real variable methods in fourier analysis (north-holland mathematics studies 46)

REAL VARIABLE METHODS IN FOURIER ANALYSIS

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NORTH-HOLLAND MATHEMATICS STUDIES 46

Notas de Matematica (75) Editor: Leopoldo Nachbin

Universidade Federal do Rio de Janeiro and University of Rochester

Real Variable Methods in Fourier Analysis

MIGUEL DE GUZMAN Universidad Complutense de Madrid Madrid, Spain

1981

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK OXFORD

0 Nortli- Holland Publishing Compuny, 1481

All rights reserved. No part of this publication may be reproduced. stored in a retrievalsystem. o r transmitted, in any form or by any means, eleclronic, mechunical, photocopying, recording

or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 86124 6

publisher^:

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Sole distributors for the U.S.A. and Canurlu:

5 2 VANDERBILT AVENUE, NEW YORK, N.Y. 10017 ELSEVIER NORTH-HOLLAND, INC.

Library of Congress Cataloging in Publication Data

Guzdn, Miguel de, 1936- Real variable methods in Fourier andysis.

(Notas de m t d t i c a . 75) (North-Holland

Bibliography: p. Includes index. 1. Fourier analysis. 2. Functions of real

mathematics studies ; 46)

variables. 3. Operator theory. I. Title. 11. Series. W.N86 no. 75 LQ403.51 510s [515'.24331 8022545 ISBN 0-444436124-6

PRINTED IN THE NETHERLANDS

Dedicated to

ALBERTU P. CALVERbN

and

ANTON1 ZYGMUNV

PREFACE

The work presented he re i s centered around t h e s t u d y o f some o f t h e r e a l v a r i a b l e methods newly developed i n a n a t u r a l way f o r t h e t rea tmen t o f d i f f e r e n t problems i n F o u r i e r A n a l y s i s , p a r t i c u l a r l y f o r problems r e l a t e d t o t h e p o i n t w i s e convergence of some i m p o r t a n t ope ra to rs . s tand these ques t i ons i s t h e corresponding maximal o p e r a t o r and so t h e methods presented here concern t h e genera l S te in -N i k i s h i n theo ry , t h e genera l and s p e c i a l techniques t h a t can be used t o deal w i t h d i f f e r e n t types o f o p e r a t o r s , t he c o v e r i n g methods o r i g i n a t e d i n d i f f e r e n t i a t i o n ' t heo ry , methods connected w i t h t h e theo ry of s i n g u l a r i n t e g r a l ope ra to rs , F o u r i e r m u l t i p l i e r s , . . .

t r y t o desc r ibe , i n a c o n t e x t as s imp le as p o s s i b l e , some o f t h e main i deas around a p a r t i c u l a r t o p i c . h a u s t i v e i n g i v i n g r e s u l t s . On t h e o t h e r hand we have t r i e d t o p r e s e n t those methods i n a c t i o n and i t i s under t h i s l i g h t t h a t t h e a p p l i c a t i o n s o f those methods t h a t we show as samples i n t h e book have t o be understood. The main aim o f o u r e x p o s i t i o n t h e r e f o r e i s t h a t t h e reader who f o l l o w s our work can l o c a t e t h e r i g h t p l a c e which each one o f t h e techniques and methods we p r e s e n t occupies i n t h e modern F o u r i e r A n a l y s i s . t ime he w i l l be a b l e t o a c q u i r e a f i r s t f a m i l i a r i z a t i o n w i t h those techniques by see ing some o f t h e i r most i m p o r t a n t a p p l i c a t i o n s .

I n t h e f i e l d we a re go ing t o e x p l o r e t h e r e a r e many i n t e r e s t i n g open problems. I have t r i e d t o emphasize some o f t h e ones t h a t a r e connected w i t h t h e aspects o f t h e t h e o r y we s h a l l s tudy . i n t h e t e x t i s g i v e n a t t h e end.

perhaps be meaningless f o r t h e t o t a l l y n o n - i n i t i a t e d , b u t t hey may be o f some use f o r t h e reader who i s acqua in ted w i t h t h e fundamentals o f r e c e n t F o u r i e r Ana lys i s .

Chapter 1 cons ide rs i n an a b s t r a c t way t h e most i m p o r t a n t problem we deal wit- o f t h e p o i n t w i s e convergence o f a sequence of o p e r a t o r s . The Banach p r i n c i p l e , which i s a p a r t i c u l a r form o f t h e u n i f o r m boundedness p r i n c i p l e , i s t h e s t a r t i n g p o i n t o f o u r s tudy . The f i n i t e n e s s a.e. o f t h e assoc ia ted maximal ope ra to r leads t o t h e convergence a.e. o f t h e sequence o f ope ra to rs ,

t he modern m m e n t s o f N i k i s h i n , Maura is and G i l b e r t . T h e i r work i s more e a s i l y understood under t h e l i g h t o f i t s g e n e t i c g rowth and so we p re - sen t f i r s t t h e r e s u l t s o f A. Calderbn, S t e i n and Sawyer, acco rd ing t o which

The key t o under-

Our work has an i n t r o d u c t o r y c h a r a c t e r . I n each chap te r we s h a l l

Our goal i s t o p r e s e n t methods, n o t t o be ex-

A t t h e same

A l i s t o f t h e ones ment ioned

The f o l l o w i n g i n d i c a t i o n s about t h e con ten ts o f t h e whole work w i l l

I n Chapter 2 we s h a l l f o l l o w t h e l i n e o f t hough t which has l e a d t o

v i i

v i i i PREFACE

the f in i t eness a.e. of the maximal operator i s equiva len t , under some par- t i c u l a r circumstances, t o the weak type of the same maximal opera tor . The r e s u l t s of Nikishin, Maurais and Gi lber t extend and simplify the previous theorems i n t h i s di rec t ion .

study of t h e m a l opera tor , such as those of covering and decomposition of functions, i n t e rpo la t ion , ex t rapola t ion , majorization, 1 i nea r i za t ion , summation, ... Some of them a r e of constant use i n t h i s type of Analysis. The method of i n t e rpo la t ion , i n pa r t i cu la r , has developed i n t o a f u l l branch of Analysis. We present here some of t h e most important results and r e f e r t o the specialized modern monographs f o r fu r the r information.

allow the use of a pa r t i cu la r method which seems t o be of i n t e r e s t . order t o see whether the maximal onerator i n question i s of weak type (1 , l ) i t su f f i ces t o study i t s ac t ion on f i n i t e sums o f Dirac de l t a s . This i s the main theorem of Chapter 4 , where some consequences and extensions a re given.

For the type (2,2) of an operator there a r e special techniques ava i lab le , such a s the Fourier transform and the lemma of Cotlar. Also the method of ro t a t ion i s useful i n order t o extend a one-dimensional i n - equa l i ty t o more dimensions. These methods a r e presented in -- Chapter 5 .

t a i n very bas i c operators, t he Hardy-Littlewood maximal operator and i t s var ian ts . other operators of grea t i n t e r e s t , such a s the Calderbn-Zygmund opera tors and the diverse operators o f approximation of t he i d e n t i t y . Also their behaviour i s intimately r e l a t ed t o the d i f f e r e n t i a t i o n of i n t eg ra l s .

c o n n e c t i o n m e n coverings, d i f f e ren t i a t ion and several extensions of the Hardy-Littlewood maximal operator.

a t i on p r o p m f some bases of i n t e rva l s and rec tangles i n R2. This study has been g rea t ly enriched by the important recent cont r ibu t ions of Cbrdoba, R. Fefferman, Stromberg and o thers .

measurable s e t s , f i r s t developed by Besicovitch, t h a t a r e most re levant f o r the study of some of t he problems t h a t a r i s e i n a natural way i n d i f f e ren - t i a t i o n theory and i n other a reas of Fourier Analysis.

i d e n t i t y , viewedin t h e i r re la t ionship w i t h d i f f e r e n t i a t i o n theory.

in tegra l operators. The methods presented i n previous chapters a r e succes s fu l ly p u t t o work i n order t o obta in , i n a very easy way, the c l a s s i ca l results about the Hilbert transform and the Calderbn-Zygmund theory.

the poss ib i l i t y of applying the Fourier transform t o ce r t a in problems

Chap te r 3 considers some of the general techniques which ease the

Convolution operators , of paramount importance i n Fourier Analysis, In

Chapters 6 throuqh 9 a r e c lose ly connected w i t h the study of cer-

Their importance stems from the f a c t t h a t they control many

Chapter 6 shows the most important general r e s u l t s about the

Chapters 7 and 8 deal w i t h the special covering and d i f f e r e n t i -

Chapter 9 describes some of the fea tu res of t he theory of l i n e a r l y

Chapter 10 deals w i t h d i f f e r e n t types of approximations of the

Chapter 11 unfolds the main theorems i n the theory of s ingular

The r ecen t work of Nagel, RiviGre, S te in and Wainger have shown

PREFACE i x

r e l a t e d t o d i f f e r e n t i a t i o n and t o some analogues o f t h e H i l b e r t t r a n s f o r m a long curves i n n-dimensional Euc l i dean space. T h e i r methods, o f which some examples a r e presented i n Chapter 12, a r e of g r e a t i n t e r e s t .

F i n a l l y , Cha t e r 13 p resen ts some a p p l i c a t i o n s o f t h e methods of d i f f e r e n t i a t i o n t h e o r y -5 o Chapter 8 t o s o l v e some problems about F o u r i e r m u l t i p l i e r s : r e s u l t s o f Cbrdoba and R. Fefferman.

There are, o f course, many o p i c s o f c u r r e n t F o u r i e r A n a l y s i s which have been l e f t out , such as H b spaces, f u n c t i o n s o f bounded mean o s c i l l a t i o n (8MO) , we igh t t heo ry , A.P. Ca lde rbn ' s theorem on t h e Cauchy i n t e g r a l ... monographs and some o t h e r s seem t o be s t i l l i n a v e r y f l u i d shape, which makes t h e i r e x p o s i t i o n r a t h e r d i f f i c u l t .

C. Fef ferman's theorem on t h e u n i t d i s k and t h e more r e c e n t

Some o f these t o p i c s have been r e c e n t l y t r e a t e d i n competent

T h i s book i s e s s e n t i a l l y s e l f - c o n t a i n e d f o r t hose who know t h e fundamentals o f t h e Lebesgue i n t e g r a l and o f F u n c t i o n a l A n a l y s i s . I have t r i e d t o make i t a c c e s s i b l e and easy t o read. The background and t h e m o t i v a t i o n i s l o c a t e d , o f course, i n t h e modern F o u r i e r Ana lys i s . A s h o r t i n t r o d u c t i o n t o it, l i k e Hardy and Rogosinsk i [19441 w i l l s u f f i c e t o under- s tand t h i s m o t i v a t i o n . It i s , however, q u i t e c l e a r t h a t t h e more t h e reader knows o f works such as Zygmund 119591, Ste in-Weiss [19711, S t e i n [19701, t h e more he w i l l p r o f i t f rom t h i s book.

T h i s work i s t h e f r u i t o f severa l courses and seminars o rgan ized a t t h e Un ive rs idad Complutense de Madrid. I wish t o acknowledge t h e h e l p and s t i m u l u s I have rece ived , among so many hours o f work and d i s c u s s i o n , f rom my f r i e n d s and co l l eagues : M.T. C a r r i l o , A. Casas, A. Cdrdoba, P. Ci fuen tes , J. Garcia-Cuerva, S . Garcia-Cuesta, A. G u t i & r r e z , M.T. Manlrguez, B.Lz. Melero, R. Moreno, R. Moriydn, I. Pera l , E. RodrTguez, J..L. Rubio de F ranc ia , B. Rubio Segovia, A. Ruiz, A. de l a V i l l a , M. Wal ias. I thank a l s o P i l a r A p a r i c i o f o r h e r h e l p i n t y p i n g my manuscr ip t .

MIGUEL DE GUZMAN

TABLE OF CONTENTS

DEDICATION V

PREFACE v i i

CHAPTER 1 : P O I N T W I S E CONVERGENCE OF A SEQUENCE OF OPERATORS

1.1. F i n i t e n e s s a.e. and c o n t i n u i t y i n measure o f t h e maximal o p e r a t o r

1 . 2 . C o n t i n u i t y i n measure a t 0 E X o f t h e maximal o p e r a t o r and a.e. convergence

CHAPTER 2 : FINITENESS A.E. AND THE TYPE OF THE MAXIMAL OPERATOR

2 .1 . A r e s u l t o f A.P. Calder6n on t h e p a r t i a l sums o f t h e F o u r i e r s e r i e s o f f E L 2 ( T )

2.2 Commutat iv i ty o f T* w i t h m i x i n g t r a n s f o r m a t i o n s . P o s i t i v e ope ra to rs .

2.3. Commutat iv i ty o f T* w i t h m i x i n g t r a n s f o r m a t i o n s . The theorem o f S t e i n

2.4. The theorem o f N i k i s h i n

The theorem o f Sawyer

CHAPTER 3 : GENERAL TECHNIQUES FOR THE STUDY OF THE MAXIMAL OPERATOR

3:l . Reduct ion t o a dense subspace

3.2. Cover i ng and decomposi t i on

3.3. Kolmogorov c o n d i t i o n and t h e weak t y p e o f an o p e r a t o r

3.4. I n t e r p o l a t i o n

3.5. E x t r a p o l a t i o n

3.6. M a j o r i z a t i on

3.7. L i n e a r i z a t i o n

3.8. Summation

CHAPTER 4 : ESPECIAL TECHNIQUES FOR CONVOLUTION OPERATORS

4.1. The t y p e (1,l) o f maximal c o n v o l u t i o n o p e r a t o r s

4.2. The t ype (p,p), p > l , o f maximal c o n v o l u t i o n o p e r a t o r s

1

8

11

13

14

19

23 29

35 35 39 50 54 60 63 66 68

73 74 88

x i

x i i TABLE OF CONTENTS

CHAPTER 5 : ESPECIAL TECHNIQUES FOR THE TYPE (2,2)

5.1. F o u r i e r t r a n s f o r m

5.2. C o t l a r ' s lemma

5.3. The method o f r o t a t i o n

CHAPTER 6 : COVERINGSy THE HARDY-LITTLEWOOD MAXIMAL OPERATOR AND DIFFERENTIATION. SOME GENERAL THEOREMS.

6.1. Some n o t a t i o n

6.2. Cover ing lemmas i m p l y weak t y p e p r o p e r t i e s o f t h e maximal o p e r a t o r and d i f f e r e n t i a t i o n

6.3. From t h e maximal o p e r a t o r t o c o v e r i n g p r o p e r t i e s

6.4. D i f f e r e n t i a t i o n and t h e maximal o p e r a t o r

6.5. D i f f e r e n t i a t i o n p r o p e r t i e s i m p l y c o v e r i n g p r o p e r t i e s

6.6. The h a l o problem

CHAPTER 7 : THE B A S I S OF INTERVALS

7.1. The i n t e r v a l b a s i s 4 2 does n o t have t h e V i t a l i

7.2. D i f f e r e n t i a t i o n p r o p e r t i e s o f g 2 . p r o p e r t y .

i n e q u a l i t y f o r a b a s i s which i s t h e C a r t e s i a n p roduc t o f another two

7.3. The h a l o f u n c t i o n o f 6 ) ~ . Saks r a r i t y theorem

7.4. A theorem o f B e s i c o v i t c h on t h e p o s s i b l e va lues o f t h e upper and lower d e r i v a t i v e s w i t h r e s p e c t

7.5. A theorem o f Marst rand and some g e n e r a l i z a t i o n s

7.6. A problem o f Zygmund so l ved by Mor i ydn

7.7. Cover ing p r o p e r t i e s o f t h e b a s i s o f i n t e r v a l s . A theorem o f Cdrdoba and R. Fef ferman

7.8. Another problem o f Zygmund. S o l u t i o n by Cdrdoba

It does n o t d i f f e r e n t i a t e L 1 Weak t y p e

t o 82

CHAPTER 8 : THE B A S I S OF RECTANGLES

8.1. The Perron t r e e

8.2. A lemma of Fefferman

8.3. The Kakeya problem

8.4. The B e s i c o v i t c h s e t

8.5. The Nikodym s e t

8.6. D i f f e r e n t i a t i o n p r o p e r t i e s o f some bases o f r e c t a n g l e s

8.7. Some r e s u l t s concern ing bases o f r e c t a n g l e s i n lacunary d i r e c t i o n s

9 1

9 1

92

96

103

104

105

114

118

136

149

159

160

160

165

171

177

182

184

193

199

201

207

209

210

215

224

233

TABLE OF CONTENTS x i i i

CHAPTER 9 : THE GEOMETRY OF LINEARLY MEASURABLE SETS

9.1. L i n e a r l y measurable s e t s

9.2. Dens i t y . Regular and i r r e g u l a r s e t s

9.3. Tangency p r o p e r t i e s

9.4. P r o j e c t i o n p r o p e r t i e s

9.5. Sets o f p o l a r l i n e s

9.6. Some a p p l i c a t i o n s

CHAPTER 10: APPROXIMATIONS OF THE IDENTITY

10.1. Radi a1 k e r n e l s

10.2. Kernels non - inc reas ing a long rays

10.3. A theorem o f F. Zo

10.4. Some necessary c o n d i t i o n s on t h e k e r n e l t o d e f i n e a good approx ima t ion o f t h e i d e n t i t y

CHAPTER 11: SINGULAR INTEGRAL OPERATORS

11.1. The H i l b e r t t r a n s f o r m

11.2. The CalderBn-Zygmund o p e r a t o r s

11.3. S i n g u l a r i n t e g r a l ope ra to rs w i t h g e n e r a l i z e d homogenei t y

CHAPTER 12: DIFFERENTIATION ALONG CURVES. A RESULT OF STEIN AND WAINGER

12.1. The s t r o n g t y p e (2,2) f o r a homogeneous cu rve

12.2. The t y p e (p,p) l<p<m o f t h e maximal o p e r a t o r

12.3. An a p p l i c a t i o n . D i f f e r e n t i a t i o n by r e c t a n g l e s determined b y a f i e l d o f d i r e c t i o n s

CHAPTER 13: MULTIPLIERS AND THE HARDY-LITTLEWOOD MAXIMAL OPERATOR

13.1. The c h a r a c t e r i s t i c f u n c t i o n o f t h e u n i t d i s k . A theorem o f C. Fefferman

13.2. Polygons wi th i n f i n i t e l y many s i d e s

13.3. The maximal o p e r a t o r w i t h r e s p e c t t o a c o l l e c t i o n o f r e c t a n g l e s . A theorem o f A. CBrdoba and R. Fef ferman

REFERENCES

A LIST OF SUGGESTED PROBLEMS

241

242

245

252

258

268

276

281

282

286

292

296

305

306

313

327

337

338

348

350

359

362

368

371

379

389

391 INDEX

CHAPTER 1

POINTWISE CONVERGENCE OF A SEQUENCE OF OPERATORS

There a r e q u i t e a number o f i m p o r t a n t problems i n F o u r i e r A n a l 1

s i s and i n o t h e r areas i n which a sequence ( o r g e n e r a l i z e d sequence ) o f

ope ra to rs a r i s e s i n a n a t u r a l way.

i) I f f E L1([0,2x)), f p e r i o d i c o f p e r i o d 2 ~ r , t h e p a r t i a l

sums

F o u r i e r s e r i e s can be i n t e r p r e t e d as t h e a p a l i c a t i o n o f t h e oDera to r

t o t h e f u n c t i o n f. E q u i v a l e n t l y Sk f ( x ) = Dk * f ( x ) = 1 where Dk(x) , t h e D i r i c h l e t k e r n e l , i s d e f i n e d by

Sk

f ( x - y ) D k ( y ) d y TI

-Ti

s i n ( k t T ) x 1

Dk(X) = X TI s i n - 2

ii) I n an analogous way, t h e Cesaro sums

S,f(x) -t S l f ( X ) -t .,. -t S k f ( X ) O k f ( X ) =

k + l

can be i n t e r p r e t e d as t h e a p p l i c a t i o n o f t h e o p e r a t o r ak t o f. I f fk

i s t h e F e j 6 r k e r n e l , d e f i n e d by

1

2n( k + l ) Fk (x ) =

1

2 1. POINTWISE CONVERGENCE OF O P E R A T O R S

We know t h a t A r f ( x ) = Pr * f ( x ) where Pr(x), t h e Poisson k e r n e l of parameter r , i s

1 1 - r2

2n 1 - 2 r cos x + r 2 Pr(x) = -

i v ) If f E L1(Rn) , B(0,r) = Cx e Rn : 1x1 6 r I

i s t h e mean va lue o f f ove r B (x , r ) which i s cons ide red i n t h e t h e o r y

o f d i f f e r e n t i a t i o n o f i n t e g r a l s .

v ) I n a more genera l way , i f k E L1(Rn)),

-n x \ k ( y ) d y = 1, kE(x ) = E k(,) f o r E > 0, then

a r e t h e opera to rs considered i n t h e s t u d y o f t h e approx ima t ions o f t h e

i den t i t y . i f 1x1 2 E

0 , i f 1x1 < E

v i ) I f f E L’(R”) and hE(x) =

1.0. INTRODUCTION

then

i s t he truncated ( a t E ) Hilbert transform.

v i i ) If k i s a complex valued function defined on Rn--COI such t h a t

then, f o r f 8 L 1 ( R n ) , Calder6n-Zygmund transform of the function o f s ingular in tegra l opera tors .

KEf(x) = k E * f ( x ) i s the truncated ( a t E )

f , considered i n t h e theory

The most natural question in a l l these s i t u a t i o n s i s :

( A ) Tv dind vu.t W h d h a , Oh u n d a which a d d i t i v n d nvn &vial

c v n d i t i v a vn 5 v h an t h e vpehatvlrn Th ,the cornuponding bequence Tk6 (x ) cvnuagen 6vh e v a y x O h denvot eVt?hy x and whdt ahe t h e p m ~ Weh ad the,& h L i 2 .

This i s , i n a l l the cases we have considered, a t the same time the most d i f f i c u l t question. In many of them, because of t h i s d i f f i c u l t y , the theory s t a r t e d w i t h a l e s s ambitious program:

( B ) Tv 5ind o u t w h e t h a , V R u n d a which a d d i t i v n d nvn .&ivial

cvnd&..LvnA vn 4 ah On t h e vpadtvhn T k . t h e CvntLupanding oeyuence 04 ~ u n C ; t i v a ~~6 cvnuenga i n Avme opace LP.

Question B has in many of t he above presented s i t u a t i o n s a r a the r simple so lu t ion . Suppose t h a t we know t h a t T k i s l i n e a r , t h a t

4 1. POINTWISE CONVERGENCE OF OPERATORS

we can show, f o r example, t h a t , i f

1 1 T k f 1 1 6 c Ilf 1 1 w i t h c independent o f f and k , and t h a t f o r

g E t o (Rn) t o , say, Tg. L e t us prove t h a t { T k f l i s t hen a Cauchy sequence i n L2(Rn)

and so T k f w i l l converge i n L2(Rn) . We can w r i t e , f o r f ego (Rn),

f E L2(Rn), Tk f E L2(Rn)

converges i n L2(Rn)

and t h a t

we a r e a b l e t o show t h a t Tkg

Hence, g i v e n E > 0, we can f i r s t choose f E t o m n ) such t h a t

2c ] I f - g 112 6 E/2 and, once g i s f i x e d , k o such t h a t , i f

p,q > k o , L2 (R') .

1 1 Tpg - Tqg 6 4 2 . So I T k f } i s a convergent sequence i n

Th is would s o l v e q u e s t i o n (B) and would l e a v e unanswered q u e s t i o n

L e t us t a k e a c l o s e r l o o k

d l i k e t o (A ) . What can we do t o th row some l i g h t on it?

a t i t s meaning. Assume, as be fo re , t h a t Tk i s l i n e a r . We wou

be a b l e t o prove, f o r example,that, f o r and X > 0, f E L1(Rn)

[ A ( f , X ) I = ( t x e R n : l i m sup I T f ( x ) - T q f ( x ) ( > X PY9 -+

P 1 = o

T h i s would g i v e us t h e convergence o f { T k f ( x ) ) a t a lmost eve ry x E Rn.

Assume t h a t we know t h a t , f o r g e

converges. Then, if h = f - g,

0(Rn) and f o r each x E Rn , { T p g ( x ) l

and so t h e problem i s reduced t o prove t h a t A(h,X) i s o f sma l l measure

i f h i s o f sma l l L' - norm. Assume t h a t we can p rove t h a t , f o r each

f i x e d X > 0,

T h i s would s o l v e ou r prob em.

1.0. INTRODUCTION 5

However, the s e t A(f,A) has a r a the r unhandy s t ruc tu re and so one can think of subs t i t u t ing i t by some o ther eas i e r t o handle.

I t i s qu i t e c l ea r t h a t

l A ( f , A )

and t h a t t h e oper t o r T* defined by T*f(x) = s u p lTkf (x ) [ has a r a t k k

e r simple s t ruc tu re . We may hope tha t we wi l l be able t o prove now t h a t I A * ( f y A ) I + 0 as 1 1 f 1 1 1 -f 0 , and t h i s wi l l as well give us our desired almost everywhere convergence of {Tkf} .

So we a r e led t o consider t he operator T* defined by

which i s ca l led the maxim& a p e h a t a h associated t o {Tk } . I f {Tk} i s a n ordinary sequence, k = 1,2,..., T*f i s c l ea r ly measurable. I f k i s not countable one has t o prove t h a t T*f i s anyway measurable or e l s e t o deal with the outer measure o f tT*f > A I . The operator T* i s such t h a t f o r each f and x, T*f(x) a 0 and, i f the Tk a r e l i n e a r , we can wr i t e

The relevance of t he operator T* stems from the r o l e i t plays in the pointwise convergence proofs, as ind ica ted , and in the information i t furnishes about the l i m i t , when i t e x i s t s .

Assume, f o r example, in the l a s t mentioned s i t u a t i o n , t h a t we can prove t h a t f o r each f 6 L 1 ( R n )

with c independent of f 6 L’(Rn)). Then we obtain f o r each X 0 ,


and so ICT*f > X 11 -f 0 as

where convergence r e s u l t . Furthermore i f t h e l i m i t i s Tf,

II Tf III C IIT*f IIi X c I1 f I I 1 .

1) f ] I 1 -f 0. Thus we o b t a i n t h e a lmos t e v e r y

O f course, in o r d e r t o o b t a i n t h e a lmost everywhere convergence,

c o n d i t i o n (*) i s somewhat super f l uous and sometimes f a l s e . I t i s good

enough t o know t h a t

f o r each f e L1@) and f o r each X > 0 , w i t h c independent of f and X . O r even j u s t t o know t h a t f o r each 31 > 0

When (**I ho lds one says t h a t T* i s o f weak t ype (1,l) .

C o n d i t i o n (***) j u s t says t h a t T* i s cont inuous i n measure

a t o f rom L t o ?V .

Observe t h a t c o n d i t i o n (**) can be e q u i v a l e n t l y expressed by

s a y i n g t h a t

I n f a c t (**) t r i v i a l l y i m p l i e s (**) ' and, i f we have (**I' and

1 1 f 1 1 1 > 0, we can w r i t e

Our f i r s t t a s k w i l l be t o e s t a b l i s h some equiva lences between

a.e. - convergence and p r o p e r t i e s of T* and t o c l e a r up a - l i t t l e th.e r o l e

t h e f u n c t i o n @ ( A ) p l a y s i n t h e whole bus iness.

The genera l s e t t i n g i n which we w i l l p l ace ou rse l ves i s t h e

Genahae o e t t i n g . ( a ) We c o n s i d e r (Q,F,p) , a measure space

f o l l o w i n g :

t h a t w i l l be i n some cases o f f i n i t e measure and i n some o t h e r s + f i n i t e .

1.0. INTRODUCTION 7

(b ) We denote by “I t h e s e t of r e a l ( o r complex) va lued measuc

a b l e func t i ons d e f i n e d on n , t h a t a r e f i n i t e u-a.e.

( c ) Wi th

f rom Q t o R ( o t t o t ).

X we denote a Banach space of measurable f u n c t i o n s

( d ) The sequence { T k I w i l l be an o r d i n a r y sequence of opera-

t o r s f r o m X t o . I n many cases t h e r e w i l l be no problem i n assum-

i n g k t o be a cont inuous parameter.

j u s t t o s a t i s f y t h e f o l l o w i n g cond

we have

ITk(X1 fl ’ 1 2 f 2 ) 6 1 x 1 1

(e) Each T k w i l l be assumed t o be l i n e a r and i n some cases

t i o n : f o r fl, f 2 e X , X 1 , 1 2 E lR ,

( f ) With T* we des ign t h e maximal o p e r a t o r , i . e . f o r f E X

and x e fi ,

(9) We denote by T t h e 1 imi t o p e r a t o r , i .e.

T f = l i m T k f k+m

when i t e x i s t s i n some sense.

(h) F i n a l l y f o r X > 0, $(A) w i l l be

$ ( A ) = sup u { x E R : T * f ( x ) } f ex

1.1. FINITENESS A.E. AND CONTINUITY I N MEASURE OF THE MAXIMAL OPERATOR

The f i r s t i m p o r t a n t r e s u l t we s h a l l s tudy i s a genera l p r i n c i p l e

due t o Banach. Roughly s t a t e d : The com%ukty .in meanme ad each one 06 the ope ha to^ oa a @miey YJRu a ~LnCteneoh cmclump-tivn on the cahhanpond i n g maxim& opehatoh -impfie0 the continLLity .in meanwre at 0 0 6 the maL i m & opeha tuh .itnd4. T h i s s tatement , o f course, has a l l t h e f l a v o u r o f

a u n i f o r m boundedness p r i n c i p l e , and so i t i s .

s imp le a p p l i c a t i o n o f t h e genera l u n i f o r m boundedness theorem and t h i s i s

t h e way we f o l l o w here.

[1970 , pages 1-4 1 .

It can be o b t a i n e d by a

Fo r an a l t e r n a t i v e p r o o f one can see A.Gars ia

I n o r d e r t o p r e s e n t t h e theorem as a p a r t i c u l a r case o f t h e

u n i f o r m boundedness p r i n c i p l e , we endow t h e l i n e a r space '"I (a) p-measurable p-a.e. f i n i t e f u n c t i o n s (where f u n c t i o n s t h a t c o i n c i d e

p-a.e. a r e t h e same) w i t h a d i s t a n c e t h a t w i l l d e f i n e i n "I ( 0 )

t h e topology o f t h e convergence i n measure.

p(Q) < m and f o r f E ??I (R) l e t us s e t

o f a l l

More s p e c i f i c a l l y , l e t

I t i s an easy e x e r c i s e t o check t h a t d:/)n(R) + [ 0 , m ) i s a quasi-norm

i n t h e sense o f Yosida [1965 , p.30 I . A l s o i t i s easy t o show t h a t f o r

a sequence { fnI E and f e "m , we have fn -f f (p-measure) i f and

o n l y i f d ( f - f n ) -f 0.

For a s u b l i n e a r o p e r a t o r T f r o m a normed space X t o 'hl (R) one a l s o shows e a s i l y t h a t t h e f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t .

(a ) The o p e r a t o r T : X -' M(R) i s cont inuous a t 0 E X

(b ) The f u n c t i o n 4 : (0,m) -f [O,m) d e f i n e d by

1.1. FINITENESS A.E. AND CONTINUITY 9

tends to 0 as X tends to a.

Of course, if T is linear, then (a) is equivalent to the

continuity of T on X .

ng statements are

I > h } + O a s X t m

Likewise, let (TcOaeA be a family of sublinear operators from

X to W (R). Then one easily shows that the follow

a1 so equi Val ent

For the theorem that follows we shall use the following form of

the uniform boundedness principle, that can be seen in Yosida [1965, p.681:

L e I ( X , i l 1 1 ) be a Banach npace and (Y,d) a quai-named

k%mu~ A ~ U C ~ . L ~ A be a ~amieq o p e h a t a u &horn X to Y A&

dying doh each f, g e X -

d(Ta(Af)) = d(ATaf)

7 6 the A & {Taf : a e A} c Y bounded doh each f E X, ,then

lim d(Taf) = 0 u n i ~ a m L q in a e A. I l f l l - . 0 We recall here that the fact that a set ScY is bounded means that for

each neighborhood U of the origin there is an E > 0 such that E S c U.

With these preliminaries the proof of the following theorem is

straightforward.

1.1.1. TtlEOREM. lei {TkIF=l be a bequence o d hubfineah op- e l d u r n dhom X, a Banach hpace, t o "@l (a) w i t h u ( Q ) < m. h w n e

t h a t each Tk LA conLLnuvlLs and that t h e maxim& opeha2vh T* dedined

doh f E X and x E R an

A huch thud T * f 6 %I doh each f, i .e. T* f A ~.ivLite u-a.e. Then

T* A d o continuouA at 0 , and thenedahe

- Phaod. For n = 1,2,3,. . . we de f ine t h e t r u n c a t e d ( a t n) max-

ima l ope ra to r T i i n t h e f o l l o w i n g way. For f E X and x e n

C l e a r l y T i i s s u b l i n e a r , cont inuous f r o m X t o and s i n c e

0 & T;f(x) c T * f ( x ) f o r each x e a , we have d(T*,f) c d ( T * f ) f o r

each f.

Therefore t h e uni form boundedness p r i n c i p l e a p p l i e s and {T*,) i s equi-

cont inuous a t 0, i . e .

Observe t h a t , i f p(R) = m , we can s e l e c t c R w i t h

~(6) < - and, i f we d e f i n e

&i) = sup p 1 x E 6 : T * f ( x ) > X 1 I l f I1 6 1

then w i t h t h e same hypo thes i s of t h e theorem we a l s o o b t a i n

1.2. CONTINUITY AND A.E. CONVERGENCE 11

1.2. CONTINUITY I N MEASURE AT 0 E X OF THE MAXIMAL OPERATOR AND A.E. CONVERGENCE

We a l r e a d y know t h a t t h e f i n i t e n e s s a.e. o f t h e maximal oper-

a t o r i m p l i e s i t s c o n t i n u i t y i n measure a t 0 e X . We s h a l l now see t h a t

t h i s c o n t i n u i t y i m p l i e s t h e closedness i n X o f t h e s e t o f elements f

o f X i n which t h e sequence T k f converges a.e. I n most i n t e r e s t i n g

cases i t i s easy t o e s t a b l i s h such a convergence f o r some s e t dense i n X

and so we o b t a i n t h e a.e. convergence f o r a l l f u n c t i o n s i n X. For t h e

theorem t h a t f o l l o w s i t i s n o t necessary t o assume t h a t X i s complete.

1 .2 .1 . THEOREM. L e A I T k ) be a sequence 06 &hean. opehatom 6hom X, a named Apace, t o 'hl (n). h b w n e t ha t t h e ma&& opehdtoh T*

h cow%nuouh in memwte &am X t o at 0 e X . Then t h e be;t E ad

d e m e h f 06 x dolt which {Tkf} CUnvmgtA at a.e. x e R iA doseed i n X .

Phood. L e t f E X and cons ide r g E E. Then f o r any a > 0 - we have

P I X e R : l i m sup ITmf(x) - T n f ( x ) [ > CL 1 = m,n -+a

= PIX E R ; l i m sup \ T m ( f - g ) ( x ) - T n ( f - g ) ( x ) J > ~1 } G m,n +a

G 211 I x 8 R :

Since @ [h) J- 0

f e E.

as I ] f - g 1 1 J- 0 we see t h a t i f f E t hen

1.2.3. COROLLARY. 16 ,the space X 06 t h e p h e c e h g Theohem .h a nomed subspace oh %'t and id doh each g e E Me have Tkg(x ) -+g (x )

at a . e . x e R , then we &o have doh each f E , T k f ( x ) -+ f ( x ) at

1. P O I N T W I S E CONVERGENCE OF OPERATORS

LI C X E fi : lirn sup k + -

= FIX E R : I i r n s u p k +-

a < ~ C X 6 0.: T * ( f - g ) ( x ) > 7 ) + V { X e n : I ( f - g ) ( x ) l > a 1 2

b u t t h i s tends t o cero as g -f f ( h ) ,

CHAPTER 2

FINITENESS A.E. AND THE TYPE OF THE MAXIMAL OPERATOR

As we have seen i n Chapter 1, the mere f a c t t h a t , f o r each f E X ,

T * f ( x ) x E R , can g i v e us t h e a.e. conver-

gence r e s u l t i n many cases. However, once we cons ide r t h e o p e r a t o r T* i t

i s o f i n t e r e s t i n many c i r cuns tances t o have more i n f o r m a t i o n about i t ,

f i r s t o f a l l i n o r d e r t o g a i n some more knowledge about t h e l i m i t o p e r a t o r

T.

i s f i n i t e f o r a lmos t eve ry

Acco rd ing t o t h e o b s e r v a t i o n a f t e r Theorem 1.1.1. we know t h a t

i f T * f ( x ) i s f i n i t e a t a lmost each x e R f o r each f e X, then, even

when < ~0 , and we s e t , f o r X > 0, u(R) = m , l'f we f i x E c . n , u(c)

we have $(A,;) -+ 0 as + m . Now we s h a l l see t h a t , i f we assume a l i t t l e more about t h e

o p e r a t o r s

f e X

h a v i o u r o f t h e f u n c t i o n $(A) and so about t h e t ype o f t h e o p e r a t o r T*

(hence, about t h a t o f t h e l i m i t o p e r a t o r T ) .

CTk) , then t h e a lmost everywhere f i n i t e n e s s o f T*f , f o r each

,, p e r m i t s us t o deduce a more q u a n t i t a t i v e knowledge abou t t h e be-

We r e c a l l t h a t f o r a s u b l i n e a r o p e r a t o r S f r o m nn? ( Q ) t o

W(R) ( i . e . such t h a t f o r f l y f n e m(Q), h l , X p e R , S ( A l f l + Xz fz ) ( x ) 6 I h l I I S f l ( x ) l + IX21 ISf,(x) l) , we say t h a t i t i s o f

( s t r o n g ) t ype (p,q) , 1 6 q c 00,

we have S f e Lq(R) and

1 c q 6 m, when f o r each f e LP(Q)

II Sf II q c c I I f l l p

13

14 2. FINITENESS AND THE TYPE

w i t h c > 0 independent o f f. I t i s s a i d t o be o f weak t ype (p,q) a

1 6 p c m , 1 6 q < m , if t h e r e e x i s t s c > 0 such t h a t f o r each X > 0

and each 'f E Lp one has

u C x e R :

Type (p,qf , q < m i m p l i e s weak t y p e (p,q), s i n c e f o r X >O, A X = I l S f l > 11

But t h e converse i s n o t t r u e i n genera l .

A l l f ou r s e c t i o n s o f t h i s Chapter f o l l o w t h e same p a t t e r n . Some

leads us t o u s e f u l i n f o r - a d d i t i o n a l assumption about t h e o p e r a t o r s

m a t i o n about T*. S e c t i o n 2.1., a l i n e o f t hough t i n i t i a t e d i n a theorem

o f A.P. CalderBn, serves as m o t i v a t i o n f o r t h e f o l l o w i n g ones. S e c t i o n 2.2.

p resen ts a theorem of Sawyer, m o d i f i c a t i o n o f t h e one o f S t e i n p resen ted

i n Sec t i on 2.3. Very r e c e n t l y N i k i s h i n has o b t a i n e d a q u i t e genera l and

power fu l v e r s i o n o f t h e r e s u l t s o b t a i n e d p r e v i o u s l y i n t h i s d i r e c t i o n . I n

S e c t i o n 2.4 we p resen t s imp le p r o o f o f one o f t h e main theorems o f N ik i sh in .

TI:

2.1. A RESULT OF A.P. C A L D E R ~ N ON THE PARTIAL SUMS OF THE FOURIER SERIES OF f E L 2 ( T ) .

Some y e a r s b e f o r e the s o l u t i o n by Car leson [19661 of t h e c o n v e r

gence problem f o r t h e F o u r i e r s e r i e s o f a f u n c t i o n f o f L2 (T ) Zygmund

[ 1959 , Iap.165 ] p resen ted an i n t e r e s t i n g r e s u l t o f CalderBn abou t t h e

p o i n t w i s e convergence o f t h e p a r t i a l sums o f t h e F o u r i e r s e r i e s o f a f u n 2

t i o n f e L 2 ( T ) . The i d e a behind i t i s t h e k e r n e l o f t h e i m p o r t a n t g e n e r

a1 theorem o f E.M.Stein [ 19611 and o f t h e theorem o f Sawyer [ 19661 pre-

sented i n t h e f o l l o w i n g s e c t i o n s .

2.1. A RESULT OF A.P. CALDERON 1 5

2.1.1. THEOREM ?oh f pehio&c 06 pehiod 2a and i n N 1 ckeikX be t h e N-th pan t i ae nm 06

-N L2([0,2a]) Le2 SNf (x ) =

? v w ~ L e h benien. L e 2 S* be t h e comapunding maximal opehatoa, i . ~ .

S * f ( x ) = sup I S N f ( x ) l .Then SNf(x) conwagen a.e. a ~ s N -+ m i6 and

only .id S* LA 06 weak t y p e . (2,Z). N

- Pkoud. From t h e genera l theorem 1.1.1. o f Chapter 1, we e a s i l y

deduce t h a t convergence a.e. o f SNf f o r each f and a.e. f i n i t e n e s s

o f S * f ( x ) f o r each f a r e e q u i v a l e n t . The a d d i t i o n a l i n f o r m a t i o n here

r e f e r s t o t h e t y p e of S*. Of course, if S* i s of weak t y p e (2 ,2) t h e n

S * f ( x ) < m a.e. f o r each f e L2 . The d i f f i c u l t y c o n s i s t s i n p r o v i n g

t h a t if S*f(x) < m a.e. f o r each f E L 2 then S* i s . o f weak t y p e (2,2).

Assume t h a t S* i s n o t of weak t y p e (2,2) , i .e. f o r each C > 0

t h e r e e x i s t s fC, I [ fC1I2 = 1 and A c > 0 such t h a t

we can choose CN * m , pN t r i g o n o m e t r i c po l ynomia l s w i t h 1 1 p N ( ( 2 = 1,

AN > 0, such t h a t

We have > - cN 2Tr and so A N -f m ,

We a r e go ing t o c o n s t r u c t , by means of {Ak} a new sequence

{ A N r w i t h Nk nondecreasing such t h a t k k = l

To do t h i s we f i r s t choose C > Z1 . We have CN: / 1;: 6 2a. N:

16 2. F IN ITENESS AND THE TYPE

We choose kl copies o f t h e same number, k l be ing such t h a t

< 2n. The k l f i r s t terms o f t h e new sequence a r e going CN:

1 < k l 72- N:

t o be

N;

N; Now we choose N; > NP such t h a t CN* 1 z 2 and cons ide r ~2 < 2 .

we choose k2 copies o f t h e same number, k2 be ing such t h a t

< 2~ . The f o l l o w i n g k2 terms w i l l be N*2

C

1 c k 2 ~ A:

,k, N;

A , A , . * a , A N; N;

And so on. We e a s i l y check t h a t f o r t h i s sequence t X N 1 k

We s h a l l now a p p l y t h e f o l l o w i n g lemma t h a t we p rove l a t e r .

2 . 1 . 2 LEMMA. L e 2 {Ek) be a hequence o d meanwLabLe bubo&

u s T nuch t h a t C l E k l = a. Thcn thehe exA& { x k l C T ouch t h a t

demoht L V U L ~ x e Y L& in LndiniteRy many 0 6 the h& Exk + E k ) .

We choose xk accord ing t o t h e lemma. We a l s o choose n a t u r a l

numbers mk SO r a p i d l y i n c r e a s i n g t h a t t h e t r i g o n o m e t r i c po l ynomia l s

eimkX pN ( x ) do n o t have t r i g o n o m e t r i c monomia k

f o r m t h e s e r i e s

s o f t h e same o r d e r . We

1

l N k

-

2.1. A RESULT OF A.P. CALDERON 17

and so t h e above s e r i e s i n t h e F o u r i e r s e r i e s o f a f u n c t i o n f i n L 2 .

If x - X k e Ek , t h e n S*pNk ( X - x k ) > k N k by t h e d e f i n i t i o n o f E k y

and so some complete b l o c k a r b i t r a r i l y advanced i n t h e t a i l o f t h e F o u r i e r

s e r i e s S i s i n abso lu te va lue b i g g e r than 1. Since a lmos t each x i s

i n i n f i n i t e l y many o f t h e s e t s xk f Ek x t h e s e r i e s S i s n o t convergent . T h i s proves t h e theorem.

t h i s means t h a t a t a lmost ever,y

P m o d ad Lemma. 2.1.2. L e t XI x z . . . , XkY . . . L e t US

Such p o i n t s a r e

e s t i m a t e t h e measure o f t h e p o i n t s which a r e i n f i n i t e l y many o f t h e s e t s

xk f E k .

m m m

IJ (xk + Ek) ] ' = I J 0 ( x k f E n ) ' n = l k=n n = l k=n

We w ish t o prove t h a t f o r some p a r t i c u l a r s e l e c t i o n o f t h e p o i n t s

t h i s s e t i s o f n u l l measure. F o r t h i s i t i s s u f f i c i e n t t o p rove t h a t f o r

some sequence {xk3 we have, f o r each n

{ x k }

m

I 0 (xk f E k ) ' I = 0 k=n

L e t x l k be t h e c h a r a c t e r i s t i c f u n c t i o n o f E;C . Consider t h e f u n c t i o n D. , I

X ( t ) c h a r a c t e r i s t i c f u n c t i o n o f f' ( x k f E k ) ' . We have k = l

, t as v a r i a b l e s and w r i t e xP

Consider X I , X Z , ...

TI

* I",- x --TI i;' t=-TI x l ( t + x ) * + * x P i ( t + x Fi ) d t d x l . . .dx P1 xp 1- 1 =-TI

So there e x i s t s a p 1 s u f f i c i e n t l y b i g so t h a t

So we can choose X I , X Z , ... X p

P We now choose i n the same way x

, such t h a t

1 < ?

+ly ... , x such t h a t P 2

And so on. SO f o r each n , I I' ( x k + E k ) ' 1 = 0 k = n

2.2. THE THEOREM OF SAWYER 19

2.2. COMMUTATIVITY OF T* WITH M I X I N G TRANSFORMATIONS. POSIT I V E OPERATORS, THE THEOREM OF SAWYER.

CalderBn's theorem has been extended i n s e v e r a l d i r e c t i o n s , f i r s t

by S t e i n [1961] and then by Sawyer 1119661 . Sawyer's theorem i s con-

c e p t u a l l y s i m p l e r t han S t e i n ' s and so w i l l be presented f i r s t . Once t h e

p a t t e r n o f t h e p r o o f o f CalderBn's r e s u l t has been understood, t h e theorems

o f Sawyer and o f S t e i n a r e more t r a n s p a r e n t .

The s e t t i n g here w i l l be t h e f o l l o w i n g :

( a ) (R,F,p) w i l l be a measure space w i t h u ( R ) = 1.

(b ) {TkI i s a sequence o f l i n e a r ope ra to rs f r o m some Lp(s2),

1 i p c m, t o W(R) , t h a t a r e cont inuous i n measure.

( c ) Each T k i s assumed t o be p o s i t i v e , i . e . i f f > 0 , then Tkf 2 0.

( d ) We assume t h a t t h e r e i s a f a m i l y of mappings

f rom R t o R t h a t a r e measure p rese rv ing , i,e,, i f

A c. R , A € 3 , then S i l (A ) E > mappings i n t h e f o l l o w i n g sense:

( S ) o a s I

and u(S,l(A)) = u(A). ( e ) We a l s o assume t h a t ( S a ) cI I i s a mixing damily o f

I f A,B 6 > and p > 1, then t h e r e e x i s t s Sa such t h a t

u (A (1 S i l ( B ) ) 4 P ~ ( A ) u ( B ) .

(Observe t h a t i f Sa were such t h a t

u(A o Shl ( 5 ) ) = u(A) u ( B )

t h e r e A and Sil(B)

r e q u i r e so much. The f a m i l y Sa "mixes" t h e measurable s e t s o f R i n

t h e above sense).

would be p r o b a b i l i s t i c a l l y independent. We do n o t

( f ) We a l s o assume t h a t t T k l and (Sa) a E I commLLte i n

t h e f o l l o w i n g sense:

I f f E LP(R) and Saf(x) = f (Sax) , t h e n f o r each

Tk , Sa, Tk Sa = Sa Tk , i . e . f o r each f 6 LP(n) and x E R ,

With these n o t i o n s we can s t a t e and prove Sawyer's theorem.

( a ) T* A 06 wuLk t y p e (p,p)

(b ) FUR. each f E LP(R) T * f ( x ) < , a.e.

Fo r t h e p r o o f o f t h e theorem we s h a l l f o l l o w t h e p a t t e r n of t h e

p r o o f o f Ca lde rbn ' s theorem.

1 emma.

So we f i r s t prove t h e same t y p e o f a u x i l i a r y

Pmod 06 t h e lemma 2.2.2. Consider t h e s e t s S i l ( A k ) , where

t h e mappings S k a r e y e t t o be chosen. The s e t o f p o i n t s i n f i n i t e l y

many o f t h e s e t s S i l ( A k ) i s

m

Our goal i s t o prove t h a t f o r each f i x e d n p ( f'i

prov ided the Sk a r e c o n v e n i e n t l y chosen. S ince 1 p (Ak) = m ,

we can choose p1 such t h a t

S i l ( A ' k ) ) = 0 k=n oo

n= 1

2 . 2 . THE THEOREM OF SAWYER 21

By the m i x i n g property of ( S a ) a 6 I , we can choose S 1 , S p y . . . ,S , such t h a t P1

We then choose p 2 such tha t

a n d then S , ... , S , such t h a t pi p2

And so on. Clearly we have for each n

This proves the lemma.

Phaad ad ,the. Theaheni 2 . 2 . 1 . Assume T* i s not of weak type (p,p). Then, i f we f i x a sequence ck 4 00 , c k > 0 , t he re e x i s t s a sequence { f k I C L p , f k a 0 and a sequence X k > 0 such tha t

, A k = {T*gk > 1 I . We can wr i t e fk Let us ca l l g k =


L e t hk be a n a t u r a l number such t h a t 1 6 hk u(Ak) c 2. We t a k e

hk copies A i , A; , ... , A;k o f Ak.

Thus

and so, by t h e lemma, t h e r e a r e

j = 1,2,,..,

o f t h e s e t s

SJk B (Sa) a B I , k = 1,2,...,

such t h a t a lmost each x 6 R i s i n i n f i n i t e l y many hk ’ . (SJk)-’ (A;) .

D e f i n e t h e f u n c t i o n

. . F(x) sup “k ’Jk gJk (x)

k = 1 , 2 , . . . j=1,2,,. . h k

Where gJk f gk and ak > 0 i s go ing t o be chosen i n a moment.

Then

Thus, s i n c e t h e SJk a r e measure p r e s e r v i n g

4 m

6 2 1 - . k = l ‘k

If ck = Zpk and ak = Zk” , we g e t ak 4 m and \IF \ I D i m .

Because o f t h e p o s i t i v i t y o f each Tk , we have , f o r each

k , j , x i

2 . 3 . A THEOREM OF STEIN

. . T*F(x) a T* ak SJk gJk ( x ) = ak T* S: g: ( x )

By t h e commutat v i t y o f { T k I w i t h (Sa) a 6 I , we eas

23

-1 I f x 6 [ Sjk ] [ A d ] , then T*F(x) > ak . Since a lmos t

each x E R belongs t o i n f i n i t e l y many o f t h e s e t s [ Sjk1-l [ A d ] and

ak 1. m , we g e t

T*F(x) = , a.e.

The o t h e r i m p l i c a t i o n o f t h e theorem i s obv ious

2 . 3 . COMMUTATIVITY OF T" WITH M I X I N G TRANSFORMATLONS. THE THEOREM OF STEIN.

The f o l l o w i n g r e s u l t o f S t e i n [1960 i s on t h e one hand more

genera l t han t h e p reced ing one, s i n c e t h e o p e r a t o r s we c o n s i d e r a r e n o t

p o s i t i v e , and on t h e o t h e r hand l e s s genera l s i n c e i t s a p p l i c a b i l i t y i s

r e s t r i c t e d t o LP(n ) w i t h 1 < p c 2. The techn ique o f p r o o f , t h e

use o f t h e Rademacher f u n c t i o n s , i s q u i t e i n t e r e s t i n g and p e r m i t s us t o

d ispense w i t h t h e p o s i t i v i t y o f t h e opera to rs .

m

The Rademacher f u n c t i o n s { r k ( t ) } a r e d e f i n e d on [0,1) k= 1

i n t h e f o l l o w i n g way

24 2. F IN ITENESS AND THE TYPE

. . . . . . . . . . . . . . . . . . . . . .

The properties we are going to use here of these functions are the following:

1 i6 k = j

0 0 i d k # j

1

(1) J r , ( t ) r j ( t ) d t =

doll t i n a b e t o 1 2 ’ -

Property

n c I k = l

pk r k ( t ) be a n y . t i n e a h camb&aLLon 06

Bk e R , and 1 c h 5 n . Then

- (1) i s obvious. For ( 2 ) i t i s sufficient t o ob serve t h a t 5 Bk r k ( t ) has the same sign as @h rh ( t ) in a

k h 1 se t of measure n o t less that .

2.3.1. THEOREM . l e i (n,&,u) be a meanme npace wLth

~ ( n ) = 1. L e t CTkI be a hequence 06 f i n e m opmato~h 6 h a m OOme

2.3. A THEOREM OF STEIN 25

Pfiood. We j u s t need t o prove t h a t ( a ) i m p l i e s ( b ) , s i n c e t h e

o t h e r i m p l i c a t i o n i s obv ious. Our e x p o s i t i o n f o l l o w s t h a t o f A.Garsia

[1970, ch.1 ] .

Assuming t h a t T* i s n o t o f weak t y p e (p,p) we s h a l l p rove

t h a t $(A) 1. 0 as A > Q) , where

Thus, by t h e theorem 1.1.1, ( a ) cannot h o l d .

We f i r s t assume p = 2 . The case 1 & p & 2 i n v o l v e s a m ino r

t e c h n i c a l d i f f i c u l t y t h a t w i l l be d e a l t w i t h a t t h e end.

I f T* i s n o t of weak t ype (2,2,) , g i v e n H > 0 , t h e r e

e x i s t s f > 0 , j/f l l z s 1 and A > 0 s x h t h a t

L e t

t h e f u n c t i o n f by d i f f e r e n t mappings S so t o i n c r e a s e t h e measure

o f t h e s e t where T* o f t h e f u n c t i o n so ob ta ined i s b i g g e r than X

Since we would l i k e t o m a i n t a i n smal l t h e norm o f t h e f u n c t i o n , we d i v i d e

such sums by a cons tan t M t h a t w i l l be c o n v e n i e n t l y f i x e d l a t e r . I f

t h e o p e r a t o r s T k were p o s i t i v e , as i n Sawyer‘s theorem, t h e T* o f

t h e sums would be n o t l e s s than t h e sums o f t h e T* o f t h e added f u n 5

t i o n s . Since t h i s i s n o t t h e case, we i n t r o d u c e t h e Rademacher f u n c t i o n s

i n o r d e r t o s h u f f l e t h e f u n c t i o n s t o be added. I n t h i s way, by p r o p e r t y

( 2 ) o f t h e Rademacher f u n c t i o n s , we hope t o a r r i v e a t t h e same r e s u l t as

i n t h e p r o o f o f Sawyer’s theorem.

A = { x : T * f ( x ) z A }. Our aim i s t o add up many t rans fo rms o f

So we proceed i n t h e f o l l o w i n g way.

26 2, FINITENESS AND THE TYPE

With some mappings Sk , k = 1,2,3,...,n t h a t w i l l be chosen

i n a moment, we d e f i n e

f k

Observe t h a t i f x 6 Ak

x ) = f ( S k x ) and Ak = S i l ( A )

t hen Sk x 6 A and so

T * f k ( x ) = T * f (Skx) > A

We now d e f i n e , f o r x E R, t E [O,l),

where M i s t o be f i x e d l a t e r .

Because o f t h e o r t h o g o n a l i t y o f Erk(t)l we can w r i t e

and so

Hence, i f J- = 1 we g e t M2

3 and s o t h e r e i s B [D,1) , IBI z a such t h a t , i f t E B , I1 F ( * , t ) 112 1.

Observe now t h a t if x E Ah , 1 c h 6 n, then T* fh (x ) z X and so, by p r o p e r t y ( 2 ) of t h e Rademacher f u n c t i o n s , f o r x E A,, t h e r e

e x i s t s a s e t

T*F(x,t) > . L x c [O,l) , l L x l > 1 , such t h a t if t E Lx , then

x

2.3. A THEOREM OF STEIN 27

n

k= 1 Hence we can a l s o say t h a t , i f x e A* = IJ Ak , t h e r e e x i s t s

1 Ix = B f I Lx [0,1) , / I x ( a , such t h a t , i f t e Ix , we have

simul t aneous ly

n L e t us t r y t o e s t i m a t e p (A* ) . We have p(A*) = 1 - 1-1

But, i f we choose c o n v e n i e n t l y S k , we know t h a t

Hence

1 So, i f we take n such t h a t 2 a n p (A ) > 1, then p(A*) > . Thus, i t i s c l e a r t h a t t h e measure, i n t h e p roduc ts space R x l 0 , l ) o f t h e s e t

{ ( x , t ) : x e A* , t e Ix I

i s b i g g e r than . There fo re t h e r e e x i s t s t* e l 0 , l ) 1

such t h a t PIX e A* : t* e I x 1 > . I f we cons ide r F ( * , t * ) we have 1 1 F ( - , t * ) I I z c 1 and a t

t h e same t ime

x 1 1 - 1 t X 6 R : T*F(x,t*) > R ) > .

L e t us observe t h e r e l a t i o n s we have between t h e d i f f e r e n t

cons tan ts .


X J F T So we o b t a i n A > i . e

1 Thus i t i s c l e a r t h a t $(q) > 8 f o r each TI > 0 and t h i s i s what

we wanted t o prove.

F i n a l l y , when 1 & p 6 2 , t h e process o f t h e p r o o f i s t h e same

u n t i l we a r r i v e t o t h e d e f i n i t i o n o f

NOW, i f 1 &

, we can 2 - and - P 2-P

p L 2 , by H o l d e r ' s i n e q u a l i t y w i t h exponents

w r i t e

l n P 6 - c I f k W

Mp k = l

and t h e remainder o f t h e p r o o f con t inues as be fo re . (For t h e l a s t ine-

q u a l i t y i n t h e above chain, one proves t h a t i f al B a2 > ... 2 ak > > ... > an > 0 and c1 > B > 0 , then

and one takes CI = 2, f3 = p ) .

2.4. THE THEOREM OF NIKISHIN 29

2.4. THE THEOREM OF NIKISHIN.

In 1970 Nikishin published a very general extension of t he theorem of S te in . Like the theorems previoulsy presented in t h i s Chapter the N i k i s h i n theorem gives a weak type r e s u l t f o r t he maximal operator s t a r t i n g from i t s f i n i t eness a . e . The theory has been fu r the r developed by Maurey [1974] . For a c l e a r and thorough exposit ion of this recent theory we r e f e r t o a forthcoming monograph by Gi lber t . present a version of one of t he main theorems of Nikishin. i s inspired in t h a t of Gi lber t J.L.Rubio de Francia [1979] , in a very luc id paper.

Here we sha l l Our exposit ion

119791 , w i t h some modifications due t o

Let (X,u) , (Y,w) two o - f i n i t e (pos i t i ve ) measure spaces. We sha l l consider operators T : L p ( X , u ) + h ( Y , w ) from Lp(X,p)

t o the space (Y ,w) of a . e . f i n i t e measurable functions from Y

t o R endowed with the metric of the convergence i n measure.

We sha l l say tha t T : L p ( X , u ) +w(Y,w) i s &niMeahizable I b u p W n e n n in Nik ish in ' s terminology), when f o r each f o e Lp(X) there i s a fineat operator such t h a t

" f 0

I u f o f o I = I Tfol w-a.e. and

f o r each f a L p ( X ) , I u f I c lTfl w-a.e. 0

That T i s l i nea r i zab le means therefore t h a t t he re i s a family ( u ) of l i n e a r operators such t h a t T majorizes

each one of them a n d , f o r each f o , T coincides in absolute value with the corresponding u p rec ise ly a t f o .

f 0

foa L q X )

"Li neari zabl e" imp1 i e s "absol u te ly homogeneous" , i . e . /T(hf)l = 1x1 / T f / , since

M o t i v a t i o n and t y p i c a l example o f t h i s d e f i n i t i o n i s t h e t run -

ca ted maximal o p e r a t o r TG o f a sequence { T k I o f l i n e a r o p e r a t o r s

f rom Lp(X) t o w ( Y ) . For a f i x e d g E Lp(X) we choose

t h e measurable f u n c t i o n such t h a t 6, : Y + [1,N] ,

T i g(Y) = IT$N(y) g ( y ) l

o p e r a t o r u f ( y ) = T g

and d e f i n e , f o r f 6 Lp!X), t h e l i n e a r

f ( y ) f rom LP(x) t o ‘yh ( Y ) . We $N(Y)

6 T i f (y) . c l e a r l y have 1 u g g ( y ) l = ] T i g ( y ) [ and I llg f ( y )

There fo re Ti i s l i n e a r i z a b l e .

The N i k i s h i n theorem can now be s t a t e d i n the f o l l o w i n g terms.

2.4.1. THEOREM ( N i k i s h i n ) . L e t T : Lp(X,p) + w ( Y , v )

1 c p a, be e-inemizable and conLLnuau6 -in m m m e at 0. LeA

q = i n f (p,2) . Then t h e m e&& Cp E m ( Y , v ) , Cp 0 a.e.,nuch

t h a t h u t e m h f E Lp(X,p) and 60/r each X > 0

Remmh. The theorem means t h a t if we weigh t h e space Y by

means o f 0 , i .e . i f we change i t s measure element dv by ch dv

then t h e c o n t i n u i t y i n measure f rom Lp(X) t o m ( Y ) i m p l i e s t h e

weak t ype (p,q) of T.

P t a u d . Since Y i s a - f i n i t e , one e a s i l y sees t h a t i t

s u f f i c e s t o prove t h e theorem under t h e c o n d i t i o n

s i m p l i c i t y we assume

v(Y) < m . For

v(Y) = 1.

The p r o o f w i l l be performed i n t h r e e s teps .

( i ) The c o n t i n u i t y i n measure o f T a t 0, as we know,implies

t h e f o l l o w i n g r e l a t i o n : There e x i s t s c ( h ) G 0 as h .f such t h a t

2.4. THE THEOREM OF N I K I S H I N 31

( 1 f 11 6 1 i m p l i e s v { y E Y : ( T f ( y ) ( > A 1 6 c (X) . From t h i s

c o n t i n u i t y we s h a l l deduce t h e ( a p p a r e n t l y s t r o n g e r ) c o n d i t i o n : Z h a e

and A > 0 . Set A = I y E Y : sup ( T f k ( y ) ( > X 1 . l t k 6 M

I f y a A then t h e r e i s some fko such t h a t lT f k ( y ) l > A . L e t 0

u = u be t h e corresponding l i n e a r o p e r a t o r (T i s l i n e a r i z a b l e )

0 f k

such t h a t

For y E A we s e t I ( y ) 2 I t 6 [0,1) : I U (1 r k ( t ) f k ( Y ) ) I > X

M I ( y ) C (t E [O,1) : IT( 1 r k ( t ) f k ( Y ) ) I > 1 ' I * ( Y )

1

a r e t h e Rademacher f u n c t i o n s . We s h a l l t ry rk 1

where t h e f u n c t i o n s

t o p rove t h a t ( I * ( y ) l > F . We d e f i n e :

C l e a r l y

I I l ( Y )

I I ( Y ) I we have

[0,1) = I l ( y ) u I z ( Y ) u 1 3 ( y ) , I,i fl I. = 0 if i # j and

= I I2 (y ) l . On t h e o t h e r hand I ( y ) t I z ( y ) U I 3 ( y ) and so 1 1 > 7 . There fo re , s i n c e f o r each y E A (I*(y)l 2 ,

by F u b i n i ' s theorem, i f m i s t h e Lebesgue measure on [0,1),

J

32 2. FINITENESS A N D THE TYPE

1

v (B m ( E l ) G A -'I2 1 / / C r k ( t ) f I P d t 6 M p Y q p / 2

0 ' P

where the l a s t inequal i ty holds by v i r tue of the property t h a t t he Rademacher functions s a t i s f y , a s we sha l l see a t t he end of the proof.

A1 so

because of the continuity of T . I f we s e t ;(A) = 2Mp,q h-p /2 t 2 ~ 0 , ~ ' ~ )

we obtain the inequal i ty (*) t h a t we wished.

( i i ) We now sha l l prove tha t there e x i s t s EC Y , v(E) > 0,

such t h a t T E defined for f e L p ( X ) ( v- weighted) weak type

a s TEf = (Tf) XE i s o f

( p , q ) , i . e . a

This i s &OAZ the inequal i ty of the theorem.

1 Take R > 0 such t h a t E(R) < 7 , where ;((A) i s the function defined in step ( i ) . Assume t h a t (**) does not hold. For each F C Y , with v (F ) > 0 there e x i s t s then F c F and g e L p

-

2.4. THE THEOREM OF N I K I S H I N 33

- such t h a t v(T) > Rq 1 1 glI; and I T g ( y ) l > 1 on F. By Z o r n ’ s

lemma, t h e r e e x i s t s a d i s j o i n t sequence {Fjl J and C g j l c Lq such t h a t

, v(Fj) > 0 , C v(F.) = 1

Since C Rq l / g j 11; < 1 we o b t a i n , by s t e p (i)

( i i i ) By us ing a g a i n Zo rn ’ s lemma and ( i i ) we o b t a i n

E j C Y , U E . = Y , E j d i s j o i n t such t h a t J

m

We now d e f i n e @(y) = 1 c 1 *-j XE ( y ) and t h i s f u n c t i o n s a t i s - 1 J j

f i e s t h e s tatement o f t h e theorem.

For t h e i n e q u a l i t y about t h e Rademacher f u n c t i o n s t h a t we have

used i n s t e p ( i ) one can appeal t o t h e K h i n c h i n e ’ s i n e q u a l i t y

(See S t e i n [1970] , Appendix 0 ) . With t h i s we have

I f p > 2, then q = 2 and we have, by M inkowsk i ' s i n t e g r a l

i n e q u a l i t y :

I f p < 2 , then q = p and we have (by t h e i n e q u a l i t y

CHAPTER 3

GENERAL TECHNIQUES FOR THE STUDY OF THE MAXIMAL OPERATOR

I n t h e p reced ing chap te rs we have seen t h a t under t h e c o n d i t i o n

o f t h e f i n i t e n e s s a.e. o f T * f f o r each f E X we s o l v e t h e a.e.

convergence problem and t h a t i f something more i s known abou t t h e opera-

t o r s Tk , we a r e even a b l e t o determine t h e t ype o f t h e o p e r a t o r T*.

I n t h i s chap te r we s h a l l t ry t o p r e s e n t some genera l methods t o

s i m p l i f y t h e s tudy o f t h e maximal ope ra to r .

t o t h e s tudy o f i t s a c t i o n on f u n c t i o n s wi th a much s i m p l e r s t r u c t u r e .

I n S e c t i o n 2 we p r e s e n t some methods t o deal d i r e c t l y w i t h some b a s i c

ope ra to rs by means o f cove r ings and decomposi t ions.

d i t i o n i n Sec t i on 3 c o n s t i t u t e s a n o t t o o wellknown b u t v e r y n i c e t o o l

t o s tudy t h e t y p e o f an opera to r .

I n S e c t i o n 1 we reduce i t

The Kolmogorov con-

The common f e a t u r e i n t h e techniques o f i n t e r p o l a t i o n and e x t r a

p o l a t i o n i s t h e f o l l o w i n g .

behaves w e l l on some spaces o f a c e r t a i n f a m i l y o f f u n c t i o n spaces. Can

one say any th ing about i t s behaviour on t h e i n t e r m e d i a t e spaces o f t h a t

f a m i l y ( i n t e r p o l a t i o n ) o r on t h e extreme cases o f t h a t family ( e x t r a p o l -

a t i on) ?

Assume t h a t we know t h a t an o p e r a t o r T

I n t h e techniques o f m a j o r i z a t i o n , l i n e a r i z a t i o n and summation

one t r i e s t o reduce t h e s tudy o f a d i f f i c u l t and comp l i ca ted o p e r a t o r t o

t h a t o f some o t h e r s t h a t a r e s i m p l e r o r b e t t e r known.

3.1. REDUCTION TO A DENSE SUBSPACE.

I t i s o f t e n t h e case t h a t t h e s t u d y of t h e maximal o p e r a t o r

i s much e a s i e r t o c a r r y o u t on f u n c t i o n s w i t h a s i m p l e s t r u c t u r e T* adapted t o t h e o p e r a t o r i n ques t i on . The f o l l o w i n g theorem shows t h a t

35

36 3. GENERAL TECHNIQUES

i n many cases i t i s s u f f i c i e n t t o o b t a i n t h e t ype o f T* r e s t r i c t e d t o

such func t i ons i n o rde r t o have i t over an ampler domain o f f u n c t i o n s .

3.1.1. THEOREM. L e A (Q,F,p) be a meaute npace, (Q)

t h e neX a6 memutabkk 4eal ( o h cvmpLex) valued &nct ionb, X a n a m e d bpaCC v d 6unc~viont, i n %I (Q) and S a denbe nubdpace 06 X . LeR:

and

Then

mmuhe. L e A T* be thein. maximal opehha-tu4. F O J L

( a ) $ ( A ) = $,(A) d o 4 each A > 0.

( b ) Tn pcvLticuRan, 4 T* iA ob w a k t y p e (p,p) v u m S

6 ~ 4 dome p, 1 c p L m, Lt iA a6 w a k t y p e ( p , p ) (vum X).

( c ) Id T* iA v 6 t y p e (p,p) o v u S 6ua dome p, 1 c p 6 W,

Lt 0 06 t y p e (p,p) ( o v a X 1 .

P4ood. (a ) We have, o f course $ ( A ) 5 $ S ( A ) . We w i s h t o

prove $ ( A ) L ~ I ~ ( A ) . L e t a 2 0 and $(A) > c1 . We s h a l l show

t h a t $ s ( X ) > c1 . I n f a c t , if $ ( A ) > c1 , t h e r e e x i s t s then f e X , Ilf 1 1 6 1 , such t h a t

Consider T*N , def ined f o r h E X by T*Nh(x) = suplTkh(x) l

l & k h N

3.1. REDUCTION TO A DENSE SUBSPACE 37

L e t us assume f i r s t t h a t t h e r e i s one N such t h a t

(*) > p { x E R : T*Nf (x ) > A 1 > ci

Choose Cgk} c S , Ilg,Jl c 1, gk + f ( X ) . Since

1 'd{x E R : T*iif(x) > 11 = l i m p ( x 6 R : T*Nf (x ) > A t I j + m J

and

we have f o r a s u f f i c i e n t l y b i g j

Since each Tk

member tends t o ze ro as k - fa, arid so f o r a s u f f i c i e n t l y b i g k

i s cont inuous i n measure, t h e f i r s t t e rm i n t h e l a s t

If t h e assumption ( * ) does n o t hold i t i s because f o r each N

we have e i t h e r

(1) p { x E R : T X N f ( x ) > 1 } c ci

o r e l s e

( 2 ) { x E R : T X N f ( x ) > A } = t o

I f we have (1) f o r each N , then


and t h i s i s excluded. I f f o r some N we have (2), l e t us c o n s i d e r N o ,

t h e f i r s t N f o r which ( 2 ) ho lds . Take now a subse t 6 o f R such

t h a t

and proceed as be fo re .

(b)

( c ) L e t f a LP(o)

The s tatement (b) i s j u s t an a p p l i c a t i o n o f ( a ) .

and (gh I c s , gh -+ f(LP).

Then g, - gh -+ O(Lp) as s,h -f m and we have

Thus we o b t a i n

Since \ I T*(gh - g s ) ( [ + 0 as h,s -f 9 {T*gh-)

i s a Cauchy sequence i n Lp and so converges i n Lp t o a f u n c t i o n G. By

C a n t o r ’ s d iagonal process we can choose a subsequence Cyh> o f Egh)

such t h a t s imu l taneous ly

- and f o r each k, T k f ( x ) = l i m Tk gh(x) (a.e.)

So f o r each k we have a t a lmost eve ry x 8 R

Hence T * f ( x ) G G(x) a.e. and so

where C i s ’ t h e t ype cons tan t o f T* ove r S.

3.2. COVERING AND DECOMPOSITION 39

3.2. COVERING AND DECOMPOSITION.

Cover ing and decomposi t ion techniques a r e among t h e most b a s i c

ones i n t h e s tudy o f t h e t ype o f t h e problems we a r e d e a l i n g w i t h . Cover-

i n g techniques a r e p a r t i c u l a r l y u s e f u l f o r t h e t rea tmen t o f t h e Hardy - L i t t l e w o o d maximal ope ra to r , one o f t h e most fundamental i n modern Analys is

and o f i t s g e n e r a l i z a t i o n s . We f i r s t p r e s e n t here i n paragraph A t h e

ve ry u s e f u l and i m p o r t a n t c o v e r i n g lemma o f B e s i c o v i t c h and r e f e r t o

Guzmdn [1975]

i n g lemmas.

f o r g e n e r a l i z a t i o n s o f i t and f o r some o t h e r types o f c o v e r

I n B we p resen t seve ra l examples o f t h e use o f t h e p r o p e r t i e s

o f t h e dyad ic c u b i c i n t e r v a l s f o r t h e p r o o f o f v e r y i m p o r t a n t r e s u l t s such

as Whi tney 's c o v e r i n g lemma, and t h e CalderBn - Zygmund decomposi t ion l e m

ma.

I n C we examine a c o v e r i n g theorem f o r convex s e t s o f wh ich

we s h a l l make use l a t e r on.

A. Bedicvvixch cvvcxing Lemma and t h e weak t y p e (1,1) a6 t h e

H ~ d y - L ~ e w v v d maximal v p a a t v h .

3.2.1. THEOREM. L d A C Rn be a bounded 6&. Fvh each

x e A we a t e given a dobed ba l l B ( x , r ( x ) ) wLth cedeti at x and

nadiud r ( x ) > 0. Then, 6hvm t h e coUecLivn ( B ( x , r ( x ) ) x A vne

cun chvvbe a sequence 0 6 b a L h { B R I buch t h a t

( i ) A c UBk

(ii) . I B k I can be d i6MbLLted into cn dequenced {BiI , Cn C B i I , ... CBk I each vne v a d i n j v i n t b m . Hehe cn h a c o w d a d

depending only vn n.

( i i i ) One h a 1 xB ( x ) c cn at each x e R n , i . e . t h e k

v v d a p v d t h e bad22 0 6 ( B k I h un i6vmly bvunded b y cn.

Pkvo6. We choose { B k I i n t h e f o l l o w i n g way. I f

a. = sup { r ( x ) : x e A 1 = - , then a s i n g l e b a l l w i t h s u f f i c i e n t l y

l a r g e r a d i u s i s enough. L e t us then assume a. > 00. We then t a k e

3 xle A such t h a t r ( x l ) > B a n and B1 = B(xL,r(xl)). L e t us now

cons ide r al = sup { r ( x ) : x e A - B i I . We t a k e then x2e A - B,

such t h a t r ( x 2 ) > T al and B2= B ( x 2 , r ( x 2 ) ) , and so on. I n t h i s

way we o b t a i n a sequence {Bk) , t h a t can be f i n i t e o r i n f i n i t e . I f f i

n i t e , it i s so because A c UBk and so {Bk} s a t i s f i e s (i). I f i n f j

n i t e , we have r ( x k ) + 0 as k -+ m . I n f a c t , o the rw ise we would

have an i n f i n i t e number o f k's w i t h r ( x k ) > 01 > 0. I f we observe

t h a t t h e b a l l s

them a r e i n a bounded s e t I z e R n : d(z,A) c a. 3 , we e a s i l y see t h a t

t h i s i s impossib le . The re fo re r ( x k ) -+ 0 as k + m . I f

s e A - UBk # 0 , then r ( s ) > 0 and B ( s , r ( s ) ) has been unduly

over looked i n o u r s e l e c t i o n process. Hence A - LIBk = 0 and {Bk) s a t

i s f i e s ( i ) .

3

1 1 B(xk, 5 r ( x k ) ) = 7 Bk a r e d i s j o i n t and t h a t a l l o f

I n o r d e r t o prove ( i i ) l e t us f i x a Bh and ask ou rse l ves how

many Bk ls w i t h k < h i n t e r s e c t Bh. There a r e some such B k l s

w i t h cen te r xk such t h a t d(xh,xk) c 3 r ( x h ) , l e t us c a l l them o f t y p e

1, and t h e o the rs , o f t ype 2, such t h a t d(xh,xk) > 3 r ( x h ) . A l l o f them

a r e such t h a t r ( x k ) > r ( x h ) s i n c e k < h. Fo r a Bk o f t y p e 1 we

cons ide r Bk c o n c e n t r i c w i t h Bk and w i t h r a d i u s rfx,). For a Bk o f t ype 2 we j o i n i t s c e n t e r xk t o xh and on t h i s segment we take

t h e p o i n t x i a t d i s t a n c e 3 r ( x h ) f rom xh. We then cons ide r t h e b a l l 1

t h e b a l l s 5 Bk a r e d i s j o i n t and a l l con ta ined i n t h e b a l l B(xh,4r

There fo re they a r e i n number l e s s than

3

1 Y

ik o f c e n t e r x i and r a d i u s r ( x h ) . It i s now easy t o observe t h a t 1 -

'n. 4'/(+)" = 42n =

P r o p e r t y ( i i i ) i s , o f course, an i nmed ia te consequence o f (

There a r e many i n t e r e s t i n g v a r i a n t s o f t h i s lemma o f Besicov

For some o f them t h e reader i s referred t o Guzman [1975] . Wi th t h e ideas

o f t h e p r o o f o f t h e p rev ious theorem he shou ld t r y h i s hand a t t h e f o l l o w

i n g s i m i l a r statement.

3.2.2. THEOREM. Le , t A c R n be a bounded Foh each

x e A we m e given a cloned i v L t a v d I ( x ) , ( I ( x ) ) " f 0 , c e n t a e d at x i n nuch a &~m t h d id x e A , y e A t h e i n t a u & I ( x ) , I ( y )

me cornpahabee i n n i z e , i . e . id XhamLated t o be c e n t a e d CLZ 0 one .& confairzed in t h e otheh. Then, 6 m m t h e cokXeeection (I(X))~€A one can

choone a nequence { I k ) nuch t h a t

( i ) A c UIk

( i i ) The nequence { Ik} can be cL&txLbuZed i n t o pn A &

quenca C I ~ 1 , 11; I , ... { I E ~ I , each o d them ad d b j o i n t i n t a u & . Hae pn depend o n l y on t h e dimenilion.

( i i i ) OW han 1 X I ( x ) ,< pn at M C ~ x E R ~ . k

Observe tha t the f a c t t ha t A i s bounded has been only used i n the proof of t o cope w i t h the case a. = sup { r (x) : x E A 3 = m

the theorem. One can permit A t o be unbounded assuming a. < . Also the f a c t t h a t the b a l l s i n the f i r s t theorem or the in t e rva l s i n the second a r e closed i s ra ther i r r e l evan t . One can assume them open o r w i t h p a r t of the boundary.

In order t o mark the way fo r the appl ica t ion of t h i s important lemma of Besicovitch we sha l l now show how the weak type (1,l) f o r t he Hardy-Littlewood maximal operator i s an easy consequence of i t .

One of t he var ian ts of the n-dimensional Hardy-Littlewood op e ra to r can be defined in the following way. For f E L’(Rn) and x a R n we s e t

Mf(x) = SUP

where Q runs over a l l open cubic in t e rva l s containing the point x. I t i s easy t o see t h a t Mf

i n g property : If f l , f 2 E L1(Rn) , A I , A 2 E R , i s measurable and t h a t i t s a t i s f i e s the follow-

then

We want t o show tha t there e x i s t s c > 0 such t h a t f o r each X > 0 and each f E LNn)


I f x E A , then t h e r e e x i s t s a c u b i c i n t e r v a l Q c o n t a i n i n g

x such t h a t

I f we cons ide r t h e minimal open c u b i c i n t e r v a l Q* cen te red a t

x and c o n t a i n i n g Q, we have

Where cx depends o n l y on t h e dimension n.

I t i s a l s o easy t o see t h a t , s i n c e f a L' and X > 0, t h e

x 6 AX , i s f i n i t e . supremum o f t h e d iameters o f t h e cubes Q* , when

We app ly B e s i c o v i t c h lemma and o b t a i n CQ*,I such t h a t

Then we can w r i t e

T h i s proves t h a t M i s o f weak t ype (1 , l ) . Since M i s t r i v i a l l y o f

t y p e ( m , ~ ) , Marc ink iew icz i n t e r p o l a t i o n theorem t e l l s us t h a t M i s

of t ype (P,P).

(Observe t h a t i n p a r t i c u l a r we g e t

Th is f a c t w i l l be used l a t e r ) .

The f a c t t h a t M i s o f weak t y p e (1,l) t o g e t h e r w i t h t h e

t r i v i a l obse rva t i on t h a t , i f g e gNn) we have, for each x E Rn

3.2. COVERING A N D DECOMPOSITION 43

and each sequence I Q k ( x ) I of cubic in t e rva l s containing x , such t h a t s(Q,(x)) -t 0

gives us the c l a s s i ca l theorem o f Lebesgue on d i f f e r e n t i a t i o n of i n t eg ra l s .

3.2.3. THEOREM. LeX f E L1(Rn) . T h e m &xh& a heX oh m m -

uht zmv Z c R n huch t h a t , each 2 $ Z and u c h hCqUencC { Q k ( x ) )

a6 cubic ivLtem& covLtaivcing x w L t h 6 ( Q k ( x ) ) -f 0 , one h a

Prroal;. We wish t o prove tha t fo r each A > 0

B u t , i f f = g + h w i t h g E e o (R') , we have

With C independent of f , g , h , A . Thus, given E > 0 , we choose h such t h a t

one proves

CJlhl[ < E . This proves t h a t / A A \ = 0 . In the same way A


B. The dyadic cubeil and b#me uppficaA;iun6, Wkitney'o Lemma.

C d d a b n - Z ygmund decompob&on.

The use o f t h e dyad ic cub ic i n t e r v a l s i s a powerfu l t o o l f o r

many d i f f e r e n t purposes i n r e a l a n a l y s i s , as we s h a l l now see.

For t h e i n t r o d u c t i o n o f t h e dyadic C u b a i n R n , we f i r s t

1

w i t h i n t e g r a l coo rd ina tes . We

i s t h e un ion o f Zn d i s j o i n t cubes

cons ide r i n Rn t h e f a m i l y D O o f a l l h a l f - open c u b i c i n t e r v a l s

(open t o t h e r i g h t and c losed t o t h e l e f t ) o f s i d e l e n g t h equal t o

hav ing v e r t i c e s a t a l l p o i n t s of Rn

now s u b j e c t D o t o a homothecy o f c e n t e r 0 and r a t i o 2, f o r k E Z

and so o b t a i n Dk . Each cube o f D

The cubes o f D have s i d e l e n g t h 2J . I t i s c l e a r t h a t o f Dj-l . i f Qls D j and Q 2 € D k w i t h j s k , then e i t h e r Q1 i7 Q2 = 0 o r e l s e Q 1 c Q 2 .

j

j

We s h a l l use a l s o t h e f o l l o w i n g s imp le p r o p e r t y o f t h e dyad ic

cubes.

3.2.4. THEUREM . LeL (9,) clcA be u g i v e n coUecLLon 06

dyadic c u b i c i n t a w a h . C 1 $ C2s .. . a d C u b a od (9,) ,cA 6ivLite. Then t h e muximd cubeil {Q,} 06 each u c e n d i n g c h a i n 0 6 (Q,) ahe d i n j o i n t and b a U 6 y UQ, = UQa .

The proof i s a t r i v i a l consequence o f t h e f a c t t h a t

Q j , Q,

Annunie t h a t ecrch abcending c h a i n

,€A

Pkood.

f o r each conple

d i s j o i n t o r e l s e one i s s t r i c t l y con ta ined i n t h e o t h e r .

o f d i f f e r e n t dyad ic cubes, e i t h e r t h e y a r e

AppLicaA;ian I . Wkitney'o covehing Lemma.

As a f i r s t a p p l i c a t i o n we prove t h e f o l l o w i n g usefu l c o v e r i n g

lemma due t o Whitney [1934].

3.2.5. .THEOREM . L e L G c Rn be a n open bt?X , G # R n , G # 0. Then t h a e exha2 a d i n j o i n t nequence {QkI 0 6 cubeil t h c d ahe obRdined by A t a n 6 W o n ad dyadic c u b i c in . tehv&, nuch t h a L

(i) G = U Q,


d ( Q k y aG) whehe

2 6 T 6 y (ii) F o t ~ each k,

d denotec, t h e Euclidean dintance , 8G ,iA t h e boundmy 06 G and &(a,) ,the diameXeh o d 0,.

P t ~ o o d . We can assume, by pe r fo rm ing a t r a n s l a t i o n , i f necessary,

t h a t 0 E aG . For each x E G we t a k e t h e g r e a t e s t dyad ic cube

Q(x ) such t h a t x E Q ( x ) and

F o r Q ( x ) we c l e a r l y have

and i f Q* (x ) i s t h e " f a t h e r " o f Q ( x ) i n t h e dyad ic g e n e r a t i o n we have

d(x , aG) 6 3 6 ( Q * ( x ) = 6 6 ( Q ( x ) )

There fo re we can w r i t e

The (Q(X)),,~ s a t i s f y t h e f i n i t e ascending c h a i n c o n d i t i o n o f

s i n c e t h e cubes o f any i n f i n i t e ascending c h a i n f i n i s h t h e Theorem 2.4,

by be ing a t ze ro d i s t a n c e f rom 0 and t h i s c o n t r a d i c t s d ( Q ( x ) , aG) > k 2 6 ( Q ( x ) ) .

theorem.

We now app ly Theorem 3.2.4. and o b t a i n t h e s tatement o f t h e

I f we a p p l y i n t h e same v e i n t h e c o v e r i n g lemma 2.4. t o p rove

t h e weak t y p e o f

cubes we e a s i l y o b t a i n a r e v e r s e i n e q u a l i t y .

t h e Hardy -L i t t l ewood maximal o p e r a t o r r e l a t e d t o dyad ic

3.2.6. THEOREM . L e X f e Limn) and

Mf(x) = sup & I a ( x ) I f [ , whehe t h e sup A ,tatahen oweh u l l

dyadic C u b a Q ( x ) c o n t a i n i n g x . Then, 6 a h each X > O , we have,i6

AX = { x : Mf(x) > A}

Phou6. For x E A h there i s a l a r g e s t dyadic in te rva l Q(x) If1 > A . Clearly hi Q ( x )

containing x such tha t

1 l Q ( x ) l < x 1 1 f 1 1 1 and so i t i s obvious t h a t f i n i t e ascending chain condition. We apply3.2.4. obtaining Q, d i s jo in t such tha t Ah = U Q, . Observe t h a t

( Q ( X ) ) ~ ~ ~ s a t i s f y the

where Qt i s the f a t h e r of Q,. W i t h these inequa l i t i e s the s t a t e m n t i s obvious.

A i y f i c a t i o n 2. C d d m 6 n - Zyqmund decompob&on lemma.

The following r e s u l t of CalderBn and Zygmund 119521, used by them in their c l a s s i ca l paper on s ingular i n t e g r a l s , has become a very important t oo l , useful i n many d i f f e r e n t contexts. I t can be given many d i f f e r e n t forms. Here we present the or ig ina l one, which r e f e r s t o the dyadic cubic in t e rva l s . For other l e s s geometrical var ian ts one can see Guzmdn [ 1975 , p. 16-17 3 .

3.2.7. THEOREM . L e X f E L1(Rn) , f 2 0 and A > 0 . Then

Rhme e h a 2 a bequence 0 6 d i b j o i n t dyadic c u b i c intmvab {Q,} duch t h d

( i i ) f ( x ) 6 A a t a . e . x 4 UQ,

f ( x ) = g ( x ) t h ( x ) (Calder6n - Zygmund decomposi t ion)

we have

( a ) g ( x ) < PA f o r x E UQ,

( b ) g ( x ) 6 x f o r a.e. x 4 UQ,

P m o d . L e t A X be t h e s e t o f p o i n t s x a R n such t h a t

f > A where I . Q,(x)) is t h e sequence f ( x ) = l i m

o f decreas ing dyad ic cubes c o n t a i n i n g x. F o r each x E l e t Q ( x )

be t h e l a r g e s t dyad ic cube c o n t a i n i n g x such t h a t

1 a(Q,(x)) + O I Q k ( x ) l 'Qk(X)

The cubes

s i n c e I Q ( x ) l c 1 1 f / I 1 . We app ly Theorem 3.2.4. and o b t a i n Q, I s a t i s f y i n g ( i ) and ( i i ) .

(Q(x) lxeAX s a t i s f y t h e f i n i t e ascending c h a i n c o n d i t i o n ,

1

Remmk . Observe t h a t t h e same process o f t h e p r o o f i s v a l i d

t o o b t a i n t h e f o l l o w i n g v a r i a n t o f t h e theorem.

3.2.8. THEOREM . L e t Q be a cubic i r z t a v d ud Rn , f 6 L'(Q),

& /Q f 6 A * f L 0 , X > 0. AbbWe t h a t

Then . them exint a nequence 0 6 dyadic Aubcubu 0 6 Q , C Q, t h d dhe dinjoint and A a - t i A d y

( i i ) f ( x ) 6 A a t a.e. x 6 Q - UQ,

48 3 . GENERAL TECHNIQUES

C . A c a v d n g theahem 6vh n u t & canvex b e h .

Later o n , when dealing with s ingular in tegra l operators i n ChaL t e r 11, we sha l l make use of the following in t e re s t ing covering r e s u l t .

3 .2 .9 . THEOREM , LeL (K,) be a ~a.mi.ly a4 campact canvex

bLd.5 0 6 Rn w s h nan-empty i n t d a t r and w i t h c e n t e ~ at t h e v h i g i n . Abbume

t h a t they m e n u t e d , i. e. 6vh any &a a6 them K,, , K,, , & h a

Kol, Ka, K,, Ka,.

L e t B c R n be any campact b& and ahume t h a t 6 a t each

i . e . ,the f i a m l a t i o n t o x a6 t h e b d

x e B we m e g iven an index a ( x ) e A . Camidetr, doh each x E B , xhe O& Ka(x). '

Cx = x + K

Then, @am ,the g iven caUecaXon ( C x ) x a B One can chavbe a

bQqUQMCe {C, 1 thcLt

t h a t may be 6ini te oh i n d i n i t e , 0 6 d i n j v i n t b&tA h u h k

B C U 5 C k 'k

whme 5 C x Lh t h e b e t abahined 6hom C x by a hamvthecy 0 6 ha.tLo 5 and

cedm x, t h e centeh 0 6 b y m m e h y 0 6 C,.

Phvro06. We sha l l give a sketch of t he proof. I t will be easy f o r the reader t o f i l l i n the d e t a i l s .

Assume f i r s t t h a t the index s e t A i s M and t h a t j h e NI,

J 'k j > h , implies K. C K h . We proceed t o choose our s e t s C . Take

x, such t h a t ,(XI) is as small as possible, i . e . C x l as big as pos- s ib l e . Take now X Z E B - 5 C, such t h a t ~ ( x z ) i s a s small as pos-

s i b l e , then x 3 e 6 - 1.J 5 C such t h a t a ( x 3 ) i s as small as POL

sible,and so on. t ha t B c U 5 C and t ha t C fl C, = !?j i f i # j .

'i 'i j The case of a general index s e t A can be eas i ly reduced t o the

2 1

i :1 'i I t i s e a s i l y proved, as in the previous covering resu l t s ,

previous one.

3.2. COVERING A N D DECOMPOSITION 49

The following consequence of the preceding theorem i s i n t e r e s t ing and useful f o r the d i f f e ren t i a t ion of i n t eg ra l s and f o r the study o f

the approximations of the i d e n t i t y .

3.2.10. THEOREM. LQt (Ka) aaA be a l;amLLy 06 compact conuex he& a i n t h e pkecedincj theokem. F o l ~ each x 8 R n and a E A L e t UA

denote

K a ( x ) = x + Ka

and c o a i d a , d o t f E L j O c (Rn) , x 8 R n t h e maximd opetrcLtoh M

I f 1 K,(x)

Mf(x) = sup clEA

Then M i~ 0 4 weak Xype (1,l) w s h a t y p e cov~5Xant 5n .I& dependent oQ t h e 6~~nLLiey ( K a ) acA .

Ptrooh. Let A > 0 , f E L1(Rn) and l e t B be any compact subset of

C x e R n : Mf(x) > A }

For each x E B there i s a n a(X) such t h a t , i f we have

+ K C A X )

We apply the preceding theorem and obtain a d i s j o i n t sequence I C(x , ) 1 such tha t B c U 5 C(x,) . So we have

50

There fo re

3. GENERAL TECHNIQUES

3.3. 'KOLMOGOROV CONDITION AND THE WEAK TYPE OF AN OPERATOR.

Weak t ype i nequal i ti es p r e s e n t c e r t a i n i m p o r t a n t disadvantages

w i t h respec t t o those o f s t r o n g type. The l a t t e r p e r m i t summation, i n t e

g r a t i o n and comparison processes t h a t cannot be c a r r i e d o u t w i t h t h e f o r

mer ones. T h i s o b s e r v a t i o n w i l l be b e t t e r understood w i t h some examples.

Assume t h a t i s a f a m i l y o f s u b l i n e a r o p e r a t o r s f rom

some LP(n) t o LP(n) , 1 6 p c m , which a r e o f s t r o n g t y p e (p,p) , w i t h constants ca . We cons ide r t h e o p e r a t o r T d e f i n e d on each

f B LP(n)

i s o f s t r o n g t ype (p,p).

i n e q u a l i t y t o o b t a i n

(Ta) , 0 < cx < 1,

by T f ( x ) = i [ T a f ( x ) [ da and we want t o s tudy wheter T 0

I t may be p o s s i b l e t o a p p l y M inkowsk i ' s i n t e g r a l

so, i f 1' ca da < m , then T i s o f s t r o n g t y p e (p,p). I f we o n l y

know t h a t each T

cable.

0

i s o f weak t y p e t h i s procedure i s n o t d i r e c t l y a p p l i -

Another i n t e r e s t i n g example, t h a t we s h a l l l a t e r use, i s t h e

f o l l o w i n g . Assume t h a t a c e r t a i n s u b l i n e a r o p e r a t o r L f rom some

Lpmn) , 1 < p < m t o (Rn) i s r e l a t e d t o ano the r one S i n

t h e f o l l o w i n g way. For each x E Rn and each f E Lp(Rn) t h e r e e x i s t s

an annulus:

3.3. KOLMOGOROV

Q ( X ) = { z E R ~ : r

CONDIT I ON 51

such t h a t f o r each y f Q ( x ) we have I L f ( x ) l c ISf(y)l . Assume

t h a t S i s known t o be of s t r o n g t y p e (p,p). Can we say a n y t h i n g

about t h e t ype of L ? Yes, and ve ry e a s i l y . We i n t e g r a t e t h e above

i n e q u a l i t y ove r y 8 Q ( x ) and d i v i d e by I Q ( x ) l . Thus we g e t

where c depends o n l y on t h e dimension n and M i s t h e Hardy - L i t t l e w o o d maximal o p e r a t o r over b a l l s which i s o f t ype (p,p) . Hence

and so L i s of s t r o n g t y p e (p,p).

What can be s a i d if we j u s t know t h a t S i s o f weak t y p e (p,p)?

We s h a l l now see i n cases l i k e t h e p rev ious ones we can proceed i n t h e

same way s u b s t i t u t i n g t h e weak t y p e i n e q u a l i t y (p,p) by some o t h e r e q u i v

a l e n t s t r o n g i n e q u a l i t y which we s h a l l c a l l , f o l l o w i n g C o t l a r [1959] , t h e K a ~ o g o h o u con&an.

3.3.1. THEOREM . L e X T be a n u b f i n e a t o p e h d o h dhom (n) ,to (n). annume thcLt T A 0 6 weak .type (p,s) , 1 c p , s G

w a h con?s,tatarzt c . Then, . id 0 < 0 < s and A A any m e a o ~ ~ a b l e nubneX

0 6 f 6 h'l (n) , ,the ~ a U o w L n g

(Ko4hugotrov'n J LnequaLLty,

w s h @IJ& meanme, we have, d o h each

0 < a < s , and d a h each f E LP(R) and each A c R wLth u(A) < m , Rhen T ad weak .type (p ,s) .

Phtood. L e t T be .of weak t y p e (p , s ) w i t h cons tan t c, i . e . f o r

each f 6 “ I ( R ) and each A > 0,

L e t 0 < a < s A c R p(A) < m and f o r g , measurable

f u n c t i o n d e f i n e d on R , c a l l

t h e d i s t r i b u t i o n f u n c t i o n o f g . Then

lTf l ‘ = u Ad-l W l T f l X (A) dA = u[ ION -t ] < A N 0

I l f IIS cs 3 dh - - im A S

c u p(A) dA + u N

I 1 f I 1 u(A)

I f we choose N = c t h a t makes minimal t h e l a s t member, we

o b t a i n Kolmogorov’s i n e q u a l i t y (*) ,

Assume now t h a t T s a t i s f i e s (*) f o r u < s . L e t f e %(a), h > 0, K c Q any measurable s e t o f f i n i t e measure con ta ined i n

{ I T f ) > A I . Then, i f we app ly t h e i n e q u a l i t y (*) , we g e t

3.3. KOLMOGOROV CONDITION 53

Hence T i s o f weak t ype (p,s) . The r e l a t i o n between t h e cons tan ts o f

t h e weak t ype and t h e Kolmogorov i n e q u a l i t i e s a r e c l e a r f rom t h e compu-

t a t i o n s i n t h i s proof .

With this theorem one can e a s i l y hand le t h e p r e v i o u s examples,

even i f we s u b s t i t u t e t h e s t r o n g t ype f o r t h e weak t ype i n e q u a l i t y .

I n t h e f i r s t one, i f each T, i s o f weak t y p e (p,p) , 1 < p < , w i t h cons tan t c, , we can w r i t e , f o r each A w i t h v (A) < m y

i f l < ~ < p , ~ E L P

i .e.

Hence, s i n c e T f ( x ) = I' IT,f(x) I dx , 0

The second example, t o g e t h e r w i t h some o t h e r s , w i l l be t r e a t e d

i n S e c t i o n 6 when d e a l i n g w i t h t h e technique o f m a j o r i z a t i o n .

3.4. INTERPOLATION.

The i n t e r p o l a t i o n methods have been e x t e n s i v e l y s t u d i e d and t h e r e

a r e severa l r e c e n t monographs about them. We s h a l l s t a t e here, w i t h o u t

p r o o f , two o f t h e most b a s i c r e s u l t s , t h e R iesz -Thor in theorem and t h e

Marc ink iew icz i n t e r p o l a t i o n theorem. F o r a more complete t rea tmen t t h e

reader i s r e f e r r e d t o Sadosky [1979] , t o Bennet and Sharp ley 1119791

and t o a fo r thcoming monograph by t h e two l a s t mentioned au tho rs . A l so

t h e book by S t e i n and Weiss [1971] c o n t a i n s a ve ry good chap te r on i n t e r -

po l a t i on.

We g i v e a l s o i n t h i s paragraph t h e p r o o f o f an i n t e r e s t i n g the-

[1959] , which i s a t t h e same t i m e a good example orem by S t e i n and Weiss

o f t h e technique o f l i n e a r i z a t i o n presented i n 3.7.

A . The R ienz - Thohin theorrun.

1 1 1 1 1 1 - = ( 1 - t ) - t t - , - = ( 1 - t ) - t t - P t Po P1 q t 90 q1

FwLthehmurre.,

l n ~ L V U X C W , i6 pt < m t h e opetratoh T can be uvLiyudy

extended t o t h e whoLe npaw LPt(fil) phenmving t h e l a ~ t i nequu ldy .

For t h e p r o o f we r e f e r t o Zygmund [1959] .

3.4. INTERPOLATION 55

B . The Marrcinkiewicz themem.

3.4.2. THEOREM. LeA ( a l , Y l y ~ l ) and (QZ,S2,p2) be Awa mean -

W L ~ bpaCU und T

L "(a,) t L '(al). Annume t h a t T A v2; weah Z y p ~ ( p o , q o ) and (pl,ql)

ttrhetre

u nubadditcue opehcLtvh t h d A de6ined vn ,the npace P P

1 6 Po 6 - Y Po c P I . go # 91.

F O h O < t < l , L d

1 - ( 1 - t ) t - 1 t q1

- ( 1 - t ) + - t , P t Po P I 9 t qo

Then T A &a 0 2 ; t y p e (pt,qt)

For t h e proof we r e f e r t o Zygmund [1959] o r S t e i n [1970 1. I n t h e l a s t r e f e r e n c e one can see some r e c e n t g e n e r a l i z a t i o n s .

C. A thev tm vl; S t e i n and W ~ A .

For many opera to rs t h a t a r i s e i n a n a t u r a l way i t i s r e l a t i v e l y

easy t o prove t h a t t h e y v e r i f y a weak t y p e i n e q u a l i t y when r e s t r i c t e d t o

c e r t a i n types o f f u n c t i o n s (smooth, s imple, c h a r a c t e r i s t i c f u n c t i o n s o f

sets , . . . ) . For t h i s reason i t i s i n t e r e s t i n g t h e f o l l o w i n g theorem o f

E.M.Stein and G.Weiss [1959] t h a t a l l o w s t o i n t e r p o l a t e between such

types o f i n e q u a l i t i e s i n t h e s t y l e o f t h e Marc ink iew icz theorem.

L e t (n ,F ,p ) be a measure space and T : W(n) -+ ,W(n) a s u b l i n e a r ope ra to r . We say t h a t T i s o f hc%t.kicted weah type (p,q),

1 G p & m , 1 & q 6 m , when t h e r e e x i s t s a cons tan t

f o r each c h a r a c t e r i s t i c f u n c t i o n xE of a measurable s e t E c R

w i t h p ( E ) < and f o r each A < 0 we have

c > 0 such t h a t


When one t r i e s t o i n t e rpo la t e , using the technique of t he proof of Marcinkiewicz theorem, i n order t o obtain an intermediate ha-tt icted type from the knowledge of two r e s t r i c t e d weak types, there i s no problem a t a l l , as the following lemma shows.

P4006. Let E be measurable, E t . We t r y t o prove -

w i t h At independent of E. We know

If we s e t .,(A) = i-~ { x e n

Because of t he inequa l i t i e s above

Xqt w ( A ) -+ 0 as A -P 0 and as A --t m.

TXE

Thus

If we integrate and s e t N = u(E)’ with the value

which makes minimal the l a s t term, then

However, when we try t o obtain the nonrestricted strong type by means of the preceding technique, the process does not work, since we need an estimate for w T f ( X ) for f E L P ( Q ) t h a t we do n o t have, a t least so direct ly as above. The diff icul ty can be obviated going over t o the dual space i n the following manner.

Assume now that the operator T : ’l/Yl (a) -t %(a) i s f i n m and of res t r ic ted type ( p t , q t ) ( as in the $onclusion o f the preceding lemma), 1 & pt < m , 1 < q t < m . On L q t ( Q ) , dual space of L q t , ye are going t o define an operator T* in the following way. Let f E ~ q t ( n ) anf for E c , V(E) i , we s e t

We then have

Hence vf(E) is a s igned measure con t inuous w i t h r e s p e c t t o p . So by

t h e Radon - Nikodym

h z T* f e L1 such t h a t vf(E) = JE h . So, f o r each f e Lqt (n)

and each measurable E, u (E ) < m we have

theorem t h e r e is an e s s e n t i a l l y un ique f u n c t i o ?

Si.nce T i s l i n e a r f o r each s imp le f u n c t i o n s

1 ( T s ) f = \ sT*f

and so T* i s a s o r t o f a d j o i n t o f T. Now we can s t a t e t h e f o l l o w i n g

1 emma.

3.4.4. LEMMA. L d T be a in Lemma 3.4.3 and benididen

Lineah. LeR: T* be dedined an i n Ahe phecedcng &na. Then T* Lb 06

weah type. (q;; , p; 1 -

Then

1

The s e t EX i s t r e a t e d i n t h e same way and so we g e t t h e lemma.

These two lemmas p e r m i t us t o o b t a i n v e r y e a s i l y t h e f o l l o w i n g

theorem.

Prrood. We take to,tl such t h a t 0 < to < s < tl < 1. Then,

by Lemma 3.4.3, T i s o f r e s t r i c t e d t ype ( p t o y q to ) and (ptly qtl) . By Lemma 3.4.4, T* i s o f weak (qi, , p i , ) and (qC,,pil). By t h e

Marc ink iew icz theorem T* i s t hen o f s t r o n g t y p e ( q i , p:). Therefore

i t s a d j o i n t T* i s w e l l d e f i n e d and i s o f s t r o n g t y p e (ps,qs) . I f f E Lq5 and s i s a s imp le f u n c t i o n

j ( T * f ) s = j f ( T s ) = f (T**s) J

The re fo re T**s = Ts f o r s s imp le and so T i s o f s t r o n g t y p e (ps,qs).

The theorem o f S t e i n and Weiss has a drawback. I t r e q u i r e s t h a t

T be l i n e a r and so cannot be d i r e c t l y app l i ed , f o r example, t o maximal

o p e r a t o r s .

method t h a t we s h a l l see l a t e r , pe rm i t s us t o extend t h e r e s u l t t o t h i s

s i t u a t i o n .

The f o l l o w i n g c o n s i d e r a t i o n , an example of t h e l i n e a r i z a t i o n

m 3 . 4 . 6 . TffEOREM. L d {Tk} k=l be a sequence 0 6 f ineah opehatoh5

@om (a) t o 'm ( Q ) . ld T* be t h e i t maxim& opeh&oh. &Arne

t h a t T* ,LA oh rrena7hted weah t g p e n (po ,qo ) and (pl,ql) , w a h

1 6 P O ,i qo < m 1 6 p1 c q1 4 m , q o f q l . Then T* LA a15 s&vng Rgpe 1

91 - ( 1 - s ) t - s . 1 - ( 1 - 5 ) + - s , (ps,qs) w4xh 0 < s < 1 , - - -

Ps P o P1 9, 90

Prrood. L e t $ : Q -f N be any a r b i t r a r y measurable f u n c t i o n

F o r g E W(R) and x E R we d e f i n e Ti) g ( x ) = T $ ( x ) g ( x ) . The

60 3. G E N E R A L TECHNIOUES

operator T so defined i s obviously l i nea r from ()"l (Q) t o ,h (n). We c lea r ly have, f o r each x E ~2 and g E % (Q),

$

T*. Hence T i s of r e s t r i c t e d weak types same constants as T* , i . e . with constants

by Theorem 3.4.5. T i s of strong type + J,

( p S , q , ) with constants independent of + .

Let now f E L p s . We choose @ : G. -f M measurable such t h a t T*f(x) c 2 IT f ( x ) I fo r each x 6 G. . ( t o do t h i s def ine @ ( x ) on the s e t I T j f ( x ) l > Z k ) .

with c independent of f . So T* i s o f strong type (ps,qs).

f i r s t consider Tfi defined by Tfif(x) = sup ITkf (x ) l . We obtain the

r e s u l t f o r Tfi obtain i t f o r T*.

L Cx 6 G. ' : 2 ktl > T*f(x) > Z k 1 as t he f i r s t j such t h a t

Thus we have 1 1 T*fl l c 2 1 1 TOfllqs 6 c I l f l /ps

In general we can

q S

Here we have t a c i t l y assumed T * f ( x ) # m . 1s k&N

and then a passage to the l i m i t as N + m allows us t o

3.5. EXTRAPOLATION

The aim of the extrapolation technique can be understoood in t h e following concrete example. Let T : k ( n ) + h ( n ) be a subl inear operator and assume t h a t we k n o w t h a t i t i s , f o r ins tance , of strong type ( p , p ) , P O < p < P I , with a constant depending on p , c ( p ) , i . e . f o r each f 6 Lp(n) ,

Assume tha t we have some more information about c ( p ) f o r example t h a t c ( p ) L A / ( p - p o ) ' f o r p c lose t o p o . Can we ex t r ac t

3.5. EXTRAPOLATION 61

f rom t h i s s i t u a t i o n more i n f o r m a t i o n about t h e t ype of t h e o p e r a t o r T?

As an i l l u s t r a t i o n of t h e thechnique we p r e s e n t a r e s u l t of Yano

The method does n o t seem t o have been e x p l o i t e d v e r y e x t e n s i v e [ 1951 1. 1Y.

3.5.1. THEOREM. L e R M be a bubadditive, panLtLve, panL?Xue-

Ly homogeneow a p e h a t o h 64om %I ( n ) t o (n) . knwne t h a t

1 1 M f I l o o 6 1 1 fll, do4 each f E: Lm and t h a t thetre exint con&tm,tb c,

s > 0 nuch t h a t 604 each p E (1,2) and d o t each mmu/rabLe be,t 0 6 bounded meanme E , we have

Then, do4 each f E: L ( l + log+Ls) and do4 each mmwrabbe be,t 06 bounded m m m e X C we have

w i t h clrc2 independenX ad f, X.

As we can see, t h i s i s a s o r t of Kolmogorov c o n d i t i o n r e l a t e d

t o t h e space L ( l + log+L)' .

P4aud. We know t h a t f o r each K c n w i t h u (K ) <

we have

L e t us take f E L ( l + log+L)' , f > 0 , and c a l l

E o = IX f X : 0 < f ( X ) < 1)

Ek = { x 6 x : 2k-1 & f ( x ) < 2k1 f o r k = lY2,3,...

6 2 3. GENERAL TECHNIQUES

We can w r i t e

The sum o f t h e terms i n t h e s e r i e s above corresponding t o those

ek f o r w ich ek 6 3-k i s f i n i t e s i n c e

I f ek > 3-k t hen we have

and so

Hence

k+ 1 ~

2 k ( k + 1)' ek kt2 < 2 k ( k + l ) s ek3

By r e f i n i n g t h e p rev ious methods MoriyBn [1978 ] has ob ta ined

a r e s u l t o f e x t r a p o l a t i o n f o r p o > 1 s i m i l a r t o t h e p reced ing one o f

Yano.

3.6. MAJORIZATION 63

3.5.2. TIfEOREM . L e t M be a d u b a d d i t i v e , p u ~ . i t i w e , ponLi5ve - Ly hamugeneaw ap-o4 6 m m i)n (a) ,to ,%‘I (n). Anbwne M a t

1 1 M f [ I oo < l l f l [ w c > 0 , and anbume ,that 6 0 4 each

K a6 baunded m m m e , we have

each f E Lm . L e t 1 < po < m, E > O , s > O , p E ( p o , p o + E ) and 6 0 4 each K c n,

Then, 6 0 4 each t > p o ( s + 1) - 1 Rhehe e x h ~ 2 ,that don each K C n 0 6 bounded mmute and don

- - c = C ( t , p o , E , S , C ) bUCh

each f 2 0 , f E M(n)

3.6. MAJORIZATION.

I f T , S a r e subl inear operators from k ( n ) t o w(n) and

ITf (x ) / L I S f ( x ) / , i t i s q u i t e c l e a r t h a t i f S i s , f o r example , f o r each f e ?X (0) and each x E R one knows t h a t

of weak type ( p , p ) , them so i s T .

Sometimes , and we sha l l l a t t e r s ee important examples, when dealing with s ingular in tegra l operators, this t r i v i a l majorization does not work, and one has t o appeal t o some o ther sub t l e r procedures. Here , as we sha l l see, the Kolmogorov condition plays an important ro l e . We t r y t o give the flavour o f the technique with two concrete b u t c h a r a c t e r i s t i c examples .

3.6.1. THEOREM L c t T and S be n u b f i n e a r apehaXo4~ 64om

%7 (Rn) w(’Rn) . AAAWAC t h a t T .& majahized by S -in ,the

6uUawLng heme. Foh each f a Wmn) and each x E Hn thehe exAi2

a hphetLicaX nheRe Q ( x ) = { z e Rn : r 6 I z - x ( c 2 r l w a h r depend-

i n g on x and f duch t h a t d o t each y 8 Q ( x ) one hm

I T f ( x ) l 6 l S f ( y ) l

Then, i6 S LA 0 6 w a k t y p e ( p y p ) ( o h dome p ,1 c p < m , T i.6 &a 04 weak .type (p,p).

I

L

t e g r a t i n g over y 8 Q ( x ) and d v

have i f f e Lp(Rn)

Where Q* (x ) i s t h e min imal c u b i c i n t e r v a l cen te red a t x and conta in-

i n g Q x ) M i s t h e Hardy -L i t t l ewood maximal ope ra to r , and c i s a con-

s t a n t ndependent o f f and x. The re fo re y i f K i s any s e t o f f i n i t e

measure and A > 0 .

If we now r e c a l l t h e

t y p e i n e q u a l i t y (1,l) f o r M

[HA( < m , such t h a t AX

remark a t t h e end o f t h e p r o o f o f t h e weak

C HA and

i n 3.2.A , we see t h a t t h e r e e x i s t s H A Y

w i t h c independent o f HA, A, f , K. I f we now a p p l y Kolmogorov's ine-

q u a l i t y t o S q i t h exponent u

3.6. MAJORIZATION

Hence

6 5

C** l A X l I H X I - ll f l l ( :

XP

w i t h c** independent o f f, X ,K . So T i s o f weak t y p e ( p , p ) .

3.6.2. THEUREM L& T and S be Xiuv n u b f i n e m o p e m i t v ~ dhom

(Rn) tv (Rn) . Annume Lhcd T A majvhized by S i n t h e dvL - Lowing oenhe, F 0 4 eclch x E Rn and edch f thehe exint ;two

cubic i n - t e t r u a h centmed CLt x, Q ( x ) and Q*(x ) , Q * ( x ) w a h diam&.te/r 4 .tima t h d t u,'J Q ( x ) , nuch t h d t do4 each y E Q ( x ) we have

Then, i d S vd weak t y p e ( p , p ) dvh bVme p , 1 < p < a, T

d e n v 06 weak type ( p , p ) .

Phovd . We proceed as be fo re , i n t e g r a t i n g ove r y E Q ( x ) and

d i v i d i n g by l Q ( x ) l , a f t e r hav ing taken t h e a - t h power o f t h e above

i n e q u a l i t y ,

I f we app ly t o S t h e Kolrnogorov's inequa

T f ( x ) l a c c 1Q(x)

U

i t y , we g e t

Th is proves t h e theorem.

3.7. LINEARIZATION.

As we have o f t e n seen, many i n t e r e s t i n g o p e r a t o r s a r i s i n g i n a

n a t u r a l way i n t h e a.e. convergence t h e o r y a r e n o t l i n e a r . I m p o r t a n t re-

s u l t s of f u n c t i o n a l a n a l y s i s a re n o t a p p l i c a b l e t o them. The l i n e a r i z a t i o n

technique c o n s i s t s i n s u b s t i t u t i n g t h e n o n l i n e a r o p e r a t o r under s tudy by

another l i n e a r one t h a t i n a c e r t a i n sense m a j o r i z e s it. The use of per-

t i n e n t techniques o f l i n e a r f u n c t i o n a l a n a l y s i s may then p e r m i t us t o

o b t a i n t h e i n e q u a l i t y we l o o k f o r . We have a l r e a d y seen t h i s techn ique

work.ing i n t h e p r o o f o f N i k i s h i n ' s theorem and a l s o i n t h e e x t e n s i o n t o

maximal ope ra to rs of o f S t e i n and Weiss.

Now we sha

n ique which appears

t h e theorem on r e s t r i c t e d weak t ype i n t e r p o l a t i o n

1 p r e s e n t a s imp le example of t h e use o f t h i s tech-

n a paper o f Cdrdoba r1976 ]. A lso we g i v e some

re fe rences f o r more e l a b o r a t e a p p l i c a t i o n s o f t h e same technique.

If 18 i s a c o l l e c t i o n o f open s e t s i n Rn w i t h bounded measure,

we d e f i n e the Hardy -L i t t l ewood maximal o p e r a t o r M r e l a t i v e t o d3 i n

t h e f o l l o w i n g way. For f E Lloc (Rn)

I f ( y ) l d y if x e II B B B E 8 M f ( x ) =

I o

3.7.1. THEUREM . L e Z be a cof lect ion ad A & an above.

hbume t h a t t h e opehtratoh M o d type (p,p) doh A U t w p with

1 < p < a, i.e. doh f 6 Lp(Rn) , ( 1 M f l I p 6 c ( ( f ( l p w L t h c indepe~dent 06 f .

3.7. LINEARIZATION 67

Then b b&&4& t h e dok%wLng covehing p k o p d y : Given any

dinAXe coReection C B k I k = l ol; be& 0 6 8 , AX pobbibee 20 choobe

@om them u hequence { R k } k = l 4 U C h t h a t :

H M

k = l k = l ( i ) I 1J Bk [ 6 c1 [ 0 Rk I , ( i . e . t h e Rk 's coven u good poh -

c o n 0 6 mhat 2he Bi( s coveh) .

o n m a i n Lq-noxm). ~ e t r e q = P p- l . The co~n;trcna2 c 1 ,c2 depend onLy on c and p .

Pmol ; . We choose R 1 = B1 . For R 2 we choose t h e f i r s t Bk 1

among BZyB3,...,B" such t h a t IBk 0 R1 1 c 7 I Bk I . Assume i t i s

67 = RZ . For R 3 we choose t h e f i r s t Bk f rom B s y B 9 , ... B,, such t h a t

I B k 0 ( R 1 11 R 2 ) 1 c 7 l B k l . And so on. So we g e t {Rk) k = l . For

each B t h a t has been l e f t o u t we have IB 0 1J R k 1 > IBI and so

1 N

1 N

Thus we have ( i ) .

We now p rove (ii). Observe f i r s t t h a t i f Ek = Rk - R j < k j '

1 lEkl 2 7 l R k l . then

t h e f o l l o w i n g way . For

We d e f i n e now a l i n e a r o p e r a t o r T : L p + L p i n

f B Lp ,

Observe t h a t I T f ( x ) l < M f ( x ) . There fo re T i s bounded w i t h a norm

m a j o r i z e d by c. I t s a d j o i n t w i l l be a l i n e a r o p e r a t o r S : Lq -f Lq

68 3 , G E N E R A L TECHNIQUES

whose norm i s a l so majorized by c . B u t S can be eas i ly wr i t t en in e x p l i c i t form

We have

and so

This i s ( i i ) .

Other nice appl ica t ions of the l inea r i za t ion technique can be seen i n CBrdoba [1977] and a l so i n C.Fefferman [1973].

3.8. SUMMATION.

Later on and in d i f f e r e n t contexts we sha l l f ind ourselves i n We want t o study the type of an op- s i t ua t ions l i ke the following one.

e r a t o r T t h a t i t can be majorized by a sum of operators 1 ck T k with a simpler s t ruc tu re whose type we can e a s i l y determine. the type of T? I n general , the strong type does not o f f e r any d i f f i c u l t y , s ince 1 1 1 ‘Ck T k l I p 6 1 l C k l ~ ~ T k ~ ~ p . For the weak type we can a l so

sometimes obtain sa t i s f ac to ry r e su l t s .

and by means o f geometric or ana ly t i c considerations we f ind

What can we deduce about

k k

Let, f o r example T = 1 ck T k , and assume t h a t each T k i s uniformly of weak type (1,l) , i . e . , f o r each f e L ’ ( R n ) , h > 0 and

3.8. SUMMATION

k = 1,2, ...

69

and t h e r e f o r e

The p rev ious example i s a p a r t i c u l a r case o f t h e f o l l o w i n g

theorem on summation o f weak t y p e i n e q u a l i t i e s . I t i s due t o E.M.Stein

and N.Weiss [1969] . We s t a r t w i t h a lemma.

3.8.1. LEMMA.

Suppode t h a t doh j = 1 2 3,. . . , on Ph doh which I I x : g j ( x ) > s } l 6 I

a nonnegative dunc-tition

d a h each s > 0 . Le,t { c . } J

gj

be a dequence od p o n h X v e numbem w a h C c j = 1 and J

m

Then, doh each s > 0 ,

Phood. F o r each j = 1,2,3, ... l e t us d e f i n e

g j b f if g j ( x ) < 7 S

S if g j ( x ) > 7 v . ( x ) = J


S g j ( x ) if g j ( x ) > - 2 c j

if g j ( x l 6 - 2c S u . ( x ) = J

m.(x) = g . ( x ) - v . ( x ) - u j ( x ) J J J

L e t v ( x ) = c c . v . ( x ) , u ( x ) = C c . u . ( x ) , m(x) = c c . m . (x ) . J J J J J J Observe t h a t , for each x , we have v ( x ) i 4 and t h a t

L e t us w r i t e Aj(y) = I { x : g j ( x ) y I } f o r y > 0 . Observe t h a t

y X j ( y ) 6 1. Then we have, s i n c e -3.- > m . ( x ) J > S a t each x, j

2c

J

From t h i s we g e t { x : m(x) > - S 1 6 ___ 2(Kt1) , and f i n a l l y , 2 1 S

With t h i s lemma we e a s i l y a r r i v e t o t h e f o l l o w i n g theorem.

3.8.2. THEOREM. LeA {Tk}E=l be a Aequence 06 hubadd i t i ve

o p a a t o / r o &,om L ’ (Rn) t o r/l (Rn) t h a t me uni6omly 06 weak t y p e (1 , l ) .

3.8. SUMMATION 71

7

CHAPTER 4

ESPECIAL TECHNIQUES FOR CONVOLUTION OPERATORS

Many of the operators of i n t e r e s t i n Fourier Analysis a r e oper - a tors o f convolution type. the Introduction t o Chapter 1 as motivation fo r t h i s work a r e of such type.

I n pa r t i cu la r a l l operators which appear i n

We consider a sequence or generalized sequence of functions C k . 1 C L ’ ( Q ) ( ke rne l s ) where R wil l be here e i t h e r the n-dimensional

torus TTn o r R n (Q could be as well a l oca l ly compact group) and f o r a function f 6 L P ( Q ) , 1 < p 6 00, we def ine

J

K.f(x) = k . * f ( x ) J J

We ask about the convergence of K.f in L p o r pointwise. In order t o J t r e a t the convergence in L p we a r e to consider I I K . f l l as explained in the Introduction t o Chapter 1, and to study the pointwise convergence we a r e led t o inves t iga te the behaviour of

J P ’

K* defined by

K*f(x) = s u p I b . * f ( x ) l J j

The f i r s t and second sec t ions of this Chapter reduce the problem to the study of the action of the operators K j o r K* over f i n i t e sums of Dirac de l t a s concentrated a t d i f f e r e n t points of R ( f o r the weak or strong type (1 , l ) ) and over l i n e a r combinations of Dirac de l t a s ( f o r t he weak or strong type c re t i za t ion of the operators i n question, as we sha l l show l a t e r .

( p , p ) , 1 < p < m ) . This reduction permits t he dis-

A previous r e s u l t in t h i s d i rec t ion has been obtained by K.H.Mg and i s presented here as Theorem 4.13. The idea of charac te r - on [1976]

izing the weak type fo r the maximal convolution operator by means of t he Dirac de l t a s as i t appears in Theorem 4.1.1. belongs t o the author. The

74 4. CONVOLUTION OPERATORS

extens ions and r e f i n e m e n t s t h a t f o l l o w have been o b t a i n e d by M . T . C a r r i l l o

[1979] .

4.1. THE TYPE ( l Y 1 ) OF MAXIMAL CONVOLUTION OPERATORS.

Convo lu t i on opera to rs a c t i n g over a sum o f D i r a c d e l t a s g i v e

a d i s c r e t e sum which i s , i n many cases , r a t h e r easy t o handle.

I f k 6 L ’ (R) , a1,a2,.. .,a,, B R , and 6h denotes t h e

D i r a c d e l t a concentrated a t ah then , i f

H H

h = l h = l K f ( x ) = K * ( 1 6 h ) ( X ) = 1 k ( X - a h )

We s h a l l say t h a t K 06 weak .type (1,l) aveh ~$Linite 6wnn

0 6 V&ac d& when t h e r e e x i s t s c > 0 such t h a t f o r each 1 > 0

and each

H

we have

For a maximal o p e r a t o r K* o f c o n v o l u t i o n opera to rs Kj o f

t h e k i n d desc r ibed i n t h e i n t r o d u c t i o n of t h i s Chapter one can w r i t e , if

4.1. THE TYPE (1,l)

H

75

K * f ( x ) = sup I K . f ( x ) j J

and so K* i s o f weak t ype (1 , l ) over f

t h e r e e x i s t s c > 0 such t h a t f o r each

H f = 1 Ish

h = l

H = sup I 1 k . ( x - a h ) l

j h = l

n i t e sums o f D i r a c d e l t a s when

and each X > 0 ,

The main theorem i n t h i s s e c t i o n i s t h e

t i o n of t h e o r d i n a r y weak t ype (1 , l ) of K*.

f o l l o w i n g c h a r a c t e r i z g

Then K* & ad weak .type ( 1 , l ) i 6 and anLy .id K* ih a6 weak

.type (1 , l ) o v a & h X e nwnh ad QitLac dcLtuh.

l n o t h m W o h h (dahge;U;ing a b u t P h c delakhl K* ih 06 weak

.type (1,l) id and o n l y id t h e m ex.Ath c 0 nuch t h a t , d m each

dinite heL ad di6deh'en.t pointh a1,a2, ... , aH 6 R and d m each X > 0 ,

we have

P m a d . (A) We f i r s t prove i n f o u r s teps t h a t i f K* i s o f

weak t y p e (1 , l ) t ype (1,1).

over f i n i t e sums o f D i r a c d e l t a s , t hen i t i s o f weak


(1) Assume t h a t t h e r e e x i s t s c > 0 such t h a t f o r each

H

h = l f = 1 and X > 0 we have

H We want t o p rove i n t h e f i r s t p l a c e t h a t i f f = h=l 1 ch 6h

w i t h ch E Nl , then

H

I f f o r a f i x e d n a t u r a l number N we c a l l K;f(x) = sup K . f ( x ) l 6 j G N

then, s i n c e c l e a r l y

m

IJ { X : K$f(x) > X} = { x : K * f ( x N = l

i t is c l e a r t h a t i t w i l l s u f f i c e t o prove t h a t f o r each f i x e d N

w i t h c independent o f N . So we f i x an N .

Now f o r each k , 1 6 j G N , we take g E eD(Q) such t h a t j j

I l k j - g j l l l G n t e r . For each p o i n t ah 6 fi we choose ch p o i n t s b i , b t Y ... hh , a l l of them d i f f e r e n t . We then can w r i t e f o r each j

where TI > 0 w i l l be c o n v e n i e n t l y chosen a l i t t l e la ‘h

4.1. THE TYPE (1,l) 71

Now f o r each c1 such t h a t 0 < c1 < A , we have

l { x :

c I I X : sup I c j c N

H Lh

By t h e hypo thes i s

H

‘h h = l c c - - - - - - - A - a

I f we prove t h a t , f o r a r b

can choose b;h and g

s tep (1).

t r a r y E > 0 , and a w i t h 0 < a < A , we

so t h a t I P < E , we a r e t h e n f i n i s h e d w i t h

Observe t h a t we can w r i t e

Thus we can s e t

78

Now

4. CONVOLUTION OPERATORS

Hence , g iven E > 0 , c1 > 0 , we f i r s t choose g 6 t o ( ( n ) j

E H

j so t h a t l ] g j - k j I I 6 rl and - ‘‘ 1 ch & 7 . Observe t h a t t h e g

c I 1

a r e un i fo rm ly cont inuous. Once t h e g

c l o s e t o ah t h a t

have been f i x e d , we take b;h so j

Thus we g e t

( 2 ) From (1) we s h a l l e a s i l y prove t h a t i f f = 1 ch bh

I 2 < E and so we conclude t h e p r o o f o f s t e p (1). H

h= 1

w i t h ch > 0 then f o r each X > 0 we have H

T h i s i s obvious i f ch 6 Q . If ch E R , c h > 0 we can take H

F = 1 dh bh w i t h dh = ch + rh , rh smal l , dh 6 Q . Then, i f O < c 1 < X Y

h = l

4.1. THE TYPE (1 , l ) 79

H

H

By choosing a p p r o p r i a t e l y rh one ge ts (*).

( 3 ) We want t o prove now t h a t K* i s of weak t y p e ( 1 , l ) o v e r

l i n e a r combinat ions o f c h a r a c t e r i s t i c f u n c t i o n s o f dyad ic i n t e r v a l s . I f

d = 1 ch xI w i t h Ih dyad ic i n t e r v a l and ch > 0 we want t o p rove

t h a t f o r each X > 0 and f o r any such d

H

h = l k

H

As b e f o r e , i t w i l l be s u f f i c i e n t t o prove t h a t if N i s f i x e d ,

F i r s t of a l l observe t h a t we can assume t h a t t h e s i z e o f each

Ih i s as smal l as we please. Otherwise we s u b d i v i d e each Ih and t h e

i n e q u a l i t y we want t o ' p r o v e i s independent of t h e number H o f dyad ic

i n t e r v a l s we have.

We now proceed as i n s t e p (1). For each k , 1 G j G N we j t ake g j E '$ o!n) such t h a t 11 k . - g.111 G n , where n > 0 w i l l be

c o n v e n i e n t l y chosen l a t e r . L e t

J J

H

h = l f = 1 'h 'h

where 6h i s t h e D i r a c d e l t a concen t ra ted a t ahy t h e l e f t extreme

80 4. CONVOLllTION OPERATORS

p o i n t o f Ih.

Then, i f 0 < c1 < A , we can w r i t e

A l l we have t o do now i s t o prove t h a t t h e second te rm i n t h e

l a s t member can be made sma l l by an a p p r o p r i a t e c h o i c e o f g j and 1 Ih/.

We can w r i t e

There fo re we can s e t

4.1. THE TYPE (1,l)

I n t h e same way

81

A1 so

Given E > 0 we f i r s t choose g such t h a t / / k . - g . l l l c rl , w i t h

so smal l t h a t

j J J

Then we choose

v i r t u e o f the above inequa l ty. Thus we g e t

I h so smal l t h a t I { x : Ag(x) > :I[ &:,what can be made i n

f o r d l i n e a r combinat ion o f c h a r a c t e r i s t i c f u n c t i o n s o f dyad ic i n t e r v a l s .

(4 ) We know (Theorem 3.1.1.) t h a t t h e r e s u l t i n ( 3 ) a l r e a d y

i m p l i e s t h a t K* i s of weak t ype (1,l). T h i s concludes t h e p roo f o f ( A ) .

H ( B ) Assume now t h a t K* i s of weak t y p e (1 , l ) . L e t f = fjh.

h = l

We take d i s j o i n t dyad ic i n t e r v a l s Ih c o n t a i n i n g t h e p o i n t s ah. L e t

. We know t h a t 1

and want t o prove t h a t f o r each f i x e d N

We w r i t e , f o r 0 < ct < x ,

For t h e l a s t term we proceed as before i n (1).

t h a t

If g j 6 g,,(O), i s such

I l k j - g j ( l l b TI , we w r i t e

As before,one f i r s t chooses g j 6 9 ,,(n) c l o s e t o k j

Ih s u f f i c i e n t l y smal l . So we g e t l { K * f > A 3 1 C c lP= c: . i n L' and t h e

+

The method o f p r o o f o f t h e p reced ing theorem can be a p p l i e d t o

many o t h e r i n t e r e s t i n g s i t u a t i o n s . We j u s t s t a t e a t y p i c a l theorem t h a t

can be ob ta ined w i t h it.

4 .1 . THE TYPE (1 , l ) 83

4.1.2. THEOREM . L&t C k j ? y = l be a nequence 0 6 &uatLtionn in

L ' ( f i ) and d o t f E L1(Q) dedine K.f(x) = k . * f ( x ) . J J

Then t h e open&^

a&y id they me unidotmLy 05 weak .type d-.

K j ahe uni~vhmly 06 weak t y p e (1 , l ) id and

(1,l) oven 6ivLite nmn a 6 Dhac

and X > 0 we have d a f i tach

I{x : lKjg

id and only i d d o t each f = H 2 6h we have

h = l

In the preceding theorems one can change weak type (1,l) f o r strong type (1 , l ) .

The theorem of K.H.Moon mentioned in the introduction of t h i s Chapter i s as follows.

THEOREM 4.1.3. kj*f(x) , K*f(x) = s u p I k . * f ( x ) / .Then K* LA ad weak t y p e (1 , l )

L e i C k j l y = l c L1(Q) and, d o t f 6 L ' ( f l ) , K . f ( x ) =

id and on ly id

nite uniann od dyadic inte,twP~.

J j J

K* LA ad W M ~ t y p e o v m c h a h a c t e ~ A L i c duneLivnn 04 8.i-

Ptoo6. After Theorem 4.1.1. a l l we have to do is t o show t h a t i f K* i s of weak type (1,l) over c h a r a c t e r i s t i c functions of f i n i t e un- ions o f dyadic in t e rva l s , then K* i s of weak type (1 , l ) over f i n i t e sums of Dirac de l t a s . B u t t h i s is e a s i l y done as i n (B) o f t he proof of the Theorem 4.1.1. by taking there the s e t s Ih of the same s i ze .

- The previous theorems r e fe r t o the weak type (1,l) of t he maxi ma1 operator of an ordinary sequence o f convolution operators. In many

84 4 . CONVOLUTION OPERATORS

cases, however, one has t o deal with the maximal operator of a whole fam - i l y of convolution operators indexed, f o r example,by the s e t of rea l numbers. Such i s the case, f o r instance, of the maximal Hi lber t the Hardy-Li ttlewood maximal operators , the maximal Calder6n e ra to r s , . . .

The natural question i s then: Can one charac te r ize (1 , l ) of the maximal operator by means of i t s weak type (1,l

transform , Zygmund op-

the weak type over f i n i t e

sums of Dirac de l tas as we have done i n the case o f an ordinary sequence?

The answer f o r t he general case i s negative,as the following sim ple example shows

Let

1 i f x = 1,2,3,4,. . .

i f x e R - M k ( x ) =

and, f o r E > 0 , k E ( x ) = E - ~ k(:) . For f E L’m) , l e t K E f ( x ) = k E * f

and K*f(x) = sup IKEf(x) l . E>O

Then, f o r each E > 0 , we have K f z 0 and so K*f = 0 . E

Therefore K* i s of weak type (1 , l ) . However, f o r each x E ( O , l J , n e W, X i f E~ - - - , we have

n n n k ( x ) = - k ( - . x ) = - X X > n X E X

Therefore

and so K* i s not of weak type (1 , l ) over f i n i t e sums of Dirac de l t a s .

However, by imposing some mild conditions on the kernels we can s t i l l recpver the same kind of charac te r iza t ions . For example, i f

4.1. THE TYPE (1 , l ) 85

k 6 L j o c ( Rn - (0) ) , f o r E > 0 , x a R n

and, f o r f E L ’ N n ) , KEf(x) = kE * f ( x ) , and K*f (x) = sup IKE f (x ) l , R s E > O

then we e a s i l y o b t a i n

and so one can app ly t h e p rev ious r e s u l t s . Observe t h a t t h e Calder6n-

Zygmund maximal ope ra to rs f a l l under t h i s t y p e .

Also, i f B i s any measurable subse t o f Rn w i t h p o s i t i v e

and f o r each sequence measure so t h a t f o r each

we have x,.~ + xEB J

E > 0 { E ~ } , E~ -f E

a.e. then, if, f o r f E L1(Rn) , we s e t

t hen we e a s i l y o b t a i n

sup I K E f ( x ) l = sup I K E f W I R a 0 0 Q3 E > O

and so one can a p p l y t h e p rev ious theorems.

ope ra to r ,

The Hardy -L i t t l ewood maximal

f o r example, f a l l s under t h i s ca tegory .

I n t h i s c o n t e x t t h e f o l l o w i n g genera l theorems a r e of i n t e r e s t ,

e s p e c i a l l y f o r some r e s u l t s on approx imat ions o f t h e i d e n t i t y t h a t we s h a l l

s tudy i n Chapter 10.

4.1.4. LEMMA . L e t k E L’ Lw(Rn) and f E L1(Rn) . L& A be a d e u e hubb& 06 (0,m) . Then,id doh E > 0 ,

kE(x) = E - ~ k (:) and KEf (x ) = kE * f ( x ) , we have

86 4. C O N V O L U T I O N OPERATORS

Given q > 0 we f i r s t choose g 6 @ o m n ) such t h a t t h e f i r s t

t e rm o f t h e l a s t member o f t h e c h a i n o f i n e q u a l i t i e s i s l e s s t h a t q/2 . Then we choose ci E A so c l o s e t o E t h a t t h e second term i s l e s s than

n/Z . I n t h i s way we o b t a i n t h e lemma.

Wi th t h e p reced ing lemma t h e f o l l o w i n g theorem i s s imple.

4.1.5. THEOREM . L e L k E L’ 0 Lm(Rn) and doh E > 0 ,

kE(x) = E-nk(:). Le, t UA dedine, doh f 6 L’(Rn),

Then:

(a ) K*R LA ad weak ,type (1 , l ) i6 and anLy .i6 K*Q LA 06

(b) K*R can be 06 weak t y p e (1,l) oven ~ u n C . t i u ~ d wLthuu t

w u k t y p e (1,l) oveh 6 i n i t e numb 06 VhAc deetad.

being 06 w& type (1,l) oveh din.& bum5 od PhAc d u .

4.1. THE TYPE (1,l) 87

Phovd. The p r o o f o f (a ) i s i nmed ia te f rom t h e Lemma 4.1.1. and Theorem 4.1.1. For ( b ) l o o k a t t h e example shown a f t e r t h e p r o o f

o f Theorem 4.1.2.

I n t h e p reced ing theorem k E L’ 1’1 Lamn). I f we o n l y have

k E L’(R”) , then we can s t a t e t h e f o l l o w i n g r e s u l t .

4.1.6. THEOREM. L e Z k E L’(Tln) , k a 0 and K* = K*R be

dedined a.4 i n Theahem 4.1.5. Then, 4 K* Lb 06 weak t y p e (1,l) vveh 6inite numb 0 6 D*ac deetan, Lb vtj w u k t y p e (1 , l ) .

Phvvd. For j = 1 , 2 , 3 , . . . l e t us w r i t e

0 , i f k ( x ) > j

and K? f ( x ) = sup I k: * f ( x ) 1

Since

f o r each j

O < E E R 3

K* i s of weak t ype (1 , l ) over f i n i t e sums o f O i r a c d e l t a s we have

w i t h c independent o f j, ah , A . Since k j a L’ 1’1 Lm, by Theorem

4.1 .5 . we know t h a t K* i s o f weak t y p e ( 1 , l ) w i t h a c o n s t a n t i ndepend

e n t o f j. By pass ing t o t h e l i m i t as j -f OD , we see t h a t K* i s o f

weak t y p e (1,l) .

J

I n a s i m i l a r way we a l s o o b t a i n t h e f o l l o w i n g r e s u l t f o r a

k E L’ (1 $(Rn).

4.1.7. THEOREM . L e A k e L’ ( ) t ( R n ) , and leA K* = K* R be dedined an i n Thevmn 4 .1 .5 . Then K* 0 06 weak t y p e ( 1 , l ) 4 and o n l y i d Lt Lb 06 weak t ype (1,l) vveh ~ i n L t e bumb 013 V h a c d-.


4.2. THE TYPE (p,p) ,p > 1, OF MAXIMAL CONVOLUTION OPERATORS.

A l s o t h e t ype (p,p) , p > 1 , o f t h e maximal o p e r a t o r of a

sequence o f c o n v o l u t i o n opera to rs can be s t u d i e d by l o o k i n g a t i t s a c t i o n

ove r t h e O i r a c d e l t a s . However one cannot o b t a i n he re a necessary and

s u f f i c i e n t c o n d i t i o n .

m 4.2.1. THEOREM. le,t {kjIjzl c L1(R) be an ohdifiany b e -

qucnce. 0 6 ~unc,t io~n and C K . 1 t h e nequence 0 6 convolution o p e h a t o ~ 5 a ~ n g J cicLted t o a. L e t K* be t h e cohhuponding maxim& opehatan. 1eL p > 1.

Annwne t h a t dotr each X > 0 and doh tach ~ivLite n e t a6 didde& ent poi& al ,a2, .. . , aH 4 R we. have , don a c > 0 independent 0 6

a A, j y

Then K* .LA 0 6 W M ~ type. (p,p).

Phoo6. The p r o o f is o b t a i n e d f o l l o w i n g t h e same s t e p s o f t h e

p r o o f o f Theorem 4.1.1.

I f KG f ( x ) = sup ] k j * f ( x ) l one f i r s t o b t a i n s f o r j=l,. . . ,N

From here one ge ts t h e same i n e q u a l i t y f o r a genera l s e t

and f i n a l l y , approx ima t ing by means o f d i s j o i n t d y a d i c i n t e r v a l s

II,...,IH one o b t a i n s

c ~ , . . . , c ~ 8 R

P

There fo re K; i s o f weak t ype (p,p) w i t h c independent o f N. Hence

K* i s o f weak t ype (p,p).

4.2. THE TYPE (p,p) 89

L e t us now observe t h e f o l l o w i n g . L e t B = B(0, l ) Rn and

and f o r f E Lp(Rn) , 1 1 kj = / ~ 3 j X B ~ B . = - B y j = 1 , 2 , 3 ,...,

J J l < p < m ,

K * f ( x ) = sup I k j * f ( x ) \ J

I f M i s t h e o r d i n a r y Hardy -L i t t l ewood maximal o p e r a t o r K * f ( x ) c M f ( x )

and so K* i s o f weak t y p e (p,p) , 1 < p < a. However, as we s h a l l

now see, i t i s n o t o f weak t ype over f i n i t e sums o f D i r a c d e l t a s . I n f a c t ,

i f i 2 1 ,

and so, i f i t were , f o r some c < m,

and t h i s i s a c o n t r a d i c t i o n f o r i s u f f i c i e n t l y b i g . The re fo re , f o r p >I, t h e K* o f t h e Theorem 4.2.1. can be o f weak t y p e (p,p) w i t h o u t b e i n g

s o . o v e r f i n i t e sums o f D i r a c d e l t a s .

A l s o one shou ld observe t h a t t h e same t ype of c o n s i d e r a t i o n s

we have made a f t e r Theorem 4.1.3. a r e v a l i d i n t h i s case p > 1.

The r e s u l t s and methods we have presented i n t h i s chap te r can

be extended, o f course, t o t r e a t t h e u n i f o r m s t r o n g t ype o f a sequence

o f ope ra to rs . We s t a t e h e r e a t y p i c a l r e s u l t .

4.2.2. TffEOREM. LeA {kj}ycL’(n) . FOX f E LP(Q), LeA

K . f ( x ) = k . * f ( x ) . k b u m e tkdt t h e apehatom K j ate uni~omnLg 06

weak type (1,l) o v p f ~ ,5ivLite numb 06 DhacdctYan.

06 weak t y p e ( 1 , l ) .

J J Then t h e y me uni60hmLg


One should a l so observe t h a t some of the r e s u l t s of t h i s chanter can be used i n order t o deduce useful and in t e re s t inq qeometric pronerties re la ted to ce r t a in operators. In f a c t , from ana ly t ica l considerations we may know t h a t a ce r t a in maximal convolution operator i s of weak tvoe (1,l). Then, usingTheorem 4 .1 .1 . we deduce t h a t i t i s of weak tvne ( l , l , ) over f i n i t e sums o f Dirac de l t a s . i n an in t e re s t ing geometric way, giving us a r e s u l t t h a t sometimes i s f a r from easy t o obtain i n a d i r e c t wav.

B u t t h i s nronertv can of ten be in te rnre ted

For example, we know t h a t the Hardy-Littlewood onerator i n Rn over Euclidean ba l l s i s of weak type (1,l) . So i t i s of weak tyne (1,l)

over f i n i t e sums of Dirac d e l t a s . r i c language, we get the following in t e re s t ing covering propertv.

I f we t r a n s l a t e t h i s f a c t i n t o geomet-

4.2.3. THEOREM. 1eA a1 ,a2 ,..., aH 6 R n , v > 0 . Fvk j = 1,2 ,..., H , Leet AJ be t h e n e t ad p u i n t ~ 0 6 R" i n at LeanX 5 v b

t h e b d h B j ,Ba ,.. ., B i , centetred at a1 ,... , aH, heApecfiv&9 and H .

0 6 volume j v . Let A, = I I A J . Then j=1

uhetrhe c LA a con~akn t t h a t depencb v n t y vn t h e dimennivn.

Likewise, as we sha l l see i n ChaDter 11, the maximal s inqular in tegra l operators t rea ted there a r e shown t o be of weak type (1,l).

The reader should t r y t o obtain the geometric meaninq of t h i s f a c t .

CHAPTER 5

ESPECIAL TECHNIQUES FOR THE TYPE (2,2)

I n o r d e r t o s tudy t h e t ype o f an o p e r a t o r one can r e s o r t t o t h e

i n t e r p o l a t i o n technique, f o r which one has t o know a l r e a d y t h e t ype o r weak

t ype of t h e o p e r a t o r i n some space. T h i s i s t h e case, f o r example, o f t h e

Hardy -L i t t l ewood maximal o p e r a t o r f o r which one can o b t a i n d i r e c t l y t h e

weak t y p e ( l , l ) , by means o f a c o v e r i n g lemma, and t h e t y p e (-,m) which i s

t r i v i a l . However i n some o t h e r cases t h e weak t y p e ( 1 , l ) o r t h e t ype (my-)

a r e n o t a v a i l a b l e and one has t o t ry t o show more o r l e s s d i r e c t l y t h e t ype

o f t h e opera to r . There a r e n o t many s tandard techniques a v a i l a b l e f o r t h i s

purpose.

theorem enable us t o t r e a t t h e t y p e (2,2) o f c o n v o l u t i o n o p e r a t o r s i f we

can e s t i m a t e t h e Lm-norm o f t h e F o u r i e r t r a n s f o r m o f t h e co r respond ing

k e r n e l s . T h i s i s t h e easy way presented i n S e c t i o n 1.

The use o f t h e F o u r i e r t ransform and t h e Parseva l -P lanchere l

I n o r d e r t o handle t h e L 2 - t h e o r y o f t h e H i l b e r t t r a n s f o r m and

t h e Calder6n-Zygmund opera to rs C o t l a r [1959] i n t r o d u c e d another d i f f e r e n t

method. T h i s i s presented i n S e c t i o n 2 . I t has been used l a t e r f o r many

d i f f e r e n t purposes.

The r o t a t i o n method o f CalderBn and Zygmund was i n t r o d u c e d by

them [1956 ] i n o r d e r t o t r e a t t h e i r s i n g u l a r i n t e g r a l ope ra to rs . I t can

a l s o be used f o r h a n d l i n g c e r t a i n problems i n app rox ima t ion t h e o r y and i n

d i f f e r e n t i a t i o n of i n t e g r a l s . T h i s method i s presented i n S e c t i o n 3.

5.1. FOURIER TRANSFORM

The easy s tandard t o o l i s t h e Parseval -P lanchere l theorem t h a t

can be used as i n t h e f o l l o w i n g theorem.

91

92 5. THE TYPE (2 ,2 )

5.1.1. THEOREM . lel { k . } be a oequence a Q 6unc.tLtiu~n i n J

c > 0 nuch t h a t dvn each j

F m f E L2(Rn) C e l K j f ( x ) = k * f ( x ) . bourne t h a t t h m e j

Phavd. By t h e Parseva l - P l a n c h e r e l theorem

5 .2 . COTLAR'S LEMMA.

The p r e s e n t a t i o n o f t h e lemma f o l l o w s t h a t o f F e f f e r m a n [19741.

5.2.1. THEOREM . 1eL H be a H i L b m A npacc? and Tl,T2, ..., TN

a 6ivLite he.quencc? 06 vpehaLvhA Qhvm H tv H . Adhume t haL c : 4 +[O,M)

LA u bunct ivn ouch t h a t

d e n a k a the. a d j a i n t 0 6

co

1

Ti ,

( c ( k ) ) ' " c A < w and XhCLt, 45 T t k=-m

Then

P4ovQ. We can w r i t e

5.2. COTLAR’S LEMMA 93

And r e p e a t i n g t h e p r o c e s s

Thus

N ?k N

1 I / Ti TF Ti TC ... Ti T? II

2 k - 1 2k 1 2 3 4 II c T i II = c ,

i ,i2,. . . ,i k=l

Each t e r m of t h e l a s t sum i s m a j o r i z e d by

and a l s o by

P = /ITil T*i211 ... 1 1 Ti T4 1 1 G c ( i l - i 2 ) . .. ~ ( i ~ ~ - ~ - i2k) 2 k - 1 2k

T h e r e f o r e i t i s a l s o m a j o r i z e d by t h e i r g e o m e t r i c means. Thus

Hence

94 5. THE TYPE (2,2)

N and making k -f m , we g e t 1 1 Ti [ [ c A.

1

With t h i s theorem one can e a s i l y o b t a i n t h e u n i f o r m s t r o n g t y p e

o f t h e t r u n c a t e d H i l b e r t t r a n s f o r m and o f t h e Calder6n-Zygmund s i n g u l a r

i n t e g r a l ope ra to rs .Th is was t h e f i r s t a p p l i c a t i o n o f t h e lemma which ap-

pears i n C l o t a r [1959].

For j = 0, 21, k2, ... we d e f i n e

and f o r f F: L 2 ( R 1 ) , T . f ( x ) = h j * f ( x ) I

hi(x) = hi(-x),

J - We have, w i t h

- - T? f (x ) = hi * f ( x ) , T? T . f = (hi * h . ) * f J J

Observe now t h a t i h i ( x )dx = 0 and so

5.2. COTLAR'S LEMMA 95

c Ih i ( t ) l I h j ( x - t ) - h j ( x ) ) dx d t t X

But h . ( x ) = 2 - J h o ( 2 - j x ) and so J

t 1 Now, i f I- I 2 T , then we have 2J

t 1 1 I f 1-1 < T , and Jo = Cx : G 1x1 6 13 2J

t A = { x : x E J O , X - - E J ~ } 2J

2 j B = C X : X B J O , X - - B J o }

C = I x : e i t h e r x e Jo o r t e JoI

then we can argue as follows

If x e B then

If x e A then

If x E C t hen

A lso we have [ A ] 6 l J o l =

t - 7) - h o ( x ) l L 2 2J

1 , ICI c 8 1-1 t . T h e r e f o r e 2J

t t / ( h o ( x - - - h o ( x ) ( dx c 20 1-1 2J 2J

96 5. THE TYPE ( 2 , 2 )

and so

I n t h i s way we g e t

and so t h e hypotheses o f C o t l a r ' s lemma a r e s a t i s f i e d .

f o l l o w i n g theorem.

We thus o b t a i n t h e

The f a c t t h a t we have chosen a d i s c r e t e t r u n c a t i o n i s r a t h e r

i r r e l e v a n t . One o b t a i n s e a s i l y t h e same r e s u l t f o r

5.3. THE METHOD OF ROTATION.

I n some cases, t h e s tudy o f t h e t y p e o f a c o n v o l u t i o n o p e r a t o r

i n Rn can be reduced t o t h a t o f some known one-dimensional o p e r a t o r a&

s o c i a t e d t o i t i n a n a t u r a l way, by i n t r o d u c i n g p o l a r coo rd ina tes . We j u s t

" r o t a t e " t h e one-dimensional t ype i n e q u a l i t y we a l r e a d y have. T h i s i s t h e

b a s i c i d e a o f t h e r o t a t i o n method i n t r o d u c e d by Calder6n and Zygrnund[19561.

We desc r ibe f i r s t t h e genera l frame o f t h e method and then a p p l y i t t o

5.3. THE METHOD OF ROTATION

Some p a r t i c u l a r cases.

97

Assume we want t o s tudy t h e t ype o f some o p e r a t o r s o f t h e f o r m

K . f ( x ) = k j * f ( x ) J

- L e t c = I ~ E R ~ : 171 = 11 , y = ry , o < r < m y ye c , i .e .

- f o r y e Rn , y # 0 , 7 w i l l denote t h e p r o y e c t i o n o f y ove r C ( r , y

a r e t h e p o l a r coo rd ina tes of y ) . We can then w r i t e

Assume t h a t we can w r i t e k j ( r 7 ) = g ( y ) h j ( r ) .

Then

K . f J

- m

For a f i x e d 7 e C , l e t K: f ( x ) = \ 0

x - r a r " ' d r dy

h . ( r ) f ( x - r a r n - ' d r . J

If Y i s t h e hyperp lane through t h e o r i g i n o r thogona l t o 7 , we have, f o r x a R n , x = z + sy , w i t h s e R , z E Y , and so

- - K: f ( x ) = K: f ( z + sy) = h j ( r ) f ( z + ( s - r ) y ) rn- l d r

0 -

I f f o r f i x e d

w r i t e , f o r s E R

e C , z e Y , we s e t f ( z + t y ) = f;(t) t hen we can

- m - ( K j Y Y f ) z ( s ) = 1 h j ( r ) rn- l f; ( s - r ) d r E H . fy ( s )

J Z 0

where H j i s t h e o p e r a t o r d e f i n e d by t h e above exp ress ion .

I t can happen, as we s h a l l see i n t h e examples, t h a t

known f a m i l y o f onedimensional ope ra to rs , and t h a t we a l r e a d y have f o r

some p, 1 G p < my i f v a LP(R')

H j i s a

98 5. THE TYPE (2,2)

w i t h C = C ( p ) independent o f v and j .

Then we can w r i t e , w i t h z v a r y i n g over Y

- 1 K i f ( x ) l p dx = I,,, I ( ( f); ( s ) l p ds dz =

-m

There fo re

-

II K; f l l p 6 c l l f l l p'

Now, by Minkowsk i ' s i n t e g r a l i n e q u a l i t y ,

So, i f I g ( y ) ( d y < m , we g e t t h a t t h e o p e r a t o r s K . a r e u n i f o r m l y o f J c t y p e (P,P).

The i d e a o f t h e method i s c l e a r . Before pass ing t o t h e some

conc re te a p p l i c a t i o n s l e t us make an i m p o r t a n t remark. Suppose t h a t f o r

H . we j u s t know t h a t they a r e u n i f o r m l y o f weak t y p e (p,p). The method J does n o t w o r k . f o r two main reasons. f i r s t we do n o t know how t o i n t e g r a t e

a weak - type i n e q u a l i t y and so we cannot a r r i v e t o t h e weak t y p e i n e q u a l i t y

f o r KY . Second, even i f we had t h e u n i f o r m weak t y p e i n e q u a l i t y f o r - J

K$ , we should s t i l l make t h e l a s t s t e p work, i . e . t h e use o f M inkowsk i ' s

i n t e g r a l i n e q u a l i t y , and t h i s does n o t seem easy f o r a weak t ype inequal -

i t y . Here t h e Kolmogorov c o n d i t i o n may h e l p .

5.3. THE METHOD OF ROTATION 99

As a straightforward application of the rotation method let us

first prove an easy version of the Lebesgue differentiation theorem in Rn without any covering lemma.

inequality (p,p) , 1 < p < , for the one-dimens onal Hardy-Littlewood maximal operator, we obtain the strong type inequa ity (p,p) for the n- dimensional Hardy-Littlewood maximal operator over Euclidean balls. For

f E LP(R~), we set

Assuming that we already know the strong type

For y 8 C fixed , we set - My f(x) = sup 1' If(x-y)l on-' dp

r>O r 0

Proceeding as before,

Where MI is the (onesided) onedimensional Hardy-Littlewood maximal

operator and

- f p ) f(z+sY) = f(x)

Therefore

W jRn lMy f(x)lp dx = I(My f)! (s)Ip ds dz G i,,, JL

100 5 . THE TYPE (2,2)

So we a r r i v e a t / I M n f l / c 'd Ilfll as be fo re , u s i n g P

M i nkows k i ' s i n t e g r a l i nequal i ty . We s h a l l see l a t e r , when d e a l i n g w i t h approx imat ions o f t h e

i d e n t i t y i n Chapter 10 , some more examples o f t h e same t ype . Now we

app ly the method t o t h e s tudy o f a p a r t i c u l a r case o f t h e Calderdn - Zygmund s i n g u l a r i n t e g r a l ope ra to rs (odd k e r n e l s ) .

L e t k E Limn - { O l ) be a k e r n e l which i s

( a ) homogeneous o f degree -n , i . e . f o r each 1 > 0, x e R n -{O},

k(Xx) = X- " k ( x ) ,

(b ) i n t e g r a b l e ove r 1 and w i t h mean v a l u e zero ove r 1 , i . e .

J c ' ( c ) odd , i . e . k ( - x ) = - k ( x )

For f e Lp(Rn) , 1 < p < a, and 0 < E < n , we d e f i n e

f ( x ) = k k f ( x ) , where KE ,rl E ,rl

k ( x ) , i f E 1x1 6 rl

0 o t h e r w i s e

kE,n(x) =

Tha t i s ,

f kc ,n

We t ry t o prove, assuming

x ) = k

t h a t we a l r e a d y know t h a t t h e t r u n c a t e d H i l b e r t

transforms a r e u n i f o r m l y of s t r o n g t y p e (p,p) , 1 < p < a

above t runca ted Calderdn-Zygmund opera to rs K E Y n a r e u n i f o r m l y o f s t r o n g

t y p e (P,P).

, t h a t t h e

To do t h i s we w r i t e u s i n g ( a ) and ( c ) ,

5.3. THE METHOD OF ROTATION 101

Now f o r a f i x e d 7 E C we w r i t e

I f , as b e f o r e , we s e t f o r x e R n , x = z + sy , z a Y , s B W , we g e t

- f ( z + ( s - p ) F ) - f ( z + ( s + p ) j q dp =

P K;,' = f ( x ) =

~

where we w r i t e f:(t) = f ( z + t y ) and f o r g e Lp(R' ) ,

i s t h e t runca ted Hi 1 b e r t t rans fo rm.

Since we assume t h a t we know I \ H E y n g l l 6 c I l g l I , w i t h

c independent o f E, q , g, we can proceed as b e f o r e and g e t

CHAPTER 6

COVERINGS, THE HARDY - LITTLEWOOD MAXIMAL OPERATOR AND DIFFERENTIATION

SOME GENERAL THEOREMS

Many problems i n r e a l a n a l y s i s can be reduced t o a geomet r i c

s tudy o f t h e c o v e r i n g p r o p e r t i e s o f c e r t a i n f a m i l i e s o f s e t s . We have

a l r e a d y seen how t h e Lebesgue d i f f e r e n t i a t i o n theorem i s an easy conse-

quence o f t h e c o v e r i n g theorem o f B e s i c o v i t c h .

t o o l s as t h e Whitney cove r ing lemma and t h e Calder6n-Zygmund decomposi-

t i o n lemma have been e a s i l y deduced f rom t h e fundamental c o v e r i n g prop-

e r t y o f t h e dyad ic cubes.

A l so such fundamental

The c o v e r i n g p r o p e r t i e s o f a f a m i l y o f s e t s a r e s t r o n g l y r e l a t e d

t o t h e p r o p e r t i e s o f t h e corresponding Hardy-Li t t l e w o o d maximal o p e r a t o r .

A h i n t o f t h i s connec t ion i s t h e theorem o f Co'rdoba we have seen i n Chap-

t e r 3 when d e a l i n g w i t h t h e technique o f l i n e a r i z a t i o n . As we s h a l l show

one can go f u r t h e r i n t h i s d i r e c t i o n .

The Hardy-Li t t l e w o o d maximal o p e r a t o r i s t h e maximal o p e r a t o r

r e l a t i v e t o t h e f a m i l y o f ope ra to rs one s t u d i e s i n t h e problem o f d i f -

f e r e n t i a t i o n o f i n t e g r a l s . I t s s tudy , o f course, f u r n i s h e s v a l u a b l e

r e s u l t s i n t h e s o l u t i o n o f such problem, b u t i t i s a l s o t r u e t h a t , i n

t h e o p p o s i t e d i r e c t i o n , any i n f o r m a t i o n one can a c q u i r e r e l a t i v e t o

t h e d i f f e r e n t i a t i o n p r o p e r t i e s o f a d i f f e r e n t i a t i o n b a s i s p rov ides l i g h t

f o r t h e s tudy o f t h e maximal ope ra to r which can be used i n many o t h e r

f i e l d s , such as t h a t o f t h e s i n g u l a r i n t e g r a l ope ra to rs , i n which t h e

Hardy -L i t t l ewood o p e r a t o r p l a y s a v e r y i m p o r t a n t r o l e .

I n t h i s Chapter we s h a l l t r y t o b u i l d up a genera l frame i n

which t o p resen t t h e i n t e r a c t i o n o f these t h r e e elements so s t r o n g l y

i n te rconnec ted . We p resen t o n l y t h e most i m p o r t a n t genera l r e s u l t s , and

r e f e r t h e reader t o t h e monograph by Guzmin [1975 ] f o r a more s p e c i a l -

i z e d knowledge o f t h e f i e l d . We s h a l l use t h i s o p p o r t u n i t y t o complete

103

104 6. C O V E R I N G S y HARDY-LITTLEWOOD AND DIFFERENTIATION

t h a t monograph by p r e s e n t i n g t h e most i m p o r t a n t r e c e n t r e s u l t s i n t h i s

f i e l d t h a t has been g r e a t l y expanded by t h e work o f many people, espe-

c i a l l y those work ing i n t h e f i e l d o f F o u r i e r A n a l y s i s .

6.1. SOME NOTATION

I n genera l a didde,tevLticLtio~ b a d i n Rn w i l l be a c o l l e c -

t i o n (I3 = I! @ ( x ) o f bounded measurable s e t s w i t h p o s i t i v e measure x eRn

such t h a t f o r each x e R " t h e r e i s a s u b f a m i l y @ ( x ) o f s e t s o f iB so t h a t each B e B ( x ) con ta ins x and i n O ( x ) t h e r e a r e s e t s o f

a r b i t r a r i l y smal l d iameter .

As an example, (B can be t h e c o l l e c t i o n of a l l open b a l l s i n

Rn. For each x e R n & ( x ) i s t h e subfami ly o f a l l b a l l s c o n t a i n i n g

x . Another example : For each x e R n @(x) w i l l be t h e f a m i l y o f

a l l t h e open b a l l s cen te red a t x and B = \I @(x) . X E R"

A Bu.bemann-Fe,Ue,t didde,tnevl.tiation b a d 8 i s a d i f f e r e n t i a -

t i o n bas i s such t h a t each B e B i s open and i f x e B e & , then

B e B ( x ) .

I f f o r each x E A c Rn we a r e g i ven a c o l l e c t i o n o f bounded

measurable s e t s w i t h p o s i t i v e measure

L m e u J w a o d muximd ope,khatatr ( r e l a t i v e t o 5" = (I y ( x ) ) i n t h e

y ( x ) we can d e f i n e t h e ffahdy-

xsA

f o l 1 owi ng way :

I f f E

M f ( x ) =

0 i f x f 11 S S€ Y

For a Busemann-Feller b a s i s ( f o r s h o r t , a 8-F b a s i s ) i t i s

obvious t h a t { M f > A ] i s an open subse t o f Rn and so M f i s meas-

6.2. COVERINGS, IMPLY TYPE AND DIFFERENTIATION

u rab l e.

105

I f we a r e g i v e n a d i f f e r e n t i a t i o n b a s i s i n R n we can

d e f i n e , f o r f 6 Lloc (Rn) ,

D ( f x ) = l i m sup & I f (y)dy ( t h e uppeh detLivative o f f a t x ) - ' b ( B ) + O B

1 D( f , x ) = l i m i n f i f ( y ) d y ( t h e L o w t den iwat ive o f f a t x ) i S(B)+O B -

B&(x)

I f D ( f , x ) = ( f , x ) we say t h a t t h e d e r i v a t i v e D( if.x) - I a t x e x i s t s and i f t h i s happens a t a lmos t each x e R n and

D( f , x ) = f ( x ) a.e. we say t h a t dS d L d d W e W a f. I f t h i s i s

t r u e i f o r each f i n a c l a s s o f f u n c t i o n s X we say, a b b r e v i a t e l y , t h a t

P dcd,4jWevLtiatQc\ x .

I n genera l we s h a l l s t a t e most o f t h e r e s u l t s i n d i f f e r e n t i a -

t i o n f o r a B-F b a s i s . Fo r such a b a s i s M f ( * ) , D( f, .) ,D( f,.)

a r e e a s i l y shown t o be measurable. Many o f t h e r e s u l t s a r e e x t e n s i b l e

t o o t h e r types o f d i f f e r e n t i a t i o n b a s i s w i t h o u t d i f f i c u l t y . However

one shou ld n o t t h i n k t h a t t h e f a c t t h a t i n a genera l d i f f e r e n t i a t i o n

b a s i s each s e t B 6 @ ( x ) i s i n some way "anchored" t o t h e p o i n t x

i s devo id o f importance.

-I I

6.2. C O V E R I N G LEMMAS IMPLY WEAK PROPERTIES OF THE MAXIMAL OPERATOR AND DIFFERENTIATION.

We have a l r e a d y seen how t h e B e s i c o v i t c h c o v e r i n g p r o p e r t y

o f t h e maximal o p e r a t o r . leads i n an easy way t o t h e weak t y p e (1 , l )

106 6 . COVERINGS, HARDY-LITTLEWOOD AND DIFFERENTIATION

We sha l l f i r s t s t a t e here a couple of easy general r e s u l t s i n th i s direc t ion . Then we sha l l see how the Besicovitch theorem gives us a very general and useful form of the Vi ta l i covering lemma and f i n a l l y we sha l l see how t o use these lemmas in order t o obtain r e s u l t s on d i f f e r - en t i a t i on .

p h o p e h t g : 7 6 d o t each x e A then we can chaohe 6hom ( S ( X )

\ A ( , 6 C l U S k l and c X (X 'k

c R~ we m e given S ( X

a oequence i s k ) huch XEA

h H at each x e R~ . Whehe c

and H depend o n l y on (9 , nat an A a h ( S ( x ) ) ~ ~ ~ .

Then t h e maximal VpPhCLtoh h&%t.ive t u 9 u a weak .type (1 , l ) .

phoo6. Let, fo r A > 0 a A X = tMf (x ) > A ) . I f x e AX we

I f ( > A . We apply P7i-T I S ( x ) choose S(x) E q ( x ) such t h a t

the covering property t o A A and ( S ( X ) ) x e ~ A and obtain CS,) . We can then wr i te

and obtain the r e s u l t

The same method i s applicable to the weak type ( p , p ) , 1 < p < m.

6.2. COVERINGS, IMPLY TYPE AND DIFFERENTIATION 107

Then ,the maximal ape.hatoh ~ . & ~ 2 v e t o 0 6 weak ,type ( q , q ) , P

q = p - l *

Phoaij. L e t f E Lq , X > 0 and AX = ( M f > A }

I f 1 > and T d T j S(x )

For x 6 AX we choose S ( x ) such t h a t

app ly t h e cove r ing p r o p e r t y , o b t a i n i n g { sk } . We can w r i t e

and t h i s proves t h e theorem.

The c l a s s i c a l V i t a l i lemma i s an easy consequence o f t h e

B e s i c o v i t c h p r o p e r t y t h a t has been proved i n Chapter 3

can s t a t e a more genera l r e s u l t v a l i d f o r an a r b i t r a r y measure w i t h t h e

same e f f o r t .

f o r b a l l s . We

6.2.3. THEOREM . L e L y = u ( x ) be a nybXeJn 0 6 x sRn

camp& 0e. t~ ulith poniaXve m w m e nuch t h a t doh each x and S E v ( x ) , we have x E S . Annume t h d t each (/’ ( x ) containh nets 06 ahb&ahiey

nm& diameLteh.

(BesicovLtch) covehing p h a p e h t y : 16 doh each x E A c Rn we ahe given S(x) e Y ( x ) then we can choose &&om ( S ( X ) ) ~ € ~ a nequence I S k )

nuch Ahat:

L a ~n nuppone a h a t h a t y ve14ie.h t h e ~o.Uawing

108 6 . C O V E R I N G S y HARDY-LITTLEWOOD AND DIFFERENTIATION

Ptiuol;. I t i s easy t o see t h a t t h e r e i s no l o s s o f g e n e r a l i t y

i f 'we assume 0 < pe(E) < m . For each x a E we choose an a r b i t r a r y

S(x) E { Q k ( x ) } c v(x) and a p l l y t h e B e s i c o v i t c h cove r ing p r o p e r t y

o f 9 o b t a i n i n g I S k } s a t i s f y i n g ( i ) and ( i i ) . For one o f t h e se,-

quences { S i 3 , ... , { S f } ( L e t us assume i t i s { S t } ) , we have

I n f a c t , o t h e r w i s e

1 e Pe(E) C p('sk) h 1 U( I1 Sjk C Ue(E)

j=1 k

and t h i s i s a c o n t r a d i c t i o n . L e t us t a k e a p-measurable s e t P 3 E ,

p(P) = pe (E) and a f i n i t e subsequence o f { S c } , say IS: } Y h l

k= 1

such

P

t h a t

The f i r s t hl elements o f t h e sequence {Rk) t h a t we a r e l o o k i n g f o r ,

w i l l be { S 3 , S l y ... S ; l l . So we g e t

h, Fo r each x e E - !I Rk = E l we t a k e S(x) a { Q k ( x ) I c y ( x )

such t h a t S (x ) n (I Rk = 0 and we proceed w i t h El as we have done

w i t h E o b t a i n i n g now { R k }

h i 1

such t h a t h2

h,+l

I n t h i s way we o b t a i n {Rkl and ue(E - URk) = 0

I t i s i n t e r e s t i n g t o observe t h e f o l l o w i n g r e l a t i o n s h i p bet -

ween t h e V i t a l i p r o p e r t y and t h e B e s i c o v i t c h p r o p e r t y o f a system o f s e t s .

For many purposes i n A n a l y s i s , i n p a r t i c u l a r i n d i f f e r e n t i a t i o n

theo ry , t h e f o l l o w i n g p r o p e r t y o f a system 9 = 11 q ( x ) of compact x eRn

measurable s e t s i n Rn i s good enough : 7Q don u c h x E A we me given

S(x) E y(x) , t h e n One can chaohe dtvm ( S ( X ) ) ~ ~ ~ u hequence IS,} nuch

< 0 . (Weak B e s i c o v i t c h p r o p e r t y ) . t h a t A c USk and 1 xsk

O f course, i f Sp 6.2.3. , then i t s a t i s f i e s t h

B e s i c o v i t c h p r o p e r t y i s s t r

s a t i s f i e s t h e B e s i c o v i t c h p r o p e r t y o f Theorem

s one, s i n c e 1 xsk < 0 . The weak

c t l y weaker than t h a t o f Theorem6.2.3. The

f o l l o w i n g example i s due t o B.Rubio (unpubl ished) .

6.2.4. THEOREM. T h a e LA a hynteni y = LI 0 ( x ) aQ com - x ERn

p a c t he,th h a t i n d y i n g t h e weak BenicavLtch p'rop&y WLthoLLt h a t i n d y i n g

t h e hi7~VKge.R ane 06 Thevheni 6.2.3.

PXaad. We take 1 = b l > bp > b3 > .. . , b k + 0 9

2 = B1 > B2 > . .. , Bk -f 0 and f o r each x E R2 and k = 1,2 ,... we

cons ide r t h e symmetri,c c losed cross Q k ( x ) cen te red a t x i n d i c a t e d i n

the p i c t u r e

F i g u r e 6.2.1.

y = U ,(J(x) s a t i s - x ER k = 1 , 2 , . , . , The system y ( x ) = ( Q k ( x ) )

f i e s t h e weak B e s i c o v i t c h p r o p e r t y . L e t Q be any square o f d iamete r 2,

A c Q and f o r each g i ven .

We choose as S1 one of t h e s e t s Q ( x ) , x 6 A w i t h min imal k ( x ) .

x 6 A - S1 w i t h minimal k ( x ) , Then as SP one of t he s e t s Q

and so on. I f t h e process s tops, i t i s because A - USk = 0 . I f no t ,

n e c e s s a r i l y S(S,) + 0 , I n f a c t o t h e r w i s e , i f b(Sk) > c1 > 0 , s i n c e the t h i r d s o f t h e c u b i c i n t e r v a l s a t t h e c e n t e r s o f t h e crosses

sk i n s i d e a bounded s e t and t h i s i s imposs ib le . But i f

A - USk = 0 . Otherwise i f x 6 A - usk , we have ove r looked

Q,(,)(x) i n o u r s e l e c t i o n process.

process o f t h e s e t s Sk

2 x 4 s e t s sk . I t s u f f i c e s t o c o n s i d e r how many crosses c o n t a i n i n g z

have t h e i r cen te rs i n t h e f i r s t o f t h e c losed quadrants determined by z .

They a r e a t most 2.

x e A assume t h e r e i s one S ( x ) = Q,(,)(x)

k ( x )

k ( x ) ’

a r e d i s j o i n t , we would have i n f i n i t e l y many d i s j o i n t congruent cubes

6(Sk) + 0 , then

Because o f t h e fo rm o f t h e s e l e c t i o n

i t i s e a s i l y seen t h a t no p o i n t i s i n more than

Therefore 1 xsk c 8 .

Now i f t h e system would s a t i s f y t h e s t r o n g B e s i c o v i t c h p r o p e r t y

i t would s a t i s f y t h e V i t a l i p rope r t y , acco rd ing t h Theorem 6.2.3. But

one can choose bk , Bk so t h a t t h i s i s n o t t r u e .

I n f a c t l e t Q be t h e must cube. D i v i d e t h e cube d y a d i c a l l y

i n f o u r p a r t s and f o r each p o i n t i n each one o f them t a k e a c ross cen-

t e r e d a t t he p o i n t such t h a t t h e l e n g t h , B1 , o f t h e arm i s t w i c e as l a r g e

as t h e s i d e l e n g t h o f each o f t h e dyad ic cubes and t h e w i d t h o f t h e arm

o f t h e cross, bl, i s smal l enough so t h a t t h e area o f t h e c ross i s l e s s

than --< o f t h a t o f t h e d y a d i c cube o f t h i s d i v i s i o n . We d i v i d e dyad-

i c a i l y again and t a k e crosses centered a t each p o i n t i n t h e same way so

t h a t t h e area o f t h e c ross i s l e s s than

cube o f t h i s d i v i s i o n . It i s easy t o see t h a t i f we s e l e c t any system od

d i s j o i n t crosses we can t a k e a t most one of t h e crosses o f .each s tage and

so they can n o t cover a lmos t a l l t h e cube Q .

1 2

4x2 2 o f t h a t o f t h e d y a d i c

Here we have a l s o c o n s t r u c t e d a b a s i s t h a t d i f f e r e n t i a l s L’

b u t does n o t s a t i s f y t h e V i t a l i lemma. As we s h a l l see l a t e r t h i s i s

imposs ib le f o r a B-F b a s i s t h a t i s i n v a r i a n t by homothecies.


The i n e q u a l i t i e s f o r t h e maximal o p e r a t o r o f t h e t y p e o b t a i n e d

i n 6.2.1. l e a d t o a d i f f e r e n t i a t i o n theorem f o r L1, as we have shown i n

Chapter 3. The one i n 6.2.2. l eads i n t h e same way t o d i f f e r e n t i a t i o n

o f Lq .

Now we s h a l l show how f rom t h e V i t a l i lemma o f 6.2.3. one can

a l s o deduce t h e d i f f e r e n t i a t i o n p r o p e r t y o f a b a s i s .

6.2.5. THEOREM . L e R 1-1 be a n e t dunction dedined on diniAe

u n i u ~ n ad &abed cubic in tehv& a6 Rn . Ab~ume t h a t 1-1 LA nvnnegative, manoiane, ~ i n i t e L q addi t ive and dini te an each cube. Then at almost evetry

( i n t h e Lebe~gue A C U ~ ) p a i n t vne h a , dah each Aeyuence { Q k ( x ) } ad &ohEd cubic in tehv& c e n t a e d at x and cant'iaoting t o x , th lLt t h e L h L t

x 8 Rn

exddh, i~ dini te and i~ independent 0 6 t h e neyuence {Q,fx)) .

P m o d . I n o r d e r t o prove t h i s r e s u l t , d e f i n e f i r s t

We t ry show t h a t

t e r v a l Q and a c o n s t a n t M > 0 . For each x E A, 0 Q we have

I Q,(x) 1 c o n t r a c t i n g t o x such t h a t

\A,] = 0 . L e t us t a k e an a r b i t r a r y c l o s e d c u b i c i n - 0

0

and Qk(x ) C Q . We app ly Theorem 6.2.3. o b t a i n i n g a d i s j o i n t se - quence IS,} f r om such cubes so t h a t

n

112 6 . COVERINGSy HARDY-LITTLEWOOD AND DIFFERENTIATION

So we g e t

0 Since M i s a r b i t r a r y and u ( Q ) < m we o b t a i n IA,n Q ) = 0

and so / A m [ = 0.

L e t us now take an a r b i t r a r y c losed c u b i c i n t e r v a l Q and de-

f i n e , f o r r z s > 0

We t r y t o prove t h a t \ A r s ( = 0 . L e t E > 0 a r b i t r a r y and G an open

s e t c o n t a i n i n g A,, such t h a t ( G I 6 ( A r s I e + E . We a p p l y V i t a l i ' s

theorem t o Ars w i t h t h e cubes Q{(x) , t a k i n g o n l y those con ta ined i n

G and we o b t a i n a d i s j o i n t sequence { S t } o f such cubes so t h a t

l A r s - II S g [ = 0 . We c l e a r l y have k

Observe t h a t , i f

0

C = A,, II ( II S i k

we have 1C/ , = l A r s l e . For each x 8 C t h e r e i s an S* such t h a t

x 6 S* and so , s i n c e x E Ars t h e r e i s a l s o a sequence Q k ( x ) + x

such t h a t Q,(x) C S* and

J 0

J 0

J

U(Qk(x ) )

14krx)(> r .

We now apply t h e V i t a l i theorem aga in t o C w i t h ' t h e s e cubes and so

6 . 2 . COVERINGS, IMPLY TYPE AND DIFFERENTIATION 113

o b t a i n a d i s j o i n t sequence { S , } such t h a t each Sk i s i n some S* and

1C - ( J S,] J

= 0 . Thus we have, t a k i n g i n t o account t h e above i n e q u a l i -

t i e s

I A r s

< r S

Thus

Since E i s a r b i t r a r i l y smal l we o b t a i n [ A r s [ = 0 . Wi th t h i s one eas-

i l y concludes t h e p r o o f o f t h e theorem.

We o b t a i n i n p a r t i c u l a r t h e f o l l o w i n g . I f P i s an a r b i t r a r y

p l ( Q ) = \ Q [I P i e , s e t o f R" we o b t a i n t h a t t h e l i m i t

and we s e t f o r each closed c u b i c i n t e r v a l

e x i s t s and i s f i n i t e a t a lmost eve ry x .

We can a l s o t a k e u,(Q) = / Q 0 P I i , i . e . t h e i n t e r i o r meas

ure o f Q 0 P , and so

e x i s t s and i s f i n i t e a t a lmost every X .

I f u ( Q ) = 1, f w i t h f nonnegat ive and l o c a l l y i n t e g r a b l e ,

we g e t t h a t


e x i s t s , every x E R",

i s f i n i t e and independent o f t h e p a r t i c u l a r {Q,(x)I a t a lmost

6.3. FROM THE MAXIMAL OPERATOR TO COVERING PROPERTIES.

As an example o f t h e use o f t h e method o f l i n e a r i z a t i o n we have

seen i n Chapter 3 how a s t r o n g t y p e (p,p) p r o p e r t y f o r t h e maximal

o p e r a t o r i m p l i e s a q - t ype cove r ing p r o p e r t y . However one more o f t e n knows

t h a t t h e maximal ope ra to r s a t i s f i e s a weak t ype p r o p e r t y , and t h e theorem

can be extended t o t h i s s i t u a t i o n . T h i s r e s u l t be longsindependent ly t o

C.Hayes [1976]

o f Hayes and ou r v e r s i o n i s a l i t t l e more genera l t han t h e one h e presents.

and A.CBrdoba [:1976] . The p r o o f we p resen t f o l l o w s t h a t

6.3.1. THEOREM . L e t ( Q , F , p ) be a meuhme npace and

@. = ( R a ) a E A a cvUecaXvn v6 meanaabke. AU~APLA 0 6 R Mlith 6 i n i t e m e a n - me. We cvnniden t h e ( H c v r d y - L U e w v o d ) maximal opmahh M [belated

tv A ) i n t h e 6vUvwing dvhm. 7 6 f 6 L(Q) and x E 0 , we n e t

LeA + : [0,w) -+ [O,w) be an inmeaning 6unctivn wLth

@(O) = 0 and U A A W ~ ~ t h a t doh A > 0 and doh each f E L(Q) we have

h h w n e t h a t I) : [0,w) -+ COY-) a nondechming ~ u n u X v n duch t h a t dotl each p > 0 t h e m e . x A a 2 k (p ) > 0 auch t h a t , 6vh each

6.3. FROM THE MAXIMAL OPERATOR TO COVERINGS 115

u > 1 , Lhre hccwe

I) R k 1

N

.__ Remwlh. I n t e r e s t i n g f u n c t i o n s assoc ia ted as t h e f u n c t i o n s 4 and $ of t h e s tatement o f t h e theorem can be e a s i l y found. Examples:

, c > o

Ptloo6. We can assume t h a t has a f i n i t e number o f (R&B s e t s . Otherwise we take a f i n i t e s u b c o l l e c t i o n such t h a t t h e measure o f

i t s un ion i s s u f f i c i e n t l y c l o s e t o t h a t o f II R a . BEB

The cho ise o f t h e s e t s R k i s made as f o l l o w s . Take R 1 a r b i -

t r a r i l y . Assume R1,R2, ..., Rm have been chosen so t h a t

We f i x T- such t h a t 0 < rl < 1 , c E , and ask 1-T-

ourse lves whether t h e r e i s among t h e s e t s o f (R ) which have n o t a P E B

116 6. COVERINGS, H A R D Y - L I T T L E W O O D AND D I F F E R E N T I A T I O N

been chosen a s e t W such t h a t we have s imu l taneous ly

If t h e r e a r e such sets , we t a k e one o f them as o u r Rmtl . If

n o t , we a r e f i n i s h e d w i t h t h e s e l e c t i o n process.

We have

N So we s t o p i n a f i n i t e number o f s t e p t s . L e t R1,RZ, ..., R

be the chosen se ts . for t h e se ts W i n (RB)BEB which have n o t been

chosen, we have a t l e a s t one o f t h e f o l l o w i n g i n e q u a l i t i e s

6.3. FROM THE MAXIMAL OPERATOR TO COVERINGS 117

Because of t h e hypo thes i s of t h e theorem on t h e maximal o p e r a t o r ,

we have t h a t t h e union o f a l l such se ts W v e r i f y i n g (1) has measure l e s s

than o r equal t o

IJ R k 1

IJ R k 1

Also t h e un ion o f t h e s e t s W v e r i f y i n g ( 2 ) has a measure l e s s

than o r equal t o

Hence R1, R 2 , ... , RN s a t i s f y ( a ) and ( b ) .

It i s an i n t e r e s t i n g open problem t o f i n d t h e e x a c t l i m i t s o f

t h i s t y p e o f theorem, i n t h e f o l l o w i n g sense. Assume t h a t one knows t h a t

t h e maximal o p e r a t o r s a t i s f i e s an i n e q u a l i t y o f t h e t y p e appear ing i n t h e

s tatement w i t h $ ( u ) = u . What i s t hen t h e b e s t c o v e r i n g p r o p e r t y one

can deduce f rom t h i s ? Can one t a k e $ (u ) = eu2 ? O r , assume t h a t f

$ ( u ) = u ( l + l o g u ) . Can one t a k e + (u ) = eu ?

118 6 . COVERINGS , HARDY-LITTELWOOD AND DIFFERENTIATION

6.4. DIFFERENTIATION A N D THE MAXIMAL OPERATOR.

When of a d i f f e ren t i a t ion bas is B one knows t h a t i f s a t i s f i e s L p ( l < p < m),

one can apply the general theorems of Chapter 1

a d i f f e ren t i a t ion property such as t h a t i t d i f f e r e n t i a t e s then, since i t i s c l ea r t h a t the corresponding maximal operator i s bounded a .e . fo r each and obtain weak type proper t ies for the maximal operator.

f E L p ,

However t h i s type of r e s u l t s can be obtained by d i r e c t methods t h a t a r e simpler by f a r . re la ted to individual d i f f e ren t i a t ion proper t ies t h a t a r e not covered by the abs t r ac t theorems. Busemann and Fe l le r r19341 and some o thers in Hayes and Pauc [1955] . We present here a sample of r e s u l t s of t h i s type. For more d e t a i l s and fu r the r information one can consult the monograph Guzmdn [1975] .

Moreover, by such methods one can ge t r e s u l t s

Some of the r e s u l t s we present o r ig ina t e in

A .Den&Ltq p m p e t L t i ~ . As a f i r s t r e s u l t we prove t h a t the d i f f e r e n t i a t i o n o f the

c h a r a c t e r i s t i c functions of measurable s e t s f a c t equivalent t o the apparently s t ronger property of d i f f e r e n t i a t i o n

( devlni2q p m p W y ) i s in

of L”(FP).

Phoad. Since the d i f f e ren t i a t ion of j f a t x i s a local property , i . e . depends only on the behavior of f in a neighborhood of x , we assume tha t f has compact support A . We a l so can assume without loosing genera l i ty t h a t f o r every x , 0 G f ( x ) c H < a. By Lusin’s theorem, given E > 0 , there exists a compact s e t K i n . A such t h a t ! A - K I c E and f i s continuous on K. Let f K = fXv ,

~ A - K = ~ x A - K - We f i r s t prove O( jf, ,x) = f K ( x ) a t almost every x a R’ .

In f a c t , assume R k E 6j ( x ) , R k + x as k -+ a . We can wr i t e

6.4. DIFFERENTIATION AND THE MAXIMAL OPERATOR 119

I f x 6 K , then f,&y) -f f K ( x ) as y -f x , y E K and so t h e above

exp ress ion tends t o zero as k -f m . I f x 6 K , then f k ( x ) = 0 and

the f i r s t member i s ma jo r i zed by

proper ty , tends t o ze ro f o r a lmos t a l l such p o i n t s x .

D( fk ,x) = f K ( x ) a lmos t everywhere i n Rn.

IRk (7 K I . T h i s , by t h e d e n s i t y

Hence T

With t h i s , f o r an a r b i t r a r y cx > 0 , we can s e t

i

p r o p e r t y . So D ( f , x ) = f ( x ) f o r a lmost each x B Rn . f,x) = f ( x ) a lmos t everywhere and so t h e theorem i s proved

- I The f o l l o w i n g c h a r a c t e r i z a t i o n o f a d e n s i t y b a s i s belongs t o

Busemann and F e l l e r c1934 3 .

120 6 . COVERINGS, HARDY-LITTLEWOO0 AND DIFFERENTIATION

( a ) LA a demLtity b u d , .

( b ) FUR. each A, 0 A < 1, S o t each nvndecAwbing bequence

{Ak} U S bvunded meanwrabLe be,tb buch t h a t I A k ( J. 0 and 604 each nvn - i n a e u i n g sequence Irk) 0 6 heat numbem buch thctt rk + 0 we have

whehe, ~ V R . each k, h, xh = xAh , and

Phvvd. That ( a ) i m p l i e s ( b ) i s easy. L e t 0 < A < 1 and

{Ak ) as i n (b ) . F i x an Ah. Fo r a lmost each x 6 Ah we have, i f

(a ) i s t r u e , D(\ xh,x) = 0 , and so , if k i s s u f f i c i e n t l y b i g ,

Mkxh(x) < A . Hence

CMkXk

by t h e d e f i n i t i o n o f

1 i m k-

Since

d e n s i t y

A h [ -t 0 we g e t (b ) .

We now prove t h a t n o t - ( a ) i m p l i e s n o t - (b ) . I f '@ i s n o t a

b a s i s , t h e r e i s a measurable s e t A , w i t h I A l > 0 , such t h a t

I n f a c t , assume t h a t f o r each measurable s e t 0 , we have, a t

a lmost each x 6 P , - D( j x p , x ) = 0 , i . e . ,x) = 0 = ~ ( ~ x p , x ) .


If we app ly t h i s t o t h e complement P ' o f P , i f I P ' I > 0 , we o b t a i n

t h a t a t a lmost each x 6 P ' , i . e . a t a lmost each x E P , we have

Observe now t h a t

and so we have , a t a lmost each x E P

and t h e r e f o r e '@ would be a d e n s i t y bas i s .

L e t us then t a k e A measurable, w i t h J A I > 0, such t h a t

There e x i s t s then a measurable s e t C, w i t h C c A ' , I C I > 0 , such

t h a t a t each x E C we have E( x A 7 x ) > A . L e t { G k I a sequence

o f non inc reas ing open s e t s such t h a t G k ~ C, J G k - C I -+ 0 and l e t

Ak = G k 0 A. C l e a r l y { A k } i s non inc reas ing and IA,:I + 0 s i n c e

A k C Gk - C . any non inc reas ing sequence o f r e a l numbers {r,]

such t h a t rk + 0. We s h a l l prove t h a t {Mkxk > XI 3 C f o r each k .

I Take

I n f a c t , l e t x E C and k be f i x e d . S ince E( x A 7 x ) > A t h e r e

i s a sequence { R h I C B ( x ) w i t h Rh -+ x such

Hence

122 6. COVERINGS, HARDY-LITTLEWOOD AND DIFFERENTIATION

and M k x ( x ) > A . Th is proves C CMkxk(x) > A 1 f o r each k and,

s i n c e I C I > 0 t h i s shows t h a t n o t - ( b ) holds, T h i s concludes t h e p r o o f

o f t he theorem.

Ak

When 8 i s a B - F b a s i s t h a t i s i n v a r i a n t by homothecies,i .e.

when 6 i s such t h a t i f R 6 63 t hen any s e t homothet ic t o R w i t h

any r a t i o and any cen te r o f hornothecy i s a l s o i n '@ (and so i n p a r t i c u l a r

any t r a n s l a t e d s e t o f R ) ,

p l e r form, as the f o l l o w i n g theorem proves.

then t h e p reced ing c r i t e r i o n r e c e i v e s a sim-

6.4.3. THEOREM. L e t @ be a B-F b a d RhaX d inwahiant by h v r n v f h e ~ ~ . Then .the ZWV 6 v U v d n g p 4 " r a p ~ ~ t - L ~ c ~ z e e q d v d e n t :

( a ) '63 & a det&.ty b t ~ . . ~ i ~ .

( b ) Fvlz each A, 0 < A < 1, t h m e exim.2 a ponLCLwe cvn~Aant c ( A ) < m ouch RthCLt ~ V R . each bounded rneauhabLe+ n e t A vne h a

Ptrvo6. That (b ) i m p l i e s ( a ) i s a s imp le consequence o f Theorem

6.4.2., s i n c e ( b ) i m p l i e s c o n d i t i o n ( b ) o f t h a t theorem.

I n o r d e r t o prove t h a t ( a ) i m p l i e s ( b ) we s h a l l use t h e f o l

1 owi ng 1 emma.

6.4.4. -- LEMMA. L e X G be any baunded vpen A ~ R -in R" and LeX

K a cLthjvi& sequence { K k } 0 6 .be& hamaRh&c t o K conta ined in

/ G - (I Kkl = 0 and S ( K k ) < r.

be any cvrnpaot A & w a h pvnLCLve rneame. L& r > 0 . Then t h e m A

P R V V ~ v 6 Rhe Lemma. The lemma i s an easy consequence p f t h e

f a c t t h e bas i s o f a l l s e t s homothet ic t o K s a t i s f i e s t h e theorem

of V i t a l i . However a s imp le p r o o f o f i t can be g i v e n i n the f o l l o w i n g way.

0

L e t A be a ha l f -open c u b i c i n t e r v a l such t h a t K t A and

l e t c t l A l = I K I w i t h €I < ct < 1. We p a r t i t i o n G i n t o a sequence

o f d i s j o i n t ha l f -open c u b i c i n t e r v a l s {Ah} o f d iameter l e s s than r. For

each Ah l e t Ph be t h e homothecy t h a t c a r r i e s A i n t o Ah and l e t

K* = PhK. We can keep a sequence {KiI:Il of these s e t s such t h a t , i f h

then G1 i s open and

by t a k i n g N1 s u f f i c i e n t l y b i g . We now s e t Kh = K* , h = l , Z , ..., Nl

and proceed w i t h G1 as we have done w i t h G, o b t a i n i n g now CKh)h:N +1 such t h a t

N h

And so on. So we o b t a i n t h e sequence CKhI s a t i s f y i n g t h e lemma.

We now c o n t i n u e w i t h t h e p r o o f o f t h e theorem. Assume t h a t (b )

does n o t ho ld . The t h e r e e x i s t s a p o s i t i v e number A > 0 such t h a t f o r

each i n t e g e r k > 0 t h e r e i s a bounded measurable s e t Ak such t h a t , if x k = XAk 3

t h e r e i s a l s o a p o s i t i v e number rk such t h a t

. L e t C k be a compact subset o f {M x > A} such Mrk k j where Mk means

t h a t [ C k [ > 2 [ A k [ . By t h e p rev ious lemma we can cove r t h e open

u n i t cube Q

o f s e t s homothet ic t o Ck such t h a t i f c1 i s t h e {cJ,} i=1,2,. . . r a t i o o f t h e homothecy Pkj c a r r y i n g Ck i n t o C i we have akjrk < Z-k

k+ 1

almost comp le te l y by means o f a d i s j o i n t sequence

k j

f o r each j and k .

s e t s AJk , k = 1,2, ..., j = 1,2 ,... L e t Pkj Ak = AJk and l e t A be t h e union o f a l l

We then have

- We s h a l l now prove t h a t a t a lmost each x 6 Q we have

1 Since I A l < 7

f o r A .

I)(] x A y x ) a A > 0.

t h i s w i l l p rove t h a t t h e d e n s i t y p r o p e r t y i s n o t t r u e

F i x k and l e t x E C k . There i s t hen R e a ( x ) , w i t h

6(R) < rk such t h a t

I R f'l A k l > A

I R I For each j, t h e image R* o f R by t h e homothecy P i s such t h a t

k j 6(R* ) < 2-k and

IR* o A1 > A .

I R* I

S ince f o r each f i x e d k a lmos t every p o i n t x o f Q i s i n some CJk , i t r e s u l t s t h a t f o r almost each x o f Q t h e r e i s a sequence R k o f

elements of a3 ( x ) c o n t r a c t i n g t o x such t h a t

Thus D( xA ,x) > A a lmos t everywhere i n Q.

I f one knows t h a t a d e n s i t y b a s i s d9 d i f f e r e n t i a t e s t h e i n t e -

g r a l of a f u n c t i o n f e L , t h e one can a f f i r m t h a t t h e maximal ope ra to r

assoc ia ted t o @ s a t i s f i e s a c e r t a i n weak t y p e p r o p e r t y . Th i s i s es-

s e n t i a l l y t h e con ten ts o f t h e main theorem i n t h e s e c t i o n . I n o r d e r t o

p rove i t we s h a l l make use of another i m p o r t a n t theorem t h a t a s s e r t s t h a t

the d i f f e r e n t i a t i o n o f i n t e g r a l s o f f u n c t i o n s by a b a s i s @ i s t r a n s m i t -

t e d t o s m a l l e r f u n c t i o n s .

L1955J. I t i s based on a i d e a of Jessen used by Papou l i s [1950] f o r a d i f f e r e n t .

purpose.

and P a w [1955] , equ iva lence (a )<=>(c ) .

T h i s l a s t theorem i s due t o Hayes and Pauc

The p r o o f we p resen t here i s c o n s i d e r a b l y s h o r t e r and s i m p l e r .

The main theorem ,6.4.G., i n t h i s s e c t i o n i s p a r t l y due t o Hayes

P4ood. For a f i x e d N > 0 d e f i n e

f ( x ) if f ( x ) < N

if f ( x ) I N f N ( X ) =

N and l e t fN be such t h a t a t each x e Rn. By

hypo thes i s D ( f , x ) = f ( x ) a t a lmost eve ry x e Rn and a l s o , s i n c e

f ( x ) = f ( x ) + f N ( x )

N

I 'p, i s a d e n s i t y bas i s , by Theorem 6.4.1.,0( f N y x ) = f N ( x ) a t a lmos t

eve ry x E R~ . so we g e t a t a lmost every E R~ , D ( / fN,x) = f ( x ) .

L e t us now de f ine

and g* such t h a t g ( x ) = g,(x) + g*(x) a t each x o Rn. Then we have

D( 1 g,,x) = g,(x) a lmost everywhere. S ince l g * l L fN , we have a t a l -

mosz every x f o r each sequence { R k ( x ) ) c Q ( x ) c o n t r a c t i n g t o x.

I g * ( x ) I < N a t each x e Rn and so , aga in by theorem 6.4.1.,

w i t h t h i s we have D ( g,x) = g ( x ) a lmos t everywhere. Analogously

I)( g,x) = g (x ) a lmost everywhere and t h i s proves t h e theorem. -_ - J

The f o l l o w i n g theorem c h a r a c t e r i z e s t h e d e r i v a t i o n by 8 o f t h e

i n te rm which a r e s i m i l a r t o those o f t h e den- i n t e g r a l o f a f u n c t i o n f

s i t y theorem o f Busemann and F e l l e r . I t i s v a l i d f o r a genera l b a s i s .

6.4.6. THEOREM . L t L 63 be a b u h w a h t h e d e a a y p h o p e h -

ty. L e t f > 0 , f E L1(Rn) . Then t h e & o U v w i n g t h e e condi t ionn me - eq iLivaeevLt :

( a ) 9 di66mentiaten

( b ) doh each A > 0 , each nequence Cf,}, w L t h fk E L', f k < f f ,

f k ( x ) G 0 CLt d m o & t a c h x E Rn and doh each numeAhaP. ACQUenCe Cr,}

w L t h rk G 0, we have.

whme Mk

by t a k i n g @om @ jui t h e e..temenx2 wLth diamQ;tm Lenn t h a n ,the maximal a p e h a t a h u n v c i a t e d t o .the b a d % rk vbRained

rk . ( c ) Fv/z each 1 > 0 , each n o n i n c h m i n g nequence 0 5 meuw -

abRe n u 2 { A k } w L t h \ A k [ + 0 , and each numehicd neyuence irk} w L t h

rk G 0 , we have

PaoaQ. I n o r d e r t o prove t h a t ( a ) i m p l i e s ( b ) we t a k e an

a r b i t r a r y open c u b i c i n t e r v a l Q and E > 0. We have f k + 0 p o i n t w i s e

on Q and so, by Egorov ’s theorem, t h e r e i s a measurable s e t A, w i t h

J A J < E , such t h a t fk + 0 u n i f o r m l y on Q - A . Hence, g i v e n A > 0,

t h e r e e x i s t s a p o s i t i v e i n t e g e r h such t h a t f k ( x ) < X i f k > h

and x 6 Q - A. S ince we a r e assuming t h a t 6 d i f f e r e n t i a t e s If, by

t h e p reced ing theorem, % d i f f e r e n t i a t e s

each x B Q - A and f o r each sequence { R j ( x ) } C ( x ) c o n t r a c t i n g

t o x we have, as j +

J Ifk f o r each k . Hence, f o r

There fo re

l i m { x E Q : M~ f h ( x ) > c A k-

I t i s c l e a r t h a t , i f k 2 h, s i n c e fk < fh , we have

and so

l i m I { x 6 Q : Mk f k ( x ) > A 1 ) e 6 ] A ) < E k-m

Since Q and E a r e a r b i t r a r y , we g e t (b ) .

That ( b ) i m p l i e s ( c ) i s t r i v i a l by t a k i n g fk = f X A k .

I n o r d e r t o prove t h a t ( c ) i m p l i e s (a) , l e t

Ak = C f ;r k3 f o r k = 1,2,...

Since f E L ~ ( R ” ) , 1 ~ ~ 1 + 0.


We have, c a l l i n g f X A = fk , f = fk t f . Since i s k

assumed t o be a d e n s i t y bas i s , f o r a lmos t eve ry x, D( J f k * x ) = f k ( x ) So f o r each X > 0,

The f i r s t t e rm i n t h e l a s t member of t h i s c h a i n o f i n e q u a l i t i e s tends t o

ze ro by hypo thes i s and k * m. The second one because f e L1(Rn) . So

we g e t i( f ,x) = f ( x ) a lmos t everywhere. S i m i l a r l y D(( f,x) = f ( x )

a lmost everywhere. I

T h i s concludes t h e p r o o f o f t h e theorem.

With t h e p rev ious theorems i t i s ve ry easy t o g i v e a cha rac te r -

i z a t i o n o f b a s i s d i f f e r e n t i a t i n g L'(R")

i n t h e s t y l e o f Busemann and F e l l e r .

i n terms o f t h e maximal o p e r a t o r

6.4.7. THEOREM. 1eX fi be a did6etrentiation banA i n Rn.

Thcn t h e Aktlee I;oUuwing canditioMn ahc eqlLivaLent:

( c ) F O X each X > 0 , each f e L1(Rn) , each nonirzc/rea,&uj

btqUc?nCe 0 6 meanmabLe A& t A k ) nuch t h a t ] A k [ -+ 0 , a d each numehicd


P m o d . I f any o f t h e t h r e e cond

i s a d e n s i t y bas i s , by Theorem 6.4.1.

consequence o f Theorem 6.4.6.

0

t i o n s ( a ) , (b

The theorem

, ( c ) ho lds , t hen

i s t hen a d i r e c t

When one assumes t h a t t h e b a s i s f i i s i n v a r i a n t by t r a n s l a t i o n s

o r by homothecies, t h e p reced ing c h a r a c t e r i z a t i o n takes a s i m p l e r form.

6.4.8. THEOREM . L e A @ be a B - F b a d t h d d invahiant by XhanbLatium. Then t h e A.va doUawing conditiam ahe qU.ivdevLt:

whme

S U P

I n t h e p r o o f o f t he theorem we s h a l l make use of t h e f o l l o w i n g

lemma due t o A.P.CalderGn, which has been a l r e a d y presented i n Chapter 2

6.4.9. LEMMA . L e A C A k l be a bequence ad meanwrabLe n e A

Q c R n . poi& in R n and a net S lu i th

cantdined in a dixed cubic i n t e h v d Then t h m e LA a nequence Cxkl 0 6 p V b i , t i V e meanwre cantdined i n Q buch .thcLt each s e S LA i n i n @ h X d y

many n e A 06 t h e dahni

and nuch thcLt C I A k \ =

xk + Ak.


Phuoi) 0 6 t h e Theohem 6.4.8. Tha t ( b ) i m p l i e s (a ) i s a

s imp le consequence o f Theorem 6.4.7.

( b )

I n o rde r t o p rove t h a t ( a ) i m p l i e s

l e t us prove f i r s t t h a t ( a ) i m p l i e s t h e f o l l o w i n g :

(b*) F o ~ each dixed cubic intehwd Q t h e h e txht paniaXwe

co1z~tant5 c = c ( Q ) r = r ( Q ) nuch , that doh each non negative f 6 L1

w a h nuppoht in Q arid M C ~ > 0 we have

Assume t h a t

t h a t for each p a i r o f

fk E L1 suppor ted i n

(b*) does n o t ho ld . Then t h e r e is a f i x e d Q such

cons tan ts c k Y rk > 0 t h e r e i s a nonnegat ive

Q and a l s o a X k > 0 such t h a t t h e s e t

Ek = { x 8 Rn : M f k ( x ) > X k } rk

S a t i s f i e s 1 ~ ~ 1 > ck ,/ L e t us take a sequence i r k } rk + 0, such t h a t a l l numbers rk

a r e l e s s than the s i d e - l e n g t h o f Q y and l e t ck = 2k . We c a l l gk = f k / X k

and Q* t h e c u b i c i n t e r v a l w i t h t h e same c e n t e r as t h a t o f Q and t h r e e

t imes i t s s i z e . C l e a r l y E k C Q * and

We can choose f o r each k a p o s i t i v e i n t e g e r hk such t h a t

1 Q * 1 G h k l E k l G 21Q*1. So we have

m

We cons ide r t h e sequence {Ah}

by r e p e a t i n g hk t imes each Ek , i.e.

m

o f s e t s con ta ined i n Q* ob ta ined

t h e f o l l o w i n g sequence:

h, h h3 E i , E:, .. . El , E:, Et,.. . , E2' E:, E:, .. . , E 3 , Eiy. ..

i where Ek = Ek f o r each j w i t h 1 G j $ hk . S ince


m m

and a l l se ts a r e con ta ined i n Q* we can a p p l y Lemma 6.4.9. We thus o b t a i n

t h e p o i n t s

h 2 h3 x2 , x i 7 x;, ..., x g , x; ,... x i , x;, ... , x, , x i , x;, ..., h l

and a s e t S w i t h p o s i t i v e measure con ta ined i n Q* such t h a t each p o i n t

o f S

f o r each k and each j = 172,...,hk,

i s i n i n f i n i t e l y many o f t h e s e t s Ejk . We d e f i n e t h e f u n c t i o n s ,

and f i n a l l y t h e f u n c t i o n

where ak z 0 w i l l be chosen i n a moment

We have

k and,since h k [ E k / g 2 [Q* l and I E k ( > 2 [ Igk[ l l , we g e t

L e t R B @ . We can o b v i o u s l y w r i t e

Each

L e t t hese s e t s be

s 6 S belongs t o an i n f i n i t e number o f s e t s o f t h e fo rm EJk .

By t h e d e f i n i t i o n of t h e s e t s E; , t h e r e i s t hen a sequence { R h l C @ ( s ) ,

w i t h such t h a t , because o f t h e above e q u a l i t y , Rh -f s,

f > f o r m = 1,2, ..

L e t us choose now ak so t h a t ak + and a t t h e same t ime

< a . ak

k = l Zk c -

For example, l e t us s e t ak = Zk” . Then we o b t a i n f 8 L’ and a t each

(b * ) .

s e S we have 6 ( f ,s ) = f m . T h i s c o n t r a d i c t s ( a ) . Hence ( a ) i m p l i e s

We have now t o deduce ( b ) f r o m (b*) . F i r s t of a l l i t i s c l e a r ,

b y the i n v a r i a n c e by t r a n s l a t i o n s o f a o f (b*) do n o t depend on t h e p lace i n Rn where Q i s l oca ted .

, t h a t t h e cons tan ts c ( Q ) , r ( Q )

I t i s a l s o c l e a r t h a t Mr,2f 6 Mrf and so we assume i n (b*)

t h a t r ( Q ) i s l e s s than h a l f t he l e n g t h of t h e s i d e o f Q. Assume now

t h a t f a 0 i s a f u n c t i o n i n L1 w i t h suppor t con ta ined i n i n f i n i t e l y

many d i s j o i n t cub ic i n t e r v a l s {QjIjZ1 each’one o f them equal i n s i z e

t o Q and such t h a t t h e d i s t a n c e between any two o f them i s a t l e a s t

equal t o t h e s i d e l e n g t h of 4. Then, if r i s , as we have assumed,less

than h a l f t h e s i d e l e n g t h o f Q, we c l e a r l y have

m CU

and t h e se ts H a r e d i s j o i n t . Hence j


a ( n ) Now, f o r an a r b i t r a r y f E L 1 , f a 0 , we can s e t f = 1 fh where

h = l

each fh i s o f t h e

suppor ts and a ( n )

t ype a l r e a d y t r e a t e d , t h e f u n c t i o n s fh have d i s j o i n t

depends o n l y on t h e dimension. Thus

The r e s t r i c t i o n f > 0 i s t r i v i a l l y removed and so we o b t a i n t h e theorem.

The theorem of Busemann-Fel l e r f o r a b a s i s t h a t i s homothecy

i n v a r i a n t i s now an easy c o r o l l a r y o f Theorem 6.4.8.

6.4.10. THEOREM. 1e.L be a B - F b a A t h a A ' A homothecy i n v u t i a n t . Then t h e Awo 6 o ~ Y o w i n g ConditioMn me eqLLiudevCt:

( b ) The maximd o p e ~ a A o f i M 06 i~ 06 weak t y p e ( l , l ) , i . e . thetre exL.02 a cavl0.tunt c > 0 nuch t h a t 6ofi each f E L' and each A > 0 one h a

- P h d . I t i s s u f f i c i e n t t o prove t h a t f o r t h e homothecy i n v a r -

i a n t b a s i s

t h i s theorem.

, c o n d i t i o n ( b ) o f Theorem 6.4.8. i m p l i e s c o n d i t i o n ( b ) o f

The re fo re , we assume t h a t t h e r e e x i s t c > 0 and r > 0 such

t h a t f o r each f 6 L' and A > 0 we have

Take a number p > 0 and a f u n c t i o n @ E L ' . D e f i n e a new f u n c t i o n f

by s e t t i n g , f o r x E Rn ,

134 6 . COVERINGS, HARDY-LITTLEWOOD A N D DIFFERENTIATION

Observe tha t 447 P x )

f ( x ) I dx = ~ ( r - ) ~ 1 I d z . c \ 1-1 A dx = c i l ~ P

In pa r t i cu la r , of course, f e L ’ . I f s e t

y e R n and R e ?3,(y) , we can

1 I \ f ( $ z ) \ d z = I$ R I ( F ) n R

- - 1 I f (x ) l (F)n d x = 1

1; R I ( F I n R

r MP(P (y) = M f ( r y ) , s ince - R e 8 (“ y ) and when R

Pr P Pr This provesthat runs over a l l % ( y ) , the s e t - R runs over a l l ‘13 (- y ) .

P P r P

I C Y e R n : Mp $ ( y ) > XI1

/{!Z : Mrf(z) > X)l

= I C Y e R n : M f ( T y )

= (F)n I { z : Mrf(z) > A

r P

=

c (by hypothesis) G

This proves t h a t f o r any p > 0 and any 4 8 L’ we ge t

with the same constant. Hence f o r each $ E. L ’ ,

and t h i s concl udes the proof of the theorem.

The type of Theorems presented in this Section C s a r i l y connected w i t h the d i f f e ren t i a t ion of a f i n e m space. and Peral [1974] have obtained r e s u l t s concerning weak type proper t ies

i s not neces Rubio [1971]

f o r t h e maximal o p e r a t o r when one knows t h a t t h e co r respond ing b a s i s d i f -

f e r e n t i a t e s a space

t y p e due t o Rubio.

$ ( L ) . We s h a l l c o n s i d e r here one theorem o f t h i s

6.4.11. THEOREM. L e t $ : [0,.3] + [O,m] be a n&tio tey inmean -

ing aunctian wLth $ ( O ) = 0 and buch thaR $ ( u ) ud m d e h ghe&eh

than oh equal t o t h e o/rde,t ol; u when u -+ m . Le,t $ ( L ) be t h e coL-

Lecfion 06

1eL @ be a humothecy invahiuvct B - F b a & thcLt d i Q d e ~ e r t t i a Z e ~ $ ( L ) .

then t h e m e u h 2 a c o ~ h t ~ ~ v c t c > 0 buch thaR Qvh each A > O and each

f E $ ( L ) , f 2 0 one h a

meanmubLe 6unctioMh f : Rn -+ R nuch t h a t I $ ( I f 1 ) < m.

Phood. Assume t h a t t h e theorem i s n o t t r u e . Then, f o r each

ck > 0 t h e r e e x i s t fk E $ ( L ) , fk > 0 and > 0 such t h a t

. We t a k e a sequence { c k } , ck > 0 f k L e t us c a l l gk =

such t h a t

There e x i s t s a sequence (rkl , rk > 0 , such t h a t

There i s a l s o a compact subset Ek o f iM gk > 11 such rk

t h a t l E k l > ck $ (gk ) . We cons ide r t h e open u n i t cube Q and a f i x e d

k . By us ing Lemma 6.4.4. we cover a lmost comp le te l y Q by means o f a

o f s e t s homothet ic t o Ek con ta ined i n Q d i s j o i n t sequence E E k l h,l be t h e homothecy t h a t c a r r i e s and o f d iameter l e s s than l / k . L e t

Ek i n t o Ek . We d e f i n e t h e f u n c t i o n gk by s e t t i n g

h

pk h h


D e f i n e then

and f i n a l l y f = sup Sk . One e a s i l y g e t s f E $ ( L ) and k

However, D ( f , x ) a 1 a t a lmos t each x e Q . Since @ d i f f e r e n t i a t e s f $ ( L ) , we g e t f ( x ) > 1 a t a lmost each x E Q and so

,

Th is c o n t r a d i c t i o n proves t h e theorem.

6.5. DIFFERENTIATION PROPERTIES IMPLY C O V E R I N G PROPERTIES.

Since t h e beg inn ing o f t h e d i ' f f e r e n t i a t i o n theo ry , seve ra l

i n t e r e s t i n g theorems have been fo rmu la ted t h a t p e r m i t t o deduce u s e f u l

cove r ing p r o p e r t i e s f rom d i f f e r e n t i a t i o n p r o p e r t i e s .

o f R. de Posse1 [19361 and t h e ones o f Hayes and Pauc [19551. More

r e c e n t l y Hayes [19761 and a l s o Cordoba and R. Fef ferman [19771 have

amply extended the scope o f t h e o r i g i n a l theorems.

t h e method o f p r o o f o f such theorems seems q u i t e n a t u r a l .

o b t a i n an economical cove r ing f rom a g iven, perhaps h i g h l y redundant,

cove r o f a set,one chooses t h e b i g g e s t p o s s i b l e s e t s among those whose

o v e r l a p w i t h t h e a l r e a d y chosen ones i s smal l i n some sense.

cove r i s then shown, us ing the d i f f e r e n t i a t i o n p r o p e r t y , t o cover t h e

Such a r e t h e r e s u l t s

As one can observe,

I n o r d e r t o

T h i s sparse

6.5. DIFFERENTIATION IMPLIES COVERING PROPERTIES 137

o r i g i n a l s e t .

We f i r s t p resen t t h e theorem o f t h e Posse1 c h a r a c t e r i z i n g den-

s i t y bases by means o f a cove r ing p r o p e r t y .

Hayes and P a w [1955]

p a r t i c u l a r f u n c t i o n .

a c t e r i z i n g the bases t h a t d i f f e r e n t i a t e

a c o v e r i n g p r o p e r t y .

t h a t f o r a B - f b a s i s t h a t i s i n v a r i a n t by homothecies, t h e d i f f e r e n t i a

t i o n o f L 1 m n ) i s e q u i v a l e n t t o t h e V i t a l i p r o p e r t y .

Then we show a theorem o f

t h a t concerns a c o v e r i n g p r o p e r t y r e l a t e d t o a

I n t h e t h i r d p l a c e we s h a l l p r e s e n t a r e s u l t char-

F i n a l l y we o f f e r a r e s u l t o f Mor i ydn [1975] p r o v i n g

L P m n ) y 1 < p < mY’n terms o f

I n o r d e r t o s t a t e more e a s i l y t h e f o l l o w i n g theorems, g i v e n a

s e t A and a d i f f e r e n t i a t i o n b a s i s 9 , we s h a l l say t h a t a sub fami l y

‘2 of @ i s a VLtitaei c o v a o f A i f f o r each x E A t h e r e i s a

sequence { B k ( x ) } c % such t h a t S ( B k ( x ) ) -f 0,

Prrovd. L e t us assume t h a t i s a d e n s i t y b a s i s . We t ry t o

prove p r o p e r t y ( P ) . L e t G be open, w i t h G 3 A such t h a t ) G - A [ c E

w i t h o u t loss o f g e n e r a l i t y we can assume t h a t a l l elements o f 2 a r e

con ta ined i n t h e s e t G. Otherwise we keep o n l y those elements o f ? t h a t s a t i s f y t h i s p r o e e r t y .

( b ) . L e t us t a k e a w i t h 0 < c1 < 1, t h a t w i l l be chosen c o n v e n i e n t l y

i n a moment. We d e f i n e

I n t h i s way we a u t o m a t i c a l l y o b t a i n p r o p e r t y

PI= SUP {[R[ : R E % , \ A n R ( > a(R(1

138 6. COVERINGS, HARDY-LITTLEWOOD AND D I F F E R E N T I A T I O N

Since / A / > 0 a n d f o r each I R k ( x ) 3 c @ ( x ) w i t h R k ( x ) + x we have

i t i s c l ea r t h a t p1 > 0 . We take R1 8 "t such t h a t

Let us ca l l A = Al and AP = A1 - R1 . I f / A 2 [ = 0 the process of se lec t ing Rk i s f in i shed . Otherwise we define

a n d we s e l e c t RL E 2. such t h a t

Define now A 3 = A 2 - R 2 and so on. We obtain a sequence { R k I k , l f i n i t e or i n f i n i t e .

In order t o see t h a t { R k 3 s a t i s f i e s ( b ) , we f i r s t observe (Rk (1 A k ) 1'1 ( R . I1 A . ) = 4 i f k # j , and so we can wr i te J J

If {R,} i s f i n i t e , we c l ea r ly have \ A - II R k l = 0 . Assume t h a t

I R k } i s an i n f i n i t e sequence. Since 1 l R k l < a we have l R k l -, 0

and so pk < 3 \ R k J i s such t h a t pk + 0 . Let us ca l l A,=A - I) R k . k

Assume \A,/ > 0 . Then, i f we define

k

k 4

6.5. DIFFERENTIATION IMPLIES C O V E R I N G P R O P E R r I E S 139

we c l e a r l y have p, > 0 , p, ,c pk f o r each k . T h i s c o n t r a d i c t i o n

proves t h a t I A - (I R k l = 0.

For t h e p r o o f of

we have ob ta ined , I A l 2 c1

( c ) we can w r i t e , because of t h e i n e q u a l i t y

( R k l , and because o f ( a ) ,

1 Hence, i f we choose c1 s o t h a t (; - 1) I A l G E , we o b t a i n ( c ) .

The second p a r t o f t h e theorem i s easy. Assume t h a t 9 s a t i s -

f i e s p r o p e r t y ( P ) f o r each measurable s e t A w i t h 0 < \ A \ < m.

We want t o prove t h a t % i s a d e n s i t y b a s i s s i n c e

t o Theorem 6 . 4 . 1 . , t o d i f f e r e n t i a t i o n of L".

t h i s i s e q u i v a l e n t

L e t M be a measurable s e t . For a A > 0 and H > 0 we

d e f i ne

So, f o r each x f A t h e r e e x i s t s C R k ( x ) f c @ ( x ) , such t h a t Rk(x) + x

and

IRk (x ) ('1 M I

IR,(x)l > A .

We s h a l l prove t h a t I A l = 0 . I f no t , we t a k e an a r b i t r a r y E > 0 and

app ly p r o p e r t y (P) t o A w i t h t h e V i t a l i c o v e r i n g

'r- = (Rk (X) )xeA , k = 1,2,2 ,...

and w i t h E > 0 . We o b t a i n { R k ) s a t i s f y i n g (a) , (b ) , ( c ) . Hence ,hav ing

140 6 . C O V E R I N G S , HARDY-LITTLEWOOD AND DIFFERENTIATION

i n t o account t h a t M c A ' we can w r i t e

Since E i s a r b i t r a r y , I A l = 0.

So we o b t a i n f o r a lmost a l l x E M ' D ( x M,x) = 0 . I f we

app ly t h i s r e s u l t t o N = M I , we have , f o r a lmos t a l l x e N '

D ( x N Y x ) = 0 . But t h i s i m p l i e s D(xM,x) = 1. Hence i s a d e n s i t y

bas i s ,

The f o l l o w i n g theorem can be viewed as an e x t e n s i o n o f t h e

Posse l ' s theorem t o a measure t h a t i s cont inuous w i t h r e s p e c t t o Lebesgue

measure.

6.5.2. -__ THEOREM. L e i be a B - F b a h wLth t h e deMnLty

p'lopehty and teX f E L f > 0

and nuddicient cvndition i n o t d a t h a t 8 ~ ~ ~ e ~ e ~ c ~ t ~ lvwing :

be a hixed dunctivn. Then a necunwry

f .LA t h e h u t - I ( E ) Given a bvunded meurnab le b&t A , g iven E > 0 and a

A , t h e m ex-&& a bequence { R k } buch t h d , denvang ViAaLL c o v a 5 06

xk = xRk k R = 0 Rk , we have

P'laod. Assume t h a t @ d i f f e r e n t i a t e s f. L e t A , E, , I be as i n p r o p e r t y (E) o f t h e s tatement o f t h e theorem. L e t T-I z 0 be

a f i x e d cons tan t t h a t we s h a l l choose c o n v e n i e n t l y i n a moment. L e t

f o r k 6 h . Since 7 4 d i f f e r e n t i a t e s f, we have, f o r a lmos t eve ry

x E Ak , i

a sequence

such t h a t

We can assume

(1 t ri ) k IRh (x ) l < \ f 6 ( l + q ) k ' l I R h ( x ) I f o r each h = 1 , 2 , ... Rh

We s h a l l d i s r e g a r d t h e n u l l s e t o f

and a l s o t h e n u l l s e t where f may be i n f i n i t e . We a p p l y t h e P o s s e l ' s

theorem t o

{Rh(x)} and w i t h an ck + 0 t h a t w i l l be c o n v e n i e n t l y chosen l a t e r . We

Ak where the p reced ing i s n o t v a l i d

Ak w i t h t h e V i t a l i cove r ing o b t a i n e d by means o f t h e s e t s

thus o b t a i n a sequence

such t h a t , i f we denote

k (i) IAk - s

e x t r a c t e d f rom (Rh(x))xaA , h = 1,2 ,... , k

xsk f xk and Sk = S j , we have

'j'j>l k

J j J

= o

Observe t h a t we can a l s o w r i t e

So c o n d i t i o n (iii) can be w r i t e n

142 6. COVERINGSy HARDY-LITTLEWOOD AND DIFFERENTIATION

where yk > 0 can be choosen i n advanced a r b i t r a r i l y sma l l .

can be chosen as t h e f a m i l y we a r e k We now t h a t ' CSjl k y j

l o o k i n g f o r i n o rde r t o prove p r o p e r t y (E). Observe t h a t

and so we have ( a ) . A lso we have

and so we have ( b ) i f we choose E~ so t h a t 1 ck < E .

For each k we can w r i t e

so we can s e t

i f o n l y yk c y f o r each k. Hence

There fo re , b y choosing q y y and E~ c o n v e n i e n t l y we o b t a i n (a ) , (b )

and ( c ) o f p r o p e r t y (E ) . T h i s concludes t h e f i r s t p a r t o f t h e theorem.


Assume now t h a t ( E ) ho lds . L e t us t r y t o show t h a t %?I d i f -

f e r e n t i a t e s . We can assume t h a t f has compact suppor t w i t h o u t loss o f g e n e r a l i t y . For each r > s > 0 we c o n s i d e r t h e s e t

if

The s e t A i s bounded and f o r each x 6 A t h e r e e x i s t s a sequence

C R k ( x ) } c @ ( x ) , Rk(x ) -f x such t h a t

We app ly ( E ) and e x t r a c t from (Rk (X) )xeA , k=l,2,... a se-

quence C T k I s a t i s f y i n g (a ) , ( b ) , ( c ) . We can w r i t e , c a l l i n g

T = 0 Tk 3

k

(For t h e second i n e q u a l i t y we have used

Given n > O we.can choose E > 0 f o r t h e a p p l i c a t i o n o f (E)

such t h a t

i . e . ( r - 5 ) l A / c i) . Hence \ A / = 0. So we have proved t h a t t h e s e t

where Is( f , x ) i s n o t f ( x ) i s o f n u l l measure. I n t h e same way one

proves t h a t O( f , x ) = f ( x ) a lmos t everywhere . T h i s concludes t h e

p roo f o f t h e theorem. I I

The f o l l o w i n g theorem c h a r a c t e r i z e s those d i f f e r e n t i a t i o n

bases t h a t d i f f e r e n t i a t e Lp , 1 O aRd g i v e n a V i a k L L c v u a 2 0 6 a memuhub& 9

he,t A, 0 < / A 1 < w, t h e m d a nequence {Rk Ika lc 2 huch LhaX

( a ) ] A - u R k I = 0

P m v d . Assume t h a t 63 d i f f e r e n t i a t e s L q . We t r y t o prove

(P,). F i r s t o f a l l we prune 2 G > A and I G - A1 c E and keeping o n l y those s e t s i n ? con ta ined

i n G . We keep c a l l i n g *‘t t h e rema in ing cover o f A. We now observe

t h e f o l l o w i n g : I f { R k ) i s any sequence ( f i n i t e o r i n f i n i t e ) of

elements o f 7 such t h a t f o r 0 < ci < 1 we have

by t a k i n g an open s e t G such t h a t

The reason f o r (1) i s t h a t @ i s a d e n s i t y b a s i s

c a ( 1 - a ) ( W 1 .

and f o r ( 2 ) t h a t

6 . 5 . DIFFERENTIATION IMPLIES COVERING PROPERTIES 145

fl d i f f e r e n t i a t e s L p and the function in brackets i s in L p and i s 0 on A - O R k .

Therefore we can wr i te

( i i ) ' (I-a) (1

= I A

r

G [ I W ( ) ( A -

This suggests how we can proceed i n the se l ec t ion of CRkI . We f i x a, 0 < a < 1, such t h a t alGl < E , and choose f i r s t R I E ? such t h a t I R 1 I 2 3/4 sup { [ V l : V st}. I f I A - R I I = 0 we a r e f in i shed . Otherwise IR1) s a t i s f i e s ( i ) , ( i i ) and ( i i i ) . Call w1 the co l lec t ion of a l l s e t s W 8 2 sa t i s fy ing (1) and ( 2 ) corresponding to t h i s sequence C R 1 1 as above. Choose R2s w1 such t h a t

{ I W l : W E w1 I . If IA - f~ R k [ = 0 we a r e f in i shed . Otherwise ER1, R z } s a t i s f i e s ( i ) , ( i i ) , and ( i i i ) , Call w2 the collec- t ion of a l l s e t s W E '% as above corresponding t o t h i s sequence. Choose R3a w z such t h a t I R 3 1 2 3/4 sup I W I : W E w2}. And so on. In t h i s way we obatin {RkI . [ A - O R k l = 0 . I f i t i s i n f i n i t e and / A - u R k [ > 0 then there e x i s t s W E 2 sa t i s fy ing (1) and ( 2 ) . B u t c l ea r ly , since (1 - a) 1 I R k ] s ] G I , we have l R k l -f 0 . There i s a f i r s t R k such

I R z 1 2 3/4 sup 1

If i t i s f i n i t e , then i t i s so because

t h a t l R k l G 3/4 IWI and t h i s c o n t r a d i c t s t h e cho ice o f R k . There fo re

[ A - i iRkl = 0 and C R k I s a t i s f i e s (a) , (b ) and ( c ) .

Assume now t h a t @ s a t i s f i e s (P,) . L e t f E Lp and f o r

ol 7 0 and M > 0 l e t

For an a r b i t r a r y ri > 0 l e t us choose g E @ such t h a t i f h = f - g

we have llhll 6 rl. We can w r i t e

sa M = t x E R ~ : 1x1 < M, l i m sup 1 6 \ h (y )dy - h ( x ) I > c11 xERk“(x) 5 S(Rk) + o

and a l s o

I n o r d e r t o e s t i m a t e S!” we use (P,) w i t h an E > 0. Fo r

each

such t h a t

x E S!” t h e r e i s a sequence { R k ( x ) ) c U 3 ( x ) c o n t r a c t i n g t o x

We can assume t h a t S ( R k ( x ) ) 4 1. We app ly p r o p e r t y (P,) t o t h i s

s i t u a t i o n o b t a i n i n g { S , } C ( R k ( x ) )

such t h a t

xeS:”, k = l ,2 ¶ . . .

t h a t

6.5. DIFFERENTIATION IMPLIES C O V E R I N G PROPERTIES 147

- -

Using H o l d e r ' s i nequa l ty, p r o p e r t y

, we o b t a i n

sk

c ) o f (P,) and t h e f a c t

I n 1

M Since I ( h / ( c rl and 0 i s a r b i t r a r i l y sma l l , we see t h a t l S a l = 0

f o r each ci and M. T h i s proves t h a t d i f f e r e n t i a t e s L p . P

As we have a l r e a d y shown i n Theorem 6.2.4. a d i f f e r e n t i a t i o n

b a s i s can d i f f e r e n t i a t e L ' w i t h o u t hav ing t h e V i t a l i p r o p e r t y . However,

as MoriyBn [1975] has proved , i f t h e b a s i s @ i s a B - F b a s i s

i n v a r i a n t by homothecies, t hen t h i s i s n o t p o s s i b l e . I f t h e b a s i s 8 i s a d e n s i t y b a s i s then (B has t h e V i t a l i p r o p e r t y .

6.5.4. THEOREM. L& '@ b~ a B - F ba&ih t h a t A i n v d a n t

by hvmcdhwien. Then 3 o!il;l;e~enLia.t~ L ' il; and a d l j id h a t h e

v u pfi3hOpuLty.

P m a d . We need o n l y t o prove t h a t i f fB d i f f e r e n t i a t e s L '

We s t a r t by c o n s i d e r i n g t h e s e t

t hen i t has t h e V i t a l i p r o p e r t y .

and we prove t h a t I K I < co .

We know, acco rd ing t o Theorem 6.4.3, t h a t t h e maximal o p e r a t o r

r e l a t e d t o i s o f weak t y p e (1 , l ) . L e t Bk = B(O, l /k) ( b a l l c e n t e r - ed a t 0 and r a d i u s l / k ) and

Kk = II CR 6 @ ( 0 ) : I R I 6 1 , R 3 B k I

C l e a r l y K k c K k + l and K = I", Kk . On t h e o t h e r hand

Next we show t h a t I K / < a, i m p l i e s t h a t K i s bounded. Sup-

pose t h a t K i s unbounded. We can then choose, by t h e i n v a r i a n c e by

homothecies , C R k l c @ ( O ) , R k C S K , I R k ( = 1, 6(Rk) + m ,so

t h a t each R k has some p o i n t xk e Rk w i t h j x k / 2 8 6 ( 0 R j ) . k - l

1

We now choose a

S1 1 R 1 . I f

then S 2 = - X Z

The s e t s Lk =

sequence o f s e t s C S k l i n t h e f o l l o w i n g way. Take

1 1 I R 2 (7 S 1 l & then S 2 = R2 . If I R 2 17 S 1 I > 7

R Z . If

1 > then

K (by t h e

> 7 (by 1

k - 1 sk - IJ

1

2 1 R 3 17 ( y S j ) / 6 7 t hen

nva r iance by homothecies) and

n t e g r a t i o n a long l i n e s p a r a l l e l t o oxk).

S j a r e d i s j o i n t . T h e r e f o r e I K I = .

S 3 = R 3 . If

S 3 = - x3+ R 3 . And so on. I t i s easy t o

Hence, i f 8 d i f f e r e n t i a t e s L ' , then K i s bounded and

so we can enc lose K i n a c losed c u b i c i n t e r v a l Q cen te red a t 0.

L e t / Q / = aIK1 . For each B E @ (0) w i t h IBI = 1 we see t h a t

B C Q and 141 = (aIKI) I B I . I t i s then c l e a r , by t h e i n v a r i a n c e

by homothecies, t h a t f o r each

Val Q cen te red a t x such t h a t 101 = alKl I B I . The b a s i s @ i s s a i d

t o be heglLeah w i t h r e s p e c t t o t h e b a s i s c f cub ic i n t e r v a l s cen te red a t

t h e corresponding p o i n t s .

r e g u l a r i t y i m p l i e s t h e V i t a l i p r o p e r t y f o r 8

B e @ ( x ) t h e r e i s a c losed c u b i c i n t e r

I t i s an easy e x e r c i s e t o show t h a t t h i s

.

6.6. THE HALO PROBLEM

6.6. THE HALO PROBLEM.

149

L e t % be a B - F b a s i s i n Rn t h a t i s i n v a r i a n t by homo-

t h e c i e s and s a t i s f i e s t h e d e n s i t y p r o p e r t y .

6 .4 .3 . the re e x i s t s a f u n c t i o n +* : (1P) -f [ O P ) such t h a t each bound

ed and measurable s e t A and f o r each u e ( 1 , ~ ) , one has

Accord ing t o t h e theorem

T h i s suggests we def ine, f o r any B - F b a s i s % , even i f i t

i s n o t i n v a r i a n t by homothecies and does n o t have t h e d e n s i t y p roper t y ,

t h e f o l l o w i n g f u n c t i o n + t h a t will be c a l l e d t h e halo 6uncxXun 0 6 a . For each u E (1,m) we s e t

We now can say t h a t , if 6 i s i n v a r i a n t by homothecies then

B i s a d e n s i t y b a s i s i f and o n l y i f $ i s f i n i t e a t each u c: ( 1 , ~ ) .

If % i s a d e n s i t y b a s i s , then , f o r each u > 1 we have 1 l{MxA > > ] A 1 f o r each A measurable w i t h I A l > 0 and t h e r e f o r e ,

$ ( u ) > 1. We can extend $ t o [O,m) by s e t t i n g

$ ( u ) = u f o r u E [0,1]

We have a l r e a d y seen bases whose h a l o f u n c t i o n s behave r a t h e r

d i f f e r e n t l y . I n ’ f a c t , t h e h a l o f u n c t i o n $ l (u) o f t h e b a s i s 3 1 of cub ic i n t e r v a l s i n Rn behaves l i k e u, i . e . t h e r e e x i s t two cons tan ts

c 1 and c z independent o f u such t h a t

The h a l o f u n c t i o n $ 2 ( u ) o f t h e b a s i s 74 o f i n t e r v a l s i n Rn behaves

l i k e u ( l + log’u)”’. I n f a c t , we s h a l l see i n Chapter 7,

$2 (u ) c c u ( l + log+

150 6 . COVERINGSy HARDY-LITTLEWOOD AND DIFFERENTIATION

The other inequal i ty r e s u l t s very e a s i l y by considering Mx4 where Q i s the unit cubic in t e rva l . One e a s i l y f inds

The halo function 4 1 ~ o f

f i n i t e a t each u > 1, as we sha

x c* u ( l f log+ U ) n - l

the bas is 53 1 see in ChaDter 8.

of a l l rectangles i s i l l

On the other hand 3, d i f f e r e n t i a t e s L 1 ¶a2 d i f f e r e n t i a t e s L ( l t log' L)"' ( R n ) i s t i c functions of measurable s e t s .

and 8 does not d i f f e r e n t i a t e s a l l the character-

I t seems c l ea r t ha t the order of growth of $ a t infin.i ty can give important information about the d i f f e r e n t i a t i o n proper t ies of 8 . So a r i s e s the following question : Knowing the halo function @ of 33 invar ian t by homothecies i n order t o ensure tha t conjecture, looking a t the p ic ture described above, seems t o be t h a t i f 8 i s invar ian t by homothecies and q~ i s i t s halo function, then 13

d i f f e ren t i a t e s $ ( L ) . We sha l l ca l l this the "halo conjecture".Perhaps Gl;63 2 y % have a very pa r t i cu la r geometric s t r u c t u r e in order t o jus t i f y the conjecture. open.

f ind out a minimal condition on f e L l o c ( R n ) d i f f e r e n t i a t e s I f . More prec ise ly , t he natural

The problem suggested by the halo function i s s t i l l

I t will be useful t o look a t the problem from another point of view. We know t h a t the maximal operator M of 9 i s of r e s t r i c t e d weak type $ in the following sense: For each u E ( 1 , ~ ) and each A bounded measurable, with ( A 1 > 0 , one has

@ ( u ) A . We want to.prove tha t M s a t i s f i e s a l so a non-restricted weak type

@ inequal i ty , i . e . f o r each f 6 L l o c and f o r each A, > 0 one has

being the best possible constant s a t i s fy ing t h i s f o r a l l such s e t s

6.6. THE HALO PROBLEM 151

In what follows we sha l l present some r e s u l t s r e l a t ed t o t he halo problem. F i r s t we deduce some easy proper t ies of t he halo function. In ( B ) we present a r e s u l t of Hayes 119661 , t h a t i s r a the r general and in ( C ) another one due t o Guzmdn [1975] t h a t gives a b e t t e r r e s u l t f o r some cases. Finally we sha l l o f f e r some remarks t h a t might be useful in order t o a t tack the problem.

We consider a B - F basis t h a t i s homothecy invar ian t and s a t i s f i e s the dens i ty property. From the de f in i t i on

u , i f u E [0,1']

: A bounded, measurable, ] A / > 0) M u ) =

we see t h a t 8 i s non decreasing.

When @I i s a basis of convex o r star-shaped s e t s , one e a s i l y sees t h a t @ ( u ) > u . In f a c t , l e t u E ( 1 , ~ ) . We take any s e t B

Let B* be a s e t homothetic t o B such t h a t B * 3 B and

1 Then CMXB > ; 1 3 B* and therefore

Since E > 0 i s a r b i t r a r y $ ( u ) 2 u .

The following property i s more in t e re s t ing from the point o f

view of the d i f f e ren t i a t ion theory.

6.6.1. THEOREM. L e R '@ be a B - F ba&h t h a t .LA inulVLiAnt by hornathec ia and oati06iecl t h e d e r k t q pkvpt%tq. 1eA o:[O,m) +[O,m) be a nvndecfieixbing ~uncLLvn ouch t h a t .-$# + d o t u + m, $


being t h e halo duncfion 0 6 6 3 . Then @ doen n o t diddehentLaLte u ( L ) .

P m u d . Accord ing t o 6.4.10 , if 4 d i f f e r e n t i a t e s u(L) , then we have, f o r each f B L and each X > 0

w i t h c independent o f f and X.

+ ( u o )

> c. L e t us choose uo such t h a t

Then t h e r e e x i s t s a s e t A, measurable and bounded, w i t h IAl> 0,

such t h a t

1 uo . and t h i s c o n t r a d i c t s t h e p reced ing i n e q u a l i t y t a k i n g

Therefore "4 cannot d i f f e r e n t i a t e u ( L ) .

f = xA, h = -

B. A henub! 0 6 ha ye^.

The f o l l o w i n g theorem c o n s t i t u t e a good approx ima t ion t o t h e

It i s e s s e n t i a l l y due t o Hayes [1966] i n a c o n t e x t a h a l o c o n j e c t u r e .

l i t t l e more genera l and a b s t r a c t than t h e one we s e t here.

6.6.2. THEOREM. L e t @ be a B - F b a h [ n o t n.eccennahiey -__ i n v a h i a n t by homoZhecied). Let 4 be t h e h a l o dunct ion oh 'p3 , A A A W I ~ ,that + h &buXe on [O,m) (hememba ,that @ ( u ) = u 5o.k u B [0,1]). LeA u : [O,W) + [O,-) be a nun d e m e a i n g dunct ion nuch t h a t a(0) = 0 ,

and doh Aome a > 1 , we have

Then, doh each d u n c ~ o n f E L and doh each A > 0 , we have

6.6. THE HALO PROBLEM 153

Phuod. Assume f i r s t t h a t f 2 0 . For X > 0 l e t us d e f i n e

Then we have

We s h a l l now prove t h a t , i f g i s a f u n c t i o n such t h a t i t s

values a r e e i t h e r 0 o r b i g g e r t h a t 1, then we have

I f f i s n o t n e c e s s a r i l y non-negative, t hen we can s e t

I n o r d e r t o prove (*), l e t c1 > 1 be such t h a t

and l e t us c a l l , f o r k = 1,2,...,

We can w r i t e

a,

f o r each k = l , Z , . . . , 1 s ince, i f x i s such t h a t MXk(X) < ~

then, f o r each B E @ ( x ) , we have

a ( a k - l )

Therefore

and a p p l y i n g t h e f a c t f o r each k we have, f o r A > 0,

we o b t a i n

Th is concludes t h e p r o o f o f t h e theorem.

From t h e theorem we have proved we e a s i l y g e t some i n t e r e s t i n g

d i f f e r e n t i a b i l i t y r e s u l t s . L e t f o r example be + ( u ) 6 cu . We can

t a k e a(u) = u ( l t log' u)'+€ and we o b t a i n

T h i s r e s u l t , by r o u t i n e methods, shows t h a t t h e co r respond ing b a s i s

d i f f e r e n t i a t e s L ( l + l o g * 1 ) I t E .

L e t now $(u) s c u ( l t l o g t u ) . With t h e same u as b e f o r e

6.6 . THE HALO PROBLEM 155

we get

and so a d i f f e r e n t i a t e s L ( l + log’ L)’+€

As one can see, Theorem 6.6.2. does not give in these cases the bes t possible r e s u l t . For 8 1 in R 2 we have $ ( u ) c cu and

d i f f e r e n t i a t e s L . For 8 , i n R 2 , + ( u ) c cu (1 + log’ u ) and 8 2 d i f f e r e n t i a t e s L ( l + log’ L ) .

In the next paragraph we sha l l use another method t h a t , f o r cases indicated above, gives a f i n e r r e s u l t .

C . An appficatian 0 6 t h e e x h a p o l a t i o n mdhod 0 6 Yano.

A straightforward appl ica t ion o f t he ex t rapola t ion method of Yano presented in 3.5.1. gives us the following r e s u l t . MoriyBn “781

has re f ined i t i n order t o deal with the ex t rapola t ion t o pa > 1 , u s i n g his theorem presented i n 3.5.2.

6 .6 .3 . THEOREM. LeR 63 be a B - F d i ~ ~ ~ e v L t i a t i 0 n b a A

i n R n and f ~ 2 + be .iA h d a 6uncaXon. Annume t h a t , 6011. name 6ixed

s > 0 and ha& each p , u X h 1 < p < 2 , we. have

D. Some. /rematrhn on t h e h d o pkoblm.

The following remarks a r e perhaps of i n t e r e s t f o r the so lu t ion of the halo problem, s ince they suggest some possible ways o f handling i t .

6 .6 .4 . THEOREM. ( a ) 16 Ahehe e x h d a denbag B - F b a O

t h a t A hamothecy invahiant and buch t h a t 6011. ~32 h d o ~uncLLon 0 we have

t h e n t h e h u b conjeotwre A @.!-be.

Pfioud. (a ) I f t h e h a l o c o n j e c t u r e were t r u e , @ would d i f - L

2 f .

f e r e n t i a t e $ ( L ) and t h e r e f o r e a l s o a ( L ) = c + ( ~ ) , s i n c e $3 d i f f e r -

e n t i a t e s If i f and o n l y i f 70 d i f f e r e n t i a t e s f But we have

f o r u -f m . Therefore, by what we have seen i n A , @ does n o t d i f f e r -

e n t i a t e o ( L ) . T h i s c o n t r a d i c t i o n proves (a ) .

( b ) Accord ing t o (a ) , i f t h e h a l o c o n j e c t u r e i s t r u e , t hen

f o r '8 one has

where c i s a cons tan t independent o f u. There fo re , s i n c e @ ( u ) i s non decreasing, we have, i f k i s an i n t e g e r b i g g e r than 1,

k Hence, if c = 2p , we g e t @ ( Z ) < Zpk , and so, i f Zk- ' c u < Z k , we

o b t a i n

Accord ing t o Theorem 6.6.2., a p p l i e d w i t h

t h a t 8 d i f f e r e n t i a t e s a t l e a s t LP+' f o r each > 0.

~ ( u ) = u logl+"(l+u) we g e t

6.6. THE HALO PROBLEM '1 57

Therefore, i n o r d e r t o d i sp rove t h e h a l o c o n j e c t u r e , i t would

be s u f f i c i e n t t o e x h i b i t a d e n s i t y B - F b a s i s , t h a t i s i n v a r i a n t by

homothecies and does n o t d i f f e r e n t i a t e any Lp w i t h p i m . One can

c o n s t r u c t a d e n s i t y b a s i s t h a t does n o t d i f f e r e n t i a t e any

p < m , b u t t h i s b a s i s i s n o t o f t h e t ype r e q u i r e d here. Fo r t h i s c o n s t r u g

t i o n o f Hayes

L p w i t h

[1952, 19583 one can a l s o see Guzmhn [1975].

For a counterexample t o t h e h a l o c o n j e c t u r e one c o u l d t r y t o

c o n s t r u c t a B - F b a s i s i n v a r i a n t by homothecies and such t h a t i t s h a l o

f u n c t i o n behaves a t i n f i n i t y l i k e eu .

Tha t t h e h a l o c o n j e c t u r e i s t r u e i n case $ ( u ) - u a t i n f i n i t y

i s an easy consequence o f t h e r e s u l t o f MoriyBn [1978] p resen ted i n 6.5.4.

CHAPTER 7

THE BASIS OF INTERVALS

I n t h i s Chapter we s h a l l ana lyze some i n t e r e s t i n g c o v e r i n g and

d i f f e r e n t i a t i o n p r o p e r t i e s o f t h e b a s i s o f i n t e r v a l s i n Rn . For each

x 6 Rn we cons ide r as @ ( x ) t h e f a m i l y o f a l l open bounded i n t e r v a l s

c o n t a i n i n g x, and B = \I ,@(x). T h i s b a s i s w i l l be denoted as 8, x a R

and i t s maximal o p e r a t o r w i l l be c a l l e d M P . H i s t o r i c a l l y i t was t h i s b a s i s t h e one w i t h which s t a r t e d t h e

expansion of t h e modern t h e o r y o f d i f f e r e n t i a t i o n l o n g a f t e r t h e Lebes-

gue d i f f e r e n t i a t i o n theorem.

has t h e d e n s i t y p r o p e r t y ( s t r o n g d e n s i t y theorem), a l l t h e e f f o r t s t o

extend t h e c o v e r i n g and d i f f e r e n t i a t i o n p r o p e r t i e s o f t h e b a s i s o f c u b i c

i n t e r v a l s ( V i t a l i p r o p e r t y and Lebesgue d i f f e r e n t i a t i o n p r o p e r t y ) t o

s i g n i f i c a n t l y d i f f e r e n t systems o f s e t s were f r u i t l e s s . I n 1924 Banach

proved t h a t 8, does n o t have t h e V i t a l i p r o p e r t y , a r e s u l t a l s o o b t a i n e d

by H.Bohr about t h a t t ime, as appears i n an appendix o f t h e work o f

Carath6odory [1927] . I n 1927 N i kodym ob ta ined ano the r i n t e r e s t i n g resu l t ,

r a t h e r d i s c o u r a g i n g f rom t h e p o i n t o f view o f d i f f e r e n t i a t i o n , f r o m wh ich

Zygmund deduced t h a t t h e b a s i s o f a l l r e c t a n g l e s i n R2 t h e d e n s i t y p r o p e r t y . T h i s r e s u l t o f Nikodym w i l l be p resen ted i n t h e n e x t

Chapter.

U n t i l 1933 , when Banach proved t h a t B ,

does n o t have

A f t e r t h e s t r o n g d e n s i t y theorem o f Saks p 9 3 q t h e p o s i t i v e

r e s u l t s s t a r t e d p i l i n g up w i t h t h e work o f Zygmund [1934] , Busemann and

F e l l e r [1934] , Jessen, Marc ink iew icz and Zygmund [1935], de Posse1 [1936]

y . . .

I n o u r e x p o s i t i o n we s h a l l f o l l o w a more o r l e s s c h r o n o l o g i c a l

o rde r , s t a r t i n g w i t h t h e e a r l y n e g a t i v e r e s u l t s and end ing w i t h t h e more

r e c e n t and i n t e r e s t i n g r e s u l t s o r i g i n a t i n g m a i n l y i n some problems pro-

159

160 7. THE B A S I S OF INTERVALS

posed by Zygmund and so l ved o n l y r e c e n t l y by Marst rand, MoriyBn, CoFdoba, ...

7.1. THE INTERVAL BASIS B 2 DOES NOT HAVE THE VITAL1 PROPERTY. I T DOES NOT DIFFERENTIATE L ' .

Accord ing t o Theorem 6.5.4. o f M o r i y h , f o r a B - F b a s i s

t h a t i s i n v a r i a n t by homothecies, d i f f e r e n t i a t i o n o f L' is e q u i v a l e n t

t o the V i t a l i p r o p e r t y and e q u i v a l e n t a l s o t o t h e r e g u l a r i t y o f t h e b a s i s

w i t h respec t t o t h e b a s i s o f c u b i c i n t e r v a l s . Obv ious l y

s a t i s f y t h e l a t t e r p r o p e r t y .

B2 does n o t

Even i f we c o n s i d e r the minimal B - F

{Ik)

b a s i s i n v a r i a n t by

o f open bounded i n t e r - homothecies t h a t con ta ins a g iven sequence

v a l s we can a f f i r m t h e same i f t h e e c c e n t r i c i t y o f

o f t h e l onger s i d e and t h e s m a l l e r one) tends t o i n f i n i t e . I k ( i . e . t h e r a t i o

7.2. DIFFERENTIATION PROPERTIES OF $2. WEAK TYPE INEQUALITY FOR A BASIS WHICH I S THE CARTESIAN PRODUCT OF ANOTHER TWO.

Though B2 does n o t d i f f e r e n t i a t e L ' m " ) , i t i s a d e n s i t y

bas i s , as Saks [1935] has proved. Moreover i t d i f f e r e n t i a t e s

L ( l t log' L)"' (Fin) , as t h e theorem o f Jessen-Marcinkiewicz-Zygmund

[1935] a f f i r m s . We s h a l l o b t a i n t h i s theorem by c o n s i d e r i n g t h e b a s i s

B2 as t h e i t e r a t e d C a r t e s i a n p r o d u c t o f t h e i n t e r v a l b a s i s o f R' . T h i s

method o f p r o o f belongs t o Guzma'n [1974] . A p rev ious r e s u l t i n t h i s d l

r e c t i o n was ob ta ined by B u r k i l l [1951] .

7.2. DIFFERENTIATION PROPERTIES 16 1

7 . 2 . 1 . THEOREM. Let M 2 be ,the muximde opehcLtoh a6ocicLted Rv

t h e i v t t e h v d b a b dB2 in R 2 . Then, doh each f e Lloc (R’) and each

A > 0 we have

whehe c ~2 a ~ O A L C ~ V ~ condtant, independent v d f and A , and l o g a = 0 id 0 6 a c 1 and l o g a = l o g a -id a > 1.

+ +

P4vv 6. t h e easy, b u t t e d

We p resen t here a proof o f t h e theorem d i s r e g a r d i n g

ous , measurabi 1 i t y problems t h a t a r i s e i n i t .

I n t h e p r o o f we s h a l l i n d i c a t e by / P I 1 and 1Q/ t h e Lebesgue

measure o f t h e measurable s e t P t R1 and Q C R 2 r e s p e c t i v e l y . For t h e

sake o f c l a r i t y we s h a l l denote by Greek l e t t e r s

t h e dummy v a r i a b l e coo rd ina tes which appear i n t h e i n t e g r a l s and d e f i n i -

t i o n s .

(El,<’) , (n1 , r12) , . . .

L e t f > 0 , f 6 Lloc(R2) . Fo r (xl, x 2 ) e R2 we de f ine

1 f(E,1,x2)dE,1 : J i n t e r v a l o f R1,x’e J} K J Tl f (x1,x2) = sup{

f o r A > 0 we cons ide r t h e s e t

and a g a i n we d e f i n e , fo r (x1,x2) Q R 2

T 2 f ( x 1 , x 2 ) = sup{ xA(x1,n2) Tlf(xi,n2)dn2: H i n t e r v a l o f R’,x26HI

We s h a l l f i r s t prove t h e

B = { (5’ , C 2 ) 6 R 2 : M z f ( E 1 , E ’ ) > A

re1 a t i o n

16 2 7. THE B A S I S OF INTERVALS

Take a f i x e d p o i n t (x1,x2) o f B. We w i s h t o p rove t h a t

(x1,x2) a C . Since (x1,x2) a B , t h e r e i s an i n t e r v a l I = J x H of

R2 such t h a t x1a J , x2e H and

We now p a r t i t i o n t h e i n t e r v a l I i n t o two s e t s C1, C z , each one b e i n g

a un ion o f segments o f t h e s i z e of J p a r a l l e l t o t h e a x i s Ox’ i n t h e

f o l l o w i n g way. L e t J x {yz} be one o f such segments. I f f o r each p o i n t

(z ’ ,yz) E J x {y2} we have Tlf (z’,y2) > 2 we s e t J x { y 2 } c C1.

Otherwise, i . e . i f t h e r e i s some p o i n t

T l f (z’,y2) c we s e t J x { y 2 } c C 2 . Observe t h a t J x {y2}CCz

i m p l i e s i n p a r t i c u l a r t h a t

A

(z1,y2) E J x {yz} such t h a t

x

and so, i n t e g r a t i n g t h i s i n e q u a l i t y ove r t h e s e t G o f a l l 4’ i n H such t h a t J x {c21 c C Z , we g e t

Since

We can a l s o w r i t e , by v i r t u e o f t h e d e f i n i t i o n o f T2 and o f T, ,

7.2. DIFFERENTIATION PROPERTIES 163

By t h e d e f i n i t i o n o f C 1 and A, i f (q1,r12) E C1 t hen (xl,rlz) E A

and so t h e l a s t member o f t h e above c h a i n of i n e q u a l i t i e s i s

T h i s concludes t h e p r o o f t h a t B C C. We now prove t h a t C s a t i s f i e s

t h e i n e q u a l i t y we a r e l o o k i n g f o r .

t We can assume f E L ( l t l o g L ) , s i n c e o t h e r w i s e t h e r e i s

n o t h i n g t o prove. I n t h e f o l l o w i n g argument c w i l l be an a b s o l u t e

c o n s t a n t n o t always t h e same i n each ocurrence, independent i n p a r t i c y

l a r o f f and A . By v i r t u e o f t h e weak t ype (1 , l ) f o r t h e un id imens iona l b a s i s

9 1 o f i n t e r v a l s , f o r a lmos t each f i x e d x’e R we can w r i t e

Hence, i f we i n t e g r a t e over a l l such X’E R and in te rchange t h e o r d e r o f

i n t e g r a t i o n , we g e t

x I f ( t1 ,c2) 6 A , then Tlf(<1,<2) > 7 , and so i f 0 < u & 1,

we have

164 7 . THE BASIS OF NTERVALS

R’ , we get

I n order t o estimate Sz we define f o r a fixed 0 > 0

and f*(t1,t2: 5 ) such t h a t

For brevity l e t us wr i te f = f z + f: . I t i s c l ea r t h a t

A 5 and T l f z 6 Hence,

7.3. SAKS RARITY THEOREM 165

Adding up we get

and this implies the inequality of the theorem.

For some generalizations of this type of results one can see

Guzm6n [1975] .

7.3. THE HALO FUNCTION OF 8 2 . SAKS RARITY THEOREM.

The halo function of B2 can be easily estimated from below from the following geometric observation which will also be useful in the proof of the rarity theorem of Saks [1935] .

A u U a / r y cvnhn;ttruotivn. Let H be an integer bigger that 1

and consider in R 2 the collection of open intervals 1 1 , 1 2 , ..., IH obtained as ind icated in Figure 7.3.1. (where H = 3).

Figure 7.3.1.


Each I j i s an open i n t e r v a l w i t h a v e r t e x a t 0 , a s i d e on t h e pos i -

t i v e p a r t of Ox w i t h l e n g t h j, and ano the r on t h e p o s i t i v e p a r t o f Oy

w i t h l e n g t h - . Hence t h e area o f I i s H, t h a t o f t h e i n t e r s e c t i o n

E= 0 I j i s 1 and t h a t o f t h e un ion

H j j

H

j=1

From t h i s c o n s t r u c t i o n we o b t a i n

1 CM2 XE > %I 3 JH

s i nce . Hence f o r each H

As we have a1 ready seen,

Therefore t h e h a l o f u n c t i o n @ z of & ( R 2 ) s a t i s f i e s

C l U ( 1 + l o g + u) h @2(u) G czu( 1 + l o g + u)

and analogously i n Rn

C lU(1 + l o g + G @ Z ( U ) h c2u(1 + l o g +

Hence, acco rd ing t o t h e c o n s i d e r a t i o n s o f Theorem 6.6.2 we

deduce t h a t ?Bz

L ( l t log' L)"' (R'), i . e . i f I) : [0,m) +.

does n o t d i f f e r e n t i a t e any space worse.than

[O,m) i s such t h a t

7.3. SAKS RARITY 167 THEOREM

as U + m

f o r i n s t a n c e +(u) = ' u ( 1 + log t uln- '-€ , then &mn) does n o t d i f

f e r e n t i a t e +(L).

Saks [1935] has proved a s t r o n g e r r e s u l t . F o r "a ln iost a l l "

f u n c t i o n s f o f L 1 , "almost a l l " i n t h e sense o f B a i r e ' s ca tegory , one has [I( I f , x ) = + m a t each x 6 Rn . The p r o o f o f Saks uses a c o n s

t r u c t i o n of H.Bohr . Here we s h a l l make use o f t h e a u x i l i a r y c o n s t r u c t -

i o n o f Sec t i on 1.

7.3.1. THEOREM . The o e t F 06 &~nction.4 f i n L ' ouch

t h a t D ( / f , x ) < 00 at borne po in t x e Rn LA 0 6 t h e &Oi~,t ca;tegohq i n

L' .

Ptrao6. For t h e sake o f c l a r i t y , we s h a l l p r e s e n t t h e p r o o f f o r

n = 2 . We show t h a t F i s t h e un ion of a coun tab le c o l l e c t i o n o f nowhere

dense s e t s i n t h e f o l l o w i n g way. Fo r k = 1,2,3, ... we d e f i n e Fk

as t h e s e t o f f u n c t i o n s f i n L 1 such t h a t f o r some p o i n t x e R 2 , 1 w i t h 1x1 c k , i t happens t h a t f o r a l l I e-od,(x) w i t h 6 ( I ) < T;

r m

we have 11; 1 c k l I ( . We c l e a r l y have F =

t h a t each Fk i s nowhere dense, o r , what i s e q u i v a l e n t t h a t Fk has no

i n t e r i o r p o i n t s .

(I k = l

Fk . We now prove

For each k we have Fk = Fk . I n f a c t , assume t h a t f -f f i n j

j y L1 and f . 6 F k f o r j = 1,2,... Fo r each j t h e r e i s some p o i n t x

w i t h J x . 1 c k, such t h a t i f I e%,(x j ) and & ( I ) < T; , then 1 J J

I f , f j / c k l I / . S ince B(O,k ) i s compact, one can e x t r a c t f r o m ( x . 1

a convergent subsequence. We can assume, changing n o t a t i o n i f necessary,

t h a t x j

i s such t h a t 6 ( I ) < we can w r i t e

J

converges t o a p o i n t x . C l e a r l y , 1x1 < k and i f I e Q 2 ( x ) 1

168 7 . THE B A S I S OF INTERVALS

Since I i s open, x E I and x . + x , we have I ( x j ) f o r j

as j -f m , and we g e t 1 f l h k l 11 , p r o v i n g t h a t f e Fk. Therefore

pk = F

J s u f f i c i e n t l y l a r g e , and so (1, f j ( G k l I l . Fur thermore ,

i, k '

I n o r d e r t o p rove t h a t Fk we s h a l l use t h e f o l l o w i n g lemma which c o n s t i t u t e s t h e k e r n e l o f t h e p r o o f

o f t h e theorem. The lemma j u s t means t h a t f o r each neighborhood V o f

t h e o r i g i n of L ' and f o r each k = 1,2, ... t h e r e i s a f u n c t i o n $

i n t h a t neighborhood which i s n o t i n

does n o t c o n t a i n any i n t e r i o r p o i n t

k ,V Fk . I t s p roo f i s g i ven l a t e r .

7 . 3 . 2 . LEMMA. Foh each Matwlae numbm N ,thehe a nonnaja-

LLwe 6unCtion +N huch ,that

With t h i s lemma t h e f a c t t h a t F k does n o t have any i n t e r i o r

q z 0 p o i n t s i s e a s i l y obta ined.

t h e r e i s a f u n c t i o n g E L1 - F k such t h a t ( ( g - f ( ( , g q. L e t

h e Lm o L ' be such t h a t (If - h i l l 6

where $N i s t h e f u n c t i o n o f t h e lemma wi th an N t h a t w i l l be chosen

i n a moment. We can w r i t e

L e t f c Fk . We prove t h a t f o r each

q /2 . We d e f i n e g = h + +N

Accord ing t o (b ) o f t h e lemma, f o r each x 6 B(O,N), t h e r e 1 e x i s t s an i n t e r v a l I 6 B z ( x ) , w i t h 6 ( I ) < such t h a t

7 . 3 . SAKS RARITY THEOREM 169

1 We now choose N such t h a t N x k , c , N - I l h l Im > k . Then we have

119 - f / I i c n and

Hence g d Fk as we wanted t o prove.

P m v d v d t h e Lemma 7.3.2. For t h e proof of t h e lemma we s t a r t

o f t h e beg inn ing o f t h i s S e c t i o n w i t h t h e s imp le a u a u h y c v M n ~ u c 2 i v n w i t h an H t h a t w i l l be c o n v e n i e n t l y f i x e d i n a moment. By us ing lemma

6.4.4. we can cover a lmost

d i s j o i n t sequence I S k } o f

i a r y c o n s t r u c t i o n con ta ined

L e t R = B(O,N) - (I S k .

we can t a k e an open s e t G

co

1

comp le te l y t h e b a l l B(O,N) by means o f a

s e t s homothet ic t o t h e s e t J,, o f t h e a u x i l

i n B(0,N) and w i t h d iameter l e s s than 1/N.

We have IRI = 0 and so, f o r each E > 0

c o n t a i n i n g R and such t h a t [GI b E . For

each x E R we t a k e an open c u b i c i n t e r v a l I ( x ) cen te red a t x w i t h

d iameter less than 1/N con ta ined i n G. We a p p l y t o ( I ( X ) ) ~ € ~ t h e

theorem o f B e s i c o v i t c h o b t a i n i n g {Ik} so t h a t

0 b e i n g an abso lu te cons tan t .

L e t us c a l l E k t h e s e t ob ta ined f rom E o f t h e a u x i l i a r y

c o n s t r u c t i o n by t h e same homothecy t h a t c a r r i e s J H i n t o S k . We now

d e f i n e t h e f o l l o w i n g f u n c t i o n s

0 i f x 6 Ek I

Then

and so

1

4N2@ i f E < - ,

Now i f x E B(O,N) i s i n some s k , then i t w i l l be in some o f

the intervals I: , j = 1,2,...y H composing Sk and IJk has diameter

less t h a n 1/N. So we get

+ R . I f we choose H so that 1 2 where a ( H ) = 1 + - + .. .

I f x B Sk then x € I k for some k and by the definit ion Of Pk and ON 7

7.4. A THEOREM OF BESICOVITCH 171

@N > l l k l * 'k

Th is concludes t h e p r o o f o f t h e lemma.

7.4. A THEOREM OF BESICOVITCH ON THE POSSIBLE VALUES OF THE UPPER AND LOWER DERIVATIVES WITH RESPECT TO B2.

We cons ide r t h e b a s i s @I 2 i n R2 . L e t f e L ' ( R 2 ) . S ince

8 c 6, and @, d i f f e r e n t i a t e s / f, i t i s easy t o see t h a t w i t h r e s p e c t

t o t h e b a s i s 9 2 we have

a t a lmost eve ry

can happen t h a t t h e s e t s

x e R2 . However, acco rd ing t o t h e p rev ious s e c t i o n s , i t

C X 6 R 2 : f ( x ) < D ( / f , x ) }

{ X e R 2 : - D ( 1 f , x ) < f ( x ) }

have p o s i t i v e measure.

posed t h e f o l l o w i n g ques t i on : Can any 0 4 the. ne&

With r e s p e c t t o t h i s s i t u a t i o n Saks [1934] pro-

C X E R 2 : f ( x ) < iS(1 f , x ) < m}

Cx e R 2 : -a < - D ( 1 f , x ) < f ( x ) }

be 06 puoLt iwe rneaute.? The n e g a t i v e answer i s due t o B e s i c o v i t c h [1935]. Here we p resen t t h e r e s u l t of B e s i c o v i t c h .

show how t h e theorem can be somewhat extended.

I n t h e remark a t t h e end we

7.4.1. THEOREM . We comida t h e i n t a u a k ? b a d f12 i n R 2 .

LeZ f e L' (R2) be a dixed dunc t i an . Then t h e &a neA

{ X E R' : f ( x ) < II( if,.) < a)

have m u m e z a a .

P m o d . We s h a l l c a r r y o u t t h e p r o o f f o r t h e f i r s t o f t hese se ts .

For t h e o t h e r s e t one can p u t and app ly t h e r e s u l t f o r t h e f i r s t

s e t t o g.

g = -f

I t w i l l be enough t o prove t h a t f o r a , B , y r a t i o n a l such

t h a t 0 < a < B i y , t h e s e t

E(ii,B,y) = { x 6 R2 : f ( x ) -t a < D ( f , x ) < f ( x ) -t p , - y < f ( x ) < y ) f has measure zero.

L e t us assume t h a t t h e r e a r e t h r e e numbers cx , B ,y as above

so t h a t t h e s e t E = E ( a , B , y ) i s o f p o s i t i v e measure.

f = f -t fy where

L e t us s e t

Y

Since ' @ z d i f f e r e n t i a t e s La and f e Lm, we have Y

D ( f ,x) = f ( x ) a lmost everywhere. L e t us c a l l t h e subse t o f E

where D(Jfy,x) = f y ( x ) . We have $ 1 = \El > 0. f Y Y

I t x E we have

and t h e r e f o r e , hav ing i n mind t h e d e f i n i t i o n o f E, i f x e E c E we have

7.4 . A THEOREM OF BESICOVITCH 173

< f y ( X ) + fY(X) + B

and -y < f ( x ) < y , i . e . f y ( x ) = 0

- Hence, i f x E E we can w r i t e

a < a ( / f Y , X ) < B

and f y ( x ) = 0 . Thus t h e s e t

has p o s i t i v e measure. We now i n t e n d t o prove t h a t t h i s l eads t o a con t ra -

d i c t i o n and t h i s w i l l conclude t h e p r o o f o f t h e theorem.

A

For each s, w i t h 0 < s < 1 , we de f ine E, as t h e subse t o f

p o i n t s x o f E such t h a t f o r each I S @ ~ ( X ) , w i t h 6 ( I ) < s , one

has

A

A

The s e t i s t h e union o f a l l E s w i t h s 6 Q , s > 0, and so t h e r e must

be some s* > 0 such t h a t Es* i s o f p o s i t i v e e x t e r i o r measure. L e t E* be a subse t o f Es* w i t h f i n i t e p o s i t i v e e x t e r i o r measure . L e t H be an

open s e t such t h a t H 3 E* , I H I 6 ( l + n ) I E * l e w i t h an 17 > 0 t h a t

h

w i l l be c o n v e n i e n t l y chosen l a t e r .

For each x 6 E* one can choose I ( x ) = I C H, I = I ( X ) €@2(X),

w i t h 6 ( I ( x ) ) < s* such t h a t

We s h a l l now app ly the f o l l o w i n g lemma whose p r o o f i s p resen ted a t t h e end.

7.4.2. LEMMA. 1e.Z I be any

t h u t 604 u c h x E I we have & h a fy

UA M h U m C Ah&

open intmvd and

x ) = 0 o 4 I f q x

fy E L' nuch

I > y . L e i

Then one can chaooe a d i n j a i n t bequence

i n I ouch t h a t each k

{Ik} 0 6 open i n t e h v d contcuned

To each I ( x ) we app ly t h i s lemma, o b t a i n i n g { I k ( x ) } s a t i s -

f y i n g I k ( x ) c I ( x ) C H,

f o r each k .

L e t now be G = I J I ( x ) XEE*

U = U { I k ( x ) : x E E* , k = 1,2, ... 3

1 ?.

and l e t E = G - U . C l e a r l y E i s measurable, s i n c e G and U a r e open.

Furthermore E 3 E*. I n f a c t , f o r each , y E E * C ES, and each i n t e r

Val I E @ 2 ( y ) w i t h 6 ( I ) i s* we have, by t h e d e f i n i t i o n of ES, ,

- h

h

However, f o r each one o f t h e s e t s I k ( x ) composing U we have

6 ( I k ( x ) ) < s* and

I

So y 1 U. T h i s i m p l i e s E*c G - U = E.

and so

Hence, by

on a/y ,

each I ( x

(1

7.4. A THEOREM OF BESICOVITCH

Now f o r each I(x) w i t h x e E* we have

G c { x e R 2

175

Theorem 6.4.3. we have GI s c ) U \ . Where c depends o n l y - -. Therefore ( E l c c [ U ( . We c l e a r l y have U U E C H s i n c e

i s i n H. So we g e t

1 T h i s i s imposs ib le i f we choose II < c . T h i s c o n t r a d i c t i o n conlcudes

t h e Droo f o f t he theorem.

P m u B vd t h e Lemma. 7.4.2. We t a k e t h e i n t e r v a l I and per form

on i t t h e f o l l o w i n g process P. If d i s t h e s m a l l e r s i d e and ( d l 1 i t s

l e n g t h , D i s t h e b i g g e r s i d e and \ D J 1 i t s l e n g t h , we d i v i d e D i n t o

equal p a r t e s o f l e n g t h between '!!L and

i n g p o i n t s , we draw through them l i n e s p a r a l l e l t o d. We g e t a p a r t i t i o n

o f I i n t o a c e r t a i n number o f p a r t i a l i n t e r v a l s {IT . 1; ,..., 1; 1

such t h a t t h e r a t i o between t h e i r b i g g e r s i d e s and t h e i r s m a l l e r ones is between 2 and 4. I f t h e mean o f fy on each one o f t h e p a r t i a l i n t e r v a l s

i s l e s s than y, then t h e process P on I i s f i n i s h e d . I f t h e r e i s some

p a r t i a l i n t e r v a l It such t h a t t he mean o f f on 1: i s b i g g e r t h a n o r

equal t o y, we t a k e a maximal i n t e r v a l con ta ined i n I and c o n t a i n i n g

13 such t h a t t h e mean o f fy on i t i s e x a c t l y y. I n t h i s way we have

p a r t i t i o n e d I i n t o a f i n i t e d i s j o i n t c o l l e c t i o n Q o f i n t e r v a l s on

which t h e mean o f fy i s y and another f i n i t e d i s j o i n t c o l l e c t i o n %?,* c o n s t i t u t e d by t h e i n i t i a l p a r t i a l i n t e r v a l s o r by those p a r t s o f

them ob ta ined i n t h e process o f c o n s t r u c t i o n o f t h e i n t e r v a l s o f a;! . On t h e i n t e r v a l s o f '5Lf t h e process P on I .

. Once we have t h e d i v i d

J

IJ

t h e mean o f fy i s l e s s than y . T h i s f i n i s h e s

176 7 . THE BASIS OF INTERVALS

We now keep t h e i n t e r v a l s o f ;R and on each i n t e r v a l of ?R * we pe r fo rm t h e same process P. I n t h i s way we g e t a sequence {Ik) of

d i s j o i n t i n t e r v a l s , w i t h (I I k c I and such t h a t

On t h e o t h e r hand, i f f e I - (J I k , then t h e r e e x i s t s a sequence of i n t e r v a l s H k ( x ) e @ 2 ( x ) c o n t r a c t i n g t o x such t h a t t h e r a t i o

between t h e b i g g e r s i d e an t h e s m a l l e r s i d e i s between 2 and 4 such t h a t

Since F e L , and the sequence IHk(x) I i s r e g u l a r w i t h r e s p e c t t o

squares, we g e t

f o r almost eve ry x B I - I! I k and, t h e r e f o r e , f y ( x ) c y and so

f Y ( x ) c o a t a lmost eve ry x e I - L J I ~ . SO we can w r i t e

p l I J = I, f Y = f y ' I,,, k f Y h I,,, f y S Y I \ J I k I '1- I J Ik k

T h i s concludes t h e p r o o f o f t h e lemma.

For an ex tens ion o f t h e p rev ious theorem one can see Guzmin

e 9 7 5 1 .

7.5. A T H E O R E M OF MARSTRAND 177

7.5. A THEOREM OF MARSTRAND AND SOME GENERALIZATIONS.

According to the r a r i t y theorem we have seen in 7.3. there i s a function f e L1(R2) such t h a t 6( f , x ) = + m a t a . e . x with respect t o 8 2. Zygmund proposed the following problem (see Guzma'n and We1 land [1971] ) : f e L1(R'). I s i t pos- s i b l e t o choose a p a i r of rectangular d i rec t ions so t h a t t he bas i s of a l l rectangles in those d i rec t ions d i f f e ren t i a t e s f a t a , e . x e R 2 ?

I Suppose we a re given a function

I That the answer i s negative was proved f o r the f i r s t time by

Marstrand 119771, , who found a function f e L1 (R2) such t h a t f o r every o r i en ta t ion o f the axes y we have B ( f , x ) = t m a t a.e. x with respect t o the rectangles in d i rec t ion y. Later on E l Helou [19781 found such an f belonging t o (1 L ( l t log' L)'. More recently Bernard0

L . Me1 ero [I9781 has general i zed the method of Marstrand showing t h a t i t works f o r any t r ans l a t ion invar ian t B - F bas i s and t h a t the s i t u a t i o n i s then s imi la r t o t h a t o f the Saks' r a r i t y theorem. Here we sha l l f i r s t present the r e s u l t o f Marstrand. which Melero obtains from t h i s method.

i O<a<l

Then we ind ica t e the general r e s u l t

7.5.1. JHEUREM. Thehe A a ljunotian f E L' @lz) buch t h a t

I ~.

doh ewmy dine.Otion y we. have 6,( f , x ) = +m n . e . w L t h hapec t t o

t h e heotangben ,in t h a t dihecLLon.

Phoolj. We shal l cons t ruc t f E L ' ( Q ) , Q u n i t square such t h a t o r ( f , x ) = +- a t a . e . statement. Let , f o r 0 < r < 1,

x E Q . From t h i s one eas i ly obtains the above

where B(0 , r ) means the open ball of center 0 and radius r . The s e t H(r) looks l i k e F i g . 7.5.1. shows.

F i g u r e 7.5.1. I t i s n o t d i f f i c u l t t o g e t t h e e s t i m a t e f o r smal l r

H ( r ) 2 cr (1 + l o g r 1 )

Fo r each k we f i x an i n t e g e r Mk > 0 t h a t w i l l be conven-

i e n t l y chosen l a t e r on and we d e f i n e a f u n c t i o n

way. Consider Q ( k , j ) , j = l y 2 , ..., 2 k , t h e ZMk dyad ic subsquares

o f o r d e r Mk o f Q. L e t B ( k , j ) be t h e b a l l c o n c e n t r i c w i t h Q ( k , j ) ,

t angen t t o Q ( k , j ] . rk , 0 < rk < 1, t h a t w i l l a l s o be chosen

l a t e r , we take H(k, j) homothet ic t o H ( r k ) w i t h t h e same homothecy

t h a t sends B (0 , l ) t o B ( k , j ) . L e t C (k , j ) be t h e b a l l homothe t i c

t o B(0yr-k) by t h e same homothecy. (See F i g . 7 . 5 . 2 . , where

fk i n t h e f o l l o w i n g M

Wi th an

Mkl = 2 , Mk2 = 4 ) .

F i g u r e 7.5.2.

7.5 . A THEOREM OF MARSTRAND 179

The f u n c t i o n fk w i l l have cons ta t i t v a l u e X k > 0 ( t o be

chosen l a t e r ) on each C ( k , j ) and w i l l be 0 o u t s i d e t h e i r un ion . Fo r

any f i x e d d i r e c t i o n y l e t H(k, j ,y ) be t h e s e t c o n c e n t r i c w i t h H ( k , j )

ob ta ined by r o t a t i n g H ( k , j ) .

I f x e H(k , j , y ) t h e r e i s a r e c t a n g l e I i n d i r e c t i o n y such t h a t

From now on we f i x t h i s d i r e c t i o n .

We have I C ( k ) l = C rt , w i t h c a b s o l u t e cons tan t . A l so

Take now f = sup f k . Then

SO if we want f e L1 i f s u f f i c e s t o s e t 1 akrk < m .

Observe now t h a t i f x belongs t o i n f i n i t e l y many s e t s K(k,y)

and ak -f m we o b t a i n , by t h e r e l a t i o n (*) above, 0 ( Y

So we w i l l t r y t o ar range our cons tan ts so t h a t

l l i m sup K ( k , y ) \ = 1 k - t m

Since

co m

Q - l i m sup K(k,y) = (2 - (1 I J K(k,y) = IJ f) (Q - K(k,y)) k + m 1=1 k h l 1=1 ka1

we need I f) ( Q - K ( k , y ) ) l = 0 f o r each 1. We achieve t h i s by means of k a l

t h e f o l l o w i n g lemma t h a t w i l l be proved l a t e r .

180 7. THE BASIS OF INTERVALS

LEAiMA. 7.5.2. L e t A,S be -two meawlabEe b&tA i n the. uvLit

bquatle Q und E > O . Covlcriden t h e 2k d y a d i c bubhquaheh Q ( k , j )

j = 1,2,...,2k 06 o t d e h k 0 6 Q . l e t S(k ) be t h e union a d t h e 2k

S ( k , j ) C Q ( k , j ) , whene S ( k , j ) & homotheLLc t o S by ,the hume

homvthecy t h u f CWU Q i n t o Q ( k , j ) . Then id k A budl;icievLtey b i g

With t h i s lemma we lsroceed as . f o l l o w s . Take MI = 2 . App ly

t h e lemma w i t h A = Q - K(1,y) S = Q - H ( r 2 , y ) , where H ( r 2 , y ) i s

H ( r 2 ) i n d i r e c t i o n y . So i f MZ i s b i g enough we g e t

We now a p p l y a g a i n t h e lemma w i t h

and o b t a i n , if M 3 i s b i g enough,

3 3

Observe t h a t t h e second member i s i ndependen t o f y . So, i f we a r e a b l e

t o choose rk so t h a t

we g e t l l i m sup K(k, - { ) I = 1 as d e s i r e d .

I n o r d e r t o a c h i e v e

k +

7.5. A THEOREM OF MARSTRAND 181

Pnmd 06 ,the Lemma. We take k b i g enough so t h a t t h e r e i s

G = II Q ( k , j ) , G 3 A I G I X (1 f E ) I A l j BJ

Then

For a b a s i s @3 i n Rn t h a t i s i n v a r i a n t by homothecies we

can d e f i n e f o r 0 < r < 1

Then t h e genera l r e s u l t ob ta ined f o l l o w i n g t h e same p a t t e r n o f p r o o f i s

as f o l l o w s .

-_ THEOREM. 7.5.3. Let Y(1,m) -f (0,m) be an incneaning convex 6unotion nuch t h a t

L.Melero s t u d i e s i n t e r e s t i n g p a r t i c u l a r cases. The r e s u l t o f 1 Y ( u ) = u l o g u(1og l o g u ) - , E l Helou i s ob ta ined f rom t h i s one by s e t t i n g

u > e.


The r a r i t y r e s u l t i s ob ta ined by showing t h a t , f o r eve ry

b,s > 0, t h e s e t o f f u n c t i o n s

each y we have

f E Y ( L ) w i t h t h e p r o p e r t y t h a t f o r

con ta ins a dense open subse t o f Y ( L ) , and t h a t eve ry f i n a c e r t a i n

denumerable i n t e r s e c t i o n o f s e t s o f t h i s t ype i s as bad as t h e theorem

s t a t e s .

7 . 6 . A PROBLEM OF ZYGMUND SOLVED BY MORIY~N.

Consider i n R 2 t h e f o l l o w i n g b a s i s 3. For each x e R 2 , B ( X ) w i l l be t h e c o l l e c t i o n o f a l l open bounded i n t e r v a l s c o n t a i n i n g x and

such t h a t , if d i s t h e l e n g t h o f t h e s m a l l e r s i d e and D i s t h e l e n g t h

o f t h e l a r g e r s i d e one has What a r e t h e d i f f e r e n t i a t i o n

p r o p e r t i e s of t h i s b a s i s B? O f course one knows t h a t $3 d i f f e r e n t i a t e s

L ( l f l o g L ) b u t one c o u l d perhaps expec t something b e t t e r . Tha t i t

i s n o t so has been proved by MoriyGn [1975] .

D2 6 d G 0 6 1.

i.

Phood. Tha t @ d i f f e r e n t i a t e s L ( l f l o g t L i s obvious,

s i n c e J3 i s a subbasis o f 8 2 , t h e b a s i s o f a l l i n t e r v a l s . I n o r d e r

t o prove t h e second p a r t o f t h e theorem we proceed as f o l l o w s .

For each x e ( 0 , l ) , Q, w i l l denote t h e f o l l o w i n g i n t e r v a l

of R 2 : Qx = (0,~~'~) x (0,x ) and cx w i l l be t h e e q u i l a t e r a l hyper-

7 .6 . A PROBLEM OF ZYGMUND 183

bola passing through (x,x2) , as in the following picture

1 I x3/2x

Figure 7 .6 .1 .

Clearly, each rectangle o f 4 with a vertex a t (0,O) and the opposite one on cx contains Q, and has the same measure, i . e . i f

0 6 [x3”,x] , Q,,, = (O,,) x ( 0 , x 3 0 - l ) then / Q X y e / = x 3 . A l s o

S ( Q x , , ) & 2x . Hence, since

we get for each x E ( 0 , l ) and each 0 E [x3/’,x] ,

where M r , for r 0 , means the maximal operator associated t o the se t s of “a3 with diameter less t h a n r . So we have


I f @ d i f f e r e n t i a t e s O(L), accord ing t o Theorem 6.4.8. t h e r e

e x i s t r > 0 and c > 0 such t h a t f o r each measurable f u n c t i o n f and

f o r each A > O

I C Y E R’ : Mrf(y) > All & c i @(8)ds x (**)

r Hence, if x 6 x Q = min(1,T) , hav ing i n t o account ( * ) and

I f we s e t I ~ x - ~ ” = 1 , then we g e t f o r h >lo = 16 ,

~ ( h ) > k l ( 1 -+ log’ A )

and t h i s o b v i o u s l y i m p l i e s t h e s tatement o f t h e theorem

7.7. CGVERING PROPERTIES OF THE B A S I S OF INTERVALS. CORDOBA AND R. FEFFERMAN.

A THEOREM OF

The maximal o p e r a t o r a s s o c i a t e d t o t h e b a s i s o f i n t e r v a l s i n

R 2 s a t i s f i e s the f o l l o w i n g weak t y p e i n e q u a l i t y

7.7 . C O V E R I N G PROPERTIES 185

Accord ing t o Theorem 6.3.1. v a l s i n R 2 s a t i s f i e s a good cove r ing p r o p e r t y :

of C6rdoba and Hayes, t h e system o f i n t e r

Given any c o l l e c t i o n o f i n t e r v a l s (Ia)aeA such t h a t 1 ~ ~ 1 ~ 1

we can choose a f i n i t e sequence 11,) f rom ( Ia)asA s a t i s f y i n g

w i t h c1,c2 independent o f ( Ia)aeA . That i s , t h e s e l e c t e d {Ik} cover

a good p a r t o f LJI, and they have a v e r y sma l l ove r lap .

+ However, observe t h a t t h e i n v e r s e f u n c t i o n o f @ ( u ) = ( l + l o g LI),

u 2 0 behaves a t i n f i n i t y l i k e t h e f u n c t i o n $(u) = eu and n o t l i k e

e

r i g h t c o n j e c t u r e seems t o be o b t a i n e d by s u b s t i t u t i n g ( b ' ) by

,1/2 . So one c o u l d expect a s t i l l b e t t e r c o v e r i n g p r o p e r t y f o r $ 2 . The

and so i t was fo rmu la ted i n Guzmdn [1975,

1B2 i n Rn t h e r i g h t o v e r l a p p i n g i nequa l

i n t e r v a

p r e s e n t

We cons

p.1651 . I n a s i m i l a r way f o r

t y i s

The p r o o f t h a t t h i s was indeed t h e r i g h t c o v e r i n g theorem f o r

s was ob ta ined by Cdrdoba and R.Fefferman [1976] . Here we s h a l l

t h e easy geometr ic p r o o f t h a t they g i v e of t h i s theorem f o r R? der the system o f dyad ic i n t e r v a l s of R 2 , i . e . t h e i n t e r v a l s

o b t a i n e d by t h e C a r t e s i a n p roduc t of dyad ic i n t e r v a l s o f R ' . There i s

no fundamental d i f f e r e n c e i n what t h e c o v e r i n g p r o p e r t y respec ts and

t h i s system i s e a s i e r t o handle. We s h a l l make use o f t h e weak t ype

i n e q u a l i t y f o r t h e maximal o p e r a t o r M w i t h respec t t o t h i s system of i n t e r v a l s .


where A i s any measurable s e t o f R ? By means o f i t we e a s i l y p r o v e

t h e f o l l o w i n g lemma.

7.7.1. LEMMA. L e Y {Bk } be. a 6 i n i i c crcyuence 0 6 dyadic i n -

tehvden ad R 2 .Then we can creXe.ot 6hvm them C R k I no th&

1 (b ) For each k , I R k 0 L J R j l 6 / R k ] j # k

ffem c an abno&u;te coM2atant.

P ~ o a d . We choose f i r s t R: = B, , and l o o k a t B, . 1 (1) If I B 2 II R:] L 5 1B21 , then we s e t R; = B Z .

, then we l e a v e B Z as ide . 1 (2 ) If ( B z " R:l > 5 ( 8 2 1

Assume (1) happens. We l o o k a t

1 (1) I f I B 3 I\ (I R:l L 5 I B 3 I , then R'; = B3

B3. 2

j=1 1 2

( 2 ) I f I B 3 I ) I! R? > 5 I B 3 1 , then we leave B 3 as ide . Assume j=1 J

now t h a t ( 2 ) happens. We l o o k a t BI, 2

1 (1) I f (BI, 11 1J R j \ c 5 ( B 4 ( , then R: = B4 j =1

1 2 (2) I f (B, 11 ( I R j 1 > I B 4 1 , we leave B, as ide .

L I n t h i s way we o b t a i n tRi} k=l . For each B t h a t has been l e f t

as ide, we have

j = 1

j

7 . 7 . COVERING PROPERTIES 187

and so ( fo r the chosen ones t h i s i s obv ious)

On the other hand, f o r each k

- L -" - We now consider the sequence f R k l k = l where RI= R r R P = R t - l , . . . ( i . e . we reverse the order of { R E 1 ) and proceed with C R k I k z 1 - have done with { B k l obtaining i R k ) t h a t s a t i s f y

as we

We now prove the following easy lemma f o r dyadic in t e rva l s i n R1 .

7 . 7 . 2 . LEMMA. L e I S = {I , ) be a &hi.Xe oequence 0 6 diddehent

dyadic inte~vu.42 0 4 R1 nuch t h a t 404 each I k we have

Then auk each r = O Y l y 2 , 3 , ... we have

whme c an a b n a U e cvnbAant.

P400d. We can c l ea r ly wr i te

A, = { x : lx ( x ) 2 r + 11 = IJ { x E Ik : x belongs t o a t l e a s t r Ik k


r 1 s e t s r j con ta ined i n I k l f IJ I k , and we t r y t o prove I I C I 6 - 1 1 I . k 5r

1 I f r = 1 we have = ]Ik') I I 1.1 c 5 ] I k ] by the I jCI , J

hypothes is .

any k and f o r r = 1,2 ,..., h . L e t r = h +1 and l o o k a t 1;" as

a subset o f 1; . The s e t 1; i s a d i s j o i n t un ion o f elements of

S = { I k } . L e t

Assume now t h a t t h e i n e q u a l i t y I I [ ~ G -+ l I k j i s t r u e f o r 5

N

1=1 r; = II 17

From t h e d e f i n i t i o n s we c l e a r l y have

I'i f l 1;+1 = I'ih

Therefore, a p p l y i n g t h e i n d u c t i v e hypo thes i s ,

Adding up over k

But, if we assume t h a t t h e I k a r e o r d e r e d by s i z e 1111 >, 1121 2 ... ,

7.7. COVERING PROPERTIES 189

C Therefore IA,I 6 - 1 1 ) Ikl .

!jr

With these two lemmas t h e f o l l o w i n g theorem i s easy.

Ptiood. F i r s t we t a k e a f i n i t e sequence EB,} f rom (Ba)acA

1 w i t h measure g r e a t e r t han

o b t a i n i n g a f i n i t e sequence

1uB,I, Then we app ly lemma

{ R k } s a t i s f y i n g

7 . 7 . 1 . t o { B k }

and

Observe t h a t , by t h e preceding i n e q u a l i t y , no s e t of { R k }

i n ano the r o f {Rk).Let us c a l l ak ,bk t h e l e n g t h o f t h e s i d e s o f Rk

p a r a l l e t t o O x and Oy r e s p e c t i v e l y . L e t us take any l i n e t p a r a l l e l

t o Ox and c a l l

i s con ta ined

Hence t h e j * I f I j c Ik t hen ak r a and so bk < b

j two-dimensional i n e q u a l i t y

190 7. THE BASIS O F INTERVALS

1 I 0 Ij11 5 l l k l l r j I J

and therefore , applying lemma 7 . 7 . 2 . ,

C I{x E t : CxI (XI r + 111 6 - 1 0 1 ~ 1 1

k 5r

and so

An aeteAulative pmod ad Theohm 7.7.3. We now present another proof of the covering theorem f o r dyadic in t e rva l s in R 2 t ha t i s in te r - es t ing in i t s e l f and will give us a method t o solve another r e l a t ed pro blem.

Let (Ba)aEA be a co l lec t ion of dyadic in t e rva l s of R2 w i t h . We choose f i r s t {Bkj! so t h a t 1 LIB,\ 6 2 1 i~ B k / . Let us M

I [,Ba( <

ca l l t h e i r side-lengths a,&,. We can assume tsl r 6 ? k... 6, , and also t h a t there i s no 8 k contained i n another one. We choose now

R , = B1 . Assume t h a t R 1 ,.. . , Rh have already been chosen a n d l e t R . = El . We then choose as Rhtl the next Bk in the sequence J

where ri wi l l be fixed i n a moment.

7.7. COVERING PROPERTIES 191

H So we o b t a i n { R j l l w i t h s i d e s a b t h a t s a t i s f y

b l a bZ 2 ... a b H , al i( a 2 G . . . G aH ,

j ' j '

and a l s o j -1

CXRK

I R . 1 7 I J R I = - e dx 6 e e e l+ ' l e I R j I ' 1 0 R k k < j Rj k < j

k < j 'I R . 0 IJ R k

The re fo re f o r each j

and so

Fur thermore we have

Then

h h h 2oe I ;J R j l + e ( l+q ) [Rhtll L 20el IJ R j l+ e(l+rl) [Rh+l - 11 R . 1 c c

1 i J 1- l+rl e

ti i f we choose rl s u f f i c i e n t l y small . So t h e tR.1 s a t i s f y

J i

H

IJ Rj

t i H H We now try to prove I LJ Bk - LJ Rj I c cI IJ R. I .

1 i J

Take any B E iBkIM that has not been chosen.

intervals with bk > 6 then we have

1

Let us intersect B y R1,R Zy...,RH by a line

If R1 ,.. .,R1 are the

paralell t Ox and let the intersections be called two-dimensional inequality

S , I ,... ,IH. Since a < ak , 1 < k c 1, the 1

1

transforms into the one-dimensional one

and so, if M, is the maximal operator with respect t o intervals of s ,

we have

1

There fore

7.8. ANOTHER PROBLEM OF ZYGMUND 193

H

If we integrate over all lines s H

7.8. ANOTHER PROBLEM OF ZYGMUND. SOLUTION BY C6ROOBA.

According to a result of Zygmund l19651, if we consider the

system of all intervals in R 3 then the corresponding maximal operator M satisfies the inequality

such that one of their bases is a square,

(1 i. logf

and so this basis differentiates L ( l f log' L ) ( R 3 ) .

In a similar way, if we consider a system of intervals in R 3

such that there is some reasonable constraint between their three different side-lengths , it is to be expected that this system will behave again like the two-dimensional basis of intervals, i.e. its maximal

operator will satisfy the same inequality as above,


The f o l l o w i n g theorem of CGrdoba [1978] i s t h e s o l u t i o n t o a

problem o f Zygmund i n t h i s d i r e c t i o n .

v e r s i o n , g i v i n g t h e corresponding c o v e r i n g lemma f r o m which t h e weak

t y p e i n e q u a l i t y f o r t h e maximal o p e r a t o r i s an easy consequence.

We w i l l p r e s e n t i t i n t h e dyad ic

i~ a dixed duncfitian nandemeaning i n t h e ,two vatLiable? 4epmaXcLy.

?'hood. The p r o o f i s p a t t e r n e d a f t e r t h e a l t e r n a t i v e proof we

have g i ven i n 7.7. of t h e cove r ing theorem 7.7.3. f o r i n t e r v a l s i n

R and w i l l be b e t t e r understood under i t s l i g h t .

M We f i r s t choose a f i n i t e sequence {Bk )k= l such t h a t

M I o B a ( r; 21 IJ B k ( . L e t us c a l l

We can assume t h a t no

ak , 6, , ck t h e s i d e - l e n g t h s of Bk . cl 1

Bk i s con ta ined i n another one and t h a t

a C 2 > c j .... 2 EM , By t h e m o n o t o n i c i t y c o n d i t i o n o f 4 we

have , i f t, a c, , t h a t e i theh iik 6 ii, O h fik C b1 .

7.8. ANOTHER PROBLEM OF ZYGMUND 195

We now choose the { R j } exactly in the same way as in the

above mentioned proof o f the covering theorem for intervals in R z , i.e. we choose first R 1 = B 1 and them R P as the next Bk such that

and so on. We obtain { R j I Y satisfying (see that proof)

(i) c1 c2 a . . . 2 cH

( t i ) If Ck > c1 then either a k G al or bkX bl

(iv) If B e {B,IM - { R j } Y and if R 1 , R 2 , ..., R1 are the

intervals with ck> c , then

1

By means of properties (i) - (iv) we shall now try to prove M H H

that I I) Bk - II R j l c cI II R . 1 1 i J in order to obtain (a).

We intersect B and R 1 , R 2 , ..., R1 of (iv) by a plane u

orthogonal to Oz obtaining S, I I z,. . . , I 1 . Due to the fact that c1 2 cz 2 ... a c1 > e , the three-dimensional inequality of transformed into the two-dimensional one

1

(iv) is


For each I we have e i t h e r a a a o r b 6. Assume t h a t

11,12, ... Ih a r e such t h a t a . h a , b . c 6 and t h a t Ih+l , .. . , I , a r e such t h a t a . 6 a, b . a 6 t h e s ides c ) , The s i t u a t i o n i n the p lane o i s t h a t o f t h e f i g u r e

be1 ow

j j j

J J (we can now f o r g e t about t h e o r d e r of

J J j

F i g u r e 7.8.1.

L e t us c a l l P,Q t h e p r o y e c t i o n s o f S over OX, Oy, and J K

those of I j over t h e same axes Ox, Oy. We can w r i t e j’ j

7.8. ANOTHER

+ & J J 1 h s f) ( ' J I j ) 0 ( IJ I . ) h + l 1 J

PROBLEM OF ZYGMUND

h 1

Therefore we can write

having set


Hence, i f p = min ( 3 ,G) , we have e i t h e r al>p or aL,, Q a

Therefore

= E I J F

B u t , u s i n g the weak type (1,l) f o r t he onedimens maximal operator and in tegra t ing we obtain

Therefore

J

d s > p } =

onal Hardy-Littlewood

CHAPTER 8

THE B A S I S OF RECTANGLES 1 B s

The bas i s B 3 o f a l l r e c t a n g l e s i n R2 r a i s e s a l a r g e number

o f i n t e r e s t i n g and amusing ques t i ons . Some o f them were handled, r a t h e r

l a b o r i o u s l y , a t t h e v e r y beg inn ing o f t h e t 5 e o r y o f t h e Lebesgue measure,

some o t h e r s have been so l ved v e r y r e c e n t l y and many, as we s h a l l see,

a r e s t i l l w a i t i n g f o r an answer.

I n 1927 Nikodym , m o t i v a t e d by t h e i n t e r e s t t o understand

t h e geometr ic s t r u c t u r e o f Lebesgue measurable se ts , c o n s t r u c t e d a r a t h e r

pa radox ica l s e t . The Nikodym s e t N i s a subset o f t h e u n i t square i n R 2 o f f u l l measure ( i . e . I N 1 = 1 ) , such t h a t through each one of i t s

p o i n t s x E N t h e r e i s a s t r a i g h t l i n e l ( x ) so t h a t l ( x ) 0 N = {XI. One can say t h a t t h e v e r y t h i n complement Q - N i n Q o f t h e t h i c k s e t

N has i n some sense many more p o i n t s than N i t s e l f . Zygmund ( C f . t h e

remark a t t h e end o f Nikodym's paper) p o i n t e d o u t t h a t t h i s i n m e d i a t e l y

i m p l i e s t h a t t h e b a s i s o f a l l r e c t a n g l e s i n R 2 i s v e r y bad i n what c o ~

cerns d i f f e r e n t i a t i o n p r o p e r t i e s . The b a s i s &is does n o t even d i f f e r e n -

t i a t e t h e c h a r a c t e r i s t i c f u n c t i o n s o f a l l measurable s e t s , i n p a r t i c u l a r

o f an a p p r o p r i a t e subset o f t h e Nikodym s e t .

Ten yea rs e a r l i e r Kakeya [1917] had proposed a v e r s i o n o f what

i s now c a l l e d " t h e Kakeya problem" o r " t h e needle problem": What i s t h e

in f imum o f t h e areas o f those s e t s i n R2 such t h a t a needle o f l e n g t h 1

can be c o n t i n u o u s l y moved w i t h i n the s e t so t h a t a t t h e end i t occupies

t h e o r i g i n a l p lace b u t i n i n v e r t e d p o s i t i o n ?

Almost s imu l taneous ly B e s i c o v i t c h [19181 had s o l v e d an i n t e r - e s t i n g q u e s t i o n concern ing t h e Riemann i n t e g r a l : Assume f

i n t e g r a b l e f u n c t i o n i n R2. i s a Riemann

Is i t then t r u e t h a t t h e r e i s a p o s s i b l e

199

200 8. THE B A S I S OF RECTANGLES

cho ice o f o r thogona l axes Ox,Oy f o r which t h e f u n c t i o n f ( * , y ) i s

Riemann i n t e g r a b l e f o r each y and f ( x , * ) d x i s Riemann i n t e g r a b l e ?

To answer t h i s ques t i on he c o n s t r u c t e d a compact s e t B i n R 2 of

two-dimensional n u l l measure c o n t a i n i n g a segment o f l e n g t h one i n

each d i r e c t i o n . Such a t y p e o f s e t we s h a l l c a l l a B e s i c o v i t c h s e t . Wi th

t h i s s e t B one can i n m e d i a t e l y see t h a t t h e answer t o B e s i c o v i t c h ’ s

q u e s t i o n i s nega t i ve . I n f a c t , we can assume t h a t B con ta ins no v e r t i -

c a l o r h o r i z o n t a l segment w i t h a r a t i o n a l coo rd ina te . L e t F be t h e

subse t o f B o f p o i n t s w i t h a r a t i o n a l coo rd ina te . Then t h e d i s c o n t i -

n u i t y p o i n t s o f xF a r e i n B and so xF i s Riemann i n t e g r a b l e i n R2. But i n each d i r e c t i o n t h e r e i s some segment con ta ined i n

I

B a long which

xF i s n o t Riemann i n t e g r a b l e .

As i t was r e a l i z e d much l a t e r [1928] , t h e s e t B g i v e s a l s o

a s o l u t i o n t o t h e needle problem: The in f i n imum o f t h e areas on which t h e

needle can be i n v e r t e d i s zero.

The c o n s t r u c t i o n of t h e B e s i c o v i t c h s e t was s i m p l i f i e d by Per-

r o n C19281 and l a t e r on by Radeniacher [1962] and Schoenberg [1962] . I t s connec t ion w i t h d i f f e r e n t i a t i o n theo ry was b rough t t o l i g h t f i r s t by

Busemann and F e l l e r [1934] who used i t i n o r d e r t o g i v e a s i m p l e r p r o o f

( n o t based i n t h e e x i s t e n c e o f t h e Nikodym s e t ) o f t he f a c t t h a t P, i s

n o t a d e n s i t y bas i s . (Nikodym’s c o n s t r u c t i o n o f h i s s e t was e lementary

b u t e x t r a o r d i n a r i l y c o n p l i c i t e d ) . L a t e r on Kahane [1969] gave an i n t e r -

e s t i n g c o n s t r u c t i o n o f a B e s i c o v i t c h s e t .

had e s t a b l i s h e d t h e connec t ion o f such ’a t y p e o f s e t s w i t h t h e t h e o r y

developed by him o f t h e geometr ic of l i n e a r l y measurable s e t s i n R2 ( s e t s o f Hausdor f f dimension 1).

Before t h a t B e s i c o v i t c h [1964]

Very much a t t e n t i o n has been p a i d t o c o n s t r u c t i o n s connected

w i t h t h e B e s i c o v i t c h s e t and t h e Nikodym se t , among o t h e r s e s p e c i a l l y

by Davies [1953] and Cunningham [1971,1974] . And r i g h t l y so, s i n c e

they p r o v i d e ve ry much l i g h t i n o r d e r t o g e t a deeper ‘understanding o f

impor tan t geometr ic and measure - theo re t i c p r o p e r t i e s r e l a t e d w i t h t h e

c o l l e c t i o n of rec tang les i n R2 .

8.1. THE PERRON TREE 20 1

As we sha l l see , most of what we sha l l present i n t h i s Chapter depends on w h a t we sha l l ca l l t he Perron t r e e ( the construction proposed by Perron [1928] Nikodym set can be most ea s i ly b u i l t by means of i t . For t h i s reason we sha l l present f i r s t t h i s fundamental construction and from i t we sha l l draw a good number of conclusions and r e s u l t s of h i g h i n t e r e s t . Then we sha l l present several recent r e s u l t s connected with subbases of 8 , such as those of Stromberg [1977] , C6rdoba and Fefferman [I9781 and CBrdoba [1976] . We s t a t e a l s o some in t e re s t ing open problems in t h i s area.

i n order t o simplify t h a t of Besicovitch). Even a

8.1. THE PERRON T R E E .

The construction we present here of the Perron t r e e follows of Rademacher [1962] , w i t h some s l i g h t modifications t h a t wi l l make i t more useful f o r our purposes.

8.1.1. THEOREM. Comidm .in R 2 t h e 2n open M a n g l u

C A h ) h = l 2n o b h u k e d by joining Xthe paid ( 0 , l ) w d h t h e p o i & ( O , O ) ,

( 1 , 0 ) , ( 2 , 0 ) , ( 3 , 0 ) , ...,( 2",0). LeA A h be t h e M a n g l e w a h ueh'tice~

( 0 , l ) (h-1,O) , ( h , O ) . Then, given

t o make u p w i a e R e l L t u n h U n ad

pOhi,thn A h h o XhcLt one h a

2n I u A h / c

h = l

Pkood. The theorem wi l l be obtained by r epe t i t i on of the f o l

lowing process t h a t , fo r reference purpose, we sha l l ca l l t he basic con s t r uc t i on.

202 8 . THE B A S I S OF RECTANGLES

&mic euvl,l,?huotion. Consider two a d j a c e n t t r i a n g l e s T1,T2 w i t h

b a s i s on Ox, w i t h t h e same b a s i s l e n g t h b and w i t h h e i g h t l e n g t h h,

as i n F i g u r e 8.1.1. L e t 0 < c1 < 1. Keeping T1 f i x e d we s h i f t TP

towards T 1 t o p o s i t i o n T: i n such a way t h a t t h e s i d e s t h a t a r e n o t

p a r a l l e l meet a t a p o i n t a t d i s t a n c e ah f rom Ox as i n F i g u r e 8.1.2.

F i g u r e 8.1.1.

F i g u r e 8.1.2.

The union of TI and T: i s composed by a t r i a n g l e S ( n o t shaded por-

t i o n i n F i g . 8.1.2.) homothet ic t o T1 IJ T2 p l u s two “excess t r i a n g l e s “

A l , A 2 . One can e a s i l y g e t

I S / = ~ t ’ / T 1 (I TP/

and so

8.1. THE PERRON TREE 203

We s h a l l now app ly t h i s b a s i c c o n s t r u c t i o n t o t h e s i t u a t i o n o f

t h e theorem. Consider t h e 2"' p a i r s o f a d j a c e n t t r i a n g l e s

( A 1 , A z ) , ( A 3 , A & ) ,..., (A

c o n s t r u c t i o n w i t h t h e same ci g i v e n i n t h e s tatement o f t h e theorem. We

o b t a i n t h e t r i a n g l e s S 1 , S z Y . . . , S and t h e excess t r i a n g l e s

,A n ) . To each p a i r we a p p l y t h e b a s i c 2"l 2

2"l

A;,A;;A:,A~: ...; , . We now s h i f t S 2 a long Ox towards S 1

so t h a t i t becomes s 2 ad jacen t t o S1. Then we s h i f t S3 t o p o s i t i o n s 3 a d j a c e n t t o S1 Ij S 2 , and so on. I n these mot ions each Sh must c a r r y

h h w i t h i t t h e two excess t r i a n g l e s a l ,A2 , so t h a t what we a r e i n f a c t do ing

i s e q u i v a l e n t t o s h i f t i n g t h e t r i a n g l e s

p o s i t i o n s A2, A 3 , ..., A Consider now A 1 IJ AP 0 x3 ...(I A

T h i s f i g u r e i s composed by

i s o f area

,.,

A2,A3 , . . .yA2n , t o some new - - - - - zn * -. 2n* 2 n - l

t r i a n g l e s S 1 , S P Y . . . ,S2,,-1 , whose un ion

p lus s h i f t e d excess t r i a n g l e s , whose un ion i s o f area n o t l a r g e r than

The 2"' t r i a n g l e s S l , i 2 , ? 3 y . . . , S Z n - l a r e i n t h e same s i t u a t i o n as

t h e i n i t i a l t r i a n g l e s A1,A2,A3,. . . ,A2n. One s u b j e c t s them t o t h e same

process, always c a r r y i n g t h e excess t r i a n g l e s so t h a t i n f a c t one moves

t h e e n t i r e t r i a n g l e s A2,A3, ... ,Azn. T h i s process i s repea ted n t imes

and a t t h e end one o b t a i n s a f i g u r e A1 0 A P A 3 ... A

which i s composed by a t r i a n g l e H homothet ic t o Al (J A2 0 ... II A

o f area

- 2n

2"

p lus a d d i t i o n a l t r i a n o l e s whose un ion has an area n o t l a r g e r than

Hence, i f we s e t A1 = T i l , we g e t

T h i s concludes t h e p r o o f o f t h e theorem.

I t i s c l e a r t h a t one can perform an a f f i n e t r a n s f o r m a t i o n i n

t h e s i t u a t i o n o f Theorem 8.1.1. i n o r d e r t o g i v e i t a more f l e x i b l e

s t r u c t u r e . P a r a l l e l l i n e s keep be ing p a r a l l e l a f t e r t h e t r a n s f o r m a t i o n

and r a t i o s between areas o f f i g u r e s do n o t change.

t o the f o l l o w i n g r e s u l t .

So one e a s i l y a r r i v e s

8.1.2. THEOREM . LeL A B C be u XkLangRe o d meu H. Given

uny E > 0 Lt A pobbibL?e t o pah&Xivn t h e baA B C i n t o 2" p W I I ,Iz ,I3 , . . . , I ( n dependn on E ) und .to ~kil;t t h e &LungLen

T I ,T2 , T3 , ..., T w L t h baA 1 1 , 1 2 , 1 3 , ..., ' a n d common ventex 2n

2n I 2 n

A d o n g B C ZCJ pVb&UMn f l Y f 2 , f 3 , ... ,TZn b~ t h a t

Pkoud. We f i r s t t a k e a so t h a t 0 < a < 1, 211-a) < € 1 2 , E

and then take n so t h a t aZn < 7 . We now c o n s i d e r t h e r e s u l t o f

Theorem 8.1.1. w i t h t h i s n and a, and an a f f i n e t r a n s f o r m a t i o n p t h a t c a r r i e s ( 0 , l ) t o A , (0,O) t o B and (2",0) t o C . Then

p(Ah) = Th and p(Ah) = ih f o r h = 1,Z ,...,Zn.

t h e r a m i f i c a t i o n s due t o t h e excess t r i a n g l e s .

As we s h a l l see l a t e r t h e Per ron t r e e has p l e n t y o f a p p l i c a t i o n s

t o v e r y many d i f f e r e n t problems. It would be of i n t e r e s t t o have a h i g h

8.1. THE PERRON TREE 205

degree o f f l e x i b i l i t y t o c o n s t r u c t d i f f e r e n t types o f Pe r ron t r e e s adapted

t o m o d i f i c a t i o n s o f t h e problems we a r e go ing t o be a b l e t o s o l v e w i t h t h e

c o n s t r u c t i o n we have performed. Fo r t h i s reason i t i s i n t e r e s t i n g t h e f o l -

l o w i n g o b s e r v a t i o n which pe rm i t s us t o o b t a i n a Perron t r e e once we a r e

g i ven a B e s i c o v i t c h s e t . We s h a l l see t h a t t h e r e a r e d i f f e r e n t methods t o

c o n s t r u c t B e s i c o v i t c h se ts .

Assume B i s a compact n u l l s e t formed by t h e u n i o n o f c e r t a i n

segments of l e n g t h one whose d i r e c t i o n s f i l l a c losed ang le o f 60" (see

F i g u r e 8.1.3.)

F i g u r e 8.1.3.

L e t E > 0 and t a k e an open s e t G such t h a t G 3 B and [GI < E . For

each u n i t segment 1 o f B we take an open t r i a n g l e con ta ined i n G and

c o n t a i n i n g 1 i n i t s i n t e r i o r so t h a t t h e angles a t t h e upper extreme

p o i n t s o f each 1 a r e equal . S ince B i s compact we can cove r B w i t h

a f i n i t e number o f such t r i a n g l e s . We then t r a n s l a t e p a r a l l e l y these

t r i a n g l e s t o have t h e upper v e r t e x a t t h e same p o i n t and so we see, re-

v e r t i n g t h e c o n s t r u c t i o n , t h a t , g i v e n E > 0 and a c l o s e d e q u i l a t e r a l

t r i a n g l e ABC o f h e i g t h ha b i g g e r equal t o 1 / 2 , i t i s p o s s i b l e t o

d i v i d e ABC i n a f i n i t e number of t r i a n g l e s TI, ..., Th and t o t r a n s l a t e

them p a r a l l e l y t o p o s i t i o n s Tl,...,Th so t h a t 10 thl 6 E

However, t h e c o n s t r u c t i o n of t h e Per ron t r e e we have per formed

has some a d d i t i o n a l f e a t u r e s t h a t make i t e s p e c i a l l y i n Theorem 8.1.2.

notewor thy.

206 8. THE B A S I S OF R E C T A N G L E S

REMARK I . The t r i a n g l e s T I , . , . ,f o f t h e c o n s t r u c t i o n of

Theorem 8.1.2. end up w i t h t h e upper v e r t i c e s i n reve rsed o r d e r w i t h

respec t t o t h e i r bases, i . e . if t h e b a s i s o f 7 of T i t hen t h e upper v e r t e x of 7 i s t o t h e l e f t of t h a t of Ti .

2"

i s t o t h e r i g h t o f t h a t j

j

Thmel(ohe, i d we extend t h e LtiungLen fh ubove the,& u p p u vehticen t h e ~ e e.xten.bivru uhe. d in jo in t (See F i g . 8.1.4)

F i g u r e 8.1.4.

REMARK 2. 16 we extend t h e L h h n g L a 7, b d o w the& b a a

thene exten.bion.b WVQA on t h e h-thip pat&& t o ttkin b a d ! 0 6 w i h t h at &at u &an& egud t o t h e ohiginal one A B C LU indicated .in

F i g . b . I , 5 . no m & u how we have -taken CI and n i n t h e co~n. i%uotion 06 t h e P m v n O ~ e e 06 Theoxem

ha

8.1.2.

A

t ha

B C

\ ha /

\ I /

F i g u r e 8.1.5.

8.2. A LEMMA OF FEFFERMAN 20 7

___ REMARK 3 . I n Z h e i h ~ i n d p v b U o n t h e uppeh uemXca v6

fh vd t h e bub.in V/J ABC.

neweh yet &mtheh tv t h e Re@ v d .that v d f , by rnvhe -than t h e Length

8.2. A LEMMA OF FEFFERMAN.

F o r an i m p o r t a n t p r o b l e m i n t h e t h e o r y o f F o u r i e r m u l t i p l i e r s ,

t h a t we s h a l l examine l a t e r on, C.Fef fe rman [1971] used a lemma c o n c e r n

i n g t h e s t r u c t u r e o f r e c t a n g l e s . T h i s lemma can b e v e r y e a s i l y o b t a i n e d

by means o f t h e c o n s t r u c t i o n we have p e r f o r m e d o f t h e P e r r o n t r e e .

Y

fiehe Rh denoten t h e bhaded pvh-tivn ub t h e &7~cne 8.2.1 \

F i g u r e 8.2.1.

208 8 . THE BASIS OF RECTANGLES

-- P m a d . The proof i s straightformard from Theorem 8.1.2. with the Remark 1. For each one of the t r i ang le s Ph we perform the cons t rug t ion indicated in Fig. 8 . 2 . 2 taking as R h the rec tangle indicated and as E the Perron t r e e 10 f h l . The R h a r e d i s j o i n t according to Remark 1. The area of t h e i r union i s a good portion of t h a t of the or ig ina l t r i ang le ABC with which we s t a r t e d and so we can arrange everything so t h a t 1 E l < rl 1 1 R h l . On the other hand R h 0 E 3 t h and f, i s a good portion of i i h . SO j~~ E I > __

- 1 - 100 l R h l *

Figure 8 . 2 . 2 .

8.3. THE KAKEYA PROBLEM

8.3. THE KAKEYA PROBLEM.

209

The s o l u t i o n o f t h e needle problem i s a l s o i nmed ia te w i t h t h e

Perron t r e e .

Pkvvl; . F i r s t o f a l l we show t h a t one can c o n t i n u o u s l y move a

segment f rom one s t r a i g h t l i n e t o another one p a r a l l e l t o i t sweeping

o u t an area as smal l as one wishes. I t i s enough t o observe i n F i g u r e

8.3.1. t h a t one can move A B t o A4 BI, sweeping o u t t h e area o f t h e

shaded p o r t i o n which can be made as small as one wishes t a k i n g

A4 64 A3 B2=B3

F i g u r e 8.3.1.

A BS s u f f i c i e n t l y l a r g e .

We now t h a t A B can be moved t o a s t r a i g h t l i n e f o r m i n g an

angle o f 60 "w i th i t s o r i g i n a l p o s i t i o n w i t h i n a f i g u r e o f area l e s s than

n /6 . S i x r e p e t i t i o n s o f t h e same process w i l l g i v e us t h e f i g u r e F o f

t h e theorem. L e t M N P be an e q u i l a t e r a l t r i a n g l e o f area equal t o 10

p laced so t h a t A B i s i n t h e i n t e r i o r o f M N. Observe t h a t t h e h e i g h t

o f M N P i s b i g g e r than 1. To M N P we a p p l y Theorem 8.1.2. t a k i n g

as b a s i s N P and w i t h an E such t h a t 10 E < f-)- . We o b t a i n t h e

t r i a n g l e s 71, 7 2 , ... ,$2n . The segment A B can be c o n t i n u o u s l y moved


w i t h i n 71 f rom M N t o t h e o t h e r s i d e o f 7, n o t on N P. From t h e r e

one can move t h e segment t o t h e s i d e o f f p p a r a l l e l t o i t sweeping an

area l e s s than ' . Now we move i t aga in w i t h i n TP t o t h e -

12 x zn o t h e r s i d e o f T2

process i s l e s s than n / 6 , and t h e needle i s a t t h e end on a l i n e fo rm ing

an angle o f 60" w i t h t h e o r i g i n a l p o s i t i o n .

n o t on N P, and so on. The area swept o u t i n t h i s

8.4. THE BESICOVITCH SET.

From t h e Perron t r e e o f 8.1.2. we o b t a i n a B e s i c o v i t c h s e t as

f o l l o w s .

8.4.1. THEOREM. Thehe A u cornpct he,t F i n R2 ad niLee rneaute containing u h t p U L t 0 6 ui& Length i n euch dihecfion.

Paood. I t i s enough t o produce a compact n u l l s e t F t h a t

con ta ins a segment o f u n i t l e n g t h i n each

T h i s i s s t r a i g h t f o r w a r d f rom t h e f o l l o w i n g lemma whose p r o o f i s presented

a t t h e end.

d i r e c t i o n o f an ang le o f 45:

8.4.2. LEMMA. Given a cLobed puh&LeLogtrarn P 0 6 bididen

a,b,c,d and ri > 0 thehe A CL &uXe c o U b t L o n 06 dahed pah&eLogtlurnh

R = { w l , w2,. .., uH I thcLt

w d h one hide on a and anathe,k one on c hUCh

( 2 ) E a c h h e p e n t joining u paint 0 6 a t o anotheh po in t ad

j' c adrnh.2 a p a h a l l e l thaMn&aA;ion thud ca/LILien t o \ I w

We s t a r t a p p l y i n g t h i s lemma t o t h e c l o s e d u n i t square

Q = ABCD w i t h r11 = 1/2 o b t a i n i n g { wl, up,... , w I . Observe t h a t H:

8.4. THE BESICOVITCH SET A

211

H i L, = u uj i s a compact s e t of area n o t g r e a t e r t han 1/2 con ta ined

j=1

i n Q and c o n t a i n i n g segments o f u n i t l e n g t h i n each d i r e c t i o n o f an

ang le o f 450, namely A t B .

To each o. we app ly aga in t h e lemma w i t h an n2 so sma l l t h a t J H i 02 c - , o b t a i n i n g { w ( j , l ) w ( j , 2 ) )...) w ( j , H j ) l . The s e t

2 2

H i H;

j=1 r = l L z = ( J ! J w ( j , r )

i s a compact s e t con ta ined i n L1 w i t h area n o t g r e a t e r t han 1/2’ and

c o n t a i n i n g segments of u n i t l e n g t h i n each d i r e c t i o n o f A e 8 . And m

so on. The s e t F = I1 L . i s a compact n u l l s e t c o n t a i n i n g segments J j = 1

i n each d i r e c t i o n o f A ^B C as r e q u i r e d .

Prrovd 0 6 Lemma. . 8.4.2. f i r s t of a l l we t a k e two s t r i p s

w1 = ASTD and w2= DLBT as i n d i c a t e d i n t h e f i g u r e 8.4.1. and such

t h a t

A l so we cons ide r t h e p o i n t V ob ta ined as i n d i c a t e d i n t h e f i g u r e , where

B V i s u a r a l l e l t o L T.

A a L B

F i g u r e 8.4.1. F i g u r e 8.4.2.

212 8. THE BASIS OF RECTANGLES

Then we d i v i d e SB i n t o a f i n i t e number o f equal segments o f

s m a l l e r l e n g t h than AS . We j o i n V t o t h e d i v i d i n g p o i n t s and cons ide r

To each one o f them we app ly t h e

c o n s t r u c t i o n o f t he Per ron t r e e o f 8.4.2. w i t h an E so sma l l t h a t t h e

area o f t h e un ion o f a l l t h e Perron t r e e s ob ta ined i n t h i s way i s l e s s

than n/4. Observe t h a t by Remark 3 of 8.4. t h e upper v e r t i c e s o f t h e

small t r i a n g l e s ob ta ined i n those Per ron t r e e s never go t o t h e l e f t o f d.

We now proceed s y m m e t r i c a l l y s t a r t i n g from t h e s i d e BC . F i n a l l y we

s u b s t i t u t e each one o f t h e i n t e r s e c t i o n s w i t h P o f t h e smal l t r i a n g l e s

o f such Perron t r e e s by p a r a l l e l o g r a m s as r e q u i r e d i n t h e s ta temen t o f

t h e theorem as i n d i c a t e d i n F i g u r e 8.4.2.

each one o f t h e t r i a n g l e s VMi M i + l .

We have p o i n t e d o u t b e f o r e t h a t t h e r e a r e seve ra l d i f f e r e n t

ways t o o b a t i n B e s i c o v i t c h se ts . I n t h e n e x t Chapter we s h a l l see how t o

o b t a i n them very s imp ly by means o f t h e geometr ic t h e o r y o f l i n e a r l y

measurable s e t s . Here we s h a l l p r e s e n t another s i m p l e way due t o Kahane

[1969] . As a m a t t e r o f f a c t t h e c o n s t r u c t i o n o f Kahane, as p o i n t e d o u t

by Casas [1978] , i s a p a r t i c u l a r case o f t h e c o n s t r u c t i o n i n d i c a t e d

above by means o f l i n e a r l y measurable s e t s . However i t w i l l be i n s t r u c t i v e

t o show i t w i t h o u t appea l i ng t o t h a t t heo ry .

8.4.2. THEOREM. On t h e negment joining 0 = (0,O) t o

A = (1,O) c . o a i d a a pehdect neA La Cantox dividing OA i n t o law e . q d d o b e d h e g r n e d , .taking t h e Xwo e x h m e o n u , dividing again each

one i n r v d o u h e q d d o s e d negmena , and so on. 1e.X CO be t h e beA no

0 bXained.

On t h e segment j u i n i n g B = ( 0 , l ) t o D = (1/2,1) coaidm

u neA C1 h u m o t h d c t o CO.

LeL F be t h e u n i o n 0 6 & &abed segmenh joining a point 06 C o to anotheh p o i n t 0 6 C 1 .

8.4. THE BESICOVITCH SET 213

Phoah. Tha t F c o n t a i n s i n f a c t segments i n a l l those d i r e c -

t i o n s i s easy.

T

F i g u r e 8.4.3.

Observe t h a t F can be viewed as t h e i n t e r s e c t i o n of a l l t h e compact

s e t s Kk ob ta ined by j o i n i n g t h e p o i n t s o f t h e k - t h phase o f t h e

c o n s t r u c t i o n o f C o t o p o i n t s o f t h e k - t h phase o f t h e c o n s t r u c t i o n o f

CI . We have K k + l ~ Kk and each Kk preserves t h e p r o p e r t y o f hav ing

p a r a l l e l t r a n s l a t i o n s o f each segment j o i n i n g a p o i n t o f OA t o ano the r

o f B D . It s u f f i c e s t o check t h i s by mere i n s p e c t i o n f o r K1 , s i n c e K P

i s e s s e n t i a l l y o b t a i n e d from KI as K1 i s ob ta ined f rom OADB. Observe

t h a t TL i s p a r a l l e l t o OR BL i s p a r a l l e l t o DS and DA p a r a l l e l

t o RS. So K1 covers segments o f a l l t h e d i r e c t i o n s above mentioned.

The f a c t t h a t F i s of n u l l measure can be proved i n t h e f o l -

l o w i n g way. Observe t h a t o f we i n t e r s e c t t h e s e t F by a l i n e p a r a l l e l

t o OA a t h e i g h t u, 0 6 1-1 6 1, t h e p o i n t s so ob ta ined have absc issae 1

~.rx + (1 - u )x ' where x 8 C1= 7C0 x ' 6 C o . So i t i s s u f f i c i e n t t o

prove t h a t f o r a lmos t each X 6 [O,-) we have

ICO + X C O l 1 = I t x + Ax' : x € C o y x ' E CoI I1= 0

To do t h i s we c o n s i d e r two s e t s l i k e C o y one on y = 0 and t h e o t h e r

on y = 1 . L e t us c a l l them D o D1 and l e t G be t h e un ion o f a l l


t h e segments j o i n i n g a p o i n t o f D O t o another one o f D1 . The s e t G

i s t h e union o f f o u r se ts Gl, , Glr , Grl , Grr each a f f i n e t o G by

an a f f i n i t y o f r a t i o 1 /4 . The s e t Glr i s t h e un ion o f segments j o i n -

i n g p o i n t s of t h e Lee@ h a l f o f Do t o t h e gigk kt h a l f of D1.

F i g u r e 8.4.4.

We have t h e r e f o r e lGll 1 = lG l r l = lGrl I = I G r r / = 1/4 1 G I and

G = Gll (I Glr (J Grl ( J Grr . So we have lGll 1’1 Girl = 0 . I f we

pe r fo rm t h e a f f i n i t y t h a t c a r r i e s Gll t o G , we see t h a t Gll 0 Glr

i s c a r r i e d t o t h a t p a r t o f G which i s below t h e d o t t e d l i n e d i n t h e

f i g u r e , and t h i s p a r t o f G has n u l l measure . By symmetry a l s o t h e p a r t

o f G under f has n u l l measure. There fo re t h e r e i s an a > 0 such

t h a t

I C o + A C ~ I ~ = o f o r a lmos t every x E cola]

1 1 Now t h i s i m p l i e s 1 7 C O + ACoIl= 0 s i n c e - C o c C O f o r each n.

There fo re I C o + A4n C 0 l 1 = 0 f o r a lmos t each A E [0,4’a] . Hence

/ C o + XColl= 0 a.e. X E L0,m)

a.e. A 6 lR and so we have shown t h a t t h e u n i o n o f t h e whade fins

j o i n i n g p o i n t s o f C O and C 1 i s a c l o s e d s e t o f p lane measure zero.

4 4”

I n t h e same way we see than \ C O + X C o l i = O

8.5. THE NIKODYM SET 215

8 .5 . THE NIKODYM SET.

The c o n s t r u c t i o n o f t h e Nikodym s e t i s r a t h e r easy once we have

t h e f o l l o w i n g lemma which i s q u i c k l y ob ta ined by means o f t h e Perron t r e e .

Observe i t s analogy t o Lemma 8.4.2. which gave us t h e B e s i c o v i t c h se t .

8.5.1. LEMMA. L e t R be t h e c loned tectangLe ABCD od Figme 8.5.1. and S t h e one ABEF obtained b y dtLCW-ing a p d t l & d f i n e 1 t o

t h e b a A AB . I!& q > 0 be given. Then AX i~ punnib& .to &CW a SL-

i t i t e numbm od pat~&dogtramn

and a n o t h a one i n DC nu thcLt t h e y t u u a ABEF ' and

{ wl, w2, . . . , wH 3 w i t h one baA on AB

D r

A B

F i g u r e 8.5.1.

P m o d . The p r o o f i s performed r e c a l l i n g t h e Remark 2 i n 8.4 r e l a t e d t o t h e c o n s t r u c t i o n o f t h e Perron t r e e i n 8.4.2.

We f i r s t t a k e a s t r i p w1 = AJKD p a r a l l e l t o AD o f area l e s s

than n/8 . A lso we take a l i n e G H s l i g h t l y above 1 and p a r a l l e l t o 1,

so t h a t I F E H G l 6 q/8 . On JK and above DC we take a p o i n t V so

h igh t h a t lVLl > I L J I and then we t a k e t h e t r i a n g l e VLN w i t h


1 \ L 1 4 \ l = . I f we apply the construction o f the Perron t ree of 8 .4 .2 . t o V L N , according to the Remark 2 in 8.4, the extension below LN o f the small triangles o f the Perron t ree will cover PQSJ where SL

i s parallel t o V N .

D

G

F

I

Figure 8.5.2.

Through Q we draw a l ine parallel t o AD and take V 1 and the tr iangle V L l N1 . If we apply the construct on of 8.4.2. t o V L l N 1 we cover with the extensions o f the small t r angles the s e t P I Q I S I J 1 . So we can advance in a f i n i t e number of steps until i s beyond the midpoint o f

EF. The Perron trees for the t r ian les V . L . N are taken with E 50

small t h a t their union i s of area less than o / 8 . We proceed now sym- metrically s tar t ing from the side C B . So we get two s t r ip s w ~ , w ~ and many small triangles R1 ,R2 , . . . , R k . Their union covers ABEF and by

choosing the E of the Perron trees small enough we get

Q, J J j


Now f o r each t r i a n g l e R . we can s u b s t i t u t e i t s i n t e r s e c t i o n w i t h R by

a f i n i t e number o f s t r i p s con ta ined i n R . as r e q u i r e d i n t h e Theorem as J i n d i c a t e d i n F i g u r e 8.5.3. and t h i s f i n i s h e s t h e c o n s t r u c t i o n . ( R e c a l l

t h a t by Remark 3 o f 8.4. t h e v e r t i c e s o f t h e smal l t r i a n g l e s a r e t o t h e

r i g h t o f AD and t o t h e l e f t o f CB).

J

F i g u r e 8.5.3.

From t h e preceding lemma we e a s i l y o b t a i n t h e f o l l o w i n g one

8.5.2. LEMMA. L e t R1,R2 be a3.1~ dvned pm&&vgtam i n

R2 ouch LhaA R1 R Z . LeL E > O and L e t w be vne v d t h c Awv cloned n M p o dehhmined by Rl., . Then .thehe u &.42e coUeeectivn a 6

clvne.d n-thiph R = Cwl, W Z , . . . , wk 1 A U C ~ thaA

(1) F a t u c h i = l y 2 , . . . , k , w i 11 R1 c w 0 R2

k

k

2 18 8. THE B A S I S OF RECTANGLES

Pkwd. T h i s lemma i s t h e p reced ing one i f R1 and R 2 a r e

r e c t a n g l e s as i n F i g . 8.5.4.

R 2 i

F i g u r e 8.5.4.

We now proceed t o remove the r e s t r i c t i o n s on R1 and R 2 . An

a f f i n e t r a n s f o r m a t i o n shows t h a t t h e r e s t r i c t i o n imposing t h a t

a r e rec tang les can be e a s i l y removed.

ho lds i f R 1 and R 2 a r e two p a r a l l e l o g r a m s as i n F i g u r e 8.5.5.

R1 and R 2 There fo re we know t h a t t h e lemma

Assume now t h a t R1 and R 2 a r e t h e p a r a l l e l o g r a m s A B C D and E F G H o f F i g u r e 8.5.6.

F i g u r e 8.5.5.

F G N

M E A D H

F i g u r e 8.5.6.

8.5. THE N I K O D Y M SET 219

We r e p l a c e R 2 by R S = M F N H such t h a t M F i s p a r a l l e l t o A B y

A D i s on M H and R 2 c R; . We a l r e a d y know t h a t t h e lemma i s

v a l i d f o r R 1 and R: . I t i s e a s i l y seen t h a t t h e same s t r i p s we o b t a i n

s a t i s f y ( l ) , ( 2 ) and ( 3 ) f o r R 1 and R 2 .

Assume now t h a t R 1 and RZ a r e as i n F i g u r e 8.5.7. , w i t h

A B p a r a l l e l t o E F and C D p a r a l l e l t o G H . We a p p l y t h e lemma

t o R: = M B C N and R, w i t h an ~ / 2 . Each one

F P 0 G

E M N H

F i g u r e 8.5.7.

- o f t h e s t r i p s

F i g u r e 8 .5 .8 .

GI, G z , ..., wk we g e t i s i n t h e s i t u a t i o n i n d i c a t e d i n

F P Q G

I H j

w E

F i g u r e 8.5.8. -

So we can now app ly t h e lemma t o each one o f t h e p a r a l l e l o g r a m s wi 0 APQD

and R 2 w i t h ~ / 2 k , amd we g e t for each i = 1,2 ¶ . . . , k t h e s t r i p s

220 8 . THE BASIS OF RECTANGLES

4 } j = 1 , 2 , . . . , r i . The co l lec t ion of a l l these s t r i p s u?; i s e a s i l y

seen t o s a t i s f y (l), ( 2 ) and (3) .

Fina l ly , i f R 1 and R Z a r e i n the general s i t u a t i o n of t he lemma one can subs t i t u t e R 2 by another parallelogram R: , R: 3 R,,

with s ides para l le l t o those of R l and apply the lemma t o R1 and R: . The s t r i p s we obtain a re a l so va l id for R 1 and Rz.

The second lemma we a r e going t o use is an easy consequence o f

t he previous one.

8 .5 .3 . LEMMA. L e t R1 and R 2 be Awu doded pamUehg/Zam buch t h a t R 1 c R Z . L e t R be a &Lrtite coUecLLon a6 cloned h-thiph,

R 1 . L e t E > 0 be given.

Then, do& each h. th ip u i y i = 1,2, ... , k

coUecLLon o d cloned h i x i p d W: , W; , . . . , u J i huch Zha.2, id we c a l l

= {ol ,UP,. .. , uk 3 , whobe union c o v m

one $an conbahuc2 anotheh dini te

i R* = { W! : i = 1 ,2 ,... , k j = 1 , 2 , . . , , j i 1 , we have:

j ( 2 ) Fon. each i and j , wi 0 R z c w i .

From the foregoing lemmas we obtain the following r e s u l t ,

8.5.4. THEOREM. Thehe AA i n R2 a b e t K 06 nURe memme huch t h a t doh each x E R2 thehe AA a na%thcLigkt f i n e r ( x ) paAning

t h a u g h x ha t h a t r ( x ) c K 1 J c x 3 .

The r e s u l t of Nikodym i s of course, an easy consequence of this theorem. In f a c t i f Q i s the uni t square and N = Q - K , then I N 1 = 1 and f o r each x 6 N the l i n e r (x) s a t i s f i e s r(x) 0 N = 1x1,

THE NIKODYM SET 221 8 . 5 .

Y m o A O X ;the Theohem 8

For H > 0 l e t Q(H)

5.4.

be t h e c l o s e d square i n t e r v a l cen te red

a t 0 and o f s i d e - l e n g t h 2H. L e t us c a l l f o r b r e v i t y Q(1) = Q . We app ly Lemma 8.5.2. t o R1 = Q and R 2 = Q(2) w i t h an s1/4 > 0 t h a t

w i l l be f i x e d l a t e r . We o b t a i n a c o l l e c t i o n o f s t r i p s i l l , . We d i v i d e Q

i n t o f o u r equal c losed square i n t e r v a l s each one h a l f t h e s i z e o f Q. L e t

us denote them by Qf , i = 1,2,3,4. F i x an i and a p p l y Lemma 8.5.3.

w i t h

c o l l e c t i o n R* o f c l o s e d s t r i p s t h a t we s h a l l c a l l

R 1 = Q j , R2 = Q(3) , n = nl , E = ~ ~ / 4 ’ > 0. So we o b t a i n a

Rl . L e t us s e t

4

R2 = 0 n: i = l

We now d i v i d e each Qj one h a l f t h e s i z e of Ql . So we o b t a i n 4’ squares Q2 i = l,2,...y42.

F i x an i and app ly aga in Lemma 8.5.3. , w i t h R 1 = Q2 ,R2 = Q(4) , R = Q2 , E = ~ ~ / 4 ~ . So we o b t a i n t h e c o l l e c t i o n R* o f t h e lemma

t h a t we s h a l l denote a’, and we w r i t e

i n t o f o u r equal c losed square i n t e r v a l s , each i

i

And so on.

i k , t h e un ion o f a l l s t r i p= Rk

covers t h e square Qk-l . For each w E Rk we d e f i n e B = w - Qk-l

and l e t

acco rd ing t o Lemma 8.5.3.

Observe t h a t f o r a f i x e d i i i

i Rk Kk = 11 CB : w E R k l . We have, by t h e c o n s t r u c t i o n o f

and so we g e t I K k f ) Q ( k ) l c sk f o r each E~ . We now d e f i n e

W W

K* = l i m i n f Kk = 0 Kk. h = l k=h

We choose E~ -f 0 and so / K * I = 0 . I n f a c t , i f we f i x N and h and

then we t a k e j , j > h, j > N we o b t a i n


Hence

Since t h i s ho lds f o r each N, we g e t 1 f ) K k l = 0 f o r each h and so k=h

I K * / = 0.

We now show t h a t f o r each x E Q t h e r e e x i s t s a s t r a i g h t l i n e

r ( x ) pass ing through x and con ta ined i n K* LJ {XI. L e t x E Q be

f i x e d and l e t Q:(xyn) n = l Y 2 , 3 , ... be a c o n t r a c t i n g sequence o f

t h e squares we have c o n s t r u c t e d so t h a t x E Q i ( x y n ) f o r each n. For

n = 1 we t a k e a s t r i p w 1 of R1 J(xyl)

i s some s t r i p o2

w z ( 7 Q ( 2 ) ,= w1 and so on. For n = k t h e r e i s some s t r i p wk o f

Qk c o n t a i n i n g x and such t h a t wk (1 Q ( k ) c w ~ - ~ . So t h e r e e x i s t s

a sequence o f l i n e s { r k ( x ) } pass ing through x and such t h a t r k ( x ) c w r

Since t h e w i d t h o f t h e s t r i p s uk tends t o ze ro (because of t h e f a c t

t h a t E~ -f 0 ) and one has uk I1 Q ( k ) c w ~ - ~ , one has t h a t t h e d i -

r e c t i o n s of t h e l i n e s {r,(x)} converge t o t h e d i r e c t i o n o f a l i n e r ( x )

through x.

c o n t a i n i n g x. For n = 2 t h e r e o f r 2 z j ( x 3 2 ) c o n t a i n i n g x and such t h a t

We now prove t h a t r ( x ) c K* 0 { x } L e t y E r ( x ) y f x.

Then t h e r e i s a n a t u r a l number N such t h a t i f n a N we have

Y 6 Q i ( x y n ) and y E ;(n)

L e t us take a sequence of p o i n t s

There i s an M such t h a t i f k > M, we have

{yk} such t h a t yk 6 r k ( x ) , yk + Y.

I f i > n 2 max(M,N) we can w r i t e


Since yi d Q i ( x y N ) we a l s o have yi d Q A ( x y n ) . Hence yi E 6,. So we

have proved t h a t f o r a f i x e d n > max (M,N) we have yi E Gn f o r each

i > n. Since 6;, i s c losed, we g e t y E in . Hence y E K* and t h i s

proves r ( x ) c K* I) { x } .

Observe now t h a t t h e above process can be per formed on any

g i ven square i n t e r v a l Q n o t n e c e s s a r i l y equal t o Q ( 1 ) . That i s ,

g i ven Q t h e r i s K* such t h a t I K * l = 0 and f o r each x E Q t h e r e

i s a s t r a i g h t l i n e r ( x ) through x so t h a t r ( x ) c K* II {XI. We

app ly t h i s t o Q 1

K: , K; ,... ,K; ,... We now d e f i n e

t h e s tatement o f t h e theorem.

= Q ( 1 ) , Q 2 ( 2 ) ,... , Q k ( k ) ,... and we o b t a i n

k=l K = 8 K i and t h i s s e t s a t i s f i e s

The f o l l o w i n g r e s u l t can be e x t r a c t e d q u i t e e a s i l y f rom t h e

p reced ing p r o o f . It w i l l be q u i t e use fu l f o r t h e c o n s i d e r a t i o n s t h a t

f o l l o w , i n t h e n e x t Sec t i on (8 .6 .4 . ) .

8.5.5. THEOREM . LeX Q be t h e d m e d ~ q w e in;tmvd pen - t a e d at Q and w a h A i d e LengZh 2 . The t h m e e & t ~ a AubAeA M 0 6 Q ob 61LeR meanme, i . e . l M l = I Q I and a h& K * c R2 0 6 null meu-

une huch thuX don each x E M t h m e LA a A i X a i g k t f i n e r ( x ) pun ing

thnvugh x and contained i n K* I) {XI i n huch a way XhaX t h e d i h e d a n

0 6 r ( x ) v&en i n a rneanmabRc way.

--

Pnood. L e t us r e t u r n t o t h e p r o o f o f t h e theorem 8.5.4.

subset M o f Q i s go ing t o be t h e complement i n Q o f t h e un ion o f

t h e boundar ies o f a l l s t r i p s we have s e l e c t e d i n t h a t process. C l e a r l y

I M I = I Q I . L e t us denote a l s o by r k ( x ) E [ O , ~ T ) t h e ang le a s s o c i a t e d

t o t h e l i n e r k ( x ) . We s h a l l show t h a t a t each s t e p k o f t h e c o n s t r u c

t i o n we can make a s e l e c t i o n o f l i n e s r k ( x ) f o r x 6 M such t h a t t h e

f u n c t i o n i s a measurable f u n c t i o n . S ince we a l s o

have r k ( x ) + r ( x ) a t each x E M as k + m we see t h a t r ( x ) i s

measurable on M.

The

x E M -f r k ( x ) E [O,T)


Consider the s t r i p s w:, ui, w:, ... s e l e c t e d i n t h e f i r s t s tep .

To t h e p o i n t s i n W: 0 M

t h e p o i n t s i n (ui - u: ) 0 M we a s s i g n t h e d i r e c t i o n of t h e s t r i p w:.

To t h e p o i n t s i n (u: - LO'.) 0 M t h e d i r e c t i o n o f u:. And so on.

So we o b t a i n r l ( x ) on M t h a t i s a s t e p f u n c t i o n .

we ass ign t h e d i r e c t i o n o f t h e s t r i p w:. To

I', j=1 J

Consider now w i f ) M and t h e s t r i p s o f t h e second s t e p c o v e r i n g

ui ( 7 Q . They a re such t h a t t h e i r i n t e r s e c t i o n s w i t h Q can proceed t o ass ign d i r e c t i o n s as above.

(u: - uj) 0 M and t h e s t r i p s of t h e second s t e p c o v e r i n g LO: we can

proceed i n t h e same way . And so on. The second assignment r 2 ( x ) i s

a l s o a s tep f u n c t i o n on M. I n t h i s way we see t h a t r ( x ) i s measurable.

a r e i n ui. We

When we now c o n s i d e r

The s e t K* o f Theorem 8.5.4. s a t i s f i e s t h e s tatement o f

ou r theorem.

8.6. DIFFERENTIATION PROPERTIES OF SOME BASES OF RECTANGLES.

L e t us now e x t r a c t some i n f o r m a t i o n about t h e d i f f e r e n t i a t i o n

p r o p e r t i e s o f B 3 and o f some subbases of B 3 from what we have a l r e a d y

seen.

From t h e e x i s t e n c e o f t h e Nikodym s e t , as we p o i n t e d o u t b e f o r e

Zygmund observed t h a t B 3 cannot even be a d e n s i t y b a s i s . I t i s n o t neces

sa ry t o go so f a r t o o b t a i n t h i s f a c t . Wi th t h e c o n s t r u c t i o n o f t h e Perron

t r e e of 8.1.2.

which t h i s f a c t i s an easy consequence.

we a r e go ing t o be a b l e t o prove a s t r o n g e r r e s u l t f r o m

8.6.1. THEOREM . CoaLdm t h e B - F diddmenLLaLion baA


bT invahiant by homothecia genetra-ted by all U n g L a

Th h a LJ~JLLLCCA ( 0 , l ) ,(h-1,O) ,(h,O) . Then BT 0 not u der&Ltg

b a d .

{Thl;=l , whetre

P t o o ~ . L e t MT be t h e maximal o p e r a t o r assoc ia ted t o BT . I f E i s a Perron t r e e c o n s t r u c t e d f rom {Thlh,, 2n

we can w r i t e

as i n Theorem 8.1.1.

where 7, i s t h e e x t e n s i o n o f Th below t h e b a s i s o f Th between

y = 0 and y = -1 . There fo re acco rd ing t o t h e Remark 2 o f S e c t i o n 8.1.

we g e t

Therefore, acco rd ing t o t h e c r i t e r i o n o f Busemann and F e l l e r f o r d e n s i t y

bases, BT cannot be a d e n s i t y b a s i s .

Fo r each Th o f Theorem 8.6.1. l e t Rh be t h e r e c t a n g l e

i n d i c a t e d i n t h e F i g u r e 8.6.1.

F i g u r e 8.6.1.


and l e t BR be t h e B - F b a s i s i n v a r i a n t by homothecies generated by

{ R h I . If MR i s t h e corresponding maximal o p e r a t o r we have w i t h an

absol Ute cons tan t c

M T f ' 6 C M f R

and so &lR i s n o t a d e n s i t y b a s i s . T h i s o f course i m p l i e s t h a t B3 i s

n o t a d e n s i t y b a s i s .

8.6.2. COROLLARY. The. b u d oh hec&ngLen B3 d nv.t a

dennity b u i h . So we see t h a t n o t o n l y 63 i s a v e r y bad d i f f e r e n t i a t i o n

b a s i s , b u t a l s o t h a t a r a t h e r smal l subbasis o f 63 such as

t a i n i n g r e c t a n g l e s i n a smal l s e t o f d i r e c t i o n s and f o r each d i r e c t i o n

a small subset o f a l l t h e p o s s i b l e r e c t a n g l e s i n t h a t d i r e c t i o n i s a

ve ry bad d i f f e r e n t i a t i o n b a s i s . T h i s r a i s e s a number o f i n t e r e s t i n g

ques t i ons .

B R , con-

PROBLEM 1 . L e t 0 be a subse t o f [0,2n) . Consider t h e b a s i s

Ba o f a l l r e c t a n g l e s i n d i r e c t i o n s (I E Q , i . e . one o f whose s i d e s

forms an ang le (I Q 0 w i t h Ox. How shou ld @ be d i s t r i b u t e d i n o r d e r

t h a t BQ have some good d i f f e r e n t i a t i o n p r o p e r t i e s ?

We s h a l l soon see t h a t i f @ i s any s e t such t h a t i t s c l o s u r e

i s o f p o s i t i v e Lebesgue measure, t hen BQ i s a v e r y bad bas i s . But

t h e r e a re bases BQ w i t h 0 denumerable, such as BR above, t h a t a r e

a l s o bad. However, f o r s e t s 0 t h a t a r e lacunary, f i r s t Strornberg [1976]

and l a t e r CBrdoba and R.Fefferman [1977] have ob ta ined p o s i t i v e d i f f e r e n

t i a t i o n p r o p e r t i e s f o r BQ

s u l t s . The genera l problem i s s t i l l unsolved. It i s even unknown whether

Ba i s o r n o t a d e n s i t y b a s i s when 0 i s t h e coun tab le s e t o f endpoints

o f a l l t h e chosen i n t e r v a l s a r i s i n g i n t h e succes ive phases o f t h e con-

s t r u c t i o n o f t h e Cantor s e t i n [0,1] .

. I n t h e n e x t s e c t i o n we w i l l s t u d y such re-


PROBLEM 2 . Even when B;P i s a d i f f e r e n t i a t i o n bas i s , i t s

p r o p e r t i e s can improve when we r e s t r i c t ou rse l ves t o c o n s i d e r f o r each

4 E 0 n o t CLee t h e r e c t a n g l e s i n t h a t d i r e c t i o n b u t o n l y those homothe

t i c t o t h e ones o f a f i x e d c o l l e c t i o n R o f r e c t a n g l e s i n d i r e c t i o n 4 . So we o b t a i n a new B - F b a s i s B(Q,R) generated by ( 0 R4)+€@ . The p r o p e r t i e s o f t h i s k i n d o f bases have n o t been e x p l o r e d so f a r . One

o n l y knows some r a t h e r t r i v i a l r e s u l t s . Fo r example, i f @ = [0,2n) and

f o r each 4 E 0 , R4 i s j u s t a square, t hen B(0.R) s a t i s f i e s t h e

V i t a l i p rope r t y , d i f f e r e n t i a t e s L’, ... Bu t even i f f o r example @ = {O)

and R o i s a sequence o f i n t e r v a l s { I k } w i t h e c c e n t r i c i t y t e n d i n g t o

m one does n o t know wheter, by an a p p r o p r i a t e cho ice o f { I k } y t h i s

b a s i s w i l l have b e t t e r d i f f e r e n t i a t i o n p r o p e r t i e s than those o f t h e

b a s i s o f a l l i n t e r v a l s .

4

The case i n which @ i s t h e s e t o f a l l d i r e c t i o n s and f o r each

I$ E Q we c o n s i d e r a l l r e c t a n g l e s R$ i n d i r e c t i o n I$ w i t h e c c e n t r i c i t y

n o t exceeding a f i x e d number H independent o f I$ has o f course v e r y good

d i f f e r e n t i a t i o n and c o v e r i n g p r o p e r t i e s (has t h e B e s i c o v i t c h p r o p e r t y , t h e

V i t a l i p rope r t y , . . . ) . I t s maximal o p e r a t o r i s o f weak t y p e ( 2 , 2 ) w i t h a

c o n s t a n t c(H) t h a t increases w i t h H t o i n f i n i t y . C6rdoba [1976] has

ob ta ined a measure o f t h e s i z e o f c (H) .

PROBLEM 3. Consider t h e b a s i s o b t a i n e d i n t h e f o l l o w i n g way.

For each x e R 2 t a k e a d i r e c t i o n d ( x ) E [ O , Z . r r ) and c o n s i d e r t h e

c o l l e c t i o n

x . (The b a s i s

r e n t i a t i o n p r o p e r t i e s o f Bd ?

Bd(x) o f a l l t h e open r e c t a n g l e s i n d i r e c t i o n d ( x ) c o n t a i n i n g

Bd i s n o t a Buseman-Feller b a s i s ) . What a r e t h e d i f f e -

How does t h e cho ice o f d ( x ) a f f e c t them?

I n what f o l l o w s o f t h i s S e c t i o n we s h a l l examine c e r t a i n nega -

t i v e r e s u l t s concern ing some of these ques t i ons . I n t h e n e x t s e c t i o n we

s tudy t h e r e s u l t o f Stromberg and o f C6rdoba and R.Fefferman and l a t e r

on i n Chapter 12 some theorems o f S t e i n and Wainger concern ing t h e ques-

t i o n s around Problem 3.

The f o l l o w i n g Theorem has been ob ta ined by M.T.Men2rguez and

t h e a u t h o r and i s p u b l i s h e d h e r e f o r t h e f i r s t t ime.


8 . 6 . 2 . THEOREId. L e A 0 c [0,2n) be a b e i whabe cRobwe h a pobi,tive meanwe. Then t h e B - F b a b BQ 0 4 a l l tre.etangLec?n i n di-

teCtion $ e 0 0 not a denbLty b a 0 .

Ptrood. Observe f i r s t t h a t i f R i s a r e c t a n g l e i n d i r e c t i o n

d, e such

then t h e r e

l i l 6 21R

Therefore

t h a t

s a r e c t a n g

, and so

e i n d i r e c t i o n $ e Q such t h a t 6 '3 R ,

MQf (x ) 6 Mm f ( x ) 6 2 MQ f ( x ) . Hence we can assume, i n

o r d e r t o prove t h e Theorem, t h a t Q = and t h a t = p > 0 .

$ 0

t r e e

B k

i n d

We can a l s o assume w i t h o u t l o s s o f g e n e r a l i t y t h a t each p o i n t

Q i s a d e n s i t y p o i n t o f 0 . Wi th an E > O we c o n s t r u c t a Perron

PE as i n Theorem 8.1.2. s t a r t i n g f rom a t r i a n g l e ABC w i t h

n/2 and so t h a t t h e s i d e BA i s i n d i r e c t i o n n/4 and CA i s

r e c t i on 3 ~ 1 4 .

For any p o i n t d, e Q n o t c o i n c i d i n g w i t h any o f t h e d i r e c t i o n s

we have a sequence o f nondegenerate o f t h e s ides o f t h e t r i a n g l e s Th i n t e r v a l s Ik(d,) = [d, - ak , $ + Bk] , 0 G ak , Bk so t h a t

(b ) I k ( d , ) t r i a n g l e s T,, a t A

i s i n s i d e one ang le o f those determined by t h e

A p p l y i n g V i t a l i ' s theorem we can s e l e c t a f i n i t e number o f


such s e t s I k ( 4 ) , c a l l them { E l , E 2 ,..., EH} , s a t i s f y i n g

(i) The s e t s E a r e d i s j o i n t j

( i i ) I I J E ~ I >

(iii) One endpo in t o f each Ek i s i n @ .

( i v ) Each E j i s i n s i d e one ang le of those determined by

t h e t r i a n g l e s Th a t A .

Now i n t h e c o n s t r u c t i o n o f t h e Per ron t r e e PE of Theorem 8.1.2.

we c o n s i d e r t h e t r i a n g l e S w i t h base on BC and v e r t e x a t A w i t h

angle a t A determined by E j and make S j s o l i d a r y w i t h Thy t h e

t r i a n g l e t h a t c o n t a i n s S . A f t e r t h e t r a n s l a t i o n s t h a t t a k e Th t o 7, t h e t r i a n g l e S . goes t o t h e p o s i t i o n 3 . . L e t us c a l l ? t h e t r i a n -

g l e symmetric t o Sj w i t h r e s p e c t t o i t s v e r t e x t r a n s l a t i o n of A. I f we

c o n s i d e r t h e B - F b a s i s Bs i n v a r i a n t by homothecies generated by

s1 YS2,. . . ,SH and c a l l MS t h e co r respond ing maximal o p e r a t o r , we have

j

J ' J J j

- Therefore, s i n c e t h e t r i a n g l e s sj a r e d i s j o i n t , by Remark 1

o f 8.1. , we have

where u i s a f i x e d number t h a t depends o n l y on p . Hence

Now i t i s easy t o see, l o o k i n g a t F i g u r e 8.6.2. t h a t

and so Msf c c M @ f


F i g u r e 8.6.2.

w i t h c independent o f t h e t r i a n g l e s S j .The re fo re

S ince E i s a r b i t r a r i l y sma l l we o b t a i n t h a t BQ cannot have t h e d e n s i t y

p r o p e r t y .

Tha t one b a s i s B, i s a bad d i f f e r e n t i a t i o n b a s i s i s n o t neces

s a r i l y due t o t h e f a c t t h a t 0 has t o o many d i r e c t i o n s as i n t h e p r e -

ceding theorem. It can be mo t i va ted by t h e d i s t r i b u t i o n o f t h e d i r e c t i o n s

con ta ined i n @ . As we have seen i n Theorem 8.6.1. i f

cp = I a r c t g 1, a r c t g 2, a r c t g 3, ... 1 t hen B, i s n o t a d e n s i t y b a s i s .

I n t h e same way i t can be shown t h a t i f 0 = I a r c t g 1, a r c t g PP,

a r c t g 3p,...3 then B has n o t t h e d e n s i t y p r o p e r t y . However i f P

QP @ = { a r c t g 2, a r c t g 2', a r c t g Z 3 , . . . I t h e n B, i s a good d i f f e r e n -

8.7. I n b o t h cases Imll= 0. t i a t i o n bas is , as w i l l be proved i n S e c t i o n

L e t now cp be t h e s e t o f d i r e c t i o n s determined by j o i n i n g

t h e p o i n t ( 0 , l ) t o t h e p o i n t s on Ox o f absc issae

1 , 2 , 3 , 3x2+1 , 3x2+2 , 32,32x2. 3%2+1,...32x2+3, 2+2 , 33,. .. k A f t e r 3k we t a k e

g e t 3 k x 2 + 3 k - 1 x 2 + 3 k - 2 x 2 + ... + 3 l x 2 + 3Ox 2.

number i s

3 x 2 and then we add t h e p r e v i o u s numbers u n t i l we

The f o l l o w i n g

k+ 1 3k x 2 + 3k -1 x 2 + ... + 31 x 2 + 30 x 2 f 1 = 3

and now we con t inue . I t i s n o t known whether

b a s i s BQ B o * ob ta ined when we t a k e as O* t h e s e t o f d i r e c t i o n s determined by

j o i n i n g t h e p o i n t

success ive phases of t h e c o n s t r u c t i o n o f t h e Cantor s e t i n t h e u n i t i n t e r

Val o f Ox.

Ba i s a d e n s i t y b a s i s . T h i s

has t h e same d i f f e r e n t i a t i o n p r o p e r t i e s as those o f t h e b a s i s

( 0 , l ) t o t h e endpoints o f t h e i n t e r v a l s taken i n t h e

The bases Bd considered i n PROBLEM 3 a r e n o t B - F bases.

I t i s q u i t e easy t o see t h a t i f one can c o n s t r u c t a Nikodym s e t N

t h a t f o r each p o i n t x E N t h e d i r e c t i o n o f t h e l i n e l ( x ) so t h a t

l ( x ) f ) N = {XI c o i n c i d e s w i t h d ( x ) t hen Bd i s n o t a d e n s i t y b a s i s .

We can f o r m u l a t e t h i s f a e t a l i t t l e more p r e c i s e l y .

such

8.6.3. ___. THEOREM. Ah~ume thcLt N d a nQ.t 0 6 ponUve rneauhe i n R 2 nuch thcLt doh. each xsN t h e m d u L i n e l ( x ) Zhotaughxbatin6qing

l ( x ) II N = {XI. LeX d be a d i d d ol( ditrecLLo~o buch tha t d o t mch

xsN t h e f ine 1 ( x ) h a t h e d4,teotion d ( x ) . Then t h e bmd Bd AA not

a deMnMq b a h .

Ptoozf . We take a compact subset F o f N such t h a t

I F / > 3/4 I N 1 . For each x c F t h e l i n e l ( x ) i n t e r s e c t s F j u s t

a t x. So, us ing t h e compactness o f F, we can draw r e c t a n g l e s R o f

d i r e c t i o n d ( x ) , i . e . t h a t of l ( x ) , c o n t a i n i n g x and c o n t r a c t i n g

t o x so t h i n t h a t

< 1/3 I R I

The re fo re t h e l ower d e n s i t y o f F a t each o f i t s p o i n t s x w i t h r e s p e c t

t o Bd i s l e s s than 1 and so Bd i s n o t a d e n s i t y b a s i s .

We have a l r e a d y seen i n 8.5. t h a t one can c o n s t r u c t a Nikodym

s e t N such t h a t t h e d i r e c t i o n o f t h e l i n e l ( x ) v a r i e s i n a measurable

way. T h i s l eads t o t h e f o l l o w i n g r e s u l t .


8.6.4. THEOREM. Them ~~~2 a covLtinuauh &ieRd 0 6 ditrecfionh d a n R ouch t h a t Bd 0 no2 a dennay b a d .

Ptraod. L e t N be a s e t o f p o s i t i v e measure such t h a t t h e

d i r e c t i o n d ( x ) o f t h e l i n e l ( x ) through x such t h a t N 0 l ( x ) = Cxj

v a r i e s i n a measurable way. By L u s i n ' s theorem we can take a compact

subset F o f p o s i t i v e measure such t h a t d (x ) v a r i e s c o n t i n u o u s l y . We

extend t h i s f i e l d o f d i r e c t i o n s c o n t i n u o u s l y t o R2 , o b t a i n i n g d. The

ng t o t h e p reced ing theorem cannot be a d e n s i t y bas i s . b a s i s Bd accord

We sha l l a t e r see i n Chapter 9 t h a t a c c o r d i n g t o a r e s u l t

o f Casas [1978 1 , when d ( x ) i s a L i p s c h i t z f i e l d o f d i r e c t i o n s then

i t i s n o t p o s s i b l e t o c o n s t r u c t a Nikodym s e t N x E N

l ( x ) has d i r e c t i o n d ( x ) . I t i s n o t known y e t whether t h e L i p s c h i t z

c o n d i t i o n o f d i s s u f f i c i e n t t o make of Bd a d e n s i t y b a s i s . I n

Chapter 12 we s h a l l p r e s e n t some p o s i t i v e r e s u l t s o f S t e i n and Wainger

[I9781 f o r some smooth f i e l d s o f d i r e c t i o n s .

such t h a t f o r

When t h e f i e l d d has a coun tab le number o f va lues EdhlF=l

t h e b a s i s Bd d i f f e r e n t i a t e s ~ ( 1 + l o g + L ) .

f E L (1 t l o g L ) (R2) and Eh = { x : d ( x ) = d h l . L e t Bd be t h e

b a s i s o f a l l r e c t a n g l e s i n d i r e c t i o n dh . We know t h a t Bd d i f f e r e n :

t i a t e s i f . There fo re t h e s e t o f p o i n t s o f Eh where Bd does n o t

d i f f e r e n t i a t e f i s o f n u l l measure . Since t h e r e i s a denumerable

c o l l e c t i o n of s e t s Eh , t h e s e t o f p o i n t s o f R2 where Bd does n o t

I n f a c t , l e t +

h

h

I f i s o f n u l l measure.

8.7. LACUNARY DIRECTIONS 233

8.7. SOME RESULTS CONCERNING BASES OF RECTANGLES I N LACUNARY DIRECTIONS.

The problem d e a l t w i t h h e r e i s t h e f o l l o w i n g . Assume @ i s

t h e f o l l o w i n g s e t o f d i r e c t i o n s Q, = { 2- , 2L , i!L , 2- ,... I . Consider 2 2 2 3 2 4 2 5

t h e b a s i s

What a r e t h e d i f f e r e n t i a t i o n p r o p e r t i e s o f t h e b a s i s

[1976] proved t h a t t h i s b a s i s d i f f e r e n t i a t e s

any E > 0 . t h a t BQ, d i f f e r e n t i a t e s L2( T h i s i s e q u i v a l e n t , s i n c e B@ i s i n v a r i a n t

by homothecies, t o t h e f a c t t h e maximal o p e r a t o r MQ, a s s o c i a t e d t o

i s o f weak t y p e ( 2 , Z ) ) . The methods used by Stromberg and a l s o by C6r-

doba and R.Fefferman i n v o l v e pu re r e a l v a r i a b l e c o n s i d e r a t i o n s o f t h e

t ype we have been h a n d l i n g i n t h i s and t h e p reced ing two chap te rs . L a t e r

on S t e i n and Wainger [1979 1, by methods o f F o u r i e r a n a l y s i s , have im-

proved these r e s u l t s .

8@ o f r e c t a n g l e s w i t h one s i d e i n one o f t hese d i r e c t i o n s .

L 2 ( log' L)4+E (R2) f o r

B o ? Stromberg

Wi th e a s i e r means C6rdoba and R.Fefferman [1977] proved

Bo

Here we s h a l l examine t h e method o f C6rdoba and R.Fefferman

w i t h a s l i g h t l y mod i f i ed v e r s i o n o f t h e b a s i s

e a s i e r t o handle. From t h i s r e s u l t t h e same c o v e r i n g theorem and t h e

weak t y p e (2,2)

f o r t h e above b a s i s .

B@

f o r t h e co r respond ing maximal o p e r a t o r can be o b t a i n e d

t h a t w i l l be a l i t t l e

We s h a l l c o n s i d e r t h e b a s i s &i o f a l l p a r a l l e l o g r a m s R

s a t i s f y i n g :

( a ) Two o f t h e i r s i d e s a r e p a r a l l e l t o Oy.

(b ) The o t h e r p a i r o f s i d e s have one o f t h e d i r e c t i o n s

T I 7 1 {-, - , ... 1

2 2 2 3

( c ) The p r o j e c t i o n p(R) o f R ove r Ox i s a dyad ic i n t e r v a l .

(d ) Each R i s so t h i n t h a t i f i i s t h e min imal i n t e r v a l

c o n t a i n i n g R we have l R l / l i l G 1/8 .

F o r t h i s b a s i s & we s h a l l prove t h e f o l l o w i n g c o v e r i n g re-

s u l t .


H

H

(Here and i n the proof, t he constantc i s a pos i t i ve absolu te constant n o t depending on the co l l ec t ion a t each occurrence )

(B,)aeA , not always the same

Ptuo6. We can f i r s t s e l e c t a f i n i t e sequence C B 1 , ..., BN 3 - -- from so t h a t

N I ( ' B,I c 21 y Bk l

We can assume t h a t B1 , B 2 , . . . have been so ordered t h a t

b ( B j ) = length of projection of 9-. over Ox > length o f J proyection of B j t l over Ox = b ( B . ) J+1

Also we can assume t h a t no 9 . i s contained i n another one. J

We s t a r t choosing the R s e t t i n g R1 = B1 . We examine B 2 . I f

j

then we s e t R2 = B2 . Otherwise we leave B2 as ide . Assume R 1 = B1 , R 2 = B2 . Examine B3 . I f

2

1 9 1'31 1 I B3 0 R. J I = XB, ii, X R j i =1

then R B = B 3 . Otherwise we leave B3 as ide . Assume t h a t R 1 = B 1 ,

R P = B1 , B 3 has been l e f t . Consider B1, . I f

8.7. LACUNARY D I R E C T I O N S 235

1 ! xBl, ( R1 + XR,) 6 7 1 8 4 1

then R 3 = B 4 . Otherwise we l e a v e B4 as ide . And so on. So we f i n i s h

ou r s e l e c t i o n i n a f i n i t e number of s teps o b t a i n i n g CR.1: t h a t s a t i s f y : J

( a ) b ( R j ) b ( R j + l )

1 h (b ) I ( XRj) XR ‘ 7 I R h + l l

h+ 1

( c ) I f Bi has n o t been chosen , then

P r o p e r t y ( i i ) i s e a s i l y ob ta ined . I n f a c t

But we have

There fo re l I R j 1 c 2 I OR j I and we g e t ( i i ) . To prove p r o p e r t y ( i ) we use ( c ) , ( d ) and t h e l a c u n a r i t y o f

t h e sequence as f o l l o w s . I f Bi has n o t been chosen, then

There fo re we have


or

or

where d ( R j ) ( A ) i s true are in the si tuation o f the figure

means the direction 4 of R j . The se ts Bi for which

Figure 8.7.1.

If we intersect by a vertical l ine 1 , x X ilnd call M1

the unidimensional maximal operator w i t h respect t o intervals of get

1 we

1 1 0 Bi C {(Xyy) : M 1 ( l X R j ) (1,~) > g 1

Therefore the union of a l l se ts Bi f o r which (A ) h o l d s has a measure n o t exceeding

8.7. LACUNARY DIRECTIONS 237

Consider now a s e t Bi f o r which (B) i s t r u e .

J R j i n t e r e s t i n g Bi such t h a t d(R.) > d(Bi) and b(Rj ) > b

Draw t h e minimal c losed i n t e r v a l Bi c o n t a i n i n g Bi . We sha

- IBi 0 R . 1 [Bi 0 R . 1

J (*I > c l j I I B j l

So we have

ake one

B i ) . 1 p rove

- Therefore Bi c Bi c Ix : MZ 1 x R . ( x ) > 2 j

3 where MZ i s t h e maximal o p e r a t o r w i t h r e s p e c t t o i n t e r v a l s o f R2. Thus

we o b t a i n t h a t t h e union of t hose B, f o r which (B) i s t r u e has a rea

1 ess than

I n o r d e r t o prove (*) we cons ide r t h e f o l l o w i n g f i g u r e

I I

4

A I 1 I I I l a I I I I

4

F i g u r e 8.7.2.

8. THE B A S I S OF RECTANGLES

j We have, whatever i s t h e s i t u a t i o n o f R

Bu t i t i s now easy t o show, because o f t he l a c u n a r i t y o f Q

f a c t t h a t Bi i s t h i n , t h a t 6 c. So we have

and t h e a

F i g u r e 8.7.3.

For a s e t Bi so t h a t ( C ) i s t r u e a s i m i l a r c o n s i d e r a t i o n

holds. So we o b t a i n

and t h e theorem i s proved.

8.7.2. COROLLARY. The maxim& op&on M comenponding t o

B ,455 ad weak .type (2,2).

Phood . L e t f E L 2 and A = I M f > A > 0 . If K i s any

compact subset of A and x E K t h e r e i s R, f B c o n t a i n i n g x such

t h a t

8.7. LACUNARY D I R E C T I O N S 239

we a p p l y t h e theorem , o b t a i n i n g E R j I . So we have To ( R X ) X E K

CHAPTER 9

THE GEOMETRY OF LINEARLY MEASURABLE SETS

The geometric theory of l i nea r ly measurable s e t s i n R2 was developed mainly by Besicovitch. His fundamental papers on t h i s subjec t were wr i t ten i n 1928, 1938, 1939, 1964. The whole theory i s i n i t s e l f qu i t e i n t e re s t ing and beautiful and does not seem to have been suf f ic ien- t l y exploited from the point of view of i t s connections w i t h the real var iab le theory. I n the au thor ' s opinion i t shows promising signs of becoming a very useful tool t o handle some of the problems a r i s i n g in areas where one has t o look carefu l ly a t the geometric s t r u c t u r e o f s e t s of R 2 w i t h two-dimensional measure zero o r of co l l ec t ions of f igures t h a t in some sense can be assimilated t o them. e r in Chapter 8 co l lec t ions of th in rectangles of d i f f e r e n t nature. As we have seen, ce r t a in f igures associated to two-dimensional s e t s of Le- besgue measure zero,such as the Besicovitch s e t o r the Nikodym s e t can shed a powerful l i g h t on them. As we sha l l l earn in t h i s Chapter, Besicovitch's theory of l i nea r ly measurable s e t s will help us t o understand b e t t e r some of these s i t u a t i o n .

We have been led t o consid

The theory, of which we are going t o give a glimpse here, abounds in problems a n d theorems tha t look q u i t e elementary, The t r ea t - ment of them, however, i s often qu i t e complicated and the re a re many questions s t i l l open in the f i e l d . I t would be very des i r ab le t o have more straighforward proofs o f many o f the r e s u l t s we sha l l study. There i s in the l i t e r a t u r e no complete systematic exposit ion of t h i s beaut i fu l portion of the geometric measure theory, t h o u g h some of the r e s u l t s can be presented as pa r t i cu la r cases o f the general theory of Federer ' s book [1969].

In what follows we sha l l developed some of t he f a c t s t h a t a r e needed t o a r r i v e t o some in t e re s t ing appl ica t ions , espec ia l ly t o

241

242 9. GEOMETRY AND LINEAR MEASURE

t h e tvDe o f problems we have been h a n d l i n g i n t h e l a s t Charjter. Our

e x p o s i t i o n i s s t r o n g l y i n s p i r e d i n t h e work o f who has

improved and S i m D l i f i e d some p o r t i o n s o f t h e t h e o r y o f B e s i c o v i t c h .

A.Casas [1978]

9.1. LINEARLY MEASURABLE SETS.

The Hausdor f f measure As o f dimension s , 0 c s c 2 , i n R 2 i s d e f i n e d i n t h e f o l l o w i n g way, Fo r E c R 2 and p > 0

one f i r s t cons ide rs

The q u a n t i t y ASCE1 , i nc reases as p decreases and we c a l l P

A S * ( E ) = l i m A; (E) . Then As* i s an o n t e r measure, Cara th6odorv ' s P'O

process g i v e s us t h e assoc ia ted measure A s . and each B o r e t s e t i s As-rneasurable, Fur thermore t h e measure AS i s r e q u l a r . For each s e t E t h e r e i s a s i n g l e number t, 0 c t C 2

such t h a t f o r s i t we have AS(E) = m and f o r each s > t, A s ( E ) = O .

The s e t E i s then s a i d t o be of Hausdor f f dimension t. It can, of

course , s t i l l happen A (E) = 0 o r At (E) = m . B e s i c o v i t c h [1928,

1938, 1939, 1 9 6 4 1

f i n i t e l i n e a r measure, i . e . o f those s e t s E such t h a t 0 c A 1 ( E ) < m . We s h a l l r e s t r i c t our a t t e n t i o n t o such s e t s and so we w i l l w r i t e A f o r

A' . I n t h i s chapter , un less o t h e r w i s e e x D l i c i t l y s t a t e d , "meanukable" w i l l mean

T h i s measure i s complete

t

s t u d i e d t h e geometr ic p r o o e r t i e s o f t h e s e t s o f

'I A-meaukable w a h dinite t n m u h e " .

Before we l o o k a t t h e d e n s i t y p r o o e r t i e s o f such s e t s we s h a l l

prove a u s e f u l form o f t h e V i t a l i theorem f o r t h e measure A .

9.1. LINEARLY MEASUREABLE SETS 243

9.1.1. THEOREM. 1eR: E be a meau/Lab.ee b d . Fok each x 6 E a heqUenCe { C k ( x ) > a6 c&ohed CitLdeA centeked CLt x WLth 6 ( C k ( x ) ) + 0

A given . Let E > 0 . Then o ~ e CUM choahe a sequence {D;} ad & j o i n t J

c & c l e ~ @om (Ck (x ) IxeE such thcLt

k=l ,2 ¶ . . .

A ( E - D.) = rl J (i 1 j

( i i ) A C E ) c 1 6 ( ~ ~ ) t E j

I n o r d e r t o o b t a i n ( i i ) we prove f i r s t t h e f o l l o w i n g n r o p e r t y

t h a t , f o r re fe rence , we s t a t e s e n a r a t e l y as a lemma.

9.1.2. LEMMA. 1eX E be meanuhable and E > 0 . Then thehe

e x i ~ a 2 p > 0 , p = ~ ( E , E ) , nuch LhcLt d n f i each coReection (Ba)aeA

06 B u t e l b& such t h d 0 < 6 (Ba) 0. From t h e

d e f i n i t i o n o f h(E), g i ven E > 0 t h e r e e x i s t s P O such t h a t f o r each

sequence { A j } w i t h 6(Aj ) < P O such t h a t ( J A , D E we have

k l e t , f o r each k = 1 , Z ¶... { A , } be a sequence o f s e t s whose u n i o n J k covers E - !I Bu w i t h 0 < 6(Aj) < P O and such t h a t

1 AS) + A ( E - 1) B ~ ) as k + m

J 1 6(Aj ) ?; A ( E ) - E. Assume P 6 P O . L e t us c o n s i d e r E - ! J B, and

j k Then [ A j } IJ {Ba) i s a cover of E w i t h d iameter l e s s than p o . T h e r e

f o r e

But i f we make k t e n d t o m then we g e t

244 9. GEOMETRY AND L I N E A R MEASURE

P4ood ad Theo4em 9.1.1. We f i r s t choose p acco rd ing t o

t h e lemma we have j u s t proved and cons ide r c i r c l e s

l e s s than

( i ) , t hen

C,(x) o f d iameter

p , So we now have t o ca re o n l y about ( i ) , s i nce , i f we g e t

A ( E ) = A(E 0 I! D s ) 6 1 6 ( D j ) + E

Fo r each x E E we choose f i r s t one such c k ( x ) and aoo ly B e s i c o v i t c h

cove r ing theorem 3.2 .1 . o b t a i n i n g a f a m i l y I S k } o f c i r c l e s whose un ion

covers E t h a t can be d i s t r i b u t e d i n t o 5 d i s j o i n t sequences

{ S i I ,..., 1 . For one o f them, say { S i l , we must have

s i n c e o the rw ise

and t h i s i s c o n t r a d i c t o r v .

Now we have

A ( E - ( IJ S i ) ) 6 tL A ( E ) 5

We keep a f i n i t e number o f s e t s o f { S i l , c a l l i n g them {D j ) and g e t

H1 A(E - D ~ ) 6 + A ( E ) = X A ( E ) , A < 1

1 5+ 7

H1 We can now oroceed w i t h El = E - IJ D j as we have done w i t h E o and

o b t a i n IDjlHH:+l so t h a t 1

H2 H2 A ( E - 1) D j ) 6 X A ( E - I! D j ) 6 X 2 A ( E )

H1+1

9.2. REGULAR AND IRREGULAR SETS

I n t h i s way we o b t a i n C D j I s a t i s f y i n g ( i ) and ( i i ) .

9.2. DENSITY , REGULAR AND IRREGULAR SETS.

L e t E be a measurable s e t , 0 L A ( E ) < m. Fo r x 8 R 2 we

cons ide r t h e l i m i n f and t h e l i m sup as r + 0 o f

A ( E 0 B(x, r ) ' - 2 r

where B(x,r) i s a c losed c i r c l e cen te red a t x and o f r a d i u s r > 0 .

They a r e r e s p e c t i v e l y t h e l o w t and t h e uppm devm3ity ( t h e derm%q , i f they c o i n c i d e ) o f E a t x. We denote them by D(E,x) , D(E,x) (D(E,x) if t h e y a r e e a u a l ) .

- We s h a l l f i r s t prove t h a t t h e f u n c t i o n s - D ( E , * ) , D(E,*) a r e

measurable. Fo r t h i s l e t 1 > 0 and F = I x 6 R 2 : D(E,x) > A } . L e t

p > 0, E > 0 be two r a t i o n a l numbers and l e t

p o i n t s x on which

F ( P , E ) be t h e s e t o f

- A ( E 17 B(x,r)) 2 r

A + €

f o r each r 6 p . We have c l e a r l y

We now prove t h a t each F ~ , E ) i s a c losed s e t . I f { x k 1 c F ( p , € ) , x k + x , then f o r each r b p t h e r e i s an i ndex ka such t h a t i f

k a k a we have Xk B B(x, r ) . L e t B(xk,rk) be t h e c i r c l e cen te red

a t xk and o f maximal r a d i u s con ta ined i n B ( x , r ) , We have rk c r 4 p

and l i m rk = r. We a l s o have

245


There fo re A(E 11 B(x , r ) ) >, 2 r ( X + E and so x 6 F(P,E) . I n a

s i m i l a r way one ge ts t h e m e a s u r a b i l i t y o f 6(E,.).

By means o f t h e cove r ing lemma 9.1.1. we e a s i l y o b t a i n t h e

f o l l o w i n g p r o p e r t i e s f o r t h e d e n s i t y .

9.2.1. THEOREM . L e X E be a measwrabLe b e X . Then at cLemont

each x 6 E we have D(E,x) = 0 .

P m o d . L e t us f i x c1 t 0 and de f i ne __

We t r y t o Drove t h a t ACHl = 0 . L e t E > 0 and F be a compact subset

o f E s a t i s f y i n g A(E - F ) 6 E , For each x e H we have a sequence

o f c losed c i r c l e s (Dk(x)) cen te red a t x and c o n t r a c t i n g t o x such

t h a t

We app ly t h e cove r ing lemma 9.1.1. w i t h n z 0 and o b t a i n a d i s j o i n t

sequence {Ski o f c i r c l e s such t h a t

There fo re

9.2. REGULAR AND IRREGULAR SETS

1 A ( H ) 6 cl 1 A(E (1 s k ) + q = 1. A ( E (1 ( s k ) ) t r7 = c1 k

Since E and q a r e a r b i t r a r i l y small , we have A ( H ) = 0.

9.2.2. THEOREM . L & t E be meaukable. Then CLt &oh2 each

x E E we have

1 6 D(E,x) 6 1

P t l a a d . Let us f i x c1 > 0 and def ine - F = { x e E : 6 ( E , x ) > l t a )

For each x 6 F there centered a t x cont rac t ing t o

According t o the previous r e s u l t

247

i s a sequence C C k ( x ) ) of closed c i r c l e s x such t h a t

we know t h a t we have a t almost each x E F

Therefore, f o r almost each x E F , we have

We apply the covering theorem 9.1.1. w i t h E 0 obtaining a seauence f S k } o f d i s j o i n t c i r c l e s such t h a t

248 9 . GEOMETRY AND LINEAR MEASURE

t E = - A ( F ) t l+a

Hence A(F) and s i n c e E i s a r b i t r a r i l y sma l l , A(F) = 0.

Hence D(E,x) 1 a t a lmost each x 6 E.

I n o r d e r t o prove t h a t D(E,x) 2 1 / 2 a t a lmost each x E E, l e t 11s f i x P < 1/2 and l e t

For r > 0 l e t Gr be t h e s e t o f t hose p o i n t s x 6 G such t h a t i f

C(x) i s a c l o s e d c i r c l e cen te red a t x w i t h r a d i u s l e s s t h a n r we

have

I t i s easy t o see t h a t Gr i s c losed and

Gr G =

O < r e Q

We t r y t o prove t h a t A(G,) = 0 . Assume A(Gr ) > 0. Acco rd ing t o

t h e d e f i n i t i o n o f

s e t s { A ? of diameter l e s s than r / 4 so t h a t

A(Gr) , f o r any rl > 0 t h e r e e x i s t s a sequence o f

Fo r a t l e a s t one o f these s e t s Ak we have t h e r e f o r e

L t x e A k ( 7 G r

9 . 2 . REGULAR AND IRREGULAR SETS

6(Ak) < (1 + Q) A ( G r Ak)

nd l e t C ( x ) be the minimal closed c i r c l e c a t x containing Ak. We have

1 6 ( c ( X ) ) 6 & ( A k ) < (1 -t Q) A ( G r (1 A k ) 6 (1 + rl) h ( G r

SO

249

ntered

we have D(A,x) = n (B ,x ) , and - D(A,x) = - D(B,x ) a t almost each x E B We can s t a t e t h i s more generally.

9 .2 .3 . THEOREM. L c L I E k ) bc a dcljucnce od meawlable bC.65

nuch ,that E = 0 Ek A &o 06 6inLf-e i\-meaAme. Then at dmob, t

each. p in t x 06 each Ek we have

Contrarily t o what happens with the two-dimensional Lebesgue measure, there e x i s t A-measurable s e t s E such t h a t the dens i ty D ( E , * ) e x i s t s and i s s t r i c t l y between 0 and 1 a t each point of a subset o f E of pos i t ive measure . This makes the whole theory more in t e re s t ing and a l so more complicated.

250 9. GEOMETRY AND L INEAR MEASURE

9 . 2 . 4 . DEFSNTTTON. Let E be measurable. A point x e E wil l be called a treguRatr p o i a t of E I f D(E,x) = 1 . Otherwise x i s s a id t o be an i m e g u h t p o i a t of E . A s e t E i s ca l led treguRan when almost a l l of i t s points a r e regular. A s e t E i s sa id t o be i m e g u h t when almost a l l of i t s points a r e i r r egu la r .

By Theorem 9.2.3. t h e b e t a 6 tregul?atr p o i n t h o d E a xeg.guRuJc

b e t und t h e b e t 06 L m e g d a n point2 b an LmeguRcv~ A & .

d i s j o i n t . Both s e t s a r e

So the study of the s t r u c t u r e of E can be reduced t o the study of regular s e t s a n d o f i r r egu la r s e t s . Regular and i r r egu la r s e t s have sharply contrasting geometric, p roper t ies , as we sha l l see.

Before we go on t o consider some of them we prove a useful ex-

tension of the covering theorem 9 .1 .1 .

9.2.5. THEOREM. L e A E be any meabutrabLe set. Fotr each x e E

Let {Hk(x)} be a bqUenCe a4 rneccnutluble ne,t% con,taining x , conttuc - Ling t o x ( i . e . 6 ( H k ( x ) ) -f 0 ) , aMd nuch Ahat

with c1 independent o h X . Le,t E > 0.

Ptrood. By Lemma 9.1.2, i f we consider only s e t s Hk(x) t ha t a r e su f f i c i en t ly small , i t wi l l be s u f f i c i e n t t o prove ( i ) . f o r any ri , 0 c n < - , we have a t almost each x e R 2 , for any sequence of su f f i c i en t ly small c i r c l e s c k ( x ) contracting to x and centered a t x,

We know t h a t c1

4

9.2. REGULAR AND IRREGULAR SETS 251

Therefore, f o r a lmost each x B E we can s e l e c t a sequence

CHk(x)1 so t h a t , if C,(x) i s t h e s m a l l e s t c i r c l e cen te red a t x

c o n t a i n i n g Hk (x ) (and t h e r e f o r e 7 1 B(Ck(x)) 6 6 ( H k ( x ) ) c b(Ck(x ) ) ,

We app ly t h e c o v e r i n g lemma 9.1.1 and o b t a i n a d i s j o i n t sequence

{ c k j so t h a t

{ H k } we have

A ( E - I J C ~ ) = 0. If we t a k e t h e co r respond ing s e t s

A ( E - IJ H k ) = A ( E (1 ( IJ (Ck - H k ) ) ) =

We can take a f i n i t e number o f s e t s Hk so t h a t

h l 1 t q - g * A ( E ) = A A ( E ) , A 1 1 M. A ( E - 0 Hk) 6

1 1 t + - 2)

h l We can r e p e a t t h e process w i t h El = E - 11 Hk and so on. I n t h i s

way we o b t a i n f i n a l l y t h e s tatement o f t h e theorem. 1

9.3. TANGENCY PROPERTIES

From t h e d e f i n i t i o n s i t i s easy t o show t h a t i f S i s a r e c t i -

f i a b l e curve (more p r e c i s e l y , t h e graph o f a r e c t i f i a b l e cu rve ) o f f i n i t e

l eng th , then S i s measurable and i t s A-measure c o i n c i d e s w i t h i t s

l e n g t h . On t h e o t h e r hand i f S i s such a curve, x i s one o f i t s p o i n t s

which i s n o t an endpo in t o f t h e cu rve and C(x) i s any c losed c i r c l e

cen te red a t x we have

when C(x) i s s u f f i c i e n t l y s m a l l . T h e r e f o r e D(S,x) >, 1 and so we

have O(S,x) = 1 a t a lmos t each x e S. Hence we have t h e f o l l o w i n g

r e s u l t .

9.3 .1 . THEOREM. E v a y tLecti&iabLe cuhve 06 dinite LengRh 0

U hegdUh b d .

We know t h a t a r e c t i f i a b l e cu rve has a tangen t a t a lmos t each

o f i t s p o i n t s .

l i k e a r e c t i f i a b l e curve and an i r r e g u l a r s e t i s something comp le te l y

o p p o s i t e t o a r e c t i f i a b l e cu rve .

widen a l i t t l e t h e n o t i o n o f tangent , an i r r e g u l a r s e t has a tangen t a t

a lmost none o f i t s p o i n t s .

We s h a l l see t h a t i n many respec ts a r e g u l a r s e t behaves

We s t a r t by showing t h a t , even i f we

9.3.2. PEFTNTTIUN. L e t E a measurable s e t and x e E be

such t h a t 6(E,x) > 0. The tangent dt x t o E i s d e f i n e d as t h a t

un ique s t r a i g h t l i n e t, i f i t e x i s t s , such t h a t f o r each, E , 0 < E < i, t h e d e n s i t y a t x o f t h e s e t E - R( t ,E) i s zero, where R(t,E) i s t h e

shaded c losed s e c t o r i n t h e f i g u r e below.

9.3. TANGENCY PROPERTIES 253

We s h a l l now prove a r e s u l t f rom which t h e non-ex i s tence o f t h e

tangen t a t a lmost each p o i n t of an i r r e g u l a r s e t i s i nmed ia te .

9.3.3. THEOUEM. Le.2 E be an &egulah A&, a (0) a dixed dobed angle a d arnpLLtude ct , 0 < ct < T , with v M e x . c d 0 and L e i

8*(0) denote t h e VppUAih? angle. FOR each x e R z leA: a ( x ) and

u * ( x ) denote t h e ;thaMnlcuXaMn oh a(0) and a*(O) t o x . Then cd

& o ~ t each x e E we have

1 a ( E 0 a(x) ,x) + D ( E 0 a*(x),x) x 5 s in ct

Phood. We can assume E t o be bounded. We t ry t o p rove t h a t _I_

t h e s e t o f p o i n t s x e E such t h a t

1 (*) D(E ('I a ( x ) , x ) + 6 ( E o a*(x) ,x) < 4 s i n a

To do t h a t i t s u f f i c e s t o show t h a t i f we f i x E] , E ~ , 0 < €1 , 1 4 0 < s p , E I + E Z < - sen ct t hen t h e s e t E ( E ~ , E ~ ) o f p o i n t s x e E

such t h a t

i s o f n u l l measure. Assume i t i s n o t so L e t E ( E I , E 2 , r O ) be t h e subse t

o f E(E],Ez) o f p o i n t s x 8 E ( E ~ , E z ) such t h a t f o r each r < r o we

254

have

9. GEOMETRY AND LINEAR MEASURE

A ( E f7 u(u) f? C ( x , r ) ) < E12r

A ( E 17 u* (x ) f Y C (x , r ) ) 6 ~ ~ 2 r

where C(x, r ) means t h e c losed c i r c l e o f c e n t e r x and r a d i u s r. I t i s

easy t o show t h a t E ( E ~ , E ~ , ~ ~ ) i s measurable. And we have

E(EI ,E~) = \ I E ( E ~ , E ~ , ~ ~ ) I J Z , A ( Z ) = 0 O < r o e Q

and so some E ( E I , E ~ , ~ o ) has p o s i t i v e measure. From t h e d e f i n i t i o n o f

E ( E ~ , E ~ , ~ ~ ) by c losed convex s e t s { A k } such t h a t

A ( E ( ~ 1 ~ ~ 2 , r - o ) ) we see t h a t i f q > 0 t h e r e must be a cove r ing o f

There fo re t h e r e i s some A k y l e t us c a l l i t A, such t h a t

We can take a compact subset F o f E ( E ~ ,E2 , r0 ) 0 A such t h a t

and f o r each p o i n t x o f F we have, if r < r o

We s h a l l now t r y t o prove by geometr ic c o n s i d e r a t i o n s t h a t t h i s contra-

d i c t s El+ E~ < T s i n CI . 1

We t a k e a1,bl E F , al E a*(bl), bl E o(al) such t h a t

d(al,bl) = sup d(a,b) : a E F 11 a * ( b ) , b E F 17 o ( a ) 1

Th is can be done i f t h e s e t i n c u r l y b racke ts i s n o t e m p t y - s i n c e F i s

compact. (If t h a t s e t i s empty,what we a r e go ing t o do by t h i s process

we s t a r t now i s t r i v i a l ) .

From t h e e l e c t i o n o f al,bl i t i s c l e a r t h a t

and s i n c e d(al

A ( F (1 a(al

A ( F 11 o*(bl

bl) < r o we have

If we c a l l P I t h e i n t e r i o r o f a ( a l ) II a * ( b l l f1 C (a l ,d (a l ,b l ) ) (1

1 ) C(bl,d(al,bl)), we have A(F (1 PI) s ( € 1 + ~ 2 ) 2 d ( a l , b 1 )

If

s i b l e , a2,b2 E Q I (1 F , a2 Q o*(bz) , b2 E o(a2) such t h a t

O1 = R 2 - P I , then O1 I1 F i s compact . L e t US now take , i f pos-

D e f i n e as b e f o r e

The s e l e c t i o n process can be f i n i t e o r i n f i n i t e . I n any case observe

t h a t if we p r o j e c t t h e segments a . b

P2,02 and c o n t i n u e i n t h i s way as l o n g as p o s s i b l e .

on one o f t h e s i d e s o f t h e ang le

p a r a l l e l y t o t h e o t h e r t h e p r o j e c t i o n s a r e d i s j o i n t and so, s i n c e J j

a(0) &(A) < ro we have d (a . ,b . ) -+ 0 i f t h e r e a r e i n f i n i t e segments 1

J J a . b There fo re i n any case we have t h a t J j ‘

F - 0 P j = F 11 0 Qj j>l j>l

cannot c o n t a i n two d i f f e r e n t p o i n t s a,b such t h a t a E &(b ) , b Q .(a).

256 9 . GEOMETRY AND L INEAR MEASURE

Hence f o r each two points a ,b i n F - II P i the segment ab forms an j > l

angle w i t h the b issec tor l i n e of o(0) of amplitude biqger than a n d

so F - P j i s contained in a Lipschitz ( the re fo re r e c t i f i a b l e ) curve.

Since F i s assumed t o be i r r egu la r , we necessar i ly have j>l

A ( F - IJ P . ) = 0. j > l J

We now estimate

projections of a . b over other s ide 1" and a" b"

J j

j j

We can then wr i te

A ( F 0 ( IJ P j ) ) , Let us c a l l a! J J b! the i > l

the s ide u l ' of a(0) para l l e ly t o the the ones over 1 " pa ra l l e ly t o 1 ' .

O n the other hand

4 , 6 ( A ) s i n c1

Since n > O i s a r b i t r a r i l y small we get

9.3.4. 'TtEfJREM, L e t E be an ih 'w jda t r beX. Then ,the b& - 06 poi& x a E at urkich t h e tangent t o E e x A a 2 0 6 nu& m m u m .

The proof is obvious from Theorem 9 . 3 . 3

Besicovitch defined a Z - A ~ as a measurable set whose in te r - On t he o ther hand

Y-6e t i s a measurable s e t contained i n a conmutable union of r ec t i - sec t ion w i t h any r e c t i f i a b l e curve has measure zero. a f i abl e curves.

I t i s obvious t h a t any i r r egu la r set i s a Z-set and one can e a s i l y observe t h a t the statement and proof of Theorems 9.3.3. and 9 .3 .4 . a r e va l id i f we merely assume E t o be a Z-set .

Besicovitch [19381 proved a l so the following important f a c t .

9.3.5. THEOREM. I6 E b a 2 - h d , ,then D(E,x ) s 314 at

a h a o t each x 6 E .

From t h i s theorem we e a s i l y obtain the following charac te r iza- t i o n of i r r egu la r s e t s .

T h i s gives us e a s i l y the charac te r iza t ion of regular s e t s

9.3.7. THEOREM. A rneuuhable A & E b hegdan i d and o n l y

id AX A a h o b t a Y - A ~ ( i . e . id thehe. e.xhd.4 N c E , A ( N ) = 0

ouch ,that E - N h a Y-set).

P h O O d Of course any s e t E t h a t i s almost a Y-set i s regular . - We have A ( E ) < .

Let us now assume A ( € ) > 0 and E regular . Let cil = S U P I A( y ( X E ) : y r e c t i f i a b l e curve 1 . Since E i s not a

Z-set we have cil > 0 and the re e x i s t s a r e c t i f i a b l e curve y1 such t h a t ACE I1 r l ) > . Consider E l = E - y1. If A ( E l ) = 0 then we ge t t he statement o f the theorem. I f A ( E 1 ) > 0 , since E l i s again n o t a Z-set we can f ind a r e c t i f i a b l e curve y2 such t h a t

a1

1 1 ACEI (7 y2) > scx2 = 7 sup A ( E l (1 y ) : y r e c t i f i a b l e curve)

H

1 J And so on. I f the process is f i n i t e , c l e a r l y A ( E - LJ y . ) = 0. I f

i n f i n i t e , then c1 -f 0 . Call Em = E - II yj and assume A(Eoo) > 0 .

Then we can find a r e c t i f i a b l e y such t h a t A(Em I1 y ) = ci > 0 and this cont rad ic t s t he e lec t ion of the ci f o r CI s u f f i c i e n t l y small .

j

j j

m

j . There fo re h(Em) = 0 and E - E,c " y

We a r e n o t go ing t o p resen t he re t h e p r o o f o f Theorem 9.3.5.,

s i n c e i t i s p r e t t y l o n g and compl icated. We o n l y remark t h a t as a con-

sequence o f t h e p rev ious theorems we o b t a i n t h e f o l l o w i n q c h a r a c t e r i z a -

t i o n o f r e g u l a r and i r r e g u l a r s e t s i n terms o f tangency p r o n e r t i e s . The

p r o o f i s s t a i g h t f o r w a r d s t a r t i n g f rom 9.3.6. and 9,3.7.

9.3.8. TffEOREM. A m~anutra6Le he,t E 4eguLa.k 4 and vnP.rr

id Lt han a 1ange.d aR: demvh-t each 0 6 L t b poha%.

9.4. PROJECTION PROPERTIES.

A r e c t i f i a b l e cu rve ( O f p o s i t i v e l e n g t h ) has o r thogona l

p r o j e c t i o n o f p o s i t i v e l i n e a r measure on eve ry s t r a i g h t l i n e wi th t h e

p o s s i b l e excep t ion o f t hose l i n e s i n a s i n g l e d i r e c t i o n . T h i s prop-

e r t y , t o g e t h e r w i t h t h e c h a r a c t e r i z a t i o n o f a r e g u l a r s e t as a lmos t

an Y-set, p e r m i t s us t o e s t a b l i s h e a s i l y t h a t t h e same p r o o e r t y i s

shared by t h e r e g u l a r se ts o f p o s i t i v e measure ( B e s i c o v i t c h 1119281

p. 426).

Our main o b j e c t i v e here w i l l be t o show t h a t f o r an i r r e g u l a r

i n t h e sense o f t h e Lebesgue measure on t h e u n i t c i r c l e ) t h e pro-

s e t t h e oppos i te i s t r u e , namely f o r a lmos t each d i r e c t i o n

here

j e c t i o n i n t h a t d i r e c t i o n eve r any s t r a i g h t l i n e i s o f Lebesgue u n i d i -

mensional measure zero. T o prove t h i s we g i v e f i r s t some n o t a t i o n s and

d e f i n i t i o n s .

( "a lmos t "

9.4.1. NOTATION. I f x 6 R 2 0 c e l < O 2 < T , r > rl then

(a) d(x,81) w i l l mean t h e l i n e through x i n d i r e c t i o n 81.

9.4. PROJECTION PROPERTIES 259

( b ) a(x,a1,e2) w i l l mean t h e b i s i d e d open s e c t o r o f v e r t e x

x and extreme s i d e s i n d i r e c t i o n s e1,02, i . e .

a(x,e1,e2) = ~ ~ C d ( x , e ) : e l < e < e 2 3 - E X )

0

( c ) o(x,el,B2,r 1 = o(x,el,e2) (I B ( x , r )

(d ) 1-1 w i l l mean t h e o r d i n a r y Lebesgue measure on [ O , T I ) .

(e ) If A C [O,T) t hen d(x,A) w i l l mean t h e un ion o f

t hose l i n e s through x whose d i r e c t i o n i s i n A, i . e .

d(x,A) = ( J I d ( x , e ) : e E A 1

9.4.2. DEFINITION. L e t E be a measurable s e t , x 6 E and

e E [O,T). We say t h a t e i s a d k e c t i u n ad CandenhatLon oh &h&t

ondeh 0 6 E at x when x i s a l i m i t p o i n t o f E II d(x,0) We sav

t h a t 8 e ( 0 , ~ ) i s a di)Lec.tion ad candenhation 0 6 hecand mdeh oh E

at x when g i v e n any t h r e e D o s i t i v e numbers q,p, E t h e r e e x i s t r,

0 < r i p, and elYe2, w i t h o s a1 < 0 < e 2 < ~ r , 0 2 - < E such t h a t

We*say t h a t x b a hadiCLtion p o i n t ad E when 1-I-almost each d i r e c -

t i o n i s a d i r e c t i o n o f c o n s i d e r a t i o n

E a t x. ( o f f i r s t o r second o r d e r ) o f

Observe t h a t i f E i s an i r r e g u l a r s e t we have

A(E 0 o(xyely 0 , ) ) = A(E 17 a(x,B1,82)) and ana logous ly

A(E I7 o(x,01,ez, r ) ) = A(E 1) o(x,B1,02,r)) s i n c e t h e i n t e r s e c t i o n

of E w i t h any s t r a i g h t l i n e o r c i r cumfe rence i s o f ze ro measure.

- Observe a l s o t h e f o l l o w i n g way o f o b t a i n i n g t h e s e t o f d i r e c

t i o n s o f condensat ion of t h e second o r d e r t h a t w i l l be use fu l f o r t h e

p r o o f o f t h e n e x t theorem. If q , p , ~ a r e t h r e e p o s i t i v e numbers t h e

s e t H(x,q,p,~) of d i r e c t i o n s c1 f ( 0 , ~ ) such t h a t t h e r e e x i s t s r ,

e l < c1 < e2 < TI , e2 - e l < E so t h a t 0 < r < p, and a1,e2, 0 c

A(E (1 a

i s c l e a r l y an open subset o f


i s

o f second o r d e r o f E a t x .

p-measurable and i s p r e c i s e l y t h e s e t o f d i r e c t i o n s o f condensat ion

Our n e x t goal w i l l be t o p rove t h a t a lmos t each p o i n t o f an

i r r e g u l a r s e t E i s a r a d i a t i o n p o i n t of E. F i r s t we s t a t e two easy

1 emmas.

9.4.3. LEMMA. L c t E b e a r n c a n w m b l e b a and x any p o i n t - 0 6 R 2 . Then t h e h u b h d 0 6 joining x t o t h e pointn a 06 E , a # x , a v - r n U u h a b l e O&.

[Q,T) 06 ~ c ~ o ~ de;tenminc?d by a%& f i n e n

- Phood. Assume f i r s t t h a t t h e d i s t a n c e from x t o E i s

p o s i t i v e , and t h a t A(E) = 0. Then,by p r o j e c t i n g f r o m x t h e c o v e r i n g

d e f i n i n g

by x and t h e p o i n t s o f E i s o f v-measure zero. I f we have h(E)=O we o b t a i n t h e same r e s u l t by c o n s i d e r i n g t h e subsets

1 Ei = { z E E : d(x,z) a 1 , i = 1,2,3 ,...

A*(E) = 0 , we e a s i l y see t h a t t h e s e t o f d i r e c t i o n s determined

I f E a compact measurable s e t , t hen t h e s e t o f d i r e c t i o n s

o f t h e s tatement , i f x B E , i s a l s o compact and so w n e a s u r a b l e .

Analogously we can remove t h e c o n d i t i o n x B E as b e f o r e .

Any measurable s e t E i s a coun tab le u n i o n o f comoact subsets

p l u s a s e t o f ze ro measure. This proves t h e lemma.

9.4.4. LEMMA. L e A E be a meanutmbLe and x 8 E . TCrcn t h e

n e t 06 d-ihe&an?) a6 CandeVLbation ad &ht ohdeh ad E ax x L$

p-meuuhabLe.

Phoad, L e t G(x,r) f o r r > 0 be t h e s e t o f d i r e c t i o n s

determined by j o i n i n g x t o t h e p o i n t o f E 0- B(x,r) - {XI . The

s e t G(x,r) i s p-measurable acco rd ing t o t h e p rev ious lemma , We have

t h a t

1 m

G(x ) = I\ G(x,F) n = l

i s t h e s e t o f d i r e c t i o n s o f condensat ion o f f i r s t o r d e r o f E a t x . So

9.4. PROJECT I ON PROP ERT I ES

G(x) i s measurable.

26 1

9.4.5. IHEOREM. AenasR each po in t od a n LW~eguRan b e t E LA

a t i ad id t ion p a i n t .

We sha l l f i r s t prove the following lemma from which the theorem i s an easy consequence.

9.4.6. LEMMA . L e t E be an h h e g d a n ne.2, Abbume t h a t - x 8 E batib6ie.h t h e 6oUowing condition:

L e t c1 e ( 0 , ~ ) be a dihecLion buch tha2

Pnood oh t h e Lma. 9.4.6. Since 0 (G(x) ,a ) = 0 we can T u

take e 1 , e 2 , 0 < e l < ~1 < e 2 < T , 0 2 - e l < min(E, ) such t h a t

1 and s ince G(x) = 0 G(x, E) , we have a l so a p ' < p such t h a t n = l

26 2 9 . GEOMETRY A N D LINEAR MEASURE

We k n o w from condition ( * ) t h a t t he re e x i s t s r q 2r(e2 - 8 , ) then c1 8 H(x,q,p,~) and we have the statement. Assume

The s e t G ( x , p ’ ) t’r ( e l ,e ,) i s u-measurable and so we can find a n open s e t I ( therefore I i s a union o f d i s j o i n t oDen i n t e r v a l s ,

I = \ J (@;,$; ) ) such t h a t i

Let us d i s t r i b u t e {(@;,I$!) i n to two co l l ec t ions in the following way:

We then have

We a l so have ( r eca l l 9 . 4 . 1 . ( e )

Therefore, by condition ( * ) and

9.4. PROJECTION PROPERTIES

Hence

263

Now we know t h a t , f o r each i 8 Al ,

and a l so , f rom (**I,

For each i E Al l e t ( Y ' Y ; ) c ( e l , O , ) be t h e u n i o n of a l l t hose

open i n t e r v a l s (m, , m 2 ) c o n t a i n i n g ($;,$;) and con ta ined i n ( e , , 0 2 )

such t h a t

j y

(***) A ( E o(x,ml,m2,r)) q 2r(m2 - ml)

I t i s easy t o see t h a t t h e (Y!,Y'!) J J S O ob ta ined a r e d i s j o i n t and

A lso i t i s c l e a r t h a t i f J = IJ ( Y ! , Y ' ! ) each p o i n t o f J, by (***),

i s i n H(x,q,p,&) and so

j J J

There fo re we have

T h i s means, s i n c e €I1 < c1 < 0 2 , and e 2 - e l i s a r b i t r a r i l y sma l l , t h a t

264 9. GEOMETRY AND L I N E A R MEASURE

P m a 6 o6 Theonem. 9.4.5. Let E be an i r r egu la r s e t . Ac- cording t o Theorem 9.3.3. we know t h a t i f we f i x t j l , 0 2 , 0 < e l < e , <IT, e 2 - e l < - we have a t almost each x e E 71

Therefore we can a l so say t h a t a t almost each x e E we have f o r each

& s couple (el ,e , ) , e l € Q, e 2 E Q , 0 < e l < e 2 < T , e2- e l <

Since E is i r r egu la r , the function

A ( E (1 u(x ,e l ,e2 , r ) ) of e 1 , e 2 , r

i s continuous and so we can omit i n the preceding paragraph the condi- t i on B Q , 0, a Q . By excluding a subset o f E of n u l l measure we can assume t h a t we have the condition 9.4.6. f o r each x 6 E.

(*) of the statement of Lemma

Now i f p(G(x)) = ~ i , x i s c l e a r l y a point of r ad ia t ion , As-

sume p(G(x)) < IT . We have t h a t p-almost each point c1 6 [O,T) - G ( x ) s a t i s f i e s D (Gfx), a ) = 0 . If we apply the lemma 9.4.6. we have

point a e [ O , I T ) - G(x) i s i n H ( x , q , p , ~ ) f o r any fixed q,p,E ,

Since we have t h a t p-almost each point i n t0.r) - G ( x ) i s in H(x) and so

p(G(x) 1J H ( x ) ) = IT . Hence p a l m o s t each c1 i s a condensation d i rec t ion o f E a t x , and so x i s o f rad ia t ion of E .

1.1 c1 e H ( x , q , p , E ) o r DpCH(x,q,p,E), a ) > 0. Therefore p-almost each

H(x) = 0 (H(x,q,p,c) : 0 < q E M , 0 < p 6 Q , 0 < E B Cp 3

From theorem 9.4.5. we e a s i l y obtain the following important proyection property.

9.4. PROJECTION PROPERTIES 265

The proof will be straightforward from the following four lemmas.

- 9.4.8. LEMMA. LeA E be an i m ~ e g d a h .bat. Then, doh u-al

moht each dineotion e 6 [O,IT) , d m a h t each x E E A nuch t h a t 0 A ad condmhation 04 E at x.

Pnood. Consider R2 x 10,~) with the measure A x u , By

theorem 9.4.5. we know that, for A-almost each x B E c R 2 , u-al-

most each direction e a [ O , T ) is a condensation direction of E at

x . Therefore, by Fubini’s theorem, for p-almost each A-almost each x 6 E is such that 8 is a condensation direction of

E at x. This is the lemma.

8 6 [O,r),

9.4.9. LEMMA. LeA E be any memumble beA oa ze,ko m m u h e .

Then i,t~ phoyeotion o v a any nLt;ltaigkt f i n e Lib 0 6 null meanme.

Phood . If r ( K ) means the projection of K we have

that d(.rr(K)) 6 S ( K ) and SO A*( T C E ) ) < A * ( E ) . Hence, if A(E) = 0 we have A(IT(E)) = 0 .

9.4.10 LEMMA . LeA E be a nubbeA oh a m m w l a b l e he2 A,

and Re2 I T ( E ] be Rhe a h t h o g o d ! phoyect.ion 0 4 E ove,k Ox . Ahhwne

t h a t each p a i n t a E n(E) A phojection 06 i n ~ i n i t d y many poivztn 0 6

-

A . Then A(T(E)) = 0

P m o 6 . We can assume that A is in the unit cube

Q = i(x,y) : 0 6 ,x i 1 , 0 6 y < 1 1 , Let N be a natural number,

We partition Q into 2N dyadic strips parallel to Ox, Set

-

N for r = 0,1,2, . . ., 2 -1.

Let E(r,N)I = E f’r S(r,N) , A(r,N) 0 i m a 2 -1 let Tm be the collection of a

numbers

N = A f’l S(r,N) . 1 m-tuples of

For

nteger

N k = ( k l , k 2 , ..., k m ) , 0 c k l < k z < ... < k, c 2 -1

We define

m

t h a t i s , a p a i d P 0 6 [0,1) h i n F(m,N) when P h phvjeOtia~ ad

CLt L m t m pa in tb 06 A LoccLted in m didde,tent bxXLpb ad t h e 2N we have.

Sa we cRecvtey have

m m

n ( E ) c f ) F ( m , N ) m=l N=[liog,m]tl

Let us estimate the measure o f F ( m , N ) . description of F ( r n , N ) , t ha t

We have c l e a r l y , from the above

2N- 1

and in tegra t ing

1 Therefore A ( F ( m , N ) ) c A ( A )

Observe t h a t F ( m , N ) c F ( m , N t l ) and so

Therefore A(n(E)) = 0

9.4.11. LEMMA, L e t E be a nubbet 06 a rneabu.tabRe b& A,

and aSbUe t h d t doh each

06 conden5at;ian 0 6 becond a h d a 06 E at a . L e t n(E) be t h e

a 6 E t h e v e h t i c d dine&an 0 a dLteOtion

9.4. PROJECTION PROPERTIES 26 7

The easy n r o o f o f t h e Lemma Dresented here i s due t o R.Moreno.

P/rvad. We can assume t h a t A i s con ta ined i n t h e s t r i p

{ (x,y) : 1 6 y < 2 I . L e t a 6 E and .rr(a) i t s p r o j e c t i o n ove r Ox.

u(a,e ' ,e" , r ) w i t h 4 i < ~ r / 2 < ei; , 'di - O i J. 0 , rkJ. 0, such t h a t

We know t h a t f o r each f i x e d q > 0 t h e r e i s a sequence o f s e c t o r s

k k k

Therefore, i f I k = n( 0 ( a k , e i Y e i , r k ) ) , we have w i t h a cons tan t c in-

denendent o f A,a,k, A ( I k ) 6 c 2 r k { 4; - 0;) c

We e a s i l y see t h a t I k c o n t r a c t s t o a n ( a ) . So we can a n o l y

V i t a l i ' s lemma t o n ( E ) and o b t a i n a sequence EJiI o f d i s j o i n t i n -

t e r v a l s such t h a t

There fo re

A*( TI ( E ) ) c A * ( T ( E ) - (I Ji) t A ( IJ J i ) =

and so A ( IT ( E ) ) = 0 .

Phoolj ad ,the Theohem. 9.4.7. Acco rd ing t o Lemma 9.4.8.

t h e r e i s a p - n u l l s e t N c [ O , T I ) such t h a t , i f 0 E [ O , T I ) - N , f o r

L e t us f i x 0 E [O ,? r ) - N . Then

A-almost x 6 E t h e d i r e c t i o n 4 i s a condensat ion d i r e c t i o n o f E a t x.

9. GEOMETRY AND L I N E A R MEASURE

where

E o = Cx E E : 8 i s not of condensation o f E a t x 1

El = {x E E : 0 i s o f condensation of f i r s t order of E a t XI

EZ = { x e E : 0 i s o f condensation o f second order of E a t x )

By Lemma 9 . 4 . 2 . , since ~ ( E o ) = 0 , the projection in d i r ec tion e i s o f measure zero.

By Lemma 9.4.10, A ( n ( E , ) ) = 0 and by Lemma 9.4 .11 . A ( n ( E 2 ) ) = 0 . Therefore the projection n ( E ) o f E i s a lso of zero-

measure.

9.5. SETS OF POLAR L I N E S .

Let A be a subset of R 2 and l e t C = C ( 0 , l ) be the circumference of radius 1 centered a t 0. For each x E R2 l e t p ( x ) be the polar l ine of x with respect t o C a n d l e t us denote by ( p ( A ) the union of the collection of polar l ines of points of A, i . e .

In th i s Section we shall be concerned w i t h some of the two- dimensional theoretic properties of the s e t p(A) related with the geometric and A-measure-theoretic properties o f the s e t A .

denated by A.

F i r s t of a l l l e t us announce t h a t , as we shall inmediately prove, the choice of the c i rc le C perties we are going t o s t u d y .

i s rather irrelevant for the pro-

9.5. SETS OF POLAR L I N E S 269

I n t h e f i r s t p lace, observe t h a t i f A i s any con t inuous c u r v e

w i t h endpo in ts a,b, a # b, t h e n p(A) i s a X-measurable s e t o f i n f i - n i t e A-measure.

We s h a l l f i r s t p rove t h a t , as expected, when A i s an i r r e g u l a r

se t , t h e oppos i te w i l l ho ld . The p r o o f o f t h i s f a c t i s a u i t e s t r a i a h t -

f o rward f rom Theorem 9.4.7.

9.5.1. THEOREAd. Lei E be a n L w u g u R a h h e i . Then n(E)

ih a A - n & h e i .

Pmad . From theorem 9.4.7. we know t h a t f o r p-a lmost each - 8 B [CI,.) t h e i n t e r s e c t i o n o f t h e s e t (I d(x,O) (he re d(x,e)

means t h e l i n e th rouqh x i n d i r e c t i o n e ) w i t h any l i n e i s o f XEE

A-measure zero. There fo re we see,by p o l a r i t y , t h a t f o r u-a lmost each

8 E [O,IT) t h e l i n e g ( e ) , o r thogona l t o d(0,B) th rough t h e o r i q i n ,

i n t e r s e c t s t h e s e t n(E) i n a s e t o f A-measure zero. T h e r e f o r e meas-

u r i n g A (p (E) ) i n p o l a r coo rd ina tes , we have A(D(E)) = 0

For r e g u l a r s e t s we s h a l l f i r s t p rove t h e f o l l o w i n g f a c t ,

9.5.2. THEOREM. L e i E be a hegduJL hQ,t. Then , doh

A - a h o h t each x 6 E we have, doh each z E p ( x ) and doh each r > 'Y

We s h a l l deduce t h i s theorem f rom t h e f o l l o w i n g a u x i l i a r y

The f i r s t one o f them i s q u i t e i n t e r e s t i n g i n o r d e r t o have r e s u l t s .

more f l e x

open be.Z

iiztavLt M

b i l i t y i n h a n d l i n g r e g u l a r and i r r e g u l a r s e t s .

9.5.3. THEOREM. L e i E be a meanwlable b e i contained i n a n

G ad R 2 . L e i f : G + H be a L i p b c k i t z d u n c t i o n v d CVUA - Whohe iinvehbe f - l : H + G e u k 2 a n d h &o a Lipbck i t z

Then f ( E ) Lh & a meanwLable. Id A(E) = 0 .then

A ( f ( E ) ) = 0 . 16 E Lb hegLLeah, f ( E ) Lh hegdah. 16 E h - t e g d a h , f ( E ) &eguRm.

270 9 . GEOMETRY AND LINEAR MEASURE

P m v $ . Assume A ( E ) = 0 . Then i f {An? i s a cove r o f E,

C f ( A n ) l i s a cover o f f ( E ) and 6 ( f ( A n ) ) c M 6(An). T h e r e f o r e

A * ( f ( E ) ) c M A * ( E ) f o r any E , and i f A(E) = 0, t hen h ( f ( E ) ) = 0 . If E i s measurable, t h e n E = ( IJ K j ) ( J Z, CK.1 b e i n g a sequence

j = 1

of i n c r e a s i n g compact s e t s w i t h A ( K j ) .f A (E) and A(Z) = 0, There fo re

m

J

A ( f ( E ) ) = A ( IJ f ( K j ) ) + A ( f ( Z ) ) , A ( f ( Z ) ) = 0 and

A ( f ( E ) ) 6 M A(E) < m . Hence f ( E ) i s measurable.

If E i s r e g u l a r , E = A IJ B w i t h A ( B ) = r)

A c b yk , yk cont inuous r e c t i f i a b l e curve. There fo re

f ( E ) = f ( A ) (J f ( B ) , A ( f ( B ) ) = 0 f (A ) C ( 1 f ( y k l y f ( Yk)

r e c t i f i a b l e . So f ( E ) i s r e g u l a r .

rn k = l

k= 1

F i n a l l y , i f E i s i r r e g u l a r , we c o n s i d e r f ( E ) , We know t h a t

f ( E ) i s measurable. L e t i t s r e g u l a r p a r t be f ( B ) , Then B i s r e a u l a r

and B c E , t h e r e f o r e A ( B ) = 0 and A ( f ( B ) ) = 0. T h e r e f o r e f ( E ) i s

i r r e g u l a r .

The Theorem 9.5.3. a l l o w s us e a s i l y t o see t h e e f f e c t t o

changing t h e p o l a r i t y c i r c l e i n t h e d e f i n i t i o n of a(A) . I f i n s t e a d o f

t a k i n g p o l a r l i n e s p ( x ) of t h e p o i n t s x o f A w i t h r e s p e c t t o C ( 0 , l ) we take them, p * (x ) ,

conic C* , t h e new s e t D*(A) = \ J p*(a) i s o b t a i n e d f r o m o(Rj by xaA

a nondegenerate n r o j e c t i v e t r a n s f o r m a t i o n and so, by an a p o l i c a t i o n o f

Theorem 9.5.3. we e a s i l y see t h a t t h e Theorems 9 , 5 . 1 . , 9.5.2. nre-

serve t h e i r v a l i d i t y i f we change p b y p*.

w i t h respec t t o any o t h e r f i x e d non-degenerate

A l s o we see t h a t t h e p r o j e c t i o n r e s u l t s o f t h e p rev ious s e c t i o n

can be fo rmu la ted more g e n e r a l l y . Fo r example, by means o f a p r o j e c t i v e

t r a n s f o r m a t i o n we g e t f rom Theorem 9.4.7. t h e f o l l o w i n g f a c t .

9.5.4. THEOREM. L&t E be a n y LwegguRah n&t. L e X r be

any n,ikaigkt .&fie. Then dvh A - d m v A t e a c h p a i n t x 6 r t h e p o j e c f i v n v d E @om x aveh any aXheh f i ne nvA: panbing t h o u g h x 0 0 4 zehv

A-rneuuke.

9.5. SETS OF POLAR LINES 2 7 1

The n e x t two r e s u l t s a r e easy a u x i l i a r y lemmas w i t h which we

s h a l l p rove Theorem 9.5.2. Some o f t h e computat ions a r e e a s i e r w i t h

t h e f o l l o w i n g remark.

I f x 6 R 2 - {!)I we s h a l l c a l l q ( x ) t h e l i n e th rough x

or thogonal t o Ox. I f A c R 2 - C O 3 t hen q(A) = I I q ( x ) . Observe

t h a t , f o r a s e t A such t h a t i s bounded and con ta ined i n R 2 - 1Q1 i f A* i s t h e s e t o b t a i n e d f rom A by an i n v e r s i o n w i t h r e s p e c t t o

C(0 , l ) , then by a p p l y i n g Theorem 9.5.3. we see t h a t A i s measurable

i f and o n l y i f A* i s so, A ( A ) = 0 i f and o n l y i f A ( A * ) = 0 and

A i s r e g u l a r o r i r r e g u l a r acco rd ing t o t h e r e g u l a r i t y o r i r r e g u l a r i t y

o f A* . Fur thermore , observe t h a t p ( A ) = q(A*) .

x BA

There fo re Theorem 9.5.2. w i l l be proved i f we prove t h e same

statement s u b s t i t u t i n g p by q .

9.5.5. LEMMA. LeR: U be t h e c lon ed heotangle

U = { ( x , y ) e R 2 : a - a h x c a t a , D - d c y c 0 t d I

D i u a h ~1 > 0, !I < D , 0 < d <

V a l

, a > 0 . LeL A be Lhe cloned i n t e 4 -

Then thehe c d a 2 a cav~6Lant M M(a,D) > 0 auch . tha t d o t each p L 1 and dotr each c i h d e B(z,p) contained i n A one hcu

(See F i g . 9.5.1.).

P/roud. The p r o o f i s s t r a i g h t f o rward by obse rv ing t h a t t h e -- maximum o f I ( U t’r q(C(z,p) ) ) under t h e r e s t r i c t i o n C(z,p) t A

f o r a f i x e d p i s g i v e n by t h e c i r c l e i n d i c a t e d i n t h e f i g u r e 9.5.1.,

t h a t f o r t h i s c i r c l e

o f t h e h o r i z o n t a l s t r i p determined by U and t h a t even t h i s shaded

U 0 q(C(z,p) ) i s con ta ined i n t h e shaded p o r t i o n

p o r t i o n has an a r e a less than some c o n s t a n t M(a,D) times p .

s = (a,O)

F i g u r e 9.5.1.

9.5.6. LEMMA. L & U and A be a6 i n t h e prreceding lemma. - Let s = (a,O) . Then t h m e cdh a con6,tunt N = N(a ,d,D) > 0 and a b a l l B ( s , r ) * 0 < r < 1 con ta ined i n A buch t h a t id y 6 B ( s , r )

and sy m e a n 6 t h e A & 06 point2 06 t h e begment j o i n i n g s t o y

we have

Pko06. First we can f i x ro so small t h a t a l l l ines q ( z ) f o r

z 8 B(s , ro) intersect both h o r i z o n t a l s i d e s of U . For a f i x e d p ,

0 c p < ro, if A(SY) = p , the m i n i m u m of X(U (1 q(SY)) is g r e a t e r


p o r t i o n has an area l ess than some constant M(a,D) t imes p .

s = (a,O)

Figure 9.5.1.

9.5.6. LEWA. L e X U and A be a6 i n t h e peceding lemma. - L e t s = (a,O). Then thehe cdd a con~xixtant N = N(a,d,D) > 0 and

a b a l l B(.s,r) , 0 < r < 1 cona%Lne.sf i n A 6uch that i 6 y Q B(s,r)

and 5 m a n h t h e ~ e , t 06 pointn 06 t he btgmekLt joining s t o y

we have

w. F i r s t we can f i x ro so small t h a t a l l l i n e s q(z) f o r

z a B(.s,ro) i n te rsec t both ho r i zon ta l sides o f U . For a f i x e d p,

0 < p < r o , i f h(sy) = p , the minimum o f X(U (1 q(sy)) i s g rea ter

9.5. SETS OF POLAR LINES 273

than the area o f the shaded portion o f Figure 9.5.2. In i t the point e i s obtained as in t e r sec t ion of t he circumference C o f diameter Og

w i t h t h a t C(s,p) o f center s and radius p . This area is (2d) ' t ana and one has

& = D + d sin c1 cos 6

and cos B -f C O S T as p + 0 where T i s the angle o f the tangent a t x t o C with the ax is Ox, as indicated in Figure 9 .5 .3 .

So one has

( 2 d ) ' t an a = ( 2 d ) 2 s in c1 p = P cos c1

and i t i s c l ea r t h a t this N(a,d,D) > 0

quant i ty can be estimated from below by

Figure 9.5.2.

F i g u r e 9.5.3.

Phood 06 Theohm 9.5.2. L e t E be r e g u l a r . Acco rd ing t o

Theorem 9.5.3. t h e r e i s N c E , A(N) = 0 , such t h a t E - N c (I yj,

yj cont inuous r e c t i f i a b l e arc.For a lmost each s 6 E - N t h e r e i s one

y j , c a l l i t s imp ly y , such t h a t s 6 (E - N) f I y and s i s a

d e n s i t y p o i n t o f y and E I1 y . We s h a l l prove t h a t f o r such a p o i n t

s and f o r each b 6 q ( s ) , b # s , i f U(b) i s any neighborhood

o f b, then X(U (1 q ( E ) ) > 0

We have

A ( E (1 y fl B ( s , r ) ) = 2 r 1 i m

r+O

We can assume t h a t s i s t h e p o i n t (a,O) of t h e p rev ious

lemmas, t h a t b i s t h e c e n t e r o f t h e r e c t a n g l e U o f these lemmas,

and we t a k e as t h e neighborhood U p r e c i s e l y t h a t r e c t a n g l e . F o r a

number n, 0 < T- < 1, t h a t w i l l be c o n v e n i e n t l y f i x e d l a t e r we can

t a k e r > 0 so t h a t B (s , r ) c A, B(s,r) s a t i s f i e s t h e s tatement o f

Lemma 9.5.6. and

9.5. SETS OF POLAR LINES 275

Let y,z be the endooint of the continuous arc yrcy f1 B(s,r)

passing through s. If ysZ means the polygonal line ( J SZ we

have

The second inequality by Lemma 9.5.6.

From (**) we get

and the last one by (*) above,

We can cover y, - E with a countable union ( I K of small circles

contained in A so that j

We have by Lemma 9.5.5.

Hence

If n > % then X(U (1 q(E)) > 0. This concludes the oroof. M+ 7

L

The Lemma 9.5.5. allows us to obtain in an easy way the

following expected result.

9.5.7. THEOREM. L e t E be u A - n U n e t . Then X(p(E)) = 0

Phoo6. We can assume t h a t t h e s e t E i s i n t h e s e t A o f

Lemma 9.5.5. For E: > 0 t h e r e i s a cover of E by a coun tab le un ion

o f smal l c i r c l e s u K j c A so t h a t C G ( K j ) 6 E . Therefore, bv Lem-

ma 9.5.5.

And so A(q(E)) = 0.

I n t h e c o n t e x t o f t h e Theorems o f t h i s S e c t i o n i t i s i n t e r e s -

t i n g t o know t h a t B e s i c o v i t c h [1964] proved t h a t , i f one counts m u l t i -

p l i c i t y , i . e . if one weighs each D o i n t o f p(E) w i t h t h e number o f t imes

i t i s covered by l i n e s p ( x ) w i t h x E E , then f o r eve ry r e g u l a r s e t E

w i t h A(E) > 0, t h e l i n e s cover an i n f i n i t e a rea . More o r e c i s e l y

However Davies [1965] c o n s t r u c t e d a r e g u l a r s e t E such t h a t

A ( E ) > 0, X(p(E)) < m .

9.6. SOME APPLICATIONS.

We s h a l l now show how t o use some of t h e p reced ing theorems i n

o r d e r t o o b t a i n c e r t a i n r e s u l t s connected w i t h t h e t h e o r y o f t h e pre-

ceding Chapter 8.

F i r s t o f a l l we s h a l l g i v e a ve ry easy c o n s t r u c t i o n o f a

B e s i c o v i t c h s e t .

9.6 . SOME APPLICATIONS

Figure 9.6.1.

277

9.6.1. Cvb%i%ucaon oh a R e n i c o v ~ c h net . Fo& t h e ebbed

uvLit syuahe Q o , Let Q1 = ~ ( Q o ) be the. union 0 6 t h e dout bhuded

dyadic cLobed bquafieh 0 4 F i g . 9 .6 .1 .

we appLy ,the hame opehai ion

bquahen a h i n i n g i n tkis Way , Q2 C 0, C 0 0 . And ha on. 1 ~ 2

K = Q1 0 Q2 0 Q 3 ... Then K A a n LV~eguRah net o h p o h U v e m e u - me.

To each one a() thehe h o w l bquuten

0 . LeL Q 2 be t h e union oh t h e a2 cloned

L e t $ : R 2 + R 2 be t h e I;o.Uowiny ,OLan~dom&on :

$ ( x , y ) =((1 + x ) cos 2 n y , (1 + x ) sen 2 n y ) Then $ ( K ) i~ a&# in - &egsLLedh and

cvnttai~d CLt L e u t vne f i n e i n each di/recLion.

p ( $ ( K ) ) = B A a b e t 0 6 a3dv-dOnenniond meau&e zeho t h a t

The s e t K is comnact and, usinq the natural covers f o r K,

i , e , t he squarerof Q , we see t h a t A ( K ) s L?. Therefore K i s measurable . Since i t s projection over Ox i s of length 1, we have

A ( K ) > 0. Further, the nrojection of K over the two diaqonals o f

j

Q a i s of zero measure and t h a t over Ox and Oy has measure 1. There

fore K must be i r r e g u l a r ,

Lemma 9 .5 .3 . t e l l s us t h a t $ ( K ) i s a l so i r r egu la r of nosi - t i v e measure. Moreover, s ince K has a t l e a s t one ooin t on each l i n e

y = a , 0 5 a ,< 1 , $(K) d i f f e ren t from 0 . Therefore n($(K)) contains a t l e a s t one l i n e i n

each d i r ec t ion and

has a t l e a s t one noint over each ray from r)

A ( P ( $ ( K ) ) ) = 0

9. GEOMETRY AND LINEAR MEASURE

Take a s e t L which i s t h e un ion of some l i n e s . Can one g i v e

an easy c r i t e r i o n t o dec ide whether t h i s s e t L

l i n e s of t h e n o i n t s of an i r r e g u l a r s e t ? I n o t h e r words, l e t L = \J

and l e t

i s t h e u n i o n of t h e p o l a r

d, ,€A

E = { a : d, = ~ ( a ) l

i . e . E i s t h e s e t o f po les of t h e l i n e s d, w i t h r e s p e c t t o C ( 0 , l ) . Can one g i v e a c r i t e r i o n , so t h a t by d i r e c t i n s p e c t i o n o f one can de te r -

mine whether E i s i r r e g u l a r (and so L i s of A-measure z e r o ) ?

The f o l l o w i n g r e s u l t , due t o A.Casas [1979], answers t h i s

ques t i on i n an easy way.

9.6.2. THEOREM. L e t L = I! da w h a e each d, LA a ,€A

ha%a&ktfine. L e t

E = Ca e R 2 : p(a) = d, , a E A 1

Then E A & e g W -id and ovtey id t h e doU0wLng &zue:

W e g i x Awo f inen s , t , huch t h a t doh each a e A s f da , t # d, . On s ,take a poin t S not i n L and an ohientdtion. On t

t a k e a poivLt T not

S, = d, (I s and

&om S t o S , and

H =

i n L and ah ahienta.tLan. Foh each d, let

TT, t h a t aham T t o T, . Then t he . hQ,t

= d, ('1 t. LeX SS, be the. higned din,tance Ta

Q, = (SS,, TT, ) ER' : 01 e A 1

Pk0o.d. App ly ing Lemma 9.5.3. we can assume t h a t s and t

a r e pe rpend icu la r and t h a t S c o i n c i d e s w i t h T. L e t R, be t h e pro-

j e c t i o n o f S = T over T,S, ,

Consider t h e f i g u r e 9.6.2. and t h e mapoing

9.6. SOME APPLICATIONS 279

We can assume t h a t t he s e t o f points closure i s bounded and contained in R 2 - Exy = 01 , Then we can a m l v Lemma 9.5.3. and

{Ra : ct E A ) i s such t h a t i t s

Y y - t

X

O E S E T s_ X E S 0 I u ,

1

Figure 9.6.2.

so 10, : a B A I i s i r r egu la r i f and only i f {Rct : ct 6 A) i s i r regular B u t {Ra : a B A 1 i s i r r egu la r i f and only i f E i s i r r e g u l a r . So we have the theorem.

Now i t i s easy t o understand b e t t e r the nature of the s e t of Kahane presented i n 8 . 4 . 2 .

9.6.3. THEOREM. The neL ad LLnU L = IJ da phehnnted i n M A

8.4.2 0 nuch tthcLt E = {a 6 R 2 : p(a) = da , ( 'a 6 A ) } b anihheguLm

b&

P h O O d . By Theorem 9 .6 .2 . i t i s enough t o show t h a t the set - G o x CO , where G O is the Cantor type set on [O,l] we have cons t ruc ted in 8.4.2., i s i r r egu la r . B u t t h i s i s Droved in exac t ly the same way we have followed in 9.6.1.

The l a s t appl ica t ion we sha l l give concerns the Nikodym s e t and the problems ra i sed a t the end of 8.5.

2 80 9. GEOMETRY AND LINEAR MEASURE

We know t h a t t h e r e e x i s t s a con t inuous f i e l d o f d i r e c t i o n s

0 : R2+ [OJ) and a s e t N o f p o s i t i v e A-measure A ( N ) > 0, such

t h a t f o r each x 6 N , d(x , 6 ( x ) ) n N = { X I . By means o f t h e theorems

o f t h i s Chapter we can prove t h e f o l l o w i n g r e s u l t .

9 . 6 . 4 . THEOREM. l e i 8 : R 2 + [O ,TT) be a Lipncki tz d i d d

a6 d i n e o t i v a . T h e n , them c a n n v t be. any ne;t N 0 6 d u l l ,tLuv-dimeMnioncLt? mea4uhe. , i .e. X(R2 - N ) = 0 , nueh thcLt doh each x 6 N , d ( x , 0 ( x ) ) 17 N = {::I .

P ~ a a d . Assume t h e r e i s such a s e t N. F i x a l i n e 1 such t h a t

1 17 N i s o f f u l l one-dimensional measure. Assume 1 i s Ox. Fo r each

a e Ox - ( 0 3 t h e r e i s a l i n e d(a, ( a ) ) assigned by t h e f i e l d so t h a t

d(a, 0 ( a ) ) The p r o j e c t i o n of t h e p o l e s o f t h e

l i n e s corresponding t o p o i n t s o f N o v e r Ox i s a l s o o f f u l l measure.

There fo re t h e r e i s a subset o f such po les t h a t i s o f p o s i t i v e A-measure

and r e g u l a r .

e ( a 1 v a r i e s i n a L i p s c h i t z way. Therefore t h e s e t of po les of

forms a L i p s c h i t z curve.

Hence t h e un ion of t h e corresponding p o l a r l i n e s , t h a t i s A-a1 -

- most con ta ined i n R2 - N - Ox , has p o s i t i v e A-measure. B u t t h i s c o n t r a

d i c t s A m 2 - N) = 0 . Hence t h e theorem i s proved.

CHAPTER 10

APPROXIMATIONS OF THE I D E N T I T Y

Many aporox i rnat ion problems i n modern A n a l y s i s t a k e t h e f o l l o w i n g

t h e c o n v o l u t i o n i n t e g r a l

form. To f i n d o u t whether o r under which c o n d i t i o n s on k e L1(Rn) , /k = 1, kE * f , where k E ( x ) = E - ~ k(:) ,

E -f 0 , and f e Lp(Rn) , converges t o f . It i s r a t h e r easy t o prove

t h a t kE * f + f E + 0. I n f a c t , . i f g e g o (R'), we can w r i t e .

i n t h e LP-norm as

Hence, s i n c e 1 1 kE l l = 1 1 k l l = 1 , u s i n g M inkowsk i ' s i n t e g r a i neclual i t y ,

Given r~ > 0 we f i r s t f i x a g e to such t h a t 2 ]If - gl(, c ;.

Then we have, f o r each y E Rn, E z 0, 1 1 g ( * - EY) - q ( - ) l l and f o r each f i x e d y 6 R n , \ \ g ( - - EY) - g ( - ) \ \ -f 0 as E + 0 . Therefore, by t h e dominated convergence theorem,

c 2 ( l g ( ( ,

J

f o r E s u f f i c i e n t l y smal

A more d e l i c a t e

. T h i s proves t h a t kE * f -+ f

problem c o n s i s t s i n o b t a i n i n g t h e

LP) . p o i n t w i s e

convergence. Calderbn and Zygmund [1952] have g i v e n a r a t h e r genera l

r e s u l t f o r r a d i a l k e r n e l s t h a t i s presented in S e c t i o n 10.1., t o g e t h e r

281

282 10. APPROXIMATIONS OF THE IDENTITY

with a generalization due t o Coifman. Section 10.2. dea ls w i t h some

r e s u l t s t h a t a r e ava i lab le f o r kernels which a r e not rad ia l b u t a r e nonincreasing along each ray emanating from the o r ig in . In 10.3. we examine a general r e s u l t of F . Zo [19761 t h a t can be obtained by means of t he Calder6n-Zygmund decomposition which one can deduce many o ther useful r e s u l t s . we sha l l study some r e s u l t s o f P.A.Boo [1978] and of M.T.Carrillo p979]

concerning ce r t a in necessary conditions f o r a kernel t o y ie ld a good approximation of the i d e n t i t y in

(Lemma 3 . 2 . 7 1 , and from In Section 10.4.

L'(R").

10.1. RADIAL KERNELS.

I t i s r a the r obvious t h a t , f o r g e g o (R'), we have, i f kE(x) = E - ~ k($) k E L ' ( R n ) , i k = 1 , f o r E -f 0

k E * gcx) - g(x) a t each x e R n E'O

In f a c t ,

By the dominated E -+ 0.

tends t o zero as

Therefore i t i s s u f f i c i e n t t o prove , in order t o obtain the convergence of kE * f t o f f o r f e L p a . e .

tyne (.D,p) f o r the maximal operator K* where K * f ( x ) = sup \ k E * f (x ) . (1 & p 6 m), the weak

E>O

10.1. RADIAL KERNELS 233

K * f ( x ) = sup I kE * f ( x ) \ E > o

PhvaZj. L e t f > 0 , f E L’ be f i x e d . Consider , f o r each - j E Z , t h e s e t

By t h e hypotheses on k, C . i s a s p h e r i c a l s h e l l . I f J

N k ( x ) , i f 2-N-1 < k ( x ) 6 2 I 0 , o the rw ise

N k ( x ) =

and

K E f ( x ) N = kE N * f ( x ) , KN* f(x) = sup I k z * f ( x ) l E>O

we then have

So , i f we prove

w i t h c independent of N ,f , X , we s h a l l have t h e weak t y p e (1,l) f o r K* . Since K* i s o b v i o u s l y o f t y p e (m,m) , we

NOW, s i n c e k ( x ) i s non inc reas ing w i t h 1x1

Bj i s t h e c losed b a l l cen te red a t 0 whose r a d i u s i s

e x t e r i o r r a d i u s o f C j ’

have o u r theorem.

we can w r i t e , i f

equal t o t h e


There fo re

1 f ( x - z ) d z c M f ( x ) , where M i s t h e

j

H a r d y - l i t t l e w o o d opera to r over b a l l s . The re fo re , f o r each F > 0,

I N KN* f ( x ) b 2Mf(x) 1 2’ \ a j \ c 2Mf(x) k

j=-N

Hence t h e KN* a r e o f un i fo rm weak t ype (1,l) , as we wished

t o prove.

When k

r a d i a l majorant , def ined by

i s n e i t h e r r a d i a l n o r p o s i t i v e , one can c o n s i d e r i t s

K(x) = sup I k ( t ) l It1 c 1x1

A s u f f i c i e n t c o n d i t i o n t o o b t a i n t h e conc lus ions o f t h e above theorem

f o r k i s t h a t k , which i s now p o s i t i v e , r a d i a l and non inc reas ing

a long rays , be longs t o L1(Rn). I n f a c t , t h e maximal o p e r a t o r K* i s

ma jo r i zed by t h e one k* corresponding t o k .

T h i s c o n d i t i o n , however, i s n o t necessary as we s h a l l see i n

t h e f o l l o w i n g sec t i ons .

10.1. R A D I A L KERNELS 285

The theorem above, and its proof, remains valid if,instead of

assuming that k(x) is nonnegative and nonincreasing with 1x1, we as-

sume that for some

with 1x1 . > 0, k(x)(x(-' is nonnegative and nonincreasing

10.1.2. THEOREM. L e t k a 0, k 8 L1(Rn) be hadiae a&d auch

t h a t doh aume a > 0 k(x) 1 x / -a n v n i n c h m i n g w i t h I X I . The.&, w i t h

t h e h m e nv&un a!! in ThevLm K* ih 06 weah Rype

(1,l) aHd o d .type ( p , p ) , 1 i p c m . Hence, id /k = 1

a.e. doh each f e Lp , 1 c p < m .

10.1.1. we 5eL t h a t

kE * f -f f,

PkvvZ;. Since k(x) is radial we can define

and kN such that

if 2-N-l < k(x)lxl-' c 2N

, otherwise

N k (x) =

Consider, as before, with f 2 0,

with a independent of j

On the other hand


1 f ( x - Ey)dy 6 bMf(x) ' J B j

Where b i s independent o f j and f , and M i s t h e Hardy -L i t t l ewood

maximal oDerator . Hence

Bu t

and so K * f ( x ) 6 c M f ( x ) . We thus g e t t h e r e s u l t .

10.2. KERNELS NON-INCREASING ALONG RAYS.

When t h e approx ima t ion k e r n e l k i s n o t r a d i a l and i t s r a d i a l

n i a j o r i z a t i o n i s n o t i n L'(R") , one can s t i l l g e t some genera l p o i n t w i s e

convergence r e s u l t s w i t h s u i t a b l e c o n d i t i o n s on k . One of t h e r e s u l t s

i n t h i s d i r e c t i o n belongs t o R. Coifman and i t s p r o o f

r o t a t i o n method as f o l l o w s .

10.2.1. THEOREM. L e t k 8 L1(Rn) , k a rl , each x U L t h 1 i 1 = 1, .the ~ u n o t i v n v b r > 0 , k ( r x

cheaing i n r ~ L t h 6ume c1 independent 0 6 2 . Then

u t i 1 i zes t h e

be Ouch tha-t doh

r-' i n nunin-

t h e maximal

10.2. KERNELS NON-INCREASING ALONG RAYS 287

L e t

m

T- f ( x ) = s u p I p ln - ' k ( p j ) f ( x -Epy )dp d?, f o r 7 e Ct

f i x e d . Assume t h a t f o r each y E C+ f i x e d , we can p rove t h a t t h e

o p e r a t o r T- s a t i s f i e s . Y

&>O -a Y

w i t h c independent o f j . Then, by M inkowsk i ' s i n t e g r a l i n e q u a l i t y

c* I l f l l p

and t h e theorem will be proved.

Now, i f x = z t sy s 6 R , z e Y , Y hyperp lane o r thogona l

t o 7 , we can w r i t e

I n t h e d e f i n i t i o n o f T- we observe t h a t Y

i s non inc reas ing i n p, p > 0. Hence f o r 7 , z 6 Y f i x e d we can

app ly t o T - f ( z + s i ) t h e theorem 10.1.1, , and o b t a i n Y

m W 1 ITy f ( z + s y ’ ) l p ds c c [ f ( z + s Y ) I p ds -m J -00

Thus t h e theorem i s Droved.

As we can observe, t h e r e s t r i c t i o n t o p 1 1 a r i s e s f rom t h e

f a c t t h a t f o r P = 1 we j u s t have t h e weak t y p e (1,l) f o r t h e o p e r a t o r

I t i s an open problem, w i t h i m p o r t a n t i m p l i c a t i o n s , t o f i n d

o u t whether t h e r e s u l t c o u l d be ob ta ined f o r P = 1 , i . e . whether one

can a f f i r m , i n t h e hypotheses o f t h e theorem t h a t K* i s o f weak tyDe

(1,l).

I t i s however easy t o deduce t h a t K* s a t i s f i e s t h e i n e a u a l i t y ,

f o r x > 0, 1 x 1 < m , f 6 L ~ g + L

T h i s i s a consequence o f t h e f a c t t h a t

I1 K * f l l pcl C II f l l f o r l < p < 2

and o f t h e method o f e x t r a p o l a t i o n .

By c o n s i d e r i n g t h e l e v e l curves o f a k e r n e l t h a t i s non- increa-

s i n g a long r a y s and u s i n g t h e Theorem 3.8.1. on summation o f weak tyDe

i n e q u a l i t i e s one can g e t t h e f o l l o w i n g u s e f u l app rox ima t ion theorem, which

belongs t o M . T . C a r r i l l o [1979] .

10.2 .2 . THEOREM. L e i k e L’( Rn) , k > 0 , be non-inchu- - n i n g d o n g hayn. Annume &o t h d t t h e b&

10.2. KERNELS NON-INCREASING ALONG RAYS

A = C x : k ( x ) a , ' ) , j e l

dhe cvnwex, bvunded , and t h a t 4 a j = 2' l A j l , t h e bequence { a j }

b a t i h ~ i e h t h e doUullting c o n d i t i o n

5

Then t h e maxhai! v p m u t o h K* co&fu?bponding t o t h e hefind? k b vd weak

t y p e (1,l) and ad bR;/Long type (m,m) . Thenedofie , id .fk = 1, do4

each f 6 L p m n ) , 1 6 p < m , we have kE * f + f a.e.

Pkvvd. L e t f e L1(Rn) , f a 0 and f o r N E N

k ( x ) if 2-N 6 k ( x ) 6 ilN

0 o the rw ise I kN(X) =

I f , Kfi f ( x ) = sup I k f ( x ) l , i t w i l l be enough t o Drove t h a t

Kfi i s o f weak t y p e ( 1 , l ) w i t h a cons tan t independent o f N.

E>O N , E

We have

There fo re

L e t us c a l l


Since each A i s bounded and convex , the operators M j a r e uniformly of weak type (1 , l ) (Cf. Theorem 3.2.10. ) We have f o r each E > O

j

N N

-N K;f(x) = sup k x f ( x l c 1 Z J ( A j I M.f(x) = 1 a j M j f ( x )

&>O N Y E -N J

and using Theorem 3 .8 .1 . , and the condition on the Ea.1 we ge t the weak type (1 , l ) f o r Kfi and therefore f o r K*. The type ( m p ) i s t r i v i a l .

J

Of course the Theorem 10.2.2. admits a natural extension. Instead of requiring t h a t the s e t s A be convex and s a t i s f y the entropy condition of t he previous theorem one can requi re t h a t they a r e contained in convex s e t s B j t ha t s a t i s f y t h i s condition o r in the union of a fixed number of such s e t s . We sha l l now give an example of t h i s type of exten - s ion , proving t h a t the following kernel t h a t a r i s e s i n t he s t u d y of the multiple Poisson in tegra l (see R u d i n [1969] ) y ie lds a good approximation of the iden t i ty .

j

10.2.3. APPL7CATlON. Le,t k : Rn + R be dedined by

Then t h e O p U u L t O h K* dedined by

.i~ 0 6 weak t y p e (1,l) .

P h o O 6 . Let us s e t , fo r s impl i c i ty of no ta t ion , n = 2 . We consider the level s e t s

A j = I ( x , Y ) : 1 2 Z J I , j = o , -1 , - z , . . . ( 1+x2) l l + y 2 )

We have

10.2. KERNELS N O N - I N C R E A S I N G ALONG RAYS 29 1

There fo re

The s e t s B . ( a n t l i k e w i s e t h e set; C.) a r e i n t e r v a l s and J b . = Z J 1 B . I = 4 & 2 Jh/4b . We then have

s o K* i s o f weak t y p e (1,l) .

1 b j l l g b j / < a and -02

J J

We s t a t e another two a p p l i c a t i o n s . T h e i r p r o o f s a r e l e f t as

easy exe rc i ses .

10.2.4. A P P L I C A T I O N , L e L H = { ( x , y ) 6 R 2 : l x y l c 1)

- F and k (x ,y ) = (x’ f y’) x H ( x y y ) , 1 < a < 2 . Then K* oh weak

t y p e ( 1 5 1 ) .

10.2.5. A P P L I C A T I O N . LeL 1 C . I be any hequence 0 6 bounded J convex h& c o ~ ~ t ~ r t i n 5 t h e o h i g i n and huch t h a t I C . I = 2-j . Le,t J

m

k ( x ) = 1 x (.x). Then t h e m a x h d apm.atoh K* c o ~ e ~ p o n d i n g t o k 1 c j

For a k e r n e l t h a t i s t h e p roduc t o f another two f u n c t i o n s

k(x,y) = g ( x ) h ( y )

[1979]) . we can s t a t e t h e f o l l o w i n g r e s u l t s ( M . T . C a r r i l l o

10 .2 .6 . THEOREM. L t - t k ( X I y X 2 ) = g ( X i ) h(X2) , 9 >, 0, h >, 0,

( x l , x 2 ) B Rn1+n2 u h a e

bin5 d o n 5 hayh.

g e L (R”) , h B L (R”) and both me nonincten_ Ahhume Ah& t h e h&

m e bounded and convex . L e L a j = 2’ ! A j \ , b . = 2j I B . ] and anmme

, 1 b j l l g b j l < m . Then t h e m a x h d t h a t 1 a j l l g a.1 < m

J J +m +m

t + -m J -m


The proof follows the same idea of Theorem 19.2.2. and i s l e f t as an exerc ise .

By means of t h i s Theorem one e a s i l v sees t h a t , for example, the maximal ooerator K* of

i s of weak type (1,l).

Shapiro [1977] and Di tz ian [1977] have obtained previously some s l i g h t l y l e s s general r e s u l t s of t he type of 10.2.2 and 10 .2 .6 .

I t i s s t i l l an open question t o find out whether any oos i t i ve kernel k a 0 nonincreasing along rays and in L’ defines a maximal operator K* of weak tyne (1 , l ) .

10.3. A THEOREM OF F. ZO.

The theorem we present here requi res l e s s s t r ingen t conditions than the theorem of Calderbn and Zygmund. on the decomposition lemma o f CalderBn and Zygmund we have in 3.2.7. approximation theorem 10.1.1. i s an easy consequence of .Zo ’ s theorem 119761.

The technique of proof i s based The

10.3.1. THEOREM. L e t (kO1)asI be a ~um,Zy 0 6 ~ u ~ c t i u m i n L’(Rn) duch t h a t

10.3. A THEOREM OF ZO 293

a .

) = sun I ka(x-y) - kc l ( x ) \ , .then a

s cp < , wah cp independent ad y 6 R',

x ) = sun lkcl x f ( x ) , K* 0 6 weak t upe CtE I

Phaa6 . The s t r o n g t y p e (m,m) i s t r i v i a l , s i n c e

I n o r d e r t o Drove t h e weak t ype (1 , l ) , assume

f 8 L1(Rn) , X > 0. We aon ly t h e lemma o f Calder6n-Z.vqmund (3.2.7)

t o f and 1-1 > 0 , where 1-1 w i l l be c o n v e n i e n t l y chosen l a t e r . We

o b t a i n

a lmost each x B G = ( I 0 . , f ( x ) c 1-1 , and

f > 0 ,

IQj3 , a sequence o f d i s j o i n t dyad ic i n t e r v a l s such t h a t a t

J

f = f . s 27J l - I c *I 4 j J

D e f i n e

Hence K*g(x) G v c l ( Z n + 1). We choose such t h a t wc1(Zn + 1) = 7. x Thus


Observe t h a t 1 b(x)dx = 0 s u p p b c G . L e t Gj be t h e cub ic

i n t e r v a l c o n c e n t r i c w i t h Q

We can w r i t e

Qj - - and o f s i z e f o u r t imes as b i a as Q j , G = Qi. j

Now

A1 so

But

where y i s t h e c e n t e r o f Q t h u s j j .

J

Hence

10.3. A THEOREM OF ZO 295

Therefore

Thus K* i s of weak type (1,l)

Observe t h a t , i f 1 e L1(Rn) i s nonnegative , of k’(Rn - COI)

and such tha t f o r x # 0 ,

Then the family ( lE)E,O , l E ( x ) = ~ - ~ l ( : ) s a t i s f i e s ( i ) and (ii) of the theorem. In f a c t .f lE(x)dx = / l (x )dx and

-n

Whis t h i s remark , t he following Corollary i s easy.

and

Jhen

6vhe

a . e ,

one. with

10.3.2. THEOREM. L e L 1 e L ’ ( R n ) f l %&’(Rn - l o ) ) , 1 > 0 - . L e L k 6 L1(Rn) be ouch t h u t / k ( x ) I G l ( x ) C Vl(X)l 6 ~

K* h 06 weak t y p e ( 1 , l ) 0 6 n h u n g type 1 < a < m, and thehe -

I x p

4 I k = 1 , 6vh each f e L p , 1 f D < a, lim k E * f ( x ) = f ( x ) ,

The Theorem 10.1.1. i s now an easy consequence of t he l a s t For k E L’(Rn) , k 2 0 , k rad ia l and k(x) nonincreasing 1x1 , we def ine , i f k ( l x 1 ) = k(x)

*


10.4. SOME NECESSARY C O N D I T I O N S ON THE KERNEL TO DEFINE A GOOD APPROXIMATION OF THE I D E N T I T Y .

L e t k e L’(Rn) and cons ide r , as before , t h e o p e r a t o r

K* f ( x ) = sup \ k E * f ( x ) \ . Assume t h a t K* i s o f weak t y p e (1,1) E>O

What can be deduced about t h e k e r n e l k? T h i s i s t h e t ype o f a u e s t i o n

we a r e go ing t o handle i n t h i s Sec t i on .

10.4.1. - 10.4.4., be long t o M . T . C a r r i l l o [1979]

o f p rev ious theorems o f Boo [1976] . The l a s t theorem 10.4,5 belongs

t o 800 [1976] .

The f i r s t s e t o f theorems,

and they a r e ex tens ions

m 10.4.1. THEOREM. Led k e L (R’) . L e L { ~ ~ l ~ = ~ be a Aequsrzce

huch thcLt E 4 0 and E ~ + ~ / E j -f 1. Let un w m . e j

K*f(x) = sup I k E * f ( x ) \ j j

A A A W ~ t h a t K* A 06 weak .type (1,1) . Then

ess . sup. 1x1 I k ( x ) / ~1

Phuud. Assume t h a t ess.sup. 1x1 l k ( x ) l = 40 T h i s means t h a t

f o r each b i g T 3 0 t h e r e e x i s t s E = E(T) c R1 , I E ( > 0, such t h a t

f o r each x e E we have

10.4. NECESSARY C O N D I T I O N S 297

We can assume t h a t E c (0,m) . L e t us t a k e a d e n s i t y p o i n t x 0 6 E o f

E . We have

I E (1 B(xo, r ) I 1 i m = I r+O I B ( x o , r ) I

The re fo re t h e r e e x i s t s r o > 0 such t h a t ro < x o and

3 ( E f1 B ( x o , r o I I > ( B ( x o , r o ) l L e t us s e t E* = E 0 B ( x o , r o ) and

1 e t

We can f i n d a number n o such t h a t , f o r m > n,,we have

and SO 1, 0 Im+l # fi . There fo re

m

L e t E j = E~ E* and F = (I E j . j = n o

I f x E E j > n o , then $- 6 E" E and so j j

There fo re

3 We s h a l l i n m e d i a t e l y prove t h a t IF1 >, E ( xo + ro ) and so we

s h a l l have n0


But, i f K* i s o f weak t y p e (1,l) t hen we n e c e s s a r i l y have, acco rd ing

t o Theorem 4.1.1.

C ICx : SUD \ k E ( x ) ] > XI1 c T;

j j

f o r each A, > 0 w i t h c > 0 independent o f A . T h i s c o n t r a d i c t s t h e

i n e q u a l i t y (*) f o r T s u f f i c i e n t l y b i g , and so t h e theorem i s proved.

I n o rde r t o see t h a t I F \ > & E ( x o + r o ) we Droceed i n no

t h e f o l l o w i n g way. We f i r s t choose p > n o so b i g t h a t

From t h e i n t e r v a l s Ino, Ino+l , ..., Ip we can choose two

each o f them of d i s j o i n t i n t e r v a l s seauences {JI , . . . ,Js} {TI ,TV}

such t h a t

There fo re , f o r a t l e a s t one o f them, say f o r t h e f i r s t one, we have

1 p C I J j I 7 I IJ

S

i =1 n0

3 Each Ji c o n t a i n s a s e t Ei such t h a t (Ei) > 4 IJi I and so

3 m

3 m [I I 1 > - E (xo + r o ) . = g (IJ I,- m 16 no

n0 P+ 1

The f o l l o w i n g r e s u l t i s an n-dimensional ex tens ion o f t h e

p reced ing one. The method o f p r o o f i s analogous and w i l l be omi ted .

10.4.2. THEOREM. LeR: k E L'(Rn) , n > 1 . L e L { E . } be "1 x

j ~j a hequence A O thak E .C 0 and E ~ + ~ / E~ + 1 . L e t kE ( x ) = k ( r ) ,

j

10.4. NECESSARY C O N D I T I O N S 299

K*f(x) = sup Ik * f(x)/ . L e l Un dedine t h e dunct ivn H vn t h e ul.tit

nphehe C 0 6 Rn by

Ej j

H f i ) = ess. sup. rn fk(ry

Annume &at K* A v d weak t q p e (1 , l ) . Then

buch t h a t doh each X > 0

r>O

whetre u A ,the. Lebugue meauhe a n C . When the kernel k of the preceding theorems i s continuous

one can give a somewhat simpler formulation.

10.4.3. THEOREM. L e l k e L’(R’) be cvrztinuvun and

0 0 K*f(x) = s u p (k, * f(x)l . Adoume . t ha t K* A v d weah t y p e (1,l)

Then sup 1x1 lk(x)l < m

x e R

10.4 .4 . THEOREM. L e i t k 6 L1(Rn) be ContinuvUc),

K*f(x) = sup Ik, * f(x)l nnd doh 7 6 C E>O

H(y) = sup rnl k(ry)l r>0

Abnume t h a t K* 0 6 weak .type. (1,l) . Then thehe c > 0 nuch t h a t doh each h 0

These theorems allows us to construct in a simple wayy for

example, radial kernels k e L1(Rn) operator K* i s not of weak type (1 , l ) . ( O f course k cannot be

nonincreasing , according to Theorem 10.1.1.). Take for example k e L1(R’) k continuous k(-x) = k(x) and such that for each j e Z , k(j) = 4‘1 Then K* i s not of weak type (1,l) . In R2 one can extend the preceding

such that the corresponding maximal


- - k r a d i a l l y t o k so t h a t s t i l l k E L 1 ( R 2 ) . The corresDonding maximal

ope ra to r K* i s n o t o f weak t ype (1,l) .

We have a l r e a d y seen i n Theorem 10.1.1. and t h e remark t h a t

follows i t , t h a t i f k E L1(Rn) , f k = 1 , and t h e f u n c t i o n T d e f i n e d

by E (x ) = ess sup I k ( t ) \ i s i n L1(Rn) , then f o r each f e L1(Rn) ,

k E * f ( x ) + f ( x ) a t a lmost each x €Rn. The f o l l o w i n g theorem, due t o Boo [1976] , i s a p a r t i a l converse o f t h i s r e s u l t . L e t us r e c a l l t h a t f o r

a f u n c t i o n f E L1(Rn) a D o i n t x E R” i s c a l l e d a LebQngue p o i n t 06

I t l c 1x1

We know t h a t a lmost each p o i n t o f Rn i s a Lebesgue p o i n t o f f 6 L1(Rn).

10.4.5. TffEOREM. 1eL k E L’ 17 L”(Rn), /k = 1 and dnnwne

t h a t j $ x each f E LI@P) ,

dt each Lebague p o i n t 06 f . Then t h e unction E(x) = ess sup I k ( t ) I LA in L’.

l t l 4 x l

Phood. We s h a l l prove t h a t i f !k = m , then t h e r e e x i s t s - g 8 L’(Rn) , w i t h g (0) = 0 , 0 i s a Lebesgue p o i n t o f g and s t i l l

l i m SUP € 4

L e t X be t h e subse t o f

f o r 1x1 > 1 , f ( 0 ) = 0 , f ( x ) = 0

Tha t i s

f u n c t i o n s f o f L1(Rn) such t h a t

and 0 i s a Lebesgue p o i n t o f f .

The s e t X i s a l i n e a r subspace o f L1(Rn). I f f o r f o X we d e f i n e

10.4. NECESSARY CONDITIONS 301

then ] I f ( l i s a norm i n X . We s h a l l now show t h a t X w i t h II.llx i s a Banach space.

I n f a c t , l e t { g . ) c X be a Cauchy sequence i n X . S ince J

we have t h a t Eg.1 i s a l s o a Cauchy sequence i n L'(Rn). We t a k e a

subsequence Chj l of I g j 3 such t h a t J

and Drove t h a t E h . 1 converges i n X . T h i s , of course, i m o l i e s

t h a t a l s o 19.3 converges i n X . J

J

For Chj1 we have 1 1 h j - hj+llll c IB (Q, l ) \Z - ' and so,

by F a t o u ' s lemma, we e a s i l y see t h a t I h . 1 converges a.e. and i n L ' t o a f u n c t i o n . We can s e t g(O) = 0 , g ( x ) = 0 i f 1x1 > 1.

We a l s o have

J g E L1

4 l i m i n f 1 1 h j - hi l l 2-j+l i - t w

Wi th t h i s we e a s i l y have g 6 X and h j -f g(X) . Hence X i s a Banach

space.


Observe now t h a t for a fixed E > 0 , the mapping $E defined from X t o t by

i s l inear and bounded, since

Therefore ( $ e ) E > O i s a family of bounded l inear functionals from X

t o Ic . If we can show t h a t E + L' implies t h a t there exis ts E~ + 0,

f i E X with 11 f i l l L c , such t h a t I @ E i ( f i ) l + m , then, by the uniform boundedness principle, th i s means t h a t there must exis t g E X such t h a t lim sup I$Ei(g)l = and so we obtain the contradiction

t h a t Droves the theorem. E . ' 0

1

So our goal i s t o construct for each fixed E > 0 , using t h a t /k = 00 , a function f E E X , I ( f,l( c c such tha t lim SLID l$,(f,)l=w.

Observe f i r s t tha t f F = imolies the following . Let us c a l l ,

E - t O

for s = 0,1,2,3,.., and E > 0

m r

10.4. NECESSARY CONDITIONS 303

00 ME(s) There fo re , if E + L 1 , we have 1 - + m as E + 0 ,

s=o 2ns

We now rnroceed t o d e f i n e t h e a n n r o p r i a t e f . Since E

ME(s) = ess s u p Ik,(x)l , s = 0,1,2,... we have a s e t o f

p o s i t i v e measure EE(s)

2-s-1 < E I x l < 2 - s

M E W < 1x1 < 2-’ , \ k E ( - x ) l > I E E ( s ) = { x : 2 - s - l

L e t us s e t

Now we s e t

g,, S ( X ) fE:(X) = 1 -

m

s=O zns Ns

where Ns , s = 0,1,2,.., i s chosen so t h a t N s t m , and

(Take, for example Ns = as f o r ci > 1 and c l o s e t o 1)

We see t h a t f E ( 0 ) = 0 , f E ( x ) = 0 f o r 1x1 > 1 . A lso , i f

< r 6 2-k , we have 2 - k - l

X

On t h e o t h e r hand

m

<2-s k , s ( x ) ( d x =

And so l @ E ( f E ) l -f m as E + 0. T h i s concludes t h e o r o o f o f t h e theorem,

CHAPTER 11

SINGULAR INTEGRAL OPERATORS

One of the most basic operators in Fourier Analysis i s the Hilbert transform, formaily defined through

The in tegra l value. from the ro l e i t plays in the estimation of t he pa r t i a l sums of t he Fourier s e r i e s of a function. I t s type has been studied very ea r ly by Lusin, M.Riesz , Kolmogorov among o thers . This was done i n i t i a l l y mainly th rough complex methods, as was usual in the onedimensional Fourier a- na lys i s pr ior t o the f i f t i e s . Besicovitch [1923]

pioneer of the introduction of rea l methods in t h i s area and very ea r ly he obtained the existence of the above in tegra l f o r almost every x , f i r s t when f 4 L2(R1) [1923] and then 119261 when f E L ' (R ' ) by purely rea l methods.

has t o be understood in the sense of Cauchy's principal One of the main reasons f o r the imDortance of t he operator stems

however was the

The grea t spread of the i n t e r e s t i n real methods i n Fourier Analysis came i n Zygmund in which they studied the n-dimensional analogues of the Hil- ber t transform by rea l methods, The a p p l i c a b i l i t y o f t h e i r r e s u l t s and of t h e i r methods t o the s t u d y of pa r t i a l d i f f e r e n t i a l equations or ig i - nated an increasing i n t e r e s t in t h e elaboration of purely real techniques f o r the problems of Fourier Analysis.

1952 w i t h the now c l a s s i ca l paper of CalderBn and

This Chapter i s conceived in the context of the whole book

as a t e s t i n g ground f o r the goodness of some of the methods developed in e a r l i e r Chapters. lrle f i r s t show how the general r e s u l t s permit us t o obtain w i t h ease some of the theorems which, f o r long t ime, have been

305

306 11. SINGULAR INTEGRAL OPERATORS

considered p r e t t y d i f f i c u l t and s o p h i s t i c a t e d . F i r s t we dea l w i th t h e

H i l b e r t t rans fo rm , LP- theo ry and p o i n t w i s e theo ry , and then w i t h t h e

Calder6n-Zygmund opera to rs . I n t h e t h i r d Sec t i on we deal w i t h t h e t h e o r v

o f s i n g u l a r i n t e g r a l ope ra to rs w i t h g e n e r a l i z e d homogeneity t h a t we s h a l l

use l a t e r on i n Chapter 13.

11.1. THE HILBERT TRANSFORM.

1 L e t h(x) = x f o r x # 0 , and f o r E > I), l e t

The fhunCCLtc?d H A ~ . w L ~ ~X.LZ~IA@~ ( a t E ) o f f a Lp(R1), f o r some D,

1 & p < m, i s d e f i n e d as H,f(x) = h, * f ( x ) , and t h e H i l b e r t t rans -

form i s d e f i n e d as H f = l i m H E f , whose e x i s t e n c e i n some sense we E'O

s h a l l have t o prove. The maximal H i l b e r t t rans fo rm i s d e f i n e d as

H* f ( x ) = sup I H E f ( x ) l . E'O

We s h a l l t r e a t s e p a r a t e l y and independen t l y t h e L1- t h e o r v and

t h e L 2 - t h e o r y .

l l . l . A . The L'-Theory.

The p a t h we f o l l o w i s v e r y s imp le . By means o f a lemma o f

From here, based on t h e dec reas ing

Loomis [1946]

over f i n i t e sums o f D i r a c d e l t a s .

cha rac te r o f t h e ke rne l x on R ' - 0

t h a t H* i s of weak t y p e (1,l) over f i n i t e sums o f D i r a c d e l t a s and

so of weak t y p e (1,l) , accord ing t o Theorem 4.1.1. T h i s g i v e s us e a s i l y t h e a-e'. convergence o f H E f f o r f 6 L' (R ' ) ,

we Drove t h a t t h e H i l b e r t t r a n s f o r m i s o f weak t y o e (1,l)

1 , we e a s i l y a r r i v e t o t h e f a c t

11.1.1. LEMMA, LeL a . a R , j = 1,2,3 ,..., N und X > 0. J

11.1. THE HILBERT TRANSFORM 307

N L e L f = 1 A j whehe 6 . A i h e D&c d&a concenLated CLt a . .Then

, j=1 J .?

P h a a 6 . By l o o k i n g a t t h e granh o f t h e f u n c t i o n

1 N y = c

j=1 J

N > A } ] = 1 ( v j - a,i) where i t i s q u i t e c l e a r t h a t ICx : ~

1 j=l x-a j j=1

v . j = 1,2, . . .yN a r e t h e r o o t s o f t h e eaua t ion - j y

N N . N N 1 1 x-a.- - A i . e . o f A r! ( x - a . ) = 1 r! (x-a,)

.j = 1 J j=1 " .i=1 ,i#k

From here we e a s i l y o b t a i n , by t h e Cardano-Vieta r e l a t i o n s

j=1 J 1 j=1 j= l

N N N N N 1 y . = - + 1 a j . Hence 1 (y j - a . ) = x . Thus we g e t J

Since t h e second te rm can be handled as t h e f i r s t one.

11.1.2. TffEUREM. The maximal HLLbeht opeh.aXoh H* A 06 weak

.type ( 1 9 1 ) .

Phoo6. Accord ing t o Theorem 4.1.1. i t i s s u f f i c i e n t t o n rove

t h a t H* i s o f weak t y n e (1,l) over f i n i t e sums o f D i r a c ' d e l t a s , L e t

a . E R , j = 1 , 2 , ..., N, h > 0 , and f = SLi where S. i s t h e

D i r a c d e l t a concen t ra ted a t a We have t o m o v e t h a t

-

3 j=1 J

j *


w i t h c independent o f f and X

We take an a r b i t r a r y compact s e t K con ta ined i n

{ x : H * f ( x ) > X I - { a ,a2 ,..., aN 1 . I f x 8 K, t h e r e e x i s t s E ( X ) > f l

such t h a t I H f ( x ) ! X . We t a k e a f i n i t e number o f d i s . i o i n t

i n t e r v a l s Ik = L X ~ - F. xk) , xk + € ( x k ) ] k = 1 y 2 y . , . , M , w i t h E(X)

M Xk E K such t h a t I K I C 21 ( I I k l . For each k = I , ? ,... ,M, l e t

1

f k = X I k i . e . f k i s t h e sum of t h e D i r a c d e l t a s o f f sunpor ted

by t h e i n t e r v a l I k . L e t us d e f i n e f i + f - fk , i . e . f i i s t h e

sum o f t h e D i r a c d e l t a s o f f w i t h suDDort o u t s i d e Ik. We can w r i t e

There fo re

! i f * ( t ) = k

i s decreas

Thus IHf;

i n [xk -

! H f c ( x k ) l > A . Now t h e f u n c t i o n o f t

ng over I k , s i n c e H f $ ( - ) has no s i n g u l a r i t y o v e r I k . t) I > X f o r each t i n [xk , xk + c ( x k ) ] o r f o r each t

E(Xk) xk 1 s i n c e IHf;(xk)( > A . L e t us c a l l h t h e h a l f i n t e r v a l of Ik where t h i s happens. We have then

{ I H f Z l > X 3 3 2- I k . We can a l s o w r i t e 1

We s h a l l t r y t o es t ima te t h e l a s t s e t . We have H f i = H f - H f k and

so


Hence

Therefore we can s e t using Lemma 11.1.1.,

Since IKI i s a r b i t r a r i l y c lose t o l{H*f > XI1 we get out theorem.

i s a comnact i n t e r v a l , i t i s

c l e a r t h a t Hg(x) e x i s t s a t a .e . x E R ’ . Therefore Hffx) e x i s t s f o r f e L’ a t a .e . x s R 1 , and a l so H i s of weak type (1 , l ) .

I 5 For g = 1 a j x I , where j

11.1.8. The L2-Theory.

The L2-theory of the truncated Hi lber t transform i s very simole by means of the Fourier transform. We have

w i t h c independent of E and x . Therefore , i f f E L2(R1) , . We know t h a t f o r I ( H E f ( ( 2 c I I f l ( 2 with c independent of f ,E

g = 1 a j x I j where I i s a comnact in te rva

e x i s t s a . e . By an easy d i r e c t computation one can

HEg -+ Hg(L2) as E -f 0 . From these f a c t s we sha each f E L2 the l i m i t of HEf as E + O i n L 2

N

j=l j

check t h a t

1 deduce t h a t f o r e x i s t s , In f a c t ,

t a k e a sequence {g,} o f s imp le f u n c t i o n s as above such t h a t qk -f f ( L ' ) .

Then we have

Given n > 0 one takes gk so t h a t 2c I ( f - gk112< n / 2 . Once

gk i s f i x e d , i f ~ , 6 a r e smal l enough we have 1 1 HEgk - H6gk1I2c n / 2 . There fo re H E f i s a Cauchy sequence i n L 2 and so converges t o a f u n c t i o n

H f i n L 2 . Fur thermore we have 1 1 H f l 1 2 c c I I f ( I 2 . 1 ;; With t h i s r e s u l t and t h e f a c t t h a t i s dec reas ing i n R'- I01

we s h a l l o b t a i n , as b e f o r e , t h e weak t y p e (2 ,2 ) o f H*.

11.1.3. TffEOREM. The maximal ffLLbent opehaton H* A a6 weak

N P m v d . L e t f = 1 c j x E j c . > 0,

j=1 Ej J - d i s j o i n t compact

i n t e r v a s , and X > O . We t a k e a compact s e t K con ta ned i n

e x i s t s E ( X ) > 0 such t h a t f ( x ) I > X , We t a k e a f i n i t e

number o f d i s j o i n t i n t e r v a l s I k = [ xk - E ( x ~ ) , xk + E ( x ~ ) ~ w i t h

{ H * f > XI - {endpo in ts o f t h e i n t e r v a l s E j l . Fo r each x 6 K t h e r e

M

1 Xk E K , such t h a t I K I < 2 I [I I k I . ' Fo r each k = 1 , 2 , . . . , M y l e t

f k = f X and ft = f - fk . As b e f o r e , i n t h e p r o o f o f Theorem

11.1.2, we have I k

and so I H f t ( x k ) l > X . Now t h e f u n c t i o n Hf;(-) i s non inc reas ing on I k s i n c e t h e

suppor t o f f i i s o u t s i d e Ik . Thus we can oroceed as i n t h e p r o o f o f

Theorem 11.1.2. and a r r i v e t o t h e weak t y o e (2,2) o f t h e o a e r a t o r H*.

The convergence a lmost a.e. o f

and t h e t y p e ( D , P ) , 1 < p < m,Of H* and H a r e easy consemences

of what we have proved a l ready .

H E f , f o r f e Lp , 1 6 p < m y

We add he re a couple of remarks. F i r s t , i n t h e t r e a t m e n t o f

t h e weak t y p e (2,2) o f H* we have fo l l owed a p a t h d i f f e r e n t f rom t h e

one used f o r t h e weak t y p e (1 , l ) . something 1 i ke

One c o u l d be tempted t o t r y t o p rove

and t h e apo ly Theorem 4.2.1. t o o b t a i n t h e weak t y p e (2,2) f o r H* . But t h e i n e q u a l i t y ( x ) above i s f a l s e , I n f a c t we have t h e e q u a l i t y

and so we cannot have f o r a f i x e d c < m, L c - f o r b i g X . x2 The second remark i s t h e i n t e r e s t i n q p r o o f t h a t can be o b t a i n e d

o f t h e s t r o n g t y p e (2,2) of H f o r s imo le f u n c t i o n s w i t h o u t making use

o f t h e F o u r i e r t rans fo rm. I t i s v e r y easy and makes use o f t h e f o l l o w i n g

lemma due t o B e s i c o v i t c h [1923] .

11.1.4. LEMMA. L e L tl , 1 = 0,?1,+2 ,.'. be t h e neyuence - a 6 h i g k t endpointd 06 t h e dyadic in tem& 06 R' 06 Length 2-N

x . , i = 0,?1,+2,.. . F m i ' i = o ,il ,+2, ..,

and &A

be t h e .sequence v d .the,& midpointd. Then we have id 1

A a n a h b i t m t y nequence 0 4 nvnncgaLiue nWnbe?Ld

With t h i s lemma one e a s i l y o b t a i n s t h e f o l l o w i n g r e s u l t .

11.1.5. TtlEOREM. The. o p e n a t o h H A ad n,7hong .type (2,2) u u a finem combinLttioa v 6 charcaotehinLic d u n o t i v u oh bounded i n t a v a h .

Thehedohe AX can be dedined a n L2 and AX'A ad b a u n g .type (2,2).

N Phuod. For a f u n c t i o n f = 1 a j xI. one has t o Drove

j=1 J

It i s an easy e x e r c i s e t o reduce t h i s

lemma by s u b s t i t u t i n g t h e i n t e g r a l s by

n e q u a l i t y t o t h a t o f t h e p reced ing

a p w o D r i a t e Riemann sums.

Phood 0 6 Lemma 11.1.4. It i s s u f f i c i e n t t o Drove t h a t , i f

1 - 7 , m. > 0, i = l,Z,...,n , then

1 s.= i 1

To do t h i s we can w r i t e

m. m.

1 n

1 mi m.- 1 s . - 1 ) ( s . - 17 n " 1

= 1 m i 1 - + 2 J J 1=1 1 i = l 1=1 (Si - 1 ) * l t i < j < n

But

1 n 1 ~ < 8 1=1 (S i - l I Z

< n 2 i mf and so E - 1 m

i=l i = l

On t h e o t h e r hand , i f i < j .

11.2. C A L D E R ~ N - ZYGMUND OPERATORS 313

1 n l n l c K] = 1 =1 J -'j c J1 l-si - 1 =1 J

-~ 1 n

(Si - l ) ( s . - 1 ) -

T h i s proves t h e above i n e q u a l i t y .

11.2. THE CALDER~N-ZYGMUND OPERATORS.

The Calderdn-Zygmund opera to rs , t h e n-d imensional analoques

o f t h e H i l b e r t t rans fo rm, a r i s e i n a n a t u r a l way when c o n s i d e r i n q c e r t a i n

problems r e l a t e d w i t h t h e d i f f e r e n t i a t i o n o f a Newtonian p o t e n t i a l .

t h i s m o t i v a t i o n one can see CalderBn and Zygmund [I19521 . For

1 The k e r n e l h ( x ) = o f t h e H i l b e r t t r a n s f o r m has seve ra l

i m p o r t a n t f e a t u r e s t h a t a r e r e s p o n s i b l e f o r t h e good behav iou r o f t h e

H i l b e r t t rans fo rm. F i r s t h ( 1 ) + h ( -1 ) = 0 , i . e . h has mean v a l u e

ze ro ove r t h e u n i t sphere o f R1.

x # 0 , i . e . h i s homogeneous o f degree -1. Besides h has a p r e t t v

smooth behaviour , h E e(R1 - { O } ) . T h i s , as we s h a l l see, i s n o t

necessary f o r o b t a i n i n g r e s u l t s i n Rn s i m i l a r t o those we have o b t a i n e d

about H and H* . So we s h a l l cons ide r a CalderBn-Zygmund k e r n e l k

i n Rn , i . e . a f u n c t i o n k : Rn -t R ( o r E ) such t h a t

Second h(Xx) = X-'h(x) , f o r A > 0,

ii) k h x ) = X-'k(x) , f o r X > 0 , x # 0 .

We s t i l l need some smoothness c o n d i t i o n on k t o o b t a i n a reasonable

behaviour f o r t h e o p e r a t o r s we a r e going t o d e f i n e , I t t u r n s o u t t h a t

t h e i n t e g r a l L i p s c h i t z c o n d i t i o n t h a t f o l l o w s i s a l r e a d y s u f f i c i e n t f o r

t h i s .

iii) There e x i s t s c > 0 such t h a t f o r each v e Rn

1 I X I ,4 I Y

O f course, f o r

L i p s c h i t z c o n d i t i o n

( k ( x ) - k ( x - y ) l d x h c < m.

i i ) i t i s s u f f i c i e n t t h a t k s a t i s f i e s a

I n f a c t , we t h e n have

Fo r c o n d i t i o n (*) , i n v iew o f

t o assume k 6 Q1(Rn - I01 )

t h e homogeneity o f k i t i s s u f f i c i e n t

The Calderdn-Zygmund o n e r a t o r o f k e r n e l k i s now d e f i n e d

f o r m a l l y as

K f ( x ) = k * f ( x )

I n o rde r t o g i v e a meanino t o t h i s c o n v o l u t i o n one f i r s t cons ide rs t h e

t r u n c a t e d k e r n e l s

k ( x ) , i f E 6 1x1 c n i 0 , othe rw ise

k,,-,(x) =

and, f o r f e Lp(Rn), 1 h n < m , one d e f i n e s

kE,,f(x) = kE,T) * f ( x )

The ques t i on now i s whether t h e l i m i t o f KE,T) f e x i s t s i n

some sense.

CalderBn and Zygmund, i n t h e i r now c l a s s i c a l paper o f 1952 ,

11.2. CALDERBN - Z Y G M U N D OPERATORS 315

obtained very general s a t i s f a c t o r y r e s u l t s , proving the ex is tence of t he l imi t in L p , 1 < p < m , f o r f E L p , and t h e existence of t he l i m i t a t almost each x 6 Rn f o r f 6 L p , 1 c p < m under some additional conditions. Zygmund [19671 and l a t e r on by Riviere [19731, who f i n a l l y proved tha t the simple Lipschitz in tegra l condition ( i i i ) on t he kerne ls , together w i t h the usual assumptions ( i ) and ( i i ) , su f f i ces t o obtain t h e convergence in Lp and the pointwise convergence results.

Their r e s u l t s were l a t e r refined by CalderBn, M. Weiss and

goes t h r o u g h some o f t he genera chapters. F i r s t we sha l l obtain truncated kernel s k E ,Tj the a to r s KE,TI . This r e s u l t s eas

We present here t h i s r e s u l t , following a d i f f e r e n t path t h a t theorems we have obtained in previous bv studying the Fourier transform of the

uniform boundedness in L 2 of t he oper- l y leads t o the ex is tence of

n - f w

From i t we sha l l ob ta in , using the method of majorization shown i n 3 . 6 . , the strong type (2,2) of t h e maximal operator K*

Finally we sha l l show, by means of t he Calder6n-Zygmund decom- posit ion lemma of 3.1, the weak type (1 , l ) of the maximal operator K*. From t h i s one e a s i l y obtains the almost everywhere convergence of KE,.,f f o r f e ~ P , 1 c D < m , a s ~ + ~ , q + a .

11.2.A.

11.2.1. THEOREM. l&t k : R n + R b~ a 6unc;tion ouch t h a t

i ) 1 lk(:)/di < m ,

The Uniform Boundedness i n L 2 of the Truncated Operators.

k ( i ) d i = 0 c c

i i ) For x # 0 , X > 0 , k ( X x ) = A-’k(x)

3 16 11. SINGULAR INTEGRAL OPERATORS

independent v d ~ , q . Themdohe , dux f E L2 , K f = l i m K f ( L 2 ) EYn € 4 .n-

Pxaad. We w o v e t h a t t h e r e e x i s t s c > 0 such t h a t f o r each

x E Rn , ItF rl ( x ) l

t h e n us, f o r f e L2 ,

c c < . The Parseva l -P lanchere l theorem g i v e s

i . e . , KE,q i s u n i f o r m l y o f t y p e (2 ,2 ) .

For g E @:(Rn) we have

6 a 1 I k ( y ) l l y l d y = a 1'' dp I k ( ? ) J d.7 , E l

E1GIY

where a = max I V g ( x ) l , and so one e a s i l y sees t h a t KE g ( x )

converqes a t each x and i n L 2 as E + 0, n + m , There fo re , by ap-

p r o x i m a t i n g f E L2(Rn) by f u n c t i o n s i n $' :(Rn) one ge ts t h a t K E Y q f

converges i n L2(Rn) t o a f u n c t i o n K f and t h a t 1 1 K f l 1 2 c c I I f1I2 .

YV x 6Rn

Therefore, a l l we have t o do t o prove t h e theorem i s t o o b t a i n

t h a t

11.2. CALDER6N - ZYGMUND OPERATORS 317

h

( x ) I L c < m w i t h c independent o f ~~q . I k c d l

For b r e v i t y l e t us c a l l h ( x ) = kE ( x ) . We s h a l l f i r s t p rove Y r l

t h a t

for each y, w i t h c independent o f y.

If 1x1 2 41yl , we have

Thus i f we denote

we o b t a i n

The t h i r d i n t e g r a l , over S3 , i s t r e a t e d i n t h e same way . Thus we g e t

w i t h c independent o f y .

We nex t prove t h a t t h e F o u r i e r t rans fo rm o f h i s bounded

u n i f o r m l y i n E, q . fi x = o , then \6(0)l = .r kE,n ( v )dv = 0 . Assume x f 0 I

1 , so t h a t (x,z) = . We can and cons ide r f i (x ) . Take z = - X

21x12 w r i t e

d ,v= 1 I,j . j=1

We have [ I 1 \ c c, as proved before. For 1121 we w r i t e , i f

121 IYI < 4 1 4

If I z I > 1 ~ 1 , we t a k e y 1 = $,- I z I , so t h a t Iy11 = I z I . Thus

11.2. C A L D E R ~ N - ZYGMUND OPERATORS

F i n a l l y ,

11.2.B. The Strong Type ( 2 2 ) o f t h e Maximal Operator .

11.2.2. THEOREM . L e l k be M -in 11.2.1. Then, .id , t h e maxim& openatoh K* A 0 6 W M ~

Phaaa. I n o r d e r t o p rove t h a t K* i s o f t y p e (2,2) we can, - bv Theorem 3.1.1. r e s t r i c t ou rse l ves t o n rove t h e t v n e ( 2 , 2 ) over a

dense s e t o f f u n c t i o n s . L e t g be a l i n e a r combinat ion o f c h a r a c t e r i s -

t i c f u n c t i o n s o f d i s j o i n t open bounded s e t s whose boundar ies have n u l l

measure. For such a f u n c t i o n , if x i s n o t on any o f t h e boundar ies

o f such se ts ,

~ g i x ) = l i m K ~ C X ) e x i s t s and €,I? P O

n-

g(X) = KEg(X) - Krlg(x) , where K6g(x) = l i m K s , s g ( ~ ) . rl-

Hence, f o r such x , t h e r e e x i s t s E ( X ) > 0 such t h a t

319

z ) d z ( .

0 1 Since, f o r a lmost a l l

w r i t e f o r such p o i n t s y

i f l e s s than o f equal t o

y E B (x , E ( x ) ) = BE , Kg(Y) e x i s t s , we can

t h a t t h e l a s t member o f t h e w e c e d i n g i n e n u a l i t v

Now we have, i f M i s t h e Hardy -L i t t l ewood o p e r a t o r ,

and SO, s i n c e K and M a r e of t y p e (2,2) ,

For 1 3 g , u s i n g Kolmogorov's i n e q u a l i t y f o r K w i t h p = 2 , s = 2 , 0 = 1 (Theorem 3.3.1.) we can w r i t e

and so,

The e s t i m a t e f o r I l g ( x ) i s a l i t t l e more d e l i c a t e . L e t us

d e f i ne

[ o , i f l v l < 1

Then, i f x - z = EV , x - y = EU ,

NOW, if v = rG , V E c , r > 0,

0 , i f r < l

$ ( v ) = +(rS) =

I k ( r G ) - k ( r i - u )dv , i f r 2 1

i .e . , i f u = rt ,

So 4 i s non inc reas ing a long rays i n 1x1 > 1

I f we d e f i n e

I k ( ? ) - k ( i - u ) l du , i f I v I < 1

+*(v )= $*(rV)= I k ( r V ) - k ( r i - u ) l d u , i f I v I 2 1.

Then $* o b v i o u s l y m a j o r i z e s $I , and i s non inc reas ing a long ra.vs i n

Rn. A l so we have

I k ( v ) - k (v -u ) ( d u dv =

i,v,,14* = I,,,,, I,,,,:

by c o n d i t i o n (iii) on k . And a l s o

+*(v)dv c I k ( c ) I d u dv + 1, 1 I k ( E - u ) ( d u dv I,,,,, V J d \Ul<K

T h u s I +* < - . Now

and we know a l r e a d y by t h e r o t a t i o n method t h a t t h e l a s t o p e r a t o r i s o f

t y p e (2,2) .

Hence K* i s o f weak t y p e (2,2) and t h i s concludes t h e p r o o f

o f t h e theorem.


11.2.C. The Weak Type (1,l) of t h e Maximal Operator .

11.2.3. TffEOREM. 1eA k b e ah i n Tkteahem 11.2.1. Then K*

A 0 6 weak t y p e (1,l) .

P m o 6 . L e t now f be any nonnegat ive f u n c t i o n i n L1(Rn)

w i t h compact suppor t . L e t A > 0 and app ly t o f and A t h e CalderBn-

n t Zygmund descomposi t ion lemma, o b t a

dyad ic i n t e r v a l s such t h a t

n i n g a sequence (9 .1 of d i s j o J

D e f i n e now

and l e t , f ( x ) = g ( x ) f h ( x ) . C l e a r l y supp h c ( L J Q,i) and

1 0, h ( x ) d x = 0 . A lso g ( x ) c ZnA a .e. and b o t h g and h have

J compact suppor t . We have

We a l r e a d y know t h a t K* i s o f weak t.ype ( 2 , Z ) . Therefore, x i f A i s any compact s e t con ta ined i n { x : K*g(x) > 1 , by Kolmo-

g o r o v ' s i n e q u a l i t y w i t h 0 = 1 , s = p = 2 , we have,

Hence, remembering t h a t g ( x ) c Z n I a.e, , we g e t

324

S ince

11. SINGULAR INTEGRAL OPERATORS

Therefore, a l l we have t o do now i s t o prove t h a t

As before, K* f ( x ) c 2 sun IKEh(x ) l . (Observe t h a t , s i n c e h has

compact suppor t , K h ( x ) KEh(x) - Knh(x) and b o t h KEh(x) and

K h ( x ) e x i s t and a r e f i n i t e ) . Hence f o r each

E ( X ) > 0 such t h a t

E N

EYn x 8 Rn , t h e r e i s an n

We s h a l l now show, f o l l o w i n g Calder6n and Zvqmund

f i x any a r b i t r a r y f u n c t i o n

[1973]

x E Rn + E(X) e ( 0 , ~ ) , then

t h a t i f we

and t h i s w i l l conclude t h e p r o o f o f s t e p C .

We c a l l Io j t h e c u b i c i n t e r v a l w i t h t h e same c e n t e r zi as

Qj and f o u r t imes as b i g i n d iameter . L e t us c o n s i d e r t h e f u n c t i o n

L l ( x ) = IJ 1,. [ ( h ( z ) ( + I] I k ( x -2 ) - k (x -z i ) l dz

J J where p > 0 w i l l be c o n v e n i e n t l y chosen i n a moment. We have

11.2. CALDER6N - ZYGMUND OPERATORS 325

u s i n g c o n d i t i o n (iii) on k and t h e f a c t t h a t

We now s e t

i s extended over a l l i n d i c e s j such t h a t Osj i s e n t i r e l v where r con ta ined i n

{ z : I z - x I > E ( x ) } , t h e sum lz i s extended o v e r t h e rema in inq J

i n d i c e s , and

Now, i f x d I! g j we have J

1 s i n c e \ h ( z ) d z = 0 . So I 1'1 c L l ( x ) .

O j j

C

1-1 We s h a l l i n a moment a l s o show t h a t 1 G - 11 + L l ( x ) l .

J There fo re we s h a l l t hen have

3 26 11. SINGULAR INTEGRAL OPERATORS

and t h i s w i l l conclude t h e p r o o f o f t h e theorem.

C To show t h a t 1121c IX + L l ( x ) ( , we f i r s t observe t h a t i f J

1 IP31 > 7 I Q j I then

Thus

and so

and so

11.3. GENERALIZED HOMOGENEITY

Hence

so

327

where the last written sum is extended over all indices j such that

Qj intersects Iz : Iz-xI c E(x)I . But since x t 0 6 each such Q is contained in

j

j

3 E ( X ) } 2 I z : 1 2 - X I > - 1 (1 { z : IZ-XI c 2

and so, using condition (i) on k ,

11.3. SINGULAR INTEGRAL OPERATORS WITH GENERALIZED HOMOGENEITY.

The classical operators of the CalderBn-Zygmund type that we

have studied in the preceding Section have been generalized in different

directions. to a w l y the same methods of CalderBn and Zygmund to differential operators

of parabolic type.

The motivation for such generalizations was initiallv to trv

Such generalizations have proved later also very


u s e f u l i n o r d e r t o deal w i t h s p e c i f i c Droblems i n F o u r i e r a n a l v s i s where

t h e geometry i s o f a more i n t r i c a t e n a t u r e .

a p p l i c a t i o n s w i l l be g i v e n i n Chapter 1 2 .

An example o f such t y n e o f

The f i r s t g e n e r a l i z a t i o n s i n t h i s d i r e c t i o n appeared i n t h e

Dapers o f Jones [1964] , Fabes [1966] , Fabes and

Guzmdn [1968,1970 a, 1970 b ] , and o t h e r s ,

R i v i G r e [1966,1967] ,

Much o f t h e t h e o r y we a r e go ing t o developed runs p a r a l l e l t o

c l a s s i c a l one o f CalderBn and Zygmund once we have s e t o u r Droblem i n

t h e a p p r o p r i a t e geometr ic c o n t e x t . We s h a l l e x p l a i n i t f o l l o w i n q t h e

l i n e o f t hough t o f Guzmdn [1968, 1970 a l .

The problem we a r e go ing t o handle i s t h e f o l l o w i n g . L e t A

be a f i x e d n x n m a t r i x w i t h r e a l elements. Consider, f o r A > q, t h e

mapping

x e~~ -f T ~ X = e A1ogX E R n

The t r a n s f o r m a t i o n TI i s a s o r t o f d i l a t a t i o n ( f o r A = I , TAx = Ax)

I f we assume t h a t A has e igenvalues w i t h p o s i t i v e r e a l Dar t , t h e n we

have f o r each x e Rn - I01 , TAx + 0 as A > 0 and l T A x l -f as

A + - .

We s h a l l cons ide r k e r n e l s k : Rn - I01 + R s a t i s f y i n g , w i t h

r e s p e c t t o t h e d i l a t a t i o n s TI, an homogeneity n r o o e r t y s i m i l a r t o

t h a t o f t h e CalderBn-Zygmund k e r n e l s w i t h r e s p e c t t o t h e o r d i n a r y d i l a t -

a t i o n s , i . e .

k ( x ) -tr A k (TAx) = A

We s h a l l ask ou rse l ves whether i t i s p o s s i b l e t o get,from such

ke rne ls , convo lu t i on opera to rs t h a t s a t i s f y s i m i l a r theorems as those o f

Calder6n and Zygmund ob ta ined i n 11.2.

As one c o u l d expect , i t t u r n s o u t t h a t t h e t r i c k t o do i t

c o n s i s t s i n t r u n c a t i n g a p p r o p r i a t e l y such k e r n e l s (even t h e H i l b e r t

t r a n s f o r m f a i l s t o be a good o p e r a t o r

Such a t r u n c a t i o n i s determined by t h e d i l a t a t i o n s TA.

i f t h e t r u n c a t i o n i s n o t adecuate) .

I n o r d e r t o f i n d

11.3. GENERALIZED HOMOGENEITY 329

i t we s h a l l f i r s t observe t h a t t h e r e i s a m e t r i c P , i n v a r i a n t bv

t r a n s l a t i o n s , assoc ia ted i n a n a t u r a l way t o t h e m a t r i x A, which

behaves w i t h r e s p e c t t o t h e d i l a t a t i o n s

m e t r i c I - I 11.3. A,where we s h a l l examine some o t h e r n i c e p r o p e r t i e s o f t h i s m e t r i c

t h a t w i l l enable us t o prove i n a s t r o k e some usefu l theorems on apnrox i -

ma t ion i n 11.3.B and on s i n g u l a r i n t e g r a l o p e r a t o r s i n 11.3.C.

TA e x a c t l y as t h e Euc l i dean

behaves w i t h r e s p e c t t o t h e o r d i n a r y d i l a t a t i o n s , i .e.

p (TXx) = Ap(x) f o r each A > 0 and x E Rn , T h i s w i l l be done i n

11.3.A. The M e t r i c Associated t o a M a t r i x A.

I n t h i s s e c t i o n , A w i l l denote a f i x e d n x n r e a l m a t r i x

whose e igenvalues have p o s i t i v e r e a l p a r t . Fo r t e c h n i c a l reasons t h a t

w i l l be apparent l a t e r on we s h a l l assume t h a t t h e r e a l p a r t o f t h e

e igenvalues i s b i g enough (how b i g w i l l depend o n l y on n ) . T h i s w i l l

n o t l essen t h e g e n e r a l i t y o f t h e theorems on approx ima t ion and on s i n g u l a r

i n t e g r a l s we a r e l o o k i n g f o r , Fo r A > 0, TX i s t h e mapoing

x e R n + TXx = e A 1 o g X e R n

We s h a l l i n t r o d u c e t h e m e t r i c p assoc ia ted t o A i n t h e f o l l o w i n g way.

11.3.1. LEMMA. Fotl each x a R n -I03 thehe u unique

n u m b 0 p ( x ) , 0 < P ( X ) < ~0 , nuch t h a t IT-1 X I = 1 . Let ub b e t

d o p(0 ) = 0 . The duncaXon p : Rn + COY-) bo dedined b ~ m ! ~ h & L a

t h e doU0eolLling p o p a i i a :

-

P ( X I

Thenedotle , id we dedine p*(x,y) = p ( x - y ) , the^

- p* : Rn x Rn + LO,-) a m W c i n Rn f hcd d invahiant by a%~m& L L O V l A .

Phoud. For each x e R n - { O } we d e f i n e f o r X 6 (0,m)

The f u n c t i o n Ip : ( 0 , ~ ) + (0,m) s a t i s f i e s

The m a t r i x A t A*

Therefore, f o r each z 6 Rn - { O ) , (z,(A + A*)z) > 0 and so + ' ( A ) < O

f o r each X > 0. Hence +(A) i s s t r i c t l y dec reas ing on ( 0 , a ) . It i s

c l e a r t h a t $ ( I ) -f m as X + 0 and +(A) -+ 0 as X -+ m. Hence

t h e r e i s a un ique va lue p ( x ) > 0 such t h a t + ( p ( x ) ) = 1.

is a symmetric m a t r i x w i t h p o s i t i v e e igenva lues .

Tha t p(Tllx) = y p ( x ) i s a s imp le consequence o f t h e d e f i n i -

t i o n and o f t h e m u l t i p l i c a t i v e group p r o p e r t i e s of t h e d i l a t a t i o n s

x ) . I n f a c t A,X2

T A ( i . e . TI T1 x = T 1 2

Hence p(Tllx) = p p ( x ) . P r o p e r t i e s ( i i ) and ( i i i

o r d e r t o prove ( i v ) we f i r s t observe t h a t ( T -lx( A 1 i f p(T -1 x ) = p ( x ) < 1, i . e . i f and o n l y i f

x

X I = 1

a r e simDle. I n

L 1 i f and o n l v

P ( X ) 6 A. There-

f o r e i n o rde r t o show ( i v ) we o n l y have t o prove t h a t

We have

1


where I( P I( f o r a real n x n matrix P means the Euclidean norm of P as an operator on ( R ~ , I - \ ) , i . e , l l P I \ = max{(Pxl : 1 x 1 = 11 . I t i s not d i f f i c u l t t o show t h a t ( 1 PI1 = max E eigenvalues of P*P) ) 112

Now i t i s easy t o prove t h a t i f A has eigenvalues w i t h rea l par t b i g enough (how big depends only on n ) we have, f o r each 1-1 >r)

1 1 e - A u \ ~ c e-v A+A*

In f a c t ,

m i n (eigenvalues o f 7 ) a 1.

A have real pa r t big enough . Therefore we get ( 1 e-A’ll 6 e-’

1 1 e-Aull = max { eigenvalues o f e - I c e-u, i f A+A* B u t this i s so i f t he eiqenvalues o f

f o r p > 0.

So we obtain f i n a l l y

and hence p ( x + y ) c p ( x ) + p ( y ) .

We sha l l now s t a t e and prove some proper t ies of the metric p t h a t will be useful l a t e r on.

( i l Thehe LA a C O I L A ~ ~ ~ ci > 0 nuch t h a t id p(x ) c 1 (and eqlLiwaeentey 1x1 6 1) we have

(GI Thehe LA a constutant B > 0 nuch t h a t .id p ( x ) a 1 (and equivaeentey 1x1 > 1) we have


H e m c1 and $ depend o n l y on t h e mathix A.

Pltood. Let p(x) G 1. Then (recalling that for 'CI > 0

1 1 e-Ap/l 6 e-u )

On the other hand we have , i f = e -A log P(X) x,

with c1 > 0 depending only on A. Hence (p(x))" g 1x1.

Let now p ( x ) > 1. Then, i f X = e -A log P(X) I

and so p(.x) & 1x1. On the other hand

1x1 = leA log 2 1 s 1 1 e A log = max {eigenvalues of

for some $ > 0. Therefore (x/' 6 p(x).

Associated with the metric p and the dilatations TA, A > O

one can define in a natural way a system of polar coordinates. For any x e Rn - (01 we consider

-A 109 P(X) x = T (p(x))-1 = x e c

where 1 i s the unit sphere i n Rn


The m a m i n g x 6 R n - { O l -+ (:, p ( x ) ) 6 1 x ( 0 , ~ ) d e f i n e s

a system o f p o l a r c o o r d i n a t e s .

g r a l on R"

f o l l o w i n g way

I t i s n o t d i f f i c u l t t o see t h a t any i n t e -

can be expressed i n t h i s s y s t e m o f p o l a r c o o r d i n a t e s i n t h e

tr A-1

dp h ( x ) d x = h(eA l o g P - x ) I ( A i , i ) l d i p

Jx eRn ~ D = O JieL

Here dx means t h e o r d i n a r y Lebesgue measure on t h e u n i t sphere C .

To see t h i s i t s u f f i c e s t o l o o k a t t h e Jacobian o f t h e t rans -

f o r m a t i o n ( o r e q u i v a l e n t l y a t t h e exp ress ion o f t h e volume element i n

terms o f d? and dp ) .

11.3.8. A Theorem on APDrOXimatiOn.

L e t us r e c a l l Theorem 10.1.1. I f we have a r a d i a l f u n c t i o n

and i f we s e t , f o r E > O , x e R n k e L1(Rn) , w i t h k > 0 , /k = 1,

k E ( x ) = E - ~ k(--) X

Then t h e maximal o p e r a t o r K* d e f i n e d by

K * f ( x ) = sup l kE * f ( x ) 1 E>O

i s o f weak t y p e ( 1 , l ) and so kE * f ( x ) -f f ( x ) a t a lmos t each x e R n .

The same t y p e o f theorem and a l s o t h e same t y p e o f p r o o f i s

v a l i d i f we r e p l a c e t h e Euc l i dean m e t r i c by t h e m e t r i c p a s s o c i a t e d

t o t h e m a t r i x A o f t h e t y p e cons ide red i n 11.3.A. So we a r r i v e a t

t h e f o l l o w i n g r e s u l t .

11.3.3. THEOREM. L e / t @ e L1(Rn) ,I$ > 0 , .f@ = 1 and

@ ( x ) = @(y) id p ( x ) = p (y ) ( i . e . @ h "mLddi&" w&h t a p e & t o p 1. Fuh E > O and x e Rn be2 un dedine nuw

Then t h e maxim& 0p-u~ a* dedined by

.i~ 0 6 weak t y p e (1,1), 06 Q p e (m,m) ( and t h ~ ~ e & ~ ~ e 0 6 ba%ttlong t y p e

(p,p) 1 < p 01). Hence +E * f ( x ) - + f ( x ) at &ont each x e R n 9 doh each f 6 Lp (Rn) , 1 c p < m.

Observe t h a t i f A i s any m a t r i x w i t h e iqenvalues w i t h nos i -

t i v e r e a l p a r t and E z 0 , ~ > 0, E = rlH , H > 0 we have

T h i s a l l o w s us t o assume t h a t t h e r e a l p a r t o f t h e e igenvalues o f A i s

b i g .

The p r o o f o f t h e Theorem 11.3.3. r u n s parallel t o that of

Theorem 10.1.1. One has o n l y t o observe t h a t , for 110, t h e s e t

EX = I x s R n :

volume

a r r i v e proceeding as i n 10.1.1. t o

p (x ) L XI i s an e l l i p s o i d cen te red a t t h e o r i g i n of

Atr A I E I ( and t h a t t h e s e t s EX a r e nes ted convex s e t s . We

a * f ( x ) L c S f ( x )

where S i s t h e f o l l o w i n g maximal o p e r a t o r

Bu t s i n c e E,

Theorem 3.2.10., t h a t S i s o f weak t y p e (1 , l ) . The t y p e (m,m) i s

obv ious. So one o b t a i n s t h e theorem.

i s a f i x e d f a m i l y o f nes ted convex s e t s we know, by

11.3. GENERALIZED HOMOGENEITY

11.3.C. Genera l ized S i n g u l a r I n t e g r a l Operators .

335

Once we have t h e r i g h t way o f t r u n c a t i n g t h e k e r n e l we a r e

using,one can s t a t e a theorem o f t h e Calder6n-Zvgmund t v n e f o r t h e c o r

resnondinq s i n g u l a r i n t e g r a l o n e r a t o r s . We can do i t as i n 11.2.1.,

11.2.3., 11.2.3. For examnle we have t h e f o l l o w i n g r e s u l t s .

11.3.4. THEOREM. l& k : Rn + R be u @mtiun duck - t h a t

k ( x ) -tr A (ii) Fuh x # 0 , A > 0 , k (TXx) = A

(iii) Therre exi,&b c > 0 nuch t h a t h u h each y E R n

Tl- S i m i l a r statements f o r t h e s t r o n g t y n e (2,2) o f t h e maximal

o n e r a t o r and f o r t h e weak t y p e (1 , l ) of t h e maximal o p e r a t o r can be

ob ta ined .

The n r o o f o f t hese theorems can be performed p a r a l l e l t o t h a t

o f t h e corresDonding theorems f o r t h e c l a s s i c a l case. We s h a l l o m i t

here t h e d e t a i l s and r e f e r t o t h e worksqubted a t t h e b e g i n i n g o f t h i s

Sect ion.

Observe t h a t i f k s a t i s f i e s (i), (ii), (iii) w i t h a m a t r i x

A t h e n i t s a t i s f i e s t h e same p r o p e r t i e s w i t h t h e m a t r i x HA, H > 0.


There fo re we do n o t l o s e g e n e r a l i t v by assuminq t h a t t h e r e a l p a r t o f

t h e values o f A i s big enough.

CHAPTER 12

DIFERENTIATION ALONG CURVES. A RESULT OF STEIN AND WAINGER

I n Chapter 8 we have mentioned some problems i n d i f f e r e n t i a -

t i o n t h e o r y f o r whose s tudy t h e o n l y t o o l s a v a i l a b l e u n t i l t h e p r e s e n t

t ime a r e t h e ones which t h e r e c e n t F o u r i e r A n a l y s i s has developed.

I n t h i s Chapter we present , as a sample, one o f t h e i n t e r e s t i n g

problems s u c c e s s f u l l y handled w i t h such methods f i r s t by Nagel, R i v i s r e and

and Wainger [1974, 1976 a, 1976 b l and then more comp le te l y by S t e i n and

Wai nger [ 19781.

The s t r o n g l y geometr ic c h a r a c t e r o f t h e problem c o n t r a c t s

w i t h t h e a n a l y t i c a l s u b t l e t i e s o f t h e methods used here f o r i t s s o l u t i o n

I t would be v e r y i l l u m i n a t i n g t o have a good geometr ic unders tand ing o f

t h e s i t u a t i o n and t o o b t a i n a n i c e s o l u t i o n o f t h e problem i n terms o f

t h e usual c o v e r i n g p r o p e r t i e s t h a t a r e o r d i n a r i l y used f o r such problems

as those shown i n Chapter 6 th rough 8. Besides, such a t y p e o f s o l u t i o n

would p robab ly t a k e care o f t h e l i m i t i n g case, (What happens c l o s e t o

p = l ? ) , an open problem which t h e a n a l y t i c a l methods we a r e go ing t o

use cannot handle.

The problem we a r e go ing t o s tudy he re i s t h e f o l l o w i n g . L e t

( y l ( t ) , ..., y n ( t ) ) , t e [O,m), be a f i x e d con t inuous c u r v e y ( t ) =

i n Rn w i t h y ( 0 ) = 0 . For each x 8 Rn and f o r f 6 Lp(Rn) , 1 c p c m , l e t us cons ide r

Under what c o n d i t i o n s on f and y can one say t h a t t h i s

l i m i t e x i s t s and i s f ( x ) a t a lmost each x e Rn?

337

338 12. DIFFERENTIATION ALONG CURVES

O f course, i f f E $$ (R')), then t h e above l i m i t e x i s t s and i s

f ( x ) a t each x E Rn,

maximal o p e r a t o r i s o f weak t y o e

each f B Lp(Rn) and f o r a lmost each x E Rn.

So i f we a r e a b l e t o show t h a t t h e co r respond ing

(p,p) we o b t a i n t h e same n r o o e r t y f o r

As we s h a l l see, by means o f a c l e v e r s u b s t i t u t i o n of t h e max-

ima l ope ra to r , we s h a l l be a b l e under some c o n d i t i o n s on y t o n rove t h e

t y o e (2,2) by u s i n g t h e Parseva l -P lanchere l theorem. The tvDe (p,n)

f o r 2 < p 6 m i s t r i v i a l by i n t e r D o l a t i o n between 2 and m . For t h e

t y p e (p,p), 1 < p < 2 , one embeds ou r m o d i f i e d o o e r a t o r i n an a n a l v t i c

f a m i l y and u s i n g t h e theorem o f S t e i n on i n t e r n o l a t i o n f o r such a f a m i l v

one can o b t a i n t h e t y p e

d e t a i l t h e p r o o f of t h e t y p e (2,2) which i s e a s i e r .

t y p e (p,p), 1 < p < 2 , i s much more i n v o l v e d . We r e f e r f o r i t t o t h e

DaDer o f S t e i n and Wainger [1978] .

(p,p) , 1 o , y(0) = 0 , where v i s a f i x e d v e c t o r o f t h e u n i t . sphere o f Rn and

A i s one of t h e m a t r i c e s we have considered i n 11.3. w i t h e igenvalues

w i t h p o s i t i v e r e a l Dar t . I f we make t h e s u b s t i t u t i o n t = u , H > 0

HA lg v and so we can assume w i t h o u t loss t hen r(u) = y(u ) = e

o f g e n e r a l i t y t h a t t h e e igenvalues o f A have r e a l p a r t s t h a t a r e b i q

enough. Such a cu rve w i l l be c a l l e d homogeneous.

H

H

It i s easy t o r e a l i z e t h a t f o r t h e theorem we a r e g o i n g t o prove

i t is s u f f i c i e n t t o assume t h a t t h e c u r v e y ( t ) i s n o t con ta ined i n a

hyperp lane,

t h e theorem we l o o k f o r .

Otherwise t h e same r e s u l t f o r a l ower dimension g i v e s us

I n a n a t u r a l way we s h a l l need t o c o n s i d e r t h e m e t r i c P as- s o c i a t e d t o A . t h a t we have cons ide red i n 11.3. The p r o p e r t i e s we

have proved t h e r e w i l l be v e r y u s e f u l here.

For f B Lz (Rn) and x B Rn we d e f i n e

12.1. THE STRONG TYPE (2,2) 339

Mf(x) = sup i’ / f ( x - y ( t ) l d t E?O 0

I t i s not d i f f i c u l t t o see t h a t i f f E R n + E i s a measurable func t ion , the function

i s f o r almost each x E R n a measurable function of t and so the maximal oaera tor M i s well defined a t almost each x E R n ,

For M we sha l l prove the following r e s u l t

12.1.1. THEOREM. The maxim& V p Q h a t o h M dc6ined above A 0 4 b&Vng type (2,2).

Let us f i r s t proceed h e u r i s t i c a l l y in order t o understand b e t t e r the idea behind the Droof. Assume f r 0 and wr i t e , f o r brev i ty , f t ( x ) = f ( x - y ( t ) ) . inequal i ty

One could be tempted t o wr i t e , u s i n g the Schwarz

Therefore

I f we use the Parseval-Plancherel theorem, havina i n t o account t h a t

we get

- 2 . i r i ( c , Y ( t ) ) has alwavs which, of course, leads nowhere. The f ac to r e


modulus 1 and so we cannot expect a n y t h i n g f rom ( * ) .

L e t us t r y t o modi fy ou r scheme. L e t us cons ide r , i n s t e a d o f

E E I f t ( x ) d t

0

t h e f o l l o w i n g r e l a t e d means

We have, of course, Nhf(x)

observ ing the F i g . 12.1.1. we o b t a i n

2 M f ( x ) . But, on t h e o t h e r hand,

t

- 0

t

0

F i g u r e 12.1.1.

2h

E

1 E f t ( x ) d t dh >

TE

Hence M f ( x ) 6 - 1 sup iE Nhf (x )dh l g 2 E>O 0

We have now s u b s t i t u t e d f t ( x ) by Nhf(x) and perhans we a r e l u c k i e r

w i t h Nhf if we proceed a s before. We have n

Now

I f we c a l l

X > 0 , we can w r i t e

TXx = e A 1 g ' x , T; x = e A*lg ' x , f o r x e ~ n ,

2

A Nhf(5) = ./ e -2 Ti i (Tsh v *6 )ds ;(<) = f' e-2.rri(v,TETfi<)ds i(~)= 1 1

= n(T; 5 ) ? ( < I 2

hav ing se t , f o r y E Rn , n ( y ) e -2n i ( v ,T:Y) ds

1

- Now, proceeding as i n t h e p rev ious a t tempt , v i a Schwarz i n e q u a l

i t y and Parseval -P1 anchere l theorem, we have

I f we c o u l d prove t h a t f o r each 5 w i t h 1 6

i s t h e m e t r i c assoc ia ted t o A*) we have

and

then we c o u l d w r i t e

f o r h >, 1

f o r O < h s l

1 m m

I f we s p l i t t h e i n t e g r a l , use t h e two i n e q u a l i -

t i e s (*) and (**) and undo t h e change T*.q = 5 , i n each o f t h e 2J

terms o f 1 we g e t j

2 n 2 2

II M f l l c c II f l l = c II f l l 2’

b

a The i n t e q r a l s o f t h e t y p e e2.rri f ( u ) du have been s t u d i e d

by van de r Corput (C f . Zygmund [1959] v o l . I, D. 197) and by means o f

h i s r e s u l t s we s h a l l be a b l e t o prove t h e i n e q u a l i t y (*I. The o t h e r ine-

q u a l i t y (**) i s o b v i o u s l y f a l s e , s i n c e n ( T t 5 ) r+ 1 as h -f 0 u n i f o r m l y

i n 15 : But we have i n s t e a d by t h e mean v a l u e theorem,

i f r ) < h r l , 1 c p * ( S ) c 2 ,

1 G p* (< ) c 2 I .

where c i s t h e maximum o f t h e a b s o l u t e va lue o f t h e g r a d i e n t o f n on

t h e s e t

There fo re we have

i n 11.3. between 1 . 1 and p*.

In(T{ < ) - 11 G c hB , by t h e i n e q u a l i t i e s shown

So we can again t h i n k o f m o d i f y i n g o u r m a j o r i z a t i o n o f ( 1 Y f ( 1 2

by p e r t u r b i n g

preserves i t s behaviour f o r b i g h b u t has t h e d e s i r e d behav iou r f o r h

c l o s e t o 0.

Nhf by some o p e r a t o r so t h a t t h e F o u r i e r t r a n s f o r m

TO do t h i s A

Q(0) = = 1 , and

e t us t a k e a f u n c t i o n 4 such t h a t 8 @:(Rn) , $(<) = 0 i f p*(5) >, 1. We can then w r i t e , i f

x ) f o r h > 0 ,

,-.

There fo re , if we c a l l O* f ( x ) = sup * f ( x ) ( , we h>O

Now, acco rd ing t o 11.3.3, we know t h a t 1 ) @*fjI2 G c ) ) f

the f i r s t t e rm we have

have

2

1 1 2

)I2. To e s t i m a t e

Therefore , if 1 6 p * ( < ) 6 2, we have , f o r h > 1 , p*(T; 5 ) > 1

and so +h(E) = 0 , and f o r 0 < h G 1, 1 G p*(E) 6 2 , A

Therefore we have mod i f i ed Nhf as we needed and so we o b t a

now ( I Mf112 c l l f l 1 2 - L e t us w r i t e t h e scheme o f t h i s w o o f a l i t t l e more f o r m a l l y

n

Phoad ad Theahem. 12.1.1. Let f a 0 . Consider , f o r h > 0,

We choose $ such t h a t 6 V m ( R n ) , ;(<) = 0 i f p * ( < ) > 1 and -tr A

+(T x) and $(o) = I$ = 1 . W r i t e +,(x) = h h-

@ * f ( X ) = sup \+h*f(x) I h>O

We know t h a t I ( @*f((q 6 c I ( f ( 1 , We can s e t

Now

l a s t t e rm i s l e s s than o r equal t o ' '

h

But t h e f a c t o r F(h,<) = \n( (Tt 5 ) - + ( T i 5 ) i s such t h a t , i f

1 6 P * ( C ) 6 2 Y

( * I F(h,S) = In(T; < ) I c - , w i t h ~1 > 0 , f o r h a 1 ha

(**) F(h,<) c ch' w i t h B > 0, f o r 0 < h c 1.

The i n e q u a l i t y (*)

as we have i n d i c a t e d b e f o r e ,

w i l l be proved i n t h e f o l l o w i n g lemma. Hence,

and so M i s o f s t r o n g t y p e (2,2) as we wanted t o p rove .

I n o r d e r t o p rove t h e i n e q u a l i t y (*) we have used i n t h e p r o o f

o f t h e theorem we s h a l l u t i l i z e t h e f o l l o w i n g lemma o f van de r Corout .

12.1.2. LEMMA. C o a i d e h t h e i n t e g h a l -

J a

wlzehe f 0 a heal &nct ian .in kntl( [la,b]) and ~ n w n e Aha t tioh Oame

j, 2 6 j 6 n + l

we have

we have I f ( j ) ( u ) l > aj > 0 6uh each u e [a,b] . Then

12.1. THE STRONG TYPE ( 2 , Z ) 345

whme c j dependn o n l y o n j .

The p r o o f f o r j = 2 can be seen i n Zygmund El959 , v o l . I ,p.197]. For j > 3 t h e p r o o f i s ob ta ined i n a s i m i l a r way.

T h i s r e s u l t enables us t o p rove t h e i n e q u a l i t y (*) as f o l l o w s .

12.1.3. LEMMA . WLth t h e n o t a i i o n uned i n t h e Theohem 12.1.1. and phood, connididen i h e iM-tegkal

Pkaa/,. L e t us f i x X > 1 and s e t , f o r I < \ = 1 , s > 0 ,

Now cons ide r

We s h a l l prove

w i t h 6 > 0 and c > 0 independent o f S , t .

Observe f i r s t t h a t t h e c o n d i t i o n t h a t y ( s ) i s n o t con ta ined

i n an a f f i n e hyperp lane i s e q u i v a l e n t t o t h e f a c t t h a t t h e s e t o f v e c t o r s

n - 1 B = { V , A v , ..., A v 1


is a basis in R n .

In fact if y ( s ) = e A l g 'v is in the hyoerolane (x,w) = r)

with IwI = 1 (recall that y ( 0 ) = 0) we have

(eA I g s v,w) = 0 for s > 0

Differentiating and setting s = 1 we get

(v,w) = (Av,w) = .... = (A"'v,w) = 0

An-l and so B = { v,Av, ..., v 1 cannot be a basis. Conversely if B is not a basis there is some w, IwI = 1 , such that

(v,w) = (Av,w) = ... (A"' v,w) = 0

n-1 + If zn + c1z ... + c, is the characteristic polynomial of A we

get Hence

A" + clAn-' + ... + cn,'A -t cnI = 0 and so (Anv,w) = (An+lv,w)=...=D

(eAlgSv,w) = 0 for each s > 0

and y ( s ) is in the hyperplane (x,w) = 0 .

Now observe that

... + c z + c, i s the characteristic polynomial n-1 If zn + c p -t

of A we have

and so

12.1. THE STRONG TYPE (2 ,2 ) 347

If f o r some c o # 0 and some s o we have

then g ” ( s ) = 0 f o r a l l s and so A 5 0

(eAs v , A * 2 A C 0 ) = 0 , i . e . onal t o compactness of C x 10, l g 2 1, with respec t t o X , there must e x i s t a > 0 such t h a t f o r each (5 ,s ) 8 c x LO, l g 23

y ( t ) i s i n the hyperplane orthog- A** A S o . This i s a cont rad ic t ion , and the re fo re , using the

C ={5 6 Rn : 151 = 1 3 and the l i n e a r i t y

we have

Now,for each ( s * , s * ) B C x[Oy l g 21 t he re e x i s t s a natural number j , 2 < j 6 n + l and an open ball B* i n Rn+’ centered a t ( ~ * , s * ) such t h a t f o r ecah (s,s) B B* 0 ( C x [O, l g 21 ) = I we have

By compactness we cover s e t s the the projections

1 x [ O,lg2] w i t h a f i n i t e number of such I*. Let us consider their pro jec t ions over [ 0 , l g 21 and a l l consecutive closed in t e rva l s determined by the extreme points of H

The number H depends only on our matrix A and our cons t ruc t i o n , which i s f i x e d once f o r a l l .

If

then, according t o the previous lemma, we have f o r each 5 6 C , i f s . 6 t < sj+l

3

!or X > 1 w i t h 6 > 0 and c > 0 b o t h independent o f 5, i > l , and t > 0 So we have proved , f o r I n ] a 1 ,

w i t h c independent of t > 0 and n , I r i ( > 1 .

To f i n i s h now t h e p r o o f o f t h e lemma we w r i t e , f o r 1 < h < m y

and c independent o f h > 1 . T h i s completes t h e p r o o f o f t h e lemma.

12.2. THE TYPE ( p y p ) , 1 < p 5 OF THE MAXIMAL OPERATOR.

I n t h i s Sec t i on we s h a l l s t a t e some o f t h e r e s u l t s o f S t e i n

and Wainger 119781 concern ing t h e problem d e a l t w i th i n 12.1.

12.2.1. THEOREM. L e A y ( t ) be u homogeneoun C W L U ~ i n Rn . Then, do& f e Lp( Rn) , 1 < p c m ,

3 49 12.2. THE TYPE (P,d, P > 1

L e t y ( t ) , t > 0 , be a C U / ~ V Q w a h y ( 0 ) = 0 . Abbume t h a t y ( t )

dak 0 L t L 1 fie.b in t h e finean nubbpace bpanned by t h e vectutro

{Y(j)(O)I - That , .id j = i

we have , d o t f E Ln(Rn), 1 < n i -,

For a homogeneous cu rve y ( t ) t h e r e s u l t f o r n > 2 i s ob ta in -

ed by i n t e r p o l a t i o n between p = 2 (Theorem 12.1.1) and n = ( t r i v i a l ) .

Fo r 1 < n < 2

P roo f o f Theorem 12.1.1 by

S t e i n and Wainger embed t h e o n e r a t o r Nh, d e f i n e d i n t h e

i n a f a m i l y o f ope ra to rs N: , Rez < 0 , d e f i n e d by

For t h e opera to rs N; , i d e n t i f y i n g t h e i n v e r s e F o u r i e r t r a n s f o r m o f

(p*(<)) ' and u s i n g a v e r s i o n o f t h e Theorem 10.3.1 ( Z o ' s theorem)

corresponding t o t h e g e n e r a l i z e d homogeneity, t h e f o l l o w i n g r e s u l t ho lds .

*

12.2.2. LEMMA. Them exint 0, 0 > 0 buch t h a t , L d

-u <

w i t h a &buXe numben 06 vdeueh, Re z < -3 < 0 , a n e h a , dah any meauhabkk &not ion E: Rn + (0 , - )

whem I c (211 = O ( l z l H l a P

On t h e o t h e r hand,

one can prove t h e f o l l o w i n g .

12.2.3. LEMMA. L e R E : Rn -+ (0,~) be anv meanuhabde ,$unCtivn - w L t h a ~ i n i t e numben 06 v d u e b .

-a < Re z c1 one h a

Then t h m e ex&& c1 > 0 nuch t h a t i d

Using now t h e i n t e r p o l a t i o n theorem f o r an a n a l y t i c f a m i l y o f

ope ra to rs one o b t a i n s t h e s t r o n g t y p e (p,p) f o r z = 0 and consequent ly

t h e s t r o n g t y p e (p,p) , 1 < p L m f o r M.

There a r e many d e t a i l s and t e c h n i c a l i t i e s f o r which we r e f e r t o

t h e work o f S t e i n and Wainger E19781.

The r e s u l t f o r a em curve i s however an easy consequence o f t h e

p o s s i b i l i t y of app rox ima t ing such curves near t = 0 by a homogeneous

cu rve and o f t h e r e s u l t f o r these t y p e o f curves.

12.3. AN APPLICATION. DIFFERENTIATION BY RECTANGLES DETERMINED BY A FIELD OF DIRECTIONS.

I n 8.6. (PROBLEM 3 ) we have encountered t h e f o l l o w i n g t y p e o f

ques t i on . For each x 6 R 2 l e t d ( x ) e [O,n ) . Consider t h e c o l l e c t i o n

t?,(x) o f a l l c l osed r e c t a n g l e s c o n t a i n i n g x one of whose s i d e s has

d i r e c t i o n d ( x ) i . e . forms an ang le of ,ampli tude d ( x ) w i t h t h e h o r i -

z o n t a l a x i s .

We have a l r e a d y seen t h e r e t h a t such p r o p e r t i e s can be v e r y bad, even i f

d i s a cont inuous f i e l d o f d i r e c t i o n s .

can c o n s t r u c t a Nikodym s e t assoc ia ted t o some con t inuous

t h a t Q d does n o t even d i f f e r e n t i a t e a l l t h e c h a r a c t e r i s t i c f u n c t i o n s o f

measurable s e t s , i . e . i s n o t a d e n s i t y b a s i s .

Idhat a r e t h e d i f f e r e n t i a t i o n p r o p e r t i e s of t h e b a s i s 6,?

As we have seen i n Chapter 8, one

T h i s proves d.

I t i s t a n t a l i z i n g t h e f a c t t h a t even f o r t h e a p p a r e n t l y most

s imp le case of a n o n - t r i v i a l f i e l d o f d i r e c t i o n s d, namely t h a t o f t h e

r a d i a l d i r e c t i o n s , one d i d n o t know any answer t o t h e above ques t i on ,

w h e t h e r G d i s o r n o t a d e n s i t y bas i s , u n t i l t h e problem c o u l d be handled

by means o f t h e r e s u l t s o f S t e i n and Wainger. I n s p i t e o f t h e s t r o n g

12.3. DIFFERENTIATION BY RECTANGLES 351

geometr ic f l a v o u r o f t h e problem t h e o n l y t rea tmen t u n t i l now a v a i l a b l e

goes th rough t h e a n a l y t i c comp l i ca t i ons of t h e r e s u l t s o f 12.1. and 12 .2 .

I n t h i s Sec t i on we s h a l l f i r s t dea l w i t h some conc re te examples

and then we i n d i c a t e d how one can a l s o reduce some o t h e r s i m i l a r problems

t o t h e r e s u l t s o f S t e i n and Wainger. T h i s t y p e o f a p p l i c a t i o n o f such

r e s u l t s i s due t o A.Cbrdoba, C.Fefferman and R.Fefferman (unpub l i shed) .

The f i r s t theorem we p r e s e n t i s a s t r a i g h t f o r w a r d consequence

o f t h e r e s u l t o f S t e i n and Wainger. From i t we s h a l l deduce i n a n a t u r a l

way a theorem .about a p a r t i c u l a r b a s i s ' 8 d of r e c t a n g l e s of t h e t y p e

F i g u r e 12.3.1.

P(x,a,b) = C(xl + s 1 , x 2 + s 2 + s:) : 0 c s1 6 a , 0 ,< s 2 G b3

L e R UA dedine t h e 6uUuuing maxim& openCLtuh M . Fuh f E Lp( R2) 1 O IP(x,a,b)l JyeP(x,a,b)

Then M ~2 06 ,type (p,p) , 1 < p 6 m . Themdohe doh each f 8

1 < p 6 m one han dt &ubt each x E R 2

dY

LD ,


Pt laoa , We easily compute IP(x,a,b)( = ab and also, by a change

of variables b a

where we have called

But according to Theorem 12.2.1.

and (trivially from the one-dimensional result for the Hardy-Littlewood

operator)

Therefore

and the theorem i s proved.

Let us now transform the preceding theorem in a trivial way. In R’ let us perform the following transformation

Let us observe what i s the s e t T(P(x ,a ,b) ) . We have

T(P(x ,a ,b) ) = { ( x l t s l y xz + s2 + s1 2 - (X I+ s:)'): 06s16a OrSz<bb3

= ( ( X I + s1 , X 2 t s2 - 2slX11 : O f s 1 6 a 0 6 s z c b

i s a s t r a i g h t segment of fixed length and whose d i r ec t ion i s t h e t o f the vector (1 ,-2X1).

Therefore T(P(x,a,b)) i s a parallelogram with one pa i r of ver - t i c a l s ides and the other p a i r i n the d i rec t ion o f t he vector See Fig. 12.3.2. Observe a l so t h a t the Jacobian of T i s 1.

( l y - 2 X I ) .

Figure 12.3.2.

Hence j u s t by t r ans l a t ing Theorem 12.3.1. we ge t the following one.

3 54 12. DIFFERENTIATION ALONG CURVES

1 I f(Y) I dY T(P(xya9bnl jT(P(x,a ,b)) a>o,b>O ' Nf(X) = sup

Of course this theorem solves also the differentiability problem

for the basis of rectangles 8 the collection of all rectangles containing x one of whose sides has the direction of the vector (1,-2x1) .

such that, for each x e R' , 'A,(x) is

The same procedure can be used in many other cases. Let us now try to solve the differentiability question for the basis 0,. of rectangles in radial directions.

Let us consider now the maximal operators R 1 and Rz defined in & and f = T , (-x2 ,XI the following way . For x eRZ(O1, let x' =

i,e, ? is the unit vector i n the direction of x and is obtained by

rotating 2 around 0 an angle of , I f f e Lp( R 2 ) , x e Rz-tO] we

write

If we prove that RI and RZ satisfy

then we would have, if R 0gr(.Xl

12.3. DIFFERENTIATION BY RECTANGLES 3 5 5

There fo re i f

and so N would be o f t y p e (p,p) , 1 i p < 0 9 T h i s would g i v e us t h e

Theorem.

However t h e r a d i a l n a t u r e o f R1 l e a d s one t o suspect t h a t

perhaps some k i n d o f v e r y m i l d growth o f f near t h e o r i g i n can produce

a t o o s low decrease o f Rl f (x) f o r b i g 1x1 and t h i s can cause t h e

f a i l u r e o f t h e s t r o n g t y p e (p,p) . I n f a c t , if f i s t h e c h a r a c t e r i s t i c

f u n c t i o n o f t h e u n i t d i s c , R l f ( x ) > .l+lxl and so R 7 . f I$ Lp (R') , f o r 1 < p 6 2 T h e r e f o r e RI i s n o t o f t y p e (p,p) , 1 < p f 2 .

Never the less e v e r y t h i n g works as expected i f we t r y t o keep

away f r o m t h e o r i g i n , as we s h a l l now see.

L e t o u r space be E = {x 6 R2 : 10 < 1x1 < 100)

we a r e go ing t o c o n s i d e r w i l l have suppor t i n E.For f 6 Lp(E) , 1 < p < 0 0 , l e t us cons ide r , f o r each x E E,

The f u n c t i o n s

L e t us de f i ne t h e f o l l o w i n g t r a n s f o r m a t i o n T on t h e s e t E . I f t h e

p o l a r c o o r d i n a t e s o f x B E a r e p ( x ) , 10 < p(x1 i 100, and a ( x )

we s e t

I f f i s a f u n c t i o n i n Lp(E) and we s e t g(X) = f ( x ) t hen g i s i n

Lp(T(E)) and we have

where c1 and cp a re abso lu te constants . Observe now t h a t

There f o r e

Now we have

There fo re

I n o r d e r t o deal w i t h t h e o t h e r o p e r a t o r l e t us d e f i n e on E

t h e f o l l o w i n g t r a n s f o r m a t i o n V. I f x E E i s expressed i n complex

fo rm x = p (x ) e ia(x) , 10 < p ( x ) < 100, t h e p o i n t X = Vx w i l l be

t h e p r i n c i p a l va lue o f i t s l o g a r i t h m , X = l o g p ( x ) t i a ( x ) , i . e .

x G E -+ Vx = X = l g p(x) t i a ( x )

Again, i f f 8 LP(E) and we s e t g(X) = f ( x ) f o r X = Vx t h e n

g E Lp(V(E)) and

where E l , Z 2 > 0 a r e a b s o l u t e cons tan ts .

For f 8 Lp(E) and x E E we s e t

Observe now .that

Therefore

b2

= sup 2% I fiT 1g(X + log (1 t i s ) ) \ ds = S;g(X) O<bl , b z < l

According t o Theorem 12 .2 .1 . we have

From th i s results for R; , R; we easily see t h a t i f for f E L p ( E ) we se t a t each x E E

The restr ic t ion t o E i s of course rather irrelevant. As long as E i s bounded and i t s closure i s away from 0 we get the same resul t with different constants for the type ( p , p ) . Since differentiation i s a local property we can s t a t e the following resul t .

12.3.3. - THEOREM. The b u h t3 such t h a t , doh each

x 8 R2 - 10) , @ ,(x) x

h t h e c o U e d o n 06 UU h e d n g L e b containing

one 0 6 whobe hidides h t h e d i h e d o n 06 t h e ve.ctofi X di66e,kenLikte~

LP(-R') . 1 p d "

I t i s s t i l l unknown whether there will be some positive different iabi l i ty resul ts for the space L ( l + log' L ) (TIR2) for th i s basis.

CHAPTER 13

MULTIPLIERS AND THE HARDY-LITTLEWOOD MAXIMAL OPERATOR

L e t us cons ide r a bounded f u n c t i o n m E Lm(Rn) Fo r any f u n 2

t i o n f i n L2(Rn) we can d e f i n e t h e f o l l o w i n g o n e r a t o r T, means o f

t h e F o u r i e r t r a n s f o r m

" v T m f = (m f )

C1 e a r l y , by t h e Parseval -P1 anchere l theorem

and so T, i s a bounded l i n e a r o p e r a t o r f r o m L2 t o L2 w i t h a norm

bounded by 11 m l I m .

The f u n c t i o n m E La,(Rn) i s s a i d t o be a m u l t i p l i e r on Lp , 1 c p < m, when t h e oPera to r T,,, can be c o n t i n u o u s l y extended t o Lp

and t h i s ex tens ion i s bounded f rom Lp t o Lp.

There a r e some i n t e r e s t i n g r e s u l t s s t a t i n g s u f f i c i e n t c o n d i t i o n s

under which m

p l e , some es t ima tes on t h e s i z e o f t h e d e r i v a t i v e s o f t h e f u n c t i o n

Some o f such r e s u l t s , o r i g i n a t i n g i n Marc ink iew icz , can be seen i n

Zygmund [1959] o r S t e i n [1970] .

i s a m u l t i p l i e r . These c o n d i t i o n s can i n v o l v e , f o r exam-

m.

A t ype o f m u l t i p l i e r problem which a r i s e s i n a n a t u r a l way

when one dea ls w i th d i f f e r e n t manners o f sumino up t h e terms o f a mul-

t i p l e F o u r i e r s e r i e s i s t h e f o l l o w i ' n g . Assume t h a t m i s t h e charac-

t e r i s t i c f u n c t i o n xp o f a measurable s e t P. Can one s t a t e some

c o n d i t i o n s on t h e geometry o f P t h a t ensure t h a t xp i s an Lp-

m u l t i p l i e r f o r some p # 2 ?

360 13. M U L T I P L I E R S AND MAXIMAL OPERATOR

When P is a half-plane in R2 or a polygon with finitely many sides, then one can affirm that xp is an LP-multiplier for 1 < p <m.

These results are a direct consequence of the boundedness of the one-dimensional Hilbert transformin Lp(R1) , 1 < p < m,

In fact from the definition of the truncated Hilbert transform,

for f a L*(R)

we obtain

A

Therefore lim hc(E;) = c sign 5 and so E+O

$fE) = c sign 5 ? ( E l , where sign 5 =

-I i f E < O

I f X+ is the characteristic function o f [Op) we have

'jyn " and if T, is the multiplier operator cor- X+(d 2

responding to x+ we get

Therefore

T + f = $ ( H f t f)

and so T, is of type (p,p) 1 < p 4 m as H.

Noh if P c R2 is the closed first quadrant

13.0. INTRODUCTION 36 1

P = { ( x , y ) : x > 0 , y > 01 and Tp i s t h e m u l t i p l i e r o p e r a t o r as-

s o c i a t e d t o xp we have , f o r f 8 L2(R2) ,

Hence

'iy n(T:f)(x,q)dq = T:(T: f ) ( x , y )

where T i f ( x ,q ) i s ob ta ined f i x i n g f i r s t rl , c o n s i d e r i n g t h e f u n c t i o n

6 -f f (< ,n) , a p p l y i n g t o i t t h e o p e r a t o r T, and t a k i n g t h e v a l u e a t x. P And s i m i l a r l y T:. It i s now easy t o see t h a t xp i s an L - m u l t i p l i e r

f o r l < p < m .

S i m i l a r l y , i f P i s any h a l f p l a n e , we see t h a t xp i s an

L P - m u l t i p l i e r , t h e s t r o n g t y p e cons tan t (p,p) be ing t h e same f o r a l l

h a l f p l a n e s . A lso, i f P i s any convex polygon w i t h f i n i t e l y many s i d e s .

But even i n t h e case t h a t P i s t h e u n i t c i r c l e i n R 2 t h i s

q u e s t i o n was open f o r l o n g t i m e u n t i l C.Fefferman [1971] s e t t l e d i t

p r o v i n g t h a t i f D i s t h e u n i t c i r c l e i n R 2 , m u l t i p l i e r i f p # 2. The s o l u t i o n o f t h i s problem i s c l o s e l y con-

nected w i t h t h e theorems we have developed i n Chapter 8.

r e s u l t s ob ta ined t h e r e w i l l p e r m i t us t o go a s t e p f u r t h e r t o t r e a t

n i t e l y many s ides .

a subsequence o f t h e s ides o f P converge " s l o w l y " t o a f i x e d d i r e 2

t i o n , t hen we can l i k e w i s e a f f i r m t h a t

f o r any p # 2.

xD i s n o t an Lp-

Some o f t h e

w i t h t h e same techn ique t h e case i n which P i s a po lygon w i t h i n f i - I f we impose t h e c o n d i t i o n t h a t t h e d i r e c t i o n s o f

xp i s n o t an L P - m u l t i p l i e r

T h i s t y p e o f thecrem has o f course t h e f l a v o u r o f some o f

t h e r e s u l t s on d i f f e r e n t i a t i o n we have d iscussed i n Chapter 8.

f a c t t h e y o r i g i n a t e f rom t h e same r o o t , namely t h e p o s s i b i l i t y o f

c o n s t r u c t i n g an adequate Perron t r e e such t h a t t h e s i d e s o f i t s smal l

t r i a n g l e s have t h e d i r e c t i o n s i n ques t i on .

I n

And again, t h e same t y p e

362 13. MULTIPLIERS AND MAXIMAL OPERATOR

of problems that are still open about the possibility of differentiating

with respect to the rectangles with directions in a given set of directions can be formulated in this context. We shall later mention some of them.

These connections lead one to suspect that the multiplier oper-

ators of this type and the generalizations of the Hardy-Littlewood maximal operator that are the key to understand the differentiability problems are

also deeply interrelated. light in a recent paper by CBrdoba and R.Fefferman [1977] .

That this is in fact so has been brought to

In this Chapter we first present in 13.1. the negative result of C.Fefferman [1971] about the unit disk. In 13.2. we shall examine some more negative results that can be obtained with the same techniques

about the case of a polygon with infinitely many sides, Finally in 13.3.

we present some of the results of CBrdoba and R.Fefferman on the relation

of multiplier and maximal operators. The results of 13.3 are more general

and contain, at least partially , those of 13.1. arnd 13.2. However they are more sophisticated and the whole Chapter will be more easily and pleasantly read if we present it the way we do.

13.1. THE CHARACTERISTIC FUNCTION OF THE UNIT DISK. A THEOREM OF C. FEFFERMAN.

The theorem we are going to prove is the following.

13.1.1. THEOREM. The ckwracAW;tic 6unCtiCIn v b t h e u n i t didk

D in R2 .in naA an l p - m W p f i ~ doh any p i 2.

l ? t o o ~ , Let T he the multiplier associated to XD , i.e. for

f I3 L 2 W )

Tf = ( xD f y

I f f, g 4 L~(R~) we have

13.1. A THEOREM OF C. FEFFERMAN 363

and so i f T i s an L P - m u l t i p l i e r , p # 2, and p ' = JL , we have P - 1

i . e . T i s a l s o an L P ' - m u l t i p l i e r , o b v i o u s l y w i t h t h e same norm. (These

c o n s i d e r a t i o n s are, o f course, a p p l i c a b l e t o any o p e r a t o r T, assoc ia ted

t o t h e m u l t i p l i e r m ) .

The re fo re , i n o r d e r t o prove t h e Theorem we can assume p > 2.

The p r o o f of t h e Theorem i s a combinat ion o f t h e t h r e e f o l l o w i n g

obse rva t i ons ;

(a ) (Th is o b s e r v a t i o n had been made by Y.Meyer p r e v i o u s l y t o

t h e p r o o f o f C.Fefferman). I f we d i l a t a t e any d i s k by a homothecy w i t h

cen te r a t any o f i t s boundary p o i n t s and w i t h r a t i o o f homothecy b i g enough

we o b t a i n a b i g d i s k which i s app rox ima te l y an a f f i n e h a l f p l a n e whose bor-

de r can have any d i r e c t i o n .

d i s k i s an L P - m u l t i p l i e r , 2 < p < my

remark, one can p rove t h a t i f L a r e t h e hyperp lanes

L . = { x e R 2 : (x ,v . ) 2 01, where v j = 1,2,...,k a r e u n i t v e c t o r s

and T . f . = ( X

o f f u n c t i o n s i n L (R ) one has

I f t h e c h a r a c t e r i s t i c f u n c t i o n o f t h e u n i t

w i t h norm c u s i n g t h i s t r i v i a l P'

j

J $1 j y 8.1' then, f o r any f i n i t e sequence fj, j = l , , , . , k

J J Lj I, J 2

... ( b J I f R j i s any r e c t a n g l e i n d i r e c t i o n v and R i s t h e

j j s e t i n d i c a t e d i n F i g u r e 13.1.1. then, i f f = x , one has

j R j 1

j . IT^ f j ( x ) l lQOD if x e 6

T h i s easy f a c t i s reduced t o a s imp le one-dimensional computa-

t i o n .


Figure 13.1.1.

(c ) For any small number q > 0 the re exists a measurable s e t k E c R 2

j o i n t , with the following proper t ies : and a f i n i t e co l lec t ion of rectangles {Rj}j,l , pairwise dis-

- where R denotes again t h e shaded portion o f f igu re 13.1.1.

j

This is j u s t t he Lemma 8.2.1. t h a t we have eas i ly obtained from the construction of the Perron t r e e .

With this observations ( b l and’ (c ) We e a s i l y compute, s e t t i n g f j = x ,

Rj

On the other hand, i f p > 2 , using now the

R j I

observation ( a )

13.1. A THEOREM OF C. FEFFERMAN 36 5

c ( u s i n g H o l d e r ' s i n e q u a l i t y w i t h exponents f and i t s con juga te ) 6

There fo re

and t h i s i s a c o n t r a d i c t i o n if ?-I i s sma l l .

L e t us f i n a l l y p r e s e n t t h e proof o f t h e o b s e r v a t i o n (a) i n

t h e f o l l o w i n g two lemmas.

13.1.2. LEMMA. L e A S be any f i n e m openatoh bounded @om

I I ( c" j = l

Phoaa. L e t

x = ( i L i Z Y . . . , i k ) E 1 c

p < -, w&h nomi c, . Then we have, don any

k

j = 1

lk-l = 1 . We have, f o r w E Q ,

be t h e u n i t sphere

2)1'211

o f Rk and

Since S i s bounded f rom LP(Q) t o LP(Q) we can w r i t e ,

f o r each i a 1 ,


If we write

the previous inequality is

Observe now that if a

with c independent o f

s any vector in Rk - to1 then

ct . Therefore, if we (*) over C and change the order o f integration

P c 1 ISF(w)l dw 6 c cp I F ( w )

R k p n

This can be written

ntegrate the inequality we obtain

dw P k

and this is the statement of the lemma.

- 13.1.3. LEMMA. Adbume t h a t t h e mlLetipfieh o p u ~ ~ ~ A o h T a6

Aociated t o ,the u n i t didh D Aatib6Lie6 1 1 Tf 1 1

1eA Lj 6e t h e haR6-plane

rS Cp/I f \Ip , 604 dome P p 3 2 . L e A IVj3 j=l 6e a @nite Aequence 06 u n i t veotohs i n R2.

13.1. A THEOREM OF C . FEFFERMAN 36 7

L . = x s R 2 : ( x , v . ) B 0 1 J J h

and dedine. T .fi( XL f ) ' J j

k Then, 6uh any bequence. Efj}j-l i~ Lp(R2) we have

r Phoud. Let D j be the c i r c l e of center r v and radius r > 0. j For big r , the s e t D r looks very much l i k e L j . Define T': by means o f T i = ( X

as r + m . In f a c t , i f f e (&o,we have sense. This permits us t o s h i f t the problem t o the operators T r I t wi l l s u f f i c e t o prove

j J T j

^f f V . I t i s t o be expected tha t T': will approach D; J

T i + T . f i n every des i r ab le J

j *

independently of r . na tes , t h a t

Now i t i s easy t o e s t a b l i s h , by a change of coordi

Therefore, i f (**) holds w i t h r = 1 i t holds f o r any r > 0 .

Now observe t h a t

Therefore we have

368 13. MULTIPLIERS A N D MAXIMAL OPERATOR

I f we now apply Lemma 13.1.2. one.

we conclude w i t h the proof of t he present

13.2. POLYGONS WITH INFINITELY MANY SIDES.

We have seen t h a t i f D i s the u n i t disk then xD i s not an LP-multiplier f o r any p # 2 . On the o ther hand i f J i s ha l fp lane , then xJ i s an LP-multiplier. Therefore, i f P i s a polygon t h a t can be expressed as J1 1'1 J Z 0 ... 1'1 J k where J a r e halfplanes and i f T p i s the mul t ip l i e r operator corresponding t o P and T I ,TZ ,.. . 'Tk

j

those corresponding t o J1 ,J2 ,.. . , J k , we have

'J1 'Jz . . I

A

f = ( T l T 2 . . . T k f )" x J k

Therefore T p = T I T 2 ... T k and X p i s a l so an LP-rnultiplier, l < p < m .

Assume now tha t P

(if ; + 0

i s a polygon with i n f i n i t e l y many s ides i n the sense t h a t

( i i ) For any two J . , J , the border of J i s not pa ra l l e l J k j t o the border Of J k

( i i i ) For each J j a P 1) a J j i s a segment of pos i t i ve length

I t i s easy t o cons t ruc t s e t s P of th i s type, even comac t convex s e t s of this type.

T- > $ j > 0 , $ , G 0 2 J i n those d i rec t ions a s indicated in Figure 13 .2 .1 .

For example, given any sequence of angles { $ . I ,

one can cons t ruc t a polygon P w i t h s ides J

13.2. POLYGONS WITH MANY SIDES

f $1

369

Figure 13.2.1.

The question is now whether xp for such a set, which in some sense is something between a disk and an ordinary polygon with finitely many sides will be an LP-multiplier for some p, 1 < p < m.

Positive results for some types o f sets P of this form will be obtained in the following Section 13.3,

If one l o o k s at the proof of Theorem 13.1.1. with the inten - tion o f obtaining a negative result for sets

inmediately observes that the observation (a1 is valid without any P of this class one

substantial modification.

In fact, if xp 2 c p < , and if T~ is of the side aJj 0 aP o f

LP-multiplier with the same

orthogonal to aTj I1 a P d

P ' s an LP-multiplier with norm c the translation that carries the midpoint

P to the origin, the also X-r.P is an

constant. If we call v the unit vector rected towards the interior of T . P and

J j

J

3 70 13. MULTIPLIERS AND MAXIMAL OPERATOR

L . = C x B R2 : (x,v j ) > 0 1 then, s e t t i n g J

we have, f o r any sequence i f j } o f f u n c t i o n s i n Lp(R2) , e x a c t l y as

i n Lemma 13.1.3,

I f we can c o n s t r u c t f o r t h e f a m i l y o f v e c t o r s ( v . 1 a c o l l e c J t i o n o f r e c t a n g l e s s a t i s f y i n g t h e p r o p e r t i e s of t h e o b s e r v a t i o n

t h e n we o b t a i n a c o n t r a d i c t i o n as t h e r e . ( c ) ,

T h e r e f o r e , i n o r d e r t o o b t a i n a n e g a t i v e r e s u l t f o r P, i . e .

i s n o t an L P - m u l t i p l i e r f o r any p # 2 , i t w i l l be s u f f i c i e n t t h a t xp t o prove t h a t g i ven t h e s e t o f d i r e c t i o n s f v j l , o r , what amounts t o

t h e same, t h e s e t o f d i r e c t i o n s o f t h e s i d e s o f P , f o r any TI > 0

one can c o n s t r u c t a measurable s e t E and a f i n i t e c o l l e c t i o n o f d i s -

j o i n t r e c t a n g l e s fRh)h=l , each Rh w i t h one s i d e i n d i r e c t i o n vh

so t h a t

k

2 1 -

100

as i n t h e obse rva t i on ( c ) .

One e a s i l y sees, j u s t l o o k i n g a t t h e way we have o b t a i n e d

Lemma 8.2.1.

a Perron t r e e i n t h e sense o f 8.1. w i th i t s sma l l t r i a n g l e s i n t h e

d i r e c t i o n s of some o f t hose o f 1 v . j J i s one o f t h e m o t i v a t i o n s f o r t r y i n g t o g e t d i f f e r e n t t ypes o f Pe r ron

t r e e s .

from t h e Per ron t r e e i n 8.1., t h a t i f we can c o n s t r u c t

t hen we g e t what we need. T h i s

We can s t a t e , as a sample, a theorem o f t h i s n e g a t i v e type,

deduced f rom t h e s p e c i f i c Perron t r e e we have c o n s t r u c t e d i n 8.1.1.

13.3. A THEOREM OF A. C~RDOBA AND R. FEFFERMAN 371

13.2.1. THEOREM. LeX u6 c o a i d m t h e bequence 06 chkecaXua

0 4 t h e t y p e cuuznide/ted i n tkin neot ion w d h one bide i n & not an LP-mUpUm 6uh any p f 2 .

1 J J

d e L m i n e d by 4 . = ( ~ ~ d i a n ~ ) (See Fig. 1 3 . 2 . 1 . ) . 1eA UA con~;Dtuc t any paCygon P each ClihecaXon . Then X p

13.3. THE MAXIMAL OPERATOR RESPECT TO A COLLECTION OF RECTANGLES. A THEOREM OF A. CORDOBA AND R. FEFFERMAN.

As we have seen in the preceding Section, from one single

fact, namely the possibility of constructing a Perron tree such that one side of its small triangles is in a fixed set of directions {v.},

J we have been able to deduce, on one hand, the bad properties of the

differentiation basis @ of all rectangles in directions {vj} and,

on the other hand, the bad continuity properties in Lp , I) # 2, of the multiplier operator T p associated to any polygon P with infinitely many sides i n directions Ivj} ,

The question that now arises in a natural way is whether we can say something positive, i.e., is it true that if 8 has good dif ferentiation properties then Tp is a good multiplier operator and

viceversa? The following result, due to C6rdoba and k.Fefferman [1977]

gives an affirmative answer to this question for a particular P.

Let us give ourselves a sequence of angles te,) , n J

0 < 8 . < 2 , o j + 0 , and let P be the convex set indicated in

Figure 13.3.1. J


A o

- 0 2 2 2

P

2 3

Figure 13.3.1.

A 3

The p o i n t A, i s an a r h i t r a r y point of x = 1. The other ve r t i ce s o f P a r e the points A j , j = 1,2, ... Each A j i s the point of x = ZJ such t h a t Aj,l A forms w i t h Ox an angle of amplitude e We have indicated the midpoint E j of Ajvl A j and the inner u n i t normal v t o P a t E

j * 3 . j

O n t he other hand l e t @ be t h e d i f f e r e n t i a t i o n bas is of a l l rectangles with one s i d e in one of t h e d i rec t ions { v j l .

We can s t a t e the following r e s u l t .

13.3.1. rHEOREM. LeA P be t h e deA ju - t dedined and @ t h e d . i d { ~ e n t i a t i o n ba4i.A denchibed above.

13.3. A THEOREM OF A. C~RDOBA AND R. FEFFERMAN

f E L(R2),

Then:

(a)

(6)

7 6 K O a6 hfhung type ((!)',(f)')

16 Xp O an Lp-m&2pfieti doh home p > 2 and i d we have,

nome P > 2,

then xP o an LP-mlLetiptieti

doh each mmuhabbe E C R 2

1 ICx e R2 : KXE(x) > 11 L C I E ~ then K O 0 6 weah @pc? (($1' y(5) ' )

Observe that the additional condition in (b) as we know,

holds when 6 i s known to be a density basis.

For the proof of (a) we shall use two important theorems.0ne

.is the following result of A.Co?doba and C.Fefferman [1976] .

13.3.2. TffEOREM. LeL H denote t h e H a b c k t a?~~n.hdohm and

M t h e ohdinahy fflvrdy-LLttkkwvod m a x h d o p ~ ~ n d t o h i n R'. LeL

f 6 Lp(R1) ,

1+E

g 6 Lp(R') , 1 O huch t h a t

p > 1 , LeL

Then

whme cE 0 independent 06 f and g . Observe that the fact that M is of type (p,Pl, P > 1, im-

plies that also ME is Of type (p,P)..

373

The other result we shall use i s the following theorem of

Stein [1970, Paley and Littlewood that can be seen, for instance in

p. 1041

13.3.3. THEOREM. L& ( V h j = 1 ,2 ,3 ,... E . = C(x,y) 6 R 2 :- x < 2J ) , and LeA: Sj be t h e rni&XpL&h apen_atah cornenpunding t o

J h

E j , i . e . ( S j g)* = x . g . EJ

Then, doh each g E Lp(R2) , 1 < p < m , we. have

Wi th these two r e s u l t s t h e p r o o f o f Theorem 13.3.1. i s easy.

Phaad v d t h e Theohem 13.3.1. For p a r t ( a ) we proceed as f o l -

lows. L e t F j be t h e a f f i n e h a l f p l a n e determined by Aj-l A j t h a t

c o n t a i n s P. L e t H j be t h e m u l t i p l i e r o p e r a t o r co r respond ing t o Fj,

i . e . ( H ~ g I A = x G. FJ

Observe t h a t i f T i s t h e m u l t i p l i e r assoc ia ted t o P , we P

have, f o r each j = 1 , 2 , . . .

i .e. S j Tp = H j S j .

By Theorem 13.3.3. we have

m m

II T p f I l F c II( c I S j Tp f 1 2 ) 1 / 2 1 1 ; = c I I ( E l H j s j f l 2 I 1 / 2 I I p

$1

j=1 j = 1 (5) I I n o r d e r t o e s t i m a t e t h e l a s t t e rm we examine, f o r @ 6 L (R2) , w i t h

I l @ I l 6 1 , $ 3 0

m m

1 R 2 j=1 j= l R ( 1 ( H j S j f l ' ) @ = 1 I I H j S j f 1 2 4

Now f o r each j we compute A = l l H j S j f I 2 @ b y l i n e s i n t h e

d i r e c t i o n o f v j , i . e . i f i s t h e u n i t v e c t o r ob ta ined by r o t a -

t i n g v an ang le o f

j

71 j j

Using now the

jseR

13.3. A THEOREM OF A. C6RD05A AND R. FEFFERMAN

r

I H . S . f ( s v t?.)12 @ ( s v j y t V . ) d s d t JseR j' J J

heorem 13.3.2. we have f o r almost each t e R

H . s . f ( sv . , tv . )12 @ ( s v j , t i . ) d s 6 J J J J J

375

) S j f (sv tG. ) I2 ME@ (sv t i j ) ds j 7 J j 7

where we have taken ME with respec t t o s . Therefore

where M:$ means the operator defined above b u t now on the l i n e i n direc- t ion v passing through (x,y) . Now i t i s easy t o see t h a t t he f a c t t h a t K i s of type

j ((f)',($)') implies t h a t t he operator M Z

MZ g(x,y) = SUP M: ( X , Y ) j

Therefore we ge t

and so T i s o f type ( p , p ) .

T h i s conclude the proof of par t ( a ) . We now prove ua r t (b) .

Proceeding exactly as we d i d i n Lemma 13.1.3, from the f a c t i s and LP-multiplier f o r p > 2 we a r r i v e t o the inequality, t h a t Xp

376 13. M U L T I P L I E R S AND MAXIMAL OPERATOR

for f . a Lp(R2) , j = 1,2,3, ... J

where T is as there, the multiplier operator associated to

L . = Cx a R 2 : (x,vj) 2 0 1 . j J

By means of this inequality we are going to obtain a covering lemma for the rectangles of @ from which the type

operator K is an inmediate consequence. ((:)I y(f)') of the

h Assume we are given a finite collection C R k l k = l of rectangles of the basis 8 . Let us assume they are ordered so that

- - Let us choose R1 = R 1 . The RZ will be the first Rk with

k > 1 satisfying

- - If Rk,l = R j , then Rk will be that Rh with h > j such that

3 k - 1 ,., lRh - 1' Ril ' F lRhl

- and so on, Thus we obtain {Rk3E=1 so that

Y -

On the other hand, i f R is not i n j s -

I R j - IJ R k \ k = l

that is ,

3 "if

13.3. A THEOREM OF A. CI~RDOBA AND R. FEFFERMAN 377

and t h e r e f o r e

Hence we have, acco rd ing t o t h e c o n d i t i o n we have assumed on K i n (b )

Assume t h a t R k i s i n t h e d i r e c t i o n o f v . and l e t J ( k )

- - - - - Ri = R k , R i , R i , R k be t h e r e c t a n g l e s i n d i c a t e d i n F i g . 13.3.2.

F i g u r e 13.3.2.

.., -. 5- - 1 " L e t us now s e t Ek = R k - t I RJ . S ince I E k l p l R k l

j < k we have t h a t a t l e a s t f o r h a l f o f t h e l i n e s 1 between A and B i n

d i r e c t i o n Y ~ ( ~ ) we have

( I 1 1 , means one-dimensional measure)

computat ion ( t h e same as i n (b) o f Theorem 13.1.1) we have t h a t f o r a t l e a s t h a l f o f t h e l i n e s 1 between A and B

and so , an easy one-dimensional

378 13. MULTIPL IERS AND MAXIMAL OPERATOR

- L e t us c a l l F k t h i s s e t o f p o i n t s o f Rk where t h e above h o l d s ,

i s t h e m u l t i p l i e r o p e r a t o r co r respond ing t o If ' j ( k )

= { x € R 2 : ( x , V ~ ( ~ ) ) > 0 1 we have, s i m i l a r l y , 'j ( k f

1 I ' j ( k ) 'Fk (')I ' -

a t each p o i n t o f R i . Therefore, f rom t h e i n e q u a l i t y (*) we g e t

If we now submi t t h e s e t s 6; t o t h e same k i n d o f process we have used

on t h e s e t s ik we a r r i v e t o t h e s e t s ik and so

There fo re

So we have ob ta ined f rom

t h a t ERklk=l a f i n i t e sequence {?i,lt=, such

Bu t t h i s i s known t o i m p l y ( c f . Chapter 6 ) t h a t K i s o f weak t y p e

((!)I, ($1') and p a r t (b) o f t h e theorem i s proved.

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A LIST OF SUGGESTED PROBLEMS

On t h e B e s i c o v i t c h cove r ing theorem.

on t h e dimension i n 3.2.1.? 3.2.A. How does t h e cons tan t depend

On t h e method of r o t a t i o n .

t ype i nequal i ty?

From t h e maximal o p e r a t o r t o cove r ing p r o p e r t i e s . 6.3. What a r e t h e exac t

l i m i t s o f Theorem 6.3.1. i n t h e sense o f p.117?

5.3. Can one make i t work f o r o b t a i n i n g a weak

The h a l o problem. 6.6. f i n d o u t a min imal c o n d i t i o n on f E Lloc(Rn) i n o r d e r t o ensure t h a t t h e

b a s i s o f d i f f e r e n t i a t e s I f .

On a theorem of B e s i c o v i t c h . 7.4. How f a r can one extend Theorem 7.4.1. by

t a k i n g as 8 a more genera l b a s i s ?

A problem o f Zygmund. 7.8. Can Theorem 7.8.1. be g e n e r a l i z e d t o more

dimensions?

Knowing t h e h a l o f u n c t i o n o f a d i f f e r e n t a t i o n bas i s ,

On t h e d i f f e r e n t i a t i o n p r o p e r t i e s o f some bases o f r e c t a n g l e s .

Problems 1,2 ,3 , on pages 226-227.

On t h e Nikodym s e t and d e n s i t y bases.

assoc ia ted t o a L i p s c h i t z f i e l d o f d i r e c t i o n s i n R a d e n s i t y bas i s?

On approx imat ions o f t h e i d e n t i t y . 10.2. Is t h e maximal o p e r a t o r a s s o c i a t e d

t o a k e r n e l k z 0, k E L(Rn), k non - inc reas ing a long rays , o f weak t y p e ( l , l ) ?

On d i f f e r e n t i a t i o n a long curves.

o f t h e usual cove r ing methods o f d i f f e r e n t i a t i o n ) o f theorems such as

12.1.1.

8.6. See

8.6. Is a b a s i s o f r e c t a n g l e s 2

12. To f i n d a geometr ic p r o o f ( i n terms

On m u l t i p l i e r s a s s o c i a t e d t o polygons w i t h i n f i n i t e l y many s ides .

f i n d t h e l i m i t s o f t h e theorems i n 13.2. and 13.3. 13. To

389

INDEX

Besicovitch covering lemma, 39

Besicovitch covering property, strong, 107 weak, 109

Besicovitch set, 210, 277 Busemann-Feller (B-F) differentiation

basis, 104

CalderBn-Zygmund decomposition, 46

CalderBn-Zygmund operators, 313

Condensation of first order, 259

Condensation of second order, 259

Convex sets, covering lemma, 48

Convolution operators, 73

CBrdoba and Hayes theorem, 114

Van der Corput lemma, 344

Cotlarls lemma, 92

Density basis, 118

Density property, 118

Derivative, upper, lower, 105

Differentiation, 105

Differentiation along curves, 337

Differentiation basis , 104

Extrapolation, 60

C. Fefferman ' s mu1 ti pl ier

General i zed homogeneity , integrals, 327

theorem, 362

Halo function, 149

Halo problem, 149

Hardy-Littlewood operator,

Hausdorff measure, 242

Hilbert transform, 305

41, 103, 104

Identity approximations , 281 Interpolation, 54

Irregular point, set, 250

Kahane set, 212

Kakeya problem, 209

Kolmogorov condition, 50

Linear density, 245

Linearization, 66

Linearly measurable sets, 241

Majorization, 70

Marcinkiewicz theorem on

Marstrand theorem, 177

Maximal operator, 5

Multipliers, 359

i nterpol at i on, 55

Nikishin's theorem, 29

Nikodym set, 215 singular

392 INDEX

Perron t r e e , 201

P o l a r l i n e s , 268

P r o j e c t i o n p r o p e r t i e s , 258

Rademacher f u n c t i o n s , 23

Rad ia l k e r n e l s , 282

R a d i a t i o n p o i n t , 259

Regular p o i n t , se t , 250

R iesz -Thor in theorem, 54

R o t a t i o n method, 96

Saks r a r i t y theorem, 165

Sawyer ' s theorem , 19

S t e i n ' s theorem, 23

S t e i n and G. Weiss theorem, 55

Stein-Wainger theorem, 337

Sub l i nea r ope ra to r , 7

Summation, 68

Tangency p r o p e r t i e s , 252

Tangent, 252

Type of an opera to r , 13

V i t a l i c o v e r i n g lemma, 108

V i t a l i cover , 137

Whi tney 's lemma, 44

Y-set, 256

Zo ' s theorem, 292

Z-set, 256

real variable methods in fourier analysis (north-holland mathematics studies 46)

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