = (x(t), |(t)) is a null bicharacteristic strip meeting the fibre T*(ft) we may linearly reparametrize 7 so that 7(0) = (a, |o)- Since p(ci, lo) = 0 we have d^p(a, 0) ^ 0 by hypothesis. But

and therefore dxy/dtf 0 at t = 0 for some j. Thus 7 does not remain in the fibre T t m .

If p 6 C °°(T*(fi)-(0)) is homogeneous of degree m we say that p has simple real characteristics if p(x, ) = 0 implies d^p(x, |) 7 0. That is, if the restriction of p to each fibre T J ( f l ) - (0 ) vanishes of order 1 on Z (p)n T *(n ) where

Z ip) = {(x, ) e T*(ft) - (0) I p(x, I) = 0}.

In particular it follows that Z ip) is a conic submanifold of codimension 1 in T * ( f t ) - (0 ) and, moreover, that the intersection of Z ip) with each fibre of T *(fl) - (0) is a submanifold of codimension 1 in the fibre.

Let P e have homogeneous principal symbol p. We say that Phas simple real characteristics if p does. We say that P is of real principal type if p is real and if no maximal null bicharacteristic strip of p stays over a compact subset of ft. According to Lemma 7.8 if p is real and has simple real characteristics then P is locally of real principal type.

Theorem 7.9. Let P g^J^o(^) t>e o f real principal type. Then P is semiglobally subelliptic with a loss o f 1 derivative.

Proof. Let u e^ 'iil) and assume that Pm g jFP(ft,loc). Choose Q e '^Zoi^) such that P -Q e ^ ~ " i f t ) and Q is properly supported. Then QweHc(ft). Let 7 be a maximal integral curve of Hp. Since WF^iQu) = 0 Corollary 7.5 implies that the image of 7 is disjoint from W Fs+m-iM or is contained in WFs+^_i(w). In the second case 7 is a null bicharacteristic strip and has projection contained in supp u. This possibility is ruled out by hypothesis. Thus W F^+m-iM ^ 0 *


Since the principal symbol of the formal transpose P' is p(x, -^ ) the hypotheses of Theorem 7.9 also imply that P' is semiglobally subelliptic with a loss of 1 derivative. We therefore obtain the semiglobal Fredholm alternative, the local existence, and under an additional hypothesis, the global Fredholm alternative, of Theorem 4:7.26. In view of Lemma 7.8 we also obtain local existence for operators with real homogeneous principal symbol and simple real characteristics. To avoid any fuss about proper support we state a result for differential operators.

T h eorem 7.10. Let P be a linear partial differential operator with coefficients in C "(fl). Assume that P has real principal symbol and simple real characteristics. I f f e C “(fl) and a e H then there exists u e C “(fl) such that P u = f in a neighbourhood o f a.

Proof. There is a neighbourhood of a such that (PL)' is semiglobally subelliptic with a loss of 1 derivative. Now apply Theorem 4:7.26 to obtain u g C “(o)) such that Pu=/ in a neighbourhood of a. Now cut v down to obtain u e such that Pw = / in a neighbourhood of a.

For considerably more information and some examples see Duister- maat and Hormander [1].

Under appropriate Levi conditions the proof of Theorem 7.3 may be used to obtain propagation of singularities results in the case of characteristics of constant multiplicity along the lines of Chazarain [2], but without using the theory of Fourier integral operators. The crucial point is to note that the proof of Theorem 7.3 goes through for systems in the case that the principal symbol is a scalar times the identity matrix. We will not pursue these ideas here, but instead leave them for the reader.

Bibliography

Adam s, R ., A ronszajn, N. and Sm ith, K . T .1 Theory of B essel potentials, part II , Ann. Inst. Fourier, Grenoble, 1 7 (1 9 6 7 ) 1 -1 3 5 .

A gm on, S.1 Lectures on Elliptic Boundary Value Problems, D . Van Nostrand, Princeton, 1 965 .

A tiyah, M . F.1 R esolution of singularities and division of distributions, Comm. Pure Appl. Math., 2 3 (1 9 7 0 ) 1 4 5 -1 5 0 .

B erberian , S. K .1 Measure and Integration, T he M acm illan Com pany, New Y o rk , 1965 .

Bredon , G . E .1 Sheaf Theory, M cG raw -H ill, New Y o rk , 1967 .

Brigham , E . O .1 The Fast Fourier Transform, P rentice-H all, Englew ood Cliffs, 1974 .

BIBLIOGRAPHY 347

Bronow ski, J .1 The Ascent of Man, L ittle , Brow n and C o., B oston , 1973 .

B uck, R . C.1 (Editor) Studies in Modem Analysis, Volum e 1 of Studies in Mathematics, M ath. A ssoc. A m er. 1962 .

Calderon, A . P.1 Singular integrals. Bull. Amer. Math. Soc., 7 2 (19 6 6 ) 4 2 7 ‘

Calderon, A . P. and Vaillancourt, R .1 O n the boundedness of pseudo-differential operators, J. Math. Soc. Japan, 2 3 (1 9 7 1 ) 3 7 4 -3 7 8 .

Cattabriga, L . and D e G iorgi, E .1 Sull’esistenza di soluzioni analitiche di equazioni a derivate parziali a coefffcienti costanti in un qualunque numero di variabili. Boll. U.M.I., (4) 6 (1 9 7 2 ) 3 0 1 -3 1 1 .2 U na dim ostrazione diretta dell’esistenza di soluzioni analitiche nel piano reale di equazione a derivate parziali a coefficient! constanti. Boll. U.M.I., (4) 4 (1 9 7 1 ) 1 0 1 5 -1 0 2 7 .3 U na form ula di rappresentazione per funzioni analitiche in R"", Boll. U.M.I., (4) 4 (1 9 7 1 ) 1 0 1 0 -1 0 1 4 .

Chazarain, J .1 O perateurs hyperboliques a caracteristiques de multiplicite constante, Ann. Inst. Fourier, Grenoble, 2 4 (1 9 7 4 ) 1 7 3 -2 0 2 .2 Propagation des singularites pour une classe d’operateurs a caracteristiques multiples et resobulite locale, Ann. Inst. Fourier, Grenoble, 2 4 (1 9 7 4 ) 2 0 3 -2 2 3 .

D ieudonne, J . A.1 R ecen t developm ents in the theory of locally convex vector spaces. Bull. Amer. Math. Soc., 5 9 (1 9 5 3 ) 4 9 5 -5 1 2 .

D onoghue, Jr ., W . F .1 Distributions and Fourier Transforms, A cadem ic Press, New Y o rk , 1969 .

D uisterm aat, J . J . and H orm ander, L .1 Fourier integral operators, I I , Acta Math., 1 2 8 (1 9 7 2 ) 1 8 3 -2 6 9 .

Edwards, R . E .1 Functional Analysis, H olt, R inehart and W inston, New Y o rk , 1965 .

Egorov, Y u . V .1 Subelliptic pseudo-differential operators, Soviet Math. Dokl, 1 0 (1969) 1 0 5 6 -1 0 5 9 .2 Subelliptic operators, Russian Math. Surveys, 3 0 , 2 (1 9 7 5 ) 5 9 -1 1 8 .3 Subelliptic operators, Russian Math. Surveys, 3 0 , 3 (1 9 7 5 ) 5 5 -1 0 5 .

Ehrenpreis, L .1 Fourier Analysis in Several Complex Variables, W iley -Interscience, New Y o rk , 1970 .

Enqvist, A.1 On fundam ental solutions supported by a convex cone. Ark. Mat., 12 (1 9 7 4 ) 1 -4 0 .2 On tem perate fundam ental solutions supported by a convex cone. Ark. Mat., 1 4 (1 9 7 6 ) 3 5 -4 1 .

Erdelyi, A .1 Asymptotic Expansions, D over, New Y o rk , 1956 .

Fefferm an, C. and Phong, D .1 On positivity o f pseudo-differential operators, in Harmonic Analysis in Euclidean Spaces, Proc. Symp. Pure M ath., V ol. 2 5 , part 2, A m er. M ath. Soc., Providence, R .I . , 1 979 , 1 3 7 -1 4 1 .

348 BIBLIOGRAPHY

Friedm an, A .1 Generalized Functions and Partial Differential Equations, Prentice-H all, Englew ood Cliffs, 1963 .

Friedrichs, K . O .1 Pseudo-differential Operators, An Introduction, Revised April 19 7 0 , Courant Institute, N Y U , New Y o rk , 1970 .

G arabedian, P. R .1 Partial Differential Equations, John W iley, New Y o rk , 1964 .

Garding, L .1 D irich let’s problem for linear elliptic partial differential equations. Math. Scand., 1 (1 9 5 3 ) 5 5 -7 2 .2 Transform ation de Fourier des distributions hom ogenes. Bull. Soc. Math. France, 8 9 (1 9 6 1 ) 3 8 1 -4 2 8 .3 L in ear hyperbolic partial differential equations with constant coefficients, Acta Math., 8 5 (1 9 5 0 ) 1 -6 2 .

G e l’fand, I. M . and Shilov, G . E .1 Generalized Functions, Volum e 1, Properties and Operations, A cadem ic Press, New Y o rk , 1 964 .2 Generalized Functions, V olum e 3 , Theory of Differential Equations, A cadem ic Press, New Y o rk , 1 967 .

G odem ent, R .1 Theorie des faisceaux, H erm ann, Paris, 19 6 4 .

H ayakaw a, K .1 Som e results on the Fourier image of special classes of distributions, Osaka J. Math., 9 (1 9 7 2 ) 4 9 9 -5 1 2 .

H ille, E .1 Analytic Function Theory, V olum e II, G inn, B oston , 1 962 .

H ochschild, G .1 The Structure of Lie Groups, H olden-D ay, San Francisco, 1965 .

H offm an, K.1 Banach Spaces of Analytic Functions, Prentice-H all, Englew ood Cliffs, 1962 .

H drm ander, L.1 An Introduction to Complex Analysis in Several Variables, D . V an Nostrand, Princeton, 1 966 .2 Linear Partial Differential Operators, A cadem ic Press, New Y o rk , 1963 .3 Fourier integral operators, I, Acta Math., 1 2 7 (1 9 7 1 )7 9 -1 8 3 .4 Pseudo-differential operators and non-elliptic boundary value problem s, Ann. of Math., 8 3 (1 9 6 6 ) 1 2 9 -2 0 9 .5 T h e spectral function of an elliptic operator, Acta Math., 1 2 1 (1968) 1 9 3 - 21 8 .6 On the existence and the regularity of solutions of linear pseudo-differential equations, L'Enseignement Math., 1 7 (1 9 7 1 ) 9 9 -1 6 3 .7 U niqueness theorem s and wave front sets for solutions of linear differential equations with analytic coefficients, Comm. Pure Appl. Math., 2 4 (19 7 1 ) 6 7 1 -7 0 4 .8 Pseudo-differential operators. Comm. Pure Appl. Math., 1 8 (1 9 6 5 ) 5 0 7 - 5 1 7 .9 Spectral analysis of singularities, in Seminar on Singularities of Solutions of Linear Partial Differential Equations, edited by L ars H orm ander, Princeton U niversity Press, Princeton, 1 9 7 9 , 3 -4 9 .10 T h e W eyl calculus of pseudo-differential operators. Comm. Pure Appl. Math., 32 (1 9 7 9 ) 3 5 9 -4 4 3 .

BIBLIOGRAPHY 349

11 Subelliptic operators, in Seminar on Singularities of Solutions of Linear Partial Differential Equations, edited by Lars H orm ander, Princeton University Press, Princeton, 1 9 7 9 , 1 2 7 -2 0 8 .12 Pseudo-differential operators and hypoelliptic equations, in Singular Integrals, volume 10 of Proceedings of Sym posia in Pure M athem atics (Univ. of Chicago, A pril 2 0 -2 2 , 196 6 ), A m er. M ath. Soc., Providence, 1 967 .13 O n the existence of real analytic solutions of partial differential equations with constant coefficients, Im\ Math., 2 1 (1 9 7 3 ) 1 5 1 -1 8 2 .

H orvath, J .1 Topological Vector Spaces and Distributions, Volum e I, A ddison-W esley, Reading, 1966 .

Jackson , R . J . V .1 Canonical Differential Operators and Lower-order Symbols, M em oir number 1 35 , A m er. M ath. Soc., Providence, 1973 .

K annai, Y .1 Causality and stability of linear systems described by partial differential operators, preprint.

K elley, J . K .1 General Topology, D . V an Nostrand, Princeton, 1955 .

K ellogg, O . D .1 Foundations of Potential Theory, Springer-Verlag, Berlin , 1967 .

K line, M .1 Mathematical Thought from Ancient to Modem Times, O xford University Press, New Y o rk , 1972 .

Knopp, K .1 Theory of Functions, Part II , D over, New Y o rk , 1947 .

Kohn, J . J . and N irenberg, L.1 A n algebra of pseudo-differential operators. Comm. Pure Appl Math., 1 8 (1 9 6 5 ) 2 6 7 -3 0 5 .

Kom atsu, H.1 (Editor) Hyperfunctions and Pseudo-differential Equations, Proceedings of a Conference at K atata, 1971, Lecture N otes in M athem atics, volume 2 87 , Springer-V erlag, Berlin , 1973 .

K um ano-G o, H.1 R em arks on pseudo-differential operators, J. Math. Soc. Japan, 2 1 (1969) 4 1 3 -4 3 9 .2 A lgebras of pseudo-differential operators, J. Fac. Sci. Tokyo, 1 7 (1970) 3 1 -5 0 .

L ancaster, K . E . and Petersen, B . E .1 Systems of partial differential operators with fundam ental solutions supported by a cone, Proc. Amer. Math. Soc., 7 8 (1 9 8 0 ) 2 4 7 -2 5 2 .

Lax, P. D . and M ilgram, A . N.1 Parabolic equations, in Contributions to the Theory of Partial Differential Equations, edited by L . B ers, S. B ochner and F . Jo h n , Princeton University Press, Princeton (1 9 5 4 ) 1 6 7 -1 9 0 .

L ax, P. D . and N irenberg, L .1 A sharp inequality for pseudo-differential and difference operators, in Singular Integrals, Proc. Symp. Pure M ath., Volum e 10, A m er. M ath. Soc., Providence, 1 967 , 2 1 3 -2 1 7 .2 On stability of difference schem es: a sharp form of G arding’s inequality, Comm. Pure Appl. Math., 1 9 (1966) 4 7 3 -4 9 2 .

350 BIBLIOGRAPHY

Lighthill, M . J .1 Introduction to Fourier Analysis and Generalised Functions, Cam bridge U niversity Press, 19 5 8 .

Lions, J . L. and M agenes, E .1 Non-homogeneous Boundary Value Problems and Applications, volume 1, Springer-V erlag, B erlin , 1972 .

Loom is, L . H . and Sternberg, S.1 Advanced Calculus, A ddison-W esley, R eading, 1 968 .

M ai grange, B .1 E xistence et approxim ation des solutions des equations aux derivees partiel- les et des equations de convolution, Ann. Inst. Fourier, Grenoble, 6 (1 9 5 6 ) 2 7 1 -3 5 5 .

M arden, M .1 Geometry of Polynomials, Second Edition , A m er. M ath. Soc., Providence, 19 6 6 .

M aurin, K .1 Methods of Hilbert Spaces, Polish Scientific Publishers, W arsaw, 1967.

M ellin, A .1 Low er bounds for pseudo-differential operators, Arkiv for Mat., 9 (1 9 7 0 ) 1 1 7 -1 4 0 .

M elrose, R . B .1 Local Fou rier-A iry integral operators, Duke Math. J., 4 2 (1 9 7 5 ) 5 8 3 -6 0 4 .2 M icrolocal param etrices for diffractive boundary value problem s, Duke Math. J., 4 2 (1 9 7 5 ) 6 0 5 -6 3 5 .

M eyers, N. G . and Serrin, J .1 Proc. Acad. Sci. USA, 5 1 (1 9 6 4 ) 1 0 5 5 -1 0 5 6 .

M ilnor, J . W .1 Topology from the Differentiable Viewpoint, U niversity Press of Virginia, Charlottesville, 19 6 5 .

M unroe, M . E .1 Introduction to Measure and Integration, Second Edition , A ddison-W esley, Reading, 1 971 .

N emytskii, V . V . and Stepanov, V . V .1 Qualitative Theory of Differential Equations, Princeton U niversity Press, Princeton, New Jersey , 1 960 .

Palais, R . S.1 Seminar on the Atiyah-Singer Index Theorem, Princeton U niversity Press, Princeton, 1 965 .

P eetre, J .1 U ne caracterisation abstraite des operateurs differentiels. Math. Scand., 1 (1 9 5 9 ) 2 1 1 -2 1 8 .2 Rectification a Particle ‘U ne caracterisation abstraite des operateurs differentiels’. Math. Scand., 8 (1 9 6 0 ) 1 1 6 -1 2 0 .

Petersen, B . E .1 W eak derivatives and integration by parts, Amer. Math. Monthly, 8 5 (1 9 7 8 ) 1 9 0 -1 9 1 .2 T h e total symbol of a pseudo-differential operator, Proc. Amer. Math. Soc., 3 0 (1 9 7 1 ) 3 8 8 -3 9 2 .

Piccinini, L . C.1 N on-surjectivity of the C au chy-R iem ann operator on the space of analytic functions on R ”. G eneralizations to the parabolic operators. Boll. U.M.I. (4) 7 (1 9 7 3 ) 1 2 -2 8 .

BIBLIOGRAPHY 351

Plis, A.1 A sm ooth linear elliptic differential equation without any solution in a sphere. Comm. Pure Appl. Math., 1 4 (1 9 6 1 ) 5 9 9 -6 1 7 .

Popivanov, P. R . and Popov, G . S.1 A priori estim ates and som e m icrolocal properties of a class of pseudodifferential operators, Comptes Rendus de VAcad. Bulgare des Sci., 3 3 (1980) 4 6 1 -4 6 3 .

Rudin, W .1 Real and Complex Analysis, M cG raw -H ill, New Y ork , 1966 .

Sato, M.1 Hyperfunctions and partial differential equations. C onference on Functional Analysis and R elated Topics, Tokyo, 1969 , 9 1 -9 4 .

Schwartz, L .1 Theorie des distributions, Nouvelle edition (augm entee), H erm ann, Paris, 1966 .2 Functional Analysis, Courant Institute, N Y U , New Y o rk , 19 6 4 .3 Mathematics for the Physical Sciences, A ddison-W esley, R eading, 1966 .

Seeley, R . T .1 Spherical Harm onics, Amer. Math. Monthly, 7 3 (1 9 6 6 ) supplementary issue, H erbert Ellsw orth Slaught M em orial Papers (number 11), 1 5 0 -1 2 1 .2 Integro-differential operators on vector bundles, Trans. Amer. Math. Soc., 1 1 7 (1 9 6 5 ) 1 6 7 -2 0 4 .

Sem yanistyi, V . I.1 H om ogeneous functions and som e problem s of integral geom etry in spaces of constant curvature, Soviet Math. Dokl, 2 (1 9 6 1 ) 5 9 -6 2 .

Shih, W .1 On the symbol of pseudo-differential operator. Bull. Amer. Math. Soc., 7 4 (1 9 6 8 ) 6 5 7 -6 5 9 .

Sm ith, K . T .1 Form ulas to represent functions by their derivatives. Math. Ann., 1 8 8 (1 9 7 0 ) 5 3 -7 7 .

Sm ith, K . T ., Solm on, D . C ., and W agner, S. L.1 Practical and m athem atical aspects of the problem of reconstructing ob jects from radiographs. Bull. Amer. Math. Soc., 8 3 (1 9 7 7 ) 1 2 2 7 -1 2 7 0 .

Stein, E . M.1 Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.

Stein, E . M. and W eiss, G .1 Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, 1971 .

Sternberg, S.1 Lectures on Differential Geometry, Prentice-H all, Englew ood Cliffs, 1964 .

S treater, R . F . and W ightm an, A . S.1 PC T, Spin and Statistics, and all that, W . A . B en jam in , New Y o rk , 1964.

Taylor, M.1 Pseudodifferential Operators, Princeton University Press, Princeton, 1981.

T o rre , E . D . and Longo, C. V .1 The Electromagnetic Field, Allyn and B acon, B oston, 1969 .

Treves, F.1 Topological Vector Spaces, Distributions and Kernels, A cadem ic Press, New Y o rk , 1967 .2 Basic Linear Partial Differential Equations, A cadem ic Press, New Y ork , 1975 .

352 BIBLIOGRAPHY

3 Linear Partial Differential Equations with Constant Coefficients, G ordon and B reach , New Y o rk , 1 966 .4 Locally Convex Spaces and Linear Partial Differential Equations, Springer- V erlag, New Y o rk , 1 967 .5 Introduction to Pseudodifferential and Fourier Integral Operators, Volum e 1, Pseudodifferential Operators, Plenum Press, New Y o rk , 1 980 .

U nterberger, A .1 Pseudo-differential Operators and Applications: An Introduction, Lecture N otes Series, M atem atisk Institut, A arhus U niversitet, 1976 .

Vaillancourt, R .1 A simple proof of L ax-N irenberg theorem s. Comm. Pure AppL Math., 2 3 (1 9 7 0 ) 1 5 1 -1 6 3 .

V iola , T .1 Sulle equazioni algebriche a coeflficienti reali. Rend. Accad. Sci. Napoli, 8 (1937/38) 7 6 -8 3 .

W atson, G . N.1 A Treatise on the Theory of Bessel Functions, Cam bridge U niversity Press, 1922 .

W eyl, H.1 The Theory of Groups and Quantum Mechanics, 192 8 (reprinted by D over, New Y ork).

Y osida, K.1 Functional Analysis, 5th ed., Springer-V erlag, B erlin , 1978 .

Zaidm an, S.1 Pseudo-differential operators, Ann. Mat. Pura Appl. (4) 9 2 (1 9 7 2 ) 3 4 5 -3 9 9 .

Index

a priori inequedity 230, 293 Abel-Poisson inversion formula 73 adjoint

formal 25, 193 Hilbert space 25

alibi and alias 207 amplitude function 59 asymptotic limit 213 asymptotic sum 172, 213

Banach’s theorem 276 Banach-Rota-Dieudonne-Schwartz

theorem 33Banach-Steinhaus theorem 20 band-limited (finite bandwidth) 103, 127 barrelled space 16 Bernstein’s inequality 103 Bessel potential 229 bidual 17Bochner’s theorem on tubes 145 Borel measure 8 boundary 9 bounded at infinity 11 bounded set 16

in C“(a ) 18 in C“(n) 18 in S^sCaiR^) 61

check V 47classical derivatives 24, 30, 42 classical pseudo-differential operators 199 closed range theorem 33 coercive form 319

strictly 317 commutator 200composition of pseudo-differential operators

194composition of relations 180 condition (C) 48, 49 cone 49, 146

salient 49causal, cocausal, propagation, hyper-

bolicity 142cone condition, restricted 309 conic set 146 convex hull, ch 102

closed convex hull, cch 96 convolution 5, 10, 29, 47, 87, 134 convolution algebras

L\U ) 6 S8(IR”) 10

55 C ? 55

>1,n) 99

135, 138

canonical coordinates 149, 202, 335 canonical forms 336 Cauchy integral of a distribution 45 caution 84Cesaro inversion formula 77 characteristic, multiplicity of 221, 345 characteristic set Z(P) 166, 212

derivatives of a distribution 23 Dieudonne-Schwartz theorem 17 digamma function 114 Dirac

formula 44 layer 35 measure 1, 10

353

354 INDEX

Dirichlet form 314generalized form 325 integral 315 principle 316, 318 problem 314, 315, 321

distribution 19 temperate 83

divergence theorem 13 double distributions 34 dual cone 49 dual group 122 dual space 16

elliptic operator 212, 214 uniformly strongly 319, 322

ellipticity constant 319, 322 Euler-Mascheroni constant 116 Euler’s relation 119 Euler’s vector field 337 exchange formulae 93

finite order (distributions) 31 finite part of Hadamard 39, 114 first arithmetic means inversion formula 77 Fourier series 123 Fourier transform

examples 68, 94, 113, 117, 123 integrable functions 67 L “ functions 78 measures 67temperate distributions 91

Frechet space 16 Fredholm alternative 279 Friedrich’s lemma 242, 244 Friedrichs’ symmetrization 329 fundamental group of periods 123 fundamental parallelepiped 123 fundamental solution 25, 56, 141, 296

Cauchy-Riemann operator 27, 118 causal 142Laplace operator 26, 27, 41, 122 wave operator 143

Garding’s inequality 250, 319 discussion 322 sharp 323, 334

Gauss-Weierstrass inversion formula 73 integral 107, 161 kernel 71

group of frequencies 124 group of periods 122

Hamiltonian vector field 202, 336 Hardy space 140 heat operator 218 Heaviside’s function Y 24 Hilbert transform 113, 140, 204 H51der functions 236holomorphic distribution valued functions

38, 111homogeneous distributions 46, 119 homogeneous tensor field 337 homothety Y; 46, 84 hyperbolic operator 56 hyperbolicity cone 142 hyperfunctions 45 hypoelliptic operator 215, 281

semiglobally 281 strictly 281with constant coefficients 301

integral operator 178 integration by parts 13, 37 integration over the fibre 47 invariance of pseudo-differential operators

204invariant distributions 143

Jacobi matrix 36, 206 Jacobian determinant 203

Korn trick 230, 320 Krein-Smulian theorem 277

Laplace operator 26, 27, 41, 122, 218 Laplace transform 128, 133 Lax-Milgram theorem 315 Lebesgue point 75 Leibniz’ formula 30, 37, 198 LF-space 17

subspace of 283linear transformation of a distribution 47 Lipschitz functions 236 local existence 279

real principal type 346 local property 147 localization pg 168, 171 locally convex topology 15 locally finite 12

Mackey’s theorem 16 Malgrange-John-Hormander theorem 105 Mazur’s theorem 16 Minkowski inequality 4

INDEX 355

mollifier 6, 11 Montel space 16 multi-index 3 multiple layer 34

Newtonian potential 29, 58 null bicharacteristic strip 338

oscillatory integrals 58, 107, 186, 188

pairing (• 1 •> 231Paley-Wiener theorems 98, 104, 105, 138 Pappus’ theorem 36 parametrix 215, 266, 296 Parseval-Plancherel theorem 79 partition of unity 12, 121 P(D)-convex 285

strongly 285 Peetre’s inequality 86 periodic distributions 122 phase function 59 plane wave 124 Poincare inequality 312 Poisson bracket 198, 201, 336 Poisson integral 44

kernel 44, 71Poisson summation formula 126 polynomial inequalities 99 positive type 78 principal value 21, 40 propagation cone 142 propagation of singularities 287

Hormander’s theorem 338 propagation of zeros 287 properly supported operator 180 pseudo-differential operators 1^4, 166,

176compactness criterion 254

continuity 240, 246, 251, 254, 258 pseudo-local property 147, 177, 209

strong 147, 211

Radon measure or integral 20 real principal type 345 reflexive space 17 regular domain 13 regularity theorem 219, 262 Rellich’s lemma 243 restriction of distributions 21 Riemann-Lebesgue lemma 67 Riesz-Markoff theorem 9 Riesz operators 113, 204 Routh-Hurwitz stability 141

salient cone 49 sampling theorem 127 Schwartz kernel 178, 186, 188 segment condition 309 Seidenberg-Tarski theorem 294 semialgebraic map 294 semialgebraic set 293 seminorm 15 semireflexive space 17 set "Bf 129singular convolution operator 108 smoothing operator 179 Sobolev norm \\-\\j, 225, 229

log-convexity 237Sobolev’s representation of 8 26, 43 Sobolev space H'' 228

and 232and 306

255W{Cl,\oc) 256, 263

Sobolev theorem 234 spaces

A(n) 16 ^(O) 8 B C (a) 9c(n) 16 C^(a) 6, 16 c (a) 6 c^ m 18C“ 48 Co(H) 9

CZiUn 233

si'(n) 19'(n) 22

LHn, loc) 3 89 85

82, 83S^jn, R ) 59 S {Cl,R ) 59

59S - “(n,[R'^) 59

spectrum of a distribution 124 stationary phase principle 152, 211 strong topology 16 subelliptic 217, 226, 268

equivalence of different notions 269, 288 estimate 227, 231, 270 globally 268 semiglobally 227, 268 strictly 226, 267, 301

356 INDEX

support 3, 22essential (ess. supp) 155, 209 singular 23 singular-H'‘ 259

support function 96 support of an operator 180

compactly supported 250 properly supported 180

symbol 109, 165, 183 lower order 199 map 165, 178, 197 principal 183, 199, 208 principal (of a form) 319 subprincipal 200, 202 total 199Weyl 164, 191, 194, 197, 200

temperate distributions 83 Titchmarsh-Lions theorem 103 translation 7, 46

in 241 transpose

algebraic 25formal 15, 25, 46, 169, 192

twisted distributions 153, 265

uncertainty principle 37 unique continuation property 286

variation of a measure 8

wave front set WF(u) 146, 220 examples 158, 220, 265 invariance 149, 220, 264 oscillatory integrals 156 Schwartz kernel 187 WF,(u) 259, 265

wave operator (d’Alembertian) 63, 142weak derivatives 23, 37weak topology 16weak* topology 16Weyl operator 162, 163, 191width (L-width) 312

Young’s inequality 5

introduction to the fourier transform & pseudo-differential operators

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