Real Variable Methods in Fourier Analysis (North-Holalnd 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 o r by any means, eleclronic, mechunical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 86124 6

publisher^:

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM'NEW YORK O X F O R D Sole distributors for the U.S.A.and Canurlu: ELSEVIER NORTH-HOLLAND, INC. 5 2 VANDERBILT AVENUE, NEW YORK, N.Y. 10017

Library of Congress Cataloging in Publication Data

Guzdn, Miguel de, 1936Real variable methods in Fourier andysis. (Notas de m t d t i c a . 75) (North-Holland mathematics studies ; 46) Bibliography: p. Includes index. 1. Fourier analysis. 2. Functions of real variables. 3. Operator theory. I. Title. 11. Series. W.N86 no. 75 LQ403.51 510s [515'.24331 8022545 ISBN

0-444436124-6

P R I N T E D IN THE N E T H E R L A N D S

Dedicated to ALBERTU

P. CALVERbN and

ANTON1 ZYGMUNV

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PREFACE

The work presented h e r e 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 r e a t m e n 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 o p e r a t o r s . The key t o understand these q u e s t i o n s 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 general S t e i n - N i k i s h i n t h e o r y , t h e g e n e r a 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 h e 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 h e o r y , methods connected w i t h t h e t h e o r y of 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 , F o u r i e r m u l t i p l i e r s , . .

.

I n each c h a p t e r we s h a l l 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 . t r y t o d e s c r i b e , i n a c o n t e x t as s i m p l e as p o s s i b l e , some o f t h e main i d e a s Our goal i s t o p r e s e n t methods, n o t t o be exaround a p a r t i c u l a r t o p i c . 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 haustive i n g i v i n g results. 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 r e a d e r 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 . A t t h e same t i m e 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 s e e i n g 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 r e g o i n g 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 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 problems. A l i s t o f t h e ones mentioned 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 t u d y . i n t h e t e x t i s g i v e n a t t h e end.

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 c o n t e n t s o f t h e whole work w i l l 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 h e y may be o f some use f o r t h e r e a d e r who i s a c q u a i n t e d w i t h t h e fundamentals o f r e c e n t F o u r i e r Analysis. Chapter 1 c o n s i d e r s 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 wito 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 the s t a r t i n g p o i n t o f our study. The f i n i t e n e s s a.e. o f t h e a s s o c i a t e d maximal o p e r a t o r leads t o t h e convergence a.e. o f t h e sequence o f operators, 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 h o u g h t which has l e a d t o t h e modern m m e n t s o f N i k i s h i n , Maurais 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 r o w t h and so we p r e s e n 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, a c c o r d i n g t o which vii

viii

PREFACE

the f i n i t e n e s s a.e. of the maximal operator i s e q u i v a l e n t , under some part i c u l a r circumstances, t o t h e weak type of the same maximal o p e r a t o r . The r e s u l t s of Nikishin, Maurais and G i l b e r t extend and simplify the previous theorems i n t h i s d i r e c t i o n .

C h a p t e r 3 considers some of t h e general techniques which ease t h e study of t h e m a l o p e r a t o r , such as those of covering and decomposition of functions, i n t e r p o l a t i o n , e x t r a p o l a t i o n , majorization, 1 i n e a r i z a t i o n , Some of them a r e of constant use i n t h i s type of Analysis. summation, The method of i n t e r p o l a t i o n , i n p a r t i c u l a 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 f u r t h e r information.

...

Convolution operators , of paramount importance i n Fourier Analysis, In allow the use of a p a r t i c u l a 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 ) This i s i t s u f f i c e s t o study i t s a c t i o n on f i n i t e sums o f Dirac d e l t a s . the main theorem of Chapter 4 , where some consequences and extensions a r e given. For t h e type (2,2) of an operator t h e r e a r e special techniques Also a v a i l a b l e , such a s the Fourier transform and t h e lemma of Cotlar. the method of r o t a t i o n i s useful i n order t o extend a one-dimensional i n e q u a l i t y t o more dimensions. These methods a r e presented i n Chapter 5. Chapters 6 throuqh 9 a r e c l o s e l y connected w i t h t h e study of cert a i n very b a s i c operators, t h e Hardy-Littlewood maximal operator and i t s Their importance stems from t h e f a c t t h a t they control many variants. other operators of g r e a t i n t e r e s t , such a s t h e Calderbn-Zygmund o p e r a t o r s Also their and t h e diverse operators o f approximation of t h e i d e n t i t y . behaviour i s intimately r e l a t e d t o the 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 . Chapter 6 shows t h e most important general r e s u l t s about the c o n n e c t i o n m e n coverings, d i f f e r e n t i a t i o n and several extensions of the Hardy-Littlewood maximal operator. Chapters 7 and 8 deal w i t h t h e special covering and d i f f e r e n t i a t i o n p r o p m f some bases of i n t e r v a l s and r e c t a n g l e s i n R2. This study has been g r e a t l y enriched by t h e important r e c e n t c o n t r i b u t i o n s of Cbrdoba, R. Fefferman, Stromberg and o t h e r s . Chapter 9 describes some of the f e a t u r e s of t h e theory of l i n e a r l y measurable s e t s , f i r s t developed by Besicovitch, t h a t a r e most r e l e v a n t f o r the study of some of t h e problems t h a t a r i s e i n a natural way i n d i f f e r e n t i a t i o n theory and i n other a r e a s of Fourier Analysis. Chapter 10 deals w i t h d i f f e r e n t types of approximations of the i d e n t i t y , viewedin t h e i r r e l a t i o n s h i p w i t h d i f f e r e n t i a t i o n theory. Chapter 11 unfolds the main theorems i n the theory of s i n g u l a r The methods presented i n previous chapters a r e suci n t e g r a l operators. c e s s f u l l y p u t t o work i n order t o o b t a i n , i n a very easy way, t h e c l a s s i c a l results about the Hilbert transform and the Calderbn-Zygmund theory. The r e c e n t work of Nagel, RiviGre, S t e i n and Wainger have shown the p o s s i b i l i t y of applying t h e Fourier transform t o c e r t a i n problems

PREFACE

ix

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 along curves i n n-dimensional E u c l i d e a n 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 .

-5

F i n a l l y , Cha t e r 13 p r e s e n t s 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 o Chapter 8 t o s o l v e some problems about F o u r i e r C. Fefferman'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 multipliers: 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) , w e i g h t t h e o r y , A.P. C a l d e r b n ' s theorem on t h e Cauchy 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 integral 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 .

...

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 h o s e who know t h e I have 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 . 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 A n a l y s i s . A s h o r t i n t r o d u c t i o n t o i t , l i k e Hardy and Rogosinski [19441 w i l l s u f f i c e t o understand t h i s m o t i v a t i o n . I t i s , however, q u i t e c l e a r t h a t t h e more t h e r e a d e r knows o f works such as Zygmund 119591, Stein-Weiss [19711, S t e i n [19701, t h e more he w i l l p r o f i t from t h i s book. T h i s work i s t h e f r u i t o f several courses and seminars o r g a n i z e d a t t h e U n i v e r s i d a d 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 r e c e i v e d , among so many hours o f work and d i s c u s s i o n , M.T. C a r r i l o , A. Casas, A. Cdrdoba, from my f r i e n d s and c o l l e a g u e s : P. C i f u e n t e s , 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. P e r a l , E. RodrTguez, J..L. Rubio de F r a n c i a , B. Rubio Segovia, A. Ruiz, A. de l a V i l l a , M. Walias. 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 manuscript.

MIGUEL

DE

GUZMAN

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TABLE OF CONTENTS

V

DEDICATION

vii

PREFACE CHAPTER 1 : POINTWISE 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 o p e r a t o r and a.e.

CHAPTER 2 : FINITENESS A.E.

E

convergence

X o f t h e maximal

AND THE TYPE OF THE MAXIMAL OPERATOR

2.1. A r e s u l t o f A.P. 2.2

Calder6n on t h e p a r t i a l sums o f the Fourier series o f f E L2(T) Commutativity 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 o p e r a t o r s . The theorem o f Sawyer

2.3. Commutativity 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 CHAPTER 3 : GENERAL TECHNIQUES FOR THE STUDY OF THE MAXIMAL OPERATOR

3:l. Reduction t o a dense subspace 3.2. Coveri 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. 3.5. 3.6. 3.7. 3.8.

Interpolation Extrapolation

8

11

13

14 19 23 29 35 35 39 50 54 60

Linearization

63 66

Summation

68

M a j o r i z a t i on

CHAPTER 4 : ESPECIAL TECHNIQUES FOR CONVOLUTION OPERATORS

4.1.

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 y p e (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 xi

73 74 88

xii

TABLE OF CONTENTS

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

91

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

91

5.2. C o t l a r ' s lemma

92

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

96

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

103

6.1. Some n o t a t i o n

104

6.2. Covering 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

105

6.3.

114

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

118

6.5.

136

D i f f e r e n t i a t i o n properties imply covering properties

6.6. The h a l o problem

149 159

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

160 160 165

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 v a l u e s o f t h e upper and l o w e r d e r i v a t i v e s w i t h r e s p e c t t o 82 7.5. A theorem o f Marstrand and some g e n e r a l i z a t i o n s

177

7.6. A problem o f Zygmund s o l v e d by M o r i y d n

182

7.7. Covering 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. Fefferman

184

7.8. Another problem o f Zygmund.

193

S o l u t i o n b y Cdrdoba

CHAPTER 8 : THE B A S I S OF RECTANGLES

171

199

8.1. The Perron t r e e

201

8.2. A lemma of Fefferman

207

8.3. The Kakeya problem

209

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

210

8.5. The Nikodym s e t

215

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 rectangles

224

8.7.

Some r e s u l t s concerning 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

233

TABLE OF CONTENTS CHAPTER 9 : THE GEOMETRY OF LINEARLY MEASURABLE SETS

xiii 241

9.1.

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

242

9.2.

Density.

245

Regular and i r r e g u l a r s e t s

252

9.3. Tangency p r o p e r t i e s 9.4.

258

Projection properties

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

268

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

276

CHAPTER 10: APPROXIMATIONS OF THE IDENTITY

281

10.1. Radi a1 k e r n e l s

282

10.2. Kernels n o n - i n c r e a s i n g a l o n g r a y s

286

10.3. A theorem o f F. Zo

292

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 a p p r o x i m a t i o n o f t h e i d e n t i t y

296

CHAPTER 11: SINGULAR INTEGRAL OPERATORS

305

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

306

11.2.

313

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 o p e r a t o r s 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) 12.2.

The t y p e (p,p)

f o r a homogeneous c u r v e

l 0, t h e n

a r e t h e o p e r a t o r s considered i n t h e s t u d y o f t h e a p p r o x i m a t i o n s o f t h e

id e n t it y

.

vi) If

f E L’(R”)

and

if

1x1

if

1x1
ko,

.

c

w i l l converge i n

Hence, g i v e n

L2 (R')

with

independent o f

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

L e t us prove t h a t

and so T k f

2c

11

f E L2(Rn),

112

E

11 Tpg -

Tqg

L2(Rn).

T h i s would s o l v e q u e s t i o n

k

, and

and t h a t that for

L2(Rn)

Tkg i s t h e n a Cauchy sequence

{Tkfl

We can w r i t e , f o r

f E

tomn)

f

to, i n L2(Rn)

ego

(Rn),

such t h a t

g i s f i x e d , k o such t h a t , i f I T k f } i s a convergent sequence i n

and, once 6 4 2 . So

E/2

and

converges i n

> 0, we can f i r s t choose

6

f

Tk f E L2(Rn)

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

( A ) . What can we do t o t h r o w some l i g h t on i t ? L e t us t a k e a c l o s e r l o o k a t i t s meaning. Assume, as b e f o r e , t h a t Tk i s l i n e a r . We wou d l i k e t o be a b l e t o prove, f o r example,that,

[A(f,X)I

=

for

f E L1(Rn)

( t x e R n : l i m sup I T f ( x ) P PY9

-

and

X

> 0,

Tqf(x)( >

X 1 = o

-+

T h i s would g i v e us t h e convergence o f Assume t h a t we know t h a t , f o r converges. Then, if h = f

- g,

g e

{Tkf(x)) 0(Rn)

and so t h e problem i s reduced t o prove t h a t if

h

fixed

i s o f small

X

L'

a t almost e v e r y

and f o r each

A(h,X)

x E Rn.

x E Rn , { T p g ( x ) l

i s o f s m a l l measure

- norm. Assume t h a t we can p r o v e t h a t , f o r each

> 0,

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

1.0. INTRODUCTION

5

However, t h e s e t A(f,A) has a r a t h e r unhandy s t r u c t u r e and so one can think of s u b s t i t u t i n g i t by some o t h e r e a s i e r t o handle. I t i s quite clear that lA(f,A)

defined by T*f(x) =

sup lTkf(x)[

has a r a t k k e r simple s t r u c t u r e . We may hope t h a t we w i l l be able t o prove now t h a t and t h a t t h e oper t o r T*

0 , and t h i s w i l l as well give us our desired IA*(fy A ) I 0 a s 11 f 1 1 1 almost everywhere convergence of {Tkf} +

-f

.

So we a r e led t o consider t h e operator

T*

defined by

.

I f {Tk} i s which i s c a l l e d t h e maxim& a p e h a t a h associated t o { T k } an ordinary sequence, k = 1,2,..., T*f i s c l e a r l y measurable. I f k i s not countable one has t o prove t h a t T*f i s anyway measurable o r e l s e t o deal with t h e o u t e r 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 w r i t e

The relevance of t h e operator

T*

stems from t h e r o l e i t plays

in t h e pointwise convergence proofs, as i n d i c a t e d , and in t h e information i t furnishes about t h e l i m i t , when i t e x i s t s . Assume, f o r example, in t h e 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 L1(Rn)

with

f

c

independent of

6

L’(Rn)). Then we obtain f o r each

X

0,

1. POINTWISE CONVERGENCE OF OPERATORS

6 and so

ICT*f > X

11

-f

0

as

where convergence r e s u l t .

II Tf III

C IIT*f IIi

c

X

1) f ]I1

-f

0. Thus we o b t a i n t h e a l m o s t e v e r y

Furthermore i f t h e l i m i t i s

Tf,

I1 f I I 1 .

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

(*)

condition

i s somewhat s u p e r f l u o u s and sometimes f a l s e . I t i s good

enough t o know t h a t

f e L1@)

f o r each

.

X

and

O r even j u s t t o know t h a t f o r each

When

(**I

Condition at

o

X > 0 , with

and f o r each

from

L

to

(***) j u s t says t h a t

Observe t h a t c o n d i t i o n

(**)

i s o f weak t y p e

T*

?V .

independent of

f

31 > 0

T*

h o l d s one says t h a t

c

(1,l).

i s continuous i n measure

can be e q u i v a l e n t l y expressed by

saying t h a t

I n fact

11 f 11 1 >

(**) t r i v i a l l y i m p l i e s

( * * ) ' and, i f we have

(**I'

and

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 equivalences between a.e.

- convergence and p r o p e r t i e s of

the function

@(A)

T*

and t o c l e a r up a - l i t t l e th.e r o l e

p l a y s i n t h e whole business.

The general s e t t i n g i n which we w i l l p l a c e o u r s e l v e s i s t h e following: Genahae o e t t i n g . ( a ) We c o n s i d e r

(Q,F,p)

,

a measure space

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 .

7

1.0. INTRODUCTION ( b ) We denote by a b l e f u n c t i o n s d e f i n e d on ( c ) With

Q to R

from

X

“I n ,

ing

k

X

u-a.e.

t ).

.

to

t h a t are f i n i t e

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

(ot to

( d ) The sequence t o r s from

t h e s e t of r e a l ( o r complex) valued measuc

w i l l be an o r d i n a r y sequence of opera-

{TkI

I n many cases t h e r e w i l l be no problem i n assum-

t o be a continuous parameter.

(e) Each w i l l be assumed t o be l i n e a r and i n some cases Tk 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 t i o n : f o r fl, f 2 e X , X 1 , 1 2 E lR we have

ITk(X1

fl

’

12

f2)

( f ) With and

x e fi

6 1x11

T*

we d e s i g n t h e maximal o p e r a t o r , i . e .

for

,

( 9 ) We denote by

T

t h e 1 imi t o p e r a t o r , i.e.

l i m Tkf k+m

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

(h) F i n a l l y f o r $(A)

X > 0, =

sup f ex

w i l l be

$(A)

u{

x

E

R

: T*f(x)

}

f E X

,

1. POINTWISE CONVERGENCE

8

1.1.

OF OPERATORS

AND CONTINUITY I N MEASURE OF THE MAXIMAL OPERATOR

FINITENESS A.E.

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 t u d y i s a general p r i n c i p l e

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

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

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

s i m p l e a p p l i c a t i o n o f t h e general 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

F o r an a l t e r n a t i v e p r o o f one can see A.Garsia

1-4 1 .

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 p-measurable p-a.e.

p-a.e.

m

and f o r

f E ??I (R)

i n t h e sense o f Yosida [1965 only i f

{ f n IE d(f

- fn)

and -f

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

l e t us s e t

I t i s an easy e x e r c i s e t o check t h a t

a sequence

of a l l

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 )

a r e t h e same)

t h e topology o f t h e convergence i n measure. p(Q)


\Tm(f -g)(x)

-

a > 0

Then f o r any

CL

1

=

Tn(f -g)(x)J >

~1

}

G

f

E

+a

R :

[h) 0 J-

as

COROLLARY.

I]

f -g

11 J-

we see t h a t i f

then

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 T k g ( x ) - + g ( x ) at a . e . x e R , then we &o have doh each f E , T k f ( x ) f ( x ) at 1.2.3.

16 ,the space

0

X

06

-+

1. POINTWISE CONVERGENCE OF OPERATORS

LI C X E fi :

lirn sup k + -

=

FIX E R : I i r n sup k +-




-f

a )+ 7

f ( h ),

V{X

e n : I(f-g)(x)l

>

a 1 2

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

As we have seen i n Chapter 1, t h e mere f a c t t h a t , f o r each T*f(x)

x E R

i s f i n i t e f o r almost every

, can

f

E

g i v e us t h e a.e. conver-

T*

gence r e s u l t i n many cases. However, once we c o n s i d e r t h e o p e r a t o r

it

i s o f i n t e r e s t i n many c i r c u n s t a n c e s 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. A c c o r d i n g 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

if

T*f(x)

when

u(R)

=

m

$(A,;)

we have

x e R

i s f i n i t e a t almost each

, l'f

-+

E c . n , u(c)

we f i x

0

as

+

m

f o r each


0,

we s e t , f o r

.

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 operators

f e X

,,

CTk)

,

t h e n t h e almost everywhere f i n i t e n e s s o f

$(A)

each

( i . e . such t h a t f o r

+ Xzfz)(x)

6 IhlI

ISfl(x)l +

( s t r o n g ) t y p e (p,q) , 1 6 q we have S f e Lq(R) and

c

00,

II

m(Q),

fne

fly

Sf

m,

II q

c

13

S

nn? ( Q ) eR ,

from

hl, Xp

IX21 ISf,(x)l),

1c q 6

c

T*

T ).

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(Alfl

, for

and so about t h e t y p e o f t h e o p e r a t o r

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

W(R)

T*f

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 a b o u t t h e be-

haviour o f the function

to

we say t h a t i t i s o f

when f o r each

IIfllp

X,

f e LP(Q)

2. FINITENESS AND THE TYPE

14 with

c > 0

16 p

c

m

independent o f

,

and each 'f

1

6

q
0

(p,q)

such t h a t f o r each

a

X > 0

Lp one has

E

u C x e R : Type

(p,qf

,

q
O,

A X = I l S f l > 11

B u t t h e converse i s n o t t r u e i n g e n e r a l .

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

T*.

m a t i o n about o f A.P.

Section

2.1.,

TI:

l e a d s us t o u s e f u l i n f o r -

a l i n e o f t h o u g h t i n i t i a t e d i n a theorem

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 r e s e n t s 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 r e s e n t e d i n S e c t i o n 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 general and

powerful version o f the r e s u l t s obtained previously i n t h i s d i r e c t i o n . S e c t i o n 2.4

In

we p r e s e n t s i m p l e p r o o f o f one o f t h e main theorems o f N i k i s h i n .

2.1. A RESULT OF A.P. OF f E L 2 ( T ) .

C A L D E R ~ N ON THE PARTIAL

SUMS OF

THE FOURIER SERIES

Some y e a r s b e f o r e t h e s o l u t i o n by Carleson [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 L 2 ( T ) Zygmund [ 1959 , Iap.165] p r e s e n t e d an i n t e r e s t i n g r e s u l t o f CalderBn a b o u 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 tion

f e L2(T).

a1 theorem o f

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

E.M.Stein [ 19611

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

and o f t h e theorem o f Sawyer [ 19661 pre-

2.1.

A RESULT OF A.P.

2.1.1. THEOREM

L2([0,2a])

Le2 S N f ( x )

? v w ~ L e hbenien. S*f(x) =

only

.id

?oh

sup I S N f ( x ) l N S* LA 06 weak t y p e .

S*f(x)

a.e.

k

r

pantiae

nm 06

fC,

S*

I[ fC1I2

CN *

m

SNf

of

f o r each

f e L2.

,

a.e.

finiteness

i s of weak t y p e

(2,2)

then

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

f o r each

f E L2

then

i s n o t of weak t y p e

Ac > 0

= 1 and

and

f

The a d d i t i o n a l i n f o r m a t i o n here

such

S*

(2,2)

i s . o f weak t y p e (2,2).

, i.e.

f o r each

C >0

11 p N ( ( 2 =

1,

that

pN t r i g o n o m e t r i c p o l y n o m i a l s w i t h

such t h a t

>

We have

{AN

2a and i n

(2,Z).

Of course, if S*

S*. a.e.

m

Assume t h a t

we can choose

pehiod

be t h e N-th

ckeikX

are equivalent.

f o r each

t h a t if S*f(x)
0,

-N

pehio&c

f

be t h e comapunding maximal opehatoa, i . ~ . .Then SNf(x) conwagen a.e. a ~ s N -+ m i6 and

deduce t h a t convergence

S*f(x)
Z1

.

We have

CN:

/ :;1

6

2a.

2. FINITENESS AND THE TYPE

16 kl

We choose

1 < kl

CN:




CN*

such t h a t

1

z2

and c o n s i d e r

N;

~2


0

( So ) a s I

R t o R t h a t a r e measure p r e s e r v i n g , i,e,, i f A c.R , A € 3 , t h e n Sil(A) E 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 from

>

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

u

(A

A,B 6

(1 S i l ( B ) )

4

>

and

p > 1,

Sa

u(A A

Sil(B)

and

r e q u i r e so much.

such t h a t

P ~ ( A )u ( B ) .

(Observe t h a t i f

there

Sa

then t h e r e e x i s t s

were such t h a t

o

Shl

u(A) u ( B )

(5)) =

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

The f a m i l y

Sa

"mixes" t h e measurable s e t s o f

R

in

t h e above sense). (f)

We a l s o assume t h a t

tTkl

(Sa) a E I

and

commLLte i n

t h e f o l l o w i n g sense: f E LP(R)

If

Tk

,

Sa, Tk Sa

=

Sa Tk

,

and i.e.

Saf(x) = f(Sax), f o r each

f

6

LP(n)

t h e n f o r each and

x E

R

,

2. FINITENESS AND THE TYPE

20

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

A 06 wuLk t y p e FUR. each f E LP(R)

(a) (b)

T*

(p,p) T*f(x)




an

> 0

and one takes

CI

>

and

= 2,

f3 = p ) .

c1

> B > 0 ,

then

al B

a2

>

...

2

ak

>

2.4. THE THEOREM OF NIKISHIN

29

2.4. THE THEOREM OF NIKISHIN. In 1970 Nikishin published a very general extension of t h e theorem of S t e i n . Like t h e theorems previoulsy presented i n 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 h e maximal operator s t a r t i n g from i t s f i n i t e n e s s a . e . The theory has been f u r t h e r developed by Maurey [1974] . For a c l e a r and thorough exposition of this recent theory we r e f e r t o a forthcoming monograph by G i l b e r t . Here we s h a l l present a version of one of t h e main theorems of Nikishin. Our exposition i s inspired in t h a t of G i l b e r t 119791 , w i t h some modifications due t o J.L.Rubio de Francia [1979] , in a very l u c i d paper. two o - f i n i t e ( p o s i t i v e ) measure spaces. Let (X,u) , (Y,w) We s h a l l consider operators T: Lp(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.

+w(Y,w)

We s h a l l say t h a t T : Lp(X,u) i s &niMeahizable I b u p W n e n n in N i k i s h i n ' s terminology), when f o r each f o e Lp(X) such t h a t there i s a fineat operator "f0

I u f o r each

f a Lp(X)

That

family

(u

I

fofo

,

Iu

=

I

f

I c

Tfol lTfl

w-a.e.

and

w-a.e.

0

i s l i n e a r i z a b l e means t h e r e f o r e t h a t t h e r e i s a ) of l i n e a r operators such t h a t T majorizes foa L q X ) T

each one of them a n d , f o r each f o , T the corresponding u precisely a t f0

coincides in absolute value with fo

.

"Li neari zabl e" imp1 i e s "absol u t e l y homogeneous" , i.e.

/T(hf)l

=

1x1

/Tf/

, since

30

2. FINITENESS AND THE TYPE 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 r u n -

TG

c a t e d maximal o p e r a t o r from

Lp(X)

6, : Y

+

T i g(Y)

=

to

[1,N]

.

w(Y)

,

u

g

1 ug

Ti

Therefore

o f l i n e a r operators

g E Lp(X)

, for

and d e f i n e

f(y) T $N(Y)

f(y) =

c l e a r l y have

For a f i x e d

{TkI

we choose

t h e measurable f u n c t i o n such t h a t

IT$N(y) g ( y ) l

operator

o f a sequence

g(y)l

=

from

] T i g(y)[

f 6 Lp!X),

LP(x)

I

and

the l i n e a r

‘yh ( Y ) .

to

llgf ( y )

We

.

T i f(y)

6

i s linearizable.

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

2.4.1. 1

c p

a,

THEOREM

. Then t h e m e&&

q = i n f (p,2) that hut emh

f

Remmh. means o f

(Nikishin)

Lp(X,p)

E

and

Cp 60/r

L e t T : Lp(X,p) + w ( Y , v )

, Cp

E m(Y,v)

at

LeA

0.

0

a.e.,nuch

each X > 0

The theorem means t h a t ifwe weigh t h e space

0 , i.e.

i f we change i t s measure element

t h e n t h e c o n t i n u i t y i n measure f r o m weak t y p e

.

be e-inemizable and conLLnuau6 -in m m m e

(p,q) of

P t a u d . Since

Lp(X)

to m ( Y )

by

implies

by

Y

ch

dv

the

T.

Y

is

a-finite

, 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

dv

v(Y)
A 1 6 c(X).

implies

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 )

and

A > 0

If

y a A

u = u

=I

y E Y :

(Tfk (y)( >

sup ltk6M

fko such t h a t

t h e n t h e r e i s some

fk

condition: Z h a e

X 1

.

lTfk (y)l > A .

be t h e corresponding l i n e a r o p e r a t o r

Let

(T i s l i n e a r i z a b l e )

0

y E A

I(y) C

A

Set

From t h i s

0

such t h a t

For

.

31

OF N I K I S H I N

we s e t

I(y)

( t E [O,1)

:

IT(

2

M

1

It

6

[0,1)

:

I

U (1 r k ( t ) f k ( Y ) )

r k ( t ) fk(Y))I >

1

1

t o prove t h a t

rk

u Iz(Y) u

,

> X

' I*(Y)

a r e t h e Rademacher f u n c t i o n s . 1 ( I * ( y ) l > F . We d e f i n e :

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

I

We s h a l l t r y

I,iflI.= 0 if i # j and J On t h e o t h e r hand I ( y ) t I z ( y ) U I 3 ( y ) and s o I I l ( Y ) = II2(y)l 1 1 > 7 T h e r e f o r e , s i n c e f o r each y E A (I*(y)l 2 , II(Y)I m i s t h e Lebesgue measure on [0,1), we have by F u b i n i ' s theorem, i f Clearly

[0,1)

=

.

Il(y)

.

13(y)

2. FINITENESS AND THE TYPE

32

1 // 1

v

(B

m (El)

G A -'I2

C

rk(t)f

0

I

P d t 6 MpYq 'P

p / 2

where t h e l a s t i n e q u a l i t y holds by v i r t u e of the property t h a t t h e Rademacher functions s a t i s f y , a s we s h a l l s e e a t t h e end of t h e 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 i n e q u a l i t y (*) t h a t we wished. ( i i ) We now s h a l l prove t h a t t h e r e 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 ) a s TEf = ( T f ) XE i s o f ( v- weighted) weak type ( p , q ) , i . e . a

This i s &OAZ the i n e q u a l i t y of t h e theorem.

1 , where ;((A) Take R > 0 such t h a t E(R) < 7 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 t h e r e 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

lemma, t h e r e e x i s t s a and

C g j l c Lq

Since

C Rq l / g j

,

and

1;

d i s j o i n t sequence

< 1

1

on

{Fjl , v(Fj)

, Ej

F.

>

By Z o r n ’ s

0

, C v(F.) = 1 J

lemma and ( i i )

we o b t a i n

d i s j o i n t such t h a t

m

@ ( y )=

-

we o b t a i n , by s t e p (i)

By u s i n g a g a i n Z o r n ’ s

UE. = Y J

We now d e f i n e

ITg(y)l >

such t h a t

(iii)

Ej C Y

11 glI;

v(T) > Rq

such t h a t

33

1 c 1 1

*-j

J

XE ( y ) j

and

t h i s function satis-

f i e s t h e statement 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

(See S t e i n

(i)

[1970]

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

, Appendix

0).

With t h i s we have

34

2. FINITENESS AND THE TYPE If

then

p > 2,

q = 2

and we have, b y M i n k o w s k i ' s i n t e g r a l

q = p

and we have (by t h e

inequality:

If

p < 2

,

then

inequality

CHAPTER 3 GENERAL TECHNIQUES FOR THE STUDY OF THE MAXIMAL OPERATOR

I n t h e p r e c e d i n g c h a p t e r s we have seen t h a t under t h e c o n d i t i o n o f the finiteness

a.e.

of

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 a b o u t t h e operators

Tk

, we a r e even a b l e t o determine t h e t y p e o f t h e o p e r a t o r T*. I n t h i s c h a p t e r we s h a l l t r y t o p r e s e n t some general methods t o

s i m p l i f y t h e s t u d y o f t h e maximal o p e r a t o r .

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

t o t h e s t u d y o f i t s a c t i o n on f u n c t i o n s w i t h 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 o p e r a t o r s by means o f c o v e r i n g s and decompositions.

The Kolmogorov con-

d i t i o n i n S e c t i o n 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 t u d y t h e t y p e o f an o p e r a t o r . 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 polation i s the following.

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

T

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 a n y t h i n g 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) ? 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 t u d y o f a d i f f i c u l t and c o m p l i c a t e d 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

T*

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

adapted t o t h e o p e r a t o r i n q u e s t i o n .

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 y p e o f

T*

restricted to

such f u n c t i o n s i n o r d e 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, t h e neX a6 memutabkk 4eal ( o h cvmpLex) valued &nctionb, X bpaCC v d 6unc~viont,i n %I (Q) and S a denbe nubdpace 0 6 X

mmuhe.

L e A T*

(Q)

a named

.

LeR:

be thein. maximal opehha-tu4. FOJL

and

Then (a) (b) (c)

$,(A) d o 4 each A

$(A) =

> 0.

Tn pcvLticuRan, 4 T* iA ob w a k t y p e (p,p) 6 ~ dome 4 p, 1 c p L m, Lt iA a6 w a k t y p e Id

T*

Lt 0

iA v 6 t y p e 06

type

(p,p)

(p,p)

ovu

(ova X

S

6ua dome

vum

S

( p , p ) (vum X). p, 1 c p 6

W,

1.

P4ood. ( a ) We have, o f course $ ( A ) 5 $S(A). We w i s h t o $(A) L ~ I ~ ( A ) Let a 2 0 and $(A) > c1 . We s h a l l show prove $ s ( X ) > c1 . I n f a c t , if $ ( A ) > c1 , t h e r e e x i s t s t h e n f e X , that

.

Ilf 11

6 1

, such

that

Consider

T*N

, defined

for

h E X

by

T*Nh(x) =

suplTkh(x)l l&khN

3.1. REDUCTION

TO A DENSE SUBSPACE N

L e t us assume f i r s t t h a t t h e r e i s one > p { x E R

(*) Choose

'd{x

Cgk}

E

c

R : T*iif(x)

S

, >

11

such t h a t

1

: T*Nf(x) > A

Ilg,Jl c 1,

gk

= lim

j

+

p(x

+m

6

37

>

ci

.

f(X)

Since

: T*Nf(x) > A t

R

1

J I

and

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

Since each

Tk

j

i s continuous i n measure, t h e f i r s t t e r m i n t h e l a s t

member tends t o z e r o as

k

Ift h e assumption we have e i t h e r

(1)

p { x E

(2)

{ x E

R

-fa,

(*)

arid so f o r a s u f f i c i e n t l y b i g

does n o t hold i t i s because f o r each N

: TXNf(x) > 1 } c

ci

or else

R : TXNf(x) > A

I f we have (1) f o r each

k

N

}

, then

=

t

o

38

3. GENERAL TECHNIQUES

and t h i s i s excluded. the f i r s t

N

N we have (2), l e t us c o n s i d e r N o , h o l d s . Take now a s u b s e t o f R such

I f f o r some

f o r which ( 2 )

6

that

and proceed as b e f o r e .

Then

g,

(b)

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

(c)

Let

- gh

-+

f a LP(o)

O(Lp)

(ghI c

and

as

s,h

0

as

-f

s , gh

-+

f(LP).

and we have

m

Thus we o b t a i n

Since

\ I T*(gh -

gs)([

+

i s a Cauchy sequence i n

Lp

h,s

9

-f

{T*gh-)

and so converges i n

Lp

t o a f u n c t i o n G. By

Cyh> o f

C a n t o r ’ s diagonal process we can choose a subsequence such t h a t s i m u l t a n e o u s l y

and f o r each

k,

T k f ( x ) = l i m Tk

So f o r each

Hence

where

C

T*f(x)

k

-

gh(x)

(a.e.)

we have a t almost e v e r y

G

G(x)

i s ’ t h e type constant o f

a.e.

and so

T*

o v e r S.

x

8

R

Egh)

3.2. 3.2.

COVERING AND DECOMPOSITION

39

COVERING AND DECOMPOSITION. Covering and decomposition techniques a r e among t h e most b a s i c

ones i n t h e s t u d y o f t h e t y p e o f t h e problems we a r e d e a l i n g w i t h . Coveri 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 r e a t m e n t o f t h e Hardy

-

L i t t l e w o o d maximal o p e r a t o r , one o f t h e most fundamental i n modern Analysis We f i r s t p r e s e n t here i n paragraph

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

A

the

v e r y 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 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

Guzmdn [1975] i n g lemmas. In

B we p r e s e n t s e v e r a 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 d y a d i c 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 Whitney's c o v e r i n g lemma, and t h e CalderBn

- Zygmund decomposition l e m

ma. In

C

we examine a c o v e r i n g theorem f o r convex s e t s o f w h i c h

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 vmaximal d vpaatvh.

THEOREM.

A

Rn be a bounded 6&. Fvh each 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 06 b a L h { B R I buch t h a t 3.2.1.

(i)

CBiI

L d

we a t e given a dobed b a l l

x e A

,

...

C

B(x,r(x))

A c UBk

(ii) . I B k I can be d i 6 M b L L t e d into cn dequenced {BiI , Cn 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. (iii)

1 xB ( x )

One h a

v v d a p v d t h e bad22 06 Pkvo6. a. =

(BkI

We choose

sup { r ( x ) : x e A

l a r g e r a d i u s i s enough.

c cn at each x e R n k h u n i 6 v m l y bvunded b y cn.

1

=

Let

-

{BkI

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

, then

, i.e.

the

If

a single b a l l with sufficiently

us t h e n assume

a. >

00.

We t h e n t a k e

40

3. GENERAL TECHNIQUES

xle A

3 r(xl) > B a n

such t h a t

consider

-

sup { r ( x ) : x e A

al =

such t h a t

T3 al

r(x2) >

and

way we o b t a i n a sequence

{Bk)

n i t e , it i s so because

A c UBk

nite,

+

0

as

have an i n f i n i t e number o f

k's

we have

r(xk)

1

B1 = B(xL,r(xl)).

and

.

BiI

and so on.

-+

-

so

I n f a c t , o t h e r w i s e we would

r(xk) >

with

1

If f i

s a t i s f i e s (i).I f i n f j

{Bk}

.

m

B,

I n this

can be f i n i t e o r i n f i n i t e .

and k

x2e A

We t a k e then

B2= B ( x 2 , r ( x 2 ) ) ,

, that

L e t us now

0.

>

01

I f we observe

that the balls a r e d i s j o i n t and t h a t a l l o f B(xk, 5 r ( x k ) ) = 7 Bk 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 impossible. T h e r e f o r e

s e A

-

, then

0

UBk #

r(xk)

r(s) >

-+

0

0

as

and

overlooked i n o u r s e l e c t i o n process.

k

+

m

.

B(s,r(s))

If has been unduly

A - LIBk = 0 and

Hence

sat

{Bk)

isfies (i). I n o r d e r t o prove many

Bkls

k < h

with

w i t h center

xk

(ii)

l e t us f i x a

intersect

such t h a t

Bh.

c

d(xh,xk)

Y

Bk

concentric w i t h

o f t y p e 2 we j o i n the point

xi

ik o f center

i t s center

a t distance xi

Bk xk

to

1

k < h.

xh

from

Bkls

For a

Bk

We t h e n c o n s i d e r t h e b a l l

I t i s now easy t o observe t h a t

r(xh).

4'/(+)"

T h e r e f o r e they a r e i n number l e s s t h a n

A l l o f them o f t y p e 1 we

and on t h i s segment we t a k e xh.

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 c o n t a i n e d

Property ( i i i )

such

> 3r(xh).

d(xh,xk)

F o r a Bk 1 and w i t h r a d i u s rfx,).

3r(xh)

and r a d i u s

and ask o u r s e l v e s how

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 t h e r s , o f t y p e 2, such t h a t 3 r(xh) since r(xk) >

a r e such t h a t consider

Bh

There a r e some

i n the b a l l = 42n =

B(xh,4r

'n.

i s , o f course, an i n m e d i a t e 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 r e a d e r i s referred t o

Guzman

[1975]

.

W i t h t h e ideas

o f t h e p r o o f o f t h e p r e v i o u s theorem he s h o u l d 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. x e A

THEOREM.

Le,t

AcRn

we m e given a cloned i v L t a v d

be a bounded I(x)

,

(I(x))" f 0

Foh each

, c e n t a e d at

x i n nuch a &~mt 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 ) one )~ can € A

3.2. COVERING AND DECOMPOSITION choone a nequence

(i)

41

{ I k ) nuch t h a t

A c UIk

( i i ) The nequence { I k } can be cL&txLbuZed i n t o pn A & quenca C I 1 ~, 11; I , ... , each o d them ad d b j o i n t i n t a u & . H a e pn depend o n l y on t h e dimenilion.

{IE~I

(iii)

OW

han

1XI

k

(x)

,
0 and each f E LNn)

c > 0 such t h a t f o r each

42

3. GENERAL TECHNIQUES

If x

x

E

A

,

then there e x i s t s a cubic i n t e r v a l

containing

such t h a t

Q* c e n t e r e d a t

I f we c o n s i d e r t h e minimal open c u b i c i n t e r v a l

x

Q

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

Q* , when

supremum o f t h e diameters o f t h e cubes We a p p l y B e s i c o v i t c h lemma and o b t a i n

f a L'

CQ*,I

x

and 6

AX

X

the

> 0,

,is

finite.

such t h a t

Then we can w r i t e

T h i s proves t h a t type

(m,~),

M

i s o f weak t y p e ( 1 , l )

. Since

M

i s t r i v i a l l y of

i n t e r p o l a t i o n theorem t e l l s us t h a t

Marcinkiewicz

M

is

of t y p e (P,P). (Observe t h a t i n p a r t i c u l a r we g e t

T h i s f a c t w i l l be used l a t e r ) . The f a c t t h a t t r i v i a l observation t h a t

M

i s o f weak t y p e

,if

g e

gNn)

(1,l) t o g e t h e r w i t h t h e we have, for each x E Rn

3.2. COVERING AND DECOMPOSITION and each sequence s(Q,(x))

-t

IQk(x)I

43

of cubic i n t e r v a l s containing

x

, such t h a t

0

gives us the c l a s s i c a 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 e g r a l s .

.

3.2.3. THEOREM. LeX f E L1(Rn) T h e m &xh& a heX oh m m 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(Qk(x)) 0 , one h a uht

zmv

-f

Prroal;.

But, i f

C With such t h a t one proves

f

=

We wish t o prove t h a t f o r each

g + h

with

g E

e o(R')

A > 0

, we have

independent of f , g , h , A . Thus, given E > 0 , we choose h CJlhl[ < E . This proves t h a t / A A \ = 0 In the same way A

.

3. GENERAL TECHNIQUES

44

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

B.

C d d a b n - Z ygmund decompob&on. The use o f t h e d y a d i c c u b i c i n t e r v a l s i s a powerful t o o l

for

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

Rn t h e f a m i l y

consider i n

DO o f a l l h a l f

-

, we

i n Rn

first

open c u b i c i n t e r v a l s

1

(open t o t h e r i g h t and c l o s e d t o t h e l e f t ) o f s i d e l e n g t h equal t o h a v i n g v e r t i c e s a t a l l p o i n t s of now s u b j e c t

Do

and so o b t a i n

.

of

Dj-l

if

Qls D j

o r else

Q1

Rn

w i t h i n t e g r a l c o o r d i n a t e s . We

t o a homothecy o f c e n t e r

.

Dk

c

D

.

2,

k

for

Z

E

D i s t h e u n i o n o f Zn d i s j o i n t cubes j have s i d e l e n g t h 2J It i s clear that

j with

Dk

Q2€

Q2

and r a t i o

Each cube o f

The cubes o f and

0

j

s

k

,

.

then e i t h e r

i7

Q1

Q2

0

=

We s h a l l use a l s o t h e f o l l o w i n g s i m p l e p r o p e r t y o f t h e d y a d i c cubes. 3.2.4.

a d C u b a od (9,) ,cA ucending chain 06

Pkood.

.

LeL (9,) clcA be u g i v e n coUecLLon 0 6 Annunie t h a t ecrch abcending c h a i n C 1 $ C 2 s

THEUREM

dyadic c u b i c i n t a w a h .

6ivLite.

Then t h e muximd cubeil

ahe d i n j o i n t and b a U 6 y

(Q,)

06

{Q,}

UQ,

=

...

each

UQa ,€A

.

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

o f d i f f e r e n t d y a d i c cubes, e i t h e r t h e y a r e Q j , Q, 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 c o n t a i n e d i n t h e o t h e r .

f o r each conple

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 useful c o v e r i n g lemma due t o Whitney [1934]. 3.2.5.

.THEOREM

.

LeL G

c Rn be a n open

bt?X

,G

#Rn

,

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 G #

(i) G =

U Q,

45

3.2. COVERING AND DECOMPOSITION d (Qky

(ii) F o t ~ each k, d

aG)

T

2 6

whehe

6 y

denotec, t h e Euclidean dintance , 8G ,iA t h e boundmy

06

G and

&(a,)

0,.

,the diameXeh o d

P t ~ o o d . We can assume, by p e r f o r m i n g a t r a n s l a t i o n , i f necessary, Q(x)

For

aG . F o r each

0 E

that

such t h a t

x

E

x

Q(x)

E

G

we t a k e

t h e g r e a t e s t d y a d i c cube

and

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 " of

d(x,

aG)

Q(x)

3 6 (Q*(x)

6

i n t h e d y a d i c g e n e r a t i o n we have

=

6 6 (Q(x))

T h e r e f o r e we can w r i t e

The t h e Theorem

(Q(X)),,~ 2.4,

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

by b e i n g a t z e r o d i s t a n c e from

2 6 (Q(x)). theorem.

k

If

0

and t h i s c o n t r a d i c t s

We now a p p l y Theorem 3.2.4.

d(Q(x),

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 h e weak t y p e o f

aG) >

and o b t a i n t h e statement o f t h e

t o prove

t h e H a r d y - L i t t l e w o o d maximal o p e r a t o r r e l a t e d t o d y a d i c

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 .

Mf(x) =

sup

THEOREM

&I

3.2.6.

.

a(x)

LeX

If[ ,

f e

Limn) and

whehe t h e

sup

A ,tatahen

oweh u l l

3. GENERAL TECHNIQUES

46

dyadic C u b a Q(x) containing = { x : Mf(x) > A} AX

Phou6. For x E Ah containing x such t h a t

x1 11 f

lQ(x)l


A Clearly Q(x)

.

hi

111

X > O , we

Q(x)

and so i t i s obvious t h a t ( Q ( X ) ) ~ s~ a~t i s f y t h e f i n i t e ascending chain condition. We apply3.2.4. obtaining Q, disjoint such t h a t Ah = U Q, . Observe t h a t

i s the f a t h e r of

Q,.

W i t h these

i n e q u a l i t i e s the s t a t e m n t

i s obvious.

Aiyfication

C d d m 6 n - Zyqmund decompob&on

2.

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 c a l paper on s i n g u l a r i n t e g r a l s , has become a very important t o o 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 o r i g i n a l one, which r e f e r s t o t h e dyadic cubic i n t e r v a l s . For other l e s s geometrical v a r i a n t s one can s e e Guzmdn [ 1975 , p. 16-17 .

3

3.2.7.

THEOREM

.

LeX

f E L1(Rn)

,f

2

0

and A > 0

Rhme e h a 2 a bequence 06 d i b j o i n t dyadic c u b i c intmvab

(ii)

f(x) 6 A

at

a.e.

x 4

UQ,

{Q,}

. Then duch t h d

COVERING AND DECOMPOSITION

3.2.

(Calder6n

f ( x ) = g(x) t h(x)

-

47

Zygmund decomposition)

we have g(x)

(a)

Pmod. lim a(Q,(x))

x

g(x) 6

(b)

f(x) =

PA


A where I. Q,(x)) is t h e sequence I Q k ( x ) l 'Qk(X)

o f decreasing d y a d i c cubes c o n t a i n i n g

x.

be t h e l a r g e s t d y a d i c cube c o n t a i n i n g

x

F o r each

x E

let

Q(x)

such t h a t

(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 ,

The cubes since

4

IQ(x)l c

satisfying ( i )

Remmk

1

11

/ I 1 . We

f

.

and ( i i )

.

a p p l y Theorem 3.2.4.

I

Q,

and o b t a i n

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.

f L 0

, X

3.2.8.

THEOREM

> 0.

AbbWe

.

that

Let Q

&

Then .them exint a nequence

be a cubic i r z t a v d ud Rn /Q

06

f 6 A

f(x)

6 A

at

a.e.

dyadic Aubcubu

x

6

f

*

t h d dhe dinjoint and A a - t i A d y

(ii)

,

Q - UQ,

0 6 Q , C Q,

6

L'(Q),

3 . GENERAL TECHNIQUES

48

C.

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

Later o n , when dealing with s i n g u l a r i n t e g r a l operators i n ChaL 11, we s h a l l make use of t h e following i n t e r e s t i n g covering r e s u l t .

ter

THEOREM , LeL (K,) be a ~a.mi.ly a4 campact canvex 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 bLd.5 06 Rn t h a t they m e n u t e d , i.e. 6vh any &a a6 them , K, , K,, , & h a 3.2.9.

K,,

Ka,

Kol,

BcRn

Let

x e B xhe

Ka,.

be any campact

we m e g i v e n an index

Cx = x + K

O&

'

a(x)

b&

.

e A

{C,

thcLt

k

1

i . e . ,the f i a m l a t i o n t o x (Cx)xaB

t h a t may be 6 i n i t e oh i n d i n i t e ,

B C U k whme

5 Cx

cedm

x,

Lh

t h e b e t abahined 6hom

t h e centeh

06

bymmehy

06

5

C

x

Camidetr, doh each

Then, @am ,the g i v e n caUecaXon bQqUQMCe

and ahume t h a t 6 a t each

B ,

E

a6 t h e b d

Ka(x).

One can chavbe a

06 dinjvint

b&tA

huh

'k

Cx

by a hamvthecy

0 6 ha.tLo

5

and

C.,

We s h a l l give a sketch of t h e 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 . Phvro06.

j > h

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 K. C Kh We proceed t o choose our s e t s C

, implies

such t h a t s i b l e . Take now x,

s i b l e , then

.

J

i s as small as possible,

,(XI) XZE

x3e 6

-

B - 5 C, 2

1.J

i :1

5 C

1

'i

i.e.

'k

h e NI,

. Take

such t h a t

as big as pos~ ( x z ) i s a s small as pos-

such t h a t

a ( x 3 ) i s as small as POL

Cxl

sible,and so on. I t i s e a s i l y proved, as in t h e previous covering r e s u l t s , that B c U 5 C and t h a t C fl C, = !?j i f i # j . 'i 'i j The case of a general index s e t A can be e a s i l y reduced t o t h e previous one.

3.2. COVERING AND 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 r e n t i a t i o n of i n t e g r a l s and f o r t h e study o f the approximations of t h e i d e n t i t y . 3.2.10.

LQt

THEOREM.

a i n t h e pkecedincj denote

he&

(Ka) aaA

be a l;amLLy

F o l ~ each x

theokem.

8

06 compact conuex

R n and

a

E

A

Let

UA

K a ( x ) = x + Ka

and c o a i d a , d o t

LjOc

f E

Mf(x)

(Rn) , x

8

Rn

sup

=

t h e maximd opetrcLtoh M

K,(x)

clEA

If1

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 eRn For each +

KCAX)

x

: Mf(x) >

B there i s an we have E

A

} a(X)

such t h a t , i f

We apply t h e preceding theorem and obtain a d i s j o i n t sequence

I C(x,) 1 such t h a t

B

c U 5 C(x,).

So we have

50

3. GENERAL TECHNIQUES

Therefore

3.3. 'KOLMOGOROV CONDITION AND THE WEAK TYPE OF AN OPERATOR. Weak t y p e inequal iti 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 r e s p e c 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

(Ta)

some

to

LP(n)

,0 LP(n)

. We

w i t h constants

ca

f B LP(n)

Tf(x) =

by

i s o f strong type

i s a f a m i l y o f sublinear operators from

< cx < 1, , 16 p

(p,p).

i

c

m

,

which a r e o f s t r o n g t y p e

consider the operator [ T a f ( x ) [ da 0

T

,

(p,p)

d e f i n e d on each

T

and we want t o s t u d y wheter

I t may be p o s s i b l e t o a p p l y M i n k o w s k i ' s i n t e g r a l

inequality t o obtain

so, i f

1'

ca da
0,

each

Let

< a < s

0

c R R , call

A

f u n c t i o n d e f i n e d on

p(A)


A I.

c

+ u

im

cs

N

I1 I1 f

t h a t makes

u(A) inequality

Assume now t h a t h >

(A) dA = u[

A

p(A) dA

If we choose

and f o r

m

T

m

c, i . e .

,

for

, measurable

g

. Then

g

0

c

u(A)
O , s > O , 11 M f [ I oo < l l f l [ w c > 0 and anbume ,that 6 0 4 each p E ( p o , p o + E ) and 6 0 4 each K c n, 3.5.2.

K

TIfEOREM

,

a6 baunded m m m e

Then,

604

each

,that don each

, we

have

t > p o ( s + 1) - 1

- -

Rhehe e x h ~ 2 c = C ( t , p o , E , S , C ) K C n 0 6 bounded mmute and don each f 2 0 , f E

bUCh

M(n)

3.6. MAJORIZATION. I f T , S a r e s u b l i n e a r operators from k ( n ) t o w(n) and f o r each f e ?X (0) and each x E R one knows t h a t I T f ( 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 , of weak type ( p , p ) , them so i s T . Sometimes , and we s h a l l l a t t e r s e e important examples, when dealing with s i n g u l a r i n t e g r a l operators, this t r i v i a l majorization does not work, and one has t o appeal t o some o t h e r s u b t l e r procedures. Here , as we s h a l l see, t h e Kolmogorov condition plays an important r o l e . W e try t o give the flavour o f t h e 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-

.

3. GENERAL TECHNIQUES

64

i n g on

x

and

f

duch t h a t d o t each

ITf(x)l Then, i6

T

i.6

04

&a

LA

S

weak .type

06

6

y

Q ( x ) one hm

8

lSf(y)l

w a k type

(pyp)

(oh

dome

p ,1

c p


0

.

f

and

x.

Therefore

y

if

K

and

contain-

i s a con-

i s any s e t o f f i n i t e

Ifwe now r e c a l l t h e remark a t t h e end o f t h e p r o o f o f t h e weak

type i n e q u a l i t y

[HA(

with




Assume i t i s

...B, {Rk) 1

N

such t h a t

. For

k=l

IBI

and so

(i).

We now p r o v e

lEkl

For

IBk 0 R1 1 c

such t h a t

1

.

depend onLy on c and

R1

R 3 we choose R2)1

:

( i . e . t h e R k ' s coven u good poh-

t h a t has been l e f t o u t we have

Thus we have

then

06

~etre q = P p-l

We choose

BZyB3,...,B"

RZ

be&

covehing p k o p d y : Given any 8 , AX pobbibee 20 choobe

coveh).

The co~n;trcna2 c 1 ,c2

Pmol;.

67

LINEARIZATION

.

. For

ITf(x)l < c.

(ii). Observe f i r s t t h a t i f

Ek

We d e f i n e now a l i n e a r o p e r a t o r f

B

Lp

= Rk

-

j 0 ,

we have

P m a d . (A) We f i r s t prove i n f o u r s t e p s t h a t i f K* i s o f weak t y p e ( 1 , l ) o v e r f i n i t e sums o f D i r a c d e l t a s , t h e n i t i s o f weak t y p e (1,1).

4. CONVOLUTION OPERATORS

76

c > 0

(1) Assume t h a t t h e r e e x i s t s f =

H

1

X

and

h=l

such t h a t f o r each

we have

> 0

H We want t o p r o v e i n t h e f i r s t p l a c e t h a t i f with

ch E

h=l 1 ch

f =

6h

Nl , then H

N we c a l l

I f f o r a f i x e d n a t u r a l number

K;f(x)

=

sup

K.f(x)

l6jGN

then, s i n c e c l e a r l y m

IJ { X :

N=l

K$f(x) >

X}

=

{ x : K*f(x

i t i s 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

with

c

Now f o r each Ilkj

-

N . So we f i x an

independent o f

gjlll

G

n

where

t e r . For each p o i n t

k

,

j

16 j G N

N

N.

, we

take

g

j

E

eD(Q)such t h a t

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

ah 6 fi

we choose

ch

points b i

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

, btY

...

‘h hh

la ,

4.1. Now f o r each

(1,l)

71

0 < c1 < A , we have

such t h a t

c1

THE TYPE

l{x :

c

IIX

:

sup IcjcN H

Lh

By t h e h y p o t h e s i s

H

h = l ‘h cc-------

A - a

I f we prove t h a t , f o r a r b t r a r y

can choose

bh;

and

g

E

so t h a t

s t e p (1). Observe t h a t we can w r i t e

Thus we can s e t

> 0

IP

, and
0

and

6 rl

,

‘‘

1 I

a r e u n i f o r m l y continuous. Once t h e close t o

ah

(2)

I2


0

T h i s i s obvious i f

H

h=l

dh bh

g

j

we f i r s t choose

ch &

E 7

. Observe

g

j

6 t o ( ( n )

that the

have been f i x e d , we take

bh;

g

j

so

and so we conclude t h e p r o o f o f s t e p (1). H 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 h= 1

with

F= 1

1

that

Thus we g e t

H

c

,

> 0

c1

with

E

t h e n f o r each

ch

6

dh = c h + rh

Q

,

.

X > 0 we have H

If ch

E R , ch>

rh small, dh

6

Q

0

. Then,

we can t a k e if

O 0

Ihd y a d i c i n t e r v a l and and f o r any such

ch > 0

we want t o p r o v e

d 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 i f 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 Ihi s as small as we please. Otherwise we s u b d i v i d e each 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

Ihand t h e

H

o f dyadic

i n t e r v a l s we have. We now proceed as i n s t e p (1). For each take

gj

E

'$ o!n)

such t h a t

11

k. J

- g.111 G n , J

k

,

j where

1G j G N

n

> 0

we

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

H

f = where

6h

1 h=l

'h

'h

i s the Dirac d e l t a concentrated a t

ahy

t h e l e f t extreme

80 point o f

4. CONVOLllTION OPERATORS Ih.

Then, i f

0
0

we f i r s t choose

g

j

such t h a t

//

k. -g.lll J J

c

, with

rl

so small t h a t

Then we choose

so small t h a t I { x : Ag(x) >

Ih

:I[

&:,what

can be made i n

v i r t u e o f t h e above i n e q u a l ty. Thus we g e t

for

d

l i n e a r combination 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 y a d i c i n t e r v a l s .

t h a t the r e s u l t i n (3) already ( 4 ) We know (Theorem 3.1.1.) i m p l i e s t h a t K* i s of weak t y p e (1,l). T h i s concludes t h e p r o o f o f ( A ) . (B)

Assume now t h a t

K*

We t a k e d i s j o i n t d y a d i c i n t e r v a l s

1

.

i s of weak t y p e ( 1 , l ) . L e t Ihc o n t a i n i n g t h e p o i n t s

We know t h a t

and want t o prove t h a t f o r each f i x e d

We w r i t e , f o r

0
0

we have

dafi

I{x

tach

: lKjg

id and o n l y i d d o t each

f =

H

2

h=l

6h

we have

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 i n t h e introduction of t h i s

Chapter i s as follows.

THEOREM 4.1.3. K.f(x)

, K*f(x)

Lei

Ckjly=l

c

L 1 ( Q ) and,

dot f

6

L'(fl)

,

s u p I kJ. * f ( x ) / .Then K* LA ad weak t y p e ( 1 , l ) j K* id and o n l y id LA ad W M ~t y p e o v m c h a h a c t e ~ A L i cduneLivnn 04 8.inite uniann od dyadic inte,twP~. J

=

kj*f(x)

=

Ptoo6. After Theorem 4.1.1. a l l we have t o 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 unions o f dyadic i n t e r v a l s , then K* i s of weak type ( 1 , l ) over f i n i t e sums of Dirac d e 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 h e proof of the Theorem 4.1.1. by taking t h e r e the s e t s I h of t h e same s i z e . The previous theorems r e f e r t o t h e weak type (1,l) of t h e 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 t h e maximal operator of i l y of convolution operators indexed, f o r example,by t h e s e t bers. Such i s the case, f o r instance, of t h e maximal H i l b e r t t h e Hardy-Li ttlewood maximal operators , the maximal Calder6n erators,.

..

a whole famof r e a l numtransform , Zygmund op-

The natural question i s then: Can one c h a r a c t e r i z e t h e weak type ( 1 , l ) of the maximal operator by means of i t s weak type (1,l over f i n i t e sums of Dirac d e l t a s as we have done i n t h e case o f an ordinary sequence? The answer f o r t h e general case i s negative,as the following sim ple example shows Let

1

if

x

if

xeR-M

=

..

1,2,3,4,.

k(x) =

and, f o r and

E >

0

,

kE(x)

=

E - ~k(:)

.

For

f

E

L’m) , l e t

KEf(x)=kE

*

f

K*f(x) = sup I K E f ( 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, if

E~

-

X

-

, we have k

EX

( x ) = -nX k ( - .nX x ) = -

nX > n

Therefore

and so

K*

i s not of weak type ( 1 , l ) over f i n i t e sums of Dirac d e l t a s .

However, by imposing some mild conditions on t h e kernels we can s t i l l recpver the same kind of c h a r a c t e r i z a t i o n s . For example, i f

4.1. THE TYPE ( 1 , l ) k 6 Ljoc ( Rn

- (0) ) ,

E

,

> 0

, KEf(x)

f E L’Nn)

and, f o r

for

x aRn

*

= kE

85

f(x)

, and K*f(x)

sup I K E f ( x ) l , RsE>O

=

then we e a s i l y o b t a i n

and so one can a p p l y t h e p r e v i o u s r e s u l t s . Observe t h a t t h e Calder6nZygmund

maximal o p e r a t o r s f a l l under t h i s t y p e Also, i f

B i s any measurable s u b s e t o f Rn w i t h p o s i t i v e

measure so t h a t f o r each

x,.~

we have

+

J

.

xEB

E

and f o r each sequence

> 0

a.e.

{ E ~ } , E~

-f

E

f E L1(Rn) , we s e t

then, if,f o r

t h e n we e a s i l y o b t a i n sup

Ra 0 0

IKEf(x)l

=

sup

Q 3 E>O

IKEfW I

and so one can a p p l y t h e p r e v i o u s theorems. operator,

The H a r d y - L i t t l e w o o d maximal

f o r example, f a l l s under t h i s c a t e g o r y .

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 general 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 approximations o f t h e i d e n t i t y t h a t we s h a l l s t u d y i n Chapter 10.

4.1.4. L&

A

kE(x) =

LEMMA

.

be a d e u e hubb& E - ~ k

):(

and

Let

06

k E L’ (0,m)

KEf(x) =

Lw(Rn)

. Then,id

doh

kE * f ( x ) ,

and E

> 0

f E L1(Rn)

,

we have

.

4. CONVOLUTION OPERATORS

86

Given q > 0

we f i r s t choose

g

6

@omn)

such t h a t t h e f i r s t

t e r m 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 Then we choose

n/Z

.

ci E

A

so c l o s e t o

q/2

t h a t t h e second term i s l e s s t h a n

E

I n t h i s way we o b t a i n t h e lemma. With t h e p r e c e d i n g lemma t h e f o l l o w i n g theorem i s simple.

4.1.5.

THEOREM

.

LeL

k E L’ 0 Lm(Rn)

Le,t UA dedine, doh

kE(x) = E-nk(:).

f

6

and doh

E

> 0

,

L’(Rn),

Then: (a) (b)

06

LA ad weak ,type

(1,l) o v e h 6 i n i t e numb

w u k type

being

K*R

w&

K*R

can be

06

06

( 1 , l ) i6 and anLy

.i6

K*Q

LA

06

VhAc deetad.

weak t y p e

t y p e ( 1 , l ) oveh din.&

.

(1,l) oven ~ u n C . t i u ~wd L t h u u t

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 n m e d i a t e f r o m 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 r e c e d i n g theorem

, then

k E L’(R”)

k

L’ 1’1 Lamn).

E

I f we o n l y have

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 .

L e Z k E L’(Tln) , k a 0 and K* = K*R be 4.1.6. THEOREM. dedined a.4 i n Theahem 4.1.5. Then, 4 K* Lb 06 weak t y p e (1,l) vveh Lb vtj w u k t y p e ( 1 , l ) . 6inite numb 06 D*ac deetan, j = 1,2,3,...

Phvvd. F o r

0

K? f ( x ) = sup 3 O<E ER

and

f o r each

with

c

I

k(x) >

if

k:

*

f(x)

j

1

i s of weak t y p e ( 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

K*

Since

,

l e t us w r i t e

j

independent o f

4.1.5.

we know t h a t

ent o f

j.

K*

j , ah

,A

.

Since

k j a L’ 1’1 Lm, by Theorem

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 n d e p e n d

J By passing t o t h e l i m i t as

j

-f

OD

, we

see t h a t

K*

is of

weak t y p e ( 1 , l ) . 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 y p e (1,l) vveh ~ i n L t ebumb 013 V h a c d-.

4. CONVOLUTION OPERATORS

88

,p > 1,

4.2. THE TYPE (p,p)

Also the type

OF MAXIMAL CONVOLUTION OPERATORS.

,p

(p,p)

> 1

,of

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 o p e r a t o r s can be s t u d i e d b y l o o k i n g a t i t s a c t i o n o v e r t h e O i r a c d e l t a s . However one cannot o b t a i n h e r e a necessary and s u f f i c i e n t condition. m

c L1(R) be an ohdifiany b equcnce. 0 6 ~unc,tio~nand C K . 1 t h e nequence 0 6 convolution o p e h a t o ~ 5a ~ n g J cicLted t o a. L e t K* be t h e cohhuponding maxim& opehatan. 1eL p > 1. 4.2.1.

le,t

THEOREM.

Annwne t h a t dotr each

ent poi& a

j y

.. . , aH 4 R

al ,a2,

{kjIjzl

tach ~ivLiten e t a6 didde& we. have , don a c > 0 independent 0 6

X > 0 and

doh

A,

.LA

Then K*

06

W M ~type.

(p,p).

Phoo6. The p r o o f i s 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 If

KG f ( x ) =

4.1.1. sup j=l,. ,N

..

]kj

*

f(x)l

one f i r s t o b t a i n s

From h e r e one g e t s t h e same i n e q u a l i t y f o r a general s e t and f i n a l l y

, approximating

K*

K;

i s o f weak t y p e (p,p)

i s o f weak t y p e

(p,p).

c~,...,c~

8

R

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

Therefore

for

P

with

c

independent o f

N. Hence

4.2.

THE TYPE (p,p)

89

B = B(0,l)

L e t us now observe t h e f o l l o w i n g . L e t

-1 B y

j=1,2,3

B.= J J l < p < m ,

,...,

M

/ ~ 3 Xj B ~

k j=

sup

K*f(x) = If

1

Ikj

J

*

and f o r

f

i s o f weak t y p e

K*

,

(p,p)

E

Lp(Rn)

,

f(x)\

i s t h e o r d i n a r y H a r d y - L i t t l e w o o d maximal o p e r a t o r

and so

Rn and

1< p
1.

The r e s u l t s and methods we have presented i n t h i s c h a p t e 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 y p e o f a sequence o f o p e r a t o r s . 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 a t e uni~omnLg06 J J weak type (1,l) o v p f ~,5ivLite numb 0 6 DhacdctYan. Then t h e y me uni60hmLg 06

weak t y p e ( 1 , l ) .

4. CONVOLUTION OPERATORS

90

One should a l s o observe t h a t some of t h e r e s u l t s of t h i s chanter can be used i n order t o deduce useful and i n t e r e s t i n q qeometric pronerties r e l a t e d t o c e r t a i n operators. In f a c t , from a n a l y t i c a l considerations we may know t h a t a c e r t a i n 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 d e l t a s . B u t t h i s nronertv can o f t e n be i n t e r n r e t e d i n an i n t e r e s t i n g 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. For example, we know t h a t the Hardy-Littlewood onerator i n Rn over Euclidean b a 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 . I f we t r a n s l a t e t h i s f a c t i n t o geometr i c language, we get t h e following i n t e r e s t i n g covering propertv. 4.2.3. j = 1,2

Bj

,Ba

0 6 volume j v

.

the b d h

1eA a 1 , a 2 ,..., aH 6 R n ,

v > 0 . Fvk R" i n at LeanX 5 v b , centetred at a1 ,... , a H , heApecfiv&9 and

THEOREM.

,...,H , Leet

AJ

,.. .,

Let

be t h e n e t ad p u i n t ~0 6

Bi A,

=

H

II

j=1

.

AJ.

Then

uhetrhe c LA a c o n ~ a k n t h a t depencb v n t y vn t h e dimennivn. Likewise, as we s h a l l s e e i n ChaDter 11, t h e maximal s i n q u l a r i n t e g r a l operators t r e a t e d t h e r e a r e shown t o be of weak type (1,l). The reader should t r y t o obtain t h e 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 t u d y t h e t y p e 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 y p e o r weak t y p e 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

H a r d y - L i t t l e w o o d 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 ) , trivial.

by means o f a c o v e r i n g lemma, and t h e t y p e

However i n some o t h e r cases t h e weak t y p e ( 1 , l )

(-,m)

which i s

o r the type

(my-)

a r e n o t a v a i l a b l e and one has t o t r y t o show more o r l e s s d i r e c t l y t h e t y p e o f the operator. purpose.

There a r e n o t many s t a n d a r d techniques a v a i l a b l e f o r t h i s

The use o f t h e F o u r i e r transform and t h e P a r s e v a l - P l a n c h e r e l

theorem enable us t o t r e a t t h e t y p e can e s t i m a t e t h 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

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 c o r r e s p o n d i n g

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. I n o r d e r t o handle t h e t h e Calder6n-Zygmund o p e r a t o r s

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 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 b y 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 o p e r a t o r s . 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 a p p r o x i m a t i o n 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 standard t o o l i s t h e Parseval-Plancherel 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

5. THE TYPE ( 2 , 2 )

92 5.1.1.

Fm c > 0

f

THEOREM

.

L2(Rn)

Cel

lel { k . } be a oequence a Q 6unc.tLtiu~n i n J K j f ( x ) = k * f ( x ) . bourne t h a t t h m e j nuch t h a t dvn each j E

Phavd.

By t h e P a r s e v a l - P l a n c h e r e l t heorem

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.

a 6ivLite he.quencc?

LA

.

THEOREM 06

1eL H

vpehaLvhA Qhvm

u bunctivn ouch t h a t

d e n a k a the. a d j a i n t 0 6

1

k=-m Ti

,

We c a n w r i t e

. Adhume t h a L

H tv H

co

Then

P4ovQ.

be a H i L b m A npacc? and

(c(k))'"

c A


A

I R* I Since f o r each f i x e d

k

a l m o s t every p o i n t

i t r e s u l t s t h a t f o r almost each

Thus

D( xA

a3

(x)

x

Q

x

I f one knows t h a t a d e n s i t y b a s i s

d9

f e L

Q

i s i n some CJk , Rk

of

such t h a t

Q.

@

of

t h e r e i s a sequence

everywhere i n

g r a l of a f u n c t i o n

almost

of

x

contracting t o

,x) > A

associated t o

i s such t h a t kj

IR*

elements of

P

differentiates the inte-

, t h e one can a f f i r m 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 a c e r t a i n weak t y p e p r o p e r t y .

T h i s i s es-

s e n t i a l l y t h e c o n t e n t s 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 r o v e 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

6.4. DIFFERENTIATION AND THE MAXIMAL OPERATOR

125

t h e 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 . T h i s l a s t theorem i s due t o Hayes and Pauc

L1955J.

The p r o o f we p r e s e n 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 .

I t i s based on a i d e a of Jessen used by P a p o u l i s [1950]

The main theorem ,6.4.G.,

purpose.

,

and P a w [1955]

P4ood.

N > 0

For a f i x e d

fN(X)

I

hypothesis

D( f,x)

define if

f(x)


0 , each n o n i n c h m i n g nequence 05 meuw abRe n u 2 {A k } w L t h \ A k [ + 0 , and each numehicd neyuence i r k } w L t h rk G 0 , we have

6.4. DIFFERENTIATION AND THE MAXIMAL OPERATOR

PaoaQ.

I n o r d e r t o prove t h a t

Q

a r b i t r a r y open c u b i c i n t e r v a l on

+

, such t h a t fk

E

x

6

Q

-

A.

x B Q

to

x

h

implies

- A

%

differentiates

and f o r each sequence

we have, as

j

-

Q

(b) fk

A.

we t a k e an

+

0

pointwise

6

A, w i t h

Hence, g i v e n

X

fk(x)
0. We have

u n i f o r m l y on

0

there e x i s t s a p o s i t i v e integer and

E

and so, by Egorov’s theorem, t h e r e i s a measurable s e t

Q

JAJ


k

2

c

A

f k < fh

h, s i n c e

, we

have

and so

l i m I{x k-m Since

Q

and

E

6

Q : Mk f k ( x ) > A 1 ) e

a r e a r b i t r a r y , we g e t

(c)

Ak = C f ;r k3 Since

f

E

L ~ ( R ” ),

1 ~ +~ 0.1

]A)
0,

contracting

+

lim k-

A

k > h

if

.

6 . COVERINGSy HARDY-LITTLEWOOD AND DIFFERENTIATION

128

We have, c a l l i n g

fXA

k

=

fk

, f

.

= fk t f

assumed t o be a d e n s i t y b a s i s , f o r a l m o s t e v e r y x, D( So f o r each X > 0,

J

Since

is

fk*x) = fk(x)

The f i r s t t e r m 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 z e r o by h y p o t h e s i s and we g e t

i(

I

f,x)

= f(x)

almost everywhere.

k *

m.

The second one because

a l m o s t everywhere. S i m i l a r l y

f

D((

e L 1 ( R n ) . So f,x)

= f(x)

T h i s concludes t h e p r o o f o f t h e theorem.

With t h e p r e v i o u s theorems i t i s v e r y easy t o g i v e a c h a r a c t e r ization o f basis differentiating

L'(R")

i n terms o f t h e maximal o p e r a t o r

i n t h e s t y l e o f Busemann and F e l l e 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 ) FOX each X > 0 , each f e L1(Rn) , each nonirzc/rea,&uj t A k ) nuch t h a t ] A k [ 0 , a d each numehicd btqUc?nCe 06 meanmabLe A & -+

6.4.

DIFFERENTIATION AND THE MAXIMAL OPERATOR

129

0

Pmod.

I f any o f t h e t h r e e cond t i o n s ( a ) , (b

i s a d e n s i t y b a s i s , by Theorem 6.4.1. consequence o f Theorem

,

( c ) holds, then

The theorem i s t h e n a d i r e c t

6.4.6.

When one assumes t h a t t h e b a s i s

fi

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 r e c e d i n g 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 SUP

I n t h e p r o o f o f t h e 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 . cantdined in a dixed cubic i n t e h v d Q c R n and nuch thcLt C I A k \ = poi& in R n and a n e t S l u i t h Then t h m e LA a nequence Cxkl 06 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 0 6 t h e dahni xk + Ak.

130

6 . C O VER I N G Sy HARDY-LITTLEWOOD AND DIFFERENTIATION

Phuoi) 0 6 t h e Theohem 6.4.8. That ( b ) implies (a) i s a s i m p l e consequence o f Theorem 6.4.7. I n order t o prove t h a t (a) implies (b)

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

t h e h e txht paniaXwe

Q

c o 1 z ~ t a n t 5 c = c ( Q ) r = r ( Q ) nuch ,that doh each non negative w a h nuppoht in Q arid M C ~ > 0 we have

Assume t h a t (b*) does n o t h o l d . t h a t for each p a i r o f c o n s t a n t s f k E L 1 supported i n

Ek = { x Satisfies

1 ~ > ~ ck 1

ckY rk > 0

and a l s o

Q

8

Rn : M

,/

a r e l e s s than t h e s i d e - l e n g t h o f

Q* t h e c u b i c

EkCQ*

k

We can choose f o r each G

h k l E k l G 21Q*1.

L1

is a f i x e d Q such

such t h a t t h e s e t fk(x)

rk

> Xk}

r k + 0, such t h a t a l l numbers r k and l e t

ck = 2k

.

We c a l l gk = f k / X k

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

times i t s s i z e . C l e a r l y

1Q*1

Q y

6

t h e r e i s a nonnegative

Xk > 0

a

L e t us t a k e a sequence i r k } and

Then t h e r e

f

Q

and t h r e e

and

a positive integer

hk

such t h a t

So we have m

m

We c o n s i d e r t h e sequence {Ah} o f s e t s c o n t a i n e d i n

by r e p e a t i n g

E i , E:, where

i

Ek = Ek

hk

times each

...

h, El

, E:,

f o r each

j

Ek

Et,.. with

, i.e.

.,

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

h

E:,

E2'

1G j

Q* o b t a i n e d

$

hk

E:,

.

..., Eh33 , Eiy. .. Since

6.4. DIFFERENTIATION AND THE MAXIMAL OPERATOR m

131

m

and a l l s e t s a r e c o n t a i n e d i n

Q*

we can a p p l y Lemma 6.4.9.

We thus o b t a i n

the points

... , x,h l , x i ,

x i , x;, and a s e t of

x;,

f o r each

x2

w i t h p o s i t i v e measure c o n t a i n e d i n

S

i s i n i n f i n i t e l y many o f t h e s e t s

S

h2

...,

k

and each

j

=

Ejk

.

, xi7 Q*

x;,

h3

..., x g

, x;

such t h a t each p o i n t

We d e f i n e t h e f u n c t i o n s ,

172,...,hk,

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

and,since

Let

Each

hk[Ek/

R B@

s

6

S

.

g

2 [ Q * l and

k I E k ( > 2 [Igk[ll

, we

get

We can o b v i o u s l y w r i t e

belongs t o an i n f i n i t e number o f s e t s o f t h e f o r m

L e t t h e s e s e t s be

,...

EJk

.

6. COVERINGS, HARDY-LITTLEWOOD AND DIFFERENTIATION

132

with

Rh

-f

, there

E;

By t h e d e f i n i t i o n of t h e s e t s

{RhlC@(s),

such t h a t , because o f t h e above e q u a l i t y ,

s,

f >

so t h a t

L e t us choose now ak

ck = l -akZk F o r example, l e t us s e t s

i s t h e n a sequence

e S we have 6 (

f,s)

f

and a t t h e same t i m e

ak +

< a .

ak = Zk” =

..

m = 1,2,

for

.

. This

m

f

Then we o b t a i n

8

contradicts (a).

L’ and a t each Hence ( a ) i m p l i e s

(b*). 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 , by the invariance by t r a n s l a t i o n s o f o f (b*)

a

do n o t depend on t h e p l a c e i n

I t i s a l s o c l e a r t h a t Mr,2f that

r(Q)

that

f a 0

Q

t h a t the constants where

6 Mrf

Q

L1

i s a function i n

c(Q)

, r(Q)

i s located.

and so we assume i n (b*)

i s l e s s than h a l f t h e l e n g t h of t h e s i d e o f

Q.

Assume now

w i t h support contained i n i n f i n i t e l y

each’one o f them equal i n s i z e {QjIjZ1

many d i s j o i n t c u b i c i n t e r v a l s to

, Rn

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 than h a l f t h e s i d e l e n g t h o f

4.

Then, if r

Q, we c l e a r l y have m

and t h e s e t s

H j

i s , as we have assumed,less

a r e d i s j o i n t . Hence

CU

DIFFERENTIATION AND THE MAXIMAL OPERATOR

6.4. Now, f o r an a r b i t r a r y each

f E L1

, f a 0 , we can s e t f

=

fh i s o f t h e t y p e 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

supports and a ( n )

The r e s t r i c t i o n

a(n)

1

h=l

133 fh

where

f h have d i s j o i n t

depends o n l y on t h e dimension. Thus

f > 0

i s t r i v i a l l y removed and s o 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.

THEOREM. 1e.L be a B - F b a A t h a A ' A homothecy Then t h e Awo 6 o ~ Y o w i n gConditioMn me eqLLiudevCt:

6.4.10.

invutiant.

i~ 0 6 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 The maximd o p e ~ a A o f i M

(b)

Phd.

06

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 -

, c o n d i t i o n ( b ) o f Theorem 6.4.8.

i a n t basis

implies condition (b) o f

t h i s theorem. T h e r e f o r e , we assume t h a t t h e r e e x i s t

L'

t h a t f o r each

f 6

Take a number

p > 0

by s e t t i n g , f o r

and

,

and

r > 0 such

A > 0 we have

and a f u n c t i o n

x E Rn

c > 0

@ E L'.

D e f i n e a new f u n c t i o n

f

6 . COVERINGS, HARDY-LITTLEWOOD AND DIFFERENTIATION

134

Observe t h a t c

\

1- f (Ax ) d x

In p a r t i c u l a r , of course, set

447 P x) =

c

f e L’.

If

1

-

y e Rn

I$ 1

;1 R I ( F I n

1

R

1

I dz.

R e ?3,(y)

, we

l I d x ~= ~ P( r - ) ~

i

RI(F)n

If(x)l

and

I

(F)nd x

\f($z)\dz

R

=

=

r This provesthat MP(P (y) = M f ( r y ) , s i n c e R e 8 (“ y ) and when Pr P runs over a l l % ( y ) , t h e s e t - R runs over a l l ‘13 Pr (- y ) . P P r P

I C Y e Rn =

: Mp $ ( y ) > : Mrf(z) >

/{!Z

c

X)l

=

XI1 =

(by hypothesis)

R

I C Y e Rn : M f ( T y )

(F)n I{z

r P

: Mrf(z) > A

G

This proves t h a t f o r any p > 0 and any 4

with the same constant.

can

8 L’

we g e t

Hence f o r each $ E. L ’ ,

and t h i s concl udes t h e proof of t h e theorem. The type of Theorems presented in this Section C i s not neces s a r i l y connected w i t h t h e d i f f e r e n t i a t i o n of a f i n e m space. Rubio [1971] and Peral [1974] have obtained r e s u l t s concerning weak type p r o p e r t i e s

6.4.

DIFFERENTIATION AND THE MAXIMAL OPERATOR

135

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 c o r r e s p o n d i n g b a s i s d i f f e r e n t i a t e s a space

$(L).

We s h a l l c o n s i d e r here one theorem o f t h i s

t y p e due t o Rubio.

6.4.11. THEOREM. L e t $ : [0,.3] + [O,m] be a n&tiotey inmeaning aunctian w L t h $ ( O ) = 0 and buch thaR $ ( u ) ud m d e h ghe&eh than o h equal t o t h e o/rde,t ol; u when u m . Le,t $ ( L ) be t h e coLLecfion 0 6 meanmubLe 6unctioMh f : Rn R nuch t h a t $ ( I f 1 ) < m. 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 cc t> 0 buch thaR Qvh each A > O and each f E $ ( L ) , f 2 0 one h a

I

-+

-+

Phood. ck > 0

Assume t h a t t h e theorem i s n o t t r u e . Then, f o r each

there e x i s t

fk

L e t us c a l l

E

gk =

$(L)

,

fk

.

fk > 0

> 0

and

We t a k e a sequence

such t h a t

,

{ck}

ck > 0

such t h a t

There e x i s t s a sequence

( r k l , rk > 0

There i s a l s o a compact subset that

l E k l > ck

k.

$(gk)

.

By u s i n g Lemma 6.4.4. h d i s j o i n t sequence E E k l hl,

Ek

of

,

such t h a t

iM

gk >

rk We c o n s i d e r t h e open u n i t cube

we cover almost c o m p l e t e l y

o f s e t s homothetic

.

to

Q Ek

Q

11 such and a f i x e d

b y means o f a contained i n

Q

be t h e homothecy t h a t c a r r i e s and o f diameter l e s s than l / k Let h pk h 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 b y s e t t i n g

.

136

6. COVERINGS, HARDY-LITTLEWOOD AND DIFFERENTIATION

D e f i n e then

and f i n a l l y

However, D ( $(L)

, we

f = sup Sk k

f

f,x)

get

.

One e a s i l y g e t s

a 1 a t a l m o s t each f(x)

> 1

f E $ (L)

x e Q

a t almost each

. Since x E Q

and

@ differentiates

and so

,

T h i s c o n t r a d i c t i o n proves t h e theorem.

6.5.

DIFFERENTIATION PROPERTIES IMPLY COVERING PROPERTIES.

Since t h e b e g i n n i n g o f t h e d i ' f f e r e n t i a t i o n t h e o r y , s e v e r a l i n t e r e s t i n g theorems have been f o r m u l a t e d t h a t p e r m i t t o deduce u s e f u l c o v e r i n g p r o p e r t i e s from 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

Such a r e t h e r e s u l t s

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. Fefferman [19771 have amply extended t h e scope o f t h e o r i g i n a l theorems.

As one can observe,

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 .

I n order t o

o b t a i n an economical c o v e r i n g from a given, perhaps h i g h l y redundant, c o v e 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 t h o s e 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 small i n some sense. T h i s sparse c o v e r i s then shown, u s i n g 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 , t o cover t h e

6.5.

DIFFERENTIATION IMPLIES COVERING PROPERTIES

137

original set. We f i r s t p r e s e n 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 dens i t y bases by means o f a c o v e r i n g p r o p e r t y . Hayes and P a w

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-

particular function.

LPmn)

a c t e r i z i n g t h e bases t h a t d i f f e r e n t i a t e that f o r a

B-f

L1mn)

1 < p < mY’n terms o f

y

F i n a l l y we o f f e r a r e s u l t o f M o r i y d n [1975]

a covering property. tion of

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

[1955]

proving

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 i s equivalent t o the V i t a l i property.

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 set

‘2

A

9,

and a d i f f e r e n t i a t i o n b a s i s

of

sequence

@ i s a VLtitaei c o v a

%

{Bk(x)}c

Prrovd.

of

A

such t h a t

Let

G

i f f o r each

x

S(Bk(x))

0,

-f

E

A

there i s a

i s a d e n s i t y b a s i s . We t r y t o

L e t us assume t h a t

prove p r o p e r t y ( P ) .

we s h a l l say t h a t a s u b f a m i l y

be open, w i t h

G3 A

such t h a t

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 contained i n the s e t

G.

L e t us t a k e

a

with SUP

c

E

are

?

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

0


a(R(1

6. COVERINGS, HARDY-LITTLEWOOD AND D I F F E R E N T I A T I O N

138

Since

/ A / > 0 a n d f o r each

i t i s clear that

IRk(x)3c

p1 > 0 . We take

R1

@(x)

"t

8

with

RL

E

2.

If

{R,}

and so on.

we have

=

0

t h e process of

We obtain a sequence

{RkIk,l

In order t o see t h a t {Rk3 s a t i s f i e s ( b ) , we f i r s t observe A k ) 1'1 ( R . I1 A . ) = 4 if k # j , and so we can w r i t e J

J

i s f i n i t e , we c l e a r l y have

\A

-

II

k

Rkl

= 0

.

Assume t h a t

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 k 4 and so pk < 3 \ R k J i s such t h a t pk + 0 . Let us c a l l A,=A IRk}

Assume

x

such t h a t

Define now A3 = A 2 - R2 f i n i t e or i n f i n i t e .

(Rk (1

+

such t h a t

AP = A1 - R1 . I f / A 2 [ Let us c a l l A = Al and s e l e c t i n g Rk i s f i n i s h e d . Otherwise we define

a n d we s e l e c t

Rk(x)

\A,/

>

0.

Then, i f we define

-,0

- I)

k

Rk.

139

DIFFERENTIATION IMPLIES COVERING P R O P E R r I E S

6.5.

we c l e a r l y have

p,

>

,

0

p,

pk

,c

k.

f o r each

This contradiction

proves t h a t I A - (I R k l = 0. For t h e p r o o f of c1

Hence, i f we choose

so t h a t

c1

we can w r i t e , because of t h e i n e q u a l i t y

(c)

we have o b t a i n e d , I A l 2

,

(Rkl

;(

and because o f ( a ) ,

1 - 1)

IAl G

we o b t a i n ( c ) .

E,

9

The second p a r t o f t h e theorem i s easy. Assume t h a t f i e s property

(P)

f o r each

We want t o prove t h a t t o Theorem

6.4.1.,

Let

M

A

measurable s e t

with

0

< \A\


1

>]A1

f o r each

> 1. We can extend

$(u)

A

,

f o r each

$(u) = u

by

[O,m)

(1,~).

u > 1 we have

IAl > 0

and t h e r e f o r e

setting

u

for

u c:

i s f i n i t e a t each

measurable w i t h to

$

a.

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 differently. cubic c1

and

I n ’ f a c t , the halo function

intervals i n cz

behaves l i k e

independent o f

The h a l o f u n c t i o n like

Rn

u ( l + log’u)”’.

$2(u)

u

$l(u)

o f the basis

of

such t h a t

o f the basis

74

o f intervals i n

I n f a c t , we s h a l l see i n Chapter 7, $2(u)

31

u, i . e . t h e r e e x i s t two c o n s t a n t s

c c u ( l + log+

Rn

behaves

,

150

6 . COVERINGSy HARDY-LITTLEWOOD AND DIFFERENTIATION

The other i n e q u a l i t y r e s u l t s very e a s i l y by considering i s the unit cubic i n t e r v a l . One e a s i l y f i n d s x c*

u(l

f

log+

Mx4

where Q

U)n-l

The halo function 4 1 ~o f the b a s i s 53 of a l l rectangles i s ill f i n i t e a t each u > 1, as we sha 1 s e e in ChaDter 8.

¶a2

On t h e other hand 3 ,d i f f e r e n t i a t e s L 1 differentiates L ( l t log' L)"' ( R n ) and 8 does not d i f f e r e n t i a t e s a l l the characteri s t i c functions of measurable s e t s . I t seems c l e a r t h a t t h e order of growth of $ a t infin.ity can give important information about the 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 8 . So a r i s e s t h e following question : Knowing t h e halo function @ of 33 i n v a r i a n t by homothecies find out a minimal condition on f e L l o c ( R n ) i n order t o ensure t h a t differentiates More p r e c i s e l y , t h e natural conjecture, looking a t t h e p i c t u r e described above, seems t o be t h a t i f 8 i s i n v a r i a n t by homothecies and q~ i s i t s halo function, then 13 differentiates $(L) We s h a l l c a l l this the "halo conjecture".Perhaps Gl;63 2 y % have a very p a r t i c u l a r geometric s t r u c t u r e in order t o jus t i f y the conjecture. The problem suggested by the halo function i s s t i l l open.

If.

.

I t will be useful t o look a t t h e 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 t h e following sense: For each u E ( 1 , ~ )and each A bounded measurable, with ( A 1 > 0 , one has

@ ( u ) being the best possible constant s a t i s f y i n g t h i s f o r a l l such s e t s A. We want to.prove t h a t M s a t i s f i e s a l s o a non-restricted weak type @ i n e q u a l i t y , i . e . f o r each f 6 L l o c and f o r each A, > 0 one has

151

6.6. THE HALO PROBLEM In what follows we s h a l l present some r e s u l t s r e l a t e d t o t h e halo problem. F i r s t we deduce some easy p r o p e r t i e s of t h e halo function. In ( B ) we present a r e s u l t of Hayes 119661 , t h a t i s r a t h e 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 s h a l l o f f e r some remarks t h a t might be useful in order t o a t t a c k t h e problem.

We consider a B - F basis t h a t i s homothecy i n v a r i a n t and s a t i s f i e s t h e d e n s i t y property. From the d e f i n i t i o n

u Mu)

,

if

u E [0,1']

=

: A bounded, measurable,

we see t h a t 8

] A / > 0)

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

be a s e t homothetic t o

B*

Then CMXB >

Since

E

> 0

1 ; 1 3

B*

B such t h a t

B * 3 B and

and t h e r e f o r e

i s arbitrary $(u)

2

u.

The following property i s more i n t e r e s t i n g from t h e point o f view of t h e d i f f e r e n t i a t i o n theory. 6.6.1. THEOREM. L e R '@ be a B - F ba&h t h a t .LA inulVLiAnt by hornathecia and oati06iecl t h e d e r k t q pkvpt%tq. 1eA o:[O,m) +[O,m) d o t u + m, $ be a nvndecfieixbing ~uncLLvnouch t h a t +

.-$#

6. COVERINGS, HARDY-LITTLEWOOD AND DIFFERENTIATION

152

being t h e halo duncfion

Pmud.

0 6 6 3 . Then @ doen n o t diddehentLaLte

According t o 6.4.10

t h e n we have, f o r each

f

with

f

c

independent o f L e t us choose

L

B

if

and each

X

4 differentiates

u(L)

,

> 0

X.

and

uo

,

u(L).

such t h a t

Then t h e r e e x i s t s a s e t

+(uo)

>

c.

A, measurable and bounded, w i t h

IAl> 0,

such t h a t

and t h i s c o n t r a d i c t s t h e p r e c e d i n g 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

B.

A henub!

06

f =

xA, h

=

1 uo .

u(L).

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 a p p r o x i m a t i o n t o t h e halo conjecture.

I t i s e s s e n t i a l l y due t o Hayes [1966]

i n a context a

l i t t l e more general and a b s t r a c t than t h e one we s e t here.

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 dunction 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 dunction nuch t h a t a(0) = 0 , 6.6.2.

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.

Phuod.

153

THE HALO PROBLEM

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 values a r e e i t h e r

If

f

0

o r bigger t h a t

g

i s a f u n c t i o n such t h a t i t s

1, t h e n we have

i s n o t n e c e s s a r i l y non-negative,

I n o r d e r t o prove (*), l e t

and l e t us c a l l , f o r

k = 1,2,...,

c1

t h e n we can s e t

> 1 be such t h a t

6. COVERINGS, HARDY-LITTLEWOOD AND DIFFERENTIATION

154 We can w r i t e

a,

since, i f

x

i s such t h a t

then, f o r each

B

E

MXk(X)
0,

we o b t a i n

T h i s 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 differentiability results. take

a(u) = u ( l t log'

L e t f o r example be

u)'+€

+ ( u ) 6 cu

.

We can

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 c o r r e s p o n d i n g b a s i s differentiates

L ( l + log* 1 ) I t E

L e t now

$(u) s

.

c u ( l t logt 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

differentiates

L ( l + log’ L)’+€

As one can see, Theorem 6.6.2. does not give in t h e s e cases the b e s 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 s h a 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

06 t h e e x h a p o l a t i o n mdhod

06

Yano.

A straightforward a p p l i c a t i o n o f t h e e x t r a p o l a t i o n method of Yano presented i n 3.5.1. gives us t h e following r e s u l t . MoriyBn “781 has r e f i n e d i t i n order t o deal with t h e e x t r a p o l a t i o n 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 bi 0an A

i n Rn and f ~ 2 + be .iA h d a 6uncaXon. Annume t h a t , s > 0 and ha& each p , u X h 1 < p < 2 , we. have

D. Some.

/rematrhn

6011.

name 6ixed

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 t h e s o l u t i o n of t h e halo problem, s i n c e 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. ~ 3 2h d o ~uncLLon0 we have

6. COVERINGS, HARDY-LITTLEWOOD AND DIFFERENTIATION

156

t h e n t h e h u b conjeotwre

A

@.!-be.

( 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 $ ( 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 -

Pfioud. ferentiate entiates

If

i f and o n l y i f

70

f

differentiates

2f.

But we have

u

for

-f

entiate

for

'8

where

m

.

Therefore, b y what we have seen i n A

o(L)

. This

(b)

According t o

, @ does n o t d i f f e r -

c o n t r a d i c t i o n proves ( a ) . (a), i f t h e halo conjecture i s true, then

one has

c

i s a c o n s t a n t independent o f

non decreasing, we have, i f

Hence, if c = 2p

, we

get

u.

Therefore, since

@(u) is

i s an i n t e g e r b i g g e r t h a n 1,

k

k @(Z )
0

we have

whehe

c

+ log

~2a ~ O A L C condtant, ~ V ~ independent v d

a = 0

id

P4vv

0 6 a

6.

c 1 and

+ log

f

-id

a = log a

and

A

, and

a > 1.

We p r e s e n t here a proof o f t h e theorem d i s r e g a r d i n g

t h e easy, b u t t e d ous , measurabi 1 it 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 b y measure o f t h e measurable s e t

P t R1

and

/ P I 1 and 1Q/ t h e Lebesgue 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

(El, 0

Tlf(x1,x2)

A

for

> 0

,

f 6 Lloc(R2)

= sup{

.

For

1

K J f(E,1,x2)dE,1

x 2 ) e R2 we d e f i n e : J interval o f

R1,x’e

J}

we c o n s i d e r t h e s e t

and a g a i n we d e f i n e , f o r T 2 f ( x 1 , x 2 ) = sup{

(x1,x2) Q R 2

xA(x1,n2) Tlf(xi,n2)dn2:

We s h a l l f i r s t prove t h e r e 1a t i o n

B

(xl,

= { (5’ , C 2 ) 6

R2: Mzf(E1 , E ’ ) > A

H i n t e r v a l o f R’,x26HI

Take a f i x e d p o i n t (x1,x2) a C.

R2

Since x1a J

such t h a t

(x1,x2)

, x2e H

I

i n t o two s e t s J

J x {y2} we have Tlf

(z’,y2) >

Otherwise, i . e . i f t h e r e i s some p o i n t

x

c

(z’,y2)

C1,

C z , each one b e i n g Ox’ i n t h e

parallel t o the axis

L e t J x { y z } be one o f such segments.

f o l l o w i n g way. E

B. We w i s h t o p r o v e t h a t I = J x H of

and

a union o f segments o f t h e s i z e of (z’,yz)

of

(x1,x2) a B , t h e r e i s an i n t e r v a l

We now p a r t i t i o n t h e i n t e r v a l

Tlf

INTERVALS

7. THE B A S I S OF

16 2

we s e t

J

x

A 2

we s e t

I f f o r each p o i n t J x { y 2 } c C1.

(z1,y2) E J x { y z } such t h a t

{ y 2 } c C 2 . Observe t h a t

J

x {y2}CCz

implies i n particular that

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 o v e r t h e s e t such t h a t

J

x

{c21c

G

of a l l

4’ i n H

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. By t h e d e f i n i t i o n o f

C1

DIFFERENTIATION PROPERTIES and A, i f

(q1,r12) E

C1

then

163

(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 o f 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

satisfies

C

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 . f E L ( l t log

t

L), since otherwise there 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 We can assume

.

9

By v i r t u e o f t h e weak t y p e ( 1 , l ) f o r t h e u n i d i m e n s i o n a l b a s i s 1

o f i n t e r v a l s , f o r a l m o s t each f i x e d

Hence, i f we i n t e g r a t e over a l l such

X’E

x’e R we can w r i t e

R

and i n t e r c h a n g e 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

If

we have

(t1,c2)

6

A

,

then

Tlf( 0

f*(t1,t2: 5 ) such t h a t

For brevity l e t us w r i t e

and T l f z

6

A5

Hence,

f =

fz

+ f:

.

I t i s clear t h a t

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.

7. THE B A S I S OF INTERVALS

166

Ij

Each

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

t i v e p a r t of

0

,

a s i d e on t h e posi-

Ox w i t h l e n g t h j, and a n o t h e r on t h e p o s i t i v e p a r t o f

.

H with length Hence t h e area o f I j j H E= 0 I j i s 1 and t h a t o f t h e u n i o n

is

H, t h a t o f t h e i n t e r s e c t i o n

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 CM2

.

s ince

XE

1 %I

>

JH

3

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

ClU(1

&(R2)

@ z of

+ l o g + u)

h

satisfies

@2(u) G

czu( 1 + l o g + u)

G

h

and analogously i n Rn ClU(1 + log+

@Z(U)

c2u(1 + log+

Hence, a c c o r d i n g 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 deduce t h a t ?Bz does n o t d i f f e r e n t i a t e any space worse.than L(l

t

log'

L)"'

(R'),

i.e.

if

)I

: [0,m)

+ .

Oy

[O,m)

i s such t h a t

we

7.3. SAKS RARITY THEOREM

167

as

does n o t d i f

+(L).

ferentiate

[1935] has proved a s t r o n g e r r e s u l t .

Saks

f

functions one has

&mn)

, then

+ ( u ) = ' u ( 1 + l o g t uln-'-€

f o r instance

U + m

[I( I f , x )

t r u c t i o n of

, "almost

L1

of

= +

H.Bohr

,

a l l " i n t h e sense o f B a i r e ' s c a t e g o r y

x

a t each

m

. Here we

F o r "alniost a l l "

6

. The

Rn

p r o o f o f Saks uses a c o n s

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 S e c t i o n 1.

.

7.3.1.

that

.

L'

n

=

2

D ( /f,x)

.


0

R and such t h a t [GI b E . F o r we t a k e an open c u b i c i n t e r v a l I ( x ) c e n t e r e d a t x w i t h

we can t a k e an open s e t each

B(O,N)

we can cover almost c o m p l e t e l y t h e b a l l

diameter less than

G

1/N

containing

contained i n

theorem o f B e s i c o v i t c h o b t a i n i n g

G.

We a p p l y t o

( I ( X ) ) ~ € t~h e

{Ik} so t h a t

0 b e i n g an a b s o l u t e c o n s t a n t . L e t us c a l l

Ek

E

the s e t obtained from

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

JH

o f the auxiliary into

define the following functions

I

0

if

x

6 Ek

Sk.

We now

170

7. THE B A S I S OF INTERVALS

Then

and so

if Now i f

less t h a n

If Of

Pk

,

and

=

4N2@

i s i n some s k , then i t w i l l be in some o f j = 1,2,...y H composing Sk and IJkhas diameter

So we get

1/N.

where a ( H )

1

x E B(O,N)

I:

the intervals

E < - ,

1 1+ + 2

x

B

ON

7

Sk

.. . + R . If

then

x

€

Ik

we choose H so that

for some k and by the definition

7.4. A THEOREM OF BESICOVITCH

@N >

171

llkl*

'k T h i s 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 B 2 .

We c o n s i d e r t h e b a s i s

8 c 6,and t o the basis

@,

9

a t almost e v e r y

2

differentiates

@I

/

2

.

R2

in

f e L'(R2).

Let

Since

f, i t i s easy t o see t h a t w i t h r e s p e c t

we have

x e

R2

.

However, a c c o r d i n g t o t h e p r e v i o u s s e c t i o n s , i t

can happen t h a t t h e s e t s

R2 : f ( x ) < D ( / f , x ) }

CX

6

{X

e R2 : D(1 f,x)

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]

have p o s i t i v e measure.

posed t h e f o l l o w i n g q u e s t i o n : Can a n y

CX

E R2 : f ( x )
0

f o r each

and

differentiates c > 0

O(L), a c c o r d i n g t o Theorem 6.4.8.

such t h a t f o r each measurable f u n c t i o n

A>O ICY

Hence,

E R’

if x

6

: Mrf(y)

>

All r

x Q = min(1,T)

&

c

i

x

@(8)ds

k l ( 1 -+ log’

f

and

(**)

, h a v i n g i n t o account ( * ) and

I f we s e t I ~ x =- 1~,”t h e n we g e t f o r h > l =o 16 ~ ( h )>

there

,

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 statement 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 t h e 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 A c c o r d i n g t o Theorem

6.3.1.

185

of C6rdoba and Hayes, t h e system o f i n t e r

v a l s i n R 2 s a t i s f i e s a good c o v e r i n g p r o p e r t y : Given any c o l l e c t i o n o f i n t e r v a l s we can choose a f i n i t e sequence 11,)

with

c1,c2

independent o f

from

(Ia)aeA (Ia)asA

such t h a t 1 ~ ~ 1 ~ 1 satisfying

( I a ) a e A. 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 s m a l l

overlap.

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 e

2

,1/2

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

+

@ ( u ) = ( l + l o g LI), and n o t l i k e

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

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

and so i t was f o r m u l a t e d i n Guzmdn [1975, p.1651 1B2

.

I n a s i m i l a r way f o r

i n Rn t h e r i g h t o v e r l a p p i n g i n e q u a l 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 i n t e r v a s was o b t a i n e d by Cdrdoba and R.Fefferman [1976]

.

Here we s h a l l

p r e s e n t t h e easy geometric p r o o f t h a t they g i v e of t h i s theorem f o r

R? R2 , i.e. the intervals o b t a i n e d by t h e C a r t e s i a n p r o d u c t of d y a d i c i n t e r v a l s o f R ' . There i s We cons der t h e system o f d y a d i c i n t e r v a l s o f

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 r e s p e c t s 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 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 intervals.

M

w i t h r e s p e c t t o t h i s system o f

7. THE B A S I S OF INTERVALS

186

i s any measurable s e t o f R ?

A

where

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.

LeY

LEMMA.

be. a 6 i n i i c crcyuence

{Bk}

R 2 .Then we can creXe.ot 6hvm them

tehvden ad

(b)

ffem c

(2)

I B 2 II R]:

R:l

If ( B z "

1 5

L

(1) I f

I B 3 I\

If

IB3 I)

1 5

>

(821

2

(I

j=1

Rl:

, then

= B,

, and

we s e t

R;

now t h a t ( 2 )

I!

j=1

R?

J

happens.

/Rk]

(BI,

11

(2)

(B,

11

1J

j =1 2

(I

j=1

I n t h i s way we o b t a i n aside, we have

look a t

B,

.

.

= BZ

BZ aside.

B3.

1 5 IB3I ,

>

51

IB31

R';

then

, then

We l o o k a t

2

(1) I f

we l e a v e

L

2

If

6

R:

, then

1B21

Assume (1) happens. We l o o k a t

(2)

dyadic i n -

an abno&u;te coM2atant. P ~ o a d . We choose f i r s t

(1) If

1

Rjl

k , IRk 0 LJ j#k

For each

06

C R k I no th&

we l e a v e

B3 aside.

Assume

BI,

Rj\

c

51

(B4(

, then

Rj1

>

1

IB41

, we

tRi} Lk=l

= B3

. For

R:

= B4

leave

each

B

B, j

aside.

t h a t has been l e f t

7 . 7 . COVERING PROPERTIES and so

( f o r the chosen ones t h i s i s obvious)

On the other hand, f o r each

k

- L fRklk=l

We now consider the sequence

( i . e . we reverse the order of {RE1 have done with

LEMMA.

7.7.2.

r

Then auk each

c

=

-

RP =

-

Rt-l,.. .

CRkIkz1

as we

that satisfy

LeI S

R1 nuch

= OYly2,3,

=

that

...

{I,) 404

be a &hi.Xe oequence each I k

0 6 diddehent

we have

we have

an a b n a U e cvnbAant. P400d.

A,

-"

RI= R r

) and proceed with

iRk)

{ B k l obtaining

dyadic inte~vu.42 0 4

whme

where

We now prove the following easy lemma f o r dyadic i n t e r v a l s i n

.

R1

187

{x :

lx

Ik

We can c l e a r l y w r i t e

(x) 2

r + 11

= IJ { x E

k

Ik : x

belongs t o a t l e a s t

r

188 sets

7 . THE B A S I S OF INTERVALS

rj

contained i n

If

S = {Ik}

of

.

k

= ] I k ' ) II IjCI,

r = 1 we have

k and f o r r

a subset

, and we t r y t o prove IICI

r Ik

IJ

f

Assume now t h a t t h e i n e q u a l i t y I I [ ~ G

hypothesis. any

Ikl

.

1;

= 1,2

,...,h

The s e t

1;

.

Let

r = h

1.1 c J

-+ 5

51 ] I k ]

lIkj

6

-115r1

by t h e

i s true for

+1 and l o o k a t 1";

as

i s a d i s j o i n t u n i o n o f elements of

Let

r;

=

N

II

1=1

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 h y p o t h e s i s ,

Adding up over

k

But, if we assume t h a t t h e

Ik

a r e o r d e r e d by s i z e 1111 >, 1121

2

... ,

I.

7.7. IA,I

Therefore

6

C -

!jr

11)

COVERING PROPERTIES

189

Ikl .

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 1

w i t h measure g r e a t e r t h a n o b t a i n i n g a f i n i t e sequence

1uB,I,

EB,}

Then we a p p l y lemma

from

(Ba)acA

7.7.1.

t o {Bk}

{Rk} s a t i s f y i n g

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 i n a n o t h e r o f {Rk).Let parallet to to

Ox

Ox

and

us c a l l Oy

ak ,bk

respectively.

{Rk}

i s contained

the length o f the sides o f L e t us

t a k e any l i n e

and so

bk
p or

J

=

EIJ

aL,,

Q

a

d s > p }

F

B u t , u s i n g the weak type ( 1 , l ) f o r t h e onedimens onal Hardy-Littlewood maximal operator and i n t e g r a t i n g we obtain

Therefore

=

CHAPTER 8 THE B A S I S OF RECTANGLES 1Bs

The b a s i s

B 3 o f a l l r e c t a n g l e s i n R 2 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 q u e s t i o n s . 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 b e g i n n i n g 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 s o l v e d 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

, motivated

by t h e i n t e r e s t t o understand

t h e geometric s t r u c t u r e o f Lebesgue measurable s e t s , c o n s t r u c t e d a r a t h e r paradoxical set.

The Nikodym s e t

of

(i.e.

f u l l measure

points

x E N

IN1

i s a subset o f t h e u n i t square i n R 2

N

= 1 ) , such t h a t through each one of i t s

there i s a straight l i n e

Q

One can say t h a t t h e v e r y t h i n complement N

l(x) 0N

l ( x ) so t h a t

has i n some sense many more p o i n t s than

- N

in

Q

=

{XI.

o f the t h i c k s e t

N i t s e l f . Zygmund (Cf.

the

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 ~ &is does n o t even d i f f e r e n -

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

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 y e a r s 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

infimum 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 t h e s e t so t h a t a t t h e end i t occupies the o r i g i n a l place b u t i n i n v e r t e d p o s i t i o n ? Almost s i m u l t a n e o u s l y B e s i c o v i t c h [19181 had s o l v e d an i n t e re s t i n g q u e s t i o n concerning t h e Riemann i n t e g r a l : Assume integrable function i n

R2. Is

f

i s a Riemann

i t then t r u e t h a t there i s a possible

199

8. THE B A S I S OF RECTANGLES

200 c h o i c e o f o r t h o g o n a l axes

Ox,Oy

Riemann i n t e g r a b l e f o r each y

f o r which t h e f u n c t i o n and

I

f(x,*)dx

f(*,y) i s

i s Riemann i n t e g r a b l e ?

To answer t h i s q u e s t i o n 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 . With t h i s set

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

B

q u e s t i o n i s n e g a t i v e . I n f a c t , we can assume t h a t

B

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 c o o r d i n a t e . subset o f

B

o f points w i t h

nuity points o f

xF

are i n

c o n t a i n s no v e r t i -

B

and so

xF

be t h e

i s Riemann i n t e g r a b l e i n

But i n each d i r e c t i o n t h e r e i s some segment c o n t a i n e d i n

xF

F

Let

a r a t i o n a l c o o r d i n a t e . Then t h e d i s c o n t i -

R2.

a l o n g which

B

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]

, the

set

B

gives also

a s o l u t i o n t o t h e needle problem: The i n f i n i m u m 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 b y Radeniacher [1962] and Schoenberg [1962]

I t s c o n n e c t i o n w i t h d i f f e r e n t i a t i o n t h e o r y was b r o u g h 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 h e f a c t t h a t

P,

is

n o t a d e n s i t y b a s 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 elementary 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] esting construction o f a Besicovitch set.

gave an i n t e r -

Before t h a t B e s i c o v i t c h [1964]

had e s t a b l i s h e d t h e c o n n e c t i o n o f s u c h ’ 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 geometric of l i n e a r l y measurable s e t s i n R2 ( s e t s o f H a u s d o r f f dimension 1). 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 s e 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 v e r y much l i g h t i n o r d e r t o g e t a deeper ‘understanding o f i m p o r t a n t geometric and m e a s u r e - t h e o r e 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 r e c t a n g l e s i n R2

.

8.1. THE PERRON TREE

20 1

As we s h a l l s e e , most of what we s h a l l present i n t h i s Chapter depends on w h a t we s h a l l c a l l t h e Perron t r e e ( t h e construction proposed by Perron [1928] i n order t o simplify t h a t of Besicovitch). Even a Nikodym set can be most e a s i l y b u i l t by means of i t . For t h i s reason we s h a l l present f i r s t t h i s fundamental construction and from i t we s h a 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 s h a 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 We s t a t e a l s o some i n t e r e s t i n g open problems i n t h i s area. [1976]

.

8.1. THE PERRON TREE. The construction we present here of t h e 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 w i l l make i t more useful f o r our purposes. 8.1.1. C A h )2n h=l

THEOREM.

Comidm .in

R 2 t h e 2n

open M a n g l u

o b h u k e d by joining Xthe paid ( 0 , l ) w d h t h e p o i &

(O,O),

LeA Ah be t h e M a n g l e w a h ueh'tice~ ( 1 , 0 ) , ( 2 , 0 ) , ( 3 , 0 ) , ...,( 2",0). ( 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 Ah h o XhcLt one h a 2n

I u

h=l

Ah/

c

The theorem w i l l be obtained by r e p e t i t i o n of t h e f o l lowing process t h a t , f o r reference purpose, we s h a l l c a l l t h e basic con s t r uc t i on. Pkood.

8 . THE B A S I S OF RECTANGLES

202

&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

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 b a s i s on towards

T1

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

from

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

T:

and

i s composed by a t r i a n g l e

tion i n

F i g . 8.1.2.)

Al,

One can e a s i l y g e t

A2.

homothetic t o IS/

and s o

=

~ t ’ / T 1 (I

T1 IJ T2

TP/

S ( n o t shaded por-

p l u s two “excess t r i a n g l e s “

THE PERRON TREE

8.1.

203

We s h a l l now a p p l y 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 (A1,Az)

,

(A3,A&)

,...,

(A

2"' 2"l

c o n s t r u c t i o n w i t h t h e same obtain the triangles

...;

A;,A;;A:,A~:

so t h a t i t becomes

S

.

,

and t h e excess t r i a n g l e s

2"l

We now s h i f t

adjacent t o

, . ,

To each p a i r we a p p l y t h e b a s i c

g i v e n i n t h e statement o f t h e theorem. We

ci

S1,SzY...,

s2

.

,A n ) 2

S1.

S2

along

Then we s h i f t

Ox

towards

S1

S3 t o p o s i t i o n

S 2 , and so on.

s3

I n these motions 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 d o i n g adjacent t o

S1

pairs o f adjacent t r i a n g l e s

Ij

i s equivalent t o s h i f t i n g the t r i a n g l e s positions

- -

A2, A 3 ,

-

...,A

Consider now

2n* T h i s f i g u r e i s composed by

2n-l

A2,A3

A 1 IJ

, t o some new

,...yA2n

-

AP 0

x3-.

-

...(I A

zn

.

S 1 , S P Y .. ,S2,,-1

triangles

*

, whose u n i o n

i s o f area

p l u s s h i f t e d excess t r i a n g l e s , whose u n i o n i s o f area n o t l a r g e r t h a n

The

2"'

triangles

the i n i t i a l t r i a n g l e s

A1,A2,A3,...,A2n.

process, always c a r r y i n g t h e the e n t i r e triangles

, SZn-l

Sl,i2,?3y...

A2,A3,

a r e i n t h e same s i t u a t i o n as One s u b j e c t s them t o t h e same

excess t r i a n g l e s so t h a t i n f a c t one moves

... ,Azn.

and a t t h e end one o b t a i n s a f i g u r e which i s composed b y a t r i a n g l e

H

T h i s process i s r e p e a t e d

A1 0

AP

homothetic t o

-

...

A3 Al

(J

A2

0

n

times

A

2n

... IIA 2"

o f area

p l u s a d d i t i o n a l t r i a n o l e s whose u n i o n has an area n o t l a r g e r t h a n

OF

8. THE B A S I S

204

Hence, i f we s e t

A1

=

Til , we

RECTANGLES

get

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

the situation 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 b e i n g 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.

So one e a s i l y a r r i v e s

t o the f o l l o w i n g r e s u l t .

8.1.2.

uny

E

be u XkLangRe o d meu

A B C

T I ,T2 , T3 dong

. . . , I 2n , ..., T 2n

,

B C

Theorem

n

8.1.1.

that carries p(Ah) = Th

dependn on

wLth baA

ZCJpVb&UMn

Pkoud. and t h e n t a k e

(n

so t h a t

aZn
__ 100 lRhl * -Pmad.

-

Figure 8 . 2 . 2 .

8.3. THE KAKEYA PROBLEM

209

8.3. THE KAKEYA PROBLEM. The s o l u t i o n o f t h e needle problem i s a l s o i n m e d i a t e 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 from 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 small as one wishes. I t i s enough t o observe i n F i g u r e 8.3.1.

A B t o A4 BI, sweeping o u t t h e area o f t h e

t h a t one can move

shaded p o r t i o n which can be made as small as one wishes t a k i n g

B2=B3

A3

64

A4

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 6 0 " w i t h 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 t h a n 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 p l a c e d so t h a t of

M N P

as b a s i s triangles

M N P

A B

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

i s i n the i n t e r i o r o f

i s b i g g e r than

N P 71,

and w i t h an 7 2 ,

...,$2n .

1. To M E

N

P

M N.

Observe t h a t t h e h e i g h t

we a p p l y Theorem

such t h a t

The segment

10

E


0

such

that

I C o + A C ~ =I o ~ Now t h i s i m p l i e s

ICo

Therefore

/Co a.e.

6

lR

1 C O + ACoIl= 0 s i n c e 7

+ A4n

+ XColl= 0

A

1

f o r a l m o s t every

4

C0l1

= 0

a.e. X E L0,m)

f o r a l m o s t each

1 4”

A

Co

E [0,4’a]

C O and

C1

E cola]

C O f o r each n.

. Hence

I n t h e same way we see than \ C O + XColi=O

and so we have shown t h a t t h e u n i o n o f

joining points o f

c

x

t h e whade fins

i s a c l o s e d s e t o f p l a n e measure zero.

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 o b t a i n e d 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 s e t .

8.5.1. LEMMA. L e t R be t h e cloned 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 fine 1 t o t h e b a A AB I!& q > 0 be given. Then AX i~ punnib& .to &CW a SLi t i t e numbm od pat~&dogtramn { wl, w 2 , . . , wH 3 w i t h one baA on AB

.

.

and a n o t h a one i n DC nu thcLt t h e y t u u a ABEF and '

D

r

A

B

F i g u r e 8.5.1.

Pmod.

i n 8.4

The p r o o f i s performed r e c a l l i n g t h e Remark 2

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 than

n/8

so t h a t high t h a t

. Also

IFEHGl lVLl

w1 = AJKD

we t a k e a l i n e 6

>

q/8 ILJ

. I

On

JK

GH

parallel to

s l i g h t l y above and above

DC

1

AD

o f area l e s s

and p a r a l l e l t o 1,

we t a k e a p o i n t

and t h e n we t a k e t h e t r i a n g l e

VLN

V

with

so

8. THE BASIS OF RECTANGLES

216

\L14\l 8.4.2.

.

1

=

I f we apply the construction o f the Perron t r e e of

to

V L N , according t o the Remark 2 in 8.4, the extension below PQSJ where SL i s parallel t o V N .

LN o f the small triangles o f the Perron t r e e will cover

D

G F

I

Figure 8.5.2.

Through Q we draw a l i n e parallel t o

AD a n d take V1 and the triangle 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 Q, i s beyond the midpoint o f EF. The Perron trees f o r the t r i a n l e s V . L . N are taken with E 50 J J j small t h a t t h e i r union i s of area less than o / 8 . We proceed now symmetrically s t a r t i n g from the side CB . So we get two s t r i p 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

8.5. THE NIKODYM SET

217

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 J a f i n i t e number o f s t r i p s c o n t a i n e d 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. right of

AD

t h e v e r t i c e s o f t h e small t r i a n g l e s a r e t o t h e

and t o t h e l e f t o f CB).

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 R1,R2 be a3.1~ d v n e d pm&&vgtam in 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

8.5.2.

R2 ouch LhaA

LEMMA.

R1

RZ

(1) F a t u c h k k

Let

.

i

=

l y 2, . . . , k

,

w i 11

R1 c

w

0

R2

2 18

8. THE B A S I S OF RECTANGLES

Pkwd.

T h i s lemma i s t h e p r e c e d i n g one i f

R1

and

R2

are

r e c t a n g l e s as i n F i g . 8.5.4.

R2

i

F i g u r e 8.5.4. We now proceed t o remove t h e r e s t r i c t i o n s on

R1

and

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

R2

. An R2

R1 and

a r e r e c t a n g l e s can be e a s i l y removed. holds i f

R1

and

T h e r e f o r e we know t h a t t h e lemma 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.

R2

Assume now t h a t and

E F G H

of Figure

R1

and

R2

are the parallelograms

8.5.6.

F i g u r e 8.5.5.

F

M

E

G

A

D F i g u r e 8.5.6.

H

N

A

B C

D

THE

8.5.

R2

We r e p l a c e

A D

M H

i s on

valid f o r

by

R1

RS

=

M F

N

H

R 2 c R;

and

.

and :R

s a t i s f y ( l ) , ( 2 ) and ( 3 ) Assume now t h a t

NIKODYM SET M F

such t h a t

.

219 i s parallel to

A

By

We a l r e a d y know t h a t t h e lemma i s

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

for

R1

and

R1

and

RZ

R2

.

a r e as i n F i g u r e 8.5.7.

, with

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

M

E

0

N

G

H

F i g u r e 8.5.7.

o f the s t r i p s Figure

GI, G z , ...,

-

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

wk

8.5.8.

P

F

E

w

I

j

Q

G

H

F i g u r e 8.5.8.

-

So we can now a p p l y 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

R2

with

~ / 2 k, amd we g e t for each

i = 1,2

¶...,

k

the strips

8 . THE BASIS OF RECTANGLES

220 4}j=1,2,..

. , r i . The

seen t o s a t i s f y

of a l l t h e s e s t r i p s

collection

(l), ( 2 ) and (3)

u?; i s

easily

.

F i n a l l y , i f R1 and R Z a r e i n t h e general s i t u a t i o n of t h e lemma one can s u b s t i t u t e R 2 by another parallelogram R: , R: 3 R,, with s i d e s p a r a l l e l t o those of R l and apply t h e lemma t o R1 and R: The s t r i p s we obtain a r e a l s o v a l i d f o r R 1 and Rz.

.

The second lemma we a r e going t o use i s an easy consequence o f t h e previous one. 8.5.3.

LEMMA.

Let

R1 and

R2 be

Awu d o d e d pamUehg/Zam

R 1 c R Z . L e t R be a &Lrtite coUecLLon a6 cloned h - t h i p h , , uk 3 , whobe union c o v m R 1 L e t E > 0 be given. Then, do& each h . t h i p u i y i = 1,2, ... , k one $an conbahuc2 anotheh dinite coUecLLon o d cloned h i x i p d W: , W; , . . . , u Ji i huch Zha.2, id we c a l l

buch t h a t

.

..

= {ol ,UP,.

R*

= {

W!

: i = 1 , 2 ,... , k

(2)

j = 1,2,..,,

Fon. each i and

j,

wij

ji

0 Rz

1 , we

c

wi

have:

.

From the foregoing lemmas we obtain t h e following r e s u l t , 8.5.4. THEOREM. Thehe AA i n R2 a b e t K huch t h a t doh each x E R2 thehe AA a na%thcLigkt f i n e t h a u g h x ha t h a t r ( x ) c K 1Jcx3.

06

nURe

memme

r ( x ) paAning

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 t h e u n i t square and N = Q - K , then I N 1 = 1 and f o r each x 6 N t h e l i n e r ( x ) s a t i s f i e s r(x) 0 N = 1x1,

8 . 5 . THE NIKODYM SET

221

Y m o A O X ;the Theohem 8 5.4.

Q(H) be t h e c l o s e d square i n t e r v a l c e n t e r e d 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 a p p l y Lemma 8.5.2. t o R1 = Q and R2 = Q(2) w i t h an s1/4 > 0 t h a t H > 0

For

let

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 i n t o f o u r equal c l o s e d 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

Qf ,

us denote them by

i = 1,2,3,4.

F i x an

Q

Q. L e t

i and a p p l y Lemma 8.5.3.

R 1 = Q j , R2 = Q(3) , n = nl , E = ~ ~ / 4> ’ 0. So we o b t a i n a 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 Rl L e t us s e t

with

.

4

n:

0 i=l

=

R2

i n t o f o u r equal c l o s e d square i n t e r v a l s , each i one h a l f t h e s i z e of So we o b t a i n 4’ squares Q2 i = l,2,...y42. i F i x an i and a p p l y a g a i n Lemma 8.5.3. , w i t h R 1 = Q2 ,R2 = Q(4) ,

Qj

We now d i v i d e each

R

= Q2

,E

~

=

~

t h a t we s h a l l denote

Ql .

. So / we4 a’, and

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

we w r i t e

And so on. Observe t h a t f o r a f i x e d k , t h e u n i o n o f a l l s t r i p = i i covers t h e square Qk-l F o r each w E R k we d e f i n e B = w

.

and l e t

Kk =

11

CB : w

Rkl

E

a c c o r d i n g t o Lemma

8.5.3.

and so we g e t I K k f )

Q(k)l K* =

We choose

E~

t h e n we t a k e

-f

j,

0

and s o

j > h,

.

c

lim inf

-

We have, by t h e c o n s t r u c t i o n o f

f o r each

sk

Kk =

/K*I = 0 j > N

i

.

E~

W

W

h=l

0 k=h

.

i Qk-l i

Rk

We now d e f i n e Kk.

I n f a c t , i f we f i x

we o b t a i n

Rk

N

and

h

and

BASIS OF RECTANGLES

8. THE

222

Hence

N, we g e t

Since t h i s h o l d s f o r each

IK*/

1

Kkl = 0

f)

k=h

p a s s i n g through

f i x e d and l e t

x

...

n = lY2,3,

Q:(xyn)

K* LJ {XI. Let x E Q

x

E

Qi(xyn) 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 o f r 2 z j ( x 3 2 ) c o n t a i n i n g Q(2)

w z (7

,=

containing

x

and such t h a t

a sequence o f l i n e s

{rk(x)}

0 ) and one has

-f

r e c t i o n s of t h e l i n e s through

{r,(x)}

w

each For

cw x ~

n. For

n = 2 there

and such t h a t

. So~

~

-

(because

of

wk

there exists

and such t h a t

uk tends t o z e r o

c

x.

t h e r e i s some s t r i p

wk (1 Q ( k )

uk I1 Q ( k )

rk(x)cwr

of t h e f a c t

-, one ~ has t h a t t h e d i -

converge t o t h e d i r e c t i o n o f a l i n e

r(x)

x. We now prove t h a t

r(x)

Then t h e r e i s a n a t u r a l number Y

N

There i s an

i > n

2

M

such t h a t

max(M,N)

c

if

K* 0 { x }

{yk}

and

n

y

E

r(x)

y

f x.

+

Y.

a N we have

y E ;(n)

such t h a t

k > M, we have

we can w r i t e

Let

such t h a t i f

6 Qi(xyn)

L e t us t a k e a sequence of p o i n t s

If

x

passing through

Since t h e w i d t h o f t h e s t r i p s E~

For n = k

and so on.

w1

be

be a c o n t r a c t i n g sequence o f containing

that

and so

there exists a straight l i n e

x E Q and c o n t a i n e d i n

t h e squares we have c o n s t r u c t e d so t h a t

Qk

h

= 0.

We now show t h a t f o r each r(x)

f o r each

yk

6

rk(x)

, yk

8.5.

Since

d

yi

Q i ( x y N ) we a l s o have

have proved t h a t

i > n.

Since

proves

r(x)

6;,

c

d

yi

i s closed, we g e t K*

I)

.

QA(xyn)

n > max (M,N)

for a fixed

223

THE NIKODYM SET

y E

Hence

we have

yi E 6,.

yi

in . Hence

E

y E

Gn K*

So we

f o r each and t h i s

{x}.

Observe now t h a t t h e above process can be performed on any

Q

g i v e n square i n t e r v a l

Q

given

ther i s

K*

:K ,

K;

through

= Q(1)

Q1

,... ,K; ,...

Q(1)

.

That i s ,

I K * l = 0 and f o r each x E Q t h e r e x so t h a t r ( x ) c K* II {XI. We

such t h a t

i s a straight line r(x) apply t h i s t o

n o t n e c e s s a r i l y equal t o

, Q 2 ( 2 ) ,... , Q k ( k ) ,...

We now d e f i n e

K =

t h e statement o f t h e theorem.

8

Ki

k=l

and we o b t a i n

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 r o m t h e I t w i l l be q u i t e u s e f u 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

preceding proof. follow,

i n the next Section

(8.6.4.).

8.5.5. THEOREM e pen-- . LeX Q be t h e d m e d ~ q w in;tmvd 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 meuune 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 ) p u n i n g

.

thnvugh 06

x

r(x)

and contained i n K* I) {XI i n huch a way XhaX t h e d i h e d a n v&en i n a rneanmabRc way.

Pnood. M

subset

of

The

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.

Q

Q

i s going t o be t h e complement i n

o f the union o f

t h e boundaries 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 IMI

=

IQI .

t o the l i n e

L e t us denote a l s o by

rk(x).

r k ( x ) E [O,~T) t h e angle associated

function have

x E M

rk(x)

+

measurable on

-f

r(x) M.

k

We s h a l l show t h a t a t each s t e p

t i o n we can make a s e l e c t i o n o f l i n e s rk(x) E

rk(x)

for

x

6

o f the construc

M

such t h a t t h e

[O,T) i s a measurable f u n c t i o n . Since we a l s o

a t each

x

E

M

as

k +

m

we see t h a t

r(x)

is

8. THE B A S I S OF RECTANGLES

224

Consider t h e s t r i p s w:,

ui, w:,

...

selected i n the f i r s t step.

To t h e p o i n t s i n W: 0 M we a s s i g n t h e d i r e c t i o n o f t h e s t r i p w:. To 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 So we o b t a i n

-

(u:

rl(x)

on

I', LO'.) 0 M t h e d i r e c t i o n o f j=1 J M t h a t i s a step function.

Consider now w i f )

M

u:. And so on.

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 r e 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 a r e i n ui. We can proceed t o a s s i g n d i r e c t i o n s as above. When we now c o n s i d e r

(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

proceed i n t h e same way a l s o a s t e p f u n c t i o n on The s e t

K*

. And M.

s o on.

The second assignment

I n t h i s way we see t h a t

o f Theorem 8.5.4.

satisfies

r(x)

LO:

we can

r2(x)

is

i s measurable.

t h e statement o f

o u 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 properties o f

B 3 and o f some subbases of

B3

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

B3

cannot even be a d e n s i t y b a s i s . I t i s n o t n e c e s

s a r y t o go so f a r t o o b t a i n t h i s f a c t . With 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.

we a r e g o i n g 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

which t h i s f a c t i s an easy consequence. 8.6.1.

THEOREM

.

CoaLdm t h e

B

-

F

diddmenLLaLion baA

8.6.

bT invahiant by homothecia

genetra-ted by

h a LJ~JLLLCCA( 0 , l ) ,(h-1,O) bad.

,(h,O)

Th

P t o o ~ . L e t MT If

E

225

DIFFERENTIATION PROPERTIES

all U n g L a

.

, whetre

{Thl;=l

Then BT 0 not u der&Ltg

be t h e maximal o p e r a t o r a s s o c i a t e d t o

i s a Perron t r e e c o n s t r u c t e d from

{Thlh,,2n

BT

.

as i n Theorem 8.1.1.

we can w r i t e

where

7,

y = 0

and

i s the extension o f y = -1

.

Th

below t h e b a s i s o f

Th

T h e r e f o r e a c c o r d i n g t o t h e Remark 2

between

o f S e c t i o n 8.1.

we g e t

Therefore, a c c o r d i n g 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 . F o r each

Th

o f Theorem

8.6.1.

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.

let

Rh

be t h e r e c t a n g l e

8. THE B A S I S OF RECTANGLES

226

BR

and l e t

.

{RhI

If

B

be t h e MR

absol Ute c o n s t a n t

-

F

basis invariant

i s t h e corresponding maximal o p e r a t o r we have w i t h an c MTf'

C

6

&lR i s n o t a d e n s i t y b a s i s .

and so

by homothecies generated by

M

R

f

T h i s o f course i m p l i e s t h a t

B3 i s

not a density basis.

8.6.2. COROLLARY. dennity b u i h

So we see t h a t n o t o n l y

,

oh

The. b u d

.

hec&ngLen

B3

d nv.t a

i s a v e r y bad d i f f e r e n t i a t i o n

63

B R , cont a i n i n g r e c t a n g l e s i n a small 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

basis

b u t a l s o t h a t a r a t h e r small subbasis o f

63

such as

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 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 .

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

questions.

PROBLEM 1 . Ba

Let

o f a l l rectangles i n directions

forms an a n g l e

(I Q 0

Ox.

with

(I E

, i.e.

How s h o u l d

the basis

one o f whose s i d e s be d i s t r i b u t e d i n o r d e r

@

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 ?

that

We s h a l l soon see t h a t i f t h e r e a r e bases a l s o bad.

BQ w i t h

t i a t i o n properties f o r sults.

BQ

0

.

But

BR above, t h a t a r e

t h a t a r e lacunary, f i r s t Strornberg [1976]

R.Fefferman [1977] BQ

i s a v e r y bad b a s i s .

denumerable, such as

0

However, f o r s e t s

and l a t e r CBrdoba and

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 h e n

Ba

Q

. Consider

[0,2n)

be a s u b s e t o f

0

have o b t a i n e d p o s i t i v e d i f f e r e n

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-

The general problem i s s t i l l unsolved.

i s o r n o t a d e n s i t y b a s i s when

I t i s even unknown whether

i s the countable s e t o f

0

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 s u c c e s i v e phases o f t h e cons t r u c t i o n o f t h e Cantor s e t i n

[0,1]

.

8.6.

PROBLEM

2.

227

DIFFERENTIATION PROPERTIES

Even when

B;P i s a d i f f e r e n t i a t i o n b a s 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 o u r s e l v e s t o c o n s i d e r f o r each

4

E 0

CLee

not

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

So we o b t a i n a new

B

-

F

basis

of rectangles i n d i r e c t i o n 4

R

B(Q,R)

4

generated b y

( 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 . F o r example, i f @ = [0,2n)

4

f o r each

E 0

,

R4

i s j u s t a square, t h e n

L’, ...

V i t a l i property, d i f f e r e n t i a t e s and m

Ro

i s a sequence o f i n t e r v a l s

B(0.R)

satisfies the

B u t even i f f o r example

{Ik}

and

{O)

@ =

w i t h e c c e n t r i c i t y tending t o

one does n o t know wheter, by an a p p r o p r i a t e c h o i c e 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 t h a n t h o s e o f t h e basis o f a l l i n t e r v a l s . 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 direction

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 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

V i t a l i property,...). constant

c(H)

t h a t increases w i t h

o b t a i n e d a measure o f t h e s i z e o f

PROBLEM x eR2

For each collection

.

3.

H

t o i n f i n i t y . C6rdoba [1976]

has

c(H).

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.

take a d i r e c t i o n

d(x) E

[O,Z.rr)

and c o n s i d e r t h e

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 )

containing

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 r e n t i a t i o n p r o p e r t i e s o f Bd ? How does t h e c h o i c e o f d ( x ) a f f e c t them? x

(The b a s i s

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 concerning some of these q u e s t i o n s . I n t h e n e x t s e c t i o n we s t u d y 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 c o n c e r n i n g t h e quest i o n s around Problem 3. The f o l l o w i n g Theorem has been o b t a i n e d b y 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 time.

M.T.Men2rguez

and

8. THE B A S I S OF RECTANGLES

228

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 di8.6.2.

0 not a denbLty b a 0 .

teCtion $ e 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 t h a t

s a rectang e

then there

lil

, and

21R

6

Therefore

i n direction

$ e Q

such t h a t

6

'3

R

,

so

M Q f ( x ) 6 Mm f ( x ) 6

2 MQ f ( x )

o r d e r t o prove t h e Theorem, t h a t

Q =

.

Hence we can assume, i n

and t h a t

p >

=

0

.

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 0

Q

tree

PE

$

B k

i s a density point o f as i n Theorem

. With

an

E > O we c o n s t r u c t a Perron

s t a r t i n g from a t r i a n g l e

BA

and so t h a t t h e s i d e

n/2

i s i n direction

ABC

with

n/4 and CA i s

3~14.

i n d r e c t i on

For any p o i n t d, e Q o f the sides o f the t r i a n g l e s intervals

0

8.1.2.

Ik(d,) = [d,

(b)

Ik(d,)

-

ak

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 Th

we have a sequence o f nondegenerate

, $ + Bk]

, 0G

ak

, Bk

so t h a t

i s i n s i d e one a n g l e o f those determined b y t h e

triangles

T,, at A

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

8.6. DIFFERENTIATION PROPERTIES such s e t s

, c a l l them

Ik(4)

(i) The s e t s

E

j

,..., EH},

{El,E2

satisfying

are d i s j o i n t

I I J E ~ I>

(ii)

(iii) One e n d p o i n t o f each (iv)

Ej

Each

Ek

is in

i s i n s i d e one a n g l e

the triangles

Th

at

@

we c o n s i d e r t h e t r i a n g l e

A

determined by

S

j Ej

-

of Theorem 8.1.2. and v e r t e x a t A w i t h

BC

w i t h base on and make

Sj

PE

solidary with

B

s1 YS2,.

and c a l l

F

.

basis

MS

Bs

i n v a r i a n t b y homothecies generated b y

t h e c o r r e s p o n d i n g maximal o p e r a t o r , we have

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

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.

and so

Msf

Thy the

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, J' goes t o t h e p o s i t i o n 3 . L e t us c a l l ? the trianJ j 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

consider the

.. ,SH

determined b y

A.

t r i a n g l e t h a t contains the triangle S . J g l e symmetric t o

.

of those

Now i n t h e c o n s t r u c t i o n o f t h e P e r r o n t r e e angle a t

229

c c M@f

p

that

. Hence

230

8. THE B A S I S OF RECTANGLES

F i g u r e 8.6.2. with

c

Since

independent o f t h e t r i a n g l e s

.Therefore

Sj

i s a r b i t r a r i l y s m a l l we o b t a i n t h a t

E

BQ cannot have t h e d e n s i t y

property. T h a t one b a s i s

B,

s a r i l y due t o t h e f a c t t h a t ceding theorem. contained i n cp

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 n e c e s 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 -

I t can be m o t i v a t e d b y 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

. As

@

I a r c t g 1, a r c t g 2, a r c t g 3,

=

...

1

I n t h e same way i t can be shown t h a t i f

B,

then 0

I

=

i s not a density basis. a r c t g 1, a r c t g PP,

then

.

@ =

t i a t i o n basis, as w i l l be proved i n S e c t i o n L e t now

(0,l)

the point

8.7.

I n b o t h cases

A f t e r 3k

we t a k e

number

is

k

3 x2

3 k x 2 + 3k-1x2

Imll=0.

be t h e s e t o f d i r e c t i o n s determined b y j o i n i n g

cp

t o t h e p o i n t s on

Ox

o f abscissae

, 2 , 3 , 3x2+1 , 3x2+2 , 32,32x2. 3%2+1,...32x2+3,

get

if

P 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 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 -

a r c t g 3p,...3

1

8.6.1.

we have seen i n Theorem

+

2+2

,

33,.

..

and t h e n we add t h e p r e v i o u s numbers u n t i l we

3k-2x2 +

... +

3l x 2

+

3Ox 2.

The f o l l o w i n g

8.6.

+ 3k-1

3k x 2

231

DIFFERENTIATION PROPERTIES x

+ ... +

2

31 x 2

+

30

x

2

f

1

k+ 1

= 3

and now we c o n t i n u e . I t i s n o t known whether

i s a density basis. This Ba 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 t h o s e o f t h e b a s i s

basis

BQ o b t a i n e d when we t a k e as

Bo*

(0,l)

j o i n i n g the point

t h e s e t o f d i r e c t i o n s determined by

O*

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

successive 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. The bases

considered i n PROBLEM 3 a r e n o t

Bd

B - F

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

t h a t f o r each p o i n t l(x)

f)

N

=

x E N

the direction o f the l i n e l ( x )

{XI coincides w i t h d(x)

then

Bd

bases. N

such

so t h a t

i s n o t a density basis.

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 .

THEOREM. Ah~umethcLt N d a nQ.t 06 p o n U v e 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~obuch t h a t d o t mch xsN t h e f i n e 1 ( x ) h a t h e d4,teotion d ( x ) Then t h e bmd Bd AA n o t a deMnMq b a h . 8.6.3.

___.

.

Ptoozf. IF/ at

3/4

>

d(x)

,

i.e.

t h a t of

l(x)

of

N

l(x)

such t h a t intersects

F

just

F, we can draw r e c t a n g l e s

R

of

, containing

x

and c o n t r a c t i n g

so t h i n t h a t

x




f o r which

1

g 1 (A)

h o l d s has a measure

8.7. LACUNARY DIRECTIONS

Consider now a s e t Rj

interesting

Bi

such t h a t

Bi

f o r which

d(R.) > d(Bi)

J

Draw t h e minimal c l o s e d i n t e r v a l

-

IBi

(*I

0 R.1 l

j

>

I

(B)

Bi

and

containing

237

i s true

.

ake one b(Rj) > b B i ) Bi . We sha 1 p r o v e

.

[Bi 0 R . 1 J

c

IBjl

So we have

Therefore where

Bi

c

-

Bi

c Ix

: MZ

1xR.(x) 3

>

2

j

R2. Thus

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

we o b t a i n t h a t t h e union of t h o s e

B,

f o r which (B)

i s t r u e has a r e a

1ess t h a n

I n o r d e r t o prove (*)

we c o n s i d e r t h e f o l l o w i n g f i g u r e

A I

1

I

I

I

I

4

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 whatever i s t h e s i t u a t i o n o f

We have,

R

j

B u t i t i s now easy t o show, because o f t h e l a c u n a r i t y o f a f a c t t h a t Bi i s thin , that 6 c. So we have

Q

and t h e

F i g u r e 8.7.3.

For a s e t

so t h a t

Bi

(C)

i s t r u e a s i m i l a r consideration

holds. So we o b t a i n

and t h e theorem i s proved.

COROLLARY.

8.7.2. B

,455 ad

weak .type

Phood

.

compact subset of that

Let

A

The maxim& op&on

M

comenponding t o

(2,2).

f E L 2 and and

x E K

A = IMf

there i s

R,

A

> f

>

B

0

.

If K

containing

i s any

x

such

8.7. LACUNARY D I R E C T I O N S

To

(RX)XEK

we a p p l y t h e theorem

, obtaining

239

ERjI

.

So we have

This Page Intentionally Left Blank

CHAPTER 9 THE GEOMETRY OF LINEARLY MEASURABLE SETS

The geometric theory of l i n e a r l y measurable s e t s i n R 2 was developed mainly by Besicovitch. His fundamental papers on t h i s s u b j e c t were w r i t t e n i n 1928, 1938, 1939, 1964. The whole theory i s i n i t s e l f q u i t e i n t e r e s t i n g and beautiful and does not seem t o have been s u f f i c i e n t l y exploited from t h e point of view of i t s connections w i t h t h e real v a r i a b l e theory. I n t h e a u t h o r ' s opinion i t shows promising signs of becoming a very useful tool t o handle some of t h e problems a r i s i n g in areas where one has t o look c a r e f u l l y 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 o f c o l l e c t i o n s of f i g u r e s t h a t in some sense can be assimilated t o them. We have been led t o consid e r in Chapter 8 c o l l e c t i o n s of t h i n rectangles of d i f f e r e n t nature. As we have seen, c e r t a i n f i g u r e s associated to two-dimensional s e t s of Lebesgue measure zero,such as the Besicovitch s e t o r t h e Nikodym s e t can shed a powerful l i g h t on them. As we s h a l l l e a r n i n t h i s Chapter, Besicovitch's theory of l i n e a r l y 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 . The theory, of which we a r e going t o give a glimpse here, abounds in problems a n d theorems t h a t look q u i t e elementary, The t r e a t ment of them, however, i s often q u i t e complicated and t h e r e a r e many questions s t i l l open in the f i e l d . I t would be very d e s i r a b l e t o have more straighforward proofs o f many o f the r e s u l t s we s h a l l study. There i s in t h e l i t e r a t u r e no complete systematic exposition of t h i s b e a u t i f u l portion of t h e geometric measure theory, t h o u g h some of t h e r e s u l t s can be presented as p a r t i c u l a r cases o f t h e general theory of F e d e r e r ' s book [1969]. In what follows we s h a l l developed

some of t h e f a c t s t h a t a r e

needed t o a r r i v e t o some i n t e r e s t i n g a p p l i c a t i o n s , e s p e c i a l l y t o 241

9. GEOMETRY AND LINEAR MEASURE

242

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

A.Casas [1978]

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 .

9.1.

LINEARLY MEASURABLE SETS. The H a u s d o r f f measure

i n R2

As

s,

o f dimension

i s d e f i n e d i n t h e f o l l o w i n g way,

For

E c R 2

c 2 ,

0 c s

p > 0

and

one f i r s t c o n s i d e r s

The q u a n t i t y

ASCE1 P

,

i n c r e a s e s as p

decreases and we c a l l

l i m A; (E) . Then As* i s an o n t e r measure, C a r a t h 6 o d o r v ' s PO' process g i v e s us t h e a s s o c i a t e d measure As. T h i s measure i s complete AS*(E)

=

and each

Boret

i s reqular. such t h a t f o r

For each s e t

s

it

As-rneasurable, E

t h e r e i s a s i n g l e number

we have AS(E) =

,

AS

Furthermore t h e measure m

t, 0 c t

s > t,

and f o r each

E

The s e t course

set i s

2

C

As(E) =O.

i s then s a i d t o be of H a u s d o r f f dimension t. I t can, of t 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

s t u d i e d t h e geometric p r o o e r t i e s o f t h e s e t s o f

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 A1(E)
l

Pj)),

Let us c a l l

a!J b!J

the

a . b over t h e s i d e u l ' of a(0) p a r a l l e l y t o t h e J j and a" b" t h e ones over 1 " p a r a l l e l y t o 1 ' . j

j

We can then w r i t e

On t h e other hand 4,6(A)

sin

c1

Since n > O i s a r b i t r a r i l y small we get

06

poi&

-

9.3.4. 'TtEfJREM, L e t E be an i h ' w j d a t r beX. Then ,the b & 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 ~ a s a measurable s e t whose i n t e r s e c t i o n w i t h any r e c t i f i a b l e curve has measure zero. On t h e o t h e r hand a Y - 6 e t i s a measurable s e t contained i n a conmutable union of r e c t i f i abl e curves.

9.3. TANGENCY PROPERTIES

257

I t i s obvious t h a t any i r r e g u l a r set i s a Z-set and one can e a s i l y observe t h a t t h e statement and proof of Theorems 9.3.3. and 9 . 3 . 4 . a r e v a l i d i f we merely assume E t o be a Z-set. Besicovitch [19381 proved a l s o the following important f a c t .

9.3.5. THEOREM. a h a o t each x 6 E .

I6

E b a 2-hd

, ,then

D(E,x)

s

314 a t

From t h i s theorem we e a s i l y obtain t h e following c h a r a c t e r i z a t i o n of i r r e g u l a r s e t s .

T h i s gives us e a s i l y t h e c h a r a c t e r i z a t i o n of r e g u l a r s e t s 9.3.7.

THEOREM.

A rneuuhable A & E b hegdan i d and o n l y Y - A ~ ( i . e . id thehe. e.xhd.4 N c E , A ( N ) = 0

id AX A a h o b t a ouch ,that E - N h a Y-set). P h O O d Of -

We have

A(E)
0 and t h e r e e x i s t s a r e c t i f i a b l e curve y1 such a1 that ACE I1 r l ) > . Consider E l = E - y1. If A ( E l ) = 0 then we g e t t h e statement o f the theorem. I f A(E1) > 0 , since E l i s again n o t a Z-set we can f i n d a r e c t i f i a b l e curve y 2 such t h a t

ACEI (7 And so on.

y2)

>

A ( € ) > 0 and E r e c t i f i a b l e curve 1

1 scx2 =

.

1 7 sup

A ( E l (1 y ) : y

I f the process is f i n i t e , c l e a r l y

A(E-

H LJ 1

r e c t i f i a b l e curve) y . ) = 0. J

If

c1 0 . Call Em = E - II yj and assume A(Eoo) > 0. i n f i n i t e , then j 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 c o n t r a d i c t s t h e e l e c t i o n of the ci f o r CI s u f f i c i e n t l y small. -f

j

j

258

9. GEOMETRY AND LINEAR MEASURE

h(Em) = 0

Therefore

and

E - c,E

m

" y

j

.

We a r e n o t g o i n g t o p r e s e n t h e r e t h e p r o o f o f Theorem s i n c e i t i s p r e t t y l o n g and complicated.

9.3.5.,

We o n l y remark t h a t as a con-

sequence o f t h e p r e v i o u s 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 proof i s

s t a i g h t f o r w a r d s t a r t i n g from

9.3.6.

and

TffEOREM.

A m~anutra6Lehe,t E id Lt han a 1ange.d aR: demvh-t each 06 L t b poha%. 9.3.8.

9.4.

9,3.7.

4eguLa.k

4 and

vnP.rr

PROJECTION PROPERTIES. A r e c t i f i a b l e curve ( O f p o s i t i v e length)

has o r t h o g o n a 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 e v e r y s t r a i g h t l i n e w i t h t h e possible exception o f those 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 l m o s 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 b y t h e r e g u l a r s e t s 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 s e t t h e o p p o s i t e i s t r u e , namely f o r a l m o s t each d i r e c t i o n here

("almost"

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-

j e c t i o n i n t h a t d i r e c t i o n e v e 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 definitions. 9.4.1.

(a)

NOTATION. d(x,81)

If

x 6R2

0 c

el

< O2
rl t h e n

d i r e c t i o n 81.

PROJECTION PROPERTIES

9.4. (b)

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

a(x,a1,e2) x

and extreme s i d e s i n d i r e c t i o n s a(x,e1,e2)

(c)

1

If A

o(x,el,e2)

=

C [O,T)

then

e2 3

-

EX)

(I B ( x , r )

DEFINITION.

9.4.2.

We say t h a t

E at x

when

whose

E

Let

e

is a

x

is

:

[O,TI).

w i l l mean t h e u n i o n o f

d(x,A) x

= (JId(x,e)

d(x,A)

ondeh 06


0

We know t h a t f o r each f i x e d u(a,e',e",r k k

A

We can assume t h a t

c 2 r k { 4;

Ik

-

c

in-

c

0;)

contracts t o a

and o b t a i n a sequence

n(a). EJiI

So we can a n o l y

o f d i s j o i n t in-

t e r v a l s such t h a t

Therefore

A*(

and so

A(

IT

TI

(E))

c A*(T(E)

A-almost L e t us f i x

N c [O,TI)

p-null set x

E

6

0

(I Ji)

t

A ( IJ J i )

=

(E)) = 0.

Phoolj ad ,the Theohem. there i s a

-

E

9.4.7.

A c c o r d i n g t o Lemma

such t h a t ,

if

0 E [O,TI)

9.4.8.

-

N

t h e d i r e c t i o n 4 i s a condensation d i r e c t i o n o f [O,?r)

-

N

.

Then

,

for

E a t x.

9. GEOMETRY AND L I N E A R MEASURE

where

tion

Eo =

Cx

E

E : 8 i s not of condensation o f

El =

{x

E

E : 0

EZ

{x e E : 0

=

e

By Lemma

at x 1

i s o f condensation of f i r s t order of i s o f condensation o f second order of

, since

9.4.2.

a t XI

E E

a t x)

~ ( E o )= 0 , the projection in d i r e c

i s o f measure zero. By Lemma

A(n(E2))

E

= 0

9.4.10,

. Therefore

A(n(E,))

= 0

the projection

and by Lemma 9 . 4 . 1 1 . n ( E ) o f E i s also 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 R 2 l e t p ( x ) be the polar l i n e 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 lines of points of A, i . e .

In t h i s Section we shall be concerned w i t h some of the twodimensional 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 r c l e C i s rather irrelevant for the properties we are going t o s t u d y .

9.5.

SETS

OF POLAR L I N E S

I n t h e f i r s t place, observe t h a t i f a # b, t h e n

w i t h e n d p o i n t s a,b, n i t e A-measure.

p(A)

is a

A

269 i s any c o n t i n u o u s c u r v e

X-measurable s e t o f i n f i-

A

We s h a l l f i r s t p r o v e t h a t , as expected, when

i s an i r r e g u l a r

s e t , t h e o p p o s i t e w i l l h o l d . 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 r w a r d from Theorem 9.4.7.

8

B

THEOREAd.

9.5.1. A-n&hei.

ih a

Pmad. -

From theorem

A-measure zero. E

9.4.7.

[O,IT)

(I

the l i n e

A(p(E))

i n direction

e

XEE

)

d(x,O)

g(e), orthogonal t o

n(E) i n a s e t o f

p-almost each

(here

d(x,e)

w i t h any l i n e i s o f

T h e r e f o r e we see,by p o l a r i t y , t h a t f o r

intersects the set uring

x

Then n(E)

hei.

we know t h a t f o r

the intersection o f the set

C[ ).,I

means t h e l i n e t h r o u q h

8

Lei E be a n L w u g u R a h

d(0,B)

u-almost each

through the o r i q i n ,

A-measure zero.

T h e r e f o r e meas-

A(D(E)) = 0

i n p o l a r c o o r d i n a t e s , we have

For r e g u l a r s e t s we s h a l l f i r s t p r o v e t h e f o l l o w i n g f a c t ,

9.5.2. A - a h o h t each

THEOREM. x

6

Lei

E

be a hegduJL

E we have, doh each z

E

hQ,t.

p(x)

Then , doh and doh each r > Y'

We s h a l l deduce t h i s theorem from t h e f o l l o w i n g a u x i l i a r y results.

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

more f l e x 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 .

THEOREM.

9.5.3.

open be.Z iiztavLt M

.

E

be a meanwlable

contained i n a n

bei

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 CVUAWhohe iinvehbe f - l : H + G e u k 2 a n d h &o a L i p b c k i t z

G

Then f ( E ) A(f(E))

Lei

= 0

.

tegdah , f(E)

16

Lh & a

meanwLable.

E Lb hegLLeah,

&eguRm.

f(E)

Id

A(E) = 0

Lh hegdah.

.then

16 E h

-

9 . GEOMETRY AND LINEAR MEASURE

270

Pmv$. Cf(An)l

Assume

A(E)

i s a cover o f

f(E)

A*(f(E)) c If E

M A*(E)

E

E

A(

A(f(E)) 6

M

c b

,

yk

k=l

i s regular,

E =

yk

(J

f(B)

rectifiable

.

So

,

i s measurable. B cE,

CK.1 J

Z,

(J

.f A (E)

E,

Therefore h ( f ( E ) ) = 0.

b e i n g a sequence A(Z) = 0, T h e r e f o r e

and

A(f(Z)) = 0

and

f ( E ) i s measurable.

A

B with

IJ

A ( B ) = r) Therefore rn

A(f(B)) = 0

f(A) C ( 1 k= 1

f(ykly

f ( Yk)

f(E) i s regular.

Finally, i f and

M 6(An).

continuous r e c t i f i a b l e curve.

f(E) = f(A)

f(E)

j=1

i s a cover o f

A(E) = 0, t h e n

Kj)

IJ

A(f(Z)), Hence

m

m

A(Kj)

.

A(E)
0, ! I< D

< d
0

a , D - d c y c 0

. LeL A

t d

I

be Lhe cloned i n t e 4

V a l

Then thehe c d a 2 a cav~6Lant M M(a,D) > 0 auch . t h a 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. -

maximum o f for a fixed

The p r o o f i s s t r a i g h t f o r w a r d b y o b s e r v i n g

I( U

p

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

i s given by the c i r c l e i n d i c a t e d i n the f i g u r e

that f o r this circle

U 0 q(C(z,p))

o f t h e h o r i z o n t a l s t r i p determined by

that the 9.5.1.,

i s c o n t a i n e d i n t h e shaded p o r t i o n U

and t h a t even t h i s shaded

9. 9.GEOMETRY AND LINEAR MEASUREMEASURE GEOMETRY AND LINEAR

272272

has an an area s s than constant times p times . p o rptoi rot ni o nhas a r e a l eless t h asome n some c o n s tM(a,D) a n t M(a,D)

p

.

s = (a,O)

s = (a,O)

Figure 9.5.1.

F i g u r e 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) 9.5.6. L& 06U t hand A be a6 i ns t ht e lemma. and 5 m a n h t h e LEMMA. ~ e , t 06 pointn e btgmekLt joining o prreceding y Letwe shave = (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 c o n t a i n e d 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

w.

F i r s t we can f i x r o 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 t e r s e c t both h o r i z o n t a l sides o f U . For a f i x e d p, X(U (1 q(sy)) i s g r e a t e r 0 < p < r o , i f h(sy) = p , the minimum o f

First we can f i x r o so small t h a t a l l lines q ( z ) f o r z 8 B ( s , r o ) 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, i f 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 Pko06.

9.5. SETS OF POLAR LINES

273

than t h e area o f t h e shaded portion o f Figure 9.5.2. In i t t h e point e i s obtained as i n t e r s e c t i o n of t h e circumference C o f diameter Og w i t h t h a t C(s,p) o f c e n t e r s and radius p This area is ( 2 d ) ' t a n a

.

and one has

&=D+d sin c1

and

at

cos 6

cos B C O S T as p + 0 where T i s t h e angle o f t h e tangent t o C with t h e a x i s Ox, as indicated i n Figure 9 . 5 . 3 . -f

x

So one has

(2d)' tan a = (2d)2

and i t i s c l e a r t h a t this N(a,d,D) > 0

sin

P

c1

p

cos

= c1

q u a n t i t y can be estimated from below by

Figure 9.5.2.

9. GEOMETRY AND LINEAR MEASURE

274

F i g u r e 9.5.3.

06

Phood Theorem

9.5.3.

Theohm

there i s

9.5.2.

E

Let

N c E,

be r e g u l a r . A c c o r d i n g t o

, such

A(N) = 0

yj continuous r e c t i f i a b l e arc.For c a l l i t s i m p l y y , such t h a t yj, d e n s i t y p o i n t o f y and E I1 y . We s and f o r each b 6 q ( s ) , b # s o f b, then X(U (1 q ( E ) ) > 0

almost each

s

6

- N)

(E

that

- N y and

-

E

N c (I yj,

s 6 E

t h e r e i s one

fI

s

is a

s h a l l prove t h a t f o r such a p o i n t

, if

U(b)

i s any neighborhood

We have

A ( E (1 y fl B ( s , r ) ) 2r

1i m r+O

We can assume t h a t lemmas, t h a t

b

s

i s the center

and we t a k e as t h e neighborhood number take Lemma

n,

0
0 9.5.6.

T-

< 1,

so t h a t and

=

i s the point

(a,O)

o f the rectangle U

U

of t h e p r e v i o u s

o f t h e s e lemmas,

precisely t h a t rectangle. For a

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 B(s,r)

c

A,

B(s,r)

s a t i s f i e s t h e statement o f

9.5. SETS OF POLAR LINES

275

Let y,z be the endooint of the continuous arc y r c y 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.

and the last one by ( * ) above,

From (**) we get

We can cover y, - E with a countable union contained in A so that

(I

Kj of small circles

We have by Lemma 9.5.5.

Hence

If n >

% M+ 7

then

X(U

(1

q(E))

>

0. This concludes the oroof.

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

9. GEOMETRY AND LINEAR MEASURE

276

Phoo6. Lemma

9.5.5.

For

o f small c i r c l e s

E t h e r e i s a cover of

We can assume t h a t t h e s e t E:

> 0

u K j c A

so t h a t

CG(Kj)

i s i n the set E 6 E

A

of

by a countable union

.

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] p l i c i t y , i.e.

if one weighs each D o i n t o f

i t i s covered by l i n e s with

A(E) >

0,

p(x)

with

x

E

p(E)

E , then

w i t h t h e number o f times f o r every r e g u l a r s e t

E

t h e l i n e s cover an i n f i n i t e a r e a . More o r e c i s e l y

However Davies [1965] A(E) >

proved t h a t , i f one counts m u l t i-

0, X(p(E))


0 . Further, t h e nrojection of K over t h e two diaqonals o f Q a i s of zero measure and t h a t over Ox and Oy has measure 1. There f o r e K must be i r r e g u l a r , $(K) i s a l s o i r r e g u l a r of nosiLemma 9 . 5 . 3 . t e l l s us t h a t t i v e measure. Moreover, s i n c e K has a t l e a s t one o o i n t on each l i n e y = a , 0 5 a ,< 1 , $(K) has a t l e a s t one noint over each ray from r) d i f f e r e n t from 0 . Therefore n($(K)) contains a t l e a s t one l i n e i n each d i r e c t i o n and A ( P ( $ ( K ) ) ) = 0

9. GEOMETRY AND LINEAR MEASURE

Take a s e t

L

which i s t h e u n i o n of some l i n e s .

an easy c r i t e r i o n t o decide whether t h i s s e t 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 ?

Can one g i v e

i s t h e u n i o n of t h e p o l a r

L

I n o t h e r words, l e t

L = \J

,€A d,

and l e t E = {a :

i.e.

d,

= ~ ( a ) l

i s t h e s e t o f p o l e s of t h e l i n e s

E

d,

w i t h respect 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 d e t e r mine whether

i s i r r e g u l a r (and so

E

L

The f o l l o w i n g r e s u l t , due t o

i s of

A-measure z e r o ) ?

A.Casas [1979],

answers t h i s

q u e s t i o n i n an easy way. 9.6.2.

Let

THEOREM.

L = I! ,€A

w h a e each d,

da

LAa

Let

ha%a&ktfine.

, a

Ca e R 2 : p(a) = d,

E =

Then E A & e g W

E

A 1

-id and ovtey id t h e doU0wLng

&zue:

W e g i x Awo finen s , t , huch t h a t doh each a e A s f da , On s ,take a p o i n t S not i n L and an ohientdtion. On t t a k e a poivLt T not i n L and ah ahienta.tLan. Foh each d, let S, = d, (I s and Ta = d, ('1 t. LeX SS, be the. higned din,tance t h a t aham T t o T, Then t h e . hQt, &om S t o S , and TT, t # d,

.

.

H =

Pk0o.d.

Q,

S = T

TT,

A p p l y i n g Lemma 9.5.3.

a r e p e r p e n d i c u l a r and t h a t jection of

(SS,,

=

Consider t h e f i g u r e

ER'

:

01

e A 1

we can assume t h a t

coincides w i t h

S

over S,T

)

T.

Let

,

9.6.2.

and t h e mapoing

R,

s and t be t h e pro-

9.6. SOME APPLICATIONS

279

We can assume t h a t t h e s e t o f points {Ra : ct E A ) i s such t h a t i t s closure i s bounded and contained in R 2 - Exy = 01 , Then we can a m l v Lemma 9.5.3. and

Y

y - t

X

0 I

OE SET

1

s_ u ,X

E S

Figure 9.6.2. so

10,

: a B A

I i s i r r e g u l a r i f and only i f {Rct : ct 6 A) i s i r r e g u l a r A 1 i s i r r e g u l a r i f and only i f E i s i r r e g u l a r . So we

B u t {Ra : a B have t h e theorem.

Now i t i s easy t o understand b e t t e r t h e nature of t h e s e t of Kahane presented i n 9.6.3.

8.4.2.

THEOREM. The neL

8.4.2 0 nuch tthcLt

E =

{a

ad LLnU

6R2 :

L

p ( a ) = da

= IJ

M A

,

da

( 'a6 A ) }

phehnnted i n b anihheguLm

b&

PhOOd.

Go x

ted in

By Theorem 9 . 6 . 2 .

i t i s enough t o show t h a t t h e set

C O , where G O i s the Cantor type set on 8.4.2.,

i s irregular.

we have followed in

we have c o n s t r u c B u t t h i s i s Droved i n e x a c t l y t h e same way [O,l]

9.6.1.

The l a s t a p p l i c a t i o n we s h a l l give concerns the Nikodym s e t and the problems r a i s e d a t t h e end of 8.5.

2 80

GEOMETRY AND LINEAR MEASURE

9.

We know t h a t t h e r e e x i s t s a c o n t i n u o u s f i e l d o f d i r e c t i o n s

0 : R2+ [OJ)

N

and a s e t

t h a t f o r each

x 6

, d(x,

N

o f positive 6(x))

n N

A(N) >

A-measure

= {XI

.

0, such

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. a6 d i n e o t i v a .

mea4uhe.

,

THEOREM.

8 : R2

l e i

+

T h e n ,them c a n n v t be. any ne;t N

X(R2

i.e.

d ( x , 0 ( x ) ) 17 N =

{::I

-

.

N) = 0

,

a e Ox

i s o f f u l l one-dimensional measure.

- (03

there i s a l i n e

d(a,

e ( a 1 v a r i e s i n a L i p s c h i t z way.

06

d u l l ,tLuv-dimeMnioncLt?

nueh thcLt doh each

P ~ a a d . Assume t h e r e i s such a s e t 1 17 N

be a Lipnckitz d i d d

[O,TT)

(a))

N.

x 6 N

Fix a line

Assume

1

is

,

1

Ox.

such t h a t F o r each

assigned by t h e f i e l d s o t h a t

Therefore t h e s e t of p o l e s of

d(a, 0 ( a ) ) forms a L i p s c h i t z curve. 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. T h e r e f o r e t h e r e i s a subset o f such p o l e s 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 . Hence t h e u n i o n of t h e corresponding p o l a r l i n e s , t h a t i s A-a1 most c o n t a i n e d i n R2 - N Ox , has p o s i t i v e A-measure. But t h i s contra dicts

Am2

- N)

= 0

.

-

Hence t h e theorem i s proved.

-

CHAPTER 10 APPROXIMATIONS OF THE I D E N T I T Y

Many aporoxirnation 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 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 = 1, E

-f

0

that

the convolution i n t e g r a l

,

and

kE

*

f e Lp(Rn) f

+

f

,

i n the

kE

*

converges t o LP-norm as

, where

f

E

f

.

+

0.

k

kE(x) =

e L1(Rn) ,

,

E - ~k)(:

I t i s r a t h e r easy t o prove

In fact,.if

g

e g o (R'),

we can w r i t e .

Hence, s i n c e

11 k E l l = 11 k l l

Given

= 1

,

using Minkowski's i n t e g r a

r~ > 0 we f i r s t f i x a

g e

t osuch -

Then we have, f o r each y E Rn, E z 0, 11 g ( * EY) and f o r each f i x e d y 6 R n , \ \ g ( EY) g(-)\\

-

Therefore,

-

ineclual it y ,

- q(-)ll -f

0

*

f

- gl(, c .;

2 ]If

that

as

c 2 (lg(( E

+

0

.

b y t h e dominated convergence theorem,

J

for

E

s u f f i c i e n t l y smal

.

T h i s proves t h a t

kE

-+

f LP)

.

A more d e l i c a t e problem c o n s i s t s i n o b t a i n i n g t h e p o i n t w i s e Calderbn and Zygmund [1952] have g i v e n a r a t h e r general

convergence.

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

281

in S e c t i o n 10.1., t o g e t h e r

,

10. APPROXIMATIONS OF THE IDENTITY

282

with a generalization due t o Coifman. Section 10.2. d e a l s w i t h some r e s u l t s t h a t a r e a v a i l a b l e f o r kernels which a r e not r a d i a l b u t a r e nonincreasing along each ray emanating from t h e o r i g i n . 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 h e Calder6n-Zygmund decomposition (Lemma 3 . 2 . 7 1 , and from which one can deduce many o t h e r useful r e s u l t s . In Section 10.4. we s h a l l study some r e s u l t s o f P.A.Boo [1978] and of M.T.Carrillo p979] concerning c e r t a i n necessary conditions f o r a kernel t o y i e l d a good approximation of t h e i d e n t i t y in L'(R").

10.1. RADIAL KERNELS.

k

I t i s r a t h e r obvious t h a t , f o r L ' ( R n ) , i k = 1 , kE(x) = E - ~k($)

E

kE

*

gcx)

E'O

g(x)

g

e g o (R'), for

a t each

E

-f

we have, i f 0

x e Rn

In f a c t ,

By the dominated E

-+

a.e. tyne

tends t o zero as

0.

Therefore i t i s s u f f i c i e n t t o prove , in order t o obtain t h e convergence of kE * f t o f f o r f e L p (1 & p 6 m ) , t h e weak (.D,p) f o r t h e maximal operator K* where K*f(x)= sup \ k E * f ( x ) E>O

.

233

10.1. RADIAL KERNELS K*f(x) =

sup I k E E>o

j E Z

,

PhvaZj. -

f > 0

Let

,f

E L’

*

f(x)\

.

be f i x e d

Consider, f o r each

the set

By t h e hypotheses on

N k (x) =

I

k,

i s a spherical s h e l l .

C. J

k(x)

,

2-N-1 < k ( x ) 6

if

If

2

N

, otherwise

0

and KNE f ( x )

=

kE N

*

, KN* f(x)

f(x)

=

sup E>O

Ikz

*

f(x)l

we t h e n have

So

, if

with for

c

K*

we prove

independent of

.

Since

K*

,

k(x) C

j’

(m,m)

i s nonincreasing w i t h

i s the closed b a l l centered a t

exterior radius o f

we s h a l l have t h e weak t y p e

i s obviously o f type

NOW, s i n c e

Bj

N ,f , X

0

(1,l)

, we have o u r theorem. 1x1

we can w r i t e , i f

whose r a d i u s i s equal t o t h e

10. APPROXIMATIONS OF THE IDENTITY

284

Therefore

1

f(x-z)dz

c M f ( x ) , where M

i s the

j

H a r d y - l i t t l e w o o d o p e r a t o r over b a l l s .

T h e r e f o r e , f o r each

N KN* f ( x ) b

2Mf(x)

KN*

Hence t h e

1

j=-N

2’

\aj\ c

2Mf(x)

a r e o f u n i f o r m weak t y p e

(1,l)

I

F

> 0,

k

, as

we wished

t o prove. When

k

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

r a d i a l majorant, defined b y K(x) =

A

sup Ik(t)l 1x1

It1 c

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 c o n c l u s i o n s o f t h e above theorem

for

k

i s that

k,

which i s now p o s i t i v e , r a d i a l and n o n i n c r e a s i n g

a l o n g r a y s , belongs t o m a j o r i z e d by t h e one

L1(Rn).

k*

I n f a c t , t h e maximal o p e r a t o r

corresponding t o

k .

K*

is

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 sections.

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 asis nonnegative and nonincreasing sume that for some > 0, k(x)(x(-' with 1x1 .

THEOREM. L e t k a 0, k 8 L1(Rn) be hadiae a&d k(x) 1 x /-a n v n i n c h m i n g w i t h I X I . The.&, t h a t doh aume a > 0 t h e h m e nv&un a ! i n T h e v L m 10.1.1. we 5eL t h a t K* ih 06 weah ( 1 , l ) aHd o d .type ( p , p ) , 1 i p c m Hence, id /k = 1 kE * f 10.1.2.

a.e.

doh each

PkvvZ;.

f e Lp

,

.

1c p
0.

Hence f o r

7 ,

z

6

Y fixed

we can

288

OF THE IDENTITY

10. APPROXIMATIONS

apply t o

T-f(z Y

1

+ si)

, and o b t a i n

10.1.1,

t h e theorem

m

W

ITy f ( z + s y ’ ) l p ds c

c

-m

[

J

f ( z + s Y ) I p ds -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 fact that for

p

1

1 arises from the

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 o b t a i n e d f o r

P

=

can a f f i r m , i n t h e hypotheses o f t h e theorem t h a t

1

,

K*

i.e.

whether one

i s o f weak tyDe

(1,l). I t i s however easy t o deduce t h a t

for

x

> 0,

1x1
0 , be non-inchu-

KERNELS NON-INCREASING ALONG RAYS

10.2.

A

5

=

Cx:

k(x)

,

a,')

cvnwex, bvunded , and t h a t 4 a j b a t i h ~ i e ht h e doUullting c o n d i t i o n dhe

= 2'

j e l

lAjl

, t h e bequence

{aj}

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, d o 4 each f 6 L p m n ) , 1 6 p < m , we have kE * f + f a.e.

kN(X) =

If , Kfi f ( x ) =

I

Let

Pkvvd.

sup E>O

Kfi

i s o f weak t y p e We have

Therefore

L e t us c a l l

f e L1(Rn)

k(x)

if

0 Ik

and f o r

2-N 6

k(x)

N E N 6

ilN

otherwise f(x)l

N,E

(1,l)

, f a 0

,

i t w i l l be enough t o Drove t h a t

w i t h a c o n s t a n t independent o f

N.

290

10. APPROXIMATIONS OF THE IDENTITY

Since each

A

of weak type K;f(x) =

j

i s bounded and convex (Cf.

(1,l)

sup &>O

k

x N Y E

, t h e operators M j

Theorem 3.2.10.

f(xl c

N

1-N

a r e uniformly

) We have f o r each

ZJ ( A j I

N

1

M.f(x) = J

-N

E>O

a j Mjf(x)

and using Theorem 3 . 8 . 1 . , and t h e condition on the Ea.1 we g e t t h e J weak type ( 1 , l ) f o r Kfi and t h e r e f o r e f o r K*. The type ( m p ) i s trivial. Of course t h e Theorem 10.2.2. admits a natural extension. be convex and s a t i s f y t h e entropy Instead of requiring t h a t t h e s e t s A j condition of t h e previous theorem one can r e q u i r e t h a t they a r e contained in convex s e t s B j t h a t s a t i s f y t h i s condition o r in t h e union of a fixed number of such s e t s . We s h a l l now give an example of t h i s type of extens i o n , proving t h a t t h e following kernel t h a t a r i s e s i n t h e s t u d y of the multiple Poisson i n t e g r a l (see R u d i n [1969] ) y i e l d s a good approximation of the i d e n t i t y . 10.2.3.

Then t h e O p U u L t O h

.i~ 06 weak t y p e PhoO6.

APPL7CATlON.

K*

k : Rn + R

Le,t

be dedined by

dedined by

(1,l). Let us s e t , f o r s i m p l i c i t y of n o t a t i o n ,

n

=

2

. We

consider t h e level s e t s Aj = I(x,Y) :

We have

1

( 1+x2)l l + y 2 )

2

ZJI , j

=

o , -1 , - z , . . .

10.2. KERNELS NON-INCREASING ALONG RAYS

29 1

Therefore

The s e t s

b . = ZJ 1 B . I J J so

B.

(ant

= 4 & 2 Jh/4b

We t h e n have

C.) J

1

a r e i n t e r v a l s and

bj l l g

bj/
1

/+ E~~ + 1 . L e t

.

LeL

{E.}

kE ( x ) = j

"1

.

be

x k(r) ~j

,

299

10.4. NECESSARY CONDITIONS K*f(x)

sup Ik

=

Ej

j

0 6 Rn

nphehe C

L e l Un dedine t h e dunctivn

H

vn t h e ul.tit

by

Hfi) Annume

.

* f(x)/

=

ess. sup. rn fk(ry r>O

&at K* A v d weak t q p e ( 1 , l ) X > 0

. Then

u A ,the. L e b u g u e meauhe a n

C

buch t h a t doh each

whetre

.

When the kernel k of the preceding theorems i s continuous one can give a somewhat simpler formulation.

K*f(x) Then

10.4.3. THEOREM. s u p (k, * f(x)l

=

sup x eR

0 0

1x1 lk(x)l

10.4.4.

K*f(x)

=

sup E>O


0

6

L1(Rn)

7

6

be ContinuvUc),

C

rnl k(ry)l

06 weak .type. (1,l) .

Then thehe

c > 0 nuch t h a t

These theorems allows us to construct in a simple wayy for example, radial kernels k e L1(Rn) such that the corresponding maximal operator K* i s not of weak type ( 1 , l ) . (Of 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) = 14‘ Then K* i s not of weak type (1,l) . In R2 one can extend the preceding

10. APPROXIMATIONS OF THE IDENTITY

300

k

-

radially to

so t h a t s t i l l

-

k E L 1 ( R 2 ) . The corresDonding maximal

i s n o t o f weak t y p e ( 1 , l ) .

K*

operator

k

We have a l r e a d y seen i n Theorem 10.1.1.

follows i t , t h a t i f k

E L1(Rn)

, fk

, and

= 1

and t h e remark t h a t t h e function T defined

E ( x ) = ess sup I k ( t ) \ i s i n L1(Rn) , t h e n f o r each f e L1(Rn) I t l c 1x1 k E * f ( x ) + f ( x ) a t almost each x € R n . The f o l l o w i n g theorem, due t o Boo

by

,

[1976]

i s a p a r t i a l converse o f t h i s r e s u l t .

a function

f E L1(Rn)

a Doint

x E R”

L e t us r e c a l l t h a t f o r

i s called a

LebQngue p o i n t

We know t h a t almost each p o i n t o f Rn i s a Lebesgue p o i n t o f

TffEOREM.

10.4.5.

that

j$x

each

dt each Lebague p o i n t

, Phood.

8

L’(Rn)

with

06

f

unction

Then t h e

g

,

f E LI@P)

f(0) = 0

, f(x)

X

E

L’ 17 L”(Rn),

g(0) = 0

E(x) =

ess sup I k ( t ) I ltl4xl

!k

for

6

L1(Rn).

/k = 1 and dnnwne

=

m

LA in L’.

, then

there exists

, 0 i s a Lebesgue p o i n t o f g and s t i l l SUP

€4

be t h e s u b s e t o f f u n c t i o n s = 0

f

06

.

We s h a l l prove t h a t i f

lim Let

1eL k

1x1 > 1 ,

and

0

f

of

L1(Rn)

such t h a t

i s a Lebesgue p o i n t o f

f.

That i s

The s e t

X

i s a l i n e a r subspace o f

L1(Rn).

,

If f o r

f o X

we d e f i n e

301

10.4. NECESSARY CONDITIONS

]I f ( l

then

i s a norm i n

.

X

We s h a l l now show t h a t

X

with

II.llx

i s a Banach space. I n fact, l e t

we have t h a t subsequence

X

{g.)c J

be a Cauchy sequence i n

Eg.1 i s a l s o a Cauchy sequence i n J Chjl of I g j 3 such t h a t

and Drove t h a t

E h . 1 converges i n

t h a t a l s o 19.3 J

converges i n

For

J

Chj1

we have

X

g

E

L1

.

We t a k e a

of course, i m o l i e s

X.

11

hj

- hj+llll

by F a t o u ' s lemma, we e a s i l y see t h a t t o a function

. This,

L'(Rn).

X . Since

We can s e t

c IB(Q,l)\Z-'

and so,

I h . 1 converges a.e. and i n L ' J g(O) = 0 , g ( x ) = 0 i f 1x1 > 1.

We a l s o have

4

lim inf i - t w

11 h j -

With t h i s we e a s i l y have space.

2-j+l

hill g 6 X

and

hj

-f

g(X)

.

Hence

X

i s a Banach

10. APPROXIMATIONS OF THE IDENTITY

302

Observe now t h a t for a fixed t o t by

from X

E

> 0,

the mapping

$E

defined

i s linear and bounded, since

Therefore ( $ e ) E > O i s a family of bounded linear functionals from X t o Ic . If we can show t h a t E L' implies t h a t there exists 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, t h i s means t h a t there must exist g E X such t h a t lim sup I$Ei(g)l = and so we obtain the contradiction

+

E.' 1

0

t h a t Droves the theorem.

/k =

So our goal i s t o construct for each fixed E > 0 , using t h a t , a function f E E X , I( f,l( c c such t h a t lim SLID l$,(f,)l=w.

00

E - t O

for s

m

=

r

Observe f i r s t t h a t f F 0,1,2,3,.., and E > 0

=

imolies the following . Let us c a l l ,

303

10.4. NECESSARY CONDITIONS Therefore

, if E

+ L1

, we

00

ME(s)

s=o

2ns

1 -

have

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

ME(s) =

sup

ess

Ik,(x)l

,

fE

as

+ m

.

E +

0 ,

Since

s = 0,1,2,...

we have a s e t o f

2-s-1 < EIxl

MEW

L e t us s e t

Now we s e t m

,s

where

Ns

(Take,

for example Ns

We see t h a t 2-k-l

=

< r 6 2-k

, we

S (X)

s=O

0,1,2,..,

fE(0) = 0

g,, zns Ns

1

fE:(X) =

i s chosen so t h a t

=

as

, have

for

ci

fE(x) = 0

Nst

, and

m

> 1 and c l o s e t o

for

1x1 > 1

.

1) Also, i f

I

10. APPROXIMATIONS OF THE IDENTITY

304

0.

11.1. THE HILBERT TRANSFORM

LeL f =

N

1

Aj

,j=1

whehe

Phaa6.

i h e D&c

A

6. J

N

i t i s quite clear that

v

.j = 1,2,...yN

c j=1

ICx :

x-a.1 - A

.j = 1

1

a . .Then .?

1 J

j=l

~

1 x-a j

> A}] =

N

1

j=1

( v j - a,i)

where

are the r o o t s o f the eauation

N 1

From h e r e N y. = j=1 J

concenLated CLt

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

y =

-jy

d&a

307

i.e.

J

of

A

N. N (x-a.) = 1 r! (x-a,) j=1 " .i=1 ,i#k N r!

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 N N -N + a j . Hence 1 ( y j - aJ. ) = N . Thus we g e t 1 j=1 j=l

x

1

Since t h e second t e r m can be handled as t h e f i r s t one.

11.1.2.

.type

that a. 3

( 1 9 1 )

.

TffEUREM.

Phoo6. A c c o r d i n g -

H* i s o f weak t y n e

ER,

j = 1,2,

...,N,

The maximal HLLbeht opeh.aXoh

t o Theorem

4.1.1.

H*

A

06

weak

i t i s s u f f i c i e n t t o nrove

( 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 h > 0 , and f = SLi where S. i s t h e j=1 J

Dirac d e l t a concentrated a t

a We have t o m o v e t h a t j *

11. SINGULAR INTEGRAL OPERATORS

308 with

c

independent o f

and

f

X

We t a k e an a r b i t r a r y compact s e t

fk =

{ a ,a2

-

such t h a t XIk

by t h e i n t e r v a l

F.

IKI C

i.e.

.

.

If

x

8

K, t h e r e e x i s t s

1

k = 1y2y.,.,M, with

€(xk)]

.

L e t us d e f i n e f

.

!Hfc(xk)l > A

E(X)>fl

We t a k e a f i n i t e number o f d i s . i o i n t

xk) , xk + M 21 (I I k l

sum o f t h e D i r a c d e l t a s o f

Therefore

.

in

For each

k = I , ? ,... ,M,

i s t h e sum of t h e D i r a c d e l t a s o f

fk

Ik

,..., aN 1

X

IH f(x)! E(X) Ik= L X ~

intervals Xk E K

-

XI

{x : H*f(x) > such t h a t

contained

K

fi + f

- fk ,

w i t h suDDort o u t s i d e

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

i.e.

let

sunported

f

f i i s the

Ik.We can w r i t e

t

!if*(t) = k

I k , since Hf$(-)

i s decreas ng over

IHf;

t)I >

[xk -

E(Xk)

Thus in

X

f o r each xk

t h e h a l f i n t e r v a l of

{IHfZl

>

1

since

t

i n [xk IHf;(xk)(

, xk + c ( x k ) ] >

Ik where t h i s happens.

1 X 3 3 2Ik

.

.

A

Ik

.

o r f o r each

L e t us c a l l

h

We have t h e n

We can a l s o w r i t e

We s h a l l t r y t o e s t i m a t e t h e l a s t s e t .

so

has no s i n g u l a r i t y o v e r

We have

Hfi = Hf

-

Hfk

and

t

11.1. THE HILBERT TRANSFORM

309

Hence

Therefore we can s e t

Since

using Lemma

11.1.1.,

i s a r b i t r a r i l y c l o s e t o l{H*f > XI1

IKI

For clear that f o r f e L’

g =

1 a j xI

Hg(x) a t a.e.

11.1.8.

we get out theorem.

, where I 5 i s a comnact i n t e r v a l , i t i s a . e . x E R ’ . Therefore Hffx) e x i s t s x s R 1 , and a l s o H i s of weak type ( 1 , l ) .

e x i s t sj a t

The L2-Theory.

The L2-theory of t h e truncated H i l b e r t transform i s very simole by means of t h e Fourier transform. We have

with

.

independent of E and x Therefore , i f f E L2(R1) , . W e know t h a t f o r c I I f l ( 2 with c independent of f , E

c

I( H E f ( ( 2

g =

N

1

j=l

exists

aj

a.e.

xIj

where

Ij

i s a comnact i n t e r v a

By an easy d i r e c t computation one can check t h a t

.

HEg -+ Hg(L2) as E 0 each f E L 2 t h e l i m i t of -f

From these f a c t s we sha 1 deduce t h a t f o r HEf as E + O i n L 2 e x i s t s , In f a c t ,

11. SINGULAR INTEGRAL OPERATORS

310 t a k e a sequence

{g,}

o f s i m p l e f u n c t i o n s as above such t h a t

qk

-f

f(L').

Then we have

Given gk

n

> 0

,

i s fixed in

so t h a t

gk

2c

I( f -

11 Hfl12

cII f ( I 2 .

c

With t h i s r e s u l t and t h e f a c t t h a t

11.1.3.

TffEOREM.

Pmvd. i n t e r v a s , and

XI

-

E(X)

Let

1

j=1

,

0

fk= f X

Ik

< 2

IKI

ft = f -

11.1.2,

and Ik we have

and so

IHft(xk)l > X

Theorem

c . > 0, J

Ej

H*

A

a 6 weak

d i s j o i n t compact

Ejl

.

F o r each

fk

I

=

M

[I

.

1

[ x k - E ( x ~ ), xk Ik

I

. ' F o r each

As b e f o r e

,

x

6

K

there

, We t a k e a f i n i t e

E ( x ~ ) ~with

+

k

=

1,2 ,...,My

let

i n t h e p r o o f o f Theorem

.

Now t h e f u n c t i o n fi

H*.

f(x)I > X

such t h a t

such t h a t

support o f

xEj

R'- I01

X > O . We t a k e a compact s e t K c o n t a ned i n

number o f d i s j o i n t i n t e r v a l s Xk E K

cj

{endpoints o f the i n t e r v a l s >

i s decreasing i n

The maximal ffLLbent opehaton

N

f =

1

;;

(2,2) o f

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

{H*f >

Once

~ , 6a r e small enough we have 11 HEgk - H6gk1I2c n / 2 . i s a Cauchy sequence i n L 2 and so converges t o a f u n c t i o n

L 2 . Furthermore we have

exists

.

n/2

gk112
0, i

=

11.1.4.

l,Z,...,n

I t i s s u f f i c i e n t t o Drove t h a t , i f

, then

To do t h i s we can w r i t e m.

=

n

1

i=l

mi

"

1 - + 2 1=1 (Si - 1 ) *

1

1 lti<j 0,

has a p r e t t v

T h i s , as we s h a l l see, i s n o t s i m i l a r t o t h o s e we have o b t a i n e d

So we s h a l l c o n s i d e r a CalderBn-Zygmund k e r n e l

about

H

and

H*

i n Rn

,

i.e.

a function

k : Rn

ii) k h x ) = X-'k(x)

-t

R

, for

X

We s t i l l need some smoothness c o n d i t i o n on

k

E ) such t h a t

(or

> 0

k

,

x # 0.

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 the i n t e g r a l L i p s c h i t z condition t h a t follows i s already s u f f i c i e n t f o r this.

314

11. SINGULAR INTEGRAL OPERATORS iii) There e x i s t s

1I

IY

XI4 ,

c > 0

-

(k(x)

such t h a t f o r each

k(x -y)ldx h c


, we have,

, we g e t

x

Therefore,

1 , by

Kolmo-

11. SINGULAR INTEGRAL OPERATORS

324 Since

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

compact support, K h(x)

n

E(X)

>

K

EYn

h(x)

KEh(x)

e x i s t and a r e f i n i t e ) . 0

.

sun I K E h ( x ) l

E N

(Observe t h a t , s i n c e

- Knh(x)

Hence f o r each

and b o t h x

f i x any a r b i t r a r y f u n c t i o n

x E Rn

+

E(X)

and t h i s w i l l conclude t h e p r o o f o f s t e p We c a l l

Ioj

KEh(x)

, there

and

i s an

L l ( x ) = IJ J p > 0

t h a t i f we

[1973]

e ( 0 , ~ ) , then

C.

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

and f o u r t i m e s as b i g i n diameter.

where

Rn

has

such t h a t

We s h a l l now show, f o l l o w i n g Calder6n and Zvqmund

Qj

8

h

1,.

[(h(z)(

zi

L e t us c o n s i d e r t h e f u n c t i o n + I]

Ik(x-2)

- k(x-zi)ldz

J

w i l l be c o n v e n i e n t l y chosen i n a moment.

We have

as

11.2. CALDER6N using condition

(iii) on

k

-

325

ZYGMUND OPERATORS

and t h e f a c t t h a t

We now s e t

r

i s extended over a l l i n d i c e s

where contained i n {z : Iz-xI

, the

> E(x)}

sum

lz

J

j

such t h a t

Osj

i s entirelv

i s extended o v e r t h e r e m a i n i n q

i n d i c e s , and

x d !I

Now, i f

since

J

\

Oj

h(z)dz

gj

=

we have

0

.

So

I 1'1 c j

1

Ll(x)

We s h a l l i n a moment a l s o show t h a t T h e r e f o r e we s h a l l t h e n have

. 1

J

G

-1-1C 11 +

Ll(x)l

.

11. SINGULAR INTEGRAL OPERATORS

3 26

and t h i s w i l l conclude t h e p r o o f o f t h e theorem. To show t h a t

IP31

1

> 7

Thus

and so

and so

IQjI

then

1121c J

C

IX + L l ( x ) (

, we

f i r s t observe t h a t i f

11.3. GENERALIZED HOMOGENEITY

327

Hence

so

where the last written sum is extended over all indices j such that Qj intersects Iz : Iz-xI c . But since x t 0 6j each such Q is contained in

E(x)I

j

Iz

:

12-XI

>

1 2

(1

{ z : IZ-XI c

3E(X)

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. The motivation for such generalizations was initiallv to trv to a w l y the same methods of CalderBn and Zygmund to differential operators of parabolic type. Such generalizations have proved later also very

11. SINGULAR INTEGRAL OPERATORS

328

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 An example o f such t y n e o f

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 .

w i l l be g i v e n i n Chapter 1 2 .

applications

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] Guzmdn [1968,1970 a,

, Fabes [1966] , Fabes 1970 b ] , and o t h e r s ,

and

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 g o i n g 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 geometric 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 h o u g h t o f Guzmdn [1968, 1970 a l . The problem we a r e g o i n g t o handle i s t h e f o l l o w i n g . L e t 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.

A

Consider, f o r A > q, t h e

mapping

x e~~ The t r a n s f o r m a t i o n

TI

-f

T ~ X =

eA 1 o g X

ERn

i s a sort o f d i l a t a t i o n (for

A = I , TAx =

Ax)

I f we assume t h a t A has eigenvalues w i t h p o s i t i v e r e a l D a r t , t h e n we

have f o r each

x e Rn

-

I01 , TAx

+

0

as

A > 0

and

lTAxl

-f

as

A+-. We s h a l l c o n s i d e r k e r n e l s respect t o the d i l a t a t i o n s

T,I

k : Rn

-

I01 + R

satisfying, with

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 ations, i.e. k(TAx) =

A

-tr A

k(x)

We s h a l l ask o u r s e l v e s whether i t i s p o s s i b l e t o get,from such k e r n e l s , c o n v o l u t i o n o p e r a t o r s t h a t s a t i s f y s i m i l a r theorems as t h o s e o f Calder6n and Zygmund o b t a i n e d 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

i f t h e t r u n c a t i o n i s n o t adecuate).

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 n order t o f i n d

11.3.

GENERALIZED HOMOGENEITY

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

translations, associated i n

-

,

i n v a r i a n t bv

A, which

a n a t u r a l way t o t h e m a t r i x

e x a c t l y as t h e E u c l i d e a n

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 metric

P

329

TA 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.

I I

f o r each A > 0 and x E R n , T h i s w i l l be done i n p ( T X x ) = Ap(x) 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 useful theorems on apnroximation i n

11.3.B

11.3.A.

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

The M e t r i c Associated t o a M a t r i x

I n t h i s section, A

w i l l denote a f i x e d

whose eigenvalues have p o s i t i v e r e a l p a r t .

11.3.C.

A.

n x n

real matrix

F o 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 (how b i g w i l l depend o n l y on

eigenvalues i s b i g enough

n).

This w i l l

n o t l e s s e n t h e g e n e r a l i t y o f t h e theorems on a p p r o x i m a t i o n 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 , F o r x e Rn

+

A > 0, TX

TXx =

i s t h e mapoing

eA 1 o g X

eRn

A

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 a s s o c i a t e d t o

11.3.1.

numb0

d

o

p(x) p(0) = 0

,

LEMMA. -

Fotl

0 < P(X)
0.

f o r each clear that

+

-

{O}

X

we d e f i n e f o r

6

(0,m)

satisfies

(0,m)

is a symmetric m a t r i x w i t h p o s i t i v e e i g e n v a l u e s . z 6 Rn - { O ) , (z,(A + A*)z) > 0 and so + ' ( A )

A t A*

The m a t r i x

x eRn

0 = yp(x)

-+

+ ( p ( x ) ) = 1.

such t h a t

i s a s i m p l e 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

TA( i.e.

TI

1

T1 x = 2

T

A,X2

x). I n f a c t XI

Hence

p(Tllx)

=

o r d e r t o prove if

p(T -1 x ) =

x

pp(x). (iv)

1

f o r e i n o r d e r t o show

Properties

( i i ) and ( i i i

we f i r s t observe t h a t p(x)


r )

11 In f a c t

, 11 e-Aull

m i n (eigenvalues o f A

=

e - A u \ ~c e-v

max { eigenvalues o f

A+A* 7 ) a

have real p a r t big enough

e-

A+A*

I c e-u, i f

this i s so i f t h e eiqenvalues o f

1. B u t

. Therefore

we get

(1 e-A’ll

6

e-’

for

p > 0.

So we obtain f i n a l l y

and hence

p(x + y)

c

p(x) +

p(y)

.

We s h a l l now s t a t e and prove some p r o p e r t i e s of t h e 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 c~i ~> 0~

eqlLiwaeentey

nuch t h a t id

1x1 6 1) we have

(GI Thehe LA a constutant B > 0 nuch t h a t equivaeentey 1x1 > 1) we have

.id

p(x) c 1

p(x)

(and

a 1 (and

332

11. SINGULAR INTEGRAL OPERATORS

H e m

c1

Let

Pltood.

11

e-Ap/l 6 e-u

and

depend o n l y on t h e mathix A.

$

p(x) G 1. Then (recalling that for

=

e-A log P(X)

0 depending only on A. Hence

c1 >

Let now p ( x ) > 1. Then, i f

and so p(.x)

&

1x1 =

for some

(p(x))"

X

x,

e -A log

=

1x1.

g

P(X)

I

1x1. On the other hand 21s

leA log

$ >

> 0

)

On the other hand we have , i f

with

'CI

0. Therefore

11 eA

(x/'

log

6

=

max {eigenvalues of

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

x= where

1

T (p(x))-1

i s the unit

-A 109 P(X) =

sphere i n Rn

x e c

11.3. GENERALIZED HOMOGENEITY The m a m i n g

- {Ol

x 6Rn

a system o f p o l a r c o o r d i n a t e s .

(:,

-+

p(x))

6

333

1x

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

g r a l on R " f o l l o w i n g way h(x)dx

dx

tr A-1

I

h(eA l o g P x- )

=

(Ai,i)ldi p

dp

~ D = O JieL

Jx eRn Here

( 0 , ~ ) defines

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 r a n s 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 e x p r e s s i o n o f t h e volume element i n terms o f

d?

dp ) .

and

A Theorem on APDrOXimatiOn.

11.3.8.

L e t us r e c a l l Theorem

,

k e L1(Rn)

10.1.1.

k > 0 , /k = 1,

with

K*

K*f(x)

=

(1,l)

we s e t

,

for E>O,

x eRn

d e f i n e d by sup

lkE

E>O

i s o f weak t y p e

and i f

E - ~k(--)X

kE(x) =

Then t h e maximal o p e r a t o r

I f we have a r a d i a l f u n c t i o n

and so

kE

*

*

f(x)

f(x)

-f

1 f ( x ) a t a l m o s t each

x eRn.

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 E u c l i d e a n m e t r i c by t h e m e t r i c t o the matrix

A

o f the type considered i n

11.3.A.

p

associated

So we a r r i v e a t

the following r e s u l t .

@(x) = p

1.

11.3.3.

THEOREM.

@(y) id

p(x) =

Fuh

E

> O and

Le/t @ e L1(Rn) p(y)

,I$>

0

, .f@

= 1

( i . e . @ h "mLddi&" w&h

x e R n be2 un dedine nuw

and tape& t o

334

11. SINGULAR INTEGRAL OPERATORS

a*

Then t h e maxim& 0p-u~

.i~ 0 6 weak t y p e

(1,1), 06 Q p e

Hence +E Lp (Rn) , 1 c p
~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 eigenvalues o f

A is

big. 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

EX

10.1.1.

One has o n l y t o observe t h a t , for 110, t h e

set

p ( x ) L XI i s an e l l i p s o i d c e n t e r e d a t t h e o r i g i n of A IEI( and t h a t t h e s e t s EX a r e n e s t e d convex s e t s . We

= I x sRn :

Atr

volume arrive

proceeding as i n

10.1.1.

a*f(x) L c where

S

to

Sf(x)

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

i s a f i x e d f a m i l y o f n e s t e d convex s e t s we know, by 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 But since

obvious. So one o b t a i n s t h e theorem.

335

11.3. GENERALIZED HOMOGENEITY

11.3.C.

Generalized S i n g u l a r I n t e g r a l Operators.

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 11.2.3.,

11.2.3. 11.3.4.

that

(ii) Fuh

11.2.1.,

For examnle we have t h e f o l l o w i n g r e s u l t s .

THEOREM. l & -

x # 0

,A

> 0

,

k : Rn + R

k(TXx) = A

-tr A

be u @mtiun

duck

k(x)

(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 obtained. The n r o o f o f t h e s e 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 Section. Observe t h a t i f A

then i t s a t i s f i e s

k

satisfies

(i), (ii), (iii)w i t h a m a t r i x

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 .

11. SINGULAR INTEGRAL OPERATORS

336

T h e r e f o r e 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 t u d y 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 i m e 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 t h e n more c o m p l e t e l y by S t e i n and Wai nger [ 19781. The s t r o n g l y geometric 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 geometric u n d e r s t a n d i n g 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 Besides, such a t y p e o f s o l u t i o n

as those shown i n Chapter 6 t h r o u g h 8.

would p r o b a b l y 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 g o i n g t o use cannot handle. The problem we a r e g o i n g t o s t u d y h e r e i s t h e f o l l o w i n g . L e t (yl(t),

y(t) =

i n Rn

1c p

with

c

m

..., y n ( t ) ) , y ( 0 ) = 0 . For

,let

t e

[O,m),

each

x

8

be a f i x e d c o n t i n u o u s c u r v e Rn

and f o r

f 6 Lp(Rn)

,

us c o n s i d e r

Under what c o n d i t i o n s on l i m i t e x i s t s and i s

f

and y

f ( x ) a t almost each 337

can one say t h a t t h i s

x e Rn?

338

12. DIFFERENTIATION ALONG CURVES O f course, i f

f(x)

$$ (R')),

maximal o p e r a t o r i s o f weak t y o e each

f

B

Lp(Rn)

then t h e above l i m i t e x i s t s and i s

So i f we a r e a b l e t o show t h a t t h e c o r r e s p o n d i n g

x E Rn,

a t each

f E

(p,p)

and f o r almost each

we o b t a i n t h e same n r o o e r t y f o r x E Rn.

A s 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 maxi m a l o p e r a t o 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 nrove the

by u s i n g t h e P a r s e v a l - P l a n c h e r e l theorem. The tvDe 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 .

t y o e (2,2) for

(p,n)

For the

t y p e (p,p), 1 < p < 2 , one embeds o u 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

.

(p,p) , 1 < p < a We s h a l l h e r e o r e s e n t i n 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 . The o b t e n t i o n o f t h e

one can o b t a i n t h e t y p e

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] .

12.1; THE STRONG

TYPE (2,2) FOR A HOMOGENEOUS CURVE.

We s h a l l c o n s i d e r t h e curve y(0) = 0

A

,

where

v

y ( t ) = eA lo' v,

for

t >

o ,

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

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 eigenvalues H t = u , H > 0

w i t h p o s i t i v e r e a l D a r t . I f we make t h e s u b s t i t u t i o n then r(u) = y ( u H ) = e HA lg v and so we can assume w i t h o u t loss o f g e n e r a l i t y t h a t t h e eigenvalues o f enough.

A have r e a l p a r t s t h a t a r e b i q

Such a c u r v e w i l l be c a l l e d homogeneous. I t 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

hyperplane,

y(t)

i s not contained i n a

Otherwise t h e same r e s u l t f o r a l o w e r dimension g i v e s us

t h e theorem we l o o k f o r . 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 s o c i a t e d t o A . t h a t we have c o n s i d e r e d i n

11.3.

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

P

The p r o p e r t i e s we

as-

339

12.1. THE STRONG TYPE (2,2) Mf(x) =

sup

E?O

i’ 0

/f(x

-

y(t)ldt

f E Rn

I t i s not d i f f i c u l t t o see t h a t i f

+

E i s a measura-

ble f u n c t i o n , the function

i s f o r almost each x E R n a measurable function of t and so t h e maximal o a e r a t o r M i s well defined a t almost each x E R n , For

M

12.1.1.

04

b&Vng

type

we s h a l l prove t h e following r e s u l t THEOREM.

The maxim&

VpQhatoh

M dc6ined above A

(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 t h e idea behind the Droof. Assume f r 0 and w r i t e , f o r b r e v i t y , f t ( x ) = f ( x - y ( t ) ) . One could be tempted t o w r i t e , u s i n g t h e Schwarz inequality

Therefore

I f we use t h e Parseval-Plancherel theorem, havina i n t o account

that

we get

which, of course, leads nowhere. The f a c t o r

e

- 2.iri(c, Y ( t ) )

has alwavs

12. DIFFERENTIATION ALONG CURVES

340

we cannot expect a n y t h i n g from ( * )

1 and so

modulus

L e t us t r y t o modify o u r scheme.

I

E

E

.

L e t us c o n s i d e r , i n s t e a d o f

ft(x)dt

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)

2 Mf(x).

But, on t h e o t h e r hand,

observing t h e F i g . 12.1.1. we o b t a i n t

t

0

0 F i g u r e 12.1.1.

2h ft(x)

E

E

1

d t dh

TE

Hence

Mf(x) 6

1 lg2

E>O

We have now s u b s t i t u t e d with

n

Nhf

iE

sup

Nhf(x)dh

0

ft(x)

by

if we proceed a s before.

Nhf(x) We have

and perhans we a r e l u c k i e r

>

12.1. THE STRONG TYPE (2,2)

341

Now

X

> 0

,

I f we c a l l we can w r i t e

TXx

=

, T; x = eA*lg ' x ,

eA 1 g ' x

2

Nhf(5) A

./ e -2 Ti i(Tsh

=

v * 6 ) d s ;( 0,

~1

> 0,

for

h a 1

0 < h c 1.

for

w i l l be proved i n t h e f o l l o w i n g lemma. Hence,

as we have i n d i c a t e d b e f o r e ,

M

and so

i s o f s t r o n g t y p e (2,2) as we wanted t o p r o v e . I n o r d e r t o p r o v e 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 d e r Corout. 12.1.2.

LEMMA. -

C o a i d e h t h e integhal

Ja

wlzehe j,

f

0 a heal & n c t i a n .in kntl(

2 6 j 6 n+l

we have

we have

[la,b])

and ~ n w n eA h a t tioh Oame

I f ( j ) ( u ) l > aj > 0 6uh each

u e [a,b]

. Then

dependn o n l y o n j .

whme c j

j =2

The p r o o f f o r For

345

THE STRONG TYPE ( 2 , Z )

12.1.

j > 3

can be seen i n Zygmund El959

, v o l . I ,p.197].

t h e p r o o f i s o b t a i n e d 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 r o v e t h e i n e q u a l i t y (*) as f o l l o w s .

and

.

WLth t h e n o t a i i o n uned i n t h e Theohem 12.1.1. phood, connididen i h e iM-tegkal 12.1.3.

LEMMA

Pkaa/,.

L e t us f i x

X > 1

and s e t

, for

I 0

,

Now c o n s i d e r

We s h a l l prove

with

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 contained

i n an a f f i n e hyperplane 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

B = {

V,

Av,

...,A n - 1v 1

12. DIFFERENTIATION ALONG CURVES

346

is a basis in R n . with

In fact if y ( s ) = eA l g v' is in the hyoerolane (x,w) IwI = 1 (recall that y ( 0 ) = 0) we have (eA I g s v,w)

Differentiating and setting s (v,w)

=

(Av,w)

=

=

=

for

0

= r)

s > 0

1 we get

.... =

(A"'v,w)

=

0

and so B = { v,Av, ...,An-l v 1 cannot be a basis. Conversely if B is not a basis there is some w, IwI = 1 , such that = ... (A"' v,w) = 0 n-1 If zn + c1z ... + c, is the characteristic polynomial of A we -t cnI = 0 and so (Anv,w) = (An+lv,w)=...=D get A" + clAn-' + ... + cn,'A Hence

(v,w) = (Av,w) +

(eAlgSv,w)

=

0

s > 0

for each

and y ( s ) is in the hyperplane (x,w)

=

0

.

Now observe that

If zn + c p of A we have

and so

-t

... +

cn-1z + c,

i s the characteristic polynomial

347

12.1. THE STRONG TYPE (2,2)

If f o r some

then

co #

g” (s) A50

=

0

0

(eAs v , onal t o A** A S o . compactness of C with r e s p e c t t o X (5,s)

8

c

and some s o we have

for all

s

and so

, i.e.

y ( t ) i s i n t h e hyperplane orthogThis i s a c o n t r a d i c t i o n , and t h e r e f o r e , using t h e and t h e l i n e a r i t y C ={5 6 Rn : 151 = 1 3 x 10, l g 2 , t h e r e must e x i s t a > 0 such t h a t f o r each x LO, l g 23 we have A*2AC0) = 0

1,

Now,for each ( s * , s * ) B C x[Oy l g 21 t h e r e 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 1 x [ O,lg2] w i t h a f i n i t e number of such I*. Let us consider their p r o j e c t i o n s over [ 0 , l g 21 and a l l sets the H consecutive closed i n t e r v a l s determined by t h e extreme points of

the projections

The number H depends only on our matrix A t i o n , which i s f i x e d once f o r a l l .

and our c o n s t r u c

If

then, according t o t h e previous lemma, we have f o r each s . 6 t < sj+l 3

5

6

C

, if

12. DIFFERENTIATION ALONG CURVES

348

o!r

X > 1 with

6 > 0 and c > 0 b o t h independent o f

So we have proved

with

c

,for

independent of

In] a

t > 0

1

5, i > l , and

,

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

and

c

independent o f

h

t > 0

1< 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 S e c t i o n 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 concerning t h e problem d e a l t w i t h i n 12.1.

12.2.1.

THEOREM.

Then, do& f e L p ( Rn)

,1


Let y(t)

0

-

, be a

0 L t L 1 fie.b

dak

{Y(j)(O)I

we have , d o t f

y ( 0 ) = 0 . Abbume t h a t

p=2

y(t)

the result f o r

(Theorem 12.1.1)

S t e i n and Wainger embed t h e o n e r a t o r

P r o o f o f Theorem 12.1.1

i n a f a m i l y o f operators

For t h e o p e r a t o r s

N;

3 49 y(t)

1 < n i -,

E Ln(Rn),

ed by i n t e r p o l a t i o n between

and

n > 2 i s obtainn =

(trivial).

Nh, d e f i n e d i n t h e

by

N:

,

Rez < 0

,i d e n t i f y i n g

, defined

by

the inverse Fourier transform o f

and u s i n g a v e r s i o n o f t h e Theorem

(p*( 0. j For big r , the s e t Dr looks very much l i k e L j . Define T': by means j J f V . I t i s t o be expected t h a t T': of = ( X will approach D; J Tj

Ti

^f

as r + m . In f a c t , i f f e (&o,we have T i T . f i n every d e s i r a b l e J sense. This permits us t o s h i f t t h e problem t o t h e operators T r It j * w i l l s u f f i c e t o prove +

independently of nates, that

Therefore, i f

r.

Now i t i s easy t o e s t a b l i s h , by a change of coordi

(**) holds w i t h

Now observe t h a t

Therefore we have

r = 1 i t holds f o r any

r > 0.

13. MULTIPLIERS AND MAXIMAL OPERATOR

368

I f we now apply Lemma 13.1.2. one.

we conclude w i t h the proof of t h e present

13.2. POLYGONS WITH INFINITELY MANY SIDES.

We have seen t h a t i f D i s t h e u n i t disk then xD i s not an LP-multiplier f o r any p # 2 On t h e o t h e r hand i f J i s h a l f p l a n e , 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 j and i f T p i s t h e m u l t i p l i e r operator corresponding t o P and T I ,TZ ,.. 'Tk those corresponding t o J1 ,J2 ,.. , J k , we have

.

.

.

A

'J1 'Jz Therefore T p =

TIT2

... T k

and Xp

. . I

xJk

f =

i s a l s o an

(TlT2..

.Tkf )"

LP-rnultiplier,

l < p < m .

Assume now t h a t i n t h e sense t h a t

P

i s a polygon with i n f i n i t e l y many s i d e s

(if; + 0 ( i i ) For any two J . , J , the border of J i s not p a r a l l e l J k j t o t h e border Of J k ( i i i ) For each J j a P 1) a J j i s a segment of p o s i t i v e length

I t i s easy t o c o n s t r u c t s e t s P of t h i s type, even c o m a c t convex s e t s of t h i s type. For example, given any sequence of angles { $ . I , > $ j > 0 , $ , G 0 one can c o n s t r u c t a polygon J i n those d i r e c t i o n s a s indicated in Figure 1 3 . 2 . 1 .

T2-

J

P w i t h sides

13.2. POLYGONS

f

WITH MANY SIDES

369

$1

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 intention o f obtaining a negative result for sets P of this class one inmediately observes that the observation (a1 is valid without any substantial modification.

In fact, if xp

s an LP-multiplier with norm c

P' 2 c p < , and if T~ is the translation that carries the midpoint of the side aJj 0 aP o f P to the origin, the also X-r.P is an J LP-multiplier with the same constant. If we call v the unit vector j

orthogonal to aTj

I1

a P d rected towards the interior of T . P J

and

13. MULTIPLIERS AND MAXIMAL OPERATOR

3 70

L. = J

C

x

B

R2

: (x,vj)

> 0 1

we have, f o r any sequence

then, s e t t i n g

i f j } o f f u n c t i o n s i n Lp(R2)

, exactly

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 ( c ) , 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 . Therefore,in order t o obtain a negative r e s u l t f o r

P, i . e .

xp 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 o prove t h a t g i v e n 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 IT > 0 that

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 k , each Rh w i t h one s i d e i n d i r e c t i o n vh j o i n t r e c t a n g l e s fRh)h=l

so t h a t 2

1 100

as i n t h e o b s e r v a t i o n ( 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.

from t h e P e r r o n 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

a Perron t r e e i n t h e sense o f 8.1.

with i t s small 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 h o s e o f

1 v . j t h e n we g e t what we need. T h i s 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 y p e s o f P e r r o n

trees. 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 from 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 ~ R D O B AAND R. FEFFERMAN 13.2.1.

THEOREM. LeX u6 c o a i d m t h e bequence 0 6 chkecaXua 1 ( ~ ~ d i a n(See = ~ ) Fig. 1 3 . 2 . 1 . ) . 1eA UA con~;Dtuct

d e L m i n e d by 4 . J J any paCygon P 0 4 t h e t y p e cuuznide/ted i n tkin neotion w d h one bide i n each ClihecaXon . Then Xp & not an LP-mUpUm 6uh any p f 2 .

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,) , J + 0 , and let P be the convex set indicated in

n 0 < 8 . < 2 , oj J Figure 13.3.1.

371

13. MULTIPLIERS AND MAXIMAL OPERATOR

372

P

Ao

0

A3

2

23

22

Figure 13.3.1.

The p o i n t A, i s an a r h i t r a r y point of x = 1. The o t h e r Each A j i s the v e r t i c e s o f P a r e t h e points A j , j = 1,2, point of x = ZJ such t h a t Aj,l A forms w i t h Ox an angle of We have indicated t h e midpoint E j of Ajvl A j and amplitude e j * t h e inner u n i t normal v j t o P a t E

...

3.

On t h e other hand l e t @ be t h e d i f f e r e n t i a t i o n b a s i s of a l l rectangles with one s i d e in one of t h e d i r e c t i o n s { v j l . We can s t a t e t h e following r e s u l t . 13.3.1. rHEOREM. LeA P be t h e t h e d . i d { ~ e n t i a t i o n ba4i.A denchibed above.

deA

j u - t dedined and

@

13.3. A THEOREM OF A. C ~ R D O B AAND R. FEFFERMAN

373

f E L(R2),

Then: nome P > 2,

(a) 7 6 K O a6 hfhung t y p e ((!)',(f)') then xP o an LP-mlLetiptieti (6) 16 Xp O an Lp-m&2pfieti doh each mmuhabbe E C R 2

ICx e R 2 then K

O

06

weah @pc?

:

doh home

KXE(x) > 1 11 L

(($1'

y(5)')

p > 2 and i d we have,

CIE~

Observe that the additional condition in (b) holds when 6 i s known to be a density basis.

as we know,

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. M

f

6

TffEOREM.

LeL

H denote t h e H a b c k t

t h e ohdinahy fflvrdy-LLttkkwvod m a x h d o p ~ ~ n d t oi hn Lp(R1) , g 6 Lp(R') , 1 < p < m , and doh any E > O

a?~~n.hdohmand

R'.

LeL

huch t h a t

p1+E> 1 , LeL

Then

whme cE 0 independent

06

f

and

g

.

Observe that the fact that M is of type (p,Pl, P > 1, implies that also ME is O f type (p,P).. The other result we shall use i s the following theorem of Paley and Littlewood that can be seen, for instance in Stein [1970,

13. MULTIPLIERS AND MAXIMAL OPERATOR

374 p. 1041

THEOREM.

13.3.3.

E . = C(x,y) 6 R 2 :J apen_atah cornenpunding t o Then, doh each

L&

j = 1,2,3

(Vh

x < 2J)

,...

, and LeA: Sj

x .

E j , i . e . ( S j g)* =

, 1
2 we a r r i v e t o t h e inequality,

13. M U L T I P L I E R S AND MAXIMAL OPERATOR

376

for f . a Lp(R2) J

,j=

...

1,2,3,

where T is as there, the multiplier operator associated to j L . = Cx a R 2 : (x,vj) 2 0 1 . J By means of this inequality we are going to obtain a covering lemma for the rectangles of @ from which the type y(f)') of the operator K is an inmediate consequence. ((:)I

h Assume we are given a finite collection CRklk=l of the basis 8 . Let us assume they are ordered so that

k > 1

If

-

Let us choose satisfying

-

Rk,l

= Rj

R1

R1

=

The RZ will be the first Rk with

-

k-1

,.,

- 1' -

Ril

and so on, Thus we obtain {Rk3E=1

-

Y

On the other hand, i f R IRj

that is ,

-

.

, then Rk will be that lRh

is not i n

j

-

s

-

IJR k \

k=l

of rectangles

Rh

'

3

with

F lRhl

so that 3

"if

h > j

such that

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, a c c o r d i n g t o t h e c o n d i t i o n we have assumed on

-

Ri

=

-

Rk

Assume t h a t

,

-

Ri

,

-

R i

Rk

-

, Rk

i s i n the direction o f

K i n (b)

and l e t

v J. ( 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.

L e t us now s e t

,. Ek =

-. Rk

-

tI

-5

RJ

.

Since

j