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
327 337 338 348
12.3. An a p p l i c a t i o n . D i f f e r e n t i a t i o n by r e c t a n g l e s determined b y a f i e l d o f d i r e c t i o n s
350
CHAPTER 13: MULTIPLIERS AND THE HARDY-LITTLEWOOD MAXIMAL OPERATOR
359
13.1. The c h a r a c t e r i s t i c f u n c t i o n o f t h e u n i t d i s k . A theorem o f C. Fefferman 13.2.
Polygons w i t h i n f i n i t e l y many s i d e s
13.3. The maximal o p e r a t o r w i t h r e s p e c t t o a c o l l e c t i o n o f rectangles. A theorem o f A. CBrdoba and R. Fefferman
362 368
371
REFERENCES
379
A LIST OF SUGGESTED PROBLEMS
389
INDEX
391
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1
CHAPTER
POINTWISE CONVERGENCE OF A SEQUENCE OF OPERATORS
There a r e q u i t e a number o f i m p o r t a n t problems i n F o u r i e r A n a l 1
s i s and i n o t h e r areas i n which a sequence ( o r g e n e r a l i z e d sequence ) o f o p e r a t o r s a r i s e s i n a n a t u r a l way. i)I f
f
E
L1([0,2x)),
f
periodic o f period
2 ~ r ,t h e p a r t i a l
sums
F o u r i e r s e r i e s can be i n t e r p r e t e d as t h e a p a l i c a t i o n o f t h e o D e r a t o r t o the function where
Dk(x)
f . E q u i v a l e n t l y S k f ( x ) = Dk
, the
*
f(x) =
D i r i c h l e t k e r n e l , i s d e f i n e d by
1
TI
Sk
f(x -y)Dk(y)dy
-Ti
s i n ( k t T1) x Dk(X) =
X
TI s i n 2
ii)I n an analogous way, t h e Cesaro sums
Okf(X)
=
S,f(x)
-t
Slf(X)
-t
.,. -t
Skf(X)
k + l
can be i n t e r p r e t e d as t h e a p p l i c a t i o n o f t h e o p e r a t o r i s t h e F e j 6 r k e r n e l , d e f i n e d by
Fk(x) =
1
2n( k + l )
1
ak
to
f. I f
fk
1. POINTWISE CONVERGENCE OF OPERATORS
2
We know t h a t
parameter
r
A r f ( x ) = Pr
,is
*
f(x)
Pr(x) =
i v ) If f E L1(Rn)
i s t h e mean v a l u e o f
f
over
where
1 -
2n
t h e Poisson k e r n e l of
Pr(x),
1-r2
1 - 2 r cos x + r 2
, B(0,r)
B(x,r)
= Cx
e Rn : 1x1
6
r
I
which i s c o n s i d e r e d i n t h e t h e o r y
o f differentiation o f integrals. v ) I n a more general way
\
,
k ( y ) d y = 1, k E ( x ) =
if
E
-n
k E L1(Rn)),
k),(
x
for
E
> 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 <
2 E
hE(x) = 0
,
E
1.0. INTRODUCTION
then
i s t h e truncated ( a t v i i ) If
k
E
) Hilbert transform.
i s a complex valued function defined on Rn--COI
such t h a t
then, f o r f 8 L 1 ( R n ) , KEf(x) = k E * f ( x ) i s t h e truncated ( a t E ) Calder6n-Zygmund transform of t h e function f , considered i n t h e theory o f singular integral operators. The most natural question in a l l t h e s e s i t u a t i o n s i s : ( A ) Tv dind vu.t W h d h a ,Oh u n d a which a d d i t i v n d nvn & v i a l c v n d i t i v a vn 5 v h an t h e vpehatvlrn Th ,the cornuponding bequence Tk6 ( x ) cvnuagen 6vh e v a y x O h denvot eVt?hy x and whdt ahe t h e p m ~ W e h ad the,& h L i 2 .
This i s , i n a l l the cases we have considered, a t the same time t h e most d i f f i c u l t question. In many of them, because of t h i s d i f f i c u l t y , t h e theory s t a r t e d w i t h a l e s s ambitious program: ( B ) Tv 5ind o u t w h e t h a , V R u n d a which a d d i t i v n d nvn .&ivial
cvnd&..LvnA vn 4 ah On t h e v p a d t v h n T k . t h e CvntLupanding oeyuence ~ u n C ; t i v a ~~6 cvnuenga i n Avme opace LP.
04
Question B has in many of t h e above presented s i t u a t i o n s a r a t h e r simple s o l u t i o n . Suppose t h a t we know t h a t T k i s l i n e a r , t h a t
4
1. POINTWISE CONVERGENCE OF OPERATORS
we can show, f o r example, t h a t , i f
11 T k f 11 6 c g E t o(Rn)
say,
Tg.
Ilf
p,q
]I f -
g
> 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) <
'"I(a)
f i n i t e f u n c t i o n s (where f u n c t i o n s t h a t c o i n c i d e
d:/)n(R)
, p.30 I . "m , we
f e
[ 0,m)
+
i s a quasi-norm
A l s o i t i s easy t o show t h a t f o r have
fn
-f
f
(p-measure)
i f and
0.
For a s u b l i n e a r o p e r a t o r
T
f r o m a normed space
to
X
'hl (R)
one a l s o shows e a s i l y t h a t t h e f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t .
-' M(R) i s
(a)
The o p e r a t o r
T : X
(b)
The f u n c t i o n
4 : (0,m)
-f
[O,m)
continuous a t d e f i n e d by
0
E
X
9
1.1. FINITENESS A.E. AND CONTINUITY
tends to 0 as X tends to
a.
Of course, if T is linear, then continuity of T on X .
(a) is equivalent to the
Likewise, let (TcOaeA be a family of sublinear operators from X to W (R). Then one easily shows that the follow ng statements are a1 so equi Val ent
I > h } + O
asXtm
For the theorem that follows we shall use the following form of the uniform boundedness principle, that can be seen in Yosida [1965, p.681: LeI
k%mu~
(X,il
A ~ U C ~ L.
11
~ A
dying doh each f, g e X
d(Ta(Af)) 7 6 the
(Y,d) a q u a i - n a m e d be a ~amieqo p e h a t a u &horn X t o Y A& -
) be a Banach npace and
=
d(ATaf)
{Taf : a e A} c Y d(Taf) = 0 u n i ~ a m L qi n a Ilfll-. 0 We recall here that the fact that a each neighborhood U of the origin lim
A&
bounded doh each f
E
X,
,then
e A.
set ScY is bounded means that for there is an E > 0 such that E S c U.
With these preliminaries the proof of the following theorem is straightforward.
1. POINTWISE CONVERGENCE OF OPERATORS
10
lei {TkIF=l be a bequence o d hubfineah ope l d u r n dhom X, a Banach hpace, t o "@l (a) w i t h u(Q) < m. h w n e t h a t each Tk LA conLLnuvlLs and that t h e maxim& opeha2vh T* dedined doh f E X and x E R an 1.1.1.
A huch thud T * f T*
A d
6
A ~.ivLite u-a.e. Then
%doh I each f, i .e. T*f at 0 , and thenedahe
o continuouA
Phaod. imal operator
Clearly T i 0 & T;f(x) each
TtlEOREM.
c
For
Ti
..
n = 1,2,3,.
we d e f i n e t h e t r u n c a t e d
i n t h e f o l l o w i n g way. For
f
E
i s s u b l i n e a r , continuous f r o m X t o T*f(x) f o r each x e a , we have
X
and
x e
( a t n) max-
n
and s i n c e d(T*,f) c d ( T * f )
for
f.
Therefore t h e uniform boundedness p r i n c i p l e a p p l i e s and continuous a t
~(6) <
-
0,
{T*,)
i.e.
Observe t h a t , i f
p(R) =
,
m
we can s e l e c t
c R
and, i f we d e f i n e
&i)
i s equi-
=
sup
Ilf I1 6 1
p
1 x
E
6 : T*f(x)
> X 1
t h e n w i t h t h e same h y p o t h e s i s of t h e theorem we a l s o o b t a i n
with
1.2. CONTINUITY AND A.E.
0
1.2. CONTINUITY I N MEASURE AT CONVERGENCE
E
X
CONVERGENCE
OF THE MAXIMAL OPERATOR AND A.E.
We a l r e a d y know t h a t t h e f i n i t e n e s s
a.e.
a t o r i m p l i e s i t s c o n t i n u i t y i n measure a t
0 e X
t h i s c o n t i n u i t y i m p l i e s t h e closedness i n
X
of
i n which t h e sequence
X
Tkf
11
.
o f t h e maximal operWe s h a l l now see t h a t
o f t h e s e t o f elements
converges
f
I n most i n t e r e s t i n g
a.e.
cases i t i s easy t o e s t a b l i s h such a convergence f o r some s e t dense i n and so we o b t a i n t h e
a.e.
convergence f o r a l l f u n c t i o n s i n
theorem t h a t f o l l o w s i t i s n o t necessary t o assume t h a t
For t h e
X.
X
X
i s complete.
be a sequence 06 &hean. opehatom 6hom X, a named Apace, t o 'hl (n). h b w n e t h a t t h e ma&& opehdtoh T* h cow%nuouh in memwte &am X t o at 0 e X . Then t h e be;t E ad d e m e h f 06 x dolt which {Tkf} CUnvmgtA at a.e. x e R iA doseed i n X.
we have
1.2.1.
THEOREM.
Phood. -
Let
PIX e =
R :
f
Since f e E.
X
l i m sup m,n -+a
PIX E R ; l i m sup m,n
G
E
LeA
211 I x 8
@
ITk)
and c o n s i d e r
ITmf(x)
-
g
E
E.
Tnf(x)[ >
\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 +-
<
~ C 6X 0.: T * ( f - g ) ( x )
b u t t h i s tends t o c e r o as
g
>
-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
, and
f e X, then, even
X > 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 <
f.
I t i s s a i d t o be o f weak t y p e
, if t h e r e
m
exists
c > 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 <
m
i m p l i e s weak t y p e
(p,q),
since f o r
X >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) <
there e x i s t s
a.e.
f
f o r each
m
06
From t h e general theorem 1.1.1. o f Chapter 1, we e a s i l y
r e f e r s t o t h e t y p e of
AN > 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) <
1
=
L e 2 S*
Pkoud. of
N
15
CALDERON
cN 2Tr
and so
AN
m
-f
,
We a r e g o i n g t o c o n s t r u c t , by means of
k=l
with
Nk
{Ak}
a new sequence
nondecreasing such t h a t
To do t h i s we f i r s t choose C
:N
> Z1
.
We have
CN:
/ :;1
6
2a.
2. FINITENESS AND THE TYPE
16 kl
We choose
1 < kl
CN:
<
72-
k l b e i n g such t h a t
copies o f t h e same number,
2n. The
kl
f i r s t terms o f t h e new sequence a r e going
N:
t o be
Now we choose
N;
NP
>
CN*
such t h a t
1
z2
and c o n s i d e r
N;
~2
<
2.
N; k2
we choose C
1
c
k
N*2
A:
copies o f t h e same number, k2
.
2 <~ 2~
A
The f o l l o w i n g
, A
N;
N;
,
. * a ,
terms w i l l be
k2
,k,
b e i n g such t h a t
A
N;
And so on. We e a s i l y check t h a t f o r t h i s sequence
tXN 1 k
We s h a l l now a p p l y t h e f o l l o w i n g lemma t h a t we p r o v e l a t e r .
be a hequence o d meanwLabLe bubo& u s T nuch t h a t C l E k l = a. Thcn t h e h e exA& { x k l C T ouch t h a t demoht L V U L ~ x e Y L&in LndiniteRy many 06 the h& Exk + E k ) . 2.1.2
LEMMA.
We choose numbers
mk
SO
xk
Le2
{Ek)
a c c o r d i n g t o t h e lemma.
We a l s o choose n a t u r a l
r a p i d l y increasing t h a t the trigonometric polynomials
pN ( x ) do n o t have t r i g o n o m e t r i c monomia s o f t h e same o r d e r . We k form the series
eimkX
1 lNk
2.1. A RESULT OF A.P.
17
CALDERON
and so t h e above s e r i e s i n t h e F o u r i e r s e r i e s o f a f u n c t i o n
If x - X k e Ek , t h e n
S*pNk ( X - x k )
f
in
L2.
by t h e d e f i n i t i o n o f
> kNk
Eky
and so some complete b l o c k a r b i t r a r i l y advanced i n t h e t a i l o f t h e F o u r i e r
S
series
i s i n a b s o l u t e v a l u e b i g g e r t h a n 1.
x
the series
S
Since a l m o s t each
x
is
t h i s means t h a t a t almost ever,y
i n i n f i n i t e l y many o f t h e s e t s
xk f Ek i s n o t convergent. T h i s proves t h e theorem.
.
..
P m o d ad Lemma. 2.1.2. L e t XI x z . . , XkY . L e t US e s t i m a t e t h e measure o f t h e p o i n t s which a r e i n f i n i t e l y many o f t h e s e t s xk
f
Ek.
Such p o i n t s a r e m
IJ
n=l
k=n
(xk + Ek)
]
'
=
m
m
IJ
0 k=n
n=l
(xk
f
En)'
We w i s h t o prove t h a t f o r some p a r t i c u l a r s e l e c t i o n o f t h e p o i n t s t h i s s e t i s o f n u l l measure. some sequence
{xk3
Let
xlk
For t h i s i t i s s u f f i c i e n t t o prove t h a t f o r
we have, f o r each
I
n
m
0
k=n
(xk
f
Ek)'I
be t h e c h a r a c t e r i s t i c f u n c t i o n o f
= 0
E;C
. Consider
the function
D. , I
X(t)
Consider
characteristic function o f
XI,
XZ,
...
xP
{xk}
f' ( x k
k=l
f
Ek)'.
, t as v a r i a b l e s and w r i t e
We have
2. FINITENESS AND THE TYPE
18 TI
xp 1- 1=-TI
So t h e r e e x i s t s a
So we can choose
p1
XI,
*
I",- i;' x
--TI
s u f f i c i e n t l y b i g so t h a t
XZ,
...
, such t h a t
Xp
< We now choose i n t h e same way
And so on.
SO
x l ( t + x ) * + *Px(i t + x Fi) d t d x l...dx
t=-TI
f o r each n
,I
xP
I'
k=n
+ly
1
?
...
, x P 2 such t h a t
(xk + E k ) '
1
=
0
P1
2.2. 2.2.
19
THE THEOREM OF SAWYER
COMMUTATIVITY OF T* WITH M I X I N G TRANSFORMATIONS. POSIT IV E OPERATORS, THE THEOREM OF SAWYER.
CalderBn's theorem has been extended i n s e v e r a l d i r e c t i o n s , f i r s t by S t e i n
[1961]
and t h e n by Sawyer
1119661
.
Sawyer's theorem i s con-
c e p t u a l l y s i m p l e r t h a n S t e i n ' s and so w i l l be presented f i r s t .
Once t h e
p a t t e r n o f t h e p r o o f o f CalderBn's r e s u l t has been understood, t h e theorems o f Sawyer and o f S t e i n a r e more t r a n s p a r e n t . The s e t t i n g here w i l l be t h e f o l l o w i n g : (a)
(R,F,p)
(b)
{TkI
(c)
Each
u ( R ) = 1.
w i l l be a measure space w i t h
i s a sequence o f l i n e a r o p e r a t o r s f r o m some Lp(s2), 1 i p c m, t o W(R) , t h a t a r e continuous i n measure.
Tk
i s assumed t o be p o s i t i v e
,
i.e.
if
,
Tkf 2 0.
then
We assume t h a t t h e r e i s a f a m i l y of mappings
(d)
f > 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)
<
, a.e.
F o r t h e p r o o f o f t h e theorem we s h a l l f o l l o w t h e p a t t e r n of t h e p r o o f o f C a l d e r b n ' s theorem.
So we f i r s t prove t h e same t y p e o f a u x i l i a r y
1emma.
06
Pmod t h e mappings
Sk
many o f t h e s e t s
t h e lemma
2.2.2.
Consider t h e s e t s
a r e y e t t o be chosen.
Sil
(Ak)
Sil
( A k ) , where
The s e t o f p o i n t s i n f i n i t e l y
is
m
Our goal i s t o prove t h a t f o r each f i x e d provided the
Sk
we can choose
p1
n
a r e c o n v e n i e n t l y chosen. such t h a t
p(
f'i
k=n
Since
Sil
(A'k)) = 0
oo
1
n= 1
p
(Ak)
=
m
,
2 . 2 . THE THEOREM OF SAWYER
By t h e m i x i n g property of such t h a t
We then choose
a n d then
And so on.
S
p2
(Sa) a
I
, we can choose
S1
, S p y . . . ,S
P1
such t h a t
, ... , S
pi
6
21
, such t h a t p2
Clearly we have f o r each
n
This proves t h e lemma.
Phaad ad ,the. Theaheni 2 . 2 . 1 . Assume T* i s not of weak type (p,p). Then, i f we f i x a sequence c k 4 00 , c k > 0 , t h e r e e x i s t s a sequence { f k I C L p , f k a 0 and a sequence X k > 0 such that
Let us c a l l
gk =
fk
,
Ak =
{T*gk
>
1 I.
We can w r i t e
,
2. FINITENESS AND THE TYPE
22
Let
hk
hk
copies
Ai
, A; ,
...
, A;k
u(Ak) c
1 6 hk
be a n a t u r a l number such t h a t
2.
We t a k e
Ak.
of
Thus
SJk
and so, by t h e lemma, t h e r e a r e j = 1,2,,..,
o f the sets
hk
’ .
B
(Sa) a
x
such t h a t almost each
(SJk)-’ (A;)
.
I , k = 1,2,...,
B
R i s i n i n f i n i t e l y many
6
Define the function
F(x) Where
gJk f
gk
and
sup k=1,2,. j=1,2,,. . h k
..
ak >
“k
’Jk
.
.
gJk (x)
i s going t o be chosen
0
i n a moment.
Then
Thus, s i n c e t h e
SJk a r e measure p r e s e r v i n g
4
m
6
If ck = Zpk
and
2
1
- .
k=l
ak = Zk”
‘k
,
we g e t
Because o f t h e p o s i t i v i t y o f each
k,j,xi
ak 4
m
Tk ,
and we have
\IF \ I D
, for
im
each
.
23
2 . 3 . A THEOREM OF STEIN T*F(x)
.
By t h e commutat v i t y o f
If
each ak 1.
x E m
R
.
a T* ak SJk gJk ( x ) = ak T*
x
6
{TkI
with
[ Sjk ] -1 [ A d ]
,
S:
g:
,
(x)
(Sa) a
6
then
T*F(x) >
I
we eas
belongs t o i n f i n i t e l y many o f t h e s e t s
ak
.
Since a l m o s t
[ Sjk1-l [ A d ]
and
, we g e t , a.e.
T*F(x) =
The o t h e r i m p l i c a t i o n o f t h e theorem i s obvious
2 . 3 . COMMUTATIVITY OF T" WITH M I X I N G TRANSFORMATLONS. THE THEOREM OF STEIN. [1960 i s on t h e one hand more general t h a n t h e p r e c e d i n g one, s i n c e t h e o p e r a t o r s we c o n s i d e r a r e n o t The f o l l o w i n g r e s u l t o f S t e i n
p o s i t i v e , and on t h e o t h e r hand l e s s general s i n c e i t s a p p l i c a b i l i t y i s restricted to
LP(n)
with
1< p
c
2.
The t e c h n i q u e o f p r o o f , t h e
use o f t h e Rademacher f u n c t i o n s , i s q u i t e i n t e r e s t i n g and p e r m i t s us t o dispense w i t h t h e p o s i t i v i t y o f t h e o p e r a t o r s .
} k= 1 m
The Rademacher f u n c t i o n s
i n t h e f o l l o w i n g way
{rk(t)
are
d e f i n e d on
[0,1)
24
2. FINITENESS AND THE TYPE
...................... The properties we are going t o use here of these functions are the following:
J
(1)
1 i6
1
r,(t) r j ( t ) d t
0
=
pk r k ( t )
doll t i n a
1 -
id
k # j
be a n y . t i n e a h camb&aLLon
, and
Bk e R
I
0
k=j
1c h
5
n.
06
Then
n
c
k=l
bet o
2 ’
Property serve t h a t
5
k h
(1) i s obvious.
Bk r k ( t )
s e t of measure n o t less that
2.3.1.
~ ( n =) 1. L e t
.
For ( 2 )
i t i s sufficient t o ob-
has the same sign as 1
@h r h ( t ) in a
.
l e i (n,&,u) be a meanme npace wLth CTkI be a hequence 06 f i n e m o p m a t o ~ h6 h a m OOme
THEOREM
A THEOREM
2.3.
Pfiood.
1. 0
$(A)
Our e x p o s i t i o n f o l l o w s t h a t o f A.Garsia
] .
Assuming t h a t that
25
STEIN
We j u s t need t o prove t h a t ( a ) i m p l i e s ( b ) , s i n c e t h e
o t h e r i m p l i c a t i o n i s obvious.
[1970, ch.1
OF
as
Thus, by t h e theorem
i s n o t o f weak t y p e
T*
A >
, where
Q)
1.1.1, ( a ) p=2
We f i r s t assume
(p,p) we s h a l l p r o v e
cannot h o l d .
.
1& p
The case
&
2
involves a minor
t e c h n i c a l d i f f i c u l t y t h a t w i l l be d e a l t w i t h a t t h e end.
f > 0
exists
A
Let
= {
i s n o t of weak t y p e
T*
If
,
and
H > 0
, there
A > 0 s x h that
T * f ( x ) z A }. Our aim i s t o add up many t r a n s f o r m s o f
x :
the function
j/f l l z s 1
(2,2,) , g i v e n
by d i f f e r e n t mappings
f
o f t h e s e t where
S
so t o i n c r e a s e t h e measure
X
o f t h e f u n c t i o n so o b t a i n e d i s b i g g e r than
T*
Since we would l i k e t o m a i n t a i n small t h e norm o f t h e f u n c t i o n , we d i v i d e such sums by a c o n s t a n t the operators
Tk
M
t h a t w i l l be c o n v e n i e n t l y f i x e d l a t e r . I f
were p o s i t i v e , as i n Sawyer‘s theorem, t h e
t h e sums would be n o t l e s s t h a n t h e sums o f t h e tions.
of
o f t h e added f u n 5
Since t h i s i s n o t t h e case, we i n t r o d u c e t h e Rademacher f u n c t i o n s
i n o r d e r t o s h u f f l e t h e f u n c t i o n s t o be added.
(2)
T*
T*
I n t h i s way, by p r o p e r t y
o f t h e Rademacher f u n c t i o n s , we hope t o a r r i v e a t t h e same r e s u l t as
i n t h e p r o o f o f Sawyer’s theorem.
So we proceed i n t h e f o l l o w i n g way.
2, FINITENESS AND THE TYPE
26
With some mappings
Sk
,
t h a t w i l l be chosen
k = 1,2,3,...,n
i n a moment, we d e f i n e x) = f(Skx)
fk Observe t h a t i f
x
6
Ak
then
Sk x
and 6
Ak = S i l
(A)
A and so
T * f k ( x ) = T * f (Skx) > A We now d e f i n e , f o r
where
M
x
R, t
E
E
[O,l),
i s t o be f i x e d l a t e r . Because o f t h e o r t h o g o n a l i t y
of
Erk(t)l we can w r i t e
and so
Hence, i f
J-
M2
=
B
and s o t h e r e i s
I1 F ( * , t )
1
we g e t
[D,1)
, IBI z a3
such t h a t , i f
x E Ah , 1 c h 6 n, t h e n of t h e Rademacher f u n c t i o n s , f o r
Observe now t h a t if exists a set
>
B
,
1.
112
and so, by p r o p e r t y ( 2 ) T*F(x,t)
t E
x
L x c [O,l)
.
,
lLxl
> 1 , such t h a t if
T*fh(x) z X there
x E A,, t E Lx
,
then
A
2.3.
Hence we can a l s o say t h a t , i f
1
, /Ix( a
[0,1)
Ix = B f I Lx
27
THEOREM OF STEIN
,
x e A* =
such t h a t
n
Ak
IJ
k= 1
, if
, there
exists
t e Ix , we have
simul t a n e o u s l y
L e t us t r y t o e s t i m a t e But, i f we choose c o n v e n i e n t l y
Sk
p(A*)
.
We have p(A*) = 1
- 1-1
n
1
.
, we know t h a t
Hence
So, i f we t a k e
n
2 a n p(A) >
such t h a t
1,
p(A*) >
then
Thus, i t i s c l e a r t h a t t h e measure, i n t h e p r o d u c t s space
R x l0,l)
o f the set {(x,t) i s bigger than such t h a t
. Therefore
PIX e A* :
t*
I f we c o n s i d e r
, t e IxI
: x e A*
e Ix 1 > F(*,t*)
t* e l 0 , l )
there e x i s t s 1
. 11
we have
c 1
F(-,t*)IIz
and a t
t h e same t i m e 1-1tX 6
R
: T*F(x,t*)
>
x) R
>
1
.
L e t us observe t h e r e l a t i o n s we have between t h e d i f f e r e n t constants.
2. FINITENESS AND THE TYPE
28 So we o b t a i n
AX J > F T
i.e
81
$(q) >
Thus i t i s c l e a r t h a t
f o r each
TI
>
0
and t h i s i s what
we wanted t o prove.
1 & p 6 2 , t h e process o f t h e p r o o f i s t h e same
F i n a l l y , when
u n t i l we a r r i v e t o t h e d e f i n i t i o n o f
1& p L 2
NOW, i f
2
- and P
2-P
, we
by H o l d e r ' s i n e q u a l i t y w i t h exponents
can w r i t e
ck =n I f k W l
l 6
,
-
Mp
P
and t h e remainder o f t h e p r o o f c o n t i n u e s as b e f o r e . (For t h e l a s t ineq u a l i t y i n t h e above chain, one proves t h a t i f
... >
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) <
m
.
For
v(Y) = 1.
The p r o o f w i l l be performed i n t h r e e s t e p s . (i)
The c o n t i n u i t y i n measure o f
t h e f o l l o w i n g r e l a t i o n : There e x i s t s
T
c(h) G 0
at
0, as we know,implies as
h .f
such t h a t
2.4.
(1 f 11
6 1
THE THEOREM
v { y E Y : ( T f ( 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)
,<
pn
at
M C ~
x
E R ~ .
Observe t h a t the f a c t t h a t A i s bounded has been only used t o cope w i t h t h e case a. = sup { r ( x ) : x E A 3 = m i n t h e proof of t h e theorem. One can permit A t o be unbounded assuming a. < Also t h e f a c t t h a t t h e b a l l s i n t h e f i r s t theorem o r the i n t e r v a l s i n the second a r e closed i s r a t h e r i r r e l e v a n t . One can assume them open o r w i t h p a r t of t h e boundary.
.
In order t o mark the way f o r t h e a p p l i c a t i o n of t h i s important lemma of Besicovitch we s h a l l now show how t h e weak type (1,l) f o r t h e Hardy-Littlewood maximal operator i s an easy consequence of i t . One of t h e v a r i a n t s of t h e n-dimensional Hardy-Littlewood op e r a t o r can be defined in t h e following way. For f E L’(Rn) and x a R n we s e t
Mf(x)
=
SUP
where Q runs over a l l open cubic i n t e r v a l s containing t h e point x. I t i s easy t o see t h a t Mf i s measurable and t h a t i t s a t i s f i e s the followi n g property : If f l , f 2 E L1(Rn) , A I , A 2 E R , then
W e want t o show t h a t t h e r e e x i s t s 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
<
where
Qt
x
.
Then, 6 a h each
have,i6
t h e r e i s a l a r g e s t dyadic i n t e r v a l If1 > 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
<
Let
+O
for
x E
UQ,
for
a.e.
x
UQ,
A X be t h e s e t o f p o i n t s x a R n such t h a t 1 f > 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 <
m
,
then
i s o f strong type
T
(p,p).
I f we o n l y
0
know t h a t each
T
i s o f weak t y p e t h i s procedure i s n o t d i r e c t l y a p p l i -
cable. Another i n t e r e s t i n g example, t h a t we s h a l l l a t e r use, i s t h e following. Lpmn)
,
Assume t h a t a c e r t a i n s u b l i n e a r o p e r a t o r
1< p <
t h e f o l l o w i n g way. an annulus:
m
(Rn)
to For each
x
E
L
f r o m some
i s r e l a t e d t o a n o t h e r one
Rn and each
f
E
Lp(Rn)
S
in
there exists
3.3. KOLMOGOROV CONDIT I ON Q ( X ) = { z E R ~: such t h a t f o r each that
i n e q u a l i t y over
c
we have
L ?
L
c
ILf(x)l (p,p).
Yes, and v e r y e a s i l y .
y 8 Q(x)
and d i v i d e by
depends o n l y on t h e dimension
Littlewood
and so
r
i s known t o be of s t r o n g t y p e
S
about t h e t y p e of
where
y f Q(x)
51
n
We i n t e g r a t e t h e above
M
.
Thus we g e t
i s t h e Hardy
maximal o p e r a t o r over b a l l s which i s o f t y p e
i s of s t r o n g t y p e
. Assume
Can we say a n y t h i n g
IQ(x)l
and
ISf(y)l
(p,p)
-
. Hence
(p,p).
What can be s a i d if we j u s t know t h a t
S
i s o f weak t y p e (p,p)?
We s h a l l now see i n cases l i k e t h e p r e v i o u s ones we can proceed i n t h e same way s u b s t i t u t i n g t h e weak t y p e i n e q u a l i t y (p,p) b y some o t h e r e q u i v a l e n t s t r o n g i n e q u a l i t y which we s h a l l c a l l , f o l l o w i n g C o t l a r [1959]
,
t h e K a ~ o g o h o ucon&an.
3.3.1. ,to
(n).
.
w a h con?s,tatarzt c 06
. L e X T be a n u b f i n e a t o p e h d o h dhom (n) T A 0 6 weak .type (p,s) , 1 c p , s G Then, .id 0 < 0 < s and A A a n y m e a o ~ ~ a b nubneX le meanme, we have, d o h each f 6 h'l (n) , ,the ~ a U o w L n g
THEOREM
annume thcLt
w s h @IJ&
(Ko4hugotrov'nJ
LnequaLLty,
3. GENERAL TECHNIQUES
52
0 < a < s , and d a h each f E LP(R) and each A c R Rhen T ad weak .type ( p , s ) .
Phtood. L e t T be .of weak t y p e ( p , s ) w i t h c o n s t a n t f 6 “I(R) and each A > 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) <
the d i s t r i b u t i o n function o f
lTfl‘
u
=
Ad-l
u
lTflX
W
N = c
o b t a i n Kolmogorov’s
0, K
{ ITf) >
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) <
wLth
ION]
<
-t
N
I l f IIS
3 AS
dh
-
minimal t h e l a s t member, we
(*),
s a t i s f i e s (*) f o r
u < s.
Let
f
e %(a),
Q any measurable s e t o f f i n i t e measure c o n t a i n e d
Then, i f we a p p l y t h e i n e q u a l i t y (*)
, we
get
in
3.3. Hence T
KOLMOGOROV CONDITION
.
i s o f weak t y p e (p,s)
53
The r e l a t i o n between t h e c o n s t a n t s o f
t h e weak t y p e and t h e Kolmogorov i n e q u a l i t i e s a r e c l e a r f r o m t h e comput a t i o n s i n t h i s proof. With t h i s theorem one can e a s i l y h a n d l e t h e p r e v i o u s examples, even i f we s u b s t i t u t e t h e s t r o n g t y p e f o r t h e weak t y p e i n e q u a l i t y . I n t h e f i r s t one, i f each
1< p < if
, with
constant
l < ~ < p ,~
E
L
c,
T,
i s o f weak t y p e (p,p)
,
, we can w r i t e , f o r each A w i t h v(A)
<my
P
i.e.
Hence, s i n c e
Tf(x) =
I'
IT,f(x)
I dx ,
0
The second example, t o g e t h e r w i t h some o t h e r s , w i l l be t r e a t e d i n S e c t i o n 6 when d e a l i n g w i t h t h e technique o f m a j o r i z a t i o n .
54 3.4.
3. GENERAL TECHNIQUES INTERPOLATION.
The i n t e r p o l a t i o n methods have been e x t e n s i v e l y s t u d i e d and t h e r e a r e several r e c e n t monographs about them. We s h a l l s t a t e here, w i t h o u t p r o o f , two o f t h e most b a s i c r e s u l t s , t h e R i e s z - T h o r i n theorem and t h e M a r c i n k i e w i c z i n t e r p o l a t i o n theorem. F o r a more complete t r e a t m e n t t h e r e a d e r i s r e f e r r e d t o Sadosky
, t o Bennet and Sharpley
[1979]
1119791
and t o a f o r t h c o m i n g monograph by t h e two l a s t mentioned a u t h o r s . A l s o t h e book by S t e i n and Weiss [1971] c o n t a i n s a v e r y good c h a p t e r on i n t e r p o l a t i on. We g i v e a l s o i n t h i s paragraph t h e p r o o f o f an i n t e r e s t i n g theorem by S t e i n and Weiss
[1959]
, which
i s a t t h e same t i m e a good example
o f t h e technique o f l i n e a r i z a t i o n presented i n
3.7.
A. The R i e n z - Thohin theorrun.
1
1
- = (1-t)Pt Po
t t -
1
,
P1
1
1
- = (1-t)90 qt
1
t t -
q1
FwLthehmurre.,
, i6 ln ~ L V U X C W
pt <
extended t o t h e whoLe n p a w LPt(fil)
m
t h e opetratoh T can be uvLiyudy phenmving t h e l a ~ itn e q u u l d y .
F o r t h e p r o o f we r e f e r t o Zygmund
[1959]
.
3.4. INTERPOLATION
B.
The Marrcinkiewicz themem.
3.4.2. W L bpaCU ~
L
P
FOh
THEOREM.
und
"(a,)t
ttrhetre
L
16
55
P
( a l , Y l y ~ land )
LeA
u nubadditcue opehcLtvh t h d A de6ined vn ,the npace
T
-
'(al). Annume t h a t
Po
6
be Awa mean-
(QZ,S2,p2)
Po c
Y
PI.
A v2; weah Z y p ~ ( p o , q o ) and (pl,ql)
T
# 91.
go
O < t < l , L d
-
Pt
Po
Then T A &a
( 1 - t ) + -1 t PI
02; t y p e
(1-t)
-
,
9t
t -1
qo
t
q1
(pt,qt)
For t h e proof we r e f e r t o Zygmund
[1959]
o r S t e i n [1970
1.
I n t h e l a s t r e f e r e n c e one can see some r e c e n t g e n e r a l i z a t i o n s .
A t h e v t m vl; S t e i n and W ~ A .
C.
For many o p e r a t o r s t h a t a r i s e i n a n a t u r a l way i t i s r e l a t i v e l y easy t o prove t h a t t h e y v e r i f y a weak t y p e i n e q u a l i t y when r e s t r i c t e d t o c e r t a i n types o f f u n c t i o n s (smooth, simple, c h a r a c t e r i s t i c f u n c t i o n s o f sets,...).
For t h i s reason i t i s i n t e r e s t i n g t h e f o l l o w i n g theorem o f
E.M.Stein
and G.Weiss
[1959]
t h a t a l l o w s t o i n t e r p o l a t e between such
types o f i n e q u a l i t i e s i n t h e s t y l e o f t h e M a r c i n k i e w i c z theorem. (n,F,p)
Let
a sublinear operator.
1G p
& m
f o r each with
,
1& q 6
be a measure space and We say t h a t
m
,
is of
T :
W(n)
and f o r each
xE
-+
,W(n)
hc%t.kicted weah type
when t h e r e e x i s t s a c o n s t a n t
characteristic function
p(E) <
T
c > 0
of a measurable s e t
A < 0 we have
(p,q),
such t h a t
E c R
3. GENERAL TECHNIQUES
56
When one t r i e s t o i n t e r p o l a t e , using the technique of t h e proof of Marcinkiewicz theorem, i n order t o obtain an intermediate ha-tticted type from t h e knowledge of two r e s t r i c t e d weak types, t h e r e i s no problem a t a l l , as t h e following lemma shows.
-
Let E be
P4006.
with
At independent of
Ifwe s e t
.,(A)
=
i-~ {
E.
measurable, E
t
.
We t r y t o prove
We know
x e n
Because of t h e i n e q u a l i t i e s above
Xqt Thus
w
TXE
(A)
-+
0
as
A
-P
0
and as
A
--t
m.
3.4. INTERPOLATION
If we integrate and s e t N
=
u(E)’
with the value
which makes minimal the l a s t term, then
However, when we try t o obtain the nonrestricted strong type by means of the preceding technique, the process does not work, since we need an estimate for w T f ( X ) for f E L P ( Q ) t h a t we do n o t have, a t least so directly as above. The difficulty can be obviated going over t o the dual space i n the following manner. Assume now that the operator T : ’l/Yl (a) -t %(a) i s f i n m and of restricted type (pt,qt) ( as in the $onclusion o f the On L q t ( Q ) , dual space preceding lemma), 1 & pt < m , 1 < q t < m of Lqt , ye are going t o define an operator T* in the following way. Let f E ~ q t ( n ) anf for E c , V(E) i , we s e t
.
We then have
57
3. GENERAL TECHNIQUES
58
Hence
-
t h e Radon h
i s a signed measure
vf(E)
Nikodym
and each measurable
Si.nce T
i s linear
T*
vf(E) u(E) <
E,
f o r each
1 and s o
t h e r e i s an e s s e n t i a l l y unique f u n c t i o ?
theorem
z T * f e L 1 such t h a t
c o n t i n u o u s w i t h r e s p e c t t o p . So b y
(Ts)f
JE h
=
i s a sort o f adjoint o f
\
So, f o r each
f
e Lqt (n)
we have
m
simple =
.
function
s
sT*f T.
Now we can s t a t e t h e f o l l o w i n g
1emma. 3.4.4.
Lineah. LeR: T* weah type. (q;;
LEMMA.
L d
be dedined
, p; 1 -
an
T
be a in Lemma
3.4.3 and benididen i n Ahe phecedcng &na. Then T* L b 06
Then
1
The s e t
EX
i s t r e a t e d i n t h e same way and so we g e t t h e lemma.
3.4.
59
INTERPOLATION
These two lemmas p e r m i t us t o o b t a i n v e r y e a s i l y t h e f o l l o w i n g theorem.
Prrood. by Lemma 3.4.3, By Lemma
We t a k e T
3.4.4,
i s o f r e s t r i c t e d type T*
i s o f weak
M a r c i n k i e w i c z theorem
T*
i t s adjoint f E Lq5
and
T*
(qi,,
(ptoy qto) pi,)
and
i s then o f strong type
s
and
(ptly
(qC,,pil). (qi
i s w e l l d e f i n e d and i s o f s t r o n g t y p e
,
p:).
(ps,qs)
.
qtl)
By t h e
Therefore
.
If
i s a simple f u n c t i o n
j Therefore
0 < t o < s < tl < 1. Then,
such t h a t
to,tl
T**s = Ts
j
(T*f)s =
s
for
f(Ts)
=
s i m p l e and so
J
T
f(T**s) i s o f strong type
(ps,qs).
The theorem o f S t e i n and Weiss has a drawback. I t r e q u i r e s t h a t T
be l i n e a r and so cannot be d i r e c t l y a p p l i e d , f o r example, t o maximal The f o l l o w i n g c o n s i d e r a t i o n , an example of t h e l i n e a r i z a t i o n
operators.
method t h a t we s h a l l see l a t e r , p e r m i t s us t o extend t h e r e s u l t t o t h i s situation. 3.4.6.
(a) t o
@om
t h a t T* 1 6 PO (ps,qs)
g
be a sequence
{Tk} k=l
'm( Q ) . ld
T*
0 6 fineah opehatoh5
be t h e i t maxim& opeh&oh.
&Arne
oh rrena7hted weah t g p e n ( p o , q o ) and (pl,ql) , wah q o f q l . Then T* LA a15 s&vng Rgpe 1 6 p1 c q1 4 m , qo < m 1 1s ( 1 5 ) + , (1-s)t-s. w4xh 0 < s < 1 , - - 91 Ps Po P1 9, 90
,LA
,i
Prrood. For
m
TffEOREM. L d
E
W(R)
Let
$ :
and
x
E
Q
R
-f
N
be any a r b i t r a r y measurable f u n c t i o n
we d e f i n e
Ti)
g(x) =
T $(x) g(x)
.
The
3. GENERAL TECHNIOUES
60
operator T so defined i s obviously l i n e a r from ()"l (Q) t o $ We c l e a r l y have, f o r each x E ~2 and g E % (Q),
,h(n).
T*. Hence T i s of r e s t r i c t e d weak types J, same constants as T* , i . e . with constants by Theorem 3.4.5. T i s of strong type with constants independent of .
+
+
(pS,q,)
.
Let now f E Lps We choose @ : G. M measurable such t h a t T*f(x) c 2 IT f ( x ) I f o r each x 6 G. . ( t o do t h i s d e f i n e @ ( x ) L on the s e t Cx 6 G. : 2 ktl > T*f(x) > Z k 1 as t h e f i r s t j such t h a t I T j f ( x ) l > Z k ) . Thus we have 11 T*fll c 2 11 TOfllqs 6 c I l f l / p s
'
with
c
independent of
f.
So T*
-f
qS
i s o f strong type
(ps,qs).
.
Here we have t a c i t l y assumed T*f(x) # m In general we can f i r s t consider Tfi defined by Tfif(x) = sup I T k f ( x ) l . We obtain t h e 1sk&N r e s u l t f o r Tfi and then a passage t o t h e l i m i t as N m allows us t o obtain i t f o r T*.
+
3.5. EXTRAPOLATION
The aim of the extrapolation technique can be understoood i n t h e following concrete example. Let T : k ( n ) + h ( n ) be a s u b l i n e a r operator and assume t h a t we know t h a t i t i s , f o r i n s t a n c e , of strong type (p,p) each
, f
PO < p < 6
PI
, with a constant depending on
p,
c(p)
Lp(n),
Assume t h a t we have some more information about example t h a t c ( p ) L A / ( p - p o ) ' f o r p close t o po
.
, i.e.
for
c(p) for Can we e x t r a c t
3.5.
61
EXTRAPOLATION
from t h i s s i t u a t i o n more i n f o r m a t i o n about t h e t y p e of t h e o p e r a t o r
T?
As an i l l u s t r a t i o n of t h e thechnique we p r e s e n t a r e s u l t of Yano
[ 1951 1. The method does n o t seem t o have been e x p l o i t e d v e r y e x t e n s i v e 1Y. 3.5.1. THEOREM. L e R M be a bubadditive, panLtLve, panL?XueLy homogeneow a p e h a t o h 64om % ( n )It o (n) . knwne t h a t 11 M f I l o o 6 11 fll, do4 each f E: Lm and t h a t thetre exint con&tm,tb c, s > 0 nuch t h a t 604 each p E (1,2) and d o t each mmu/rabLe be,t 06 bounded meanme E , we have
Then, do4 each f E: L ( l + log+Ls) and do4 each mmwrabbe bounded m m m e X C we have
with
clrc2
be,t
06
independenX a d f, X.
As we can see, t h i s i s a s o r t of Kolmogorov c o n d i t i o n r e l a t e d t o t h e space
.
L ( l + log+L)'
P4aud.
We know t h a t f o r each
K
c n
with
u(K) <
we have
L e t us t a k e Eo =
,f
f E L ( l + log+L)'
IX f X : 0 < f(X)
Ek = { x 6
x
: 2k-1
&
> 0 , and c a l l
<
f(x) <
1) 2k1
for
k = lY2,3,...
62
3. GENERAL TECHNIQUES
We can w r i t e
The sum o f t h e terms i n t h e s e r i e s above corresponding t o those ek
f o r wich
If
ek > 3-k
ek 6 3-k
is finite
since
t h e n we have
and so ~
2 k ( k + 1)'
k+ 1
ek kt2
<
2k(k+l)s
ek3
Hence
By r e f i n i n g t h e p r e v i o u s methods MoriyBn [1978
a r e s u l t o f extrapolation f o r Yano.
]
has o b t a i n e d
p o > 1 s i m i l a r t o t h e p r e c e d i n g one o f
3.6. MAJORIZATION
63
.
L e t M be a d u b a d d i t i v e , p u ~ . i t i w e , ponLi5ve L y hamugeneaw ap-o4 6 m m i)n (a),to ,% (n). ‘I Anbwne M a t each f E Lm . L e t 1 < po < m, E > 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
<
m
,
(p,p).
have
I t e g r a t i n g over
y
8
Q(x)
if
f e Lp(Rn)
and d v
L
Where ing
i s t h e minimal c u b i c i n t e r v a l c e n t e r e d a t x
Q*(x)
Q x ) M i s 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 , and c
stant
ndependent o f
A
measure and
> 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
<
c
(1,l)
, such t h a t
m
independent o f
quality to
for
S
M
in
AX C
HA,
A, f ,
q i t h exponent u
3.2.A
,
we see t h a t t h e r e e x i s t s
HAY
HA
and
K.
I f we now a p p l y Kolmogorov's ine-
3.6.
65
MAJORIZATION
Hence lAXl with
c**
independent o f
THEUREM
3.6.2.
(Rn) tv
(Rn)
.
C**
- llfll(:
IHXI f,
X ,K
L&
T
XP
.
T
So
and
Annume Lhcd
be Xiuv n u b f i n e m o p e m i t v ~dhom
S
T
i s o f weak t y p e ( p , p ) .
A majvhized
i n t h e dvL t h e h e exint ;two Q * ( x ) w a h diam&.te/r by
S
Lowing oenhe, F 0 4 eclch x E Rn and edch f cubic i n - t e t r u a h centmed CLt x, Q ( x ) and Q*(x) , 4 .tima t h d t u,'J Q ( x ) , nuch t h d t do4 each y E Q ( x ) we have
Then, i d denv
06
S
weak type
Phovd d i v i d i n g by
.
vd weak t y p e ( p , p )
dvh bVme
,
a f t e r h a v i n g taken t h e
a-th
S
Tf(x)la
T
t h e Kolrnogorov's inequa i t y
c
c 1Q(x) U
, we
y E Q(x)
and
power o f t h e above
inequality,
I f we a p p l y t o
a,
(p,p).
We proceed as b e f o r e , i n t e g r a t i n g o v e r
lQ(x)l
p, 1 < p <
get
3. GENERAL TECHNIQUES
66
T h i s proves t h e theorem.
3.7. LINEARIZATION.
As we have o f t e n seen, many i n t e r e s t i n g o p e r a t o r s a r i s i n g i n a n a t u r a l way i n t h e a.e.
convergence t h e o r y a r e n o t l i n e a r . I m p o r t a n t re-
s u l t s of f u n c t i o n a l a n a l y s i s a r e n o t a p p l i c a b l e t o them. The l i n e a r i z a t i o n technique c o n s i s t s i n s u b s t i t u t i n g t h e n o n l i n e a r o p e r a t o r under s t u d y by another l i n e a r one t h a t i n a c e r t a i n sense m a j o r i z e s i t . The use of pert i n e n t techniques o f l i n e a r f u n c t i o n a l a n a l y s i s may then p e r m i t us t o o b t a i n t h e i n e q u a l i t y we l o o k f o r . We have a l r e a d y seen t h i s t e c h n i q u e work.ing i n t h e p r o o f o f N i k i s h i n ' s theorem and a l s o i n t h e e x t e n s i o n t o maximal o p e r a t o r s of t h e theorem on r e s t r i c t e d weak t y p e i n t e r p o l a t i o n o f S t e i n and Weiss. Now we sha 1 p r e s e n t a s i m p l e example of t h e use o f t h i s techn i q u e which appears
n a paper o f Cdrdoba
].
r1976
A l s o we g i v e some
r e f e r e n c e s f o r more e l a b o r a t e a p p l i c a t i o n s o f t h e same technique.
If
18
i s a c o l l e c t i o n o f open s e t s i n Rn w i t h bounded measure,
M
we d e f i n e 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 t h e f o l l o w i n g way.
Mf(x)
=
f E Lloc
For
d3
in
(Rn)
B
I o
relative to
If(y)ldy
if
x e II B
B E 8
.
3.7.1. THEUREM LeZ be a coflection ad A & an above. h b u m e t h a t t h e opehtratoh M o d type (p,p) d o h A U t w p with
1< p < 06
f
a,
.
i.e.
doh f
6
Lp(Rn)
, (1 M f l I p
6
c
( ( f ( l pw L t h
c
indepe~dent
3.7.
b
Then
dinAXe coReection
b&&4&
t h e dok%wLng
CBkI k=l
ol;
@om them u hequence { R k } k = l H
I
(i)
[
1J Bk
k=l
c o n 0 6 mhat 2he
o n m a in
M
6
c 1 [ 0 Rk k=l
Bi( s
Lq-noxm).
among 67
=
IBk
each
.
For
0 ( R 1 11
B
4UCh
that
I
,
= B1
.
the f i r s t
c 7 lBkl
2
1
7 lRkl
t h e f o l l o w i n g way
Observe t h a t m a j o r i z e d by
R2
.
Bk
And so on.
p
. Bk
we choose t h e f i r s t
1
7
from
I
Bk
I.
BsyB9,
So we g e t
N
I B 0 1J R k
1
>
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
R
T : Lp+ Lp
,
. Therefore
T i s bounded w i t h a norm Lq I t s a d j o i n t w i l l be a l i n e a r o p e r a t o r S : Lq Mf(x)
-f
j
in
'
3 , GENERAL TECHNIQUES
68
whose norm i s a l s o majorized by e x p l i c i t form
c.
But
S can be e a s i l y w r i t t e n i n
We have
and so
This i s
(ii).
Other nice a p p l i c a t i o n s of the l i n e a r i z a t i o n technique can be seen i n CBrdoba [1977] and a l s o i n C.Fefferman [1973].
3.8. SUMMATION.
Later on and i n d i f f e r e n t contexts we s h a l l f i n d ourselves i n s i t u a t i o n s l i k e the following one. We want t o study t h e type of an ope r a t o r T and by means o f geometric o r a n a l y t i c considerations we f i n d t h a t i t can be majorized by a sum of operators 1 c k T k with a simpler s t r u c t u r e whose type we can e a s i l y determine. What can we deduce about t h e type of T? I n general, the strong type does not o f f e r any d i f f i c u l t y , s i n c e 11 1 ‘Ck T k l I p 6 1 l C k l ~ ~ T k . ~ For ~ pthe weak type we can a l s o k k sometimes obtain s a t i s f a c t o r y r e s u l t s . Let, f o r example T = 1 c k T k , and assume t h a t each T k i s uniformly of weak type (1,l) , i . e . , f o r each f e L ’ ( R n ) , h > 0 and
3.8.
k = 1,2,
69
SUMMATION
...
and t h e r e f o r e
The p r e v i o u s example i s a p a r t i c u l a r case o f t h e f o l l o w i n g theorem on summation o f weak t y p e i n e q u a l i t i e s . and N.Weiss [1969]
3.8.1.
.
I t i s due t o E.M.Stein
We s t a r t w i t h a lemma.
LEMMA.
Suppode t h a t d o h j = 1 2 3,.
on Ph doh which I I x : g j ( x )
> s}l
.. ,g j I
6
be a dequence od p o n h X v e numbem w a h
C
J
a nonnegative dunc-tition d a h each s > 0 . Le,t
c j = 1 and
m
Then, doh each s > 0 ,
Phood. F o r each
j = 1,2,3,
gjbf v.(x) = J
...
l e t us d e f i n e
if
gj(x) <
if
gj(x)
7 S
S > 7
{c.} J
3. GENERAL TECHNIQUES
70
gj(x) u.(x) J
S
if
gj(x) >
2cj
if
gj(xl 6
S 2c
=
m.(x) = g . ( x ) J J
- v J. ( x )
-
uj(x)
c c J. v J. ( x ) , u ( x ) = C c J. uJ. ( x ) , m(x) and t h a t Observe t h a t , for each x , we have v ( x ) i Let
L e t us w r i t e yXj(y)
6 1.
v(x) =
=
4
Aj(y) = I { x : g j ( x )
Then we have, s i n c e
yI}
-2c 3.-
j
for
.
y > 0
> mJ. ( x ) >
S
c c J. mJ. ( x ) .
Observe t h a t a t each
x,
J
From t h i s we g e t
S 1 6 ___ 2(Kt1) { x : m(x) > 2 1 S
, and
finally
,
With t h i s lemma we e a s i l y a r r i v e t o t h e f o l l o w i n g theorem.
3.8.2.
o p a a t o / r o &,om
THEOREM. L ’ (Rn)
to
LeA
{Tk}E=l
be a Aequence 06 h u b a d d i t i v e
r/l (Rn) t h a t me u n i 6 o m l y
06
weak t y p e ( 1 , l ) .
3.8. SUMMATION
71
This Page Intentionally Left Blank
7
CHAPTER 4 ESPECIAL TECHNIQUES FOR CONVOLUTION OPERATORS
Many of t h e operators of i n t e r e s t i n Fourier Analysis a r e opera t o r s o f convolution type. I n p a r t i c u l a r a l l operators which appear i n t h e Introduction t o Chapter 1 as motivation f o r t h i s work a r e of such type. We consider a sequence or generalized sequence of functions C k . 1 C L ’ ( Q ) ( k e r n e l s ) where R w i l l be here e i t h e r t h e n-dimensional J torus TTn o r R n (Q could be as well a l o c a l l y compact group) and f o r a function f 6 L P ( Q ) , 1 < p 6 00, we d e f i n e K.f(x) = k . J
J
*
f(x)
We ask about the convergence of K.f in L p o r pointwise. In order t o J t r e a t t h e convergence i n L p we a r e t o consider I I K . f l l as explained J P ’ in t h e Introduction t o Chapter 1, and t o study t h e pointwise convergence we a r e led t o i n v e s t i g a t e the behaviour of K* defined by
K*f(x)
=
sup I b . * f ( x ) l j
J
The f i r s t and second s e c t i o n s of this Chapter reduce the problem t o the study of t h e action of the operators K j o r K* over f i n i t e sums of Dirac d e l t a s concentrated a t d i f f e r e n t points of R ( f o r t h e weak or strong type ( 1 , l ) ) and over l i n e a r combinations of Dirac d e l t a s ( f o r t h e weak or strong type ( p , p ) , 1 < p < m ) . This reduction permits t h e disc r e t i z a t i o n of t h e operators i n question, as we s h a l l show l a t e r . A previous r e s u l t in t h i s d i r e c t i o n has been obtained by
K.H.Mg
on [1976] and i s presented here as Theorem 4.13. The idea of c h a r a c t er izing t h e weak type f o r the maximal convolution operator by means of t h e Dirac d e l t a s as i t appears in Theorem 4.1.1. belongs t o t h e author. The
74
4. CONVOLUTION OPERATORS
extensions and r e f i n e m e n t s t h a t f o l l o w have been o b t a i n e d b y M . T . C a r r i l l o [1979]
.
4.1. THE TYPE ( l Y 1 ) OF MAXIMAL CONVOLUTION OPERATORS.
Convolution
o p e r a t o r s a c t i n g over a sum o f D i r a c d e l t a s g i v e
a d i s c r e t e sum which i s , i n many cases
If
k
6
L’(R)
,
a1,a2,..
D i r a c d e l t a concentrated a t
Kf(x) = K
We s h a l l say t h a t
06 V&ac d&
ah
*
(
.,a,
then
, rather B
R , and 6h denotes t h e
, if
H
1
6h)(X)
h=l
K
06
=
H
1
h=l
k(X-ah)
weak .type (1,l)
c > 0
when t h e r e e x i s t s
easy t o handle.
aveh ~$Linite 6wnn
such t h a t f o r each
1 > 0
and each H
we have
For a maximal o p e r a t o r
K*
o f convolution operators
Kj
t h e k i n d d e s c r i b e d i n t h e i n t r o d u c t i o n of t h i s Chapter one can w r i t e
of
, if
4.1. THE TYPE ( 1 , l )
75
H
K*f(x) =
sup j
K*
and so
i s o f weak t y p e
there exists
c > 0
= sup
IK.f(x)
j
J
H
I 1
h=l
k.(x-ah)l
( 1 , l ) o v e r f n i t e sums o f D i r a c d e l t a s when
such t h a t f o r each f =
H
1
h=l
Ish
and each
X
> 0
,
The main theorem i n t h i s s e c t i o n i s t h e f o l l o w i n g c h a r a c t e r i z g
( 1 , l ) of
K*.
& ad weak .type ( 1 , l )
i6
t i o n of t h e o r d i n a r y weak t y p e
Then
.type
K*
and anLy
.id
K*
ih a6 weak
( 1 , l ) o v a & h X e nwnh ad QitLac dcLtuh.
l n o t h m W o h h (dahge;U;ing a b u t P h c delakhl K* ih 0 6 weak .type (1,l) id and o n l y id t h e m ex.Ath c 0 nuch t h a t , d m each dinite heL ad di6deh'en.t pointh a1,a2, ... , aH 6 R and d m each X > 0 ,
we have
P m a d . (A) We f i r s t prove i n f o u r s t e p s t h a t i f K* i s o f weak t y p e ( 1 , l ) o v e r f i n i t e sums o f D i r a c d e l t a s , t h e n i t i s o f weak t y p e (1,1).
4. CONVOLUTION OPERATORS
76
c > 0
(1) Assume t h a t t h e r e e x i s t s f =
H
1
X
and
h=l
such t h a t f o r each
we have
> 0
H We want t o p r o v e i n t h e f i r s t p l a c e t h a t i f with
ch E
h=l 1 ch
f =
6h
Nl , then H
N we c a l l
I f f o r a f i x e d n a t u r a l number
K;f(x)
=
sup
K.f(x)
l6jGN
then, s i n c e c l e a r l y m
IJ { X :
N=l
K$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 <
E
a with
,
0 < a < A
, we
we a r e t h e n f i n i s h e d w i t h
4. CONVOLUTION OPERATORS
78
Now
Hence so t h a t
l]gj
,
given
- kjII
E > 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
<
ch > 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
THE TYPE ( 1 , l )
4.1.
79
H
H
By choosing a p p r o p r i a t e l y
(3)
rh one g e t s (*).
We want t o prove now t h a t
K*
i s of weak t y p e ( 1 , l ) o v e r
l i n e a r combinations 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 . I f d =
H
1
h=l
xI
ch
with
k
t h a t f o r each
X
> 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 <
c1
< A , we can w r i t e
A l l we have t o do now i s t o prove t h a t t h e second t e r m i n t h e l a s t member can be made s m a l l b y an a p p r o p r i a t e c h o i c e o f
We can w r i t e
T h e r e f o r e we can s e t
gj
and
1 Ih/.
81
THE TYPE (1,l)
4.1. I n t h e same way
A1 so
Given
E
> 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 <
ct
<
x,
N
f =
ah.
H
h=l
Let
fjh.
For t h e l a s t term we proceed as before i n (1). that
Ilkj
-
gj(ll
b TI
, we
If g j
6
g,,(O),i s
such
write
+
As before,one
f i r s t chooses
Ihs u f f i c i e n t l y small.
gj
6
So we g e t
9 ,,(n) c l o s e l{K*f
>
to
A31 C c
kj
in
L' and t h e
lP= c:
.
The method o f p r o o f o f t h e p r e c e d i n g theorem can be a p p l i e d t o many o t h e r i n t e r e s t i n g s i t u a t i o n s . We j u s t s t a t e a t y p i c a l theorem t h a t can be o b t a i n e d w i t h i t .
4.1. THE TYPE ( 1 , l )
and d o t
L'(fi)
. L&t
THEOREM
4.1.2.
Ckj?y=l
dedine
f E L1(Q)
83
be a nequence
K.f(x) J
=
k. J
06 &uatLtionn in
* f(x).
Then t h e open&^ K j ahe uni~vhmly0 6 weak t y p e ( 1 , l ) id and a&y id they me unidotmLy 05 weak .type ( 1 , l ) oven 6ivLite nmn a 6 Dhac d-.
and
X > 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 <
a.
K*f(x) c Mf(x)
However, as we s h a l l
now see, i t i s n o t o f weak t y p e over f i n i t e sums o f D i r a c d e l t a s . I n f a c t , if
i 2 1 ,
and so, i f i t were , f o r some
c <
and t h i s i s a c o n t r a d i c t i o n f o r the
K*
o f t h e Theorem 4.2.1.
m,
i s u f f i c i e n t l y big. Therefore, f o r can be o f weak t y p e
p
>I,
(p,p) w i t h o u t b e i n g
s o . o v e r f i n i t e sums o f D i r a c d e l t a s . A l s o one s h o u l d observe t h a t t h e same t y p e of c o n s i d e r a t i o n s we have made a f t e r Theorem 4.1.3.
a r e v a l i d i n t h i s case
p > 1.
The r e s u l t s and methods we have presented i n t h i s 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
<
w
and XhCLt,
...,TN
Tl,T2,
c : 4 +[O,M)
4 5
Tt
5.2.
93
COTLAR’S LEMMA
And r e p e a t i n g t h e p r o c e s s
Thus
II c
N
1
Ti
II
?k
c
N =
i
,
..
I / Ti
, i 2 , . ,ik=l
1
TF 2 Ti
3
TC 4
...
Ti
T?
2k-1
2k
II
Each t e r m of t h e l a s t sum i s m a j o r i z e d by
and a l s o by
P
= /ITil
T*i211
... 1 1 Ti
2k-1
T4
2k
11
G
c(il
-
i2)
T h e r e f o r e i t i s a l s o m a j o r i z e d by t h e i r g e o m e t r i c means.
Hence
. ..
~
Thus
(
- ii2k)~
~
-
~
5. THE TYPE (2,2)
94
and making
k
m
-f
, we
11
get
N
Ti
[ [ c A.
1
With t h i s theorem one can e a s i l y o b t a i n t h e u n i f o r m s t r o n g t y p e o f t h e t r u n c a t e d H i l b e r t t r a n s f o r m and o f t h e Calder6n-Zygmund s i n g u l a r i n t e g r a l o p e r a t o r s . T h i s was t h e f i r s t a p p l i c a t i o n o f t h e lemma which appears i n C l o t a r [1959]. For
and f o r
f
F:
j = 0, 21, k2,
L2(R1)
,
-
hi(x)
-
T ? f ( x ) = hi
i
we d e f i n e
T.f(x) = h j J
We have, w i t h
Observe now t h a t
...
*
*
f(x)
= hi(-x),
I
-
f ( x ) , T? T . f = (hi J
hi(x)dx = 0
and so
*
h.) J
*
f
5.2.
c But
h.(x)
=
J
If
1-
t
2J
2J
<
Ihi(t)l
t
2-Jho(2-jx)
I- t I
Now, i f
2
COTLAR'S LEMMA
95
Ihj(x-t)-hj(x))
dx d t
X
and s o
T1 , t h e n we have
T1 , and J o
=
Cx :
1
G 1x1
6
A = { x : x E J O ,
X -
t -
B = C X : X B J O ,
X -
- B
C = I x : either
13
E J ~ }
2J
Jo}
2j
x e Jo o r
t
e JoI
t h e n we can argue as follows
If x e B t h e n If x e A t h e n
-
A l s o we have
[ A ] 6 lJol = 1
, ICI c
/
-
E
(ho(x -
t
2J
-
t
C then
If x
ho(x)l
7 )
2J
8
1-
t
2J
L
2
. Therefore
- h o ( x ) ( dx c 20
1-
t
2J
5. THE TYPE ( 2 , 2 )
96 and so
I n t h i s way we g e t
and so t h e hypotheses o f C o t l a r ' s lemma a r e s a t i s f i e d .
We t h u s o b t a i n t h e
f o l l o w i n g theorem.
The f a c t t h a t we have chosen a d i s c r e t e t r u n c a t i o n i s r a t h e r i r r e l e v a n t . One o b t a i n s e a s i l y t h e same r e s u l t f o r
5.3.
THE METHOD OF ROTATION. I n some cases, t h e s t u d y o f t h e t y p e o f a c o n v o l u t i o n o p e r a t o r
i n Rn can be reduced t o t h a t o f some known one-dimensional o p e r a t o r a& s o c i a t e d t o i t i n a n a t u r a l way, by i n t r o d u c i n g p o l a r c o o r d i n a t e s . We j u s t " r o t a t e " t h e one-dimensional t y p e i n e q u a l i t y we a l r e a d y have. T h i s i s t h e b a s i c i d e a o f t h e r o t a t i o n method i n t r o d u c e d by Calder6n and Zygrnund[19561. We d e s c r i b e f i r s t t h e general frame o f t h e method and t h e n a p p l y i t t o
5.3.
OF
THE METHOD
97
ROTATION
Some p a r t i c u l a r cases. Assume we want t o s t u d y t h e t y p e o f some o p e r a t o r s o f t h e f o r m
*
K.f(x) = k j J Let for
y e Rn
c
= I
,y #
0
171=
11
f(x)
,y
-
,o <
~
E
R: ~
r <
my
,
7
w i l l denote t h e p r o y e c t i o n o f
y
a r e t h e p o l a r c o o r d i n a t e s of
Assume t h a t we can w r i t e
y).
= ry
y e c , i.e. over
-
C (r,y
We can t h e n w r i t e
kj(r7) = g(y) hj(r).
Then K.f J
x-rar"'
7e
For a f i x e d
If Y
K:
f(x) =
\
m
h.(r) f(x-rarn-' J
0
,x
= z
K: f ( x ) = K: f ( z +
+
sy , w i t h
sy)
s eR
,z
If f o r fixed
e s
E
C
,
z e Y
, we
set
Y Y
1
m
-
f ( z + t y ) = f;(t)
h j ( r ) r n - l f;
0
Hj
so
t h e n we can
R
(Kj f ) z (s) = where
, and
E Y
7,
h j ( r ) f ( z + (s-r)y) rn-ldr
= 0
write, f o r
dr.
i s t h e hyperplane through t h e o r i g i n o r t h o g o n a l t o x aRn
we have, f o r
-
, let
C
d r dy
-
(s-r)dr E
-
H . fy ( s ) J
Z
i s t h e o p e r a t o r d e f i n e d by t h e above e x p r e s s i o n .
is a Hj known f a m i l y o f onedimensional o p e r a t o r s , and t h a t we a l r e a d y have f o r I t can happen, as we s h a l l see i n t h e examples, t h a t
some
p,
1G p <
my
if
v a LP(R')
98
5. THE TYPE (2,2)
with
C = C(p)
independent o f
v
Then we can w r i t e , w i t h
1 K i f ( x ) l p dx =
and j . z
v a r y i n g over
I,,,
-m
I((
Y
f);
( s ) l p ds dz
=
Therefore -
II K;
fll
p
6
c llfll
p'
Now, by M i n k o w s k i ' s i n t e g r a l i n e q u a l i t y ,
So, i f
c
Ig(y)(dy <
m
, we
get t h a t the operators
K. J
are uniformly o f
(P,P).
type
The i d e a o f t h e method i s c l e a r . Before p a s s i n g t o t h e some c o n c r e t e a p p l i c a t i o n s l e t us make an i m p o r t a n t remark. Suppose t h a t f o r
H . we j u s t know t h a t they a r e u n i f o r m l y o f weak t y p e (p,p). The method J does n o t w o r k . f o r two main reasons. f i r s t we do n o t know how t o i n t e g r a t e
-
a weak t y p e i n e q u a l i t y and so we cannot a r r i v e t o t h e weak t y p e i n e q u a l i t y for
K$
KY
J
, we
.
Second, even i f we had t h e u n i f o r m weak t y p e i n e q u a l i t y f o r
should s t i l l make t h e l a s t s t e p work, i . e .
t h e use o f M i n k o w s k i ' s
i n t e g r a l i n e q u a l i t y , and t h i s does n o t seem easy f o r a weak t y p e inequali t y . Here t h e Kolmogorov c o n d i t i o n may h e l p .
5.3. THE METHOD OF ROTATION
99
As a straightforward application of the rotation method let us first prove an easy version of the Lebesgue differentiation theorem in Rn without any covering lemma. Assuming that we already know the strong type inequality (p,p) , 1 < p < , for the one-dimens onal Hardy-Littlewood maximal operator, we obtain the strong type inequa ity (p,p) for the ndimensional Hardy-Littlewood maximal operator over Euclidean balls. For f E LP(R~), we set
For
y
8
C fixed , we set
'1
-
My f(x)
=
sup
r>O r
If(x-y)l
on-' dp
0
Proceeding as before,
Where MI is the (onesided) onedimensional Hardy-Littlewood maximal operator and -
f p )
f(z+sY)
=
f(x)
Therefore
jRn lMy f(x)lp
dx
=
i,,,
JL W
I(My !)f
(s)Ip ds dz
G
100
5 . THE TYPE (2,2) So we a r r i v e a t
/IMn f l /
M i nkows k i ' s in t e g r a l inequal it y
.
P c
Ilfll
'd
as b e f o r e , u s i n g
We s h a l l see l a t e r , when d e a l i n g w i t h approximations o f t h e i d e n t i t y i n Chapter 10
, some
more examples o f t h e same t y p e
. Now we
a p p l y t h e method t o t h e s t u d y o f a p a r t i c u l a r case o f t h e Calderdn
-
Zygmund s i n g u l a r i n t e g r a l o p e r a t o r s (odd k e r n e l s ) .
Limn
k E
Let
- {Ol )
be a k e r n e l which i s
( a ) homogeneous o f degree -n k(Xx) = (b)
X-"
Jc'
f e Lp(Rn)
For KE ,rl
f(x) =
k
E ,rl
k
1 > 0, x e R n -{O},
1
and w i t h mean v a l u e zero o v e r 1
, i.e.
k(-x) = -k(x)
,
1< p <
a,
and
0<
E
n , we
<
define
f ( x ) , where
kE,n(x)
That i s
i . e . f o r each
k(x),
i n t e g r a b l e over
( c ) odd , i . e .
,
k(x)
,
0
otherwise
if
1x1 6 rl
E
=
, kc ,n
f x) =
k
We t r y t o prove, assuming t h a t we a l r e a d y know t h a t t h e t r u n c a t e d H i l b e r t
, 1< p <
transforms a r e u n i f o r m l y of s t r o n g t y p e (p,p) above t r u n c a t e d Calderdn-Zygmund o p e r a t o r s
KEYn
using (a)
and
, that the
are uniformly o f strong
t y p e (P,P).
To do t h i s we w r i t e
a
(c)
,
5.3. THE METHOD OF ROTATION
Now f o r a f i x e d
I f , as b e f o r e s
B
W
7E
C
101
we w r i t e
x eRn
,
f(z+(s-p)F)
-
, we s e t f o r
x = z +
sy , z
,
a Y
, we g e t -
K;,'
=
where we w r i t e
f:(t)
f(x) =
f(z+(s+p)jq
dp
=
P
~
= f ( z + t y ) and f o r
g e Lp(R'),
i s t h e t r u n c a t e d Hi 1b e r t t r a n s f o r m . Since we assume t h a t we know c
independent o f
E, q
, g,
I\HEyn gll
6
c IlglI
we can proceed as b e f o r e and g e t
, with
This Page Intentionally Left Blank
CHAPTER 6
COVERINGS, THE HARDY - LITTLEWOOD MAXIMAL OPERATOR AND DIFFERENTIATION SOME GENERAL THEOREMS
Many problems i n r e a l a n a l y s i s can be reduced t o a g e o m e t r i c study o f t h e c o v e r i n g p r o p e r t i e s o f c e r t a i n f a m i l i e s o f s e t s . We have a l r e a d y seen how t h e Lebesgue d i f f e r e n t i a t i o n theorem i s an easy consequence o f t h e c o v e r i n g theorem o f B e s i c o v i t c h .
A l s o such fundamental
t o o l s as t h e Whitney c o v e r i n g lemma and t h e Calder6n-Zygmund decomposit i o n lemma have been e a s i l y deduced from t h e fundamental c o v e r i n g prope r t y o f t h e d y a d i c cubes. The c o v e r i n g p r o p e r t i e s o f a f a m i l y o f s e t s a r e s t r o n g l y r e l a t e d t o t h e p r o p e r t i e s o f t h e corresponding Hardy-Li t t l e w o o d maximal o p e r a t o r .
A h i n t o f t h i s c o n n e c t i o n i s t h e theorem o f Co'rdoba we have seen i n Chapt e r 3 when d e a l i n g w i t h t h e technique o f l i n e a r i z a t i o n . As we s h a l l show one can go f u r t h e r i n t h i s d i r e c t i o n . The Hardy-Li t t l e w o o d maximal o p e r a t o r i s t h e maximal o p e r a t o r r e l a t i v e t o t h e f a m i l y o f o p e r a t o r s one s t u d i e s i n t h e problem o f d i f ferentiation o f integrals.
I t s study
,of
course, f u r n i s h e s v a l u a b l e
r e s u l t s i n t h e s o l u t i o n o f such problem, b u t i t i s a l s o t r u e t h a t , i n t h e o p p o s i t e d i r e c t i o n , any i n f o r m a t i o n one can a c q u i r e r e l a t i v e t o the d i f f e r e n t i a t i o n properties o f a d i f f e r e n t i a t i o n basis provides l i g h t f o r t h e s t u d y o f t h e maximal o p e r a t o r which can be used i n many o t h e r f i e l d s , such as t h a t o f t h e s i n g u l a r i n t e g r a l o p e r a t o r s , i n which t h e Hardy-Littlewood operator plays a very important r o l e . I n t h i s Chapter we s h a l l t r y t o b u i l d up a general frame i n which t o p r e s e n t t h e i n t e r a c t i o n o f these t h r e e elements so s t r o n g l y interconnected.
We p r e s e n t o n l y t h e most i m p o r t a n t general r e s u l t s , and
r e f e r t h e reader t o t h e monograph by Guzmin [1975 i z e d knowledge o f t h e f i e l d .
]
f o r a more s p e c i a l -
We s h a l l use t h i s o p p o r t u n i t y t o complete
103
6. COVERINGSy HARDY-LITTLEWOOD AND DIFFERENTIATION
104
t h a t monograph by p r e s e n t i n g t h e most i m p o r t a n t r e c e n t r e s u l t s i n t h i s f i e l d t h a t has been g r e a t l y expanded b y t h e work o f many people, espec i a l l y those w o r k i n g i n t h e f i e l d o f F o u r i e r A n a l y s i s .
6.1. SOME NOTATION
t i o n (I3
=
@(x)
I!
x eRn
such t h a t f o r each
o f bounded measurable s e t s w i t h p o s i t i v e measure
x eR"
B e B(x)
so t h a t each
i n Rn w i l l be a c o l l e c -
a didde,tevLticLtio~ b a d
I n general
there i s a subfamily @(x)
contains
x
and i n O ( x )
o f sets o f
iB
there are sets o f
a r b i t r a r i l y small diameter. As an example, (B can be t h e c o l l e c t i o n of a l l open b a l l s i n
Rn. x
.
For each
x eRn
&(x)
i s t h e subfamily o f a l l b a l l s c o n t a i n i n g
Another example : For each
a l l t h e open b a l l s c e n t e r e d a t
x eRn x
and
@(x)
B =
\I XE
A
@(x)
R"
Bu.bemann-Fe,Ue,t didde,tnevl.tiation b a d 8 B e
t i o n b a s i s such t h a t each B e
w i l l be t h e f a m i l y o f
B
i s open and i f
x
.
i s a differentia-
e B e & , then
B(x). I f f o r each
x
E
A c Rn we a r e g i v e n a c o l l e c t i o n o f bounded we can d e f i n e t h e ffahdyy(x) (relative to = (I y ( x ) ) i n t h e xsA
measurable s e t s w i t h p o s i t i v e measure
L m e u J w a o d muximd ope,khatatr
5"
f o l 1owi ng way :
If
f E
Mf(x) =
0
if
For a Busemann-Feller b a s i s obvious t h a t
{Mf > A]
(for short, a
x f
11 S
S€
Y
8-F b a s i s ) i t i s
i s an open s u b s e t o f Rn and so
Mf
i s meas-
105
COVERINGS, IMPLY TYPE AND DIFFERENTIATION
6.2. u r a b l e.
I f we a r e g i v e n a d i f f e r e n t i a t i o n b a s i s define, f o r
i
-I
at
i
x
D(
e x i s t s and
f,x)
=
f(x)
t r u e f o r each
,
&I
B
i
1
f,x) = l i m inf S(B)+O B&(x)
If
D(
(Rn)
f x ) = l i m sup ' b(B)+O
D( -
D( -
f 6 Lloc
f,x) if
a.e.
B
(
=
f(y)dy
(the
uppeh detLivative o f
f a t x)
f(y)dy
(the
L o w t deniwative o f
f a t x)
f,x)
t h i s happens a t a l m o s t each we say t h a t
D(
we say t h a t t h e d e r i v a t i v e
X
b
x eRn
dS d L d d W e W a
i n a class o f functions
f
i n R n we can
f.
if.x)
and If t h i s i s
we say, a b r e v i a t e l y , t h a t
P dcd,4jWevLtiatQc\ x.
I n general we s h a l l s t a t e most o f t h e r e s u l t s i n d i f f e r e n t i a tion for a
B-F
basis.
F o r such a b a s i s
Mf(*)
-I
, D(
f,.)
I
,D(
f,.)
a r e e a s i l y shown t o be measurable. Many o f t h e r e s u l t s a r e e x t e n s i b l e t o o t h e r types o f d i f f e r e n t i a t i o n b a s i s w i t h o u t d i f f i c u l t y .
However
one s h o u l d n o t t h i n k t h a t t h e f a c t t h a t i n a general d i f f e r e n t i a t i o n b a s i s each s e t
B
6
@(x)
i s i n some way
"anchored" t o t h e p o i n t
x
i s d e v o i d o f importance.
6.2.
C O V E R I N G LEMMAS IMPLY WEAK PROPERTIES OF THE MAXIMAL OPERATOR AND DIFFERENTIATION.
We have a l r e a d y seen how t h e B e s i c o v i t c h c o v e r i n g p r o p e r t y leads i n an easy way t o t h e weak t y p e ( 1 , l ) o f t h e maximal o p e r a t o r .
6 . COVERINGS, HARDY-LITTLEWOOD AND DIFFERENTIATION
106
We s h a l l f i r s t s t a t e here a couple of easy general r e s u l t s i n t h i s dir e c t i o n . Then we s h a l l s e e how t h e Besicovitch theorem gives us a very general and useful form of the V i t a l i covering lemma and f i n a l l y we s h a l l see how t o use these lemmas in order t o obtain r e s u l t s on d i f f e r e n t i a t i on.
p h o p e h t g : 7 6 d o t each x e A c R~ we m e given then we can chaohe 6hom ( S ( X ) XEA a oequence i s k ) \A(,
6
lUSkl
C
and
cX
(X
h
H
S(X
huch
.
at each x e R~
Whehe
c
'k
and
H depend o n l y on
(9 , nat an A
Then t h e maximal phoo6. choose
S(x) E
Let, f o r q(x)
VpPhCLtoh
A > 0
such t h a t
the covering property t o AA and
a
ah
(S( x ) ) ~ ~ ~ .
h&%t.ive t u AX =
tMf(x)
P7i-T
I
9
u a weak .type ( 1 , l ) .
> A)
S(x)
.
If
If( > A
( S ( X ) ) x e ~ A and obtain
x e
. CS,)
we
AX
We apply
. We
can
then w r i t e
and obtain t h e r e s u l t The same method i s applicable t o t h e weak type ( p , p ) , 1 < p <
m.
6.2.
Then ,the maximal ape.hatoh ~ . & ~ 2 v teo P q=p-l
x
06
weak ,type ( q , q ) ,
*
Phoaij. For
107
COVERINGS, IMPLY TYPE AND DIFFERENTIATION
6
AX
Let
f E Lq
we choose
S(x)
,X
> 0
and
such t h a t
apply t h e covering property, o b t a i n i n g
AX
= (Mf > A }
TdT {sk}
.
j S(x)
If1
>
and
We can w r i t e
and t h i s proves t h e theorem. The c l a s s i c a l V i t a l i lemma i s an easy consequence o f t h e B e s i c o v i t c h p r o p e r t y t h a t has been proved i n Chapter 3 can s t a t e a more general r e s u l t
f o r b a l l s . We
v a l i d f o r an a r b i t r a r y measure w i t h t h e
same e f f o r t .
6.2.3.
THEOREM
.
LeL
y=
u
x sRn
(x)
be a nybXeJn 06
camp& 0 e . t ~ ulith poniaXve m w m e nuch t h a t doh each x and S E v ( x ) we have x E S . Annume t h d t each (/’( x ) containh nets 06 ahb&ahiey nm& diameLteh. L a ~n nuppone a h a t h a t y ve14ie.h t h e ~o.Uawing (BesicovLtch) covehing p h a p e h t y : 16 doh each x E A c Rn we ahe given S(x) e Y ( x ) then we can choose &&om ( S ( X ) ) ~ € a~ nequence I S k ) nuch Ahat:
,
6 . COVERINGSy HARDY-LITTLEWOOD AND DIFFERENTIATION
108
I t i s easy t o see t h a t t h e r e i s no l o s s o f g e n e r a l i t y
Ptiuol;. i f 'we assume
S(x) of
0 < pe(E) <
9
obtaining
quences
...
3,
{Si
For each
v(x) and
c
{Qk(x)}
E
.
m
ISk}
x a E we choose an a r b i t r a r y
a p l l y the Besicovitch covering property
s a t i s f y i n g ( i ) and ( i i ) .
, {Sf
For one o f t h e se,-
{St}
( L e t us assume i t i s
}
) , we have
I n f a c t , otherwise Pe(E) and
p('sk)
C
h
pe(E)
1
j=1
U(
I1 Sjk k
L e t us t a k e a
t h i s i s a contradiction.
p(P) =
e
C
1
Ue(E)
P
p-measurable s e t
and a f i n i t e subsequence o f
{Sc}
, say IS:
}
hl
3
k= 1
E, Y
such t h a t
P The f i r s t
w i l l be
F o r each such t h a t with
E
elements o f t h e sequence
hl
{S3, S l y
x
...
S;ll.
h,
e E - !I Rk
S(x)
n
obtaining
hi
(I 1
now
=
Rk =
{Rk)
t h a t we a r e l o o k i n g f o r ,
So we g e t
El
0
{Rk}
we t a k e
S(x) a { Q k ( x ) I c y ( x )
and we proceed w i t h h2 h,+l
such t h a t
El
as we have done
6.2.
109
COVERINGS, IMPLY TYPE AND DIFFERENTIATION
{Rkl
I n t h i s way we o b t a i n
and
ue(E
-
URk) = 0
I t i s i n t e r e s t i n g t o observe t h e f o l l o w i n g r e l a t i o n s h i p bet-
ween t h e V i t a l i p r o p e r t y and t h e B e s i c o v i t c h p r o p e r t y o f a system o f s e t s . For many purposes i n A n a l y s i s , i n p a r t i c u l a r i n d i f f e r e n t i a t i o n
9
t h e o r y , t h e f o l l o w i n g p r o p e r t y o f a system measurable s e t s i n Rn
y(x) , t h e n
S(x) E that
A
c
USk
and
One can chaohe dtvm
1 xsk
O f course, i f 6.2.3.
, then
i s good enough :
Sp
<
=
q
11
x eRn
(x)
of compact
7Q don u c h x E A we me given ( S ( X ) ) ~u ~hequence ~
IS,}
0 . (Weak B e s i c o v i t c h p r o p e r t y ) .
s a t i s f i e s t h e B e s i c o v i t c h p r o p e r t y o f Theorem
i t s a t i s f i e s t h s one, s i n c e
1 xsk
.
< 0
The weak
B e s i c o v i t c h p r o p e r t y i s s t r c t l y weaker t h a n t h a t o f Theorem6.2.3. f o l l o w i n g example i s due t o B.Rubio (unpublished). 6.2.4.
THEOREM.
T h a e LA a hynteni
y=
p a c t he,th h a t i n d y i n g t h e weak BenicavLtch p'rop&y t h e hi7~VKge.Rane
06
Thevheni
nuch
LI
x
ERn
0(x)
The
aQ com -
WLthoLLt h a t i n d y i n g
6.2.3.
. , bk+
1 = b l > bp > b3 > .. PXaad. We t a k e 2 = B1 > B2 > ... , Bk -f 0 and f o r each x E R2 c o n s i d e r t h e symmetri,c c l o s e d cross
Qk(x)
the p i c t u r e
F i g u r e 6.2.1.
and
centered a t
0
9
k = 1,2 x
,... we
indicated i n
110
6 . COVERINGSy HARDY-LITTLEWOOD AND DIFFERENTIATION
The system
y ( x ) = (Qk(x))
f i e s t h e weak B e s i c o v i t c h p r o p e r t y x e A
and f o r each
AcQ
We choose as Then as
.
Let
Q
x
Q
k(x) x
,x
(x)
and so on. necessarily
S(S,)
+
0
,
In fact
satis-
S ( x ) = Q,(,)(x) 6
A
6 A - S1 k(x) ’ I f t h e process stops, i t i s because A
SP one of t h e s e t s Q
,(J(x)
ER
be any square o f d i a m e t e r 2,
assume t h e r e i s one
one of t h e s e t s
S1
y= U
k=1,2,.,.,
otherwise , i f
given.
w i t h minimal w i t h minimal
-
USk =
0
k(x). k(x)
,
.
I f not, b(Sk) > c1 > 0 ,
s i n c e t h e t h i r d s o f t h e c u b i c i n t e r v a l s a t t h e c e n t e r s o f t h e crosses
sk a r e d i s j o i n t , we would have i n f i n i t e l y many d i s j o i n t congruent cubes i n s i d e a bounded s e t and t h i s i s i m p o s s i b l e . But i f 6(Sk) + 0 , t h e n A - USk = 0 . Otherwise i f x 6 A - usk , we have o v e r l o o k e d Q,(,)(x)
i n o u r s e l e c t i o n process.
process o f t h e s e t s
2 x 4
sets
sk
.
Sk
Because o f t h e f o r m o f t h e s e l e c t i o n
i t i s e a s i l y seen t h a t no p o i n t i s i n more than
I t s u f f i c e s t o c o n s i d e r how many crosses c o n t a i n i n g
have t h e i r c e n t e r s i n t h e f i r s t o f t h e c l o s e d quadrants determined b y They a r e a t most 2.
Therefore
1 xsk
c 8
z z.
.
Now i f t h e system would s a t i s f y t h e s t r o n g B e s i c o v i t c h p r o p e r t y i t would s a t i s f y t h e V i t a l i p r o p e r t y , a c c o r d i n g t h Theorem
one can choose
bk
But
so t h a t t h i s i s n o t t r u e .
, Bk
In fact l e t
6.2.3.
Q
be t h e must cube. D i v i d e t h e cube d y a d i c a l l y
i n f o u r p a r t s and f o r each p o i n t i n each one o f them t a k e a c r o s s cen-
t e r e d a t t h e p o i n t such t h a t t h e l e n g t h , B1 , o f t h e arm i s t w i c e as l a r g e as t h e s i d e l e n g t h o f each o f t h e d y a d i c cubes and t h e w i d t h o f t h e arm o f t h e cross, bl, i s small enough so t h a t t h e area o f t h e c r o s s i s l e s s 1 t h a n --< o f t h a t o f t h e d y a d i c cube o f t h i s d i v i s i o n . We d i v i d e dyad-
2
i c a i l y again and t a k e crosses centered a t each p o i n t i n t h e same way so t h a t t h e area o f t h e c r o s s i s l e s s t h a n cube o f t h i s d i v i s i o n .
4x2 2
o f t h a t o f the dyadic
I t i s easy t o see t h a t i f we s e l e c t any system od
d i s j o i n t crosses we can t a k e a t most one of t h e crosses o f .each s t a g e and
so they can n o t cover a l m o s t a l l t h e cube
Q
.
Here we have a l s o c o n s t r u c t e d a b a s i s t h a t d i f f e r e n t i a l s b u t does n o t s a t i s f y t h e V i t a l i lemma.
As we s h a l l see l a t e r t h i s i s
i m p o s s i b l e f o r a B-F b a s i s t h a t i s i n v a r i a n t b y homothecies.
L’
111
COVERINGS, IMPLY TYPE AND DIFFERENTIATION
6.2.
The i n e q u a l i t i e s f o r t h e maximal o p e r a t o r o f t h e t y p e o b t a i n e d i n 6.2.1. l e a d t o a d i f f e r e n t i a t i o n theorem f o r L1, as we have shown i n Chapter 3. The one i n 6.2.2. l e a d s i n t h e same way t o d i f f e r e n t i a t i o n o f Lq
.
Now we s h a l l show how from t h e V i t a l i lemma o f 6.2.3.
one can
a l s o deduce t h e d i f f e r e n t i a t i o n p r o p e r t y o f a b a s i s .
THEOREM
6.2.5.
.
L e R 1-1
be a n e t dunction dedined on diniAe
.
Ab~umet h a t 1-1 LA nvnnegative, manoiane, ~ i n i t e L qadditive and dinite an each cube. Then at almost evetry ( i n t h e Lebe~gueA C U ~ ) p a i n t x 8 Rn vne h a , dah each Aeyuence { Q k ( x ) } ad &ohEd cubic i n t e h v & c e n t a e d at x and cant'iaoting t o x , thlLt t h e L h L t u n i u ~ nad &abed cubic i n t e h v &
Rn
a6
06
exddh, i~ dinite and i~ independent
Pmod
.
We t r y show t h a t
\A,]
= 0
L e t us t a k e an a r b i t r a r y c l o s e d c u b i c i n -
M > 0
.
I Q,(x)
1 c o n t r a c t i n g t o x such t h a t
quence
IS,}
and a c o n s t a n t
.
Q
Qk(x)
0
C
Q
.
{Q,fx))
I n o r d e r t o prove t h i s r e s u l t , d e f i n e f i r s t
terval
and
t h e neyuence
.
F o r each
We a p p l y Theorem 6.2.3.
f r o m such cubes so t h a t n
x E ,A
0
0Q
we have
o b t a i n i n g a d i s j o i n t se-
112
6 . COVERINGSy HARDY-LITTLEWOOD AND DIFFERENTIATION So we g e t
Since and so
u(Q) <
i s a r b i t r a r y and
M
m
and de-
Q
r z s > 0
fine, for
\Ars( = 0
We t r y t o prove t h a t s e t c o n t a i n i n g ,A, theorem t o
Ars
II
k
such t h a t w i t h t h e cubes
and we o b t a i n
lArs -
Q) = 0
/ A m [ = 0. L e t us now t a k e an a r b i t r a r y c l o s e d c u b i c i n t e r v a l
G
0
we o b t a i n IA,n
Sg[
.
Let
.
a r b i t r a r y and
> 0
(GI 6 (ArsIe +
Q{(x)
a d i s j o i n t sequence = 0
E
,
{St}
E
.
G
an open
We a p p l y V i t a l i ' s
t a k i n g o n l y those c o n t a i n e d i n o f such cubes so t h a t
We c l e a r l y have
Observe t h a t , i f II
C = ,A,
we have 1C/, x
0
6
S* J
=
and so
such t h a t
Q,(x)
(
II
k
lArsle .
For each x
,
x E Ars
since 0
C S*
J
8
Si 0
C
there
t h e r e i s an
S* J
such t h a t
i s a l s o a sequence
Qk(x) + x
and U(Qk(x))
14krx)(>r We now apply t h e V i t a l i theorem a g a i n t o
C
. w i t h ' t h e s e cubes and so
113
6 . 2 . COVERINGS, IMPLY TYPE AND DIFFERENTIATION
o b t a i n a d i s j o i n t sequence {S,} 1C ties
= 0
S,]
(J
.
S* and J Thus we have, t a k i n g i n t o account t h e above i n e q u a l i -
such t h a t each
Sk
i s i n some
= 0
.
IArs S
< r Thus
Since
E
i s a r b i t r a r i l y small we o b t a i n
[Ars[
W i t h t h i s one eas-
i l y concludes t h e p r o o f o f t h e theorem.
We o b t a i n i n p a r t i c u l a r t h e f o l l o w i n g
set o f
.
If
and we s e t f o r each closed c u b i c i n t e r v a l
R"
P
i s an a r b i t r a r y
p l ( Q ) = \ Q [I P i e
,
we o b t a i n t h a t t h e l i m i t
e x i s t s and i s f i n i t e a t almost e v e r y x
u,(Q)
We can a l s o t a k e
ure o f
Q 0 P
, and
.
/ Q 0 PIi
=
,
i.e.
t h e i n t e r i o r meas
so
e x i s t s and i s f i n i t e a t almost every X .
If we g e t t h a t
u(Q) =
1,
f
with
f
nonnegative and l o c a l l y i n t e g r a b l e ,
6 . COVERINGS, HARDY-LITTLEWOOD AND DIFFERENTIATION
114 exists
,
every
x
6.3.
i s f i n i t e and independent o f t h e p a r t i c u l a r
{Q,(x)I a t almost
E R",
FROM THE MAXIMAL OPERATOR TO COVERING PROPERTIES.
As an example o f t h e use o f t h e method o f l i n e a r i z a t i o n we have seen i n Chapter 3
how a s t r o n g t y p e
(p,p)
p r o p e r t y f o r t h e maximal
q - t y p e c o v e r i n g p r o p e r t y . However one more o f t e n knows
operator implies a
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 weak t y p e p r o p e r t y can be extended t o t h i s s i t u a t i o n . C.Hayes [1976]
and A.CBrdoba [:1976]
, and
t h e theorem
T h i s r e s u l t belongsindependently t o
.
The p r o o f we p r e s e n t f o l l o w s t h a t
o f Hayes and o u r v e r s i o n i s a l i t t l e more general t h a n t h e one h e presents.
.
L e t ( Q , F , p ) be a meuhme npace and @. = (Ra)aEA a cvUecaXvn v6 meanaabke. AU~APLA 0 6 R Mlith 6 i n i t e m e a n me. We cvnniden t h e ( H c v r d y - L U e w v o d ) maximal opmahh M [belated tv A ) i n t h e 6vUvwing dvhm. 7 6 f 6 L(Q) and x E 0 , we n e t
@(O)
=
0
6.3.1.
THEOREM
LeA
+
and
: [0,w)
U A A W ~t ~h a t
-+
doh
[O,w)
A > 0
be an inmeaning 6unctivn wLth and doh each
f E L(Q)
we have
h h w n e t h a t )I : [0,w) COY-) a nondechming ~ u n u X v n duch t h a t dotl each p > 0 t h e m e . x A a 2 k ( p ) > 0 auch t h a t , 6vh each -+
115
6.3. FROM THE MAXIMAL OPERATOR TO COVERINGS u > 1
,
Lhre hccwe
I) 1
Rk
N
Remwlh.
I n t e r e s t i n g f u n c t i o n s a s s o c i a t e d as t h e f u n c t i o n s
.__
and
4
of t h e statement o f t h e theorem can be e a s i l y found. Examples:
$
, c > o
sets.
Ptloo6. We can assume t h a t has a f i n i t e number o f (R&B Otherwise we t a k e a f i n i t e s u b c o l l e c t i o n such t h a t t h e measure o f
i t s union i s s u f f i c i e n t l y c l o s e t o t h a t o f
II
.
Ra
BEB
The c h o i s e o f t h e s e t s trarily.
Assume
R1,R2,
We f i x
T-
...,Rm
such t h a t
Rk
i s made as f o l l o w s . Take
R1 arbi-
have been chosen so t h a t
0 < rl <
1 ,
o u r s e l v e s whether t h e r e i s among t h e s e t s o f
1-T-
a
(R )
c
E
PEB
,
and ask
which have n o t
116
6. COVERINGS, H AR D Y- LI T T LEW O O D AND D I F F E R E N T I A T I O N
been chosen a s e t
W
such t h a t we have s i m u l t a n e o u s l y
If t h e r e a r e such sets, we t a k e one o f them as o u r not,
we a r e f i n i s h e d w i t h t h e s e l e c t i o n process.
Rmlt
.
If
We have
So we s t o p i n a f i n i t e number o f s t e p t s .
be t h e chosen s e t s .
for t h e s e t s
W
in
(RB)BEB
Let
R1,RZ,
...,R N
which have n o t been
chosen, we have a t l e a s t one o f t h e f o l l o w i n g i n e q u a l i t i e s
6.3.
FROM THE MAXIMAL OPERATOR TO COVERINGS
117
Because of t h e h y p o t h e s i s of t h e theorem on t h e maximal o p e r a t o r , W
we have t h a t t h e union o f a l l such s e t s
v e r i f y i n g (1) has measure l e s s
than o r equal t o
IJ Rk 1
IJ Rk 1
Also t h e u n i o n o f t h e s e t s
W
v e r i f y i n g ( 2 ) has a measure l e s s
than o r equal t o
Hence
R1,
R2,
... , RN
satisfy
(a)
and
(b)
.
I t i s an i n t e r e s t i n g open problem t o f i n d t h e e x a c t l i m i t s o f
t h i s t y p e o f theorem, i n t h e f o l l o w i n g sense. Assume t h a t one knows t h a t t h e maximal o p e r a t o r s a t i s f i e s an i n e q u a l i t y o f t h e t y p e appearing i n t h e statement w i t h
$(u)
can deduce from t h i s ?
$(u) = u ( l + log
f
u )
= u
.
What i s t h e n t h e b e s t c o v e r i n g p r o p e r t y one
Can one t a k e
.
Can one t a k e
$(u) =
eu2 ?
+ ( u ) = eu ?
O r , assume t h a t
6 . COVERINGS , HARDY-LITTELWOOD AND DIFFERENTIATION
118
6.4. DIFFERENTIATION AND THE MAXIMAL OPERATOR.
When of a d i f f e r e n t i a t i o n b a s i s B one knows t h a t i f a d i f f e r e n t i a t i o n property such as t h a t i t d i f f e r e n t i a t e s L p ( l then, since i t i s c l e a r t h a t t h e corresponding maximal operator a . e . f o r each f E L p , one can apply t h e general theorems of and obtain weak type p r o p e r t i e s for t h e maximal operator.
satisfies < p < m), i s bounded Chapter 1
However t h i s type of r e s u l t s can be obtained by d i r e c t methods t h a t a r e simpler by f a r . Moreover, by such methods one can g e t r e s u l t s r e l a t e d t o individual 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 t a r e not covered by t h e a b s t r a c t theorems. Some of t h e r e s u l t s we present o r i g i n a t e in Busemann and F e l l e r r19341 and some o t h e r s in Hayes and Pauc [1955] We present here a sample of r e s u l t s of t h i s type. For more d e t a i l s and f u r t h e r information one can consult the monograph Guzmdn [1975]
.
.
A .Den&Ltq p m p e t L t i ~ . As a f i r s t r e s u l t we prove t h a t t h e d i f f e r e n t i a t i o n o f t h e c h a r a c t e r i s t i c functions of measurable s e t s ( devlni2q p m p W y ) i s in f a c t equivalent t o the apparently s t r o n g e r property of d i f f e r e n t i a t i o n of L”(FP).
Phoad. Since t h e d i f f e r e n t i a t i o n of jf a t x i s a local property , i . e . depends only on t h e behavior of f in a neighborhood of x , we assume t h a t f has compact support A. We a l s o can assume without loosing g e n e r a l i t y t h a t f o r every x , 0 G f ( x ) c H < a. By Lusin’s theorem, given E > 0 , t h e r e exists a compact s e t K i n . A such t h a t ! A - K I c E and f i s continuous on K. Let f K = fXv ,
~ A - K= ~ x A - K In f a c t
,
-
We f i r s t prove O( jf,,x) assume R k E 6j ( x ) , R k
= f K ( x ) a t almost every +
x
as
k
-+
a
.
x a R’
We can w r i t e
.
6.4.
If
x
6
K ,
then
DIFFERENTIATION AND THE MAXIMAL OPERATOR
f,&y)
fK(x)
-f
k
e x p r e s s i o n tends t o zero as
-f
t h e f i r s t member i s m a j o r i z e d by property,tends D(
i
fk,x)
, y E K and so t h e above . I f x 6 K , t h e n f k ( x ) = 0 and IRk (7 KI
as m
y
-f
x
T
.
T h i s , by t h e d e n s i t y
t o z e r o f o r a l m o s t a l l such p o i n t s x
= fK(x)
a l m o s t everywhere i n
With t h i s , f o r an a r b i t r a r y
p r o p e r t y . So f,x)
= f(x)
-I D(
119
f,x)
Hence
Rn.
cx > 0
=
.
f(x)
, we
can s e t
f o r almost each
x B Rn
a l m o s t everywhere and so t h e theorem i s proved
The f o l l o w i n g c h a r a c t e r i z a t i o n o f a d e n s i t y b a s i s belongs t o Busemann and F e l l e r
c1934
.
3.
6 . COVERINGS, HARDY-LITTLEWOO0 AND DIFFERENTIATION
120
LA a demLtity b u d , .
(a)
FUR. each A, 0
(b)
S o t each nvndecAwbing bequence
< 1,
A
{Ak} US bvunded meanwrabLe be,tb buch t h a t I A k ( J. 0 and 604 each nvni n a e u i n g sequence Irk) 06 heat numbem buch thctt r k + 0 we have
whehe,
~ V R .each
k, h,
Phvvd. {Ak)
as i n
(a) i s t r u e Mkxh(x)
< A
That
(b)
, D(\
.
xh
.
=
(a)
Fix
xh,x)
xAh , and
implies (b)
an
Ah.
= 0
, and
F o r almost each
so
0 < A < 1 and
i s easy. L e t
, if
k
x
6 Ah
we have, i f
i s sufficiently big
,
Hence
CMkXk by t h e d e f i n i t i o n o f 1i m kSince
Ah[
We
-t
we g e t ( b ) .
0
now prove t h a t n o t - ( a ) i m p l i e s n o t - ( b )
d e n s i t y b a s i s , t h e r e i s a measurable s e t A
, with IAl
I n f a c t , assume t h a t f o r each measurable s e t almost each
x
6
P
,
-
D( j x p , x )
= 0
,
i.e.
0
.
If
>
0
is @ ' , such
, we
not a that
have, a t
,x) = 0 = ~ ( ~ x p , x ) .
DIFFERENTIATION AND THE MAXIMAL OPERATOR
6.4.
If we a p p l y t h i s t o t h e complement t h a t a t almost each
x
6
P'
P'
, i.e.
P, i f
of
a t almost
IP'I >
each
, we
0
x E P
121
, we
obtain
have
Observe now t h a t
and so we have
and t h e r e f o r e
,
a t almost each
'@
x E P
would be a d e n s i t y b a s i s .
L e t us t h e n t a k e
A
measurable, w i t h
There e x i s t s t h e n a measurable s e t C, x E C
t h a t a t each
we have
E(
I
o f n o n i n c r e a s i n g open s e t s such t h a t
Ak = G k 0 A.
-
A k C Gk such t h a t I n fact
C
.
Take
rk + 0.
,let x
i s a sequence
Hence
Clearly
C
with
xA7x) > G
k
JAI
> 0, such t h a t
c A' , ICI
. Let
A C,~
JGk
-
{GkI
CI
i s n o n i n c r e a s i n g and IA,:I
{Ak}
0
>
a sequence and l e t
0
-+
+
0
, such
since
any n o n i n c r e a s i n g sequence o f r e a l numbers {r,] We s h a l l prove t h a t
E C
{RhIC
and B(x)
k
{Mkxk
be f i x e d . with
Rh
>
Since -+
x
XI 3 C f o r each k .
E(
such
xA7x) >
A
there
122
6. COVERINGS, HARDY-LITTLEWOOD AND DIFFERENTIATION
Mkx
and since
. This
(x) > A
Ak ICI > 0
proves
CMkxk(x) >
C
A 1 f o r each k and,
t h i s shows t h a t n o t - ( b ) holds, T h i s concludes t h e p r o o f
o f t h e theorem. When when
6
8
is a B
- F
i s such t h a t i f
b a s i s t h a t i s i n v a r i a n t by homothecies,i.e.
R 6
63
t h e n any s e t homothetic t o
any r a t i o and any c e n t e r o f hornothecy i s a l s o i n '@ any t r a n s l a t e d s e t o f
R),
R
with
(and so i n p a r t i c u l a r
then t h e p r e c e d i n g c r i t e r i o n r e c e i v e s a sim-
p l e r form, as t h e f o l l o w i n g theorem proves.
THEOREM.
6.4.3.
Let
@ be a
B-F
b a d RhaX d inwahiant by
h v r n v f h e ~ ~ .Then .the ZWV 6 v U v d n g p 4 " r a p ~ ~ t -cL~ ~z ee q d v d e n t : ( a ) '63 & a det&.ty b t ~ . . ~ i ~ .
each A, 0 < A < 1, t h m e exim.2 a ponLCLwe cvn~Aant ouch RthCLt ~ V R .each bounded rneauhabLe+n e t A vne h a
(b) c(A) <
m
Fvlz
Ptrvo6. 6.4.2.,
That ( b ) i m p l i e s ( a ) i s a s i m p l e consequence o f Theorem
s i n c e ( b ) i m p l i e s c o n d i t i o n ( b ) o f t h a t theorem. I n o r d e r t o prove t h a t
(a)
implies
(b)
we s h a l l use t h e f o l
1owi ng 1 emma.
LEMMA. -
be any baunded vpen A ~ -in R R" and LeX K be any cvrnpaot A & w a h pvnLCLve rneame. L& r > 0 . Then t h e m A a cLthjvi& sequence { K k } 0 6 .be& hamaRh&c t o K c o n t a i n e d in / G - (I Kkl = 0 and S ( K k ) < r. 6.4.4.
LeX
P R V V ~ v 6 Rhe Lemma. f a c t the basis
G
The lemma i s an easy consequence p f t h e
o f a l l s e t s homothetic t o
K
s a t i s f i e s t h e theorem
of V i t a l i . However a s i m p l e p r o o f o f i t can be g i v e n i n t h e f o l l o w i n g way. 0
Let let
ctlAl
=
A IKI
be a h a l f - o p e n c u b i c i n t e r v a l such t h a t with
€I < ct < 1.
o f d i s j o i n t half-open cubic i n t e r v a l s
We p a r t i t i o n {Ah}
G
K
t A
and
i n t o a sequence
o f diameter l e s s t h a n
r. For
each
K*h
Ah
let
= PhK.
then
Ph
123
DIFFERENTIATION AND THE MAXIMAL OPERATOR
6.4.
be t h e homothecy t h a t c a r r i e s
We can keep a sequence
{KiI:Il
A
into
Ah
and l e t
of these s e t s such t h a t , i f
G1 i s open and
N1
by t a k i n g
s u f f i c i e n t l y big.
and proceed w i t h
...,
We now s e t
G1 as we have done w i t h
Kh = K* , h = l , Z , Nl h N o b t a i n i n g now CKh)h:N +1
G,
such t h a t
And
so on. So we o b t a i n t h e sequence CKhI s a t i s f y i n g t h e lemma. We now c o n t i n u e w i t h t h e p r o o f o f t h e theorem. Assume t h a t ( b )
does n o t h o l d . The t h e r e e x i s t s a p o s i t i v e number
k > 0
each i n t e g e r if
x k = XAk
t h e r e i s a bounded measurable s e t
that
Mk
[Ck[
u n i t cube
means
.
Mrk k+ 1 > 2 [Ak[.
Q
{cJ,} i=1,2,. . .
Ak
such t h a t
,
3
rk such t h a t
t h e r e i s a l s o a p o s i t i v e number
where
A > 0 such t h a t f o r
Let
Ck
be a compact subset o f
{M x
>
A} such
k j By t h e p r e v i o u s lemma we can c o v e r t h e open
almost c o m p l e t e l y by means o f a d i s j o i n t sequence o f s e t s homothetic t o
r a t i o o f t h e homothecy
Pkj
carrying
Ck
such t h a t
Ck
i n t o C i we have akjrk < Z-k
if
c1
kj
i s the
6. C O VER I N G Sy HARDY-LITTLEWOOD AND DIFFERENTIATION
124 f o r each sets
AJk
and
j
,k
k.
1,2,
=
Pkj Ak = AJk
Let
..., j
,...
1,2
=
7
Since I A l <
be t h e union o f a l l
We then have
We s h a l l now prove t h a t a t almost each
1
A
and l e t
x
6
Q
-
we have
I)(]
xAyx) a
A > 0.
t h i s w i l l prove t h a t t h e d e n s i t y p r o p e r t y i s n o t t r u e
f o r A. Fix
k
and l e t
x
There i s t h e n
Ck.
E
R e a ( x ) , with
6(R) < r k such t h a t
IR
f'l A k l
IRI For each j , t h e image 6(R*)
<
2-k
R*
of
R
> A
by t h e homothecy
and
o
A1
.
> 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)
<
if
f(x)
I N
N f(x) = f (x) + fN(x)
= f(x)
a t almost e v e r y
'p, i s a d e n s i t y b a s i s , by Theorem 6.4.1.,0( x E R~
every
.
N
=
be such t h a t
fN
i n t h i s s e c t i o n i s p a r t l y due t o Hayes
equivalence (a)<=>(c).
f(x)
and l e t
for a different.
so we g e t a t almost every
a t each
x e Rn
x e Rn.
By
and a l s o , s i n c e
f N y x ) = fN(x) E R~
,
a t almost N D ( /fN,x) = f ( x ) .
L e t us now d e f i n e
and
g*
1
such t h a t
Ig*(x)I < N D(
g,,x)
= g,(x)
mosz e v e r y x
g ( x ) = g,(x)
a t each for
x e Rn
+ g*(x) and so
a t each
almost everywhere. Since
lg*l
each
Q(x)
sequence
x o Rn.
Then we have
, a g a i n by theorem 6.4.1.,
{Rk(x))c
L
fN
, we
have a t a l -
contracting t o
x.
6 . COVERINGS, HARDY-LITTLEWOOD AND DIFFERENTIATION
126
-J
w i t h t h i s we have
D(
-_
I)(
g,x)
= g(x)
g,x)
= g(x)
almost everywhere
a l m o s t everywhere. Analogously and t h i s proves t h e theorem.
The f o l l o w i n g theorem c h a r a c t e r i z e s t h e d e r i v a t i o n by integral o f a function
f
8
o f the
i n t e r m which a r e s i m i l a r t o those o f t h e den-
s i t y theorem o f Busemann and F e l l e r . I t i s v a l i d f o r a general b a s i s .
6.4.6.
ty. L e t
f > 0
,f
E
.
63
LtL be a b u h w a h t h e d e a a y p h o p eh L1(Rn). Then t h e & o U v w i n g t h e e conditionn me
THEOREM
-
eq iLivaeevLt :
(a)
9 di66mentiaten
doh each A > 0 , each nequence Cf,}, w L t h f k E L', f k < f f , f k ( x ) G 0 CLt d m o & t a c h x E Rn and doh each numeAhaP. ACQUenCe Cr,} w L t h r k G 0, we have. (b)
,the maximal a p e h a t a h u n v c i a t e d t o .the b a d % rk vbRained whme Mk by t a k i n g @om @ jui t h e e..temenx2 wLth diamQ;tm Lenn t h a n r k . ( c ) Fv/z each 1 > 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) <
such t h a t
S i n c e we a r e assuming t h a t
t h e p r e c e d i n g theorem, each
(a)
> 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 <
and
127
Ifk
differentiates
f o r each
(x)
{Rj(x)} C
.
k
E
It i s clear that, i f
Q
: M~ f h ( x ) >
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) <
E
(b).
That ( b ) i m p l i e s ( c ) i s t r i v i a l by t a k i n g
I n o r d e r t o prove t h a t
6
implies for
fk = f X A k
(a), l e t
k = 1,2,...
If, b y
J
Hence, f o r
Therefore {x
> 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 <
c1
< 1, t h a t w i l l be chosen c o n v e n i e n t l y
i n a moment. We d e f i n e PI=
2
Otherwise we keep o n l y those elements o f
t h a t s a t i s f y t h i s proeerty.
(b).
)G -A[
{[R[
: R
E%
, \A
n
R( >
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\ <
satism.
% i s a density basis since t h i s i s equivalent
t o d i f f e r e n t i a t i o n of
be a measurable s e t .
L".
For a
A
and
0
>
H > 0
we
d e f ine
x f A
So, f o r each
there exists
CRk(x)fc@(x),
such t h a t
Rk(x)
+
x
and I R k ( x ) ('1 M I IR,(x)l We s h a l l prove t h a t
IAl
apply property
to
(P)
= 0
A
'rand w i t h
E
> 0 . We o b t a i n
.
.
> A
I f n o t , we t a k e an a r b i t r a r y
E
> 0
and
w i t h the V i t a l i covering
=
(Rk(X))xeA {Rk)
,k
= 1,2,2
s a t i s f y i n g (a),
,... (b),(c).
Hence ,having
140
6 . COVERINGS, HARDY-LITTLEWOOD AND DIFFERENTIATION
i n t o account t h a t
Since
E
M
c A'
i s arbitrary,
we can w r i t e
IAl
=
0.
So we o b t a i n f o r almost a l l apply t h i s r e s u l t t o D(xNYx)= 0 basis
.
N = M I , we have
But t h i s i m p l i e s
x E M'
, for
D ( x M,x) = 0
almost a l l
D(xM,x) = 1.
.
I f we
x e N' i s a density
Hence
,
The f o l l o w i n g theorem can be viewed as an e x t e n s i o n o f t h e P o s s e l ' s theorem t o a measure t h a t i s continuous w i t h r e s p e c t t o Lebesgue measure.
Lei be a B - F b a h wLth t h e deMnLty > 0 be a hixed dunctivn. Then a necunwry and nuddicient cvndition i n o t d a t h a t 8 ~ ~ ~ e f ~.LA t he e h u~-t lvwing : -THEOREM. __
6.5.2.
p'lopehty and teX
f E L
f
I
A , g i v e n E > 0 and a a bequence {Rk} buch t h d , denvang
( E ) Given a bvunded m e u r n a b l e b&t ViAaLL c o v a
xk =
xRk
R
5
06
=
0
P'laod. be as i n p r o p e r t y
k
A Rk
, t h e m ex-&& , we
have
Assume t h a t (E)
@
differentiates
I
f.
Let A
o f t h e statement o f t h e theorem. L e t
, E, T-I
z0
a f i x e d c o n s t a n t t h a t we s h a l l choose c o n v e n i e n t l y i n a moment. L e t
, be
c
~
t
6.5.
for
k 6h
x E Ak
,
.
141
DIFFERENTIATION IMPLIES COVERING PROPERTIES
Since 7 4 d i f f e r e n t i a t e s
a sequence
i
f , we have, f o r a l m o s t e v e r y
such t h a t
We can assume
(1
t ri ) k
IRh(x)l
<
\
f 6 (l+q)k'l
IRh(x)I
f o r each
h=1,2,
...
Rh where t h e p r e c e d i n g i s n o t v a l i d Ak may be i n f i n i t e . We a p p l y t h e P o s s e l ' s
We s h a l l d i s r e g a r d t h e n u l l s e t o f and a l s o t h e n u l l s e t where
f
w i t h t h e V i t a l i c o v e r i n g o b t a i n e d by means o f t h e s e t s Ak and w i t h an ck + 0 t h a t w i l l be c o n v e n i e n t l y chosen l a t e r . We
theorem t o {Rh(x)}
thus o b t a i n a sequence such t h a t
,
i f we denote
(i) I A k
- sk
k 'j'j>l xsk f J
e x t r a c t e d from (Rh(x))xaA
xkj
and
= o
Observe t h a t we can a l s o w r i t e
So c o n d i t i o n
(iii) can be w r i t e n
Sk =
k
J
Sj
, we
,
h = 1,2
have
,... ,
6. COVERINGSy HARDY-LITTLEWOOD AND DIFFERENTIATION
142
where
yk > 0
can be choosen i n advanced a r b i t r a r i l y s m a l l .
We now t h a t '
k CSjl
k y j
can be chosen as t h e f a m i l y we a r e
l o o k i n g f o r i n o r d e r t o prove p r o p e r t y
and so we have ( a )
and so we have
. Also
(b)
For each
we have
i f we choose
k
(E). Observe t h a t
E~
so t h a t
1 ck
<
E
.
we can w r i t e
so we can s e t
i f only
yk c y
Therefore and ( c )
, by
f o r each
choosing
k. Hence
qy
y and
E~
c o n v e n i e n t l y we o b t a i n ( a ) , ( b )
o f p r o p e r t y (E). T h i s concludes t h e f i r s t p a r t o f t h e theorem.
6.5.
Assume now t h a t ( E ) h o l d s .
if.
ferentiates
A
r > s > 0
,
Rk(x)
We a p p l y ( E )
CTkI
T = 0 Tk k
-f
x
has compact s u p p o r t w i t h o u t loss
f
x
A
6
t h e r e e x i s t s a sequence
such t h a t
and e x t r a c t from
satisfying
%?Id i f -
we c o n s i d e r t h e s e t
i s bounded and f o r each
CRk(x)}c @ (x)
quence
L e t us t r y t o show t h a t
We can assume t h a t
o f g e n e r a l i t y . For each
The s e t
143
DIFFERENTIATION IMPLIES COVERING PROPERTIES
(a), ( b )
,
(c)
( R k ( X ) ) x e A , k=l,2,...
. We
can w r i t e
a se-
, calling
3
(For t h e second i n e q u a l i t y we have used
n > O we.can choose
Given
E
> 0
f o r t h e a p p l i c a t i o n o f (E)
such t h a t
i.e. where
(r
-5)
lA/
c
I O(I
Is( f , x )
proves t h a t
i)
.
Hence \ A / = 0.
i s not f,x)
p r o o f o f t h e theorem.
f(x)
= f(x)
So we have proved t h a t t h e s e t
i s o f n u l l measure. I n t h e same way one a l m o s t everywhere
.
T h i s concludes t h e
6 . COVERINGS, HARDY-LITTLEWOOD AND DIFFERENTIATION
144
The f o l l o w i n g theorem c h a r a c t e r i z e s those d i f f e r e n t i a t i o n bases t h a t d i f f e r e n t i a t e
Given
(P ) he,t
A,
0 <
9
/A1 <
]A
(a)
Pmvd. (P,). in
G.
and
IG
I
uRk
elements o f
7
If
2
differentiates
Lq.
by t a k i n g an open s e t
*‘t
{Rk)
06
a memuhub&
huch LhaX
We t r y t o prove such t h a t
G
and keeping o n l y those s e t s i n
E
We keep c a l l i n g
the following :
2 2
= 0
Assume t h a t 63
- A1 c
m.
>O
F i r s t o f a l l we prune
G > A
<
aRd g i v e n a V i a k L L c v u a t h e m d a nequence {RkIkalc E
w,
-
, 1< p
Lp
t h e r e m a i n i n g cover o f is
such t h a t f o r
any sequence
A.
?
contained
We now observe
( f i n i t e o r i n f i n i t e ) of
0 < ci < 1 we have
c a(1-a) (W The reason f o r (1) i s t h a t
@
1 .
i s a d e n s i t y b a s i s and f o r ( 2 ) t h a t
6 . 5 . DIFFERENTIATION IMPLIES COVERING PROPERTIES
fl
differentiates
on A
-
Lp
and the function i n brackets i s in L p
145 and i s 0
.
ORk
Therefore we can w r i t e
(1
(i i ) '
(I-a)
= I A r
G
[
I W ()(A
-
.
This suggests how we can proceed i n the s e l e c t i o n of CRkI a, 0 < a < 1, such t h a t alGl < E , and choose f i r s t R I E ? such t h a t I R 1 I 2 3/4 sup { [ V l : V st}. I f IA - R I I = 0 we a r e f i n i s h e d . Otherwise IR1) s a t i s f i e s ( i ) , ( i i ) and ( i i i ) . Call w1 the c o l l e c t i o n of a l l s e t s W 8 2 s a t i s f y i n g (1) and ( 2 ) corresponding to t h i s sequence C R 1 1 as above. Choose R2s w1 such t h a t I R z 1 2 3/4 sup { I W l : W E w1 I . If IA - f~ R k [ = 0 we a r e f i n i s h e d . 1 Otherwise ER1, R z } s a t i s f i e s ( i ) , ( i i ) , and ( i i i ) , Call w2 t h e collect i o n of a l l s e t s W E% ' as above corresponding t o t h i s sequence. Choose R3a w z such t h a t I R 3 1 2 3/4 sup I W I : W E w2}. And so on. In t h i s way we obatin {RkI . If i t i s f i n i t e , then i t i s so because [ A - O R k l = 0 . I f i t i s i n f i n i t e and / A - u R k [ > 0 then t h e r e W E 2 s a t i s f y i n g (1) and ( 2 ) . B u t c l e a r l y , since exists We f i x
(1 - a)
1 IRk]
s ] G I , we have
lRkl
-f
0 . There i s a f i r s t
Rk
such
6. COVERINGS, HARDY-LITTLEWOOD AND DIFFERENTIATION
146 that
lRkl
G 3/4
-
iiRkl
= 0
[A
IWI
and t h i s c o n t r a d i c t s t h e c h o i c e o f
and C R k I
Assume now t h a t ol 7 0
ri > 0
For an a r b i t r a r y
saM
llhll
tx
=
@
(a),
satisfies
.
Therefore
and ( c ) .
(b)
. Let
(P,)
f
E
Lp
and f o r
M > 0 let
and
we have
satisfies
Rk
6 rl.
E R :~
E
@
S(Rk)
+o
l e t us choose
g
h = f
-
g
- h(x) I
>
c11
such t h a t i f
We can w r i t e
1 x 1 < M, l i m sup xERk“(x)
5
16
\h(y)dy
and a l s o
I n order t o estimate each
x
E
S!”
S”!
we use
(P,)
with
t h e r e i s a sequence { R k ( x ) ) c U 3 ( x )
an
E
> 0.
contracting t o
such t h a t
We can assume t h a t situation such t h a t
obtaining
S(Rk(x))
4 1.
{S,}C
(Rk(x))
We a p p l y p r o p e r t y xeS:”,
k = l ,2
¶.
..
(P,)
t o this
For x
6.5.
147
DIFFERENTIATION IMPLIES COVERING PROPERTIES -
sk
Using H o l d e r ' s i n e q u a l ty, p r o p e r t y , we o b t a i n
that
c) o f
(P,)
I
I ( h / ( c rl P f o r each ci and
Since
and
M.
n
T h i s proves t h a t
[1975]
M
lSal
= 0
Lp.
differentiates
a differentiation
As we have a l r e a d y shown i n Theorem 6.2.4. as MoriyBn
1
i s a r b i t r a r i l y s m a l l , we see t h a t
0
b a s i s can d i f f e r e n t i a t e
and t h e f a c t
L ' w i t h o u t having t h e V i t a l i p r o p e r t y . However,
has proved
, i f the basis @
is a
B
-
F
basis
i n v a r i a n t by homothecies, t h e n t h i s i s n o t p o s s i b l e . I f t h e b a s i s i s a d e n s i t y b a s i s t h e n (B
6.5.4.
by hvmcdhwien.
v
u
8
has t h e V i t a l i p r o p e r t y .
'@ b~ a B - F ba&ih t h a t A i n v d a n t o!il;l;e~enLia.t~ L ' il; and a d l j id h a the
THEOREM. L& Then
3
pfi3hOpuLty.
Pmad.
We need o n l y t o prove t h a t i f
fB
differentiates
L'
t h e n i t has t h e V i t a l i p r o p e r t y . We s t a r t b y c o n s i d e r i n g t h e s e t
and we prove t h a t
IKI
<
co
.
We know, a c c o r d i n g t o Theorem 6.4.3, related to is of ed a t 0 and r a d i u s
weak t y p e ( 1 , l ) . L e t l / k ) and
t h a t t h e maximal o p e r a t o r
Bk = B(O,l/k)
( b a l l c e n t e r-
148
6 . COVERINGS, HARDY-LITTLEWOOD AND DIFFERENTIATION
CR
Kk = II K k c Kk+l
Clearly
@ (0) : IRI
6
and
Next we show t h a t I K / < pose t h a t
K
i s unbounded.
homothecies
,
t h a t each
Rk
.
a,
implies that
On t h e o t h e r hand
K
RkCS
,
1
.
R1
then
I R 2 (7 S 1 l
If = -XZ
S2
.
RZ
The s e t s
Lk =
so we can /Q/
BCQ
and
1
Q
17
S3
=
=
1 7
Sj)/ 6
-x3+ R 3
.
R2
.
then
S3
1
n t e g r a t i o n along l i n e s
sk
-
17
=
Take S1I R3
1 > 7
.
If
and
parallel t o
oxk).
Therefore I K I =
1
8
if
K
differentiates
in
aIK1
.
141
= (aIKI)
centered a t
are d i s j o i n t .
Sj
Rj).
And so on. I t i s easy t o
7
IJ
,so
1
IR2
>
(by k-1
k-l
86 ( 0
2
If
n v a r i a n c e by homothecies)
x
L'
,
then
a closed cubic i n t e r v a l
For each
IBI
b y homothecies, t h a t f o r each Val
(y 2
R3
jxk/
with
+ m
i n t h e f o l l o w i n g way.
S2
then
6(Rk)
(by t h e
enclose =
then
i s bounded. Sup-
K
Hence, Let
If
1
>
&
K
= 1,
IRk(
xk e Rk
has some p o i n t
We now choose a sequence o f s e t s C S k l S1
R 3 BkI
We can t h e n choose, b y t h e i n v a r i a n c e b y
,
CRkl c @ ( O )
,
Kk
I",
K =
6 1
B
E@
.
(0)
with
IBI
K
Q
.
i s bounded and centered a t
0.
= 1 we see t h a t
I t i s t h e n c l e a r , by t h e i n v a r i a n c e
B
such t h a t
e @ ( x ) there i s a closed cubic i n t e r 101 = alKl I B I . The b a s i s @ i s s a i d
t o be heglLeah w i t h r e s p e c t t o t h e b a s i s c f c u b i c i n t e r v a l s c e n t e r e d a t t h e corresponding p o i n t s .
I t i s an easy e x e r c i s e t o show t h a t t h i s
r e g u l a r i t y implies the V i t a l i property f o r
8 .
6.6. 6.6.
149
THE HALO PROBLEM
THE HALO PROBLEM.
Let
% be a B - F b a s i s i n Rn
t h a t i s i n v a r i a n t b y homo-
t h e c i e s and s a t i s f i e s t h e d e n s i t y p r o p e r t y .
+* :
6.4.3.there e x i s t s a function ed and measurable s e t
A
(1P)
and f o r each
T h i s suggests we define,
u
According t o t h e theorem
[OP)
-f
such t h a t each bound
e ( 1 , ~ ), one has
B
f o r any
-
F b a s i s % , even i f i t
i s n o t i n v a r i a n t by homothecies and does n o t have t h e d e n s i t y p r o p e r t y ,
+
the following function For each
u
E
that
will be c a l l e d t h e h a l o 6uncxXun 0 6
we s e t
(1,m)
We now can say t h a t , if
B
6
i s i n v a r i a n t by homothecies t h e n
i s a d e n s i t y b a s i s i f and o n l y i f
$
If % i s a d e n s i t y b a s i s , then
l{MxA
>
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) <
1
f o r each
~
a(ak-l)
k = l,Z,...,
@ ( x ) , we have
Therefore
and a p p l y i n g t h e f a c t f o r each
k
we have, f o r
A
> 0,
we o b t a i n
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 ) <
Zpk
, and
so, i f
Zk-'
c u
< Z k , we
obtain
According t o Theorem that
8
6.6.2.,
applied with
differentiates a t least
LP+'
~ ( u )= u logl+"(l+u)
f o r each
> 0.
we g e t
6.6.
'157
THE HALO PROBLEM
Therefore, i n o r d e r t o d i s p r o v e t h e h a l o c o n j e c t u r e , i t would be s u f f i c i e n t t o e x h i b i t a d e n s i t y
B - F basis, t h a t i s i n v a r i a n t by
homothecies and does n o t d i f f e r e n t i a t e any
Lp
with
p
im
m
,
can
Lp w i t h
c o n s t r u c t a d e n s i t y b a s i s t h a t does n o t d i f f e r e n t i a t e any p <
. One
b u t t h i s b a s i s i s n o t o f t h e t y p e r e q u i r e d here. F o r t h i s c o n s t r u g
t i o n o f Hayes
[1952,
19583
one can a l s o see Guzmhn [1975].
For a counterexample t o t h e h a l o c o n j e c t u r e one c o u l d t r y t o
B - F b a s i s i n v a r i a n t by homothecies and such t h a t i t s h a l o f u n c t i o n behaves a t i n f i n i t y l i k e eu .
construct a
T h a t t h e h a l o c o n j e c t u r e i s t r u e i n case
$(u)
i s an easy consequence o f t h e r e s u l t o f MoriyBn [1978]
-
u at infinity
p r e s e n t e d i n 6.5.4.
This Page Intentionally Left Blank
CHAPTER 7 THE BASIS OF INTERVALS
I n t h i s Chapter we s h a l l a n a l y z e some i n t e r e s t i n g c o v e r i n g and d i f f e r e n t i a t i o n p r o p e r t i e s o f t h e b a s i s o f i n t e r v a l s i n Rn
. For
each
Rn we c o n s i d e r as @ ( x ) t h e f a m i l y o f a l l open bounded i n t e r v a l s T h i s b a s i s w i l l be denoted as 8, c o n t a i n i n g x, and B = \I ,@(x).
x
6
xaR
and i t s maximal o p e r a t o r w i l l be c a l l e d
MP
.
H i s t o r i c a l l y i t was t h i s b a s i s t h e one w i t h which s t a r t e d t h e expansion of t h e modern t h e o r y o f d i f f e r e n t i a t i o n l o n g a f t e r t h e Lebesgue d i f f e r e n t i a t i o n theorem.
U n t i l 1933
, when
Banach proved t h a t
B,
has t h e d e n s i t y p r o p e r t y ( s t r o n g d e n s i t y theorem), a l l t h e e f f o r t s t o extend t h e c o v e r i n g and d i f f e r e n t i a t i o n p r o p e r t i e s o f t h e b a s i s o f c u b i c ( V i t a l i p r o p e r t y and Lebesgue d i f f e r e n t i a t i o n p r o p e r t y ) t o
intervals
s i g n i f i c a n t l y d i f f e r e n t systems o f s e t s were f r u i t l e s s .
8,
proved t h a t by H.Bohr
I n 1924
Banach
does n o t have t h e V i t a l i p r o p e r t y , a r e s u l t a l s o o b t a i n e d
about t h a t time, as appears i n an appendix o f t h e work
Carath6odory [1927]
.
I n 1927
of
N i kodym o b t a i n e d a n o t h e r i n t e r e s t i n g r e s u l t ,
r a t h e r d i s c o u r a g i n g f r o m t h e p o i n t o f view o f d i f f e r e n t i a t i o n , f r o m w h i c h Zygmund deduced t h a t t h e b a s i s o f a l l r e c t a n g l e s i n
R2
does n o t have
t h e d e n s i t y p r o p e r t y . T h i s r e s u l t o f Nikodym w i l l be p r e s e n t e d i n t h e n e x t Chapter. A f t e r t h e s t r o n g d e n s i t y theorem o f Saks p 9 3 q t h e p o s i t i v e r e s u l t s s t a r t e d p i l i n g up w i t h t h e work o f Zygmund [1934] F e l l e r [1934]
, Jessen,
M a r c i n k i e w i c z and Zygmund [1935],
, Busemann
and
de Posse1 [1936]
y...
I n o u r e x p o s i t i o n we s h a l l f o l l o w a more o r l e s s c h r o n o l o g i c a l o r d e r , s t a r t i n g w i t h t h e e a r l y n e g a t i v e r e s u l t s and e n d i n g w i t h t h e more r e c e n t and i n t e r e s t i n g r e s u l t s o r i g i n a t i n g m a i n l y i n some problems pro159
7. THE B A S I S OF INTERVALS
160
posed by Zygmund and s o l v e d o n l y r e c e n t l y by Marstrand, MoriyBn, CoFdoba,
7.1. THE INTERVAL BASIS B 2 DOES NOT HAVE THE VITAL1 PROPERTY. I T DOES NOT DIFFERENTIATE L'. According t o Theorem 6.5.4.
o f Moriyh, for a
B - F basis
L'
t h a t i s i n v a r i a n t by homothecies, d i f f e r e n t i a t i o n o f
is e q u i v a l e n t
t o t h e V i t a l i p r o p e r t y and e q u i v a l e n t a l s o t o t h e r e g u l a r i t y o f t h e b a s i s w i t h respect t o the basis o f cubic i n t e r v a l s . Obviously
B2
does n o t
s a t i s f y the l a t t e r property. Even i f we c o n s i d e r the minimal
B
-
F basis i n v a r i a n t by
o f open bounded i n t e r {Ik) (i.e. the r a t i o v a l s we can a f f i r m t h e same i f t h e e c c e n t r i c i t y o f
homothecies t h a t c o n t a i n s a g i v e n sequence
Ik
o f t h e l o n g e r s i d e and t h e s m a l l e r one) tends t o i n f i n i t e .
WEAK TYPE INEQUALITY FOR A BASIS 7.2. DIFFERENTIATION PROPERTIES OF $2. WHICH I S THE CARTESIAN PRODUCT OF ANOTHER TWO.
Though
B2
does n o t d i f f e r e n t i a t e
L'm")
,
i t i s a density
b a s i s , as Saks [1935] has proved. Moreover i t d i f f e r e n t i a t e s L ( l t log' [1935]
B2
L)"'
affirms.
(Fin) ,
as t h e theorem o f Jessen-Marcinkiewicz-Zygmund
We s h a l l o b t a i n t h i s theorem b y c o n s i d e r i n g t h e b a s i s
as t h e i t e r a t e d C a r t e s i a n p r o d u c t o f t h e i n t e r v a l b a s i s o f R ' . T h i s
method o f p r o o f belongs t o Guzma'n [1974] r e c t i o n was o b t a i n e d by
B u r k i l l [1951]
.
.A
previous r e s u l t i n t h i s d l
...
7.2.
16 1
DIFFERENTIATION PROPERTIES
7 . 2 . 1 . THEOREM. Let M2 be ,the muximde opehcLtoh a6ocicLted Rv t h e i v t t e h v d b a b dB2 in R 2 Then, doh each f e Lloc (R’) and each
.
A > 0
we have
whehe
c
+ 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,<’)
,
(n1,r12),
...
t h e dummy v a r i a b l e c o o r d i n a t e s which appear i n t h e i n t e g r a l s and d e f i n i tions. Let
f > 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(<1,<2)
>
x , and so 7
if
0 < u
&
1,
7 . THE BASIS OF
164
NTERVALS
R’
I n order t o estimate
and
, we get
Sz we define f o r a fixed
0
> 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)
.
<
Ptrao6.
THEOREM The o e t F 06 &~nction.4 f i n L ' ouch at borne p o i n t x e Rn LA 06 t h e &Oi~,tca;tegohq i n
00
F o r t h e sake o f c l a r i t y , we s h a l l p r e s e n t t h e p r o o f f o r i s t h e u n i o n o f a c o u n t a b l e c o l l e c t i o n o f nowhere
F
We show t h a t
dense s e t s i n t h e f o l l o w i n g way. as t h e s e t o f f u n c t i o n s with
1x1
c k,
11; r
we have t h a t each
1
Fk
f
For
.
we d e f i n e
i n L 1 such t h a t f o r some p o i n t
I e-od,(x)
i t happens t h a t f o r a l l
c klI(
...
k = 1,2,3,
We c l e a r l y have
F =
x e R2
(I
k=l
Fk
,
1
6 ( I ) < T;
with
m
Fk
. We
now prove
i s nowhere dense, o r , what i s e q u i v a l e n t t h a t
Fk has no
i n t e r i o r points.
k
For each
L1 with
If,
and f . J
6
J
Fk
= Fk
. I n fact,
j = 1,2,...
F o r each
such t h a t i f
I e%,(xj)
Fk f o r
J x . 1 c k,
we have
-f f in j j t h e r e i s some p o i n t x j y 1 and & ( I ) < T; , t h e n
assume t h a t
f
i s compact, one can e x t r a c t f r o m ( x . 1 J a convergent subsequence. We can assume, changing n o t a t i o n i f necessary, fj/
c k l I / . Since B(O,k)
converges t o a p o i n t x . C l e a r l y , 1x1 < k 1 i s such t h a t 6(I) < we can w r i t e
that
xj
and
if
I eQ2(x)
7 . THE B A S I S OF INTERVALS
168 I i s open,
Since
(1,
s u f f i c i e n t l y l a r g e , and so as
pk
j =
-f
m,
x.
x E I and
1
and we g e t
Fk '
J
+
fj(
i, f l
, we
x
klIl
G
. Furthermore
(xj)
for
j
,
, p r o v i n g t h a t f e Fk. Therefore
k l 11
h
I
have
I n order t o prove t h a t
Fk does n o t c o n t a i n any i n t e r i o r p o i n t we s h a l l use t h e f o l l o w i n g lemma which c o n s t i t u t e s t h e k e r n e l o f t h e p r o o f o f t h e theorem. t h e o r i g i n of
The lemma j u s t means t h a t f o r each neighborhood k = 1,2,
L ' and f o r each
i n t h a t neighborhood which i s n o t i n
7.3.2. LLwe 6unCtion
+N
Fk
... .
there i s a function
of
k ,V I t s proof i s given l a t e r .
Foh each Matwlae numbm
LEMMA.
V $
N
,thehe
a nonnaja-
huch ,that
With t h i s lemma t h e f a c t t h a t p o i n t s i s e a s i l y obtained. there i s a function
g E L1
Let
- Fk
f
c Fk
Fk
. We
such t h a t
does n o t have any i n t e r i o r prove t h a t f o r each ((g
-
f((,g
q.
q z 0
Let
.
-
q/2 We d e f i n e g = h + +N h e Lm o L ' be such t h a t (If h i l l 6 where $N i s t h e f u n c t i o n o f t h e lemma w i t h an N t h a t w i l l be chosen
i n a moment. We can w r i t e
According t o e x i s t s an i n t e r v a l
I
(b) 6
o f t h e lemma, f o r each
Bz(x)
, with
6(I) <
1
x
6
B(O,N),
such t h a t
there
7 . 3 . SAKS RARITY THEOREM We now choose 119
-
N
f/Ii c n
Hence
g
d
Fk
1
such t h a t N x k ,
c
169
, N - IlhlIm
> k.
Then we have
and
as we wanted t o prove.
P m v d v d t h e Lemma 7.3.2.
For t h e proof of t h e lemma we s t a r t
w i t h the simple a u a u h y cvMn~uc2ivn o f the beginning o f t h i s Section w i t h an
t h a t w i l l be c o n v e n i e n t l y f i x e d i n a moment. By u s i n g lemma
H
6.4.4.
d i s j o i n t sequence
ISk}
o f s e t s homothetic t o t h e s e t
i a r y construction contained i n Let
R = B(O,N)
-
x
E
R
by means o f a
J,,
o f the a u x i l
B(0,N) and w i t h diameter l e s s t h a n
1/N.
co
(I 1
Sk.
We have
IRI
= 0
and so, f o r each
E
> 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 ) <
Cx e R2 :
be
06
< f(x)}
-a
<
pro-
0 4 the. ne& iS(1
f,x)
D ( 1 f,x)
<
m}
< f(x)}
puoLtiwe rneaute.? The n e g a t i v e answer i s due t o B e s i c o v i t c h [1935].
Here we p r e s e n t t h e r e s u l t of B e s i c o v i t c h .
I n t h e remark a t t h e end we
show how t h e theorem can be somewhat extended.
7 . THE B A S I S OF INTERVALS
172
.
THEOREM We c o m i d a t h e i n t a u a k ? b a d be a dixed d u n c t i a n . Then t h e &a n e A
7.4.1. LeZ
f e L'(R2)
{X E
R'
: f(x)
II(if,.)
<
<
f12 i n
R2.
a)
have m u m e z a a .
P m o d . We s h a l l c a r r y o u t t h e p r o o f f o r t h e f i r s t o f t h e s e s e t s . For t h e o t h e r s e t one can p u t s e t t o g.
g
and a p p l y t h e r e s u l t f o r t h e f i r s t
= -f
a , B , y r a t i o n a l such
I t w i l l be enough t o prove t h a t f o r
0 < a < B
that
i
E(ii,B,y)
=
y , the s e t {x
6
R2 : f ( x )
f
a
-t
-t
p
,
-y
has measure zero. L e t us assume t h a t t h e r e a r e t h r e e numbers cx
, B,y
so t h a t t h e s e t E = E ( a , B , y ) i s o f p o s i t i v e measure. f = f -t fy where
as above
L e t us s e t
Y
Since ' @ z d i f f e r e n t i a t e s D(
f ,x) = f ( x )
f Y
where
Y
D(Jfy,x) It
= fy(x).
x E
La
almost everywhere. We have
$1
and
f e Lm, we have
Y
L e t us c a l l =
\El
>
the subset o f
E
0.
we have
and t h e r e f o r e , h a v i n g i n mind t h e d e f i n i t i o n o f
E, i f x e E c E we have
173
7.4. A THEOREM OF BESICOVITCH
B
< f y ( X ) + fY(X) +
and
-y < f ( x ) < y , i . e . Hence, i f
x
E
a< and
fy(x) = 0
. Thus
E
fy(x) = 0 we can w r i t e
a( / f Y , X )
< B
the set
has p o s i t i v e measure. We now i n t e n d t o prove t h a t t h i s l e a d s t o a c o n t r a d i c t i o n and t h i s w i l l conclude t h e p r o o f o f t h e theorem. F o r each points
x
of
A
E
s, w i t h
0 < s < 1 , we d e f i n e
such t h a t f o r each
A
E,
IS @ ~ ( X ) , with
as t h e s u b s e t o f 6(I) <
, one
s
has
The s e t be some
i s t h e union o f a l l
A
s* > 0 such t h a t Es*
be a s u b s e t o f
h
Es*
open s e t such t h a t
Q , s > 0, and so t h e r e must i s o f p o s i t i v e e x t e r i o r measure. L e t E*
Es
with
s
6
w i t h f i n i t e p o s i t i v e e x t e r i o r measure H
3 E*
,
IHI
6
( l + n ) IE*le
w i t h an
.
Let
H
be an
17 > 0 t h a t
w i l l be c o n v e n i e n t l y chosen l a t e r . For each with
6 ( I ( x ) ) < s*
x 6 E*
one can choose
I(x) =
IC
H, I = I ( X ) €@2(X),
such t h a t
We s h a l l now a p p l y t h e f o l l o w i n g lemma whose p r o o f i s p r e s e n t e d a t t h e end.
7 . THE B A S I S OF INTERVALS
174 7.4.2. x
t h u t 604 u c h UA M h U m C
LEMMA.
I be any open i n t m v d and
1e.Z
I we have & h a
E
fy x )
ouch t h a t
o4
Ifqx I > y
. L e i
Ik(x)
c
I(x)
{Ik}
06 open
i n t e h v d contcuned
k
each
To each
fying
0
nuch
Ah&
Then one can chaooe a d i n j a i n t bequence
in I
=
fy E L'
we a p p l y t h i s lemma, o b t a i n i n g { I k ( x ) }
satis-
I ( x ) C H,
f o r each k . L e t now be
1
and l e t
U
=
-
-U
E = G
Furthermore
I(x)
IJ
XEE*
U { I k ( x ) : x E E*
. Clearly
E 3 E*.
I E @2(y)
Val
G =
with
E
So
y
1 U.
< s*
i
, for s*
3
each
,
E*c
Ik(x)
G
y E E*C
G
and U a r e open.
ES,
and each i n t e r
h
we have, by t h e d e f i n i t i o n of
composing U we have
and
This implies
...
i s measurable, s i n c e
However, f o r each one o f t h e s e t s 6(Ik(x))
k = 1,2,
? .
In fact 6(I)
,
-U
I
=
E.
h
ES,
,
7.4. A THEOREM OF BESICOVITCH Now f o r each
I(x)
with
x e E*
175
we have
and so G c { x e R2
Hence, by Theorem on
6.4.3.-
GI s c ) U \ . Where c depends -. o n l y
we have
a/y , Therefore ( E l c c [ U ( . We c l e a r l y have I ( x i s i n H. So we g e t
UU
E C H
since
each
(1 T h i s i s i m p o s s i b l e i f we choose II <
c1 . T h i s
c o n t r a d i c t i o n conlcudes
t h e D r o o f o f t h e theorem.
P m u B vd t h e Lemma. 7.4.2. on i t t h e f o l l o w i n g process P. length
If
We t a k e t h e i n t e r v a l d
i s t h e s m a l l e r s i d e and ( d l 1 i t s and
.
i n g p o i n t s , we draw through them l i n e s p a r a l l e l t o
I
and perform
, D i s t h e b i g g e r s i d e and \ D J 1i t s l e n g t h , we d i v i d e D i n t o
equal p a r t e s o f l e n g t h between L !!'
of
I
i n t o a c e r t a i n number o f p a r t i a l i n t e r v a l s
Once we have t h e d i v i d d.
We g e t a p a r t i t i o n
{IT . 1;
,...,
1;
1
such t h a t t h e r a t i o between t h e i r b i g g e r s i d e s and t h e i r s m a l l e r ones i s between 2 and 4. I f t h e mean o f
fy on each one o f t h e p a r t i a l i n t e r v a l s
P on I i s f i n i s h e d . I f t h e r e i s some It such t h a t t h e mean o f f on 1: i s b i g g e r t h a n o r J equal t o y, we t a k e a maximal i n t e r v a l c o n t a i n e d i n I and c o n t a i n i n g 13 such t h a t t h e mean o f fy on i t i s e x a c t l y y. I n t h i s way we have p a r t i t i o n e d I i n t o a f i n i t e d i s j o i n t c o l l e c t i o n Q o f i n t e r v a l s on which t h e mean o f fy i s y and another f i n i t e d i s j o i n t c o l l e c t i o n %?,* i s l e s s than
y, t h e n t h e process
partial interval
c o n s t i t u t e d by t h e i n i t i a l p a r t i a l i n t e r v a l s them
o r b y those p a r t s o f IJ o b t a i n e d i n t h e process o f c o n s t r u c t i o n o f t h e i n t e r v a l s o f a;! .
On t h e i n t e r v a l s o f '5Lf t h e process P
on
I.
t h e mean o f
fy i s l e s s than
y
.
This finishes
7 . THE BASIS OF INTERVALS
176
We now keep t h e i n t e r v a l s o f ;R and on each i n t e r v a l of
P. I n t h i s way we g e t a sequence (I I k c I and such t h a t
we p e r f o r m t h e same process disjoint intervals, with
On t h e o t h e r hand, i f intervals
-
f e I
H k ( x ) e @ 2(x)
(J
contracting to
F e L , and t h e sequence
{ I k ) of
I k , t h e n t h e r e e x i s t s a sequence of x
such t h a t
the r a t i o and 4 such t h a t
between t h e b i g g e r s i d e an t h e s m a l l e r s i d e i s between 2
Since
?R *
IHk(x)I
i s regular w i t h respect t o
squares, we g e t
f o r almost e v e r y fY(x)
c o
-
x B I
I! I k
=
I,
fY
-
x e I
a t almost e v e r y
plIJ
and, t h e r e f o r e
=
'1-
IJ
Ik
L J I ~.
fy
'
SO
, fy(x) c
y
and so
we can w r i t e
I,, I,, kfY h
k
f y S Y I \J I k I
T h i s concludes t h e p r o o f o f t h e lemma.
e9751
.
For an e x t e n s i o n o f t h e p r e v i o u s theorem one can see
Guzmin
177
7.5. A THEOREM OF MARSTRAND 7.5. A THEOREM OF MARSTRAND AND SOME GENERALIZATIONS.
According t o the r a r i t y theorem we have seen in 7.3. t h e r e i s a function f e L1(R2) such t h a t 6( f , x ) = + m a t a . e . x with respect t o 8 2. Zygmund proposed t h e following problem (see Guzma'n and We1 land [1971] ) : Suppose we a r e given a function f e L1(R'). I s i t poss i b l e t o choose a p a i r of rectangular d i r e c t i o n s so t h a t t h e b a s i s of a l l rectangles in those d i r e c t i o n s d i f f e r e n t i a t e s f a t a , e . x e R 2 ?
I
I
That the answer i s negative was proved f o r the f i r s t time by Marstrand 119771, , who found a function f e L1 ( R 2 ) such t h a t f o r every o r i e n t a t i o n o f the axes y we have B ( f , x ) = t m a t a.e. x with respect t o the rectangles i n d i r e c t i o n y. Later on E l Helou [19781 found such an f belonging t o (1 L ( l t log' L)'. More recently Bernard0 O
i
7.5.1.
JHEUREM. ~.
doh ewmy dine.Otion
Thehe A a ljunotian
y we. have
t h e heotangben ,in t h a t dihecLLon.
6,(
I
f,x)
=
+m
L' @ l z ) buch t h a t n.e. wLth hapect t o f
E
Phoolj. W e shall construct f E L'(Q),Q u n i t square such t h a t o r ( f , x ) = +- a t a . e . x E Q . From t h i s one e a s i l y obtains the above statement. Let , f o r 0 < r < 1,
where B ( 0 , r ) means t h e open ball of center H(r) looks l i k e F i g . 7.5.1. shows.
0 and radius
r.
The s e t
7. THE B A S I S OF INTERVALS
178
F i g u r e 7.5.1.
r
I t i s n o t d i f f i c u l t t o g e t t h e e s t i m a t e f o r small H(r)
F o r each
k
2
r
cr (1 + l o g 1 )
we f i x an i n t e g e r
Mk > 0
i e n t l y chosen l a t e r on and we d e f i n e a f u n c t i o n way.
Consider
o f order
Mk
tangent t o
to
B(0yr-k)
Q.
=
l y 2 ,...,2 k
B(k,j)
With an
H(k,j)
B(0,l)
,j Let
Q(k,j].
l a t e r , we t a k e t h a t sends
Q(k,j) of
to
M
rk
the
fk
ZMk
i n the following d y a d i c subsquares
be t h e b a l l c o n c e n t r i c w i t h
,0
homothetic t o B(k,j).
,
t h a t w i l l be conven-
Let
by t h e same homothecy.
Q(k,j),
< rk < 1, t h a t w i l l a l s o be chosen
H(rk) C(k,j)
w i t h t h e same homothecy be t h e b a l l h o m o t h e t i c
(See F i g . 7 . 5 . 2 . ,
Mkl = 2 , Mk2 = 4 ) .
F i g u r e 7.5.2.
where
OF
7.5. A THEOREM
f k w i l l have c o n s t a t i t v a l u e
The f u n c t i o n chosen l a t e r ) on each any f i x e d d i r e c t i o n If
x e H(k,j,y)
We have
IC(k)l
let
SO if we want
H(k,j,y)
H(k,j).
=
rt
C
.
f e L1
,
-f
outside t h e i r union. For
I
H(k,j)
with
c
i n d i r e c t i o n y such t h a t
absolute constant.
Also
Then
i f suffices t o set x
1 akrk <
m .
belongs t o i n f i n i t e l y many s e t s
Y
So we w i l l t r y t o arrange our c o n s t a n t s s o t h a t l l i m sup k-tm
K(k,y)\
=
K(k,y)
0 (
we o b t a i n , by t h e r e l a t i o n (*) above,
m
( t o be
From now on we f i x t h i s d i r e c t i o n .
Observe now t h a t i f
ak
and
0
Xk > 0
be t h e s e t c o n c e n t r i c w i t h
there i s a rectangle
f = sup f k
Take now
and w i l l be
C(k,j)
y
o b t a i n e d by r o t a t i n g
179
MARSTRAND
1
Since
Q
-
we need
l i m sup k+m
I
f)
kal
K(k,y)
=
(Q-K(k,y))l
(2
-
= 0
co
(1
m
IJ
1=1 k h l
f o r each
K(k,y)
1.
t h e f o l l o w i n g lemma t h a t w i l l be proved l a t e r .
=
IJ
f)
1=1 ka1
(Q
-
K(k,y))
We achieve t h i s by means of
7. THE BASIS
180 7.5.2.
LEAiMA.
und
bquatle Q
otdeh
k
06
S ( k , j ) C Q ( k , j ) , whene
Q
d y a d i c bubhquaheh
2k
.
S(k,j)
homvthecy t h u f CWU Q into
INTERVALS
be -two meawlabEe b&tA i n the. uvLit
E > O . Covlcriden t h e
06
j = 1,2,...,2k
L e t A,S
OF
Q(k,j).
Then id
With t h i s lemma we lsroceed as . f o l l o w s .
t h e lemma w i t h H(r2)
A
y .
= Q
i n direction
K(1,y) So i f
S = Q
MZ
Q(k,j)
be t h e union a d t h e 2k & homotheLLc t o S by ,the hume
l e t S(k)
-
H(r2,y),
A budl;icievLtey
k
MI = 2
Take where
.
H(r2,y)
big
Apply is
i s b i g enough we g e t
We now a p p l y a g a i n t h e lemma w i t h
and o b t a i n , if
M3
i s b i g enough,
3
3
Observe t h a t t h e second member i s i n d e p e n d e n t o f y t o choose
we g e t
r k so t h a t
l l i m sup K(k,-{)I = 1 as d e s i r e d . k +
I n order t o achieve
.
So, i f we a r e a b l e
Pnmd G =
II
j BJ
Q(k,j)
06
,the Lemma.
,G3 A
181
A THEOREM OF MARSTRAND
7.5.
IGI
We t a k e X
(1
f
b i g enough so t h a t t h e r e i s
k
IAl
E)
Then
For a basis can d e f i n e f o r
@3 i n Rn
t h a t i s i n v a r i a n t by homothecies we
0 < r < 1
Then t h e general r e s u l t o b t a i n e d f o l l o w i n g t h e same p a t t e r n o f p r o o f i s as f o l l o w s .
-_ THEOREM.
7.5.3.
Let
Y(1,m)
-f
(0,m)
be an incneaning convex
6unotion nuch t h a t
L.Melero s t u d i e s i n t e r e s t i n g p a r t i c u l a r cases. The r e s u l t o f
E l Helou i s o b t a i n e d from t h i s one by s e t t i n g u > e.
Y(u)
=
1
u l o g u(1og l o g u ) - ,
7. THE B A S I S OF INTERVALS
182
The r a r i t y r e s u l t i s o b t a i n e d by showing t h a t , f o r e v e r y
f
b,s > 0, t h e s e t o f f u n c t i o n s
E Y
(L)
w i t h the property t h a t for
each y we have
c o n t a i n s a dense open s u b s e t o f
Y ( L ) , and t h a t e v e r y f
i n a certain
denumerable i n t e r s e c t i o n o f s e t s o f t h i s t y p e i s as bad as t h e theorem states.
7 . 6 . A PROBLEM OF ZYGMUND SOLVED BY MORIY~N. Consider i n R 2
x eR2, B(X)
t h e f o l l o w i n g b a s i s 3. For each
w i l l be t h e c o l l e c t i o n o f a l l open bounded i n t e r v a l s c o n t a i n i n g such t h a t , if d
i s t h e l e n g t h o f t h e s m a l l e r s i d e and
o f t h e l a r g e r s i d e one has p r o p e r t i e s of t h i s b a s i s
L(l
f
log
i.
L)
D2 6 d G 0
6
B? O f course one knows t h a t $3
J3
That
@
i s a subbasis o f
differentiates
8
2 ,
and
differentiates
b u t one c o u l d perhaps e x p e c t something b e t t e r .
Phood.
x
i s the length
1. What a r e t h e d i f f e r e n t i a t i o n
i s n o t s o has been proved by MoriyGn [1975]
since
D
That i t
.
L(l
f
logt L
i s obvious,
the basis o f a l l intervals. I n order
t o prove t h e second p a r t o f t h e theorem we proceed as f o l l o w s .
x e ( 0 , l ) , Q, w i l l denote t h e f o l l o w i n g i n t e r v a l R 2 : Qx = (0,~~'~) x (0,x ) and cx w i l l be t h e e q u i l a t e r a l hyperFor each
of
183
7 . 6 . A PROBLEM OF ZYGMUND
bola passing through
, as in the following picture
(x,x2)
I
1 x3/2x
Figure 7 . 6 . 1 . Clearly, each rectangle o f opposite one on 0
6
, Q,,,
[x3”,x]
S(Qx,,)
&
cx contains
2x.
we get f o r each
=
(O,,)
with a vertex a t (0,O) and the
has the same measure, i . e . i f ( 0 , x 3 0 - l ) then / Q X y e / = x 3 . A l s o
Q, x
4
and
Hence, since
x
E (0,l)
and each
0 E
[x3/’,x]
,
where M r , for r 0 , means the maximal operator associated t o the s e t s of “a3 with diameter less t h a n r . So we have
184
7 . THE B A S I S OF INTERVALS
If exist
@
r > 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 <
Rk
t parallel
and c a l l
If
I j c Ik t h e n ak
two-dimensional i n e q u a l i t y
r
a j
Hence t h e b j *
7. THE BASIS OF INTERVALS
190
I and therefore
0
IJ
rj
Ij11
1
5
llkll
, applying lemma 7 . 7 . 2 . , I{x
E
t
:
CxI
k
(XI
r
+ 111 6
C -
5r
1 0 1 ~ 1 1
and so
An aeteAulative pmod ad Theohm 7.7.3. We now present another proof of the covering theorem f o r dyadic i n t e r v a l s in R 2 t h a t i s i n t e r e s t i n g in i t s e l f and will give us a method t o solve another r e l a t e d pro blem.
Let
I [,Ba(
<
. We
(Ba)aEA be a c o l l e c t i o n of dyadic i n t e r v a l s of R 2 w i t h M choose f i r s t {Bkj! so t h a t 1 LIB,\ 6 2 1 i~ B k / . Let us
c a l l t h e i r side-lengths also t h a t t h e r e i s no 8 k
a,&,.
We can assume tsl r 6 ? k... 6, , and contained i n another one. We choose now
R , = B1 . Assume t h a t R1 ,.. . , Rh R . = El . W e then choose as Rhtl J
where
ri
w i l l be fixed i n a moment.
have already been chosen a n d l e t the next Bk in t h e sequence
7.7. So we o b t a i n b l a bZ
...
2
{Rjll a
H
191
COVERING PROPERTIES w i t h sides
b H , al
i(
a2
a
b that satisfy j' j'
...G
G
aH
and a l s o
IR.17
IJ
k<j
R
I
=
'I
-
R.0
T h e r e f o r e f o r each
e dx 6 Rk k<j IJ
j -1 CXRK
'1 e
,
e l +e ' l
e
0 Rk Rj k<j
IRjI
j
and so
Furthermore we have
Then
c
2oe
I
h J;
i f we choose
Rjl
rl
+
e(l+q)
[Rhtll
L
h 20el IJ R j l +
s u f f i c i e n t l y small. So t h e
e(l+rl) [Rh+l
1
1-
tR.1
ti
J i
l+rl e
satisfy
-
h 11
i
R.1 J
c
7. THE BASIS OF INTERVALS
192
H
IJ
We now try to prove
I
ti
LJ
Rj H
Bk - LJ Rj I c cI 1
H IJ
i
R. I J
.
Take any B E iBkIM that has not been chosen. If R1 intervals with bk > 6 then we have
,.. .,R1
are the
1
Let us intersect B y R1,R Zy...,RH by a line paralell t Ox and let the intersections be called S , I ,... ,IH. Since a < ak , 1 < k c 1, the 1 two-dimensional inequality
1
transforms into the one-dimensional one
and so, if M, we have
is the maximal operator with respect t o intervals of s , 1
There fore
7.8. ANOTHER PROBLEM OF ZYGMUND
193
H
If we integrate over all lines s
7.8. ANOTHER PROBLEM OF ZYGMUND.
H
SOLUTION BY C6ROOBA.
According to a result of Zygmund l19651, if we consider the system of all intervals in R 3 such that one of their bases is a square, then the corresponding maximal operator M satisfies the inequality (1
and so this basis differentiates L ( l
f
i.
logf
log' L ) ( R 3 ) .
In a similar way, if we consider a system of intervals in R 3 such that there is some reasonable constraint between their three different side-lengths , it is to be expected that this system will behave again like the two-dimensional basis of intervals, i.e. its maximal operator will satisfy the same inequality as above,
7. THE BASIS
194
OF INTERVALS
The f o l l o w i n g theorem of CGrdoba [1978] problem o f Zygmund i n t h i s d i r e c t i o n .
i s the solution t o a
We w i l l p r e s e n t i t i n t h e d y a d i c
v e r s i o n , g i v i n g t h e corresponding c o v e r i n g lemma f r o m which t h e weak t y p e i n e q u a l i t y f o r t h e maximal o p e r a t o r i s an easy consequence.
i~ a dixed duncfitian nandemeaning i n t h e ,two vatLiable? 4epmaXcLy.
?'hood.
7.7. of t h e c o v e r i n g theorem
have g i v e n i n
R
The p r o o f i s p a t t e r n e d a f t e r t h e a l t e r n a t i v e proof we
7.7.3.
f o r intervals i n
and w i l l be b e t t e r understood under i t s l i g h t .
M {Bk)k=l
We f i r s t choose a f i n i t e sequence
IoBa(
r;
M 21 IJ
cl
1
Bk(
. Let
We can assume t h a t no
a have
C2 >
, i f t,
Bk
us c a l l
ak , 6,
such t h a t
, ck t h e s i d e - l e n g t h s of Bk
i s c o n t a i n e d i n another one and t h a t
cj .... 2 EM , By t h e a c, , t h a t e i t h e h
monotonicity condition
iik
6
ii,
Oh
fik C
b1
of 4
.
we
.
7.8. ANOTHER PROBLEM OF ZYGMUND
195
We now choose the { R j } exactly in the same way as in the above mentioned proof o f the covering theorem for intervals in R z , i.e. we choose first R 1 = B 1 and them R P as the next Bk such that
and so on. We obtain { R j I Y (i)
satisfying
c2 a
c1
...
2
(see that proof)
cH
(ti)
If Ck > c1 then either a k G al or b k X bl
(iv)
If B e {B,IM
intervals with c k >
c
- {Rj}Y
and if R 1 , R 2 ,
...,R1
are the
, then 1
that
I
By means of properties (i) H H I) Bk II R j l c cI II R . 1 1 i J
M
-
- (iv) we shall now try to prove in order to obtain (a).
intersect B and R 1 , R 2 , ..., R1 of (iv) by a plane u orthogonal to Oz obtaining S, I I z,. . . , I 1 . Due to the fact that c1 2 cz 2 ... a c1 > e , the three-dimensional inequality of (iv) is transformed into the two-dimensional one We
1
7. THE B A S I S OF INTERVALS
196 For each
1 1 , 1 2 , ...
I
we have e i t h e r
j Iha r e such t h a t
a r e such t h a t the sides c be1ow
j
a
j
a
a
or
b
j
6.
Assume t h a t
a . h a , b . c 6 and t h a t Ih+l ,..., I , J J a . 6 a, b . a 6 (we can now f o r g e t about t h e o r d e r of J J ) , The s i t u a t i o n i n t h e p l a n e o i s t h a t o f t h e f i g u r e
F i g u r e 7.8.1. L e t us c a l l those of
Ij
P,Q
the proyections o f
over t h e same axes
S
Ox, Oy.
over
OX, Oy,
We can w r i t e
and
J
K
j’ j
7.8. ANOTHER PROBLEM OF ZYGMUND
+ & Js J f)
(
h
1
h+l Ij) 0 ( ' J
Therefore we can write
having s e t
h
IJ I.) 1
J
1
7. THE B A S I S OF INTERVALS
198
Hence, i f
p =
Therefore
min ( 3
,G)
, we have e i t h e r
al>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 <
with this (0,l)
and
to
p(Ah) =
11,12,13,
flYf2,f3,
We f i r s t t a k e
A
n
,
E)
H. Given
i n t o 2" p W und .to ~ k i l ;tth e &LungLen
> 0 Lt A pobbibL?e t o pah&Xivn t h e baA
I I ,Iz ,I3
A
. LeL
THEOREM
... ,TZn
B C
..., I 2 n b~
' a n d common ventex
that
a so t h a t 0 < a < 1, 211-a) < € 1 2 , E 7
.
We now c o n s i d e r t h e r e s u l t o f
and a, and an a f f i n e t r a n s f o r m a t i o n p
(0,O)
ih for
to
B and (2",0)
to
C.
Then
h = 1,Z ,...,Zn.
t h e r a m i f i c a t i o n s due t o t h e excess t r i a n g l e s . As we s h a l l see l a t e r t h e P e r r o n t r e e has p l e n t y o f a p p l i c a t i o n s t o v e r y many d i f f e r e n t problems. I t would be of i n t e r e s t t o have a
high
8.1. THE PERRON TREE
205
degree o f f l e x i b i l i t y t o c o n s t r u c t d i f f e r e n t types o f P e r r o n t r e e s adapted t o m o d i f i c a t i o n s o f t h e problems we a r e g o i n g t o be a b l e t o s o l v e w i t h t h e c o n s t r u c t i o n we have performed. F o r t h i s reason i t i s i n t e r e s t i n g t h e f o l l o w i n g o b s e r v a t i o n which p e r m i t s us t o o b t a i n a Perron t r e e once we a r e given a Besicovitch set.
We s h a l l see t h a t t h e r e a r e d i f f e r e n t methods t o
construct Besicovitch sets. Assume
B
i s a compact n u l l s e t formed b y t h e u n i o n o f c e r t a i n
segments of l e n g t h one whose d i r e c t i o n s f i l l a c l o s e d a n g l e o f 60" (see Figure
8.1.3.)
F i g u r e 8.1.3. Let
E
> 0
and t a k e an open s e t
each u n i t segment containing
1
1 of
G
G 3 B
such t h a t
and
[GI <
E
B we t a k e an open t r i a n g l e c o n t a i n e d i n
.
For
G and
i n i t s i n t e r i o r s o t h a t t h e angles a t t h e upper extreme
p o i n t s o f each
1
a r e equal. Since
a f i n i t e number o f such t r i a n g l e s
.
B
i s compact we can c o v e r
B with
We t h e n t r a n s l a t e p a r a l l e l y t h e s e
t r i a n g l e s t o have t h e upper v e r t e x a t t h e same p o i n t and so we see, rev e r t i n g t h e c o n s t r u c t i o n , t h a t , g i v e n E > 0 and a c l o s e d e q u i l a t e r a l
ABC
triangle divide
ABC
o f heigth
ha
i n a f i n i t e number of t r i a n g l e s
them p a r a l l e l y t o p o s i t i o n s
1/2, i t i s possible t o
b i g g e r equal t o
Tl,...,Th
TI,
so t h a t
...,Th and 10 thl 6 E
to translate
However, t h e c o n s t r u c t i o n of t h e P e r r o n t r e e we have performed i n Theorem 8.1.2. noteworthy.
has some a d d i t i o n a l f e a t u r e s t h a t make i t e s p e c i a l l y
8. THE B A S I S OF RECTANGLES
206
Theorem
8.1.2.
The t r i a n g l e s
Ti
7
t h e n t h e upper v e r t e x of Thmel(ohe,
7 i s t o the r i g h t o f t h a t j i s t o t h e l e f t of t h a t of Ti
if t h e b a s i s o f
r e s p e c t t o t h e i r bases, i . e .
of
T I , . , . ,f
o f t h e c o n s t r u c t i o n of 2" end up w i t h t h e upper v e r t i c e s i n r e v e r s e d o r d e r w i t h
REMARK I .
j
.
i d we extend t h e LtiungLen fh
vehticen t h e ~ ee.xten.bivru uhe. d i n j o i n t
ubove the,&
uppu
(See F i g . 8.1.4)
F i g u r e 8.1.4.
REMARK 2.
16 we extend t h e L h h n g L a
7,
b d o w the& b a a
t h e n e exten.bion.b WVQA on t h e h-thip pat&& t o ttkin b a d ! 06 w i h t h ha at &at u &an& egud t o t h e ohiginal one ABC LU indicated .in F i g . b . I , 5 . no m & u how we have -taken CI and n i n t h e co~n.i%uotion 0 6 t h e P m v n O ~ e e06 Theoxem 8.1.2. A
B
t
ha
C ha
/ I
/
F i g u r e 8.1.5.
\
\
8.2. A LEMMA OF FEFFERMAN
___ REMARK
207
.
I n Z h e i h ~ i n dp v b U o n t h e uppeh u e m X c a v6 fh neweh yet &mtheh tv t h e Re@ v d .that v d f , by rnvhe -than t h e Length vd t h e bub.in V/J ABC. 3
8.2. A LEMMA OF FEFFERMAN. F o r an i m p o r t a n t p r o b l e m i n t h e t h e o r y o f F o u r i e r m u l t i p l i e r s , t h a t we s h a l l examine l a t e r on, i n g the structure o f rectangles
C . Fe f f e r m a n [1971]
. This
used a lemma c o n c e r n
lemma c a n b e v e r y e a s i l y o b t a i n e d
by means o f t h e c o n s t r u c t i o n we h a v e p e r f o r m e d o f t h e P e r r o n t r e e .
Y
fiehe
Rh
denoten t h e bhaded pvh-tivn
ub t h e &7~cne 8.2.1
\
F i g u r e 8.2.1.
8 . THE BASIS OF RECTANGLES
208
The proof i s straightformard from Theorem 8.1.2. with the Remark 1. For each one of the t r i a n g l e s Ph we perform the c o n s t r u g t i o n indicated in Fig. 8 . 2 . 2 taking as R h the r e c t a n g l e indicated and as E the Perron t r e e 1 0 f h l . The R h a r e d i s j o i n t according t o Remark 1. The area of t h e i r union i s a good portion of t h a t of t h e o r i g i n a l t r i a n g l e ABC with which we s t a r t e d and so we can arrange everything so t h a t 1 E l < rl 1 1 R h l . On t h e -other hand Rh 0 E 3 t h a n d f, 1 i s a good portion of iih . SO j~~ E I > __ 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
<
f-)- .
8.1.2.
taking
We o b t a i n t h e
A B can be c o n t i n u o u s l y moved
8. THE B A S I S OF RECTANGLES
210 within
71
M N
from
t o the other side o f
one can move t h e segment t o t h e s i d e o f area l e s s t h a n
' zn
12
x
T2
other side o f
n o t on
process i s l e s s t h a n
an angle o f
8.4. THE
.
f p
7,
N P.
n o t on
From t h e r e
p a r a l l e l t o i t sweeping an
Now we move i t a g a i n w i t h i n
-
TP
t o the
N P, and so on. The area swept o u t i n t h i s
n / 6 , and t h e needle i s a t t h e end on a l i n e f o r m i n g
60" w i t h t h e o r i g i n a l p o s i t i o n .
BESICOVITCH SET. From t h e Perron t r e e o f 8.1.2.
we o b t a i n a B e s i c o v i t c h s e t as
follows. 8.4.1.
THEOREM.
rneaute containing u Paood.
i n R2 ad niLee 06 ui& Length i n euch dihecfion.
Thehe A u cornpct he,t
htpULt
F
I t i s enough t o produce a compact n u l l s e t
c o n t a i n s a segment o f u n i t l e n g t h i n
each
F
that
d i r e c t i o n o f an a n g l e o f 45:
T h i s i s s t r a i g h t f o r w a r d from t h e f o l l o w i n g lemma whose p r o o f i s presented a t t h e end.
8.4.2. LEMMA. Given a cLobed puh&LeLogtrarn P 06 bididen and ri > 0 t h e h e A CL &uXe c o U b t L o n 06 dahed pah&eLogtlurnh { w l , w 2 , . .., uH I w d h one hide on a and anathe,k one on c hUCh
a,b,c,d
R
=
thcLt
( 2 ) E a c h h e p e n t joining u paint 0 6 a t o anotheh p o i n t ad adrnh.2 a p a h a l l e l thaMn&aA;ionthud ca/LILien t o \I w j'
c
We s t a r t a p p l y i n g t h i s lemma t o t h e c l o s e d u n i t square
Q
=
ABCD
with
r11
= 1/2
obtaining
{ wl,
up,... , w
H:
I.
Observe t h a t
211
8.4. THE BESICOVITCH SET
uHi A
L,
=
in
j=1
uj
i s a compact s e t of area n o t g r e a t e r t h a n
t
A
B
.
To each o . we a p p l y a g a i n t h e lemma w i t h an J
02
c
- , obtaining
{ w(j,l)
22
Hi
Lz= i s a compact s e t
H;
!J
(J
j=1
r=l
contained i n
w(j,2)
The s e t
m
I1
F =
i n each d i r e c t i o n o f
j=1
A
^B
L. J
C
n2
so s m a l l t h a t
)...) w ( j , H j ) l . The
set
w(j,r)
1/2’ and
L 1 w i t h area n o t g r e a t e r t h a n
c o n t a i n i n g segments of u n i t l e n g t h i n each d i r e c t i o n o f
so on.
contained
and c o n t a i n i n g segments o f u n i t l e n g t h i n each d i r e c t i o n o f an
Q
a n g l e o f 450, namely
Hi
1/2
i s a compact n u l l s e t
A
e
8
. And
c o n t a i n i n g segments
as r e q u i r e d .
.
w1 =
Prrovd 0 6 Lemma. 8.4.2. f i r s t of a l l we t a k e two s t r i p s ASTD and w2= DLBT as i n d i c a t e d i n t h e f i g u r e 8.4.1. and such
that
A l s o we c o n s i d e r t h e p o i n t
B V
A
i s uarallel to
a
L
F i g u r e 8.4.1.
V
o b t a i n e d as i n d i c a t e d i n t h e f i g u r e , where
L T.
B F i g u r e 8.4.2.
8. THE BASIS OF RECTANGLES
212 Then we d i v i d e s m a l l e r l e n g t h than
AS
SB
. We
i n t o a f i n i t e number o f equal segments o f join
V
t o t h e d i v i d i n g p o i n t s and c o n s i d e r
.
each one o f t h e t r i a n g l e s
To each one o f them we a p p l y t h e VMi M i + l 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 o f 8.4.2. w i t h an E so s m a l l t h a t t h e area o f t h e u n i o n o f a l l t h e Perron t r e e s o b t a i n e d i n t h i s way i s l e s s
n/4. Observe t h a t by Remark 3 of 8.4. t h e upper v e r t i c e s o f t h e
than
small t r i a n g l e s o b t a i n e d i n those P e r r o n t r e e s never go t o t h e l e f t o f d. We now proceed s y m m e t r i c a l l y s t a r t i n g from t h e s i d e s u b s t i t u t e each one o f t h e i n t e r s e c t i o n s w i t h
P
BC
.
F i n a l l y we
o f t h e small t r i a n g l e s
o f such Perron t r e e s by p a r a l l e l o g r a m s as r e q u i r e d i n t h e s t a t e m e n t o f t h e theorem as i n d i c a t e d i n F i g u r e
8.4.2.
We have p o i n t e d o u t b e f o r e t h a t t h e r e a r e s e v e r a l d i f f e r e n t ways t o o b a t i n B e s i c o v i t c h s e t s . I n t h e n e x t Chapter we s h a l l see how t o o b t a i n them very s i m p l y b y means o f t h e geometric t h e o r y o f l i n e a r l y measurable s e t s . Here we s h a l l p r e s e n t another s i m p l e way due t o Kahane
[1969]
.
As
a m a t t e r o f f a c t t h e c o n s t r u c t i o n o f Kahane, as p o i n t e d o u t
, i s a p a r t i c u l a r case o f t h e c o n s t r u c t i o n i n d i c a t e d
b y Casas [1978]
above by means o f l i n e a r l y measurable s e t s . However i t w i l l be i n s t r u c t i v e
t o show i t w i t h o u t a p p e a l i n g t o t h a t t h e o r y .
8.4.2. THEOREM. On t h e negment joining 0 = (0,O) t o A = (1,O) c . o a i d a a pehdect neA La Cantox dividing OA i n t o law e . q d d o b e d h e g r n e d , .taking t h e Xwo e x h m e o n u , dividing again each one i n r v d o u h e q d d o s e d negmena, and so on. 1e.X CO be t h e beA no 0 bXained.
On t h e segment j u i n i n g u neA
C1
LeL Co
B = (0,l)
to D
=
(1/2,1)
coaidm
h u m o t h d c t o CO. F
be t h e u n i o n
to anotheh p o i n t 06
C1
.
06
& &abed
segmenh joining a point
06
8.4. THE BESICOVITCH SET
Phoah.
That
213
F c o n t a i n s i n f a c t segments i n a l l those d i r e c -
t i o n s i s easy.
T
F i g u r e 8.4.3.
Observe t h a t
Kk
sets
F
obtained by j o i n i n g t h e points o f the k-th
construction o f CI
.
can be viewed as t h e i n t e r s e c t i o n of a l l t h e compact
We have
Co t o points o f the and each
K k + l ~ Kk
k-th Kk
phase o f t h e
phase o f t h e c o n s t r u c t i o n o f the property o f having
preserves
OA
p a r a l l e l t r a n s l a t i o n s o f each segment j o i n i n g a p o i n t o f
BD.
of
I t s u f f i c e s t o check t h i s by mere i n s p e c t i o n f o r
i s e s s e n t i a l l y o b t a i n e d from that to
TL
i s parallel to
So
RS.
K1
KI as BL
OR
K1
The f a c t t h a t
OA a t h e i g h t u, 0
~.rx+ (1
- u)x'
F
where
XCOl1 =
F
y = 1
. Let
parallel
by a l i n e p a r a l l e l
6 1-1 6 1, t h e p o i n t s so o b t a i n e d have abscissae
x
C1=
8
X
6
I t x + Ax'
1 7C0 x'
Co
.
So i t i s s u f f i c i e n t t o
x'
E
CoII1= 0
6
[O,-) we have :
x
€
Co
y
To do t h i s we c o n s i d e r two s e t s l i k e C o y one on y on
DA
i s of n u l l measure can be proved i n t h e f o l -
prove t h a t f o r a l m o s t each
ICO +
and
DS
KP
Observe
covers segments o f a l l t h e d i r e c t i o n s above mentioned.
l o w i n g way. Observe t h a t o f we i n t e r s e c t t h e s e t to
K1, s i n c e
OADB.
i s obtained from
i s parallel to
t o another
us c a l l them
Do
D1
and l e t
G
= 0
and t h e o t h e r
be t h e u n i o n o f a l l
8. THE B A S I S OF RECTANGLES
214
t h e segments j o i n i n g a p o i n t o f
i s t h e union o f f o u r s e t s
Gl,
an a f f i n i t y o f r a t i o
.
1/4
Lee@
i n g p o i n t s of t h e
DO
t o another one o f
Glr
, Grl
,
half o f
, Grr
Glr
The s e t
The s e t
G
G
each a f f i n e t o
by
i s t h e u n i o n o f segments j o i n -
gigk kt
t o the
Do
.
D1
h a l f of
D1.
F i g u r e 8.4.4.
We have G = Gll
lGll
therefore (I Glr
(J
Grl
(J
1
lGlrl
=
.
Grr
=
So we have
perform the a f f i n i t y t h a t c a r r i e s i s carried t o that part o f f i g u r e , and t h i s p a r t o f
of
G
under
Gll
=
IGrr/
lGll G
has n u l l measure
has n u l l measure.
f
to
I
1’1
, we
=
1/4 1 G I and
Girl
=
0
. Gll
see t h a t
If 0
we Glr
which i s below t h e d o t t e d l i n e d i n t h e
G G
lGrl
.
By symmetry a l s o t h e p a r t
T h e r e f o r e t h e r e i s an a > 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
<
IRI Therefore the lower density o f to
Bd
set
N
i s less than
1 and so
F
1/3
a t each o f i t s p o i n t s Bd
We have a l r e a d y seen i n 8.5. way.
F
For each x c F t h e l i n e
So, u s i n g t h e compactness o f
x.
direction to
We t a k e a compact subset
IN1 .
x
w i t h respect
i s not a density basis. t h a t one can c o n s t r u c t a Nikodym
such t h a t t h e d i r e c t i o n o f t h e l i n e
This leads t o the f o l l o w i n g r e s u l t .
l(x)
v a r i e s i n a measurable
232
8. THE BASIS
OF RECTANGLES
THEOREM. Them ~~~2 a covLtinuauh &ieRd a n R ouch t h a t Bd 0 no2 a d e n n a y b a d . 8.6.4.
d
06
ditrecfionh
Ptraod. L e t N be a s e t o f p o s i t i v e measure such t h a t t h e direction
d(x)
o f the l i n e
v a r i e s i n a measurable way. subset
F
l(x)
through
x
such t h a t
o f p o s i t i v e measure such t h a t
d(x)
Bd
, obtaining
d. The
accord ng t o t h e p r e c e d i n g theorem cannot be a d e n s i t y b a s i s . We s h a l
l a t e r see i n Chapter 9 t h a t
1,
o f Casas [1978
when
d(x)
has d i r e c t i o n
condition o f
d
d(x)
.
according t o a r e s u l t
i s a L i p s c h i t z f i e l d o f d i r e c t i o n s then
i t i s n o t p o s s i b l e t o c o n s t r u c t a Nikodym s e t
l(x)
0 l ( x ) = Cxj
v a r i e s c o n t i n u o u s l y . We
extend t h i s f i e l d o f d i r e c t i o n s c o n t i n u o u s l y t o R2 basis
N
By L u s i n ' s theorem we can t a k e a compact
N
such t h a t f o r
x
E
N
I t i s n o t known y e t whether t h e L i p s c h i t z
i s s u f f i c i e n t t o make of
Bd
a density basis.
In
Chapter 12 we s h a l l p r e s e n t some p o s i t i v e r e s u l t s o f S t e i n and Wainger
[I9781 f o r some smooth f i e l d s o f d i r e c t i o n s . When t h e f i e l d the basis
Bd
+
L (1 t l o g
d
differentiates
L)
(R2) and
has a c o u n t a b l e number o f values
~ ( +1 l o g + L ) .
EdhlF=l
I n fact, l e t
Eh = { x : d ( x ) = d h l
.
Bd be t h e h b a s i s o f a l l r e c t a n g l e s i n d i r e c t i o n dh We know t h a t Bd differen: h t i a t e s i f . T h e r e f o r e t h e s e t o f p o i n t s o f Eh where Bd does n o t f
E
Let
.
differentiate
I
i s o f n u l l measure
f
i s o f n u l l measure.
f
c o l l e c t i o n of s e t s
Eh
, the
.
Since t h e r e i s a denumerable
s e t o f p o i n t s o f R2
where
Bd
does n o t
8.7.
LACUNARY DIRECTIONS
233
SOME RESULTS CONCERNING BASES OF RECTANGLES I N LACUNARY DIRECTIONS.
8.7.
The problem d e a l t w i t h h e r e i s t h e f o l l o w i n g . Assume t h e f o l l o w i n g s e t o f d i r e c t i o n s Q, = { 2-
, 2L , i!L , 2- ,... I . 23
22
@
is
Consider
25
24
o f r e c t a n g l e s w i t h one s i d e i n one o f t h e s e d i r e c t i o n s . 8@ What a r e t h e d i f f e r e n t i a t i o n p r o p e r t i e s o f t h e b a s i s B o ? Stromberg the basis [1976]
proved t h a t t h i s b a s i s d i f f e r e n t i a t e s
any
> 0
that
E
BQ,
.
L)4+E (R2)
L 2 ( log'
With e a s i e r means C6rdoba and R.Fefferman [1977]
L2(
differentiates
This i s equivalent, since
for
proved
B@ i s i n v a r i a n t
MQ, a s s o c i a t e d t o
by homothecies, t o t h e f a c t t h e maximal o p e r a t o r
Bo The methods used by Stromberg and a l s o by C6r-
i s o f weak t y p e ( 2 , Z ) ) .
doba and R.Fefferman i n v o l v e p u r e r e a l v a r i a b l e c o n s i d e r a t i o n s o f t h e t y p e we have been h a n d l i n g i n t h i s and t h e p r e c e d i n g two c h a p t e r s . L a t e r on S t e i n and Wainger [1979
1, b y methods o f
F o u r i e r a n a l y s i s , have im-
proved these r e s u l t s . Here we s h a l l examine t h e method o f C6rdoba and R.Fefferman
B@ t h a t w i l l be a l i t t l e e a s i e r t o handle. From t h i s r e s u l t t h e same c o v e r i n g theorem and t h e
w i t h a s l i g h t l y modified version o f the basis
f o r 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 can be o b t a i n e d
weak t y p e (2,2)
f o r t h e above b a s i s . We s h a l l c o n s i d e r t h e b a s i s &io f a l l p a r a l l e l o g r a m s
R
satisfying : (a)
Two o f t h e i r s i d e s a r e p a r a l l e l t o Oy.
(b)
The o t h e r p a i r o f s i d e s have one o f t h e d i r e c t i o n s T
{-,
I
22
7
... 1
-1 , 23
(c)
The p r o j e c t i o n
(d)
Each
R
R
For t h i s basis &
R
we have
over
i lRl/lil
i s so t h i n t h a t i f
containing
sult.
p(R) o f
Ox
i s a dyadic i n t e r v a l .
i s t h e minimal i n t e r v a l G
1/8
.
we s h a l l prove t h e f o l l o w i n g c o v e r i n g re-
8. THE BASIS OF RECTANGLES
234
H
H
(Here and i n t h e proof, t h e constantc i s a p o s i t i v e a b s o l u t e constant n o t depending on the c o l l e c t i o n , not always the same (B,)aeA a t each occurrence )
C B 1 , ...,BN 3
We can f i r s t s e l e c t a f i n i t e sequence so t h a t
Ptuo6. -
from
N
I(' We can assume t h a t
c
IB,
B1 , B 2 , .
..
21
y
Bkl
have been so ordered t h a t
b ( B j ) = length of projection of
proyection of
9-. over J
over Ox
Bjtl
=
Ox > length o f
b(B.
J+1
)
Also we can assume t h a t no 9 . i s contained i n another one. J
We s t a r t choosing t h e R We examine
then we s e t R2 R1
=
B1 , R 2
=
B2
. Otherwise we .
leave Examine B3 . I f
2
1 I
i =1 then
setting
R1 = B1
.
If
B2.
B2
=
j
B3 0 R.J
I
=
XB,
B2
ii,
aside.
X Rj
Assume
1
9
1'31
R B = B 3 . Otherwise we leave B3 a s i d e . Assume t h a t R P = B1 , B 3 has been l e f t . Consider B1, . I f
R1
=
B1,
235
8.7. LACUNARY DIRECTIONS
!
xBl,
then
R3
=
B4
(
R1
. Otherwise
XR,)
+
we l e a v e
6
B4
1
7
1841
aside.
And so on.
o u r s e l e c t i o n i n a f i n i t e number of steps o b t a i n i n g b(Rj)
(a)
I
(b) (c)
Property ( i i )
If
XRj) Bi
J
that satisfy:
b(Rj+l)
h
(
CR.1:
So we f i n i s h
XR
‘ h+ 1
1
7
has n o t been chosen
i s e a s i l y obtained
.
IRh+ll
, then
In fact
But we have
Therefore
lIRj
1
c 2
I ORj I
and we g e t
To prove p r o p e r t y ( i ) we use t h e sequence as f o l l o w s .
T h e r e f o r e we have
If
Bi
(c),
(ii)
.
( d ) and t h e l a c u n a r i t y o f
has n o t been chosen, t h e n
8. THE BASIS OF RECTANGLES
236
or
or
where d ( R j ) ( A ) i s true
.
means the direction 4 of R j The s e t s are in the situation o f the figure
Bi
for which
Figure 8.7.1. If we intersect by a vertical line 1 , x X ilnd call M1 the unidimensional maximal operator w i t h respect t o intervals of 1 we get 1 0 Bi
C {(Xyy) :
M1(lXRj)
Therefore the union of a l l sets n o t exceeding
Bi
(1,~) >
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) <
m
.
We s h a l l r e s t r i c t our a t t e n t i o n t o such s e t s and so we w i l l w r i t e A f o r A' . I n t h i s chapter, u n l e s s o t h e r w i s e e x D l i c i t l y s t a t e d , "meanukable" 'I A-meaukable w a h dinite t n m u h e " . w i l l mean Before we l o o k a t t h e d e n s i t y p r o o e r t i e s o f such s e t s we s h a l l prove a u s e f u l form o f t h e V i t a l i theorem f o r t h e measure
A
.
9.1.
THEOREM. 1eR: E be a meau/Lab.ee b d .
9.1.1. a heqUenCe
243
LINEARLY MEASUREABLE SETS
a6 c&ohed C i t L d e A centeked CLt
{Ck(x)>
x
A given. Let E > 0 . Then o ~ CUM e choahe a sequence c & c l e ~ @om ( C k ( x ) IxeE such thcLt k=l,2 (i1
A(E
(ii)
ACE)
6
WLth
6(Ck(x))
{D;}
ad & j o i n t
J
E
+
0
..
¶ .
-
D.)
1
rl
=
J
j
c
Fok each x
6 ( ~ t~ E)
j
I n o r d e r t o o b t a i n ( i i ) we prove f i r s t t h e f o l l o w i n g n r o p e r t y t h a t , f o r r e f e r e n c e , we s t a t e s e n a r a t e l y as a lemma.
exi~a2
06
p > 0,
Butel
If
~(E,E)
p =
such t h d
b&
P4ood. -
h(E) = 0
with
-
-
1
j Then
k
[Aj}
!I Bu
IJ{Ba)
> 0
Assume ¶...
with
AS)
E
6(Aj) < P O
A(E) E. l e t , f o r each k = 1,Z ?;
E
dnfi
(BalacA
k
1)
i s a cover of
E
tend t o
A(E)
.
> 0.
From t h e
P O such t h a t f o r each
A , D E we have J L e t us c o n s i d e r E !J (J
-
,B
k { A , } be a sequence o f s e t s whose u n i o n J k 0 < 6(Aj) < P O and such t h a t A(E
B ~ )
m
as
t h e n we g e t
and
k + m
w i t h diameter l e s s t h a n
fore
But i f we make
i s a countable c o l l e c t i o n ,
there e x i s t s
-
+
. Then t h e h e
each coReection (Ba)aeA
Assume
such t h a t
P 6 PO
E > 0
W Q have
< p
n o t h i n g i s t o be proved.
h(E), g i v e n
covers
LhcLt
We can assume t h a t
sequence
1 6(Aj)
, nuch
< 6(Ba)
0
definition of {Aj}
be meanuhable and
1eX E
9.1.2. LEMMA.
po. There
9. GEOMETRY AND L I N E A R MEASURE
244
P4ood ad Theo4em 9.1.1.
We f i r s t choose
t h e lemma we have j u s t proved and c o n s i d e r c i r c l e s
p
according t o
C,(x)
o f diameter
p , So we now have t o c a r e o n l y about ( i ) , s i n c e ,
l e s s than
i f we g e t
( i ) , then
x
F o r each
E
E
{ S i I ,...,
6
1 6(Dj)
E we choose f i r s t one such 3.2.1.
c o v e r i n g theorem covers
0 I! D s )
A(E
A(E) =
+
ck(x)
obtaining a family
ISk}
E
and a o o l y B e s i c o v i t c h o f c i r c l e s whose u n i o n
5 d i s j o i n t sequences For one o f them, say { S i l , we must have
t h a t can be d i s t r i b u t e d i n t o
1.
since otherwise
and t h i s i s c o n t r a d i c t o r v . Now we have A(E
-
( IJ S i ) )
We keep a f i n i t e number o f s e t s o f
A(E
-
H1
+
D ~ )6
obtain
IDjlHH:+l A(E -
El
{Sil,
=
H1
-
IJ
Dj
XA(E
-
I!
E
1
so t h a t H2
1)
Dj)
6
c a l l i n g them
A(E) =
5+ 7
1
We can now oroceed w i t h
tL A ( E ) 5
6
H2
H1+1
XA(E)
{Dj)
,
A < 1
as we have done w i t h
Dj)
6
X2A(E)
and g e t
E o and
245
9.2. REGULAR AND IRREGULAR SETS I n t h i s way we o b t a i n
satisfying
CDjI
(i)
and
(ii).
9.2. DENSITY , REGULAR AND IRREGULAR SETS.
E
Let
be a measurable s e t ,
consider the l i m i n f
and t h e
0 L A(E) <
l i m sup
as
r
x
8
R 2 we
of
0
+
For
m.
A ( E 0 B(x,r) -2r '
where
B(x,r)
i s a closed c i r c l e centered a t
They a r e r e s p e c t i v e l y t h e l o w t i f they coincide)
D(E,x)
of
at
E
and t h e
x.
x
We s h a l l f i r s t prove t h a t t h e f u n c t i o n s measurable.
For t h i s l e t
p > 0, E > 0
points
x
and
F = Ix
6
be two r a t i o n a l numbers and l e t
on
,
-D ( E , * ) , -D(E,*)
R 2 : D(E,x) > A } F(P,E)
,
are
. Let
be t h e s e t o f
which
A(E f o r each
1 > 0
r > 0.
( t h e derm%q
We denote them by D(E,x)
if t h e y a r e e a u a l ) .
(D(E,x)
and o f r a d i u s
uppm devm3ity
r 6 p
.
17 B(x,r)) 2r
A + €
We have c l e a r l y
We now prove t h a t each
F ~ , E )i s a c l o s e d s e t .
If
{xk1cF(p,€)
,
x k + x , t h e n f o r each r b p t h e r e i s an i n d e x ka such t h a t i f k a k a we have Xk B B(x,r) . L e t B(xk,rk) be t h e c i r c l e c e n t e r e d , We have rk c r 4 p a t xk and o f maximal r a d i u s c o n t a i n e d i n B ( x , r ) and
lim
r k = r. We a l s o have
9. GEOMETRY AND LINEAR MEASURE
246
A(E 11 B ( x , r ) )
Therefore
and so
>, 2 r ( X + E
x
6
F(P,E)
.
In a
6(E,.).
s i m i l a r way one g e t s t h e m e a s u r a b i l i t y o f
9.1.1. we e a s i l y o b t a i n t h e
By means o f t h e c o v e r i n g lemma following properties f o r the density.
9.2.1. each
x
THEOREM
6 E we have D(E,x) Pmod. __
L e t us f i x
We t r y t o Drove t h a t
of
E
. LeX
satisfying
o f closed c i r c l e s
E
be a measwrabLe b e X .
c1 t
0
and d e f i n e
.
Let
=
ACHl = 0 A(E
- F)
(Dk(x))
Then at cLemont
0.
6 E ,
E
> 0
and
x
be a compact subset
x e H we have a sequence
F o r each
centered a t
F
and c o n t r a c t i n g t o
x
such
that
We a p p l y t h e c o v e r i n g lemma sequence
Therefore
{Ski
o f circles
9.1.1. such
that
with
n
z 0
and o b t a i n a d i s j o i n t
9.2. REGULAR AND IRREGULAR SETS A(H) 6
Since
x
E
E
1
cl
1 k
and q
9.2.2. E we have
A ( E (1 sk) + q
sk))
A ( E (1 (
a r e a r b i t r a r i l y small, we have
THEOREM
1
Ptlaad. -
1c1.
=
. 6
Let us f i x
L&t
E
c1
6
t
A(H)
be meaukable.
D(E,x)
247 r7
=
=
0.
Then CLt &oh2
each
1
> 0 and d e f i n e
F = { x e E :6(E,x)>
l t a )
For each x 6 F t h e r e i s a sequence centered a t x c o n t r a c t i n g t o x such t h a t
C C k ( x ) ) of closed c i r c l e s
According t o t h e previous r e s u l t we know t h a t we have a t almost each x
Therefore, f o r almost each x E F
, we
E
have
We apply t h e covering theorem 9.1.1. w i t h f S k } o f d i s j o i n t c i r c l e s such t h a t
E
0 obtaining a seauence
F
9 . GEOMETRY AND LINEAR MEASURE
248
t E =
Hence
A(F)
and s i n c e
Hence
D(E,x)
1 a t almost each
I n o r d e r t o prove t h a t l e t 11s f i x
For C(x)
r > 0
- A(F) l+a
t
E i s a r b i t r a r i l y small,
x
E.
6
D(E,x)
A(F) = 0.
2
1/2
a t almost each
x E
E,
P < 1/2 and l e t
let
be t h e s e t o f t h o s e p o i n t s
Gr
i s a closed c i r c l e centered a t
x
with
x
6
G
such t h a t i f
radius less than
r we
have
I t i s easy t o see t h a t Gr
i s c l o s e d and
G = O < r e Q We t r y t o prove t h a t the definition o f sets
{A?
A(G,)
A(Gr)
= 0
, for
. Assume any rl > 0
of diameter l e s s t h a n
F o r a t l e a s t one o f these s e t s
Gr
Ak
r/4
A(Gr)
> 0.
According t o
t h e r e e x i s t s a sequence o f
so t h a t
we have t h e r e f o r e
9 . 2 . REGULAR AND IRREGULAR SETS < (1 + Q) A ( G r
6(Ak)
L t x e Ak at
x
1
nd l e t
(7 Gr
containing 6(c(X))
Ak.
&(Ak)
6
C(x)
249
Ak)
be t h e minimal closed c i r c l e c ntered
We have
(1 -t
<
Q) A ( G r
(1
Ak)
6
(1 + rl) h ( G r
SO
D(A,x) we have D(A,x) = n(B,x) , and We can s t a t e t h i s more generally.
THEOREM.
9.2.3.
nuch ,that
E
each. p i n t
x
=
0 Ek
06
L c L IEk)
A &o
each Ek
06
-
= D(B,x)
a t almost each
x E B
bc a dcljucnce od meawlable bC.65
6inLf-e
i\-meaAme.
Then
at
dmob,t
we have
Contrarily t o what happens with the two-dimensional Lebesgue measure, t h e r e e x i s t A-measurable s e t s E such t h a t the d e n s i t y D(E,*) e x i s t s and i s s t r i c t l y between 0 and 1 a t each point of a subset o f E of p o s i t i v e measure This makes t h e whole theory more i n t e r e s t i n g and a l s o more complicated.
.
250
9. GEOMETRY AND LINEAR MEASURE
9 . 2 . 4 . DEFSNTTTON. Let E be measurable. A point be called a treguRatr p o i a t of E I f D(E,x) = 1 . Otherwise x
x e E will
i s said t o be an i m e g u h t p o i a t of E . A s e t E i s c a l l e d treguRan when almost a l l of i t s points a r e regular. A s e t E i s s a i d t o be i m e g u h t when almost a l l of i t s points a r e i r r e g u l a r . By Theorem 9.2.3. t h e b e t a 6 tregul?atr p o i n t h o d E a xeg.guRuJc b e t und t h e b e t 06 L m e g d a n point2 b an LmeguRcv~A & . Both s e t s a r e disjoint.
So the study of the s t r u c t u r e of E can be reduced t o t h e study Regular and i r r e g u l a r s e t s have sharply contrasting geometric, p r o p e r t i e s , as we s h a l l see. of regular s e t s a n d o f i r r e g u l a r s e t s .
Before we go on t o consider some of them we prove a useful extension of t h e covering theorem 9 . 1 . 1 . 9.2.5. THEOREM. L e A E be any meabutrabLe set. Fotr each x e E Let {Hk(x)} be a bqUenCe a4 rneccnutluble ne,t% con,taining x , conttucLing t o x ( i . e . 6 ( H k ( x ) ) 0 ) , aMd nuch Ahat -f
with
c1
independent o h
X
.
Le,t
E
> 0.
Ptrood. By Lemma 9.1.2, i f we consider only s e t s Hk(x) t h a t a r e s u f f i c i e n t l y small, i t w i l l be s u f f i c i e n t t o prove ( i ) . We know t h a t c1 f o r any ri , 0 c n < 4 , we have a t almost each x e R 2 , for any sequence of s u f f i c i e n t l y small c i r c l e s c k ( x ) contracting t o x and centered a t x,
9.2. REGULAR AND IRREGULAR SETS
Therefore, CHk(x)1
f o r almost each
s o t h a t , if
containing
we can s e l e c t a sequence
i s the smallest c i r c l e centered a t
C,(x)
(and t h e r e f o r e
Hk(x)
x B E
251
7 1
B(Ck(x))
6
6(Hk(x))
x
c b(Ck(x)),
We a p p l y t h e c o v e r i n g lemma 9.1.1 and o b t a i n a d i s j o i n t sequence
A(E
{ c k j so t h a t {Hk}
-
Ifwe t a k e t h e c o r r e s p o n d i n g s e t s
I J C ~ ) = 0.
we have
A(E
-
IJ
Hk) =
A ( E (1 ( IJ(Ck
We can t a k e a f i n i t e number o f s e t s
A(E
-
hl 0 Hk)
+ *- 2) 1
M.
We can r e p e a t t h e process w i t h
El
1
1t
AA(E)
A(E) =
=
=
so t h a t
Hk
1 t q - g 6
- Hk)))
E
-
hl
11 Hk 1
way we o b t a i n f i n a l l y t h e statement o f t h e theorem.
,
A
1
and so on.
In this
252
9. GEOMETRY AND LINEAR MEASURE
9.3. TANGENCY PROPERTIES From t h e d e f i n i t i o n s i t i s easy t o show t h a t i f
S
i s a recti-
f i a b l e curve (more p r e c i s e l y , t h e graph o f a r e c t i f i a b l e c u r v e ) o f f i n i t e l e n g t h , then
i s measurable and i t s
S
l e n g t h . On t h e o t h e r hand i f
A-measure c o i n c i d e s w i t h i t s
i s such a curve, x
S
which i s n o t an e n d p o i n t o f t h e c u r v e and c e n t e r e d a t x we have
when
C(x)
have
i s s u f f i c i e n t l y small.
O(S,x)
= 1 at
C(x)
Therefore x e S.
a l m o s t each
i s one o f i t s p o i n t s
i s any c l o s e d c i r c l e
D(S,x)
>,
1 and so we
Hence we have t h e f o l l o w i n g
result.
9.3.1. U
THEOREM.
E v a y tLecti&iabLe cuhve
06
dinite LengRh 0
hegdUh b d . We know t h a t a r e c t i f i a b l e c u r v e has a t a n g e n t a t a l m o s t each
o f i t s points.
We s h a l l see t h a t i n many r e s p e c t s a r e g u l a r s e t behaves
l i k e a r e c t i f i a b l e curve and an i r r e g u l a r s e t i s something c o m p l e t e l y opposite t o a r e c t i f i a b l e curve.
We s t a r t by showing t h a t , even i f we
widen a l i t t l e t h e n o t i o n o f tangent, an i r r e g u l a r s e t has a t a n g e n t a t almost none o f i t s p o i n t s .
9.3.2. such t h a t
PEFTNTTIUN.
6(E,x)
> 0.
unique s t r a i g h t l i n e the density a t
x
E
Let
The tangent dt x t o
E
E
-
R(t,E)
shaded c l o s e d s e c t o r i n t h e f i g u r e below.
be
i s d e f i n e d as t h a t
t, i f i t e x i s t s , such t h a t f o r each,
o f the set
x e E
a measurable s e t and
i s zero, where
E
,0
<
R(t,E)
E
<
i,
i s the
9.3. TANGENCY PROPERTIES
253
We s h a l l now prove a r e s u l t f r o m which t h e n o n - e x i s t e n c e o f t h e t a n g e n t a t almost each p o i n t of an i r r e g u l a r s e t i s i n m e d i a t e .
dobed 8*(0)
u*(x)
&o~t
9.3.3. THEOUEM. Le.2 E be an &egulah A&, a ( 0 ) a dixed angle a d arnpLLtude ct , 0 < ct < T , with v M e x . c d 0 and L e i denote t h e V p p U A i h ? angle. FOR each x e R z leA: a ( x ) and denote t h e ;thaMnlcuXaMn oh a ( 0 ) and a*(O) t o x . Then cd each x e E we have a ( E 0 a(x),x)
Phood.
_ I _
the set o f points
(*)
D(E
0 < sp
, EI +
('I
We can assume x e E
a(x),x)
x
D(E 0 a*(x),x)
+
E
51
s i n ct
t o be bounded. We t r y t o p r o v e t h a t
such t h a t
+ 6(E
o
a*(x),x)
1 4
<
sin a
To do t h a t i t s u f f i c e s t o show t h a t i f we f i x EZ
1
< 4 sen ct
then the s e t
E],
E ~ ,0
E ( E ~ , E ~ )o f points
<
€1
x e E
such t h a t
i s o f n u l l measure. Assume i t i s n o t so
of
E(E],Ez)
o f points
x
8
E(E~,Ez)
Let
E(EI,E2,rO)
such t h a t f o r each
be t h e s u b s e t r < r o we
,
254
9. GEOMETRY AND LINEAR MEASURE
have
where
C(x,r)
A ( E f7 u(u) f ? C ( x , r ) )
< E12r
A ( E 17 u * ( x ) f Y C ( x , r ) )
6 ~
2
r
means t h e c l o s e d c i r c l e o f c e n t e r x
easy t o show t h a t
E(EI,E~) and s o some
~
E ( E ~ , E ~ , ~i s~ measurable. )
=
\I
O
Q
A ( E ( ~ 1 ~ ~ 2 , r - o )we ) see t h a t i f q > 0 E ( E ~ , E ~ , ~by~ c) l o s e d convex s e t s
x
And we have
From t h e d e f i n i t i o n o f
t h e r e must be a c o v e r i n g o f
{ A k } such t h a t
A k y l e t us c a l l i t A, such t h a t
We can t a k e a compact subset
and f o r each p o i n t
It i s
E ( E ~ , E ~ ,I J~ ~Z ) , A ( Z ) = 0
E ( E I , E ~ , ~ o ) has p o s i t i v e measure.
T h e r e f o r e t h e r e i s some
and r a d i u s r.
of F
F
of
E ( E ,~E 2 , r 0 ) 0
A
such t h a t
we have, if r < r o
We s h a l l now t r y t o prove by geometric c o n s i d e r a t i o n s t h a t t h i s contra1 s i n CI . d i c t s E l + E~ < T
9.3.
a1,bl E F , al E a*(bl),
We t a k e d(al,bl)
d(a,b)
sup
=
255
TANGENCY PROPERTIES bl E
o(al)
a E F 11 a * ( b ) ,
:
such t h a t
b E F 17 o ( a )
1
T h i s can be done i f t h e s e t i n c u r l y b r a c k e t s i s n o t e m p t y - s i n c e F compact. ( I f t h a t s e t i s empty,what
is
we a r e g o i n g t o do b y t h i s process
we s t a r t now i s t r i v i a l ) . From t h e e l e c t i o n o f
and s i n c e
<
d(al bl)
al,bl
it i s clear that
r o we have
A ( F (1 a(al
A(F
11 o*(bl
Ifwe c a l l 1)
If
C(bl,d(al,bl)),
O1 = R 2 - P I
sible,
a ( a l ) II a * ( b l l
PI the i n t e r i o r o f
a2,b2
E
D e f i n e as b e f o r e
we have
, then QI
(1
P2,02
F,
A(F (1 PI)
O1 I1 F
s
i s compact
,
a2 Q o*(bz)
C(al,d(al,bl))
f1
+
(€1
.
Let
(1
~2)2d(al,b1)
US
b2 E o(a2)
now t a k e , i f possuch t h a t
and c o n t i n u e i n t h i s way as l o n g as p o s s i b l e .
The s e l e c t i o n process can be f i n i t e o r i n f i n i t e . I n any case observe on one o f t h e s i d e s o f t h e a n g l e a. b J j a ( 0 ) p a r a l l e l y t o t h e o t h e r t h e p r o j e c t i o n s a r e d i s j o i n t and so, s i n c e 1 ro we have d ( a . , b . ) &(A) < 0 i f t h e r e a r e i n f i n i t e segments J J a. b T h e r e f o r e i n any case we have t h a t J j‘
t h a t if we p r o j e c t t h e segments
-+
F
-
0 Pj j>l
cannot c o n t a i n two d i f f e r e n t p o i n t s
= F 11
a,b
0
j>l
such t h a t
Qj a E
&(b),
b Q .(a).
9 . GEOMETRY AND LINEAR MEASURE
256
Hence f o r each two points a , b
in
angle w i t h the b i s s e c t o r l i n e of
-
so F
j>l
F
-
II
j>l
Pi
t h e segment
ab forms an
o(0) of amplitude biqger than
and
i s contained in a Lipschitz ( t h e r e f o r e r e c t i f i a b l e ) curve.
Pj
Since F i s assumed t o be i r r e g u l a r , we n e c e s s a r i l y have A(F
-
IJ P . ) = 0. j>l J
We now estimate projections of other s i d e 1"
A(F 0 (
IJ
i>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) <
.
course any s e t
Let us now assume
E
t h a t i s almost a
Y-set i s r e g u l a r .
r e g u l a r . Let cil = S U P I A ( y (XE) : y Since E i s not a Z-set we have cil > 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 <
w i l l mean t h e l i n e through
x
T
in
,r
> 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
<
i.e.
0
t h o s e l i n e s through
[O,T).
e
<
e1,02,
1-1 w i l l mean t h e o r d i n a r y Lebesgue measure on
(e)
E
el
~ ~ C d ( x , e :)
=
o(x,el,B2,r
(d)
e
259
e
direction i s i n E
A, i . e .
A 1
be a measurable s e t ,
x
6
E
and
d k e c t i u n ad CandenhatLon oh &h&t a l i m i t point o f
E II d(x,0)
We sav
that 8 e ( 0 , ~ ) i s a di)Lec.tion ad candenhation 0 6 hecand mdeh oh E at x when g i v e n any t h r e e D o s i t i v e numbers q,p, E t h e r e e x i s t r, 0 < r ip, and elYe2, w i t h o s a1 < 0 < e 2 < ~ r , 0 2 < E such t h a t
We*say t h a t
x
b a hadiCLtion p o i n t ad
t i o n i s a d i r e c t i o n o f consideration
E
at
of
1-I-almost each d i r e c -
I7 o(x,01,ez,r))
i s an i r r e g u l a r s e t we have
E
A(E A(E
0 o(xyely 0 , ) ) =
E
when
f i r s t o r second o r d e r ) o f
x. Observe t h a t i f
A(E A(E
E
(of
=
17 a(x,B1,82))
and a n a l o g o u s l y
1 ) o(x,B1,02,r))
since the intersection
w i t h any s t r a i g h t l i n e o r c i r c u m f e r e n c e i s o f z e r o measure. Observe a l s o t h e f o l l o w i n g way o f o b t a i n i n g t h e s e t o f d i r e c
t i o n s o f condensation of t h e second o r d e r t h a t w i l l be u s e f u l f o r t h e p r o o f o f t h e n e x t theorem. set
H(x,q,p,~)
0 < r < p,
and
If q , p , ~ a r e t h r e e p o s i t i v e numbers t h e
of d i r e c t i o n s
a1,e2,
0 c
A(E (1 a i s c l e a r l y an open subset o f
el
c1
f ( 0 , ~ ) such t h a t t h e r e e x i s t s
<
c1
<
e2
<
TI,
e2 - e l
<
E
so t h a t
r,
-
260
9. GEOMETRY AND LINEAR MEASURE
is
p-measurable and i s p r e c i s e l y t h e s e t o f d i r e c t i o n s o f condensation
E
o f second o r d e r o f
at
x.
Our n e x t goal w i l l be t o p r o v e t h a t a l m o s t each p o i n t o f an E
irregular set
1emmas. 9.4.3.
i s a r a d i a t i o n p o i n t of
LEMMA. -
0 6 R 2 . Then t h e h u b h d 06 joining x t o t h e pointn a
Phood.
06
Then,by
A(E) = 0.
x
projecting from
to x
E
is
the covering
A*(E) = 0 , we e a s i l y see t h a t t h e s e t o f d i r e c t i o n s determined
defining x
[Q,T)
Assume f i r s t t h a t t h e d i s t a n c e from
p o s i t i v e , and t h a t by
F i r s t we s t a t e two easy
b e a r n c a n w m b l e b a and x any p o i n t 06 ~ c de;tenminc?d ~ by a%& o f i n e n ~ E , a # x , a v - r n U u h a b l e O&.
E
Lct
E.
and t h e p o i n t s o f
E
is of
v-measure zero.
I f we have h(E)=O
we o b t a i n t h e same r e s u l t by c o n s i d e r i n g t h e subsets 1 Ei = { z E E : d(x,z) a 1 i = 1,2,3 ,...
,
If
E
o f t h e statement
a compact measurable s e t , t h e n t h e s e t o f d i r e c t i o n s
, if
,is
x B E
a l s o compact and so w n e a s u r a b l e .
Analogously we can remove t h e c o n d i t i o n Any measurable s e t p l u s a s e t o f z e r o measure. 9.4.4.
n e t 06 d-ihe&an?) p-meuuhabLe.
Phoad,
LEMMA.
determined b y j o i n i n g set
G(x,r)
LeA
to
E
as b e f o r e .
This proves t h e lemma.
be a meanutmbLe and
E
G(x,r) x
B
i s a c o u n t a b l e u n i o n o f comoact subsets
E
a6 CandeVLbation ad
Let
x
for
& h t ohdeh ad
r > 0
the point o f
x
E ax
be t h e s e t E 0- B(x,r)
8
.
TCrcn t h e
x
L$
E
o f directions
-
{XI . The
i s p-measurable a c c o r d i n g t o t h e p r e v i o u s lemma , We have
that m
G(x)
=
I\
n=l
G(x,F)
1
i s t h e s e t o f d i r e c t i o n s o f condensation o f f i r s t o r d e r o f
E
at
x . So
26 1
9.4. PROJECT I ON PROP ERT I ES
i s measurable.
G(x)
9.4.5.
AenasR each p o i n t od a n LW~eguRanb e t E LA
IHEOREM.
a tiadidtion p a i n t . We s h a l l f i r s t prove t h e following lemma from which t h e theorem i s an easy consequence. 9.4.6.
x
8
Let
E
LEMMA
batib6ie.h
c1
. E be an h h e g d a n ne.2, t h e 6oUowing condition:
e (0,~)
Let
be a dihecLion buch t h a 2
Pnood oh t h e Lma. 9.4.6. take e1,e2,
and s i n c e
0 <
G(x) =
Abbume t h a t
el
0
<
n=l
~1
<
e2
<
T
Since 0 (G(x),a ) u , 0 2 - e l < min(E,
1 G(x, E) , we have a l s o a
= T
p' < p
0 we can )
such t h a t
such t h a t
26 2
9. GEOMETRY AND LINEAR MEASURE
We know from condition
(*)
t h a t there exists
If h(E f’r a ( x , e , , e 2 , r ) ) > q 2r(e2 we have t h e statement. Assume
-
8,)
r < pl <
then
c1 8
p
such t h a t
H(x,q,p,~)
and
The s e t G ( x , p ’ ) t’r (el ,e,) i s u-measurable and so we can find an open s e t I ( t h e r e f o r e I i s a union o f d i s j o i n t oDen i n t e r v a l s , I =
\J
i
(@;,$;
))
such t h a t
Let us d i s t r i b u t e following way:
{(@;,I$!)
We then have
We a l s o have ( r e c a l l
9.4.1. ( e )
Therefore, by condition ( * ) and
i n t o two c o l l e c t i o n s in t h e
9.4.
263
PROJECTION PROPERTIES
Hence
Now we know t h a t , f o r each
i
E
8
,
Al
(**I,
and a l s o , from
For each
i
Al
open i n t e r v a l s
let
Y;) c jy containing
(Y'
(m, , m 2 )
(el,O,)
be t h e u n i o n of a l l t h o s e
($;,$;)
and c o n t a i n e d i n
(e,,02)
such t h a t
(***)
A(E
o(x,ml,m2,r))
I t i s easy t o see t h a t t h e
Also i t i s c l e a r t h a t i f is in
H(x,q,p,&)
q 2r(m2 (Y!,Y'!)
J
J =
J
IJ
j
SO
-
ml)
o b t a i n e d a r e d i s j o i n t and
(Y!,Y'!)
J
J
e2
- el
each p o i n t o f
J, by (***),
and so
T h e r e f o r e we have
T h i s means, s i n c e
€I1 < c1
<
02,
and
i s a r b i t r a r i l y small, t h a t
264
9. GEOMETRY AND L I N E A R MEASURE
P m a 6 o6 Theonem. 9.4.5. Let E be an i r r e g u l a r s e t . According t o Theorem 9.3.3. we know t h a t i f we f i x t j l , 0 2 , 0 < e l < e,
Therefore we can a l s o say t h a t a t almost each x e E we have f o r each couple (el ,e,) , e l € Q, e 2 E Q , 0 < e l < e2 < T , e2- e l < & s
Since E
is i r r e g u l a r , t h e function A ( E (1 u ( x , e l ,e2 , r ) )
of
e1,e2, r
i s continuous and so we can omit i n the preceding paragraph t h e condition B Q , 0, a Q . By excluding a subset o f E of n u l l measure we can assume t h a t we have the condition (*) of t h e statement of Lemma 9.4.6. f o r each x 6 E. p(G(x)) = ~ i x, i s c l e a r l y a point of r a d i a t i o n , Assume p(G(x)) < IT . We have t h a t p-almost each point c1 6 [O,T) - G ( x ) s a t i s f i e s D (Gfx), a ) = 0 . If we apply t h e lemma 9.4.6. we have 1.1 c1 e H ( x , q , p , E ) o r DpCH(x,q,p,E), a ) > 0. Therefore p-almost each G(x) i s i n H ( x , q , p , ~ ) f o r any fixed q,p,E , point a e [ O , I T ) Since H(x) = 0 (H(x,q,p,c) : 0 < q E M , 0 < p 6 Q , 0 < E B Cp 3 we have t h a t p-almost each point i n t0.r) - G ( x ) i s in H(x) and so p ( G ( x ) 1J H(x)) = IT . Hence p a l m o s t each c1 i s a condensation d i r e c t i o n o f E a t x , and so x i s o f r a d i a t i o n of E . Now i f
-
From theorem 9.4.5. proyection property.
we e a s i l y obtain t h e following important
9.4. PROJECTION PROPERTIES
265
The proof will be straightforward from the following four lemmas. 9.4.8. LEMMA. LeA E be an i m ~ e g d a h.bat. Then, doh u-alm o h t each dineotion e 6 [O,IT) , d m a h t each x E E A nuch t h a t 0 A ad condmhation 04 E at x.
Pnood. Consider R 2 x 10,~)with the measure A x u , By theorem 9.4.5. we know that, for A-almost each x B E c R 2 , u-almost each direction e a [ O , T ) is a condensation direction of E at x . Therefore, by Fubini’s theorem, for p-almost each 8 6 [O,r), A-almost each x 6 E is such that 8 is a condensation direction of E at x. This is the lemma. 9.4.9. LEMMA. LeA E be any memumble beA o a ze,ko m m u h e . Then i,t~ phoyeotion o v a any nLt;ltaigkt f i n e Lb i 06 null meanme.
.
means the projection of K we have that d(.rr(K)) 6 S ( K ) and S O A*( T C E ) ) < A * ( E ) . Hence, if A(E) = 0 we have A(IT(E)) = 0 . Phood
If
r(K)
-.
9.4.10 LEMMA LeA E be a nubbeA oh a m m w l a b l e h e 2 A, and Re2 I T ( E ] be Rhe a h t h o g o d ! phoyect.ion 0 4 E ove,k Ox Ahhwne t h a t each p a i n t a E n(E) A phojection 0 6 i n ~ i n i t d ymany poivztn 0 6 A Then A(T(E)) = 0
.
.
P m o 6 . We can assume that A is in the unit cube i(x,y) : 0 6 ,x i1 , 0 6 y < 1 1 , Let N be a natural number, We partition Q into 2N dyadic strips parallel to Ox, Set
Q =
for r
= 0,1,2,
Let
0 im
...,2N -1. E(r,N)I
=
E
f’r
S(r,N) , A(r,N)
= A f’l S(r,N)
.
For
a 2N -1 let Tm be the collection of a 1 m-tuples of nteger
numbers
9. GEOMETRY AND LINEAR MEASURE
266 k = (kl,k2,
...,k m )
, 0 c k l < kz <
... <
k,
c 2 N -1
We define m
t h a t i s , a p a i d P 0 6 [0,1) h i n F(m,N) when P h phvjeOtia~ad CLt L m t m p a i n t b 0 6 A LoccLted in m didde,tent bxXLpb ad t h e 2N we have.
cRecvtey have
Sa we
n(E)
c
m
m
f)
m=l N=[liog,m]tl
Let us estimate t h e measure o f description of F(rn,N) , t h a t
F(m,N).
F(m,N) We have c l e a r l y , from t h e above
2N- 1
and i n t e g r a t i n g
Therefore Observe t h a t and so
Therefore
c 1
A(F(m,N))
F(m,N)
A(n(E))
A(A)
c F(m,Ntl)
=
0
LEMMA, L e t E be a nubbet 0 6 a rneabu.tabRe b & A, t h e v e h t i c d dine&an 0 a dLteOtion and aSbUe t h d t doh each a 6 E 06 conden5at;ian 0 6 becond a h d a 0 6 E at a . L e t n(E) be t h e 9.4.11.
9.4.
26 7
PROJECTION PROPERTIES
The easy n r o o f o f t h e Lemma Dresented here i s due t o R.Moreno.
P/rvad. {(x,y)
I.
: 16 y < 2
Let
a 6 E
k
Therefore,
)
if
with
4 i <
Ik =
n(
denendent o f A,a,k,
0
.rr(a)
i t s p r o j e c t i o n over
ei; ,
'di - O i
J. 0 ,
Ox.
sectors
rkJ. 0, such t h a t
( a k , e i Y e i , r k ) ) , we have w i t h a c o n s t a n t
We e a s i l y see t h a t n(E)
i s contained i n the s t r i p
t h e r e i s a sequence o f
~r/2 <
A(Ik) 6
V i t a l i ' s lemma t o
and
q > 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) <
If E
A
f(Kj)) +
IJ
{An?
6(f(An)) c
, and i f
(
=
of i n c r e a s i n g compact s e t s w i t h A(f(E)) =
Then i f
and
f o r any
i s measurable, t h e n
.
0
=
therefore
i s i r r e g u l a r , we c o n s i d e r
E
L e t i t s r e g u l a r p a r t be
,
A ( f ( B ) ) = 0.
and
A(B) = 0
f ( E ) , We know t h a t
f(B)
Then
B
i s reaular
Therefore
f(E) i s
ir r e g u l a r . The Theorem
9.5.3.
a l l o w s us e a s i l y t o see t h e e f f e c t t o
changing t h e p o l a r i t y c i r c l e i n t h e d e f i n i t i o n of p ( x ) of t h e p o i n t s
taking polar lines we t a k e them, p * ( x ) ,
conic
x
of
A
a(A)
.
If instead o f
w i t h respect t o
C(0,l)
w i t h r e s p e c t t o any o t h e r f i x e d non-degenerate
C* , t h e new s e t D*(A)
=
\J
xaA
p*(a)
i s obtained from
o ( R j by
a nondegenerate n r o j e c t i v e t r a n s f o r m a t i o n and so, by an a p o l i c a t i o n o f
Theorem
9.5.3.
we e a s i l y see t h a t t h e Theorems
serve t h e i r v a l i d i t y i f we change p
by
9 , 5 . 1 . , 9.5.2.
nre-
p*.
A l s o we see t h a t t h e p r o j e c t i o n r e s u l t s o f t h e p r e v i o u s s e c t i o n can be f o r m u l a t e d more g e n e r a l l y . F o r example, by means o f a p r o j e c t i v e t r a n s f o r m a t i o n we g e t from Theorem
9.4.7.
the following fact.
L&t E be a n y LwegguRah n&t. L e X r be any n,ikaigkt .&fie. Then dvh A - d m v A t e a c h p a i n t x 6 r t h e p o j e c f i v n v d E @om x aveh any aXheh f i n e nvA: panbing t h o u g h x 0 04 zehv 9.5.4.
A-rneuuke.
THEOREM.
9.5.
271
SETS OF POLAR LINES
The n e x t two r e s u l t s a r e easy a u x i l i a r y lemmas w i t h which we s h a l l p r o v e Theorem
9.5.2.
Some o f t h e computations a r e e a s i e r w i t h
t h e f o l l o w i n g remark. x
If
orthogonal t o
A*
if
C(0,l)
-
CO3
q(x) the l i n e then
A
i s so,
q(x).
x BA
x Observe
-
1Q1
b y an i n v e r s i o n w i t h r e s p e c t t o
, t h e n by a p p l y i n g Theorem 9.5.3. A*
through
II
q(A) =
i s bounded and c o n t a i n e d i n R 2
such t h a t
i s t h e s e t o b t a i n e d from
i f and o n l y i f
A
we s h a l l c a l l
A c R 2
If
A
that, f o r a set
- {!)I
R2
6
Ox.
A
we see t h a t
i f and o n l y i f
A(A) = 0
i s measurable
A(A*)
= 0
and
i s r e g u l a r o r i r r e g u l a r according t o the r e g u l a r i t y o r i r r e g u l a r i t y
of
A*
.
, observe t h a t p ( A ) = q(A*)
Furthermore
T h e r e f o r e Theorem statement s u b s t i t u t i n g
LEMMA.
9.5.5.
U iuah
~1
p
w i l l be proved i f we prove t h e same
9.5.2.
by
q.
LeR:
U
be t h e clon ed heotangle
eR2 : a - a h x c a
= {(x,y)
,0
> 0, ! I< D
< d <
D
.
,a
t
> 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 <
r > 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))
<
m
.
constructed a regular set
E
such t h a t
9.6. SOME APPLICATIONS. We s h a l l now show how t o use some of t h e p r e c e d i n g theorems i n o r d e r t o o b t a i n c e r t a i n r e s u l t s connected w i t h t h e t h e o r y o f t h e preceding Chapter 8. F i r s t o f a l l we s h a l l g i v e a v e r y easy c o n s t r u c t i o n o f a Besicovitch set.
9 . 6 . SOME APPLICATIONS
277
Figure 9.6.1. 9.6.1.
Cvb%i%ucaon oh a R e n i c o v ~ c hnet.
Fo& t h e ebbed
uvLit syuahe Q o , Let Q1 = ~ ( Q o ) be the. union 0 6 t h e dout bhuded To each one a() t h e h e h o w l bquuten dyadic cLobed bquafieh 0 4 F i g . 9 . 6 . 1 . we appLy ,the hame opehaiion 0 LeL Q 2 be t h e union oh t h e a2 cloned
.
bquahen a h i n i n g i n tkis Way , Q2 C 0, C 0 0 . And h a on. 1 ~ 2 K = Q1 0 Q 2 0 Q 3 ... Then K A a n LV~eguRahn e t o h p o h U v e m e -u me.
Let
$ : R2 + R2
be t h e I;o.Uowiny ,OLan~dom&on
:
i~ a&# in$ ( x , y ) =((1+ x ) cos 2 n y , (1 + x ) sen 2 n y ) Then &egsLLedh and p ( $ ( K ) ) = B A a b e t 0 6 a3dv-dOnenniond meau&e zeho t h a t cvnttai~d CLt L e u t vne f i n e i n each di/recLion. $(K)
The s e t K i s comnact and, usinq t h e natural covers f o r K, i , e , t h e squarerof Q , we see t h a t A ( K ) s L?. Therefore K i s j measurable . Since i t s projection over Ox i s of length 1, we have A ( K ) > 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 <
Since k(x)
m .
is radial we can define
if
2-N-l < k(x)lxl-'
=
, Consider, as before, with f 2 0,
with a independent of j On the other hand
with
Rype
-f
and kN such that
kN (x)
auch
otherwise
c
2N
f,
10. APPROXIMATIONS OF THE IDENTITY
286
1
' Where
b
JBj
j
i s independent o f
maximal oDerator.
and
f ( x - Ey)dy 6
f , and M
bMf(x)
i s the Hardy-Littlewood
Hence
But
and so
10.2.
K*f(x) 6
cMf(x).
We t h u s g e t t h e r e s u l t .
KERNELS NON-INCREASING ALONG RAYS. When t h e a p p r o x i m a t i o n k e r n e l
niajorization i s not i n
L'(R")
, one
k
i s n o t r a d i a l and i t s r a d i a l
can s t i l l g e t some general p o i n t w i s e
k . One of t h e r e s u l t s Coifman and i t s p r o o f u t i 1izes t h e
convergence r e s u l t s w i t h s u i t a b l e c o n d i t i o n s on i n t h i s d i r e c t i o n belongs t o
R.
r o t a t i o n method as f o l l o w s .
x
10.2.1.
THEOREM.
Let
k
8
L1(Rn)
each U L t h 1 i 1 = 1, .the ~ u n o t i v nv b cheaing i n r ~ L t 6ume h c1 independent
r 06
, k a rl , be Ouch tha-t doh > 0 , k ( r x r-' i n nunin2
. Then t h e maximal
287
10.2. KERNELS NON-INCREASING ALONG RAYS
Let
T- f ( x ) = Y fixed.
c
TY
k ( p j ) f ( x - E p y ) d p d?,
Ipln-'
&>O
for
-a
Assume t h a t f o r each
operator
with
m
sup
y
C+
E
fixed
, we
7
e Ct
can p r o v e t h a t t h e
satisfies.
independent o f
j . Then, b y M i n k o w s k i ' s
integral inequality
Ilfll p
c*
and t h e theorem will be proved. Now, i f to
7 , we
x
= z t
sy
s
6
R ,z e Y
,
Y hyperplane o r t h o g o n a l
can w r i t e
I n the d e f i n i t i o n o f
i s nonincreasing i n
TY
p,
we observe t h a t
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 <
m
, f
6
K*
s a t i s f i e s the ineauality,
L ~ g +L
T h i s i s a consequence o f t h e f a c t t h a t
I1 K * f l l
pcl C
II f l l
for
l < p < 2
and o f t h e method o f e x t r a p o l a t i o n . By c o n s i d e r i n g t h e l e v e l curves o f a k e r n e l t h a t i s non-increas i n g a l o n g r a y s and u s i n g t h e Theorem
3.8.1.
on summation o f weak tyDe
i n e q u a l i t i e s one can g e t t h e f o l l o w i n g u s e f u l a p p r o x i m a t i o n theorem, which belongs t o
M.T.Carrillo
10.2.2.
n i n g d o n g hayn.
[1979]
.
- Ltehid t t heeL’( THEOREM.
Annume &o
k
b&
Rn)
,
k
> 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/ <
and
a
-02
(1,l).
i s o f weak t y p e
K*
l i k e w i s e t h e set;
.
We s t a t e another two a p p l i c a t i o n s . T h e i r p r o o f s a r e l e f t as easy e x e r c i s e s .
10.2.4.
and k ( x , y ) type
=
(x’
H = {(x,y)
APPLICATION, LeL f
y’)
-F
,1<
xH(xyy)
6
a <
R2 : lxyl
c 1)
Then K*
2.
oh weak
(151).
convex
10.2.5. APPLICATION. LeL 1 C . I be any hequence 06 bounded J c o ~ ~ t ~ r t ti hn e5 o h i g i n and huch t h a t I C . I = 2-j . Le,t
h&
J
m
k(x) =
1 x
cj
1
(.x).
Then t h e m a x h d apm.atoh K*
c o ~ e ~ p o n d i nt go
k
F o r a k e r n e l t h a t i s t h e p r o d u c t o f another two f u n c t i o n s k(x,y)
[1979])
= g(x) h(y)
.
10.2.6.
( x l ,x2)
B
we can s t a t e t h e f o l l o w i n g r e s u l t s ( M . T . C a r r i l l o
THEOREM.
Rn1+n2 u h a e
bin5 d o n 5
hayh.
that
1
-m
aj
+ llg
k ( X I y X 2 ) = g ( X i ) h(X2)
Ahhume Ah& t h e
m e bounded and convex +m
Lt-t
g e L (R”)
a.1 < J
. m
LeL
,
+m
1
-m
,h
,9
>, 0, h >, 0,
and both me nonincten_
B L (R”)
h&
a j = 2’
!Aj\ t
bj l l g
,
bjl <
b. =
J
m
.
2j I B . ] J
and anmme
Then t h e m a x h d
292
10. APPROXIMATIONS OF THE IDENTITY
The proof follows t h e same idea of Theorem
19.2.2. and i s l e f t
as an e x e r c i s e . By means of t h i s Theorem one e a s i l v sees t h a t , for example, t h e maximal ooerator K* of
i s of weak type ( 1 , l ) . Shapiro [1977] and Di t z i a n [1977]
have obtained previously
some s l i g h t l y l e s s general r e s u l t s of t h e type of
10.2.2
and 1 0 . 2 . 6 .
I t i s s t i l l an open question t o find out whether any o o s i t i v e kernel k a 0 nonincreasing along rays and in L’ defines a maximal operator K* of weak tyne ( 1 , l ) .
10.3. A THEOREM OF F. ZO. The theorem we present here r e q u i r e s l e s s s t r i n g e n t conditions than t h e theorem of Calderbn and Zygmund. The technique of proof i s based on t h e decomposition lemma o f CalderBn and Zygmund we have in 3.2.7. The approximation theorem 10.1.1. i s an easy consequence of . Z o ’ s theorem 119761. 10.3.1. L’(Rn)
duch t h a t
THEOREM.
Let
(kO1)asI
be a ~um,Zy 0 6 ~ u ~ c t i ui m n
10.3. A THEOREM a
) =
sun
s
cp <
I
a
ka(x-y)
-
kcl(x)\
sun
CtE I
lkcl
to
1-1 > 0
f
obtain
and
X
> 0.
6
R',
0 6 weak t u p e
i s t r i v i a l , since
(m,m)
, assume
f > 0,
We a o n l y t h e lemma o f Calder6n-Z.vqmund (3.2.7)
, where
IQj3 , a x B G
almost each
.then
, K*
f(x)
x
I n o r d e r t o Drove t h e weak t y p e ( 1 , l )
,
,
, w a h cp independent ad y
The s t r o n g t y p e
f 8 L1(Rn)
293
.
x) =
Phaa6.
OF ZO
1-1 w i l l be c o n v e n i e n t l y chosen l a t e r . We
sequence 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 such t h a t a t =
l-Ic
(I
0. J
, f ( x ) c 1-1
*I
4j
,
and
f = f. s J
27J
Define
Hence Thus
K*g(x)
G
v c l ( Z n + 1 ).
We choose
such t h a t
wc1(Zn + 1) =
7. x
10. APPROXIMATIONS OF THE IDENTITY
294
Observe t h a t
1
We can w r i t e
supp b
b(x)dx = 0
Qj i n t e r v a l concentric w i t h
Q
j
A1 so
But
y
j
i s the center o f
Q
thus
j .
J
Hence
Let
Gj
be t h e c u b i c
and o f s i z e f o u r t i m e s as b i a as
Now
where
c G.
Qj
- -
,G
= Qi.
10.3. A THEOREM OF ZO
295
Therefore
Thus
K*
i s of weak type (1,l)
1 e L1(Rn) i s nonnegative
Observe t h a t , i f and such t h a t f o r x # 0 ,
Then t h e family of t h e theorem.
, of k’(Rn - COI)
( l E ) E , O , l E ( x ) = ~ - ~ l ( : ) s a t i s f i e s ( i ) and (ii) In f a c t .flE(x)dx = / l ( x ) d x and
-n
Whis t h i s remark , t h e following Corollary i s easy. THEOREM. -
10.3.2.
LeL
. LeL
C
1 e L ’ ( R n ) f l %&’(Rn- l o ) ) L1(Rn) be ouch t h u t
,
1 > 0
/k(x)I
l(x)
and
Vl(X)l
Jhen
K* h 06 weak t y p e ( 1 , l ) 06 n h u n g type 1 < a < m, and thehe4 I k = 1 , 6vh each f e L p , 1 f D < a, lim k E * f ( x ) = f ( x ) ,
6vhe
6
~
I x p
k
6
G
a.e,
one. with
For
1x1
The Theorem 10.1.1. i s now an easy consequence of t h e l a s t k E L’(Rn) , k 2 0 , k r a d i a l and k(x) nonincreasing
, we d e f i n e ,
if
*
k(lx1)
=
k(x)
10. APPROXIMATIONS
296
OF
THE IDENTITY
10.4. SOME NECESSARY C O N D I T I O N S ON THE KERNEL TO DEFINE A GOOD APPROXIMATION OF THE I D E N T I T Y .
K*f(x) =
Let
k e L’(Rn)
sup
\kE
E>O
*
, as
and c o n s i d e r
.
f(x)\
Assume t h a t
before
K*
, the
operator
i s o f weak t y p e (1,1)
What can be deduced about t h e k e r n e l k ?
This i s t h e type o f auestion
we a r e g o i n g t o handle i n t h i s S e c t i o n .
The f i r s t s e t o f theorems,
10.4.1.
-
10.4.4.,
belong t o
to
800
.
[1976]
huch thcLt
E
j
.
4 0 and
E
sup I k E
f o r each f o r each
j
K* A 06 weak .type (1,1)
Phuud.
big
T
Assume 3
0
.
sup.
that
*
~ be l a~ Aequsrzce = ~
\
f(x)
. Then
1x1 I k ( x ) /
ess.sup.
there exists
x e E we have
belongs
-f
j
ess
10.4,5
m
K*f(x) =
A A A W ~t h a t
and t h e y a r e e x t e n s i o n s
The l a s t theorem
Led k e L (R’) . L e L { ~ ~ /+ E j~ 1. Let un w m . e
THEOREM.
10.4.1.
M . T . C a r r i l l o [1979]
Boo [1976]
o f p r e v i o u s theorems o f
~1
1x1 l k ( x ) l =
E = E(T)
c
R1
40
,
T h i s means t h a t
I E ( > 0, such t h a t
10.4. NECESSARY CONDITIONS
E
.
.
E c (0,m)
We can assume t h a t
297
L e t us t a k e a d e n s i t y p o i n t
x06 E
We have
IE
1i m
r+O
(1 B(xo,r)
I B ( x o, r ) I
I
= I
T h e r e f o r e t h e r e e x i s t s r o > 0 such t h a t r o< x o and 3 ( E f1 B ( x o , r o I I > (B(xo,ro)l L e t us s e t E* = E 0 B ( x o , r o )
1e t
We can f i n d a number
and S O
1,
Let
Ej =
If
x
E
0 Im+l # fi
E~
E
j
n o such t h a t , f o r
E*
and
j > no
F
.
m > n,,we
have
Therefore
m
=
Ej.
(I
j=no
, then
$- 6 j
E" E
and so
Therefore
We s h a l l i n m e d i a t e l y prove t h a t s h a l l have
IF1
>,
3
E
n0
(xo + r o )
and so we
and
of
298
OF
10. APPROXIMATIONS
But, i f
i s o f weak t y p e (1,l) t h e n we n e c e s s a r i l y have, a c c o r d i n g
K*
t o Theorem
4.1.1. ICx :
f o r each inequality
THE IDENTITY
A, > 0
(*)
with for
\ k E ( x ) ] > XI1 j j
c > 0
This contradictsthe
s u f f i c i e n t l y b i g , and so t h e theorem i s proved.
T
t h e f o l l o w i n g way. We f i r s t
{TI
&
IF\ >
choose
From t h e i n t e r v a l s
,. . . ,Js}
A.
independent o f
I n o r d e r t o see t h a t
seauences {JI
C
T;
c
SUD
E
no
> no
p
( x o + r o ) we Droceed i n
so b i g t h a t
Ino, Ino ,+ ..., l Ipwe can choose two each o f them of d i s j o i n t i n t e r v a l s ,TV}
such t h a t
T h e r e f o r e , f o r a t l e a s t one o f them, say f o r t h e f i r s t one, we have S
C
i=1 Each
Ji
contains a set
1
IJjI Ei
I
7
=
g
n0
such t h a t
m
3
p
JI
m (IJ I,n0
The f o l l o w i n g r e s u l t
I
[I
P+ 1 i s an
(Ei)
m
1
>
3 > 16
3 4
E
I
IJi
(xo
no
+
and so
ro)
.
n-dimensional e x t e n s i o n o f t h e
p r e c e d i n g one. The method o f p r o o f i s analogous and w i l l be omited
10.4.2. THEOREM. LeR: k E .C 0 and a hequence A O thak j
E E
L'(Rn) ~
,n
> 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
<
m
THEOREM.
Ik, * f(x)l
H(y) Abnume t h a t
K*
doh each h
0
.
k e L’(R’) be cvrztinuvun and Adoume . t h a t K* A v d weah t y p e ( 1 , l )
Lel
Leit
k
nnd doh =
sup r>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<2-s positive
measure EE(s)
EE(s) = { x : 2-s-l
< 2-’ ,
< 1x1
\kE(-x)l >
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
<2-s k , s ( x ) ( d x = X
On t h e o t h e r hand m
And so
l@E(fE)l
-f
m
as
E
+
0.
T h i s concludes t h e o r o o f o f t h e theorem,
CHAPTER 11 SINGULAR INTEGRAL OPERATORS
One of the most basic operators in Fourier Analysis i s t h e Hilbert transform, formaily defined through
The i n t e g r a l has t o be understood in t h e sense of Cauchy's principal value. One of t h e main reasons f o r t h e imDortance of t h e operator stems from t h e r o l e i t plays in the estimation of t h e p a r t i a l sums of t h e Fourier s e r i e s of a function. I t s type has been studied very e a r l y by Lusin, M.Riesz , Kolmogorov among o t h e r s . This was done i n i t i a l l y mainly through complex methods, as was usual in t h e onedimensional Fourier an a l y s i s p r i o r t o the f i f t i e s . Besicovitch [1923] however was t h e pioneer of t h e introduction of r e a l methods i n t h i s area and very e a r l y he obtained t h e existence of the above i n t e g r a l f o r almost every x , f i r s t when f 4 L2(R1) [1923] purely r e a l methods.
and then
119261 when
f E L'(R')
by
The g r e a t spread of the i n t e r e s t i n real methods i n Fourier Analysis came i n 1952 w i t h t h e now c l a s s i c a l paper of CalderBn and in which they studied t h e n-dimensional analogues of t h e HilZygmund b e r t transform by r e a l methods, The a p p l i c a b i l i t y o f t h e i r r e s u l t s and of t h e i r methods t o t h e s t u d y of p a r t i a l d i f f e r e n t i a l equations o r i g i nated an increasing i n t e r e s t in t h e elaboration of purely real techniques f o r t h e problems of Fourier Analysis. This Chapter i s conceived in t h e context of t h e whole book as a t e s t i n g ground f o r the goodness of some of t h e methods developed in e a r l i e r Chapters. lrle f i r s t show how t h e general r e s u l t s permit us t o obtain w i t h ease some of t h e theorems which, f o r long time, have been 305
11. SINGULAR INTEGRAL OPERATORS
306
considered p r e t t y d i f f i c u l t and s o p h i s t i c a t e d . H i l b e r t transform
, LP-theory
Calder6n-Zygmund o p e r a t o r s .
F i r s t we d e a l w i t h t h e
and p o i n t w i s e t h e o r y , and t h e n w i t h t h e I n t h e t h i r d S e c t i o n we deal w i t h t h e t h e o r v
o f s i n g u l a r i n t e g r a l o p e r a t o r s w i t h g e n e r a l i z e d homogeneity t h a t we s h a l l use l a t e r on i n Chapter 13.
11.1. THE HILBERT TRANSFORM. Let
h(x)
=
x1
for
x # 0
The fhunCCLtc?d H A ~ . w L ~ ~ X . L Z ~ I A @ ~ 1
&
p <
m,
i s d e f i n e d as
form i s d e f i n e d as s h a l l have t o prove. H*f(x) =
sup
(at
E
=
h,
H,f(x)
)
*
for
E
>
I),
let
of
f a Lp(R1),
f(x)
, and t h e H i l b e r t t r a n s -
, whose
l i m HEf
Hf =
, and
e x i s t e n c e i n some sense we
The maximal H i l b e r t t r a n s f o r m i s d e f i n e d as
IHEf(x)l
.
We s h a l l t r e a t s e p a r a t e l y and i n d e p e n d e n t l y t h e L2-theory
.
ll.l.A.
character o f the kernel
H*
By means o f a lemma o f
we Drove t h a t t h e H i l b e r t t r a n s f o r m i s o f weak t y o e ( 1 , l )
over f i n i t e sums o f D i r a c d e l t a s . that
L1- t h e o r v and
The L'-Theory.
The p a t h we f o l l o w i s v e r y s i m p l e . Loomis [1946]
D,
E O '
E O '
the
f o r some
x1
From here, based on t h e d e c r e a s i n g
, according
e a s i l y t h e a-e'. convergence o f
11.1.1.
LEMMA,
LeL
0
, we
easily arrive t o the f a c t
over f i n i t e sums o f D i r a c d e l t a s and
i s of weak t y p e (1,l)
so of weak t y p e (1,l)
-
on R '
to
HEf
T h i s g i v e s us
Theorem 4.1.1. for
a. a R J
f
6
L'(R')
,j
=
1,2,3
,
,...,N und
X > 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<
if
HEf
Therefore Hf
one takes
i s outside
Hf;(-)
i s n o n i n c r e a s i n g on
Ik. Thus we can oroceed
11.1.2. and a r r i v e t o t h e weak t y o e (2,2)
Ik
since the
as i n t h e p r o o f o f o f the oaerator
H*.
11.1. THE HILBERT TRANSFORM H E f , f o r f e Lp , 1 6 p < and H a r e easy consemences
The convergence almost a.e. o f and t h e t y p e
,
(D,P)
of what we have
1< p <
m,Of
311
H*
my
proved a l r e a d y .
We add h e r e a couple of remarks. t h e weak t y p e (2,2) o f
H*
F i r s t , i n the treatment o f
we have f o l l o w e d a p a t h d i f f e r e n t f r o m t h e
one used f o r t h e weak t y p e ( 1 , l ) . One c o u l d be tempted t o t r y t o p r o v e something 1ike
and t h e a p o l y Theorem 4.2.1.
t o o b t a i n t h e weak t y p e (2,2) f o r
But t h e i n e q u a l i t y ( x ) above i s f a l s e ,
and so we cannot have f o r a f i x e d
c <
.
H*
I n f a c t we have t h e e q u a l i t y
m,
L c
-
x2
for big
X
.
The second remark i s t h e i n t e r e s t i n q p r o o f t h a t can be o b t a i n e d
o f t h e s t r o n g t y p e (2,2) of
H
f o r s i m o l e f u n c t i o n s w i t h o u t making use
o f t h e F o u r i e r t r a n s f o r m . I t i s v e r y easy and makes use o f t h e f o l l o w i n g lemma due t o B e s i c o v i t c h [1923]
11.1.4.
a 6 h i g k t endpointd x.
1
,i
Fmi'
= 0,?1,+2,..
.
LEMMA. 06
.
L e L tl , 1 = 0,?1,+2 ,.'. be t h e neyuence t h e dyadic i n t e m & 06 R' 06 Length 2-N and &A be t h e .sequence v d .the,& midpointd. Then we have id
i = o ,il,+2,..,A a n a h b i t m t y nequence
0 4 nvnncgaLiue nWnbe?Ld
With t h i s lemma one e a s i l y o b t a i n s t h e f o l l o w i n g r e s u l t .
11.1.5. TtlEOREM. The. o p e n a t o h H A ad n,7hong .type (2,2) f i n e m combinLttioa v 6 charcaotehinLic d u n o t i v u oh bounded i n t a v a h . Thehedohe AX can be dedined a n L 2 and AX'A ad b a u n g .type (2,2). uua
312
11. SINGULAR INTEGRAL OPERATORS
Phuod.
For a f u n c t i o n
f =
I t i s an easy e x e r c i s e t o reduce t h i s
N
1
aj
j=1
xI. J
one has t o Drove
n e q u a l i t y t o t h a t o f t h e preceding
lemma by s u b s t i t u t i n g t h e i n t e g r a l s b y a p w o D r i a t e Riemann sums.
Phood 06 Lemma s.= i 1
-
1 7 , m.1 > 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
n
mi
m.1 s 1.- 1 ) ( s J.-17 J 1=1
But n
1
1=1 m
and so
i=l
~
(Si
1 < 8 -lIZ
E-
1
On t h e o t h e r hand , i f i < j .
< n 2
i
i=l
1
m.
mf
-
11.2. C A L D E R ~ N n
1
1=1
-
J
c
1
-~
-l)(s.-1)
(Si
313
ZYGMUND OPERATORS
-'j
n l c K] l-si - 1=1 J
n
J1
l
=
T h i s proves t h e above i n e q u a l i t y .
11.2. THE CALDER~N-ZYGMUND OPERATORS. The Calderdn-Zygmund o p e r a t o r s , t h e n-dimensional analoques o f t h e H i l b e r t t r a n s f o r m , a r i s e i n a n a t u r a l way when c o n s i d e r i n q c e r t a i n problems r e l a t e d w i t h t h e d i f f e r e n t i a t i o n o f a Newtonian p o t e n t i a l . t h i s m o t i v a t i o n one can see CalderBn and Zygmund [I19521 The k e r n e l
h(x) =
1
.
For
o f t h e H i l b e r t t r a n s f o r m has s e v e r a l
i m p o r t a n t f e a t u r e s t h a t a r e r e s p o n s i b l e f o r t h e good b e h a v i o u r o f t h e H i l b e r t transform.
First
h(1) +
x # 0
, i.e.
h
,
h(-1) = 0
i s homogeneous o f degree -1.
smooth behaviour,
h
E
e(R1- { O }
).
necessary f o r o b t a i n i n g r e s u l t s i n Rn
.
i.e.
h
has mean v a l u e
h(Xx) = X-'h(x)
Second
z e r o o v e r t h e u n i t sphere o f R1.
Besides
h
, for
A > 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 <
Rn
m.
i i ) it i s sufficient that
O f course, f o r
v e
k
satisfies a
Lipschitz condition
I n f a c t , we t h e n have
, i n view o f t h e homogeneity o f Q1(Rn - I01 )
k
it i s sufficient
The Calderdn-Zygmund o n e r a t o r o f k e r n e l
k
i s now d e f i n e d
F o r c o n d i t i o n (*) t o assume
k
6
f o r m a l l y as
*
Kf(x) = k
f(x)
I n o r d e r t o g i v e a meanino t o t h i s c o n v o l u t i o n one f i r s t c o n s i d e r s t h e truncated kernels
i
k(x)
k,,-,(x)
and, f o r
f e Lp(Rn),
1
=
h
n <
kE,,f(x)
=
m
0
,
if
,
otherwise
, one
kE,T)
*
E
6 1x1
c n
defines
f(x)
The q u e s t i o n now i s whether t h e l i m i t o f
KE,T) f
exists i n
some sense. CalderBn and Zygmund, i n t h e i r now c l a s s i c a l paper o f 1952
,
11.2.
CALDERBN -
ZYGMUND OPERATORS
315
obtained very general s a t i s f a c t o r y r e s u l t s , proving t h e e x i s t e n c e of t h e l i m i t in L p , 1 < p < m , f o r f E L p , and t h e existence of t h e l i m i t a t almost each x 6 Rn f o r f 6 L p , 1 c p < m under some additional conditions. Their r e s u l t s were l a t e r refined by CalderBn, M. Weiss and Zygmund [19671 and l a t e r on by Riviere [19731, who f i n a l l y proved t h a t t h e simple Lipschitz i n t e g r a l condition ( i i i ) on t h e k e r n e l s , together w i t h t h e usual assumptions ( i ) and ( i i ) , s u f f i c e s t o obtain t h e convergence in
Lp
and the pointwise convergence results.
We present here t h i s r e s u l t , following a d i f f e r e n t path t h a t goes t h r o u g h some o f t h e genera theorems we have obtained in previous chapters. F i r s t we s h a l l obtain bv studying t h e Fourier transform of t h e truncated kernel s k E ,Tj t h e uniform boundedness i n L 2 of t h e operThis r e s u l t s eas l y leads t o the e x i s t e n c e of a t o r s KE,TI
.
n - f w
shown i n
From i t we s h a l l o b t a i n , using t h e method of majorization 3 . 6 . , the strong type (2,2) of t h e maximal operator K* Finally we s h a l l show, by means of t h e Calder6n-Zygmund decom-
position lemma of 3.1, t h e weak type ( 1 , l ) of t h e maximal operator K*. From t h i s one e a s i l y obtains the almost everywhere convergence of KE,.,f f o r f e ~ P , 1 c D < m , a s~ + ~ , q + a .
11.2.A.
The Uniform Boundedness i n L 2 of t h e Truncated Operators.
11.2.1.
THEOREM.
i) ii)
1
c
lk(:)/di
For
x # 0
<
l&t k : R n + R m
,
, X > 0,
c
b~ a 6unc;tion ouch t h a t
k(i)di = 0
k ( X x ) = A-’k(x)
3 16
11. SINGULAR INTEGRAL OPERATORS
independent v d ~ , q
Pxaad.
KE,q
Themdohe , dux f
c c
.
<
g
E
@:(Rn)
a = max
IVg(x)l
Kf =
c > 0
proximating converges i n
f
E
x
1
, and
and i n
L2(Rn)
L2(Rn)
K
EYn
f(L2)
such t h a t f o r each
(2,2).
'1
Ik(y)l lyldy = a
dp I k ( ? ) J d.7
E1GIY
so one e a s i l y sees t h a t
L2
as
E +
by f u n c t i o n s
t o a function
,
El
x 6Rn converqes a t each
lim € 4 .n-
we have
6 a
where
,
L2
The P a r s e v a l - P l a n c h e r e l theorem g i v e s
i s uniformly o f type For
E
We w o v e t h a t t h e r e e x i s t s
x E R n , ItF rl ( x ) l t h e n us, f o r f e L 2 ,
i.e.,
.
in
Kf
0,
n
+ m ,
$' :(Rn) and
that
KE
YV
g(x)
T h e r e f o r e , by ap-
one g e t s t h a t
11
Kfl12 c
cII
KEYqf
f1I2
.
Therefore, a l l we have t o do t o prove t h e theorem i s t o o b t a i n that
11.2. CALDER6N h
Ikcdl
(x)I
L
c <
c
with
m
-
For b r e v i t y l e t us c a l l
independent o f h ( x ) = kE
that
for each y,
with
If 1x1
Thus
we o b t a i n
c 2
independent o f 41yl
, we
i f we denote
have
317
ZYGMUND OPERATORS
y.
Yrl
(x).
~~q
.
We s h a l l f i r s t p r o v e
11. SINGULAR INTEGRAL OPERATORS
318
The t h i r d i n t e g r a l , over
with
c
independent o f
S3 , i s t r e a t e d i n t h e same way
E,
fi x = o , t h e n and c o n s i d e r
fi(x)
.
Take
\6(0)l z =
=
.r
kE,n ( v ) d v = 0
X , so -
that
i s bounded
h
.
q
Thus we g e t
y.
We n e x t prove t h a t t h e F o u r i e r t r a n s f o r m o f uniformly i n
.
. Assume
(x,z)
=
1
x f 0
.
I
We can
21x12
write
d,v=
[ I 1 \ c c,
We have 121
IYI
If I z I >
<
as proved before.
For
1121 we w r i t e
, so
that
1
I,j
.j=1
, if
414
1 ~ 1 ,we
take
y1 =
$,- I z I
Iy11 = I z I
.
Thus
C A L D E R ~ N - ZYGMUND OPERATORS
11.2.
319
Finally,
11.2.B.
The Strong Type ( 2 2 ) o f t h e Maximal Operator.
11.2.2.
THEOREM
.
Lel
k
be
, t h e maxim&
Phaaa. -
bv Theorem
I n order t o prove t h a t
3.1.1.
Then,
11.2.1.
M -in
openatoh K*
K*
.id
A 06
i s o f t y p e (2,2)
W M ~
we can,
r e s t r i c t o u r s e l v e s t o n r o v e t h e t v n e ( 2 , 2 ) over a
dense s e t o f f u n c t i o n s .
Let
g
be a 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 i s j o i n t open bounded s e t s whose boundaries have n u l l measure.
For such a f u n c t i o n , if x
i s n o t on any o f t h e boundaries
o f such s e t s , ~ g i x )=
lim K ~ C X ) €,I? P O
e x i s t s and
n-
g(X)
=
KEg(X)
Hence, f o r such
x
- Krlg(x) , there
, where exists
E(X)
K6g(x) = >
0
lim
rl-
such t h a t
Ks,sg(~).
z)dz(
0
y E B (x,
Since, f o r almost a l l w r i t e f o r such p o i n t s
y
1
E ( x ) ) = BE
, Kg(Y)
.
e x i s t s , we can
t h a t t h e l a s t member o f t h e w e c e d i n g i n e n u a l i t v
i f l e s s t h a n o f equal t o
Now we have, i f
and
SO,
since
For
s
=
2
,0
=
M
K
i s t h e Hardy-Littlewood operator,
and
M
a r e of t y p e
(2,2)
,
1 3 g , u s i n g Kolmogorov's i n e q u a l i t y f o r
1 (Theorem 3.3.1.)
we can w r i t e
K
with
p = 2
,
11.2. C A L D E R ~ N- ZYGMUND OPERATORS
321
and so,
The e s t i m a t e f o r d e f ine
[o
Then,if
x
-
z =
NOW, if v = rG
EV
,V
E
Ilg(x)
, if
,
x
-
lvl < 1
y =
c ,r>
EU
,
0, 0
$(v) =
i s a l i t t l e more d e l i c a t e . L e t us
,
+(rS) = Ik(rG)
i.e.,
if
u = rt
r < l
if
,
-
k(ri
-
u)dv
,
if
r 2 1
11. SINGULAR INTEGRAL OPERATORS
322 So
4
1x1 > 1
i s nonincreasing along rays i n I f we d e f i n e
Ik(?)
-
+*(v)= $*(rV)= Ik(rV)
Then
obviously majorizes
$*
$I
,
if
k(ri
-
k ( i - u ) l du
, and
-
IvI < 1 u)ldu
, if
IvI
2
1.
i s n o n i n c r e a s i n g along ra.vs i n
Rn. A l s o we have
I,,,,,I,,,,:
I k(v)-k(v-u)
i,v,,14*
by c o n d i t i o n (iii) on
I,,,,,
+*(v)dv
Thus
I +* <
c
k.
=
And a l s o I k ( c ) I d u dv +
1, 1 V J d
.
( d u dv
=
I k ( E - u ) ( d u dv
\Ul
Now
and we know a l r e a d y by t h e r o t a t i o n method t h a t t h e l a s t o p e r a t o r i s o f t y p e (2,2). Hence o f t h e theorem.
K*
i s o f weak t y p e (2,2)
and t h i s concludes t h e p r o o f
11.2. C A L D E R ~ N- ZYGMUND OPERATORS
323
11.2.C.
The Weak Type (1,l) of t h e Maximal Operator.
11.2.3.
TffEOREM.
A 0 6 weak t y p e ( 1 , l ) Pmo6.
L e t now
w i t h compact s u p p o r t .
1eA k b e ah i n Tkteahem 11.2.1. Then K*
.
be any nonnegative f u n c t i o n i n
f
Let
f
A > 0 and a p p l y t o
and
L1(Rn)
t h e CalderBn-
A
( 9 . 1 of d i s j o n t
Zygmund descomposition lemma, o b t a n i n g a sequence
J
d y a d i c i n t e r v a l s such t h a t
D e f i n e now
and l e t ,
1
f(x) = g(x)
h(x)dx = 0 0, J compact s u p p o r t .
.
f
h(x)
Also
. c
g(x)
A
( L J Q,i)
supp
h
ZnA
a.e.
and b o t h
g
and and
h
have
We have
K*
We a l r e a d y know t h a t if
c
Clearly
i s o f weak t.ype ( 2 , Z ) .
i s any compact s e t c o n t a i n e d i n
gorov's i n e q u a l i t y w i t h
0 =
Hence, remembering t h a t
g(x)
1
,s
{x :
= p = 2
c ZnI
a.e,
K*g(x) >
, 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
each
, nuch
-I03
x aRn
that
IT-1
i n t h e f o l l o w i n g way.
thehe XI =
1
.
u unique
Let ub
bet
P (XI
The duncaXon
t h e doU0eolLling p o p a i i a
:
p : Rn
+
COY-)
bo
dedined b ~ m ! ~ h & L a
330
11. SINGULAR INTEGRAL OPERATORS
, id
Thenedotle
we dedine
,
the^ a m W c i n R n f h c d d invahiant by a%~m&
p* : Rn x Rn + LO,-)
p*(x,y)
=
p(x-y)
-
LLOVlA.
Phoud.
The f u n c t i o n
For each
Ip : ( 0 , ~ )
Therefore, f o r each
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
+(A) i s s t r i c t l y d e c r e a s i n g on ( 0 , a ) . I t i s as X + 0 and +(A) 0 as X m. Hence
Hence
$(I)
m
-f
-+
t h e r e i s a unique v a l u e That
p(Tllx)
p(x) > 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)
<
(iv)
1,
( T -lx( A
i . e . i f and o n l y i f
We have
a r e simDle. I n L
1 i f and o n l v
P ( X ) 6 A.
we o n l y have t o prove t h a t
1
= 1
There-
331
11.3. GENERALIZED HOMOGENEITY
where I( P I( f o r a real n x n matrix P means t h e Euclidean norm of P as an operator on ( R ~, I - \ ) , i . e , l l P I \ = max{(Pxl : 1 x 1 = 11 I t i s not d i f f i c u l t t o show t h a t ( 1 PI1 = max E eigenvalues of P*P) ) 112
.
Now i t i s easy t o prove t h a t i f A has eigenvalues w i t h r e a l p a r t b i g enough (how big depends only on n ) we have, f o r each 1-1> 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 <
(p,p)
1< p
each
f
6
01).
Observe t h a t i f t i v e r e a l p a r t and
E z
*
dedined by
( and
(m,m)
f(x)-+f(x)
t h ~ ~ e 0&6 ~ba%ttlong ~ e type each x e R n 9 doh
at &ont
m.
A
i s any m a t r i x w i t h eiqenvalues w i t h nosi-
0 , >~0, E =
rlH
,
H > 0
we have
T h i s a l l o w s us t o assume t h a t t h e r e a l p a r t o f t h e 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 ;(<)
=
f'
1
having set,
for
y
E
x e ~ n ,
e-2.rri(v,TETfi<)ds
i(~)=
1
n(T;
=
for
Rn
5) ?(
,
2
n(y)
e - 2 n i ( v ,T:Y)
ds
1
Now, proceeding as i n t h e p r e v i o u s a t t e m p t , v i a Schwarz i n e q u a l i t y and Parseval -P1 ancherel theorem, we have
I f we c o u l d prove t h a t f o r each
i s t h e m e t r i c associated t o
A*)
5 with
16
we have for
h >, 1
and for t h e n we c o u l d w r i t e
O < h s l
-
12. DIFFERENTIATION ALONG CURVES
342
1
m
m
,
I f we s p l i t t h e i n t e g r a l
t i e s (*)
and
terms o f
1
(**)
and undo t h e change
T*.q = 2J
5
use t h e two i n e q u a l i -
,in
each o f t h e
we g e t
j
II M f l l
2
n
c
II f l l
c
2
2
b
The i n t e q r a l s o f t h e t y p e
II fll 2’
e2.rri
f ( u ) du
a
( C f . Zygmund [1959]
b y van d e r Corput
c
=
vol.
I,
D.
h i s r e s u l t s we s h a l l be a b l e t o prove t h e i n e q u a l i t y q u a l i t y (**) in if
15
:
1G
i s obviously false, since p*(<)
r ) < h r l ,
where
c
c 2 I.
n(Tt
197)
have been s t u d i e d and by means o f
(*I.The
5 ) r+ 1 as h
-f
o t h e r ine-
0
uniformly
But we have i n s t e a d by t h e mean v a l u e theorem,
1cp*(S)c2,
i s t h e maximum o f t h e a b s o l u t e v a l u e o f t h e g r a d i e n t o f
n
on
the set
T h e r e f o r e we have i n 11.3. between
1.1
- 11
In(T{ < ) and
c hB
G
, by t h e i n e q u a l i t i e s shown
p*.
So we can again t h i n k o f m o d i f y i n g o u r m a j o r i z a t i o n o f
(1 Yf(12
by some o p e r a t o r so t h a t t h e F o u r i e r t r a n s f o r m Nhf preserves i t s behaviour f o r b i g h b u t has t h e d e s i r e d b e h a v i o u r f o r
by perturbing close t o
TO do t h i s A
Q(0) =
h
0.
= 1
,
e t us t a k e a f u n c t i o n
,-.
$(<) = 0
and
x)
for
such t h a t
p*(5) >, 1.
if
h > 0
4
,
8
@:(Rn)
We can t h e n w r i t e , i f
,
12.1.
T h e r e f o r e , if we c a l l
343
THE STRONG TYPE (2,2)
O*f(x) =
*
sup h>O
, we have
f(x)(
2
11 2 Now, a c c o r d i n g t o
1) @*fjI2G
11.3.3, we know t h a t
c)) f
)I2.
To e s t i m a t e
,
p*(T;
5) > 1
t h e f i r s t t e r m we have
Therefore , if A
and so
now
(I
1 6 p * ( < ) 6 2,
+h(E) = 0
,
and f o r
we have 0 < h
Therefore we have m o d i f i e d
llfl12
c
Mf112
-
G
, for
1 G p*(E) 6 2
1,
Nhf
h > 1
,
as we needed and so we o b t a n
L e t us w r i t e t h e scheme o f t h i s w o o f a l i t t l e more f o r m a l l y
Phoad ad Theahem. 12.1.1.
We choose
$(o)
=
$
I$ =
such t h a t 1
. Write
f a
0
.
Consider, f o r
V m ( R n ) , ;(<) = 0 if -tr A +,(x) = h +(T x) and h-
@*f(X)
We know t h a t
Let
6
=
I( @ * f (6( q c I ( f (1
sup
h>O ,
\+h*f(x) We can s e t
I
p*(<)
h > 0,
> 1 and
344
12. DIFFERENTIATION ALONG CURVES
Now
''
l a s t t e r m i s l e s s t h a n o r equal t o
But t h e f a c t o r
1 6 P*(C) 6 2
(*I
F(h,<)
5 )
-
h
+(Ti 5 )
i s such t h a t , i f
Y
F(h,S) =
(**)
\n((Tt
=
F(h,<)
<)I
In(T;
c ch'
with
The i n e q u a l i t y (*)
c - , with ha
B > 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<\=
1
,
s > 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 <
my
> 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<
L e A y ( t ) be u homogeneoun p c
m,
C W L U ~i
n Rn
.
12.2. THE TYPE
,t >
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*(<))'
1
.id
For a homogeneous c u r v e For 1 < n < 2
ah
>
in t h e finean nubbpace bpanned by t h e vectutro
That,
j=i
C U / ~ V Qw
(P,d, P
10.3.1
( Z o ' s theorem)
corresponding t o t h e g e n e r a l i z e d homogeneity, t h e f o l l o w i n g r e s u l t h o l d s . *
12.2.2.
-u
<
Re z <
-3
LEMMA. T h e m exint 0, 0 > 0 buch t h a t , L d < 0 , a n e h a , d a h any meauhabkk ¬ion E : Rn + (0,-)
w i t h a &buXe numben
whem
Ic
P
06
vdeueh,
(211 = O ( l z l H l a
On t h e o t h e r hand, one can prove t h e f o l l o w i n g .
12. DIFFERENTIATION ALONG CURVES
350
LEMMA. -
12.2.3.
w L t h a ~ i n i t enumben -a < Re z
c1
06
LeR
vdueb.
E
: Rn -+ (0,~) be anv meanuhabde ,$unCtivn
Then t h m e ex&&
c1 >
0 nuch t h a t i d
one h a
Using now t h e i n t e r p o l a t i o n theorem f o r an a n a l y t i c f a m i l y o f o p e r a t o r s one o b t a i n s t h e s t r o n g t y p e t h e s t r o n g t y p e (p,p)
, 1< p L
m
for
(p,p)
for
z = 0
and consequently
M.
There a r e many d e t a i l s and t e c h n i c a l i t i e s f o r which we r e f e r t o t h e work o f S t e i n and Wainger E19781. The r e s u l t f o r a
emcurve i s however an easy consequence o f t h e t = 0
p o s s i b i l i t y of a p p r o x i m a t i n g such curves near
b y a homogeneous
c u r v e and o f t h e r e s u l t f o r these t y p e o f curves.
12.3. AN APPLICATION. DIFFERENTIATION BY RECTANGLES DETERMINED BY A FIELD OF DIRECTIONS. In
8.6. (PROBLEM 3 )
q u e s t i o n . For each t?,(x)
x 6R2 l e t
we have encountered t h e f o l l o w i n g t y p e o f d ( x ) e [O,n ) . Consider t h e c o l l e c t i o n
o f a l l closed rectangles containing
direction
d(x)
zontal axis.
i.e.
x
one of whose s i d e s has
forms an a n g l e of ,amplitude
d(x)
with the hori-
Idhat a r e t h e d i f f e r e n t i a t i o n p r o p e r t i e s of t h e b a s i s 6,?
We have a l r e a d y seen t h e r e t h a t such p r o p e r t i e s can be v e r y bad, even i f d
i s a continuous f i e l d o f d i r e c t i o n s .
As we have seen i n Chapter 8, one
can c o n s t r u c t a Nikodym s e t a s s o c i a t e d t o some c o n t i n u o u s that
d.
T h i s proves
Q d does n o t even d i f f e r e n t i a t e a l l t h e c h a r a c t e r i s t i c f u n c t i o n s o f
measurable s e t s , i . e .
i s not a density basis.
I t i s t a n t a l i z i n g t h e f a c t t h a t even f o r t h e a p p a r e n t l y most
s i m p l e case of a n o n - t r i v i a l f i e l d o f d i r e c t i o n s d, namely t h a t o f t h e r a d i a l d i r e c t i o n s , one d i d whetherGd
n o t know any answer t o t h e above q u e s t i o n ,
i s o r n o t a d e n s i t y b a s i s , u n t i l t h e problem c o u l d be handled
b y means o f t h e r e s u l t s o f S t e i n and Wainger.
I n spite o f the strong
12.3. DIFFERENTIATION BY RECTANGLES
351
geometric f l a v o u r o f t h e problem t h e o n l y t r e a t m e n t u n t i l now a v a i l a b l e goes t h r o u g h t h e a n a l y t i c c o m p l i c a t i o n s of t h e r e s u l t s o f
12.1. and 1 2 . 2 .
I n t h i s S e c t i o n we s h a l l f i r s t d e a l w i t h some c o n c r e t e examples and t h e n we i n d i c a t e d how one can a l s o reduce some o t h e r s i m i l a r problems t o t h e r e s u l t s o f S t e i n and Wainger. T h i s t y p e o f a p p l i c a t i o n o f such r e s u l t s i s due t o A.Cbrdoba, C.Fefferman and R.Fefferman
(unpublished).
The f i r s t theorem we p r e s e n t i s a s t r a i g h t f o r w a r d consequence o f t h e r e s u l t o f S t e i n and Wainger. From i t we s h a l l deduce i n a n a t u r a l way a theorem .about a p a r t i c u l a r b a s i s ' 8 d
of r e c t a n g l e s of t h e t y p e
F i g u r e 12.3.1.
P(x,a,b)
LeR
=
C(xl + s 1 , x 2 + s 2 + s:)
dedine t h e 6uUuuing maxim& 1< p < and x e R 2 , L e t UA
Mf(x) =
sup
azl)
openCLtuh M 1
, b>O
: 0 c s1 6 a
IP(x,a,b)l
c
. Fuh
,
f E Lp(
JyeP(x,a,b)
0 ,< s 2 G b3
R2)
I f ( Y ) dY
Then M ~206 ,type (p,p) , 1 < p 6 m . Themdohe doh each f 1 < p 6 m one han dt &ubt each x E R 2
8
LD ,
12. DIFFERENTIATION ALONG CURVES
352
Ptlaoa,
We easily compute
IP(x,a,b)( = ab and also, by a change
of variables b
a
where we have called
But according to Theorem 12.2.1.
and (trivially from the one-dimensional result for the Hardy-Littlewood operator)
Therefore
and the theorem i s proved. Let us now transform the preceding theorem in a trivial way. In R’ let us perform the following transformation
353
12.3. DIFFERENTIATION BY RECTANGLES
Let us observe what i s t h e s e t T ( P ( x , a , b ) ) . We have T(P(x,a,b)) =
=
{ ( x l t s l y xz + s2 + s12
( ( X I + s1 , X 2 t s2 - 2slX11 : O f s 1 6 a
-(XI+
s:)'):
06s16a OrSz
0 6 szc b
i s a s t r a i g h t segment of fixed length and whose d i r e c t i o n i s t h e t o f t h e vector (1,-2X1). Therefore T(P(x,a,b)) i s a parallelogram with one p a i r of vert i c a l s i d e s and the other p a i r i n the d i r e c t i o n o f t h e vector ( l y - 2 X I ) . See Fig. 12.3.2. Observe a l s o t h a t t h e Jacobian of T i s 1.
Figure 12.3.2. Hence j u s t by t r a n s l a t i n g Theorem 12.3.1. we g e t t h e following one.
3 54
12. DIFFERENTIATION ALONG CURVES
Nf(X) =
'T(P(xya9bnl
sup a>o,b>O
1
jT(P(x,a ,b))
I f(Y) I dY
Of course this theorem solves also the differentiability problem such that, for each x e R' ,'A,(x) is for the basis of rectangles 8 the collection of all rectangles containing x one of whose sides has the direction of the vector (1,-2x1) . The same procedure can be used in many other cases. Let us now try to solve the differentiability question for the basis 0,.of rectangles in radial directions.
&
Let us consider now the maximal operators R 1 and Rz defined in and f = T(-x2 , ,XI the following way . For x eRZ(O1, let x' = i,e, ? is the unit vector i n the direction of x and is obtained by rotating 2 around 0 an angle of , I f f e Lp( R 2 ) , x e Rz-tO] we write
If we prove that RI and RZ satisfy
then we would have,
if
R
0gr(.Xl
355
12.3. DIFFERENTIATION BY RECTANGLES
Therefore i f
and so
, 1 ip
N would be o f t y p e (p,p)
T h i s would g i v e us t h e
<09
Theorem. However t h e r a d i a l n a t u r e o f
R1 l e a d s one t o suspect t h a t
near t h e o r i g i n can produce
perhaps some k i n d o f v e r y m i l d growth o f
f
a t o o slow decrease o f
1x1
Rlf(x)
for
f a i l u r e o f t h e s t r o n g t y p e (p,p) function for
,
o f the u n i t disc
1< p 6 2
Therefore
big
.
, if
In fact
f
.l+lxl and so
Rlf(x) >
RI
and t h i s can cause t h e
i s n o t o f t y p e (p,p)
i s the characteristic R7.f
,1<
I$
Lp (R')
.
p f 2
,
N e v e r t h e l e s s e v e r y t h i n g works as expected i f we t r y t o keep away f r o m t h e o r i g i n
, as
we s h a l l now see.
L e t o u r space be
E =
{x
6
R2
: 10
< 1x1 < 100)
we a r e g o i n g t o c o n s i d e r w i l l have s u p p o r t i n E.For
1< p <
00,
l e t us c o n s i d e r , f o r each
x
E
are
p(x)
6
Lp(E)
,
E,
L e t us d e f i n e t h e f o l l o w i n g t r a n s f o r m a t i o n polar coordinates o f x B E
f
The f u n c t i o n s
, 10
T on t h e s e t E . < p(x1
i 100,
and
I f the a(x)
we s e t
If f
i s a function i n
Lp(T(E)) and we have
Lp(E)
and we s e t
g(X) = f ( x )
then
g
is in
356
12. DIFFERENTIATION ALONG CURVES
where
c1 and
cp
a r e a b s o l u t e constants.
Observe now t h a t
Therefo r e
Now we have
Therefore
I n o r d e r t o deal w i t h t h e o t h e r o p e r a t o r l e t us d e f i n e on E the following transformation
, 10
p ( x ) e ia(x)
form x =
V. I f x
Again, i f g
E
Vx = X = l g p(x) t
G
E
-+
f
8
LP(E)
E
X = Vx w i l l be X = l o g p(x) t i a ( x ) , i.e.
and we s e t
ia(x)
g(X) = f ( x )
Lp(V(E)) and
where
E l ,
Z2
> 0 are absolute constants.
For f
Observe now .that
8
Lp(E)
i s expressed i n complex
< p ( x ) < 100, t h e p o i n t
t h e p r i n c i p a l value o f i t s logarithm, x
E
and x
E
E we s e t
for
X = Vx
then
12.3. DIFFERENTIATION BY RECTANGLES
357
Therefore
=
sup
O
2%
I
b2
fiT
1g(X + log (1 t i s ) ) \ ds
=
S;g(X)
According t o Theorem 12.2.1. we have
f E Lp(E)
From t h i s results for R; we s e t a t each x E E
, R;
we easily see t h a t i f
for
The restriction t o E i s of course rather irrelevant. As long as E i s bounded and i t s closure i s away from 0 we get the same Since differentiaresult with different constants for the type ( p , p ) tion i s a local property we can s t a t e the following result.
.
12.3.3.
x x
8
R2
-
-THEOREM.
The b u h
t3
such t h a t , doh each
, @ ,(x)
h t h e c o U e d o n 06 UUh e d n g L e b containing whobe hidides h t h e d i h e d o n 06 t h e ve.ctofi X di66e,kenLikte~
10)
one 0 6 LP(-R') .1
pd"
I t i s s t i l l unknown whether there will be some positive differentiability results for the space L ( l + log' L ) (TIR2) for t h i s basis.
This Page Intentionally Left Blank
CHAPTER 13 MULTIPLIERS AND THE HARDY-LITTLEWOOD MAXIMAL OPERATOR
L e t us c o n s i d e r a bounded f u n c t i o n tion
f
in
L2(Rn)
m E Lm(Rn)
we can d e f i n e t h e f o l l o w i n g o n e r a t o r
F o r any f u n 2 T,
means o f
the Fourier transform
Tmf
" f
(m
=
v
)
C1 e a r l y , by t h e Parseval -P1 ancherel theorem
and so
T,
i s a bounded l i n e a r o p e r a t o r f r o m
11 m l I m .
bounded by
The f u n c t i o n
1c p <
m E La,(Rn)
when t h e o P e r a t o r
m,
T,,,
L2
t o L 2 w i t h a norm
i s s a i d t o be a m u l t i p l i e r on Lp
can be c o n t i n u o u s l y extended t o
and t h i s e x t e n s i o n i s bounded f r o m
Lp
to
,
Lp
Lp.
There a r e some i n t e r e s t i n g r e s u l t s s t a t i n g s u f f i c i e n t c o n d i t i o n s under which
m
i s a m u l t i p l i e r . These c o n d i t i o n s can i n v o l v e , f o r exam-
p l e , some e s t i m a t e s on t h e s i z e o f t h e d e r i v a t i v e s o f t h e f u n c t i o n
m.
Some o f such r e s u l t s , o r i g i n a t i n g i n M a r c i n k i e w i c z , can be seen i n Zygmund [1959]
o r S t e i n [1970]
.
A t y p e o f m u l t i p l i e r problem which a r i s e s i n a n a t u r a l way when one d e a l s w i t h d i f f e r e n t manners o f sumino up t h e terms o f a mult i p l e Fourier series i s the followi'ng. t e r i s t i c function
xp
c o n d i t i o n s on t h e geometry o f m u l t i p l i e r f o r some
Assume t h a t
o f a measurable s e t
p # 2 ?
m
i s t h e charac-
P. Can one s t a t e some
P t h a t ensure t h a t xp i s an Lp-
13. M U L T I P L I E R S AND
360
MAXIMAL OPERATOR
When P is a half-plane in R2 or a polygon with finitely many sides, then one can affirm that xp is an LP-multiplier for 1 < p <m. These results are a direct consequence of the boundedness of the one-dimensional Hilbert transformin Lp(R1) , 1 < p < m, In fact from the definition of the truncated Hilbert transform, for f a L*(R)
we obtain
Therefore
A
hc(E;)
lim
=
c sign 5 and so
E+O
$fE) If
=
c sign 5
?(El , where sign 5
X+ is the characteristic function o f
'jyn " and X+(d 2 responding to x+ we get
if T,
=
[Op)
-I
and so T,
=
$
is of type (p,p) Noh if P
(H
is the multiplier operator cor-
f t f) 1< p
E < O
we have
Therefore T+f
if
4 m
as H.
c R2 is the closed first quadrant
13.0. INTRODUCTION P = {(x,y)
x > 0
:
xp
sociated t o
, y > 01 and , for f
we have
Tp
i s t h e m u l t i p l i e r o p e r a t o r as-
L2(R2)
8
36 1
,
Hence
'iy
where
T i f(x,q)
n(T:f)(x,q)dq
T:(T:
=
f)(x,y)
i s obtained f i x i n g f i r s t
rl
, considering
the function
and t a k i n g t h e v a l u e a t x. f(<,n) , a p p l y i n g t o i t t h e o p e r a t o r T, P And s i m i l a r l y T.: I t i s now easy t o see t h a t xp i s an L - m u l t i p l i e r
6
-f
for
l < p < m .
P
Similarly, i f
i s any h a l f p l a n e , we see t h a t
L P - m u l t i p l i e r , t h e s t r o n g t y p e c o n s t a n t (p,p) halfplanes.
Also, i f
P
xp
i s an
b e i n g t h e same f o r a l l
i s any convex polygon w i t h f i n i t e l y many s i d e s .
But even i n t h e case t h a t
P
i s the u n i t c i r c l e i n R2 t h i s
q u e s t i o n was open f o r l o n g t i m e u n t i l C.Fefferman [1971] s e t t l e d i t proving t h a t i f multiplier if
D
i s the u n i t c i r c l e i n R2
p # 2.
, xD
i s n o t an Lp-
The s o l u t i o n o f t h i s problem i s c l o s e l y con-
nected w i t h t h e theorems we have developed i n Chapter 8.
Some o f t h e
r e s u l t s o b t a i n e d t h e r e w i l l p e r m i t us t o go a s t e p f u r t h e r t o t r e a t w i t h t h e same t e c h n i q u e t h e case i n which n i t e l y many s i d e s .
P
converge
t i o n , t h e n we can l i k e w i s e a f f i r m t h a t p # 2.
i s a polygon w i t h i n f i -
If we impose t h e c o n d i t i o n t h a t t h e d i r e c t i o n s o f
a subsequence o f t h e s i d e s o f f o r any
P
xp
"slowly" i s n o t an
t o a fixed dire2 LP-multiplier
T h i s t y p e o f thecrem has o f course t h e f l a v o u r o f some o f t h e r e s u l t s on d i f f e r e n t i a t i o n we have discussed i n Chapter 8.
In
f a c t t h e y o r i g i n a t e from t h e same r o o t , namely t h e p o s s i b i l i t y o f c o n s t r u c t i n g an adequate Perron t r e e such t h a t t h e s i d e s o f i t s small t r i a n g l e s have t h e d i r e c t i o n s i n q u e s t i o n .
And again, t h e same t y p e
13. MULTIPLIERS AND MAXIMAL OPERATOR
362
of problems that are still open about the possibility of differentiating with respect to the rectangles with directions in a given set of directions can be formulated in this context. We shall later mention some of them. These connections lead one to suspect that the multiplier operators of this type and the generalizations of the Hardy-Littlewood maximal operator that are the key to understand the differentiability problems are also deeply interrelated. That this is in fact so has been brought to light in a recent paper by CBrdoba and R.Fefferman [1977] . In this Chapter we first present in 13.1. the negative result of C.Fefferman [1971] about the unit disk. In 13.2. we shall examine some more negative results that can be obtained with the same techniques about the case of a polygon with infinitely many sides, Finally in 13.3. we present some of the results of CBrdoba and R.Fefferman on the relation of multiplier and maximal operators. The results of 13.3 are more general and contain, at least partially , those of 13.1. arnd 13.2. However they are more sophisticated and the whole Chapter will be more easily and pleasantly read if we present it the way we do.
13.1. THE CHARACTERISTIC FUNCTION OF THE UNIT DISK.
C. FEFFERMAN.
A THEOREM OF
The theorem we are going to prove is the following.
D in R2
f
I3
13.1.1. THEOREM. The ckwracAW;tic 6unCtiCIn .in naA an l p - m W p f i ~doh any p i 2.
t h e u n i t didk
l ? t o o ~ ,Let T he the multiplier associated to XD L2W)
Tf = ( If
vb
f, g 4
L ~ ( R ~we) have
xD
fy
, i.e.
for
363
13.1. A THEOREM OF C. FEFFERMAN
and so i f
i.e.
T
L P - m u l t i p l i e r , p # 2, and
i s an
, we
p ' = JL P-1
have
T i s a l s o an L P ' - m u l t i p l i e r , o b v i o u s l y w i t h t h e same norm. (These
c o n s i d e r a t i o n s are, o f course, a p p l i c a b l e t o any o p e r a t o r t o the m u l t i p l i e r
T,
associated
m).
T h e r e f o r e , i n o r d e r t o prove t h e Theorem we can assume
p > 2.
The p r o o f of t h e Theorem i s a combination o f t h e t h r e e
following
observations; ( a ) ( T h i s o b s e r v a t i o n had been made by Y.Meyer p r e v i o u s l y t o t h e p r o o f o f C.Fefferman). I f we d i l a t a t e any d i s k by a homothecy w i t h c e n t e r a t any o f i t s boundary p o i n t s and w i t h r a t i o o f homothecy b i g enough we o b t a i n a b i g d i s k which i s a p p r o x i m a t e l y an a f f i n e h a l f p l a n e whose borI f the characteristic function o f the u n i t
d e r can have any d i r e c t i o n . d i s k i s an L P - m u l t i p l i e r
,
2 < p <
remark, one can p r o v e t h a t i f
L
j
my
w i t h norm
c
P' a r e t h e hyperplanes
using t h i s t r i v i a l
L . = { x e R 2 : (x,v.) 2 01, where v j = 1,2,...,k J $1 jy and T . f . = ( X 8.1' then, f o r any f i n i t e sequence J J L j I, J 2 o f f u n c t i o n s i n L (R ) one has
(bJ
If
Rj
i s any r e c t a n g l e i n d i r e c t i o n
s e t i n d i c a t e d i n F i g u r e 13.1.1. then, i f
IT^
fj(x)l
1 lQOD
if x e
6j .
f
j
=
are u n i t vectors fj, j = l,,,.,k
v
j
xR j , one
and has
..
R
j
i s the
T h i s easy f a c t i s reduced t o a s i m p l e one-dimensional computation.
364
13. MULTIPLIERS AND MAXIMAL OPERATOR
Figure 13.1.1. ( c ) For any small number q > 0 t h e r e exists a measurable s e t {Rj}j,lk , pairwise disE c R 2 and a f i n i t e c o l l e c t i o n of rectangles j o i n t , with t h e following p r o p e r t i e s :
-
where R
denotes again t h e shaded portion o f f i g u r e 13.1.1.
j
This i s j u s t t h e Lemma 8.2.1. t h a t we have e a s i l y obtained from the construction of the Perron t r e e . With this observations ( b l a n d ’ ( c ) fj=
x
Rj
We e a s i l y compute, s e t t i n g
,
RjI
On t h e other hand, i f
p > 2
, using now the observation ( a )
36 5
13.1. A THEOREM OF C. FEFFERMAN
c
f
( u s i n g H o l d e r ' s i n e q u a l i t y w i t h exponents
and i t s c o n j u g a t e )
Therefore
and t h i s i s a c o n t r a d i c t i o n if
?-I
i s small.
L e t us f i n a l l y p r e s e n t t h e proof o f t h e o b s e r v a t i o n (a) i n t h e f o l l o w i n g two lemmas.
LeA
LEMMA.
13.1.2.
p <
-, w&h
be any f i n e m openatoh bounded @om nomi
c,
c"
II(
Let
(iLiZY...,ik)
E
. Then we have, k
j=l
Phoaa.
x=
S
j=1
lk-l = 1 1 . We
don
any
2)1'211
be t h e u n i t sphere o f Rk and
have, f o r
w E Q
,
c
Since f o r each
i a
1,
S
i s bounded f r o m LP(Q)
to
LP(Q) we can w r i t e ,
6
13. MULTIPLIERS AND MAXIMAL OPERATOR
366
If we write
the previous inequality is
Observe now that if
a
s any vector
in R k
-
to1 then
c independent o f ct . Therefore, if we ntegrate the inequality (*) over C and change the order o f integration we obtain
with
c
1
R
ISF(w)l
P k
dw
6
c cp p
n
IF(w)
P k
dw
This can be written
and this is the statement of the lemma. 13.1.3. LEMMA. Adbume t h a t t h e mlLetipfieh o p u ~ ~ ~ A To h a6Aociated t o ,the u n i t didh D Aatib6Lie6 11 Tf 11 rS Cp/If \Ip , 604 dome P p 3 2 . L e A IVj3 j = l 6e a @nite Aequence 06 u n i t veotohs i n R2. 1eA Lj 6e t h e haR6-plane
13.1. A THEOREM OF C . FEFFERMAN L. =
J
and dedine.
T .fi( XL J
x s R 2 : (x,v.) B J
36 7
0 1
h
j
f)'
k
Then, 6uh any bequence. Efj}j-l
i~
Lp(R2) we have
r Phoud. Let D j be t h e c i r c l e of center r v and radius r > 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<
g E Lp(R2)
be t h e rni&XpL&h
EJ
p <
m
h
g.
, we.
have
13.3.1. i s easy.
W i t h these two r e s u l t s t h e p r o o f o f Theorem
Phaad v d t h e Theohem 13.3.1. For p a r t ( a ) we proceed as f o l Aj that lows. L e t F j be t h e a f f i n e h a l f p l a n e determined by Aj-l c o n t a i n s P. L e t H j be t h e m u l t i p l i e r o p e r a t o r c o r r e s p o n d i n g t o Fj, i.e.
( H ~g I A =
x
G.
FJ
Observe t h a t i f j = 1,2,
have, f o r each
i .e.
T
...
i s the m u l t i p l i e r associated t o
P
P, we
S j Tp = H j S j .
By Theorem 13.3.3. we have
II T p f I l F
c
II(
c
m
j=1
I S j Tp
m
f12)1/211;
=
II ( E
c
j=1
I n o r d e r t o e s t i m a t e t h e l a s t t e r m we examine, f o r
Il@Il
$1
6 1 ,
$ 3 0
1
m
R2
(
1 (Hj j=1
Sj
Now f o r each
j we compute A
direction o f
vj
ting
v
j
, i.e.
an a n g l e o f
71
if
j j
fl')
m
@
=
= l l H j Sj
1
j=l
fI2
@
I
R
sj fl 2 I 1/2 II
lHj @
6
L
(5)
IHj Sj
I
f12
p
(R2) , w i t h
4
by lines i n the
i s t h e u n i t v e c t o r o b t a i n e d by r o t a -
375
13.3. A THEOREM OF A. C6RD05A AND R. FEFFERMAN r
IH. S. f(sv
JseR Using now t h e
jseR
heorem H. J
sJ.
13.3.2. f
)Sjf
j'
@ ( s v j y t V . )d s d t J
t?.)12 J
we have f o r almost each
(sv.,tv.)12 @ ( s v j , t i . ) ds J
J
J
t eR
6
(sv t G . ) I 2 ME@ (sv t i j ) ds j7 J
j7
s.
where we have taken ME with r e s p e c t t o
Therefore
means t h e operator defined above b u t now on t h e l i n e i n direcwhere M$: tion v passing through (x,y) Now i t i s easy t o s e e t h a t t h e f a c t j t h a t K i s of type ((f)',($)') implies t h a t t h e operator M Z
.
MZ g(x,y)
=
SUP
j
M:
(X,Y)
Therefore we g e t
and so T
i s o f type
(p,p).
T h i s conclude t h e proof of p a r t ( a ) .
t h a t Xp
We now prove u a r t
(b).
Proceeding exactly as we d i d i n Lemma 13.1.3, from t h e f a c t i s and LP-multiplier f o r p > 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
direction
I 11,
A
-
IEkl and
p1 l R"k l B
in
Y ~ ( ~ we) have
( means one-dimensional measure) and so , an easy one-dimensional computation ( t h e same as i n (b) o f Theorem 13.1.1) we have t h a t f o r a t l e a s t h a l f o f t h e l i n e s 1 between A and B
378
13. MULTIPLIERS AND MAXIMAL OPERATOR
-
L e t us c a l l
Fk
t h i s set o f points o f
i s the m u l t i p l i e r operator corresponding t o
If ' j ( k )
j' ( k f
= { x
€R2
( x , V ~ ( ~ ) )> 0 1 we have, s i m i l a r l y ,
:
I at
each
point o f
Ifwe now submit
where t h e above h o l d s ,
Rk
'FkI ) ' (
'j(k)
-
Ri
.
1
'
Therefore, f r o m t h e i n e q u a l i t y (*)
6;
the sets
we g e t
t o t h e same k i n d o f process we have used
on t h e s e t s ikwe a r r i v e t o t h e s e t s ikand so
Therefore
So we have o b t a i n e d f r o m that
ERklk=l
B u t t h i s i s known t o i m p l y ((!)I,
($1')
and p a r t (b)
a f i n i t e sequence
( c f . Chapter
6)
o f t h e theorem
that
K
i s proved.
{?i,lt=, such
i s o f weak t y p e
REFERENCES
Banach,S. [1924] , Sur un th&or??me de M.Vitali, Fund. Math. 5 (1924) 1313-136. Bennet,C. and Sharpley,R. [1979] , Weak-type inequalities for tip and BMO, AMS Proc. of Symposia in Pure Math. 35 (1979) I , 201-229. Besicovitch,A.S. [1919] , Sur deux questions d'intGgrabilit6 des fonctions, J.Soc. Phys. - Math. (Perm') 2 (1919-1920), 105-123. Besicovitch,A.S. [1923] , Sur la nature des fonctions a c a r 6 sornmable measurabl es , Fund. Math. 4 (1923), 172-195. Besicovitch,A.S. [1926] , On a general metric property of summable func tions, J.London Math. SOC. 1 (1926) , 120-128. Besicovitch,A.S. [1928] , On Kakeya's problem and similar one, Math.Z. 27 (1928) , 312-320. Besicovitch,A.S. [1928'] , On the fundamental geometrical properties of linearly measurable plane sets of points, Math. Ann. 98 (1928), 422-464. Besicovitch,A.S. [1935] , On differentiation of Lebesgue double integrals, Fund. Math. 5 (1935) , 209-216. Besicovitch,A.S. [1938] , On the fundamental geometrical properties of linearly measurable plane sets of points ( I I ) , Math.Ann.115 (1938) , 296-329. Besicovitch,A.S. [1939] , On the fundamental geometrical properties o f linearly measurable plane sets of points (111), Math. Ann. 116 (1939), 349-357. 379
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llesicovitch,A.S. [1964] , On fundamental geometric properties o f plane 1 i ne-sets , 3.London Math, SOC , 39 (1964) 441-448. Bo0,P.A.
, Convergence almost everywhere of singular integrals o f dilation type,. Univ. of Umel, 1978.
[1978]
Bo0,P.A. [1978] , Necessary conditions for the convergence almost every where o f convolutions with approximation identities o f dilation type, Univ. o f Ume; , 1978. Burkill ,J.C. (19511 , On the differentiability of multiDle integrals, J.London Math. SOC. 26 (1951), 244-249. Busemann,H.and Feller,W. [1934] ,Zur Differentiation des Lebesgueschen Integrale, Fund. Math. 22 (1934). 226-256. Calder6nYA.P., Weiss,M. and Zygmund,A. [1967] , On the existence of singular integrals, Proc. Symp. Pure Math. 10 (1967) ,56-73. Calder6n,A.P. and Zygmund,A. [1952] , On the existence o f certain singular integrals, Acta Math. 88 (1952) , 85-139. Calder6n,A.P. and Zygmund,A. [1956] ,On singular integrals, Amer.J. Math. 18 (1956), 289-309. Calder6n,A.P. and Zygmund,A. [1973] , Addendum to the paper "On singular integrals", Studia Math. 46C1973) , 297-299. Carathgodory,C. [1927] , Vorlesungen iiber reelle Funktionen* (Leipzig, 1927). Carleson,L. [1966] , On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966) , 135-157. Carrill0,M.T. [1979] , Operadores maximales de convoluci6n (Tesis doctoral , Univ. Complutense de Madrid, 1979). Casas,A. [1978] , Aplicaciones de la teoria de la medida lineal (Tesis doctoral, Univ. Complutense de Madrid, 1978 ) , CBrdoba,A. [1976] ,On the Vitali covering properties o f a differentiation basis, Studia Math. 57 (1976), 91-95.
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Cbrdoba,A.
119771 , The Kakeya maximal f u n c t i o n and t h e s p h e r i c a l summ a t i o n m u l t i p l i e r s , Amer.J. o f Math. 99 (1977), 1-22.
Cdrdoba,A.
[1978]
, sxtx
$(x,t)
,
M i t t a g - L e f f l e r I n s t i t u t , Report no.9
(1978). C6rdoba,A.
and Fefferman,R.
,A
[1975]
geometric p r o o f o f t h e s t r o n g
maximal theorem, Ann. o f Math. 102(1975), 95-100. CBrdoba,A.
and Fefferman,C.
,A
[1976]
weighted norm i n e q u a l i t y f o r s i n -
g u l a r i n t e g r a l s , S t u d i a Math. 57 (1976), 97-101. Cbrdoba,A.
and Fefferman,R.
, On
11977 a]
differentiation o f integrals,
Proc. N a t l . Acad. USA 74 (1977),2211-2213. Cbrdoba,A.
and Fefferman,R.
[1977 b]
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A LIST OF SUGGESTED PROBLEMS
3.2.A.
On t h e B e s i c o v i t c h c o v e r i n g theorem.
How does t h e c o n s t a n t depend
on t h e dimension i n 3.2.1.? On t h e method of r o t a t i o n .
5.3.
Can one make i t work f o r o b t a i n i n g a weak
t y p e inequal it y ? 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 . l i m i t s o f Theorem 6.3.1. The h a l o problem.
6.6.
Knowing t h e h a l o f u n c t i o n o f a d i f f e r e n t a t i o n b a s i s ,
A
8
E
Lloc(Rn)
i n o r d e r t o ensure t h a t t h e
If.
On a theorem of B e s i c o v i t c h . t a k i n g as
What a r e t h e e x a c t
i n t h e sense o f p.117?
f i n d o u t a minimal c o n d i t i o n on f basis o f d i f f e r e n t i a t e s
6.3.
7.4.
How f a r can one extend Theorem 7.4.1.
by
a more general b a s i s ?
7.8.
problem o f Zygmund.
Can Theorem 7.8.1. be g e n e r a l i z e d t o more
dimensions? On t h e d i f f e r e n t i a t i o n p r o p e r t i e s o f some bases o f r e c t a n g l e s .
8.6.
See
Problems 1 , 2 , 3 , on pages 226-227.
Is a b a s i s o f r e c t a n g l e s 2 associated t o a L i p s c h i t z f i e l d o f d i r e c t i o n s i n R a density basis? On t h e Nikodym s e t and d e n s i t y bases.
I s t h e maximal o p e r a t o r a s s o c i a t e d L(Rn), k n o n - i n c r e a s i n g a l o n g r a y s , o f weak t y p e ( l , l ) ?
On approximations o f t h e i d e n t i t y . t o a k e r n e l k z 0, k
E
8.6.
On d i f f e r e n t i a t i o n a l o n g curves.
10.2.
12.
To f i n d a geometric p r o o f ( i n terms
o f t h e usual c o v e r i n g methods o f d i f f e r e n t i a t i o n ) o f theorems such as
12.1.1. On m u l t i p l i e r s a s s o c i a t e d t o polygons w i t h i n f i n i t e l y many s i d e s . f i n d t h e l i m i t s o f t h e theorems i n 13.2. and 13.3.
389
13.
To
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INDEX
Besicovitch covering lemma, 39 Besicovitch covering property, strong, 107 weak, 109 Besicovitch set, 210, 277 Busemann-Feller (B-F) differentiation basis, 104
Halo function, 149 Halo problem, 149 Hardy-Littlewood operator, 41, 103, 104 Hausdorff measure, 242 Hilbert transform, 305
CalderBn-Zygmund decomposition, 46 CalderBn-Zygmund operators, 313 Condensation of first order, 259 Condensation of second order, 259 Convex sets, covering lemma, 48 Convolution operators, 73 CBrdoba and Hayes theorem, 114 Van der Corput lemma, 344 Cotlarls lemma, 92
Identity approximations , 281 Interpolation, 54 Irregular point, set, 250
Density basis, 118 Density property, 118 Derivative, upper, lower, 105 Differentiation, 105 Differentiation along curves, 337 Differentiation basis , 104 Extrapolation, 60
Kahane set, 212 Kakeya problem, 209 Kolmogorov condition, 50 Linear density, 245 Linearization, 66 Linearly measurable sets, 241 Majorization, 70 Marcinkiewicz theorem on i nterpol at i on, 55 Marstrand theorem, 177 Maximal operator, 5 Multipliers, 359
C. Fefferman ' s mu1 ti pl ier theorem, 362 General i zed homogeneity , singular integrals, 327
Nikishin's theorem, 29 Nikodym set, 215
392
INDEX
Perron t r e e , 201
Tangency p r o p e r t i e s , 252
P o l a r l i n e s , 268
Tangent, 252
P r o j e c t i o n p r o p e r t i e s , 258
Type of an o p e r a t o r , 13
Rademacher f u n c t i o n s , 23 R a d i a l k e r n e l s , 282 R a d i a t i o n p o i n t , 259
V i t a l i c o v e r i n g lemma, 108 V i t a l i cover, 137
Regular p o i n t , s e t , 250 R i e s z - T h o r i n theorem, 54 R o t a t i o n method, 96 Saks r a r i t y theorem, 165 Sawyer ' s theorem , 19 S t e i n ' s theorem, 23 S t e i n and G. Weiss theorem, 55 Stein-Wainger theorem, 337 Sublinear operator, 7 Summation, 68
Whitney's lemma, 44
Y-set,
256
Zo ' s theorem, 292 Z-set, 256
