algebraic and geometric methods in mathematical physics: proceedings of the kaciveli summer school,...

MATHEMATICAL PHYSICS STUDIES
VOLUME 19
Algebraic and Geometric Methods in Mathematical Physics Proceedings of the Kaciveli Summer School, Crimea, Ukraine, 1993
edited by
and
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4663-5 ISBN 978-94-017-0693-3 (eBook) DOI 10.1007/978-94-017-0693-3
Printed on acid-free paper
All Rights Reserved © 1996 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1996
No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.
Soleil de pierre et quatre murs de chaux La veiIIe encore on avait chaud et l'on a froid maintenant
devant
Analysis and Spectral Theory 1 A.Beilinson and V.Drinfeld, Quantization of Hitchin's Fibration
and Langland's Program. . . . . . . . . . . . . . . . . . .. 3 A.Boutet de Monvel and V.Georgescu, Boundary Values of the
Resolvent of a Self-Adjoint Operator: Higher Order Estimates . . . . . . . . . . . . . . . 9
L.Boutet de Monvel, Index Theorems and Microsupport 53 G.Lebeau, Equation des Ondes Amorties. . . . . . . . . 73 N . Lerner , Oscillatory Integrals Controlling the Drift of Spectral
Projections for Pseudo-Differential Operators . . . . . . .. 111 G.Nenciu, Linear Adiabatic Theory: Exponential Estimates and
Applications. . . . . . . . . . . . . . . . . . . . . . . . . .. 127
Mathematical Problems of Quantum Field Theory and Statistical Physics 143 A.Beauville, Vector Bundles on Riemann Surfaces and Conformal
Field Theory . . . . . . . . . . . . . . . . . . . . . . 145 A.Belavin, Introduction to the Conformal Field Theory .... 167 L.Chekhov, Discretized Moduli Spaces and Matrix Models 187 L.Pastur, Spectral arid Probabilistic Aspects of Matrix Models 207 D.Petrina, On Approximation of General Hamiltonians
by Hamiltonians of the Theories of Superconductivity and Superfluidity . . . . . . . . . . . . . . . . . . . . . 243
Nonlinear Equations and Integrable Systems 261 E.Belokolos, Initial and Boundary Value Problems for the Sine
Gordon Equation . . . . . . . . . . . . . . . . . . . . . . . . 263 Yu.Daletskii, Algebra of Compositions and Nonlinear Equations. 277 N.Gonchar, Nonlinear Equations for Equilibrium Costs and their
Solvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
V.Novokshenov, Minimal Surfaces in the Hyperbolic Space and Radial-Symmetric Solutions of the Cosh-Laplace Equation. 357
J.-J.Sansuc and V.Tkachenko, Spectral Properties of Non-Selfadjoint Hill's Operators with Smooth Potentials 371
Short Communications 387 R.Bikbaev, Complex Deformations of Riemann Surfaces, Frequency
Maps and Instability Problems . . . . . . . . . . . . . . .. 389 A.Boutet de Monvel and R.Purice, A Propagation Estimate for
the Dirac Hamiltonian in the Field of an Electromagnetic Wave ............................... 395
N.Dudkin, The Commutativity of the Singularly Perturbed Self- Adjoint Operators .................. . . . . . 403
V.Dzyadyk, Approximate-Iterative Method for Solving Nonlinear Differential and Integral Equations . . . . . . . . . . . . .. 409
B.Feigin and S.Parkhomenko, Regular Representations of Affine Kac-Moody Algebras . . . . . . . . . . . . . . . . . . . . .. 415
M.Gorbachuk and V.Gorbachuk, On Boundary Value Problems for Operator Differential Equations . . . . . . . . . . 425
V.Koshmanenko, Singularly Perturbed Operators of Type -~ +),,15 . . . . . . . . . . . . . . . . . . . . . . . . . 433
L.Nizhnik, Inverse Scattering Problems for Hyperbolic Equations and their Applications . . . . . . . . . . . . . . . . . . . .. 439
A.Sakhnovich, Sine-Gordon Equation in Laboratory Coordinates and Inverse Problem on the Semi-Axis. . . . . . . . . . . . 443
W.Skrypnik, Gibbs States and Regularized Dynamics of the Chern- Simons Charged Particle System . . . . . . . . . . . 449
M.Sodin and P.Yuditski, Almost-Periodic Sturm Liouville Operators with Homogeneous Spectrum . . . . . . . 455
V.Yurko, On Higher-Order Difference Operators ..... 463 P.Zhidkov, Invariant Measures for Infinite-Dimensional Dynamical
Systems with Applications to a Nonlinear Schrodinger Equation ............................ 471
PREFACE
This volume contains the proceedings of the First Ukrainian-French Romanian School "Algebraic and Geometric Methods in Mathematical Physics", held in Kaciveli, Crimea (Ukraine) from 1 September ti1114 September 1993.
The School was organized by the generous support of the Ministry of Research and Space of France (MRE), the Academy of Sciences of Ukraine (ANU), the French National Center for Scientific Research (CNRS) and the State Committee for Science and Technologies of Ukraine (GKNT).
Members of the International Scientific Committee were: J.-M. Bony (paris), A. Boutet de Monvel-Berthier (Paris, co-chairman), P. Cartier (paris), V. Drinfeld (Kharkov), V. Georgescu (Paris), J.L. Lebowitz (Rutgers), V. Marchenko (Kharkov, co-chairman), V.P. Maslov (Moscow), H. Mc-Kean (New-York), Yu. Mitropolsky (Kiev), G. Nenciu (Bucharest, co-chairman), S. Novikov (Moscow), G. Papanicolau (New-York), L. Pastur (Kharkov), J.-J. Sansuc (Paris).
The School consisted of plenary lectures (morning sessions) and special sessions. The plenary lectures were intended to be accessible to all participants and plenary speakers were invited by the scientific organizing committee to give reviews of their own field of interest. The special sessions were devoted to a variety of more concrete and technical questions in the respective fields.
According to the program the plenary lectures included in the volume are grouped in three chapters. The fourth chapter contains short communications.
We would like to thank those who made the School possible: J.-M. Chasseriaux (Director of the Department of International Mfairs of the MRE), A.P. Shpak (Chief Scientific Secretary of ANU), J.-P. Ferrier (Head of the Mathematical Division of CNRS) and S.M. Ryabchenko (Chariman of the GKNT).
x
The School would not have been as successful as it was had it not been for the great dea1 of the organization work done by members of the Mathematica1 Division of the Institute for Low Temperature Physics of ANU, Kharkov, Ukraine (Associate Director of the Institute K. Maslov in particular), the Laboratory of Mathematica1 Physics and Geometry of the Mathematica1 Institute of Paris-Jussieu, the Mathematical Center of the ANU and the Institutes of Mathematics and Atomic Physics of the Academy of Sciences of Romania. Their work began more than a year before, culminating during the School itself.
We would like to express our persona1 gratitude to Dr D. Shepelsky who, practica1ly a1one, did a11 the technica1 editing, formatting, proof reading and even the retyping of manuscripts. '
V. Marchenko A. Boutet de Monvel
LIST OF PARTICIPANTS
Akulov, V. Institute of Physics and Technologies, Kharkov Alexeev, A. V.Steklov Mathematical Institute, St. Petersburg Anders, I. Institute for Low Temperature Physics, Kharkov Anoschenko, O. Kharkov State University Antonov, A. L.Landau Institute for Theoretical Physics, Moscow Antonyuk, A. Mathematical Institute, Kiev Babets, V. Kharkov State University Baran, A. Institute of Mathematics, Bucharest Beauville, A. Universite Paris Sud Belavin, A. L.Landau Institute for Theoretical Physics, Moscow Belokolos, E. Institute of Metal Physics, Kiev Belov, A. L.Landau Institute for Theoretical Physics, Moscow Berceanu, B. Institute of Atomic Physics, Bucharest Berezansky, Yu. Mathematical Institute, Kiev Bikbaev, R. Mathematical Institute, Ufa Boutet de Monvel, A. Universite Paris VII Boutet de Monvel, L. Universite Paris VI Buruiana, N. Institute of Mathematics, Bucharest Bykov, N. Institute for Low Temperature Physics, Kharkov Cartier, P. Ecole Normale Superiere, Paris Chaltikyan, K. L.Landau Institute for Theoretical Physics, Moscow Chapovsky, Yu. Mathematical Institute, Kiev Chekhov, L. V.Steklov Mathematical Institute, Moscow Chulaevsky, V. Institute of Math. Problems of Biology, Puschino Daletskii, Yu. Poly technical Institute, Kiev Drinfeld, V. Institute for Low Temperature Physics, Kharkov Dudkin, N. Mathematical Institute, Kiev Dzyadyk, V. Mathematical Institute, Kiev Egorova, I. Institute for Low Temperature Physics, Kharkov
xii
Feller, M. Frolovichev, S. Georgescu, V. Gerasimenko, V. Gershun, V. Gonchar, N. Gorbachuk, M. Gorbachuk, V. Gorunovich, V. Gussi, N. Iftimovici, A. Itskov, V. Kabanov, S. Kadeishvili, A. Kalyakin, I. Karol, A. Kazakov, V. Khorunzhy, A. Khoruzhenko, B. Khromov, A. Khruslov, E. Kiselev,O. Komech, A. Konovalov, V. Koshmanenko, V. Kotlyarov, V. Kozlov, M. Kravchuk, N. Krukov, S. Lashkevich, M. Lebeau, G. Lerner, N. Litvinov, E. Lukyanov, S. Malyarenko, A. Manda, H. Mandache, N. Mantoiu, M. Marchenko, A. Marchenko, V. Matveev, V. Mihalache, N.
UkrNPDO, Kiev MIEM, Moscow Universite Paris VII Mathematical Institute, Kiev Institute of Physics and Technologies, Kharkov Institute for Theoretical Physics, Kiev Mathematical Institute, Kiev Mathematical Institute, Kiev Mathematical Institute, Kiev Institute of Mathematics, Bucharest Universite Cerges-Pointoise Institute of Electrical Equipment, Moscow Saratov State University L.Landau Institute for Theoretical Physics, Moscow Mathematical Institute, Ufa St. Petersburg State University Ecole Normale Superiere, Paris Institute for Low Temperature Physics, Kharkov Institute for Low Temperature Physics, Kharkov Saratov State University Institute for Low Temperature Physics, Kharkov Mathematical Institute, Ufa Moscow State University International Mathematical Center, Kiev Mathematical Institute, Kiev Institute for Low Temperature Physics, Kharkov Institute for Electrical Equipment, Moscow Mathematical Institute, Kiev L.Landau Institute for Theoretical Physics, Moscow L.Landau Institute for Theoretical Physics, Moscow Universite Paris Sud Universite de Rennes Mathematical Institute, Kiev L.Landau Institute for Theoretical Physics, Moscow International Mathematical Center, Kiev Institute of Mathematics, Bucharest Universite Paris VII Institute of Mathematics, Bucharest Institute for Low Temperature Physics, Kharkov Institute for Low Temperature Physics, Kharkov V.Steklov Mathematical Institute, St. Petersburg Institute of Atomic Physics, Bucharest
Mikhailov, A. Minkin, A. Misiura, T. Mitropolsky, Yu. Nenciu, G. Nizhnik, 1. Nizhnik, L. Novitskii, M. Novokshenov, V. Omelyanov, G. Otwinowska, A. Parkhomenko, S. Pastur, L. Petrina, D. Popkov, V. Postnikov, V. Povzner, A. Priezzhev, V. Pugay, Va. Purice, R. Rofe-Beketov, F. Romanenko, R. Romanov, A. Rykhlov, V. Sakbaev, V. Sakhnovich, A. Sansuc, J .-J . Scherbina, M. Sharkovsky, A. Shepelsky, D. Simbirskii, M. Skripnik, 1. Skrypnik, W. Sodin, M. Sodin, O. Stukanev, A. Tkachenko, R. Tkachenko, V. Tseitlin, M. Trushin,1. Tsykalenko, T. Tuyls, P.
LIST OF PARTICIPANTS
xiii
L.Landau Institute for Theoretical Physics, Moscow Institute for Low Temperature Physics, Kharkov Mathematical Institute, Kiev Institute for Low Temperature Physics, Kharkov L.Landau Institute for Theoretical Physics, Moscow Institute for Earth Physics, Moscow Joint Institute of Nuclear Researches, Dubna L.Landau Institute for Theoretical Physics, Moscow Institute of Mathematics, Bucharest Institute for Low Temperature Physics, Kharkov Mathematical Institute, Kiev IGEM, Moscow Saratov State University Moscow Physics and Technics Institute Marine Hydrophysics Institute, Odessa Universite Paris VII Institute for Low Temperature Physics, Kharkov Mathematical Institute, Kiev Institute for Low Temperature Physics, Kharkov Institute for Low Temperature Physics, Kharkov Institute of Appl. Mathematics and Mechanics, Donetsk Mathematical Institute, Kiev Institute for Low Temperature Physics, Kharkov Saratov State University Mathematical Institute, Kiev Institute for Low Temperature Physics, Kharkov Institute for Low Temperature Physics, Kharkov IPM, St.Petersburg Saratov State University Mathematical Institute, Kiev Louvain University
xiv
van Canneyt, M. Vaksman, L. Yeller, T. Vereikina, M. Yuditskii, P. Yurko, V. Zhidkoy, P. Zhizhchenko, A. Zielinski, L. Zvyagin, A.
Louvain University Institute for Low Temperature Physics, Kharkov Institute for Low Temperature Physics, Kharkov Mathematical Institute, Kiev Institute for Low Temperature Physics, Kharkov Saratov State University Joint Institute for Nuclear Researches, Dubna Moscow State University Universite Paris VII Institute for Low Temperature Physics, Kharkov
Analysis and
Spectral Theory
AND
V.C. DRINFELD B.I. Verkin Institute for Low Temperature Physics Lenina Ave. 47, 310164, Kharkov, Ukraine
Let X be a smooth connected projective algebraic curve over C of genus 9 > 1, G a semisimple algebraic group over C, 9 its Lie algebra. Denote by BunG the moduli stack of G-bundles on X. It is a smooth algebraic stack of dimension N := (g - 1) . dim G. We are going to construct some D-modules on BunG and discuss their relation to Langlands program. We would like to mention that the idea of the construction was independently found by E.Witten but as far as we know he did not publish anything on this subject.
Our construction is based on the existence of a "big" commutative ring of twisted differential operators on BunG. This ring can be considered as a quantization of Hitchin's fibration.
2.
Let us recall some results of Hitchin (1]. He constructed a remarkable mor phism 'if from the cotangent stack T* BunG to a certain vector space V of dimension N such that for any functions <p, 'IjJ on V the functions 'if*<P, 'if*'IjJ
commute with respect to the Poisson bracket. Moreover, he showed that 'if is a Lagrangian fibration. 'if and V are constructed as follows. A point
3
A. Boutet de Monvel and V. Marchenko (eds.), Algebraic and Geometric Metlwds in Mathematical Physics, 3-7. © 1996 Kluwer Academic Publishers.
4 A.A. BEILINSON AND V.C. DRINFELD
of T* BunG is a pair (jZ", w) where jZ" is a G-bundle and wET;. Here T$ = Hl(X, g$) is the tangent space to BunG at jZ" (we denote by W$ the vector bundle corresponding to jZ" and a G-module W). Identify T; with HO (X, Ox ® g~ ). Let II, ... , ir be a set of basic homogeneous invariant polynomials on g*. Since w E HO(X, Ox 0 g~) we have h(w) E Vi where Vi := HO(X, O~ki) and ki = deg k By definition, V is the direct sum of all Vi and 1l'(jZ", w) = (lI(w), ... , ir(w)).
Remark Since there is no privileged choice of II, ... , fr one should con sider V not as a vector space but rather as an algebraic variety isomorphic to a vector space. There is a natural action of C* on V: A E C* maps v E Vi to ),iv.
3.
Denote by Bg(X) the ring of polynomial functions on V. It is a graded C-algebra and Hitchin's map 1l' (see Section 2) induces a graded morphism ifJel : Bg(X) --> {Functions on T* BunG}. We quantize ifJcl in the following sense: we construct a filtered C-algebra Ag(X) with gr Ag(X) = Bg(X) and a canonical morphism ifJ : Ag(X) -t HO(BunG, D') compatible with ifJel' Here D' denotes the sheaf of differential operators on BunG acting on O®(1/2) where 0 is the canonical bundle of BunG (one can show that O®(1/2) exists and D' does not depend on its choice).
Remark Though BunG is not a variety but only a stack D' has the usual properties: it is a sheaf (not a complex of sheaves!) and it has a filtration such that gr D' is the direct image of the sheaf of functions on T* BunG. This can be deduced from the equality dim T* BunG = 2 dim BunG, which holds under our assumption 9 > 1.
Ag(X) is, by definition, the coordinate ring of the manifold of g-opers on X where 9 denotes the Langlands dual of g. The definition of oper will be given in Section 4. Here we only mention that "oper" is a new name for an old notion. E.g., in the local situation (X replaced by a circle) this notion was introduced in [2]; an ,5((2)-oper on X is nothing but a projective connection on X (=locally projective structure on X). The term "oper" is motivated by the fact that for most of the classical 9 one can interpret g-opers as differential operators between certain line bundles on X (e.g., an ,5[(2)-oper is a Sturm-Liouville operator O®(-1/2) -+ O®(3/2)).
4.
Consider a triple (f%, \7, F) where f% is a G-bundle on a smooth algebraic curve Y, \7 is a connection on f%, and F is a flag on f%, i.e., a reduction
QUANTIZATION OF HlTCHlN'S FIBRATION & LANGLAND'S PROGRAM 5
of q to a E-bundle (E denotes a fixed Borel subgroup of G). Locally one can choose a E-trivialization of q and a local coordinate z on X. Then \7 = d~ + q(z), q(z) E g. Choose a Cartan subgroup H C E, let ga denote the root subspace corresponding to a root a and qa(z) the projection of q(z) to gao
Definition (q, \7, F) is a G-oper if q-a = 0 for a positive and non-simple and q-a(z) =1= 0 for all z if a is simple.
Remark If Y is a connected projective curve of genus greater than 1 then for a G-bundle q with a connection \7 there is at most one F such that (q, \7, F) is an oper.
Definition A g-oper is a G-oper where G is the group of inner automor phisms of g.
5.
The construction of the morphism <.p mentioned in Section 3 goes as follows. Fix some x E X and denote by BunG,x the moduli scheme of G-bundles trivialized over the formal neighbourhood of x. Let Ox be the completed local ring of x and Kx its field of fractions. Then G(Kx) acts on BunG,x. In particular G(Ox) and 9 ® Kx act on BunG,x in a compatible way. We have BunG = G(Ox)\BunG,x. Set a:= g®Kx, e:= g®Ox. Consider a and e as algebras over C and denote by U the universal enveloping algebra U a. Let I be the left ideal of U generated by e and set R = {u E U I I u c I}. We have the obvious morphism R/ I ----t {Differential operators on BunG}. Actually it is known that R/ I = 0, so we have to modify our construction.
Consider the standard Kac-Moody extension 0 ----t C ----t Ii ----t a ----t 0 and replace U by Ue := Uli/(] - c) where 1 denotes the element 1 E C c Ii and -c is equal to the dual Coxeter number of g. Define Re C Ue and Ie C Ue as above. Then precisely for the above value of c the algebra Rei Ie is "big". More precisely, according to a remarkable theorem by B.Feigin and E.Frenkel [3J Rei Ie is canonically isomorphic to Ag(Spec Ox) where the definition of Ag (Spec Ox) is quite analogous to that of Ag (X) (see Sections 3 and 4). One defines a canonical morphism Rei Ie ----t HO(BunG, D') where D' was defined in Section 3. So we have a morphism 'Px : Ag(SpecOx) ----t
HO(BunG,D'). Clearly Ag(X) is a quotient of Ag(SpecOx).
Theorem 1 'Px comes from a morphism 'P : Ag(X) ----t HO(BunG, D'), which does not depend on x.
6 A.A. BEILINSON AND V.G. DRINFELD
6.
Denote by a the Langlands dual of G. Using the homomorphism <p : Ag(X) - HO(BunG, D') one constructs a family of D-modules {c5"d on BunG parameterized by a-opers ~ on X. Let us describe this construction in the case where G is simply connected. Then G is the adjoint group corre sponding to g, so a-oper is the same as g-oper. Therefore to a a-oper ~ there corresponds a maximal ideal mt; C Ag(X). Since G is simply connected 0 181 (1/2) is unique. Consider on BunG the D-module ef := D 00 0 181(-1/2). It is also a right module over HO(BunG, D'). Set eft; := ef/ef· <p(mt;).
Theorem 2 eft; is holonomic and its characteristic variety coincides as a cycle with 11"-1(0), i.e., the zero fiber of Hitchin's morphism 1r.
7.
As explained in Section 4 a a-oper ~ is a triple (&l, \1, F). We conjecture that eft; corresponds in Langlands' sense to (&l, \1). To explain the precise meaning of these words let us consider the "big Hecke correspondence" .Yt', i.e., the stack of quadruples (x, !Jl:1, !Jl:2, ¢) where x E X, !Jl:1 and !Jl:2 are G bundles on X, and ¢ is an isomorphism!Jl:1 Ix\{x}- !Jl:2 Ix\{x}' The stack.Yt' is not algebraic but it is a union of an increasing sequence of closed algebraic substacks. In a quadruple (x, !Jl:1, !Jl:2, ¢) as above ¢ defines an element A of G(Ox)\G(Kx)/G(Ox) = P+ where P+ is the set of dominant coweightsofG. So we have a canonical stratification of .Yt' by locally closed substacks .Jt'). . .Yt'A is smooth over BunG xX of relative dimension n = (A, 2p) where 2p is the sum of positive roots of G. Usually the closure Je). is singular. Denote by ..$fA the Goreski-MacPherson D-module on Jt:'A (if j is the inclusion .Jt'). - Jt:'A then ..$fA = j,*o.Je'>-., i.e., ..$fA is the smallest D-submodule of RO j*O £>-.
such that J'..$fA = O£J. Define f : Jt:'A - BunG and h : Jt:'A -> BunG xX by f(x,!Jl:1,!Jl:2,¢) = !Jl:l, h(X,!Jl:1,!Jl:2,¢) = (!Jl:2,x). Notice that A E P+ can be considered as a dominant weight of a, so we have the irreducible a-module VA with highest weight A.
Conjecture (Heeke eigenvalue property) For any A E P+ there is a canonical isomorphism
h*(!'eft; 0 ..$fA) = efdn + 1] ~ V~
where n = (A, 2p), ~ denotes the external tensor product, and V~ is the local system on X corresponding to (&l, '\7) and VA.
Remark We have to explain the meaning of K 0 L where K and L are D modules on the stack Je)., which is usually singular. We use the following
QUANTIZATION OF HITCHIN'S FIBRATION & LANGLAND'S PROGRAM 7
definition: K ® L = tl'(K 121 L)[dJ where d = dim£). and tl : £). -; £). x £). is the diagonal morphism. On the nonsingular part of Yi>. this tensor product coincides with the usual one.
8.
Suppose that Yi>. = ~ (i.e., ,\ is a microweight of a). Then A). = o'Jf:'>,.
Theorem 3 If'\ is a microweight of a then i~hd!cf'e = cf'dnJ ® V). where ix is the embedding Buna -; Buna xX corresponding to x.
The above conjecture is supported by Theorem 3 and also by Theorem 2 and Laumon's computation [4J of the characteristic variety of a Langlands transform.
9.
Langlands program predicts that there should be D-modules on Buna that correspond (in the sense of Hecke eigenvalue property) to arbitrary pairs (!3l, \7) where!3l is a a-bundle on X with a connection \7. Such pairs depend on 2N parameters while opers on X depend on N parameters (see Section 1 for the definition of N).
Acknowledgments
The authors are grateful to V.Ginzburg, B.Feigin, and E.Frenkel for stim ulating discussions. This research was partially supported by a NSF grant and the DKNT grant No. 11.3/21.
References
1. Hitchin, N.J.: Stable bundles and integrable systems, Duke Math. Journal 54 (1987), 91~114.
2. Drinfeld, V.G. and Sokolov, V.V.: Lie algebras and equations of Korteweg-de Vries type, Journal of Soviet Mathematics 30 (1985), 1975~2035.
3. Feigin, B. and Frenkel, E.: Duality in W-algebras, Duke Math.Journal (International Mathematics Research Notices) 61 (1991), 75~82.
4. Laumon, G.: Correspondance de Langlands geometrique pour les corps de fonctions, Duke Math. Journal 54 (1987), 309~359.
BOUNDARY VALUES OF THE RESOLVENT OF A
SELF-ADJOINT OPERATOR: HIGHER ORDER ESTIMATES
ANNE BOUTET DE MONVEL AND VLADIMIR GEORGESCU Universite Paris 7 et CNRS, Paris, France
Introduction
In this article we intend to describe the regularity properties of the functions ,X f---* R('x ± iO), where R(,X ± iO) are the boundary values of the resolvent R(z) = (H - z)-l of a self-adjoint operator H. For that purpose we place ourselves in the framework of the conjugate operator theory 1.
Let.Yf' be a Hilbert space and A a self-adjoint operator in .Yf'. We shall identify .Yf' with its adjoint space .Yf'* by using the Riesz isomorphism and we shall denote by {£s} sEIR the Sobolev scale of Hilbert spaces associated to A. So, for s 2:: 0 £s is the domain of IAls equipped with the norm Ilflls = II(A)S fll, where 11·11 is the norm of.Yf' and (A) = (1 + A2)1/2, while for s < 0 £s is the completion of .Yf' for the norm II . lis given by the same expression. Then £0 = .Yf', £s c YZi if and only if s 2:: t, and Y'l's* = .Yf'-s (canonical identification). For s 2:: 0 we denote by II . II (s) the norm in the Banach space B(£S,Y'l's*), more precisely IITII(s) = II(A)-ST(A)-SII where the last norm is in B(.Yf'). Observe that for 0 < s < t we have
YZi c £S c .Yf' C Y'l's* c £t*,
continuous embeddings which decrease the norm; hence
B(.Yf') c B(£S, ~*) c B(YZi, £t*)
and IITII(t) ::; IITII(s) ::; IITII for all T E B(.Yf'). For technical reasons it is also convenient to introduce the Besov scale
{£S,p I s E JR, 1 ::; p ::; oo} associated to A. One may obtain it from
lSee [18], the paper where E. Mourre introduced the main ideas of this theory; the version that we use is presented in detail in [1].
9
A. Boutet de Monvel and V. Marchenko (eds.). Algebraic and Geometric Methods in Mathematical Physics. 9-52. © 1996 Kluwer Academic Publishers.
10 ANNE BOUTET DE MONVEL AND VLADIMIR GEORGESCU
the Sobolev scale by real interpolation: (£So, £S1 )e,p = £S,p if 80 i 81,
o < () < 1 and s = (1 - ())so + ()Sl. Note that £S,2 = £S. One may find a self-contained and elementary description of the spaces £S,p in Appendix A of [5]. In §1.1 we recall some properties of the usual Besov spaces £s,P(IR) associated to the operator A = P = -id~ in £ = L2(IE.); all these proper ties (including the Littlewood-Paley type description of the norm) remain valid for an arbitrary A in an arbitrary £.
The spaces £S,p with 8 > 0 consist of vectors in £ that have a certain degree of regularity with respect to A. It will be convenient to introduce classes of operators in B(£) which have similar regularity properties with respect to an operator .f4' acting in B(£) and naturally associated to A. The definition will be better understood with the following observation: for an arbitrary f E £ the element Af is a well defined vector in £-1 and we have f E £1 = D(A) if and only if Af E £ (which is a subspace of £-1); on the other hand the property f E £i is equivalent to the fact that the function 7 I--t eiAr f E £ is (strongly or weakly) C 1 (Lipschitz would be sufficient). In fact, one may describe all the spaces £S,p with 8 > 0 in terms of the regularity properties of the functions 7 I--t eiAr f E £. The operator A induces a group of automorphisms 1IIr of B(£), namely 1IIr [T] = e-iArTeiAr, and we formally have 1IIr = ei£#r with .f4'[T] = T A - AT == [T, A]. For each T E B(£), .f4'[T] is a well defined element of B(£i, £-d and the relation .f4'[T] E B(£) (which is a subspace of B(£i, £-d) is equivalent to the fact that the function 7 I--t T(7) == e-iArTeiAr E B(£) is of class C 1 in the strong or weak operator topology (and this is also equivalent to the property that this function is Lipschitz in norm). If this is the case we write T E C 1(A; £). So C1(A; £) c B(£) is the analogue of £i == £1,2 C £. If k 2: 2 is an integer we define Ck(A; £) (analogue of £k == £k,2) by the property that T(·) : IE. --Y B(£) be strongly of class Ck.
The classes Ck(A; £) are not really useful in our approach, we rather need classes of regularity that are analogues of £S,oo; we shall denote them by ,/&,S(A; £), where s could be an arbitrary strictly positive number. Before giving their definition it is convenient to introduce the space AS consisting of Lipschitz-Zygmund (or Holder-Zygmund; we use the terminology and the notations of [23]) functions of order 8 on IE.. Let E be an arbitrary Banach space and f : IE. --Y E a bounded function. We shall distinguish three cases:
1) 0 < s < 1: then f is of class AS if Ilf(x + E) - f(x)11 S CIEls for some constant C < (Xl and all x, E E IE.;
2) s = 1: then f is of class Al if Ilf(x + E) + f(x - E) - 2f(x)II ::; CIEI for a constant C < (Xl and all x, E E IE.;
3) s > 1: then s = k + ()" where k 2: 1 is an integer and 0 < ()" S 1 and we say that f is of class AS if f is of class BCk (i.e. k times continuously differentiable with bounded derivatives) and f(k) is of class AO".
BOUNDARY VALUES OF THE RESOLVENT 11
A better description of the spaces AS will be given in §1.2. If 0 is an open real set, we say that f : 0 -t E is locally of class AS on 0 if 8f is of class AS for each 8 E Co(O). The space Al (whose importance in Fourier analysis was stressed by A.Zygmund) is quite different from the space of Lipschitz functions (of order 1; d. §1.2).
At last, we can define the space '/fS: an operator T E B(£) is of class '/fS(A;£) if the function T f---7 e-irATeirA E B(£) is of class AS.
Assume now that a self-adjoint operator H is given in £ and let us denote by R(z) = (H - z)-l the resolvent family of H; z is a complex number not in the spectrum of H. We say that H is of class C k or '/fS
with respect to A if R(z) E Ck(A; £) or R(z) E '/fS(A; £) respectively for some z. One may then show that this property will hold for all z and that zp(H) E Ck(A; £) or zp(H) E '/fS(A; £) respectively for all zp E Co(lR) (see [3]).
Throughout this paper we shall assume that H is at least of class C l
with respect to A. If D(H) is the domain of H equipped with the graph topology, then we have dense continuous embeddings D(H) C £ c D(H)*. One may show (see [6]) that there is a unique continuous symmetric oper ator [H, iA] : D(H) -t D(H)* such that [R(z), iA] = -R(z)[H, iA]R(z) for each complex z outside the spectrum of H. Then for each real zp E Co(lR) the operator zp(H)[H, iA]zp(H) is bounded and symmetric in £. We intro duce now the set of points A E IR where a strict Mourre estimate holds:
f-l(H) = {.\ E IR I ::la > 0 and ::lzp E CO'(IR) real with
cp(A) -# 0 and cp(H)[H, iA]cp(H) 2: acp(H)2}.
This is an open real set which is not empty if A is conjugate to H at some point (this assertion may be interpreted as a definition). It was E.Mourre who understood that H has quite remarkable spectral properties in the set f-l(H), see [18].
We shall now quote a particular case of a result obtained in [4]. Assume that H is of class '/fl,l with respect to A, i.e. R(zo) E '/fl,l(A; £) for some complex zoo The class '/fl,l will be defined in §2.1; we note here only that '/fCi C '/fl,l C C l C '/fl for all a > 1. We also recall that for each s > 1/2 we have continuous embeddings:
.Yt:s C £1/2,1 C £ C ~i2,l C ~*.
In particular
with continuous embeddings. Hence one may consider the function z f---7
R(z) E B(£1/2,1, £J.i2,l) which is well defined and holomorphic in each of
the half-planes C± = {z E C I ±Imz > O}. If H has a spectral gap (i.e. its spectrum is not IR) then the preceding function has a weak* continuous extension to C±Uf..l(H); in particular, if A E f..l(H) then the symbols R(A+iO) and R(A - iO) have a well defined meaning as operators £1/2,1 -t £J.i2,1' This fact has been proved in [4] in a more general setting, but is also a consequence of the estimates that will be proved later on in this paper (see §4. 7). The result is optimal in three respects:
1) weak* continuity cannot be replaced by strong continuity (see §4.8); 2) the space .Jtl/2,1 cannot be replaced by a space Yf's,p with s < 1/2 or
s = 1/2 and p > 1; 3) the regularity class '6'1,1 cannot be replaced by a class 9ffu ,p with a < 1
or a = 1 and p > 1. The function R(·) has better continuity properties when considered as
B(Yf's, ~*)-valued if s > 1/2 and if H has better regularity properties with respect to A. The following theorem is the main result of this paper.
Theorem Let s > 1/2 be a real number. If H has a spectral gap and is of class '6's+1/2 with respect to A, then the functions
A r-; R(A ± iO) E B(Yf's, ~*)
are locally of class As-1/2 on f..l(H).
The result is optimal in the scales 9fft , Yf's, AU, as it is explained in detail in §1.4 and §1.5. For example, if s is a half-integer s = k + 1/2 with k 2': 1 integer, then the functions A r-; R( A ± iO) E B (Yf's, ~*) are of class A k but not of class C k and, if k = 1, they are not locally Lipschitz (even if H is of class '6'00).
It is easy to deduce from the Theorem the order of regularity of the maps z r-; R(z) E B(.Yf's, ~*) defined for z = A ± if..l, A E f..l(H) and f..l 2': O. Indeed, the harmonic function R(z) in the half-spaces Imz > 0 and 1m z < 0 can be recovered from its boundary values with the help of the Poisson kernel. For simplicity, we state the result only for the case when s is not a half-integer (in order to get Holder-type conditions that do not involve the second order modulus of continuity). Assume first that 1/2 < s < 3/2; then for each compact subset K of f..l(H) there is a constant C < 00 such that for all Z1, Z2 with Re Zj E K and 1m Zj 2': 0:
IIR(Z1) - R(Z2)II(s) :::; CIZ1 - z2I s - 1/2.
More generally, assume that s = k + 1/2 + a with k 2': 0 integer and 0< a < 1. Taking into account that lz:R(z) = k!R(z)k+1 for Imz i= 0 we see that for each compact set K C f..l(H) there is a constant C < 00 such that for all Z1, Z2 with Re Zj E K and 1m Zj > 0:
IIR(zl)k+l - R(z2)k+1l1(s) :::; CIZ1 - Z2r.
Indeed this is equivalent to the fact that the function z I---t R(z) E B(.7t's, ~*) is locally of class A k+a on the set of complex numbers z such that Re z E
f..L(H) and 1m z 2:: O. In particular, we obtain that limp.--++o R('\ ± if..L)k+1 := Rk+1(,\±iO) exists in the norm topology of B(.7t's, ~*), locally uniformly in ,\ E f..L(H) , the maps'\ I---t Rk+1(,\±iO) E B(.7t's, ~*) are locally Lipschitz (or
Holder) continuous of order a on f..L(H), and ~R('\±iO) = k!Rk+1(,\±iO) (derivatives in the norm topology of B(.7t's, ~*)). Notice that Rk+1{,\ ± iO) cannot be interpreted as a product of operators R('\ ± iO).
Let us review what was known before our work concerning the regular ity of the boundary values R(,\±iO) in the setting of the conjugate operator method. In [21] it was shown that'\ I---t R{,\±iO) E B(.7t's,~*) are Holder continuous of order () if 1/2 < s ::; 1 and () < 2/3{s - 1/2). In [26] this result was improved to () = {s - 1/2)(s + 1/2)-1. In both papers the regu larity conditions imposed to H are stronger than ours (they assume D(H) invariant under the group eiAT hence, according to Theorem 6.3.4 from [1], their hypotheses imply that H is of class C2 with respect to A). In [5] the assumptions on H have been weakened, but the order of Holder continuity has not been improved (when compared with [26]). The case s > 1 has first been treated in [15] (but see [20] and [13], [14] for related results). We are not going to discuss the conditions of regularity imposed on H in these papers (they are stronger than ours, being the natural extension of those of [18] and [21]). On the other hand, their results concerning the regularity of the functions ,\ I---t R('\ ± iO) are of the following type. Let n 2:: 1 be an in teger and let s > n - 1/2. Then the functions ,\ I---t R(A ± iO) E B(.7t's, ~*) are of class cn-1 and their derivatives of order n - 1 are locally Holder continuous of order (s - n + 1/2)(sn + s - n + 1/2)-1. If 1/2 < s < 3/4 this assertion cannot be true (cf. the optimality of our results, established in §1.4).
In the theorem stated above we assumed that H has a spectral gap. This condition is not really necessary (the result remains true in the setting of [4]) but is very convenient for the proof since it allows us to assume that H is a bounded operator. Indeed, let '\0 be a real number outside the spectrum of H and set R = (AO - H) -1. Then R is a bounded self-adjoint operator and for Imz -=1= 0:
By definition, R is of the same class (with respect to A) as H. Moreover, according to Proposition 7.2.5 of [1] (or see [3], [6]), if A -=1= '\0 is a real number then we have A E f..L{H) if and only if (AO - A)-l E f..L{R). Henceit is sufficient to prove the Theorem with H replaced by the bounded symmetric operator R. The preceding formula will give us the regularity properties of
the function z f---7 (H - z)-l in terms of those of ( f---7 (R - ()-1 (see the proof of Theorem 7.4.1 in [1]).
The paper is organized as follows. In Section 1, after some preliminary remarks concerning some function spaces on the real line, we show that the assertions of the Theorem are optimal. Section 2 contains several technical results needed for the proof of our main theorem but which, we think, have some independent interest. In Section 3 we prove a special case of the Theo rem by elementary means. In the final section we prove the Theorem under the assumption that H is a bounded operator. This implies the general case by the argument sketched above.
1. Preliminary Considerations
l.l. BESOV SPACES
Let Bg'P(R.) be the Besov space of order (s,p) associated to Lq(R.), where s is an arbitrary real number and p, q E [1,00] (see [19] and [23]). Special notations will be used in the cases q = 2 and q = 00:
for s E R., 1 'S= p 'S= 00,
for a > 0 and 1 'S= p 'S= 00. In this subsection some properties of the spaces .Yt's,p will be presented; several remarks concerning NY.'p can be found in §L2.
All these spaces can be described in terms of the operator P = -i d~ as follows. We denote J(P) the operator defined by
(.7 J(P)u)(x) = J(x)(.f7u)(x),
where (.f7g)(x) == g(x) = (21f)-1/2 In~. e-ixYg(y)dy is the Fourier transform of g. If J is of class Cg:' then clearly J (P) is a well defined operator on the space SC*(R.) (of temperate distributions) with values in COO(R.). Now let r.p E Cg:'(R.) with r.p(x) #- 0 if Ixl < 1 and 'lj; E Cg:'(R.) such that 'lj;(x) #- 0 if 2- 1 < Ixl < 2 and 'lj;(x) = 0 otherwise. Then .Yt's,p is the space of all u E SC*(R.) such that
where II . II is the norm of L2(R.); the space Bg,P is obtained by taking instead the norm in Lq (R.). If p = 00 the second term in the r.h.s. should be
thought as being sUPr::::11IrS'¢(P/r)ull. yt'S,p equipped with the norm 1I·lIs,p is a Banach space continuously embedded in .5"*. If p = 2 then II . Ils,2 is equivalent to the norm lIuli s = II(P)Sull where (P) = (1 + p 2)1/2; it is convenient to equip yt'S with the Hilbert space structure defined by the norm 1I.lIs 2.
If 0 is an open real set, then a function (or distribution) u on 0 is locally of class Bg'P on 0 if ()u E Bg'P for all () E C8"(O). This defines, in particular, the expressions locally of class yt'8,p or locally of class NJI.,p.
We notice that the scale of Banach spaces {yt'8,p I s E JR, 1 :S p :S oo} is totally ordered in the following sense: for arbitrary couples (s,p) and (t, q) we have either yt's,p C yt't,q or yt't,q C yt'8,p. In order to state this property in more precise terms we shall introduce a total order relation on the set JR x [l,ooJ by the following rule: (s,p) > (t,q) means s > t (then p, q are arbitrary) or s = t but P < q. Then yt'8,p C yt't,q is equivalent to (s,p) ~ (t, q). One has .5"(JR) C yt'8,p for any S,p but the embedding is
o dense if and only if p < 00. We denote by yt' 8,00 the closure of.5" in yt'8,00.
The adjoint space yt'* of yt' = L2(JR) is identified with itself as usual. This implies the following duality properties:
We discuss now the regularity properties of the elements of yt'8,p. The largest space in the scale {yt'8,P} which consists only of continuous func tions is yt'1/2,1. More precisely, yt'1/2,1 is a subalgebra of Coo(JR) (which is the space of continuous functions that tend to zero at infinity; one may easily show that the Fourier transform of a function from yt'1/2,1 is an inte grable function and then one applies the Riemann-Lebesgue lemma). This assertion is sharp in the following sense: there is a function f such that f E yt'1/2,p for all p > 1 and f is unbounded on each interval. Indeed, let 9 be the inverse Fourier transform of a function 9 E Coo(JR) such that 9 ~ 0, 9 is even and g(x) = (x log X)-1 if x ~ 2. Then 9 is a real function of class.5" outside the origin and g(x) rv log log Ixl-1 as x ---+ 0; so 9 belongs to all the spaces yt'1/2,p with p > 1, 9 is bounded from below (g(x) ~ c> -(0) and limx->o g(x) = +00. Now for each rational number r let Ar be a strictly pos itive number such that 2: Ar < 00. Then the function f(x) = 2: Arg(X - r) belongs to np>1yt'1/2,P, f(x) ~ c2:Ar > -00 and limx->rf(x) = +00 for each rational number r.
2 An elementary and self-contained presentation of the properties of the spaces yt's,p may be found in Appendix A of [5], where the context is slightly more general, namely P could be an arbitrary self-adjoint operator in an abstract Hilbert space.
1.2. LIPSCHITZ-ZYGMUND SPACES
The local regularity of the functions from ,Yes,p with s > 1/2 is best de scribed by the embedding ,Yes,p C As-1/2,p (d. Ch.III in [19]). We recall the notation Aa,p := B~P(lR) for 0: > 0 and 1 ::; p ::; 00. One has Aa,p c Aj3,Q if and only if (o:,p) 2: ({3,q).
The Lipschitz-Zygmund spaces A a := A a,oo may also be defined in terms of the modulus of continuity of order m of the function f E LOO(lR):
If 0: > 0 and f E L 00 (lR) then f E A a if and only if there is an integer m > 0:
such that Wm UJ) ::; c8a , where c is a finite constant. So, for 0 < 0: < 1, A a
is exactly the space of bounded Holder-continuous functions of order 0: on lR (defined by If(x + e) - f(x)1 ::; c8a for x E lR and lei::; 8). But if 0: = 1 the space Al is defined by the Zygmund type condition If (x + e) + f(x - e) - 2f(x)1 ::; c8 (x E lR, lei::; 8). Al contains strictly the space of Lipschitz continuous functions (defined by WI (8) ::; c8; for f E Al we only have w1(8) = O(8I1og81)). Indeed, there are functions in Al which are nowhere differentiable (see Sections 3 and 4 in Ch.II of [27]), while a Lipschitz continuous function is differentiable almost everywhere (a.e.).
If 0: > 1 and k 2: 0 is an integer such that k < 0:, then f E Aa is equivalent to f E BCk (i.e. f and its derivatives of order::; k are bounded and continuous) and f(k) E A a-k. Note that if f E A 2 for example, then f is of class C I but its derivative l' is not Lipschitz continuous in general (we have only f' E AI ).
For 1 ::; p < 00 we have A a,p c A a and the functions from A a,p have better regularity properties; for example, the functions from A l,p (p < (0) are differentiable at all points of a dense subset of lR, although this set could be of measure zero (see Theorem 3.3 and page 48 in Ch.n of [27]). If k is an integer, then A k,l C BCk and this is optimal: the elements of A 1,p are not Lipschitz continuous (in general) if p > 1. However, if f E A 1,2
then its distributional derivative is a function of class BMO (bounded mean oscillation; §6.18 in [23]).
So, for s > 1/2 we have ,Yes,p C As-1/2,p C As-1/2 for all p E [1,00] and the embeddings are optimal in an obvious sense (see Example 4 on page 50 of [19]; we mention that ,Yes,p C Bg,P(lR) if2::; q::; 00 and t = s_2- l +q-1). If 1/2 < s < 3/2, then the functions from ,Yes,p are Holder continuous of order s - 1/2. If s = k + 1/2 with k 2: 1 integer then ,Yes,l C BCk. But for example ,Ye3/2 C A 1,2 only and in general a function f from ,Ye3/2 is not Lipschitz continuous (1' is of class BMO).
1.3. HILBERT TRANSFORM
If u : JR -t C is a function such that u(x)(1 + IxD-1 is integrable, then lime-+o7l'-l ~x->'I>e(.:\ - x)-lu(x)dx exists for almost every real A. (see §5.9 in [25]). We denote u(A.) the limit whenever it exists; the (a.e. defined) function u is the Hilbert transform of u. For 1 < p < 00 the map u I--t U is a continuous linear operator in V(JR). This operator is isometric in YE = L2(JR) and its square is equal to -1; in fact we then have u = -i(sgnP)u where sgnP = PIPI-1 .
Since the unitary operator sgn P in YE commutes with the operators cp(P), 'lj;(P/r) that appear in the definition of the norm II 'lIs,p, it follows that sgn P induces an isometric bijective operator in YES'P for all s E JR, 1 :::; P :::; 00. So u E YEs,p if u E YES,p for some s > 0, 1 :::; P :::; 00. In particular, if u E YE1/ 2,1, then the almost everywhere defined function u is a.e. equal to a continuous function (since YE 1/ 2,1 C Coo). But in fact in this case the limit defining u(A.) exists for all A. E JR, uniformly in A., and the function A. I--t u(A.) so defined is continuous.
For completeness we give an elementary proof of this assertion in a slightly more general setting. Assume that u E L2(JR) n Coo(JR) is a real function and that the function u (which is defined a.e. and belongs to L2(JR); one should take here -5, = -i(sgnx)u(x) as definition of u) is a.e. equal to a function in Coo(JR). Let </J(z) = (7l'i)-l fn~. u(t)(t - z)-ldt, then </J is a holomorphic function in the region 1m z > 0. Now it follows easily from the continuity of u that limJ.t-++o 1m </J(A. + ift) exists (uniformly in A.) if and only if lime-+o 7l'-1 ~t->'I>e(A. - t)-lu(t)dt exists (uniformly in A.) and in this case they are equal (see the argument in the proof of Theorem 92 in [25]). By using Plancherel theorem and the identity i(t - A. + ift)-l = Jooo exp[i(t - A. + ift)sJds we obtain
</J(A. + ift) = _1_ roo ei>"s-J.ts2u(s)ds y'2;ff Jo
= _1_ r ei>.s-J.tlsl(u(s) + i-5,(s))ds y'2;ff J'ttt
=.!. r (A. ~2 2 (u(t) + iu(t))dt. 7l' J'ttt -t +ft
Since u + iu E Coo(JR) this clearly implies limJ.t-++o </J(A. + ift) = u(A.) + iu(A.) uniformly in A. E JR (cf. Th.3.1, Ch. I of [10]). Taking the imaginary parts we see that lime-+o 7l'-1 ~t->'I>e(A. - t)-lu(t)dt exists uniformly in A. E JR and is equal to u(A.).
If u E BC(JR) then the function u is not more than of class BMO in general (see Th.5.2 in Ch VI of [10]). However, if a > 0 and u E AOI. then u E A 01. too; on the other hand, if u is of class BC1 then u E Al but it is not
18 ANNE BOUTET DE MaNVEL AND VLADIMIR GEORGESCU
Lipschitz continuous in general (see §5.15 in [25] for the case 0 < a < 1 and Th.13.29 in Ch.III of [27] for the general case). In particular, if U E .!/t's,p
for some s > 1/2, then U and u belong to As-1/ 2 and this is optimal in the scale of Lipschitz-Zygmund spaces. If s = k + 1/2 with k ~ 1 integer, then u, u E A k but none of them is in BCk in general.
1.4. THE OPERATOR H = Q
We shall study here the boundary values of the resolvent of the operator Q of multiplication by the independent variable x in the Hilbert space .!/t' = L2(lR) , i.e. (Qf)(x) = xf(x). Then for f E .!/t' and z = A + iJ-l E <C, P, i= 0, we have
Set U = Ifl2 E L1(lR). For the real part of this expression we may use the results cited in §1.3; for the imaginary part we use Th.1, Ch.III of [23] (see also Sections 3 and 5 in Ch.I of [10] or §1.17 of [25] for simpler treatments in the one-dimensional case). It follows that the preceding expression has a limit as J-l --+ +0 for almost all A E lR and the following equality holds almost everywhere:
lim 7r-1(j, (Q - A - iJ-l)-l f) = U(A) + iU(A). (1.1) 1'--++0
Let J c lR be an open non-void set. Then from 1 Im(j, (Q-A-iJ-l)-l f)1 :s C for A E J and p, > 0 it follows that U(A) = If(A)j2 :s C /7r for a.e. A E J. Hence .!/t'1/2,1 is the largest space in the scale {.!/t's,P} with the property 1 Im(j, (Q - A - iJ-l)-ll) 1 :s CU) < 00 for A E J, J-l > 0 and f E .!/t's,p
(because .!/t'1/2,p with p > 1 contains functions that are unbounded on each interval).
The limit limp,--++o 1m (j, (Q - A - ip,) -1 I) exists locally uniformly in A E J if and only if U is a continuous function on J. But in this case the real part of (j, (Q - A - iJ-l)-l I) could diverge for a dense set of values A. However, we have shown in §1.3 that the limit in (1.1) exists uniformly in A E lR for each f E .!/t'1/2,1 (note that U = Ifl2 belongs to .!/t'1/2,1 too, this space being an algebra).
In conclusion, .!/t'1/2,1 is the largest space in the scale .!/t'S'P such that the limit (1.1) exists locally uniformly in A for each f E .!/t'S'p. We now discuss the continuity properties of the limit function 'P(A) := (f, (Q-A-iO)-l f) == 7rU(A) + 7riU(A) in terms of the scale {A/3,Q}. Note that the spaces .!/t'1/2,1
and .!/t'S'P with s > 1/2 are algebras, hence U E .!/t'S'P if f E .!/t's,p and (s,p) ~ (1/2,1). Then we have U E .!/t's,p too (cf. §1.3). So, if f E .!/t's,p
with (s,p) 2: (1/2,1), then 'P E £,s,p too (by taking f 2: 0 one sees that these assertions are optimal on the scale {£'t,q}). So for f E £,1/2,1 we have 'P E Coo(lR) (in fact 'P E B~l, which is slightly better locally but not so explicit), while for f E £,s,p with s > 1/2 we have 'P E As- 1/ 2 ,p, and this is optimal on the scale {A,8,q}.
In terms of the Lipschitz-Zygmund scale {N~}, the best one can have for the boundary value function 'P is: 'P E AS- 1/ 2 if f E £,s,p and s > 1/2. For example, if s = 3/2, then 'P E A I and (in general) 'P is not a locally Lipschitz continuous function unless p = 1, in which case 'P is of class BC I
(cf. §1.2).
1.5. THE OPERATOR H = h(Q)
Let h : lR ~ lR be a function of class C I such that h' (x) > 0 for each real x. We denote by h(Q) the operator of multiplication by the function h in £' = L2(lR). Then for f E £' and z = A + if.J" f.J, i- 0:
(f, (h(Q) - z)-I f) = r If(x)j2dx. = r If(g(x))12g'~x)dx. iJRh(x)-A-tf.J, if X-A-tf.J,
Here I = h(lR) is an open interval and 9 : I ~ lR is the inverse diffeomor phism of h. We are in a situation similar to that of §1.4 but this time the function u is defined on I by
The spectrum of the operator h( Q) is the closure of the interval I, so let us assume A E I. If we replace in (1.1) Q by h(Q) then the limit in (1.1) will exist and the equality will hold for almost every A E I (because the new function u is still in L 1 (1) ). But now the regularity properties of the function
'P(A) := (f, (h(Q) - A - iO)-l f) == 1TU(A) + 1Tiu(A), A E I
are determined not only by f but also by the degree of regularity of h. Assume first f E C(f(I), f =J O. Then u has the same regularity class as g' (on the interior of the support of fog). If g' is just continuous, then the imaginary part of'P is well behaved (it is a continuous function) but not its real part (because the Hilbert transform of a bounded continuous function is of class BMO but it could be unbounded on each interval). One may construct a CI-diffeomorphism h : lR ~ I with h E A 1,p for all p > 1 and such that U is unbounded on any open non-empty interval J c supp(f 0 g) (see Appendix 7.B in [1]). On the other hand, one can show that if h is
locally of class A 1,1 then for any f E £1/2,1 the functions u and it are continuous and the limit in (1.1) (with Q replaced by h(Q)) exists locally uniformly in A E I.
Let us consider now the higher order regularity properties that the func tion <p(A) = 7ru(A) + 7riu(A) could have; recall that u = If 0 gl2g'. Assume that f E £s,p C As- 1/ 2,p for some s > 1/2, 1 ~ p ~ 00. If h is locally of class As+l/2,p (which implies h E C1) then from the hypothesis h'(x) > 0 (\ix) we obtain that 9 is locally of class As+1/ 2 ,p on I; and this clearly im plies that g' is locally of class As-1/ 2,p on I (see Section 5.2 in [1]). Then fog will be 3 locally of class As-l/2,p hence If 0 gl2g' is of the same class (AOI.,P being algebras). So u is locally of class As-1/ 2 on I, which clearly implies that u has the same property (since s - 1/2 > 0).
In conclusion, if h is locally of class As+l/2 and f E £s,p for some s > 1/2, 1 ~ p ~ 00, then the function cp(A) = (I, (h(Q) - A - iO)-1 f) is locally of class As-l/2 on I. The preceding arguments clearly show that this assertion is optimal on the scale of Lipschitz-Zygmund spaces, even if f E C<f(lR.). By taking into account that u is locally of class AOI.,p if u has this property (for any a > 0, 1 ~ P ~ (0) one similarly sees that for h locally of class As+l/2,p, ip is locally of class As-1/ 2,p. For example, if h is locally A 3/2,1 and f E £1,1, then cp is of class C1.
2. Regularity of an Operator with Respect to a Group of Auto morphisms
2.1. CLASSES OF REGULARITY
Let A be a self-adjoint operator in a Hilbert space £ and WT = eiAT the unitary group generated by A. We shall denote by 1f/T the automorphism of the C*-algebra B(£) of all bounded operators in £ defined by 1f/TT == 1f/T[T] = W-TTWT. Then {1f/T}TEIR is a group (1f/T+a = 1f/T1f/a ) that has the following continuity property: the map T t-t 1f/T[T] is continuous in the strong operator topology for an arbitrary bounded operator T in £.
The preceding setting has a certain similarity with that of §1.2: there the role of the C*-algebra B(£) is played by the C*-algebra LOO(JR.) equipped with the group of automorphisms induced by the translation group. We shall now define the analogues of the spaces A OI.,p. It will be convenient to denote these new spaces by 't[OI.,p (A; £) and to set 't[Q == 't[0I.,00. So 't[0I. are
3The only non-trivial case is when s-1/2 = 1. But theng(x±e) = g(x)±g'(x)e+O(e"') for some O!. > 1 (in fact any O!. < 3/2), locally uniformly in x. Since wI(8,v) = 0(8Iog8) for v E AI, we get f 0 g(x ± c) = f(g(x) ± g'(x)e) + O(e'" loge"') and the last term is O(e/3) for 1 < f3 < 0:, locally uniformly in x. Now it is easy to establish that fog is locally A l,p if f is locally A l,p.
the natural version of the Lipschitz-Zygmund spaces A Q in the framework of this section.
If T is a bounded operator in .Yt' and m 2': 1 is an integer we define the modulus of continuity of order m of T with respect to A by
Wm(C:) = wm(c:;T,A):= sup II[~r -l]mTIl Irl::oe:
= sup Ilf) -l)j (rr:) e-iArjTeiArjll· Irl::oe: j=O J
Let a be a strictly positive real number and p E [1,00]. We say that T is of class ceQ,P(A;.Yt') if there is an integer m > a such that
(2.1)
If p = 00 this should be interpreted as wm{c:) ~ Cc:Q for some C E IR and all c: E (O, 1). As we said before, we write ceQ (A;.Yt') for ceQ,oo (A; .Yt').
Much of the classical theory of Lipschitz-Zygmund spaces can be ex tended to the spaces ceQ,p; one may find a rather detailed presentation of these developments in Chapter 5 of [1], we shall mention here only the facts that are relevant here. First, if (2.1) is finite for some integer m > a, then it is finite for all such integers. Moreover, one may replace wm{c:) in (2.1) by II [~ - l]mT II (cf. the comment after Theorem 3.4.6 in [1]). Each ceQ,p has a unique B-space (i.e. complete normable topological vector space) topology such that the embedding ceQ,p C B(.Yt') is continuous. The family {ceQ,P} is totally ordered, namely ceQ,p C ce{3,q if and only if (a,p) 2': (/3, q) (in the sense defined in §1.1). Finally, this family is stable under real interpolation: (ceQ,p, ce/3,Q)(},r = 7?Y,r with 'Y = (1 - (})a + (}/3, 0 < () < 1, 1 ~ r ~ 00.
Let us explain the relation between the classes <;fQ,P and the classes of operators defined by differentiability conditions. If k 2': 0 is an integer, then we say that a bounded operator T in .Yt' is of class Ck{A;.Yt') or C~(A;.Yt') if the function T I---t ~r[T] is of class Ck in the strong operator topology or norm topology, respectively (we mention that T I---t ~[Tl is strongly Ck if and only ifit is weakly Ck; cf. the proof in §2.2 of the fact that .rd is closed in the weak operator topology). The following characterizations of these two classes are also interesting: T E Ck{A;.Yt') if and only if II[~ -l]kTII ~ cc:k for a finite constant c and all c: E CO, 1), while T E C~{A;.Yt') if and only if the limit lime:--++ock[~ -:-- l]kT exists in norm in B{.Yt'). For example T E C1 is equivalent with T I---t ~r[T] being Lipschitz.
For each integer k 2': 1 we have cek ,1 C C~ C Ck C cek,oo and all the inclusions are strict (if A is not bounded). If 1 < p < 00, then the spaces
<ffk,p are not comparable with Ck and C~. In particular, the embedding Ck(Aj £) C <ffk(Aj £) is strict and optimal in the scale {<ffO!'P}.
For an arbitrary operator T E B(£) and for any integer k 2 0 we shall denote by Jdk[T] the sesquilinear form on D(Ak) defined inductively by Jdo [TJ = T, Jd [T] = [T, A] = T A - AT, Jdk+ 1 [T] = Jd [Jdk [T]]. More explicitly we have
Then the following identity holds (in the sense of sesquilinear forms on D(Ak)):
[~ - I]kT = (ic:)k k 1f!rcXk(T)dTJdk[T].
Here Xk are functions such that
(spline functions, see e.g. Proposition 3.3.5 in [1]). As an easy consequence of the preceding identity we obtain a new characterization of the class Ck(Aj £): T E Ck(Aj £) if and only if the densely defined sesquilinear form Jdk [T] is a bounded operator in £j and in this case we have i k Jdk [T] =
::;k 1f!T[T]IT=O. In particular, this justifies the notation 1f!T = eidT that will be used later on. Thus we have given a meaning to Jd as an operator in B(.Yt') with domain C 1(Aj£)j this operator is closed but not densely defined (if A is not bounded). The closure of the domain of Jd is the space C~(Aj £) consisting of the operators T such that T r--> 1f!T[T] is a norm continuous function. Then {1f!T} induces a Co-group of isometries (in fact automorphisms) of the C*-subalgebra C~(Aj£) of B(.Yt') and iJd is the infinitesimal generator of the induced group. The domain of the k-th order power of Jd is Ck(Aj £) and for T of class Ck the k-th power of the operator Jd applied to T is just the bounded operator in .Yt' associated to the sesquilinear form Jdk[T]. So, the notation Jdk[T] is unambiguous.
There is a description of the class <ffO!'P in terms of differentiability prop erties of the function T r--> 1f!T [T]. Let a > 0 be real, 1 ::; p ::; 00, and let k be the greatest integer such that k < a. So we have a = k + (J with k 20 integer and 0 < (J ::; 1. Then T E l&'O!,P(Aj.Yt') is equivalent to : T E Ck(Aj£) and Jdk[T] E l&'u,P(Aj.Yt') (see Proposition 5.2.2 in [1]). Note that there is a big difference between the case a = non-integer and a = integer. More explicitly, in the case p = 00 we have (T(k) == Jdk[T]):
(i) If a is not an integer, i.e. 0 < (J < 1, then T E <ffO!(Aj £) means T E Ck(Aj£) and 11~[T(k)]- T(k) II ::; cc:u.
(ii) If a is an integer, i.e. CT = 1 and k = a-I, then T E CCa(Aj YC) means T E Ck(AjYC) and 111fi[T(k)j + W'_e[T(k)j - 2T(k) II :; C£j but the function c: 1-7 lfi[T(k)] is not Lipschitz in general (if it is, then T belongs to Ca(Aj YC)).
An operator T E B(YC) is of class Coo with respect to A, written T E
COO(AjYC), if the function T 1-7 ~[T] is of class Coo. If this function has a holomorphic extension in a complex neighbourhood of T = 0 then we say that T is analytic with respect to A. More precisely, let w > OJ then the function T 1-7 W'r[T] has a holomorphic extension into the disc {T E C IITI < w} if and only if T E COO(Aj YC) and the radius of convergence of the series
L:k:,O ~$k[T] in B(YC) (( E C) is :2': w. We denote by C(w)(AjYC) the set of operator T having this property. It is easily shown that T E C(w)(AjYC) if and only if there is a function ( 1-7 e(.o1'[T] E B(YC) which is holomorphic in the strip I Re (I < wand such that eir.o1' [T] = e-iArTeiAr if T E JR. In this case, and if 0 < a < w, then T leaves invariant the domain of ealAI and (e(.o1'[T])f = e-(ATe(Af if IRe(l:; a and f E D(eaIA1 ).
Finally, let us notice that all the classes Ck (k :2': 1), C~, CCa,p, Coo and C(w) are dense involutive subalgebras of the C*-algebra C~.
2.2. FUNCTIONAL CALCULUS FOR $
The operator $ with domain D($) = C 1(AjYC) in the Banach space B(YC) has been defined by the rule $T == $[T] = d~~[T]lr=Oj we have $T = [T, A] and W'r = ei.o1'r in the sense explained above. On B(YC) we may consider (at least) three topologies: the norm (or uniform) topology, the strong operator (s.o.) and the weak operator (w.o.) topology. Then $ is a closed operator in the w.o. topology (hence in the s.o. and norm topology too) and it is densely defined in the s.o. topology (hence in the w.o. topology) but not in the norm topology (unless A is bounded). The density will follow easily from the functional calculus that we shall construct below (s-limep(c$)T = T and ep(c$)B(YC) C COO(AjYC) ifep E Y(JR), ep(O) = 1).
e--+O In order to show that $ is closed in the w.o. topology, we note that for T E D($)
W'rT = T + i lor W'u$TdCT, T E JR,
(the integral exists in the S.o. topology). Hence, ifTj E D($) and w-lim Tj = T, w-lim$Tj = S, then W'rT = T + i J; W'uSdCT which clearly implies T E D($).
Now we define a functional calculus for $. Let .A = .A(JR) be the Wiener algebra, i.e. the subalgebra of BC(JR) consisting of Fourier trans forms of bounded measures on R If ep(t) = In~. eitr J-L(dT) then we set IlepllA'l := IJ-LI(JR) (total variation of the measure J-L). If we equip.A with the usual oper-
ations of addition and multiplication of functions and with the norm II . liAr' then .4 becomes an abelian Banach algebra with unit. For cp E .4 we set cp(.fl1) := In~.1f/TJ.L(dr) where J.L is as above. More precisely, cp(.fl1) : B(£) ~ B(£) is defined by cp(.fl1)T = cp(.fl1)[T] = In~. e-iATTeiAT J.L(dr) (the inte gral exists in the strong operator topology). Clearly cp(.fl1) E B(B(£», Iicp(.fl1)IIB(B(.7t"» :::; IlcpIIAr and cp ~ cp(.fl1) is a unital homomorphism. More-
over, if we define cp+ by cp+(x) = cp( -x), then (cp(.fl1)[T])* = cp+ (.fl1) [T*J. The functional calculus cp ~ cp(.fl1) may be extended to larger classes
of functions but then it will produce unbounded operators in the Banach space B(£). This is of no interest here, but the following easily proven property will be useful. Let cp E .4 and k ;:::: 1 an integer such that the function CP(k)(X) == xkcp(x) belongs to .4 too (this is equivalent with re quiring that the distributional derivative of order k of the measure J.L be a bounded measure). Then cp(.fl1) improves regularity with respect to A in the following sense: cp(.fl1)Cm(A; £) C Cm+k(A; £) for all mEN (note that CO(A; £) = B(£» and for T E B(£) we have .fl1k[cp(.fl1)[TJJ = CP(k) (.fl1)[TJ. In particular, if cp E .9'(JR) then cp(.fl1)B(£) C COO (A; £). One may also get analytic operators by the following procedure: assume that cp E .4, w is a strictly positive real number and that the functions e±wxcp(x) belong to.4; then cp(.fl1)B(£) c C(w)(A;£) and for IRe(1 < w we have e(.o1[cp(.fl1)[TJ] = (e(cp) (.fl1)[TJ for T E B(£) (e( is the function t ~ e(t and e(cp E .4).
Now let cp E .4 and let c be an arbitrary real number. Then it is clear that t ~ cp(ct) is still a function in .4. We shall denote by cp(c.fl1) the operator on B(£) associated to it by the functional calculus. Since cp(c.fl1)[TJ = fIR W-e:TTWe:TJ.L(dr) it is clear that for an arbitrary bounded op erator T on £ the map c ~ cp(c.fl1)[TJ E B(£) is strongly continuous and Iicp(c.fl1)IIB(B(.7t"» :::; IlcplIAr' Moreover, cp(O.fl1) = cp(O) where cp(O) is the op erator of multiplication by the constant cp(O) on B(£). So s-lim cp(.fl1)[TJ =
e:->O cp(O)T for all T E B(£). Observe that the notation cp(c.fl1) is consistent with the following interpretation: cp(c.fl1) is the operator on B(£) associ ated by the preceding functional calculus to the self-adjoint operator cA on £.
We shall need a version of the ergodic theorem in the present context (see page 560 and Sections VII.7, VIII.7 in [8J; also Thm. 1.1 in [16]). Let ker.fl1 and ran.fl1 be the kernel and the range of the operator .fl1 in B(£). We denote ran.fl1s the closure of ran.fl1 in the s.o. topology; note that it coincides with the closure ran .fl1w ofran.fl1 in the w.o. topology (cf. Corollary 1.5 in [24]). We shall prove that
(2.2)
Let T E B(.Yt') and
TT = T-1 loT 1f/(T[T]da == (iT d)-l (eiT.oI - l)[T]
for T i= 0 (with a slightly formal notation). Since I/TTII :s IITII and the closed balls in B(.Yt') are w.o. compact there is a sequence Tn ------) 00 and Too E B(.Yt') such that w-lim TTn = Too. We have TT E D(d) and dTT = n-too (iT)-l(1f/T - l)T, hence IldTTl1 :s T-1211TII ------) 0 as T ------) 00. Since d is closed in the w.o. topology we get that Too E D(d) and dToo = 0, in other terms Too E ker d. Now let To = T - Too. Then w-lim T;l J;n 1f/(TToda = O. n-too But the formal identity
eiTm' _ 1 eiTm' - 1 - iTd 1 = iTd - iTd (iTd)2 (2.3)
can easily be made rigorous: if e(x) = (ix)-l(eiX-l) and 1](x) = (ix)-2(eix _
1 - ix), then e and 1] are functions of class At, 1](l)(X) = x1](x) belongs to At too and e(x) - i1](l) (x) == 1. Hence 1](Td)B(.Yt') C D(d) if T i= 0 and e(Td) - iTd1](Td) = 1 as operators on B(.Yt'), which is just (2.3). So we have To = e(Td)To - iTd1](Td)To. We take here T = Tn and we make n -+ 00. Since w-lim e(Tnd)To = 0 we see that To belongs to the w.o. clo-n-too sure of the range of d. This proves (2.2) because the w.o. closure is equal to the s.o. closure.
We shall make now several remarks in connection with the decompo sition (2.2). Observe first that kerd C COO(A;.Yt') and that the following three assertions are obviously equivalent:
(i) T E kerd; (ii) 1f/T[T] = T VT E IR; (iii) ep(d)T = ep(O)T Vep E At.
It is not difficult to show that the sum (2.2) is not direct in general (the identity operator could belong to randS; we thank G.Skandalis for this remark). However, if we denote by randu the closure of rand in the norm topology, then
(2.4)
is a norm closed subspace of B(.Yt') and the sum in the r.h.s. of (2.4) is a topological direct sum. Moreover, the projection :?JI of B A (.Yt') onto ker d along ran d U is given by
:?JI[T] = lim T-1 r 1f/(T[T]da = lim e(Td)[T] (2.5) T-too } 0 T-tOO
where both limits exist in norm and e is an arbitrary function in At with e(O) = 1 and such that the function e(l)(t) = te(t) belongs to At too
(note that the middle term in (2.5) corresponds to the choice e(x) = (eix -l)(ix)-l). In order to prove these assertions observe that {!(TJd)T = e(O)T = TifT E ker Ji1, while for T = Ji1[S] we have e(TJi1)T = e(TJi1)Ji1S = T-1e(1) (T.rzf)S, so Ile(T.rzf)TII ::; T-111{!(1)11..,((IISII -7 0 as T -7 00. Hence lIe(TJi1)TII -70 for all T in the norm closure of ran.rzf (because Ilu(TJi1)II ::; Ilell..,((). So the last limit in (2.5) exists for all T E BA(.Yt') and 9[T] = T ifT E kerJi1, 9[T] = 0 ifT E ranJi1u . Moreover, the norm of the operator 9: BA(.Yt') -7 BA(.Yt') is equal to 1. The norm continuity of 9 and the fact that ker Ji1 and ranJi1u are norm closed imply that BA(.Yt') is norm closed too. This finishes the proof.
We mention that the limits in (2.5) could exist (in norm, strongly or weakly) for T in spaces larger than BA(.Yt'); however there are, in general, operators T such that the first limit in (2.5) does not exist weakly.
The following observation will be useful later on: if (J E .4 has the properties (J(O) = 0 and (J' E .4, then (J(Ji1)B(.Yt') C ranJi1; in particular (J(Ji1)[T] E BA(.Yt') and 9(J(Ji1)[T] = 0 for all T E B(.Yt'). In order to prove this it is sufficient to show that the function r.p defined by r.p(0) = (J' (0) and r.p (x) = (J (x) / x if x =1= 0 belongs to .4 (because then we may use the identity (J(x) = xr.p(x)). We clearly have r.p(x) = J~ (J1(O'x)dO', or r.p =
J~ (J'adO' (with (Jla(x) == (JI(O'X)). Since IWall..,(( ::; IWII..,(( and O' f--+ (JM E.4 is weak* continuous (.4 is identified with the dual of the space of continuous functions which converge to zero at infinity) the integral J~ (J'a dO' exists in the weak* topology of .4, and so r.p E .4.
2.3. REGULARIZATION WITH RESPECT TO A
We describe here a method of approximating an operator T E B(.Yt') by operators of class Coo (or analytic) with respect to A that is similar to the regularization by convolution in the case of Loo(lR) (equipped with the translation group). Then we study the rapidity of the approximation for operators having a prescribed order of regularity.
If r.p is a function in C~(lR) and r.p(0) = 1, then for an arbitrary non zero real number c the operator r.p(c.rzf) will transform an arbitrary operator T E B(.Yt') into an analytic operator r.p(c.rzf)T and s-limr.p(c.rzf)T = T. Our
10--->0
purpose now is to determine the rapidity of this convergence under certain regularity assumptions on T. This is equivalent, in our abstract setting, to giving a Littlewood-Paley type description of the spaces N:t,p from §1.4.
Let (J E C~ (lR) and m 2: 1 an integer such that (J has a zero of order m at zero (so (J(O) = (JI(O) = ... = (J(m-l)(o) = 0). Choose a > a such that su pp (J C (-27r / a, 27r / a). Then eixa =1= 1 if x =1= 0 and x belongs to supp(J, hence the function ry(x) = (J(x) [eixa - 1tm is of class C~(lR) and (J(x) = TJ(x) [e ixa _1]m. So, for any c E lR we have (J(c.rzf) = TJ(c.rzf)[~a -1]m
and for an arbitrary T E B(£,)
(2.6)
Hence, if 0 < a < m and 1 ::; p ::; 00, then
[lolIIE-aO(EJZ1')TWE-ldE] lip::; 111]IIAl . aa [loa [Wm (::, A)r E-1dE] lip.
(2.7) In particular, for an arbitrary function 0 E C8"(lR) having a zero of order m > a at zero we have [JlIIE-aO(EJZ1')TIIPc ldE]llp < 00 ifT E CCa,P(Aj£').
For p = 00 the preceding assertion is especially simple and important for our later work, so we state it explicitly. Let a > 0 real, T E 'ifO«Aj £') and 0 E C8"(lR) such that 0 has a zero of order> a at zero. Then we have IIO( EJZ1')TII ::; CEa for all E > 0 and some finite constant C. Reciprocally, if 0,# 0, then such an estimate characterizes the class CCa(Aj £'). In order to prove this converse assertion we find it convenient to establish first an analogue of Calderon's identity (see [9]) in our context.
Let ~ E COO(lR) such that ~(O) = 0 and ~(x) = 1 near infinity. If we set 1](x) = xe(x) then 1] E C8"(lR), 1](0) = 0 and we have ~(bx) - ~(ax) =
Ji 1]( TX )T-ldT for an arbitrary b > a > 0 and x E lR. This clearly implies ~(bJZ1') - ~(aJZ1') = Ji 1](TJZ1')T- ldTj note that for T E B(£') the integral Ji 1](TJZ1')[T]T-ldT exists in B(£,) in the norm topology. On the other hand, by using the facts established in §2.2, we get s-lim ~(aJZ1')[T] = ~(O)T = 0
a-++O for each T E B(£,) and u-lim~(bJZ1')[T] = T - fJI'[T] if T E BA(£') (use
b-+oo (2.5) with (! = 1 - ~). So we have
T = fJI'[T] + s-lim lb 1] (TJZ1') [T]T-ldT a-++O,b-+oo a
for an arbitrary T E BA(£'). Observe that this result can be stated without reference to the func
tion ~. Indeed, if 1] E C8"(lR) (and this condition is, in fact, much stronger than needed), 1](0) = 0 and Jooo 1](±T)T-ldT = 1, then we may take ~(x) =
JJxl1](T sgnx)T-ldT. In conclusion, we have established the following ana logue of Calderon's identity:
Proposition 2.1 Let 1] be a function of class C8"(lR) and such that 1](0) = 0, Jooo 1](±T)T-ldT = 1. Then for each operator T E BA(£') and each number T > 0 the operator 1]( T JZ1') [T] is analytic with respect to A and the map
(0, +00) 3 T ~ 1](TJZ1')[T] E B(£,)
is of class coo (in norm). Moreover, the improper integral
exists in the s.o. topology of B(.Yt') , in other terms
exists. Finally, if 9 is the projection of B A (.Yt') onto ker Jd determined by the direct sum decomposition BA(.Yt') = ker Jd + ran Jdu (cf. (2.4) and (2.5)), then
(2.8)
Now let () E At such that ()' E At and ()(O) = 0; then 9()(Jd)[T] = 0 (see the end of §2.2) and
(2.9)
for each T E B(.Yt'), where rt(x) = "1(rx). In fact there are no problems of convergence at infinity ofthe integral, because 1I()"1TII..4I: ::; C(T) -1. To prove this for r 2: 1 we observe that
Set t = T- 1, then
B(tx)"1(X) = t 101 ()'(O"tX)X"1(x)dO" = t 101 (e'O"t"1(I») (x)dO"
with "1(1) (X) = x"1(x). Hence
lI()t"1ll..4I: ::; t 101 IWO"t"1(I) 11..41: ::; tlWII..4I: '11"1(1)11..41:
which is the estimate we were looking for. In conclusion, if () is a function in At such that ()(O) = 0 and ()' E At then there is a constant c < 00 such that 1 1 (()"1T) (Jd)TI 1 ::; cIITII(r)-1 for all T E B(.Yt') and T > 0, and we have
()(Jd)[T] = s-lim 100 (()"1T) (Jd)[T]T- l dT == 100 (()"1T) (Jd)[T]r-ldT. (2.10) a->+O a ->0
The integral over [a, 00) exists in norm in B(.Yt'), for each a > O. Moreover, if there is 0: > 0 such that T E CCD«A;.Yt') then the integral over (0,00) exists
in norm in B(£') (so that the last integral is not improper). Indeed we may assume 0 < a < 1 and then we have II(Orn(d)TII ::; IIOIIAtII1](Td)TII < C'Ta (cf. (2.7)).
One can improve the estimates near infinity if () has a zero of higher order at the origin. Assume ()(k) E .4 for k = 0,1, ... , m and ()(O) 0'(0) = ... = o(m-l)(o) = O. Then
O(tx)1](x) = t~ r1 ()(m)(utx)xm1](x)m(l- u)m-1du. m. io
Hence, if we set 1]( m) (x) = Xm1] (x), we obtain for all t E JR.:
IlotryllAt ::; '~711()(m)IIAt ·111](m)IIAt · (2.11)
We shall use this estimate with a different purpose. Let E > 0 and consider (2.9) with the function () replaced by oc(x) = O(EX). Then
11()(Ed)TII ::; 1000 II(oc1]T)(d)TIIT-1dT.
We shall estimate the integrand in two ways, according to the value of T.
If 0 < T ::; E, then clearly II(OC1]T)(d)TII ::; IIOIIAtII1](Td)TII. For T > E we shall use (2.11) but with 1] replaced by a function ( E Co(R) and such that (1] = 1] (note that (2.11) holds for an arbitrary 1] E .4 such that 1](m) E .4). Then OC1]T = ()c(T1]T = (()C/T()T1]T hence:
II(OC1]T)(d)TII = 11(()C/T()(Td)1](Td)TII::; IW/T(IIAtII1](Td)TII
::; ~ (:. ) m 11()(m) IIAt . 11((m) IIAt 111]( T d)TII· m. T
Now we can prove without difficulty the main estimate of this section.
Theorem 2.2 Let ()o, 0 be functions on R with the following properties: (1) ()o E .4(R) and there are a > 0 and b < 0 such that ()o(a) #- 0,
Oo(b) #- O. (2) There is an integer m 2: 1 such that (), 0', ... ,()(m) E .4(JR.) and
0(0) = 0'(0) = ... = o(m-l)(o) = o. Then there is a constant M < 00 such that for all T E B(£,) and all
real E > 0:
IIO(c:d)TII::; M roo min(l, (:')m)IIOo(Td)TIIT-1dT. (2.12) io T
Moreover, for a E (0, m) and p E [1,00] one has
[1000 11c:-aO(Ed)TIIPcldEf/P ::; a(:~ a) [1000 IIE-aOo(C:d)TWE-ldEf/p·
(2.13)
30 ANNE BOUTET DE MaNVEL AND VLADIMIR GEORGESCU
Proof. Taking into account the two estimates obtained before the statement of the theorem, we clearly get (2.12) with 00 replaced by T/. Now observe that it is sufficient to establish (2.12) under the supplementary hypothesis 00 E COO(lR \ {O}). Indeed, if (! E ..4l(lR) and (! is not identically zero on (0,00) and on (-00,0) then we can find such a 00 with the property that suppOo is included in the region where U(x) i- O. Then cP = Oo/{! belongs to ..4l(lR) by Wiener's theorem (see page 78 in [22]) and 00 = CPU; hence
which gives (2.12) with 00 replaced by (!.
So we may assume 00 of class Coo and with support in lR \ {O}. Let a± = Jrf IOO(±TWT-ldT. Then a± > 0 and the function T/ defined on lR by T/(±T) = (a±)-lIOo(±TW for T ~ 0 has the properties required in Proposition 2.1. So (2.12) holds with 00 replaced by T/. Since T/ = cpOo with cp(±T) = a;i/Oo(±T) for T ~ 0, and since cp E COO(JR), we obtain now (2.12) by the same argument as above.
Let Xm(cr) = min(l, crm) for cr ~ O. The inequality (2.12) may be written as
IIO(ed)TII::; M 1000 xm(cr)IIOo(~d)Tlld:. If we set fee) = IIO(ed)TII, gee) = IIOo(cd)TII, so that f,g are positive lower semicontinuous functions on (0,00), then we have
for any a > 0; in the r.h.s. we recognize the convolution product associ ated to the multiplication group (0,00) (with C1de as Haar measure). If L~ = LP((0,00);c1de) with 1 ::; p ::; 00 then by the integral Minkowski inequality we get
The last integral is finite (and equal to m/[(m - a)a]) if and only if 0 < a < m. Note that in this argument L~ could be replaced by other Banach spaces of measurable functions on (0,00). The proof of (2.13) is finished. D
Corollary 2.3 Let a be a strictly positive real number, p E [1,00] and m > a an integer. Then for T E B(.Ye) the following assertions are equivalent:
(a) T E 'i&'a,p(A; .Ye);
(b) there is () E At(IR) that is not identically zero on (0, (0) and on (-00,0) such that
(2.14)
(c) The preceding estimate holds for each () E At such that ()(k) E At for o ::; k ::; m and ()(k) (0) = 0 for 0 ::; k ::; m - l.
Observe that we may take ()(x) = (eix - l)m. The characterization of 'CO< (A; £) stated after (2.7) is a particular case of the preceding corollary. Notice also the following consequence of Corollary 2.3. Let <p E At with <p(0) = 1 and such that <p(k) E At for 0 ::; k :s; m and <p'(O) = ... = <p(m-l)(o) = O. Then for each T E 'CO< (A; £) (where 0 < a < m) there is a constant c such that II <p( e.0")[T] - Til ::; ceO<; moreover, if <p is identi cally equal to 1 neither on (0,00) nor on (-00,0), then such an estimate characterizes the property T E 'CO< (for the proof, take () = <p - 1 in the Corollary 2.3). This is an extension to our context of a theorem of Jackson and Zygmund concerning the rapidity of the approximation by trigonomet ric polynomials of periodic functions of class AO<; cf. (13.14) and (13.20) in Ch.III of [27] (see also [22]).
3. Homogeneous Hamiltonians
We recall that our purpose is to study the boundary values of the resolvent of a self-adjoint operator H (the "hamiltonian") which has a certain degree of regularity with respect to another self-adjoint operator A (the "conju gate operator"). We think it worthwile to consider first the particular case of A-homogeneous hamiltonians. In this setting the main ideas of the gen eral method (cf. Section 4) are easier to describe since several important technical difficulties can be completely avoided.
We shall say that a self-adjoint (unbounded) operator H in £ is A homogeneous if e-iAr H eiAr = er H for all real T. This is formally equivalent to [H, iA] = H, i.e. H is an "eigenvector" of the operator .0" introduced in §2.1; this is formal because H cannot be a bounded operator without being zero (indeed, if .A > 0 and (j, H J) = .A then (eiAr f, H eiAr J) = er.A - 00 as T - (0). In terms of the resolvent R(z) = (H - z)-l of H the condition of A-homogeneity is clearly equivalent to ~[R(z)] = e-r R(e-r z) for all T E IR and some (and hence all) z outside the spectrum of H. An interesting example of couple (H, A) verifying these conditions is H = ~ in L2(lRn) and A a multiple of the generator of the dilation group; see [2] for other non-trivial examples (in the first step of our argument below we follow [2]; cf. also the Introduction of Ch.7 in [1]).
We fix an A-homogeneous operator H and observe that id[R(z)] = - z :k R( z) - R( z) == - d~ (zR( z)) (take the derivative with respect to T in 'WT[R(z)] = e-TR(e-Tz)). It follows that for each z E C outside the spectrum of H we have R(z) E COO(Aj £) and for each kEN:
Let f E D(A) and F the holomorphic function defined outside the spectrum of H'by P(z) = (I, R(z)!). Then (3.1) with k = 1 gives
izP'(z) = (R(z)* f, Af) - (Af, R(z)!) - iF(z),
hence
IzP'(z) I :s: 2I1Afll'IIR{z)fll + IF{z)1 :s: (lIfll + 2I1AfIDIIR(z)fll·
Assume, for example, z = ),+ifL with fL > O. Then IIR(z)fIl2 = fL- 1 1m F(z) hence, if we denote IIfliA = IIfll + IIAfll, we get:
IzF'(z)1 :s: 2I1fIlAfL-l/2IF(z)ll/2.
If f i=- 0 then ImP(z) > 0 for Imz > 0, so F 1/2(z) == F{Z)1/2 is a well defined holomorphic function in the upper half-plane and tzFl/2(z) = ~F(z)-1/2F'{z). Hence
I ~Fl/2(Z)1 < IIfliA dz - Izl{Imz)1/2'
Imz> O. (3.2)
By a well-known theorem of Hardy and Littlewood (see the Appendix, Corollary A.2) the estimate (3.2) implies that the function p 1/ 2 is locally Holder continuous of order 1/2 in the set {z Eel 1m z ~ 0 and z f. O}. In particular, for each f E D{A) the limit limt/-->+o(l, R(), + ifL)!) exists if ), i=- 0 and, as a function of ), E lR \ {O}, it is locally of class Al/2.
By using the explicit form (3.2), one may find quite precise estimates on the boundary values of the resolvent. For this we just follow the steps of the proof of the theorem of Hardy and Littlewood. Let), f. 0 and 0 < fL < 1/.
Then the identity
and the bound
imply that limJ.!---7+o p 1/ 2(>.. + ift) == pl/2(>, + iO) exists uniformly in A if IAI 2: const. > O. Moreover, for each v > 0 we have
By taking the infimum over v > 0 of the right-hand side we obtain
1(1, R(A + iO)f) I ::; 1~111f11'llfIIA' f E D(A). (3.4)
Now let 0 < Al < A2 and ft > 0, then
pl/2(A2 + iO) - pl/2(Al + iO) = pl/2(A2 + iO) - pl/2(A2 + ift) +pl/2(A2 + ift) - pl/2(Al + ift) +pl/2(Al + if-t) - pl/2(Al + iO)
= -i fov (pl/2)'(A2 + iT)dT
Now we use
IP(A2 + iO) - P(AI + iO)1 ::; ::; 1F1/2(A2 + iO) - pl/2(Al + iO)I' (1F1/ 2(A2 + iO)1 + 1F1/2(Al + iO)1)
and the estimates (3.5), (3.4). We obtain
1(1, R(A2+iO)f) - (1, R(AI +iO)f)I::; ~~2 JA2 - Al[21Ifll·llfll~ll/2. (3.6) Al
This holds for 0 < Al < A2 and f E D(A).
It is easily shown that the estimates (3.4), (3.6) may be restated in the following terms. Denote £i = D(A) equipped with the graph norm (11/112 + IIAII12)1/2 and let £'-1 = £1*. We identify £'* = £' and £i C
£' C £'-1 as usual; in particular B(£,) C B(£i,£'-l) and IITII(l) :::; IITII where IITII(l) denotes the norm in B(£i,£'-t). Then for each A E JR, A '# 0 the limits lim/L->±oR(A + iJ..l) == R(A ± iO) exist in norm in B(£i,£'-l), uniformly in A if IAI 2: const. > 0, and we have for arbitrary non-zero real A, A1, A2:
(3.7)
IIR(A1±iO)-R(A2±iO)II(1) :::; 40IA1-A211/2max(IAll-3/2, IA21-3/2). (3.8)
High energy estimates of the form (3.7) are known in the N-body case, cf. [14] and references therein; see also [17] (we thank M. Mantoiu for correcting a previous version of (3.7)-(3.8)). The interest of our proof lies only in the fact that we did not use a modified resolvent like in the Mourre approach.
Now let k 2: 2 be an integer and let 1 E D(Ak ). Then (3.1) implies for O:::;m:::; k
(idd z)mp(z) = (f,a'm[R(z)]J) = f·(rr:)C-l)m- j (Am- j l,R(Z)Ajf). z j=O J
The operator zk ~ is a linear combination of the operators (fz z)m with 0:::; m :::; k. Hence there is a constant Ck depending only on k such that
IZk p(k)(z)1 :::; Ck L I(Aj I, R(z)Am f)1· j+m:s;k
In each term of the sum at least one of the numbers j, m is < k. Assume, for example, that j < k. Then g == Aj 1 E D(A) hence (z = A + iJ..l with J..l > 0):
1 IIR(z)gll :::; y'iL1 Im(g, R(z)g)11/2 :::; c(g)J..l-1/2
where the constant C(g) is finite and depends only on the norm of g in D(A) and on 8 if IAI 2: 8 > 0 (one can estimate c(g) by using (3.4) and (3.3) but this is of no interest here). So we get
In conclusion, for each 1 E D(Ak) and each 8> 0 there is a constant c < 00
such that for IRe zl 2: 8 and 1m z > 0:
IP(k)(z)l:::; CClmz)-1/2 = c(lmz)-k+(k-1/2).
By using the theorem of Hardy and Littlewood (Corollary A.2 in the Ap pendix) we see that the boundary value function>. .- F(>. + iO) is locally of class Ak-1/2 on R. \ {O} if f E D(Ak).
Let us state this result in slightly different terms. For 8 > 0 real let £a be the domain of IAls equipped with the graph norm (see the Intro duction). Set .JIt'-s = ~* and identify £a C .JIt' = .JIt'* C .JIt'-s. Then B(.JIt') C B(£a, .JIt'-s) and we denote II . II(s) the norm in the Banach space B(£a, .JIt'-s). By using the preceding results and the uniform boundedness principle it follows that for each integer 8 2': 1 the functions
R. \ {O} 3 >. .- R(>. ± iO) E B(£a, .JIt'-8) (3.9)
are locally of class As-1/2 (in norm). We shall now prove by complex interpolation that this assertion remains
true for all real

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