Weighted Norm Inequalities and Related Topics

WEIGHTED NORM INEQUALITIES AND RELATED TOPICS NORTH-HOLLAND MATHEMATICS STUDIES Notas de Matematica (104) Editor: Le...

Author: Jose Garcia-Cuerva | J-.L.Rubio De Francia

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WEIGHTED NORM INEQUALITIES AND RELATED TOPICS

NORTH-HOLLAND MATHEMATICS STUDIES Notas de Matematica (104)

Editor: Leopoldo Nachbin Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro and University of Rochester

NORTH-HOLLAND-AMSTERDAM

NEW YORK

OXFORD

116

WEIGHTED NORM iNEQUALlTlES AND RELATED TOPICS Jose GARCIA-CUERVA Universidadde Salamanca, Spain and

Jose L. RUBIO DE FRANCIA UniversidadAutdnomade Madrid, Spain

NEW YORK 1985

NORTH-HOLLAND -AMSTERDAM

OXFORD

Q

Elsevier Science Publishers B.V., 1985

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 87804 1 Publishers: ELSEVIER SCIENCE PUBLISHERS B.V. P.O. BOX 1991 1000 BZ AMSTERDAM THE NETHERLANDS

Sole distributors for the U.S.A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52VANDERBlLT AVENUE NEW YORK, N.Y. 10017 U.S.A.

Lihrary of Congress Cataloging in Publication Data

Oarcfa-Cuem, J o ~ . Weighted norm inequalities and related topics. (Notas de matembtica ; 104) (North-Holland mathematics studies ; 116) Bibliography: p. Includes index. 1. Hardy epaces. 2. Integral operators. 3. Inequalities (Mathematics) I. h b i o de Francia, J.-L., 194911. Title. 111. Series: Botas de m a t d t i c s (Rio de Janeiro, Erasil) ; no. 104. IV. Series: North-Holland mathematics studies ; U6. w . N 8 6 no. 104 CQA33ll 510 s C5151 85-12925 ISBN 0-444-87804-1 (Elsevier Science Pub. Co. )

.

PRINTED IN THE NETHERLANDS

T h i s book i s d e d i c a t e d t o MigueZ d e Guzmdn

f o r h i s constant e f f o r t s t o c r e a t e a s t i m u l a t i n g mathem a t i c a l atmosphere i n Spain.

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PREFACE

T h i s b o o k p r e s e n t s some o f t h e b a s i c a s p e c t s o f w e i g h t e d norm i n e q u a lities f o r the classical operators i n Fourier Analysis b u t , a s t h e t i t l e t r i e s t o indicate, t h i s i s not the only subject t o be discussed. R a t h e r , w e t a k e w e i g h t e d i n e q u a l i t i e s a s t h e c e n t r a l theme a r o u n d w h i c h o t h e r c l o s e l y r e l a t e d t o p i c s a p p e a r . Among t h e m , a g r e a t d e a l o f s p a c e i s d e v o t e d t o t h e t h e o r y o f Hardy s p a c e s .

[l], the standard A f t e r t h e c l a s s i c a l mo n o g r ap h o f A . Zygmund r e f e r e n c e s f o r t h e many i m p o r t a n t d e v e l o p m e n t s t h a t o c c u r r e d i n F o u r i e r A n a l y s i s d u r i n t h e second h a l f of t h i s c e n t u r y a r e E.M.Stein [l] a n d S t e i n - W e i s s b o t h p u b l i s h e d a r o u n d 1 9 7 0 . H ow e ve r, b o t h H a r d y s p a c e s a n d w e i g h t e d norm i n e q u a l i t i e s , t h o u g h s t u d i e d f o r many y e a r s b e f o r e 1970, have undergone a most s p e c t a c u l a r development since that date. I n t h e p r e f a c e o f t h e book by M . d e Guzmgn [2], b o t h t o p i c s were d e s c r i b e d a s b e i n g i n a v e r y f l u i d s t a t e , m a k i n g their exposition rather difficult. We f e l t t h e t i m e h a d come t o f i l l such a gap i n t h e c u r r e n t m a t h e m a t i c a l l i t e r a t u r e by w r i t i n g s u c h a n exposition.

$3,

We d o n o t i n t e n d t o c o v e r e v e r y t h i n g t h a t i s known i n t h i s a r e a , n o r t o g i v e t h e m o s t g e n e r a l f o r m o f e a c h r e s u l t . We h a v e t r i e d t o p r e s e n t t h e m a t e r i a l i n a n e a s i l y r e a d a b l e f o r m , w i t h t h e m a in i d e a s a p p e a r i n g a s n a t u r a l l y a s p o s s i b l e , s h o w i n g f i r s t how t h e s e work i n t h e s i m p l e s t c o n t e x t s , a n d b r i n g i n g t o l i g h t t h e c o n n e c t i o n s among t h e d i f f e r e n t t o p i c s . Thus, Hardy s p a c e s a r e f i r s t c o n s i d e r e d , i n c h a p t e r I , f r o m t h e c l a s s i c a l p o i n t o f v i e w o f Complex A n a l y s i s , w h i l e t h e more r e c e n t r e a l v a r i a b l e a p p r o a c h i s p r e s e n t e d i n c h a p t e r 111. A l s o , i n c h a p t e r 11, we g i v e a d e t a i l e d a c c o u n t o f t h e t h e o r y o f m a x i m a l a n d 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 t h e s p a c e s Lp(lRn) ; t h e extensions t o t h e weighted s e t t i n g o r t o v e c t o r valued f u n c t i o n s a r e s t u d i e d l a t e r , i n c h a p t e r s IV a n d V r e s p e c t i v e l y . A s a n o u t come.of o u r e m p h a s i s on t h e m u t u a l i m p l i c a t i o n s b e tw e e n d i f f e r e n t s u b j e c t s , t h e c e n t r a l r e s u l t s o f t h e t h e o r y a r e o b t a i n e d from a v a r i e t y of a p p r o a c h e s . I n p a r t i c u l a r , we g i v e t h r e e d i f f e r e n t p r o o f s ( i n c h a p t e r s I , I 1 1 a n d IV) o f C . F e f f e r m a n ' s H 1 - B M 0 duality theorem, f o u r p r o o f s o f M u c k e n h o u p t ' s c h a r a c t e r i z a t i o n o f t h e w e i g h t e d i n e q u a l i t i e s f o r t h e max i mal f u n c t i o n a n d o n e a n d a h a l f p r o o f s o f t h e Helson-Szego theorem. L i k e w i s e , c h a p t e r VI t r i e s t o s e t u p t h e b a s i s f o r a n ( a b s t r a c t ) a l t e r n a t i v e approach t o the two-weights p r o b l e m f o r g e n e r a l o p e r a t o r s , some o f whose c o n c r e t e a s p e c t s a r e d e s c r i b e d , by more c l a s s i c a l m e a n s , i n c h a p t e r IV. We w a n t e d t h e b o o k t o b e a c c e s s i b l e t o a n y o n e who i s f a m i l i a r w i t h t h e Lebesgue i n t e g r a l and t h e b a s i c f a c t s o f F u n c t i o n a l A n a l y s i s and Complex F u n c t i o n T h e o r y . We h o p e t h a t t h e r a t h e r l e i s u r e l y e x p o s i t i o n w i l l make t h e b o o k mo r e u s e f u l f o r t h o s e r e a d e r s who w a n t a n i n t r o d u c t i o n t o any of t h e a r e a s c o n s i d e r e d . This, unfortunately, vii

viii

PREFACE

a l s o made t h e book grow beyond t h e p r e v i o u s l y c o n c e i v e d s i z e . I t a l s o f o r c e d u s t o abandon t h e o r i g i n a l i d e a of i n d l u d i n g f u r t h e r developments i n t h e main s u b j e c t s , such a s w e i g h t e d Hardy s p a c e s , m a r t i n g a l e s , q u a s i - c o n f o r m a l g r o u p and C a l d e r o n ' s theorem on t h e Cauchy i n t e g r a l , harmonic measure and Hardy s p a c e s i n nonsmooth domains, e t c . In an attempt t o c o n c i l i a t e t h i s very ambitious p l a n w i t h t h e f i n i t e n e s s of o u r s t r e n g t h ( o r l i f e t i m e ) a n d of t h e e d i t o r ' s p a t i e n c e , a l a s t s e c t i o n of "Notes and F u r t h e r R e s u l t s " was added t o e a c h c h a p t e r . These s e c t i o n s i n c l u d e some h i s t o r i c a l n o t e s , c r e d i t f o r t h e a u t h o r s h i p of t h e more r e c e n t i m p o r t a n t t h e o r e m s and f u r t h e r r e s u l t s ( w i t h r e f e r e n c e s a n d / o r a s k e t c h o f t h e p r o o f ) complementing t h e m a t e r i a l c o v e r e d i n t h e main t e x t . Our p e r s o n a l t a s t e a n d , i n many c a s e s , o u r i g n o r a n c e , a r e r e s p o n s i b l e f o r t h e s e l e c t i o n of t h e r e s u l t s m e n t i o n e d , b u t we a r e aware of t h e f a c t t h a t many o t h e r cont r i b u t i o n s t o t h i s f i e l d i n r e c e n t years deserve t o appear i n t h e s e s e c t i o n s . To t h e a u t o r s of t h e s e f i n e c o n t r i b u t i o n s (whose number we e s t i m a t e t o be 10b f o r some l a r g e N ) and t o t h o s e who c o u l d c o m p l a i n f o r o u r g i v i n g i n a c c u r a t e c r e d i t t o some r e s u l t , we o f f e r o u r a p o l o g i e s and p l e a d g u i l t y i n a d v a n c e . T h i s work would have n e v e r been s t a r t e d w i t h o u t t h e encouragement o f M. de Guzmdn. We a r e most g r a t e f u l t o him and t o G . Weiss f o r t h e i r c o n t i n u o u s s t i m u l u s . Both of u s gave g r a d u a t e c o u r s e s b a s e d on t h e m a t e r i a l c o v e r e d by t h i s book i n t h e academic y e a r 1 9 8 1 / 8 2 , and have b e n e f i t e d from t h e c r i t i c i s m and encouragement of o u r c o l l e a g u e s a t t h e U n i v e r s i d a d Aut6noma d u r i n g and a f t e r t h a t c o u r s e . We must p a r t i c u l a r l y mention t h e i n v a l u a b l e h e l p o f M. W a l i a s and J . L . Torrea, who r e v i s e d t h e whole m a n u s c r i p t and made many p e r t i n e n t s u g g e s t i o n s Thanks f i n a l l y t o C a r o l i n e , Paloma and S o l e d a d who t y p e d t h e manuscript.

J . GARCIA-CUERVA

Spring 1985 J.L.

RUB10 DE FRANCIA

CONTENTS

PREFACE

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

I . CLASSICAL THEORY OF HARDY SPACES . . . . Harmonic functions . Poisson representation. Subharmonic functions . . . . . . . . . . . F . Riesz factorization theorem . . . . . . . Some classical inequalities . . . . . . . . The conjugate function . . . . . . . . . . . HP as a linear space . . . . . . . . . . . Canonical factorization theorem . . . . . . a . The Helson-Szego theorem . . . . . . . . . . 9 . The dual of H1 . . . . . . . . . . . . . . 10 . The corona theorem . . . . . . . . . . . . . 11. Notes and Further results . . . . . . . . .

CHAPTER 1. 2. 3. 4. 5. 6. 7.

Vii

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

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

1 2 26 39 52 59

. . . . . .

71 la a4

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

117 125

97

CHAPTER I1 . CALDERON-ZYGMUND THEORY . . . . . . . . . . . . . . 1. The Hardy-Littlewood Maximal function and the CalderbnZygmund Decomposition . . . . . . . . . . . . . . . . . 2 . Norm estimates for the maximal function . . . . . . . . 3 . The Sharp maximal function and the space B.M.O. . . . . 4 . Harmonic and subharmonic functions in a half-space . . . 5 . Singular integral Operators . . . . . . . . . . . . . . 6 . Multipliers . . . . . . . . . . . . . . . . . . . . . . 7 . Notes and Further results . . . . . . . . . . . . . . .

133

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

231 232 240 247 213 289

CHAPTER I11 . REAL VARIABLE THEORY OF HARDY SPACES 1 . Hp spaces for the upper half plane . . . 2 . Maximal function characterizations of Hp 3 . Atomic decompositions . . . . . . . . . . 4 . Hp spaces in higher dimensions . . . . . 5.Dualspaces . . . . . . . . . . . . . . . . 1x

. . . .

. . . . . .

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

. . . . . .

134 144 155 167 192 208 218

CONTENTS

Y

6

.

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

Interpolation of operators between

7 . Estimates for operators acting on 8

. Notes

and Further results

Hp spaces. Hp spaces .

CHAPTER IV . WEIGHTED NORM INEQUALITIES. . . . . . . . . . . . 1. The condition A . . . . . . . . . . . . . . . . . . 2 . The reverse Holder's inequality and the condition A, 3 . Weighted norm inequalities for singular integrals . . 4 . Two-weight norm inequalities for maximal operators . . 5 . Factorization and extrapolation . . . . . . . . . . . 6 . Weights in product spaces . . . . . . . . . . . . . . 7 Notes and Further results . . . . . . . . . . . . . .

.

. . . . . . . .

CHAPTER V . VECTOR VALUED INEQUALITIES . . . . . . . . . . . . . 1 . Operators acting on vector valued functions: some basic facts . . . . . . . . . . . . . . . . . . . . . . . . . 2 . A theorem of Marcinkiewicz and Zygmund. . . . . . . . . 3 Vector valued singular integrals . . . . . . . . . . . . 4 Applications: Some maximal inequalities . . . . . . . . 5 . Applications: Some Littlewood-Paley theory . . . . . . . 6 . Vector valued inequalities and A weights P 7 . Notes and Further results

. .

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

CHAPTER VI . FACTORIZATION THEOREMS AND WEIGHTED NORM INEQUALITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Factorization through weak-LP . . . . . . . . . . . . . 2 . Nikishin's theorem 3 Factorization through Lp 4 . Factorization of operators with values in Lq . The Dual Problem . . . . . . . . . . . . . . . . . . . . . . . . 5 . Weighted norm inequalities and vector valued inequalities . . . . . . . . . . . . . . . . . . . . . . . . . .

.

. . 8. 9. 6 7

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

....

The two weights problem for classical operators Weights of the form lxla for homogeneous operators Directional Hilbert transforms . . . . . . . . . . . . Notes and Further results . . . . . . . . . . . . . .

APPENDIX . . . . . . . . . . . . . A.l. Rademacher Functions . A.2. T h e m i n i m a x lemma . .

.. . .

............... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

307 315 368 385 387 395 411 422 433 450 464 475 475 482 491 497 504 518 522

527 528 533 542 549 555 558 566 571 577 581 581 583 587 601

CHAPTER I CLASSICAL THEORY OF H A R D Y SPACES

T h e t h e o r y o f F o u r i e r s e r i e s o f o n e v a r i a b l e i s d e s c r i b e d by A . Z y g muiid i n h i s f a m o u s m o n o g r a p h " T r i g o n o m e t r i c S e r i e s " a s " t h e m e e t i n g g r o u n d o f r e a l a n d c o m p l e x v a r i a b l e s " . Most o f t h e t o p i c s c o v e r e d by t h i s book h a v e t h e i r o r i g i n i n t h i s c o n n e c t i o n b e t w e e n p r o p e r t i e s o f harmonic or a n a l y t i c f u n c t i o n s i n t h e u n i t d i s c D of t h e complex plan e and t h e F o u r i e r A n a l y s i s of t h e i r boundary v a l u e s i n t h e t o r u s T = a D . No d o u b t , t h e p r o b l e m s t h a t a r o s e i n t h i s c o n t e x t w e r e i n i t i g l l y a t t a c k e d w i t h t h e h e l p of t h e p o w e r f u l m e t h o d s o f Complex Analys i s , and t h i s i s t h e a p p r o a c h f o l l o w e d i n t h i s c h a p t e r . Even t h o u g h s u c h m e t h o d s a r e o f n o u s e f o r d e a l i n g w i t h t h e s i m i l a r , b u t more g e n e r a l , s i t u a t i o n s i n l R n , w e h o p e t o g a i n some p e r s p e c t i v e f o r o u r f u t u r e w o r k by f i r s t d i s c u s s i n g t h e p r o b l e m s i n t h e i r s i m p l e s t f o r m u l a t i o n .

T h u s , t h e t h e o r y o f H a r d y s p a c e s i n t h e u n i t d i s c , w h i c h was d e v e l o p e d i n its e s s e n t i a l a s p e c t s during t h e f i r s t h a l f of t h i s c e n t u r y , occup i e s a c e n t r a l p a r t i n t h e c h a p t e r , a n d i s b a s e d on f a c t o r i z a t i o n s o f a n a l y t i c f u n c t i o n s d u e t o F. R i e s z a n d S m i r n o v , o f w h i c h no s a t i s factory analogue is a v a i l a b l e f o r harmonic f u n c t i o n s i n En. I n s p i t e o f t n i s , t h e t h e o r y h a s a n a t u r a l c o n t i n u a t i o n , by r e a l v a r i a ~ l e m e t h o d s , i n C h a p t e r 111. O n e o f t h e b a s i c e l e m e n t s o f t h e t h e o r y o f Hp s p a c e s i s t h e c o n j u g a t e f u n c t i o n , t h e s i m p l e s t example of a s i n g u l a r i n t e g r a l o p e r a t o r , whose n-dimensional a n a l o g u e s w i l l b e s t u d i e d w i t h i n t h e C a l d e r 6 n - Z y g m u n d t h e o r y d e s c r i b e d i n C h a p t e r 11. T h e s a m e can be s a i d of t h e maximal f u n c t i o n of Hardy and L i t t l e w o o d . The material c o n t a i n e d i n t h e l a s t t h r e e s e c t i o n s i s n o t so c l a s s i c a l : The Helson-Szego t h e o r e m , our f i r s t e x a m p l e o f a w e i g h t e d norm i n e q u g l i t y , m u s t be c o n s i d e r e d i n c o n n e c t i o n w i t h t h e i n e q u a l i t i e s of t h e same t y p e ( t h o u g h q u i t e d i f f e r e n t i n t h e s p i r i t of t h e m e t h o d s ) which we shall obtain i n Chapters I I andlV,while C. Fefferman'scelebrated dug l i t y t h e o r e m w i l l b e r e v i s i t e d i n s e c t i o n s DI.5 a n d lV.5 w i t h d i f f e r e n t p r o o f s ( o n e o f t h e m , by t h e w a y , r e l a t e d t o t h e H e l s o r 6 z e g o t h e o r e m ) . Althoughthe r e l e v a n t problems s t u d i e d i n t h i s chapter a r e onedimensig n a l , w e d o o b t a i n some b a s i c p r o p e r t i e s o f h a r m o n i c a n d s u b h a r m o n i c f u n c t i o n s of n r e a l v a r i a b l e s . T h i s r e q u i r e s o n l y s l i g h t l y more e f f o r t t h a n when d e a l i n g w i t h f u n c t i o n s of one complex v a r i a b l e , and p a v e s t h e way f o r l a t e r developments.

1

I. CLASSICAL HARDY SPACES

2

1.

HARMONIC FUNCTIONS, POISSON REPRESENTATION Recall that a harmonic function on a domain Q c lRn is a 2 u e C ( 0 ) (i.e. continuously differentiable up to order 2

function in

Q) and satisfying Laplace's equation

course,

A

AU =

0

in

is the Laplacian differential operator

Q , where,

A =

zof " a 1 2 axj

j=l

In this chapter we shall be concerned, primarily, with harmonic functions on a plane domain. F o r the plane we shall u s e the complex

x t iy, x and y real. There is a natural relation between harmonic and holomorphic functions in the plane. Indeed, the Laplacian operator can be factored a s coordinate

z =

The equation with

u

v

and

aF

F

ti = 0 for a complex valued function aY real, is equivalent to:

x

= u t

iv

ax - ay

which is the Cauchy-Riemann system, whose solutions

F

=

u t iv

are

precisely the holomorphic functions. Any holomorphic function i s , therefore, harmonic. Besides if

F

=

u t iv (with

u

and

is holomorphic, taking complex conjugates in the identity we s e e that

F

u - iv

=

satisfies the equation

ax

consequently, is also harmonic. It follows that both

v = (F-F)/Zi

and

-iz

ay

=

v

real)

aF

t l .aF V=

0

and,

0,

u = (FtF)/2

are harmonic functions; that is: the real and

imaginary parts o f a holomorphic function are harmonic. On the other hand, if Q c

connected domain

is a real harmonic function on a simply

u (I:

(the complex plane), the condition

(u ) - ( - u ~ )=~ 0 guarantees the existence of v such that x x dv = -u dx t ux d y , that is: v = - u and v = u Then the funcY Y Y * tion F = u t iv is holomorphic and the function v(which is harmo-

Au

=

.

nic since v = -u and xx YX conjugate of u Clearly

.

v = u ) is said to be a harmonic YY XY v is determined u p to an additive

constant. S o , we have seen that a real function

u

defined on a simply connected domain

I. 1. HARMONIC FUNCTIONS

of

C

3

is harmonic if and only if it is the real part of some

holomorphic function. Suppose

u

= ( z e C

:

is a real harmonic function on the disk D(0,R) = < R]. Then we know that u(z) = Re F ( z ) for some W holomorphic function F . Let F ( z ) = 1 ck zk be the power series k=O representarion of F . We can for

121

write

u(z)

=

-

(F(z) t F(z))2

and get a series representation

u. Let us do that using the polar form of and

r = IzI

5

-TI

8 5

=

ak = ck/2

f(

m

C

for

1 c k r k e -ike

Ckr k e iket

k=O

We conclude that any

u

with

m

=

k=O

k > 0, .a

reie

We obtain:

51,

m

u(re i8

z =

=

1 akr I k I ,ike

with

-m

Re co

and

harmonic in

-

ak = c -k / 2

D(0,R)

for

k

0.

has a series represen-

tation

converging uniformly on compact subsets of Suppose

R > 1 . Since

see that frequency

(1.1)

converges uniformly for

r = 1,

we

ak

is the Fourier coefficient, corresponding to the it k , of the function t I-+ u(e ) , that is U(eit)

Substituting this integral for

For

D(0,R).

0 5 r < 1

,-ikt ak

dt in (1.1) we get:

the series converges uniformly, its sum being

This function is the Poisson kernel for the unit disk and will be denoted by

Pr(t>.

For a function

u

harmonic in

D(O,R),

R > 1, we have obtained the

I. CLASSICAL HARDY SPACES

4

Poisson representation: 2

(1.2) u(re i9

1 - r u(eit)dt ltr 2 -2r cos(8-t)

-TI

=

it which exhibits the function ur(e ) = u(reit) as the convolution T = (eit: t e IR 1 which we identify with the (on the torus group of the functions u(eit) and Pr. interval

I-.,.])

Formula (1.2) provides the key to the solution of the Dirichlet problem for the disk. This basic problem consists in finding a and harmonic in function u continuous on D = {z e C: ( z ( 6 1) D , whose restriction to the boundary of D , u(eit), coincides on [-TI,TI] such with a previously given continuous function f(t) that f(-n) = f(n). The natural candidate for the solution will be the integral in (1.2) with f(t) in place of u ( e it ) , that i s , the

0 5 r < 1 , is the convolution of f and the Poisson kernel Pr. We write u(rei8) = Pr * f ( 8 ) or u = P(f) and say that u is the Poisson integral of f. That this function u is indeed a solution will b e seen shortly. First, w e shall show that the Poisson representation (1.2) remains valid for a much wider class of harmonic functions in the unit disk. One function

t N u(reit)

for

consequence will be the uniqueness of the solution to the Dirichlet problem. We start with the following 7H€OR€R

7. 3

. Let

u

D

ee a h a n m o n i c { u n c t i o n i n

4uch t h u t

TI

(1.4) {on

sup OSr
40me

lu(reit)IP

dt <

m

-7~

.

p > 1

Then t h e n e

i 4

f

u {unction

e

LP([-n,v])

&

that

(1.5)

u(re

i0

l T I

) = ?;i

Pr(O-t)

f(t)

dt

-TI

that

i 4 :

u

i 4

the ~ o i n n o ni n t e y n a e

o t home

LP

{unction

f.

Proof: Let rn 4 1 (i.e. rn is an increasing sequence converging to 1). Consider the functions fn(t) = u(r eit), From condition (1.4) (f,)

is a bounded sequence in

LP([-n,sr])

=

Lp'([

.,T])*,

where

P'

I. 1. HARMONIC FUNCTIONS

denotes the exponent conjugate to

p(that

is:

5

(l/p) t (l/p') = 1)

and the star * is used to denote the topological dual. Thus the sequence (f,) is in a closed ball of the normed dual to LP'([-n,nJl The Banach-Alaoglu theorem asserts that such ball is weak-+ compact and therefore, since Lp'( [-7T.T]) is separable, also metrizable ( s e e has a subsequence converg Rudin [2] pp. 6 6 - 6 8 ) . It follows that (f,) ing in the weak-+ topology to a certain for every g e ~~'([-n,n])

In

(1.6)

g(t)

fn(t)

-n

F o r each

dt

-1

n , the function

f

e

n g(t)

f(t)

dt

as

-n z

u(rnz)

,

Lp( [ - n , n ] )

n +

that is:

m

-1 D(O,rn ) , a

is harmonic in

disk of radius bigger than one. Therefore, we have the Poisson representation:

Letting

n

go to

m , the left hand side tends to

the right hand side tends t o (1.6).

In

while

Pr(e-t) f(t) d t , according to -n This yields the Poisson representation (1.5). 1 3

Let us observe that the theorem remains valid f o r ing f o r (1.4) the condition: (1.7)

u(reie)

sup IIurJlm< p c , where Obr
u

is the function

m

substitut-

t + + u(re

it

);

is bounded in D. All 1 Lm([-n,n]) = L ([-n,a])*.

or, in other words, the hypothesis that one needs to realize is that still

p =

u

A relevant question at this point in whether condition (1.4)

with

p = . I will imply a Poisson representation. The proof of theorem 1.3

does not extend to this case because L 1 ( I - n , n l ) is not a dual space (any separable dual has the Radon-Nikodym property and L1( [ - n , n ] ) However L1( [-. does not have it. S e e Diestel-Uhl [l]). can be .J) isometrically imbedded into the space y ( [ - n , n ) ) of Bore1 measures by assigning to each f e L ( [ - n , n l ) , the measure on [-n,n] dp(t) = f(t) dt. The space M ( [ - n , T ] ) is the dual of the space C( [ - v , n ] ) of continuous functions on [-n,1T] with the supremum norm. Then we can repeat the argument used in the proof of theorem 1.3 and obtain :

I. CLASSICAL HARDY SPACES

6

7HEOREfl 1 . 8 . (1.9)

u

Let

J

SUD

O j r < l 7hen,

thene

7hut in: u

wnite

Stie&eh

in a

n

lu(re

it

)Idt <

B o n e 1 meanune

P(p)).

nuch t h a t :

m

-n

i n the P o i n n o n =

D,

a hanmonic [ u n c t i o n i n

ke

on

p

inteqnal

7hene t u n c t i o n n

[-n,n]

,

nuch khat:

o & t h e meanune

u

gne o k t e n

p

(we d h a l t

c a l l e d Poinnon-

integnald.

Let u s observe t h a t c o n d i t i o n (1.9) t h e harmonic function

n

it

(u(re

)I

dt

u

=

is

1 -

20.

i 3

automatically satisfied i f

Indeed, i n t h a t case:

u(reit)

dt

=

u(0)

T h e l a s t i d e n t i y i s a n i n s t a n c e of t h e m e a n v a l u e p r o p e r t y o f h a r m o n i c f u n c t i o n s , which follows iinmediately from P o i s s o n ' s r e p r e s e n t a t i o n . We o b t a i n ,

therefore, the following

T h e m e a s u r e i s p o s i t i v e b e c a u s e i t i s o b t a i n e d a s a weak-*

l i m i t of

p o s i t i v e measures.

For

p =

m,

t h e o r e m 1.3 a n d i t s p r o o f i m p l y t h a t t h e s o l u t i o n

u

of

t h e D i r i c h l e t p r o b l e m on D w i t h boundary f u n c t i o n f i s , i f a n y , P(f), t h e P o i s s o n i n t e g r a l o f f . We s h a l l p r e s e n t l y s e e t h a t P(f) i s i n d e e d a s o l u t i o n . T h i s w i l l b e b a s e d upon t h e f a c t t h a t t h e

P o i s s o n k e r n e l g i v e s r i s e t o a n a p p r o x i m a t e i d e n t i t y . To s e e w h a t t h i s m e a n s , we e x a m i n e c l o s e l y t h e P o i s s o n k e r n e l

It is obviously a positive. Besides

2n-periodic

c o n t i n u o u s f u n c t i o n of

t. It is a l s o

7

1.1. HARMONIC FUNCTIONS

6 > 0

And finally, for any

6

Indeed for

Pr(t)

5

2

It

1 -

5 IT

sup Pr(t) &
+ 0

as

r +. 1.

is

2

2

r1+r2-2r c o s 6 1

is

5

1 - r 1-cos2 6

+ o

as

r + 1.

The last inequality is due to the fact that the denominator is minimal for

r

=

c o s 6.

In general, an approximate identity on the torus T will be a family 1 Q a of 2"-periodic functions in L ([-n,n]) , with indices a ranging over a directed set and satisfying the following three conditions:

ii)

2"

1"-"

@

a

(t) d t

=

1

a

for every

Clearly, the Poisson kernel gives an approximate identity this case k = 1 in i) and an even stronger version of a s we have seen The fact that the Poisson kernel is positive and satisfies gives converses to theorems 1 . 3 and 1.8.

m

Proof:

If the Fourier series o f

f

is

1 -m

i ke ak

Pr. In iii) holds

ii)

I. CLASSICAL HARDY SPACES

8

then

If

u , and, clearly

f is real-valued, s o is

the real part of a holomorphic function. Consequently, u in

is harmonic

D.

(1.12) and (1.13) are particular instances o f Young's inequality. They can be obtained very easily by writing

and applying Minkowski's inequality f o r integrals in Stein [I] ) :

(the dot taken).

Then -

.

J

-TI

D

~

J

und

-TI

( t h e t u n f i n t e q n u l d e n o t e n t h e t o f u 1 uuniution o{

Proof: If the Fourier series o f

then

appendix A . l

stands for the variable with respect to which noris are

i n hunrnonic i n

U(z)

(see

is

1 akeikg,

U)

that is, if

1.1. HARMONIC FUNCTIONS

As before, u

9

is clearly harmonic. Besides

0

b y Fubini's theorem.

Thus, we have seen that, in the class of harmonic functions in condition ( 1 . 4 )

Lp

integrals of

p > 1

for

D,

characterizes those which are Poisson

functions, and condition ( 1 . 4 ) for

terizes those which are Poisson integrals of

p

=

1

charac-

Bore1 measures of

finite total variation. We shall study next the boundary behaviour of Poisson intezrals, This will allow us to solve the Dirichlet problem and several variants of it by means o f Poisson integrals and will also give u s a better understanding of how

determines

u

f

in theorem 1.3 or

theorem 1 . 8 . First we study the convergence in the

Lp norm. We can

state a general result valid for every approximate identity

7HEORLfl 1 . 1 6 .

Let

in

@a*

@a L e u n u p p a o x i n a t e i d e n t i t y o n t h e t o n u n

T

Then :

J -n If,(t) 2'

L) fa

+

f

Proof:

f

i 4

- f(t)lP

dt + 0

u coniinuoun

u n i b o n m l y on

face) - f(e)

because of property

2n-peaiodic [ u n c t i o n ,

w e huue

w e - t ) - f(e))

dt

T =

1 2'

1"

m a ( t )

-71

ii) of the approximate identity. Then, Minkowslu's

inequality for integrals implies that:

I. CLASSICAL HARDY SPACES

10

for an arbitrary. 6 > 0. The second term in this sum is bounded by I@,(t)(dt, which, according to property iii) of the lazntity, tends to zero a s 0 moves in the directed set 6 is., The first term in the sum is of indices, no matter how smal bounded by @,(t)ldt)

K sup Ilf(.-t)-f[l

=

P

Itl<6

K < m(property

with

i) of the approximate identity).

But

Ilf(.-t)-flI wp(f,6) , the LP-modulus of continuity of f , b P can be made small by taking 6 small. Indeed, it is clear that su

It

Ilf(*-t)-flI * 0 a s t + 0 (note that we are taking f continuous P when p = =. F o r p < =, we just need to approximate f in the L p norm by continuous functions in order to justifythe claim) Now, to conclude the proof of the theorem, given

6 >

0

> 0 , we first choose

E

small enough to have

independently of a. Then, with this 6 fixed, all we have to do is to take a far enough in the order of the directed set of indices to render

This will suffice to have

Ilfa-f

Ilp

<

E.

0

Taking a s approximate identity the Poisson kernel

COROLLARY 1 . 1 7 u = P(f).

b) IJ r * 1.

Let

f

P r , we obtain

2 ~ - p e n i o d i c, , ! u n c t i o n on

X I und Pet

w:

f

i n continuoun

u(re

it)

+

f(t)

uni,,!oarnly i n

t

a4

1.1. HARMONIC FUNCTIONS

Thus, i n theorem 1.3, t h e f u n c t i o n functions

= u(re

ur(t)

it

),

is the l i m i t i n

f

1 < p <

11

Lp(T) of t h e

m.

P a r t b) of t h e c o r o l l a r y implies t h a t ,

c o n t i n u o u s on

f

for

T,

u = P ( f ) is, indeed, t h e s o l u t i o n of t h e classical D i r i c h l e t problem. a) implies that f o r

Of c o u r s e , p a r t l u t i o n of an

Lp

1 5 p <

u = P(f)

m,

is t h e sg

version of t h e D i r i c h l e t problem.

By d u a l i t y , t h e o r e m 1 . 1 6 i m p l i e s 7HEOl?Efl?.?8.

ge u n u p p a o x i m a t e i d e n t i t y o n t h e i o a u h

T.

7hen:

a)

f

i n

Proof:

L"([-a,

the weuk*

f

a ) We h a v e t o see t h a t $( 0) d g

+

&

n])

t o p o l o q y of

c,

JI(t)f(t)dt

f o r every

= f

it / o L l o w n t u h t

I

fol

l f a ( g ) $(g)

+ $

in

dg

converges t o

L1

f e L".

and

b) of t h e theorem.

i n t e g r a l of

has Fourier series

p

is the function

con-

0

It should be noted t h z t i f an i n t e g r a b l e function Bore1 measure

f

But

The p r o o f o f b ) i s e n t i r e l y s i m i l a r , o n l y t h i s time we t a k e tinuous and apply p a r t

+

Lm.

JI e L1:[-a,n]).

JI*@,(-*)

since

o!

1 ak eikg, u(reig)

f

o r a complex

then, the Poisson

1 a k r l k l ei k g

which,

f

or

p

f o r each fixed

r,

c a n b e viewed a s a n a v e r a g e of t h e p a r t i a l sums

=

I. CLASSICAL HARDY SPACES

12

Indeed

ur(B) = u(re ie) are called the Abel means of (the Fourier series of) f or p . Thus, every theorem about the boundary The functions

behaviour of the function

u

can be read as a theorem on the Abel

summability of the Fourier series of

f

or p . So far, we have esta-

blished the Abel summability in the

Lp

norm. Now we shall analize

the problem of pointwise summability or, in other vords, we shall study the pointwise behaviour of a Poisson-Stieltjes integral. We shall no longer obtain results for a general approximate identity. Now, the particular structure of the Poisson kernel (more specifically

Pr(t)

the fact that

is decreasing a s a function of

Itl), will be

decisive. THfORffl I.20.

&&

We know t l i u t

F

Le

u

TI

Bone1 meu4une o n

und c u l l

& u t u n c t i o n o t Lounded u u a i u t i o n u n d l

which

F'(el)

gri4i4

und i n t i n i t e .

conueageh t o

F'(8,)

5 z

tend4 t o

ut

tho4e point4

Then

u(z)

8.q t h i 4 we meun t h u t

tially.

z = reie

{ reie:

.

I e-€I,l

&y w a i t i n g

the measure functions

u. ur

F'(el)

eiel

it

&e one

u = P(0).

non-tunqen-

804 e u e q

aemuininq

c > 0 u(z) + F ' ( e l ) in t h e n e y i o n

< c( 1-r)) We 4 h u 1 1 i n d i c u t e t h i 4 u(z) -+ F'(el) u., z eiel

Proof: Note that

F'(el)

ei e l

tend4 t o

hence,

& el &

6.

u l t i n i t e ) d e n i u u t i u e u t u1mo4t eueny p o i n t

ha4

ok

p

5

.

type

o t conuengence

is, by definition, a limit of averages of

S o , it should not come a s a surprise that the

which are just another type of averages, converge to

too. The proof will be based on integration by parts. First,

just to simplify the writing, we may clearly assume that e l = 0 , and also that F'(0) = 0 (otherwise, we consider the measure dp(t) - F'(0)dt). Take c > 0. We shall show that u(reie) can be made small uniformly i n e for 101 < c(1-r), by just taking r clo!je enough to 1. Let every time that reie

E

I tl

> 0. Take 6 > 0 such that I F(t)l < E l tl < 6 . Look only at r's s o close to 1 that if

is in our region of approach, then

181 < 6 / 4 , in other words,

13

1.1. HARMONIC FUNCTIONS

let

c(1-r)

< 6 / 4 . Then, for

reie

in our region

The first term in this sum is majorized by

Clearly, this tends to 0 a s

r

-+

1 , and we just need to worry about

the other term, which we can write, after integrating by parts a s sup Pr(t) plus: something bounded b y a constant times 6/2 It1

'

Again we just need to study this last term. teness, that 0 > 0, and decompose the integral a s We look at each of the terms in this sum. < Eltl = f(-t) 5 f(6-t) in such a On the first one we use IF(t)l way that, after changing 8-t to t , w e obtain a s a bound for the absolute value of the first term: t dt

Integrating b y parts again we s e e that this quantity is bounded b y I - r2

dt

=

E 2

F o r the second term in the sum we u s e

IF(t)l

< Et

l0-tl < 0.

and

In this way, the second term is bounded by

A s for the third and last term in the s u m , we u s e

IF(t)l

<

E

t s

;~2~(t-0) and proceed exactly a s with the first term. When every

du(t)

0

=

f(t)dt

with

1

f e L ([-n,n]),

we know that for almost

is If(e+s) - f(0)lds

+

0

as

t

-+

0.

(Actually,

I. CLASSICAL HARDY SPACES

14

this fact, which i s equivalent to Lebesgue's differentiation theorem, will be proved in chapter 11. See theorem 1.9 there). Those 0 for which this holds are called Lebesgue points for f. In such a point

, clearly

0

F'(e)

COROLLAY 7 . 2 1 .

=

&

eueay L e k e n q u e p o i n t

f(e).

Therefore, we get 1

f

f

L ([-nfnJ)v

und l e t

=

p(f)

-

Then, / o n

0 as

2-

N.T.

eig

I n punticulany t h i n i n t n u e a l m o ~ te u e n y w h e n e . 0

When dp(t) = f(t)dt + do(t), where f e L 1 ( [ - n , n J ) and 0 is singular, it is known that F'(0) = f(e) for a.e.O (see Rudin [ l ) ) , s o that every Lebesgue-Stieltjes

integral has non-tangential boundary

values a.e. This applies in particular to the

harmonic functions in

D

satisfying any of the conditions (1.4),(1.7) or (1.9). contains the classical theonem of F a L o u stating that

This result

"Any t u n c t i o n hoeornonphic and kounded i n D han n o n - L a n y e n t i a l kounduny u a l u e ~u. e .

".

When a harmonic function p > 1 , then

u

u

satisfies condition (1.4) for some

can be recovered from its boundary function

Indeed, we know that

u = P(f).

However if

u

f.

just satisfies (1.9)

( i - e . , if p = l ) , then it is no longer true that u integral of its boundary function. For example, let

is the Poisson

This i s , of course, a harmonic function, and it clearly satisfies

(1.9). Its boundary function is 0. Indeed Pr(t) + 0 a s r + 1 for every t f 0 in [-n,n]. However u > 0, and i t cannot be the Poisson integral of 0 , which is identically 0. Actually, in this case u = P(6) where 6 is the Dirac delta or, i n other words, the ~ ) . difference i n the unit mass concentrated at 0 in c - 7 ~ ~ This behaviour of an LP-bounded harmonic function for p = 1 or p > 1 is a basic fact a n d , a s we shall s e e , is the natural starting point for the theory of Hardy spaces, which is the main topic in this chapter.

1.1. HARMONIC FUNCTIONS

15

Even though in this chapter we shall only deal with harmonic functions in a plane domain, which almost always will be the unit disk D , it seems natural to try to s e e to which extent our discussion of harmonic functions can be carried over to higher real dimensions o r D. We shall start by showing that

plane domains different from

harmonic functions are characterized by the mean value property.

L4t

7HEOREfl 7.22. Then

u

Be u

u

i n hunmonic i n

c o n t i n u o u n & u n c t i o n on u d o m u i n

i &and o n l y i & u

Q

Q c

Bn.

nutin&ien the

: F o n e u e n y xo e R {x e En:Ix-xol 5 r } c Q

j o l l o w i n y p n o p e n t y ( k n o w n an meun v u l u e p n o p e n t y )

r > 0

and / o n e u e n y

whene

In-1

nuch i h u t

B(xo,r)

{x e IR": 1x1 = 1)

=

L e k e n q u e meunune on

=

i n t h e u n i t nphene i n

&

B n , do

do

und

'n-1 Proof: Suppose that Let

xo e R

and

u

is harmonic in

r > 0

be such that

let f(s) stand for the average of xo and radius s , that is u(xo+sO)

a , so

A U = 0 in Q. R . For 0 < s 5 r over the sphere of center

that

B(xo,r) u

C

do

n-1

We find that the function

f

defined in the interval

(O,r]

is dif-

ferentiable and its derivative is

The integrard equals

Duu(xo+~U), which is the derivative o f xO+sa. Thsn

u

in

the direction of the outer normal in the point

is the boundary of the ball, that i s , aB(xO,s) where aB(x0,s) = C(xo,s) , the sphere of center xo and radius s ; and dos the natural Lebesgue measure on theorem we get

aB(x0,s).

By applying Green's

=

is

16

I. CLASSICAL HARDY SPACES

Thus f ( s ) is constant

for

for

* 0 and

s

f(s)

.

0 < s 5 r But, clearly, f(r) for s * r. Therefore

-+

f(s) * u(xo) f(r)

= u(xo)

and (1.23) is proved. The converse is equally easy if we assume, to

start with, that

u

is twice differentiable. If this is the case and we assume that the mean value property holds, then, with the same notation used above we have

f(s)

=

Thus u

constant on [O,r].

f'(s) = 0

Consequently

But since

u(xo),

Au

Au(xo)

and from here

is continuous,

=

0. Since this is true for every

R , we get that

xo

is harmonic.

In case

u

is j u s t a continuous fu.iction in

value property, we shall show that

u

R

satisfying the mean

is harmonic by reducing the

problem to the case of a smooth function considered previously Since the problem is local we can assume that is bounded. We shall use a

I

Cm

function

@

Q

is bounded and

with support in

u

B(0,l)

t= 1

and the approximate identity in E n ,b E , to for E > 0 . It can be = E-"@ (E-lx) which it gives rise: @,(x) and having

1R

checked quite easily that @ E is indeed an approximate identity in En.The definition is entirely similar to the one given for the torus

T

(right before theorem 1.11).

radial, that is:

@(XI

Then we approximate

=

u

We shall also require

$ '

to be

@(lxl). by the functions

(it is understood that u is extended to lRn by making it equal to 0 outside of Q). We see in the last integral that the smoothness of @ implies that uE is smooth too. Now observe that uE

1.1. HARMONIC FUNCTIONS

RE = I x e c RE, we h a v e

s a t i s f i e s t h e mean v a l u e p r o p e r t y i n

Indeed i f

u

since

B(xo,r)

s a t i s f i e s t h e mean v a l u e p r o p e r t y i n

Consequently x e

and

x o e QE

uE

is harmonic i n

RE.

17

n:

R

dist(x,

and

an)

>

€1.

B ( x O , r + ~ )c R .

But on t h e o t h e r hand f o r

RE is:

This ends t h e proof

of a g i v e n

that

is harmonic because i n a neighbourhood

u

x e R , u coincides w i t h

a l r e a d y know t h a t

u

u

E

for

E

small enough and w e

is h a r m o n i c . 0

E

I t h a s t o b e o b s e r v e d t h a t t h e mean v a l u e p r o p e r t y i n t h e o r e m 1 . 2 2 i s equivalent t o the f a c t that f o r every that

n(x,,i)j

xo e R

and e v e r y

r > 0

1

18(xg,r)l ' B ( x o , r )

(1.24)

u ( x ) dx

I n d e e d i f t h e f i r s t mean v a l u e p r o p e r t y h o l d s a n d

B(xo,r)

c 0,

have: u(xo+so)

do

n-1 for all from

0

s

to

with

r

such

c R

,

0 < s we g e t :

5

r.

Integrating both s i d e s against

sn-'

we

I. CLASSICAL HARDY SPACES

18

Thus

Which is (1.24). Conversely, if we assume the second mean value property and xo e n, we shall have, for all r in an interval to the right of

0:

The right hand side, as a function of

0:

r , will have derivative

or equivalently

do , which is (1.23).

u(xo+ro) In-1

A consequence of the mean value property is the s o called muximum

pninciple for harmonic functions, which can be stated a s follows:

COROLLAiiY 1 . 2 5 Let u l e u n e a l - v a l u e d hunmonic &unction i n a domuin n c IR". Then u cunnoi uiiuin a maximum v a l u e unle44 it i4 conniuni. u

Proof: Suppose that xo e fi

exists

such that

r > 0

such that

Since

u(x)

u(x)

< m

B(xo,r)

does attain a maximu 5

u(xo)

x

and

u

. Then

for every

m

for some

x

f

B(xo,r) < m.

would have to be A

This shows that the set open set. But

u(x)

B(xo,r)cR

B

=

R\A

is continuous. Since

A

=

of

{x

f

=

m

value, that i s , there

u(x)

=

points of R u(x) < m ]

is not empty and

u(x)

m

u

for every

over

x e B(xo,r)

where u(x) = m , is an is also open because u

a:

to be necessarily empty. Consequently

Q. Take

f

is continuous, if we had

, then the average of Thus

x

for every

Q

B

is connected, = m

for every

x

has f

Q.0

Observe that the same result is true with minimum instead of maximum.

1.1. HARMONIC FUNCTIONS

19

Here is an equivalent formulation of the maximum and minimum principles:

COROLLA/?q 1 . 2 6 . u ke u neul-uulued Function, continuouh on the clonune R of u Bounded domuin R c lRn, und hunmonic i n R.* u uttuinn it4 maximum and it4 minimum a t t h e Bounduny o!, R (only a t t h e kounduay if u in n o t u conntunt). From this we can derive a uniqueness result for the solution of the Dirichlet problem for a bounded domain. Namely:

COiiOfLARY 7.27. clonune o-&u

L e t u 1 & u2 ke t w o Lunctionn continuoun on t h e Bounded domuin R, hunmonic i n R und ~ u c ht h u t

u,(x)

=

u,(x)

f!oa

eueny

u,(x)

=

u,(x)

{on

eueny

Proof: W e may assume just need to observe mun at the boundary. u 1 - u 2 is zero all

x e a R , t h e kounduny o,! x e i7.

fi.

Then

that u 1 and u2 are real-valued. Then, we that u 1 - u 2 attains its maximum and its m i n i But in the boundary it is always zero. Hence over R .

Now we shall briefly discuss the Dirichlet problem for a ball, say

Bn

the unit ball

of

R n , that is: Bn

=

B(0.1)

=

(x e lR":Ixl

We have already solved this problem for n = 2 . The solution the Poisson integral of the boundary function f: u(re

iB

& 1"

) =

P,(B-t)f(t)dt

=

1 -r

< 1). u

was

2

f(t)

dt =

-11

W e shall s e e that for general (1.28)

u(x)

=

We shall write

n , the solution is given by:

1,

f(s) d s n-l -n (1-1x1 ) Ix-sI = P ( x , s ) , the Poisson kernel

for

the ball. The fact that ( 1 . 2 8 ) is indeed the solution of the Dirichlet problem for Bn depends upon the following properties of the Poisson kernel: a)

P(x,s)

is harmonic in

x e Bn

for each fixed

s e

I . CLASSICAL HARDY SPACES

20

b)

P(x,s) 2 0

P(x,s) ds

and

=

1

for every

x e Bn

n-1 c)

6 > 0

For every l i m r+l

I

uniforinly i n

P(rx'.s)

x'

ds = 0

> 6

Is-x'(

Zn-l* n > 2 . For

L e t u s c h e c k t h e s e three p r o p e r t i e s .

Let

a n d , t h e r e f o r e , t h e h a r m o n i c i t y of

x W P(x,s)

a ) , observe t h a t

f o l l o w s from t h e well

)x12-n i s harmonic i n X n \ { O j ( A radial f u n c t i o n E n o f t h e f o r m @ (1 x 1 ) h a s L a p l a c i a n e q u a l t o @ " ( l x l ) t t !f$ d ) ' ( l x l ) . T h u s i f w e w a n t @ ( ] X I )t o b e h a r m o n i c i n lRn\{O}, known f a c t t h a t in

a l l w e have t o do i s t o s o l v e t h e o r d i n a r y d i f f e r e n t i a l e q u a t i o n 2-n is a solution) t bl(r) = o C l e a r l y r

b"(r)

In b)

9

P(x,s) 2 0

€or every

with

r

But, clearly

i s o b v i o u s a n d , by a )

0 < r < 1.

lrx'-s/

P(rx',s)

= Irs-x'

I

and, consequently:

= P(rs,x').

Thus

which is t h e e q u a l i t y i n

b).

Finally

c)

is very simple, since f o r each

> 6 > 0

is

Irx'-sI

independent of

X I

2 c6 > 0

for every

r

s

with

e

e [0,1],

with

e ZR-l.

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

(s--x'( >

c6

1.1. HARMONIC FUNCTIONS

7H€OR€M 7 . 2 9 , function

u

f c!eLined

21

Le a continuoun Function on

7hen t h e

'n-1.

B(0,l)

i n

~

i~ c o n t i n u o u n i n

B(0,l)

B(0,l).

t h e n o l u t i o n o t t h e D i a i c h l e t pnoLlem i n

I t in, L h e a e L o n e ,

Bn w i t h L o u n d a n y & u n c t i o n

f u

P r o o f : The h a r m o n i c i t y of of

P(x,s)

in

a s a f u n c t i o n of

B(0,l)

f o l l o w s from t h e harmonicity

x ( e i t h e r by i n t e r c h a n g i n g

the Laplacian

w i t h t h e i n t e g r a l s i g n o r by c h e c k i n g t h e m e a n v a l u e p r o p e r t y d i r e c t l y by u s i n g F u b i n i ' s t h e o r e m ) . For t h e c o n t i n u i t y we j u s t n e e d t o show t h a t

r

1

+

uniformly i n

u(rx')-f(x')

=

1

u(rx')

-+

f(x')

as

We w r i t e

x' e

1 n-1

P(rx',s)(f(s)-f(x'))

ds

n-1

by u s i n g p r o p e r t y b ) o f t h e P o i s s o n k e r n e L T h e n

Since hat

f

is continuous,

given

E

> 0,

t h e f i r s t term i n t h e sum 1 s < E / 2

with t h i s choice of

6 > 0, r

6 c a n b e c h o s e n i n s u c h a way i n d e p c n d e n t l y of

s e c o n d term i n t h e s u m i s a l s o s m a l l e r t h a n x'

(by property c ) ) .

x ' .

Then,

can b e chosen so c l o s e t o 1 t h a t t h e E/2

i n d e p e n d e n t l y of

0

I f we w a n t t o s o l v e t h e D i r i c h l e t p r o b l e m f o r a b a l l c e n t e r e d a t xo

8

lRn

and having r a d i u s

R , a l l w e have t o do i s t o reduce t h e

p r o b l e m t o t h e u n i t b a l l by t r a n s l a t i o n a n d d i l a t i o n . L e t f(xotRx')

be t h e

for x' f C the function g(x') n- 1 and s o l v e t h e D i r i c h l e t problem i n t h e u n i t b a l l w i t h

boundary f u n c t i o n . Then c o n s i d e r =

f

=

I. CLASSICAL HARDY SPACES

22

boundary function v(x)

g.

The solution will be

1

=

x-xo

Then u(x) = v( let problem for

P(x,s)

g(s)

ds

1,-1

7)

will be the solution of the original Dirich-

B(xO,R).

We can write

The existence of solution

for

th e

Dirichlet problem in a

ball allows us to make the following useful observation 7H&Ol?&fl 1 . 3 0 .

S u p p o ~ et h u t

u

R

i n c o n t i n u o u n i n un open n e t

n u t i n & i e n t h e k o l l o w i n q n e e m i n y l y weuken

konm ok

&

t h e meun u u l u e

x o e Cl, t h e a e i n u 4 e ~ u e n c . eo & p o n i t i v e numkenn r. 0 (& r' s d e p e n d i n g p O h 4 i k l y o n t h e p u n t i c u l u n xo) & J j t h u t & o n euch r . . J ' Lon e u c h

pnopentg:

+

u(xo)

Then

u

u(x +r

=

n-1

xo

e

R

J

x') dx'

R

i n huamonic i n

Proof: Let

O

and suppose that

B(xO,R)c

R

,

Let

v

be the

with boundary solution of the Dirichlet problem for B(xO,R) function coinciding with u. We shall show that u = v in B(xo,R). Since x o is arbitrary, this will end the proof of the theorem. Suppose that

u-v > 0

for some point of

B(xO,R), and let m = max {u(x)-v(x): x e B(xO,R)} > 0. Since u-v = 0 on the boundary of B(xO,R), the set of points of B(xO,R) where u-v

attains the value m , will be a compact subset K o f B(xO,R). Let x 1 be a point of this compact K having maximal distance to xO' F o r each r > O small enough, at least half of the sphere aB(xl,r) is not in K. But then u(x,)-v(x,), over each of the spheres of center radii corresponding to contradiction. If we had

xl)

which is the average of u-v x 1 and radius r . (each of the

will have to be u-v < 0

J

< m , which is a

for some point in

B(xO,R)

we

would proceed exactly in the same way but using the minimum instead of the maximum. Theorem 1.30

allows us to give a very simple proof o f the following

netlection pninciple

I. 1. HARMONIC FUNCTIONS Be u d o m u i n i n R" , xn = 0. S u p p o n e t h u t u

7HEOREM 1. 3 7 . t o t h e hypenplune

in xn

Q

Q + = { x e Q : xn > 01

Q, hunmonic i n

,

t h u t in:

u(xl,

x = (xl,

eveng

.. . ,

..., xn)

x n-1 '

=

-Xn)

u

Q. 7hen,

e

Proof: We just need to show that

u

23

nymrnetnic w i t h nenpect

in u ,!unction

continuoun

and odd i n t h e uu/riuBle

...,

-U(X1, Xn-1'Xn) in hunmonic i n Q.

. & I

satisfies the mean value proper-

ty in the form appearing in 1 . 3 0 . But this is obvious, because: w e know that u is harmonic in nt a n d , consequen1) If xo e 51' tly for those r's such that B(xo,r) c Q t , u(xo) coincides with 2) If the average of u over the sphere aB(xo,r). xo e $2- = {x

...

f

n:

xn < 01, we have the same situation because for

x = (xl, xn)eQ& and this implies that

...

)

U(Xl'

. . . I

Au(x)

= 0

...,

x n-1 ' -xn) = -U(X1, also in R-. 3 ) If

Xn-1' xo = (xl,

-Xn)

...

x n-1' 0 ) , we have u(xo) = 0 and also, the average of u over each sphere centered at xo is necessarily o because u takes opposite values at a point and its symmetric in the other halfspace. (

0 As a last application of the mean value property we shall prove the

following extension of a classical theorem of Liouville:

7HEOR&fl 1. 32. 7he only Bounded hunmonic kunctionn i n lRn

une t h e

conntuntn,

Proof: Suppose every

x

e

u

R n and also

is harmonic in

Rn. Let

x1

and

x2

lu(x)l

5

M <

for

IR".

be two arbitrary points chosen in

Then, using the mean value property we have U(X1)

Take

- u(x 2 ) r

=

I

1 B(xl ' r,

I

u(x)dx 'B(xl, r )

5

Mn

-1

I

1 B(x2' r,

1

u(x)dx 'B(x2,

r)

x1 - x21 = d. Then

much bigger than

lu(~l)-u(x2)1

-

IB

is the symmetric difference o f the two balls. Then, since lB(xl,r)\B(x2,r)l

5

IB(0,l)l

(rn-(r-d)n)

''n-11

5 ___

n r n-1 d

=

11 n-1

I

rn-'d

Of course, the Dirichlet problem can also be posed f o r an unbounded

I. CLASSICAL HARDY SPACES

24

domain. As a preparation for later chapters we shall have to deal with the Dirichlet problen f o r a half-space. To set up the problem,

IRn+l

we consider the euclidean space

whose points w e denote by x = (xl, xn) e IRn and t e IR.That is, we single out the last coordinate. O u r domain will be ":RI = { (x,t) e *"+I: aIR:+l= { ( x , O ) : x e lRn 1 we shall identify : t > 0} whose boundary with IRn in the natural way. The Girichlet problem f o r IRn+l is

...

with

(x,t)

+

IR", we have to = {(x,t): x e E", t t 01, find u(x,t), continuous in Xn+l + harmonic in IRn+l and such that u(x,O) = f(x) for every x e E n . + f , a c o n m u o u s function in

the following: Given

We

observe that this problem cannot have a unique solution. Indeed

if

u(x,t)

is a solution, v(x,t)

+

u(x,t)

=

t

is a solution too.

However, we can see thpt it can hove at nost one bounded solution. We just need to prove the following iH€OR€M 7.33.

Let

ihen

u

Le u lunctior. continuoun und Lounded i n

u

IR"+' , ~ u c ht h u t

and haarnonic in

identically

i4

o

{on

eueng

IRn+l +

x e 1 ~ "

.

zeno.

in

IRn+'

u(x,t)

if

t r O

-u(x,-t)

if

t 5 0

P r o o r ' : Define a function

v(x,t)

U(X,O) =

v(x,t)

by letting

=

According to theorem 1.31,

is harmonic in

v

Bntl.

It is also, obviously, bounded. By theorem 1.32, it has to be a c o n s tant. Since v(x,O) = u(x,O) = 0, we have, necessarily v =_ 0 a n d , therefore

u E 0.

0

Next we shall seek a solution of the Dirichlet problem in nice boundary functions, say Fourier transform

F

f

f

En (see Stein-Weiss

in

Bn+l for +

Cy(IRn). O u r tool will be now the

123 ,Chapter

I), just

a s the Fourier series led us to the solution for the disk. If u(x,t)

is a solution, we shall have 2

AU(X,t)

=

AxU(X,t) t

a u 7 (x,t) at

I

u(x,O)

=

i(x),

x

f

xn.

= 0

x e.'lR", t > 0

1.1. HARMONIC FUNCTIONS

25

We take Fourier transforms with respect to the variable call

F(u(.,t))(6)

=

h(6,t)

and

Ff(6)

=

z(6) ,

x. If we

we shall have

n

For each 6 , this is a very simple linear ordinary differential equation with constant coefficients. The only solution that remains

-2n16't.

bounded for t * is h(6.t) = i(6)e transform of this function in the variable solution u(x,t) of our problem

If we call P(x) = F(e-2nl.l )(x), c,+ e - 2 n l ~ l t e 2 n i ~ * xwill be Pt(Y-X)

=

Pt(X-Y)

5

The inverse Fourier will be, hopefully,the

then the Fourier transform of

where

Pt(x> = t-"P(t-lx)

Thus, our candidate for a solution is (1.34)

u(x,t)

=

Pt is called the Poisson kernel for it i3 very simple to obtain

Then

Pt(i)

=

it

IRn+l + (or for

Xn). For

n = 1

P(x)

and the proposed solution is in this case: +x

n > 1 , the computation of P(x) is not s o simple and we refer the reader to Stein-Weiss [2] Ch. I, where it is proved that

For

I. CLASSICAL HARDY SPACES

26

IRn

T h u s , in

Pt(x)

cnt(t2+ 1x1 2)-(nt1)/2

=

The integral (1.34) is denoted a s

P(f)

(x,t)

f. We shall see that

Poisson integral of

and

and it is called the is the solution o f

P(f)

the Dirichlet problem in the 4th section of chapter I1 and we shall

IRntl the Fatou-type theorems which we have seen

try to extend to hold in the disk.

2.

SUBHARMONIC FUNCTIONS

DETINI710N 2 . 1 . A ~ u l h u n m o n i c< u n c t i o n v

function

R, w i t h

d e t i n e d on

2 v(x)<

oo

& nu ti^&-

IRn

o n u n open 4 e t

uulue4

and

i n q the Lollowinq two condition4:

uppen 4 e m i c o n t k n u o u n i n

R.

x0 e Q, t h e n e r

lull

i)

v

i i )

Fon eueny

i,5

i 4

u

4uch t h u t &on eueny

To say that

v

t e IR, the set

B(xO,r(xO))

c 0, r(xo)

> 0,

0 < r < r(xo)

is upper semicontinuous in R means that for every {x f R : v(x) < t} is open. This i s , in turn, clearly

equivalent to the following: (2.3)

For every

xo

f

R: 1im.sup.

v(x)

5

v(x,)

x+x0 in R Like continuity, uppsr semicontinuity is a pointwise property. When the inequality in (2.3) holds for a given v is upper semicontinuous at xO' Observe that the upper semicontinuity o f that

v(x)

<

m

for every

xo

R , it is said that

v , together with the fact

x e R , imply that

v

is bounded above o n

K: v(x) 2 j ] for J j = 1 , 2 , ... T h e s e K . s are compact sets and K K1 K2 ... J Since n K . = ( x 0 K: v(x) = a) = 8 by assumption, we must have every compact

K c R . Indeed, let

e

1

j=1 .I

K.

=

[x

f

1.2.

1:.

J

=

0

for some

SUBHARMONIC FUNCTIONS

< j

j , that is: v(x)

27

for every

K.

x

This is what allows u s to assign a meaning to the integrals appearing in

(2.2).

Each of these integrals is defined a s the integral of the positive part of the function

v(x0tro)

minus the integral of its negative

part. This difference makes sense because the integral of the positive part is finite, being the integral of a bounded function. A

.

priori, the integrals in ( 2 . 2 ) can be either a real number o r -CO Actually, we shall eventually show that, unless v is identically equal to

none of these integrals can be

-m,

in such a way that

-m,

is integrable over the corresponding spheres.

v

It has to be noted that, for

v(x,)

lim. sup. v(x) x+xo in Q

5

v

subharmonic, ( 2 . 2 )

implies that

a n d , consequently, we actually have equality

in ( 2 . 3 ) . Here i s a useful characterization of upper semicontinuity.

PAOPOSI7ION 2 . 4 . v i4 upne/r nemicontinuoun i n eue/ry cornpuct K c a, v i n the limit ouen

{OR

Q

it und only it,

K

o,!

u decneuniny

01 continuoun u unction^.

neyuence

Proof: First of a l l , we see that the infimum

of a family

v

v c1

of

upper semicontinuous functions, is itself upper semicontinuous. Indeed

< t)

(x: v(x)

{x: va(x) < t) , is an open s e t , since n particular, if v is the limit of a de

=bi

it is union of open sets.

creasing sequence of continuous functions in

For the converse, if

v

compact subset of

and

covering of be continuous

K

K, v

will be upper

K.

semicontinuous in

Q

is upper semicontinuous in

by balls

E

> 0

Q,

K is a

is small enough, we find a finite

B(xj,~) c

with

2 0 , with support contained in

x . e K. Then let J

B(xj,~)

@j

and such that

I

bj(x)

=

1

for every

x e K. (The

@j s

form what is known a s a

I

partition of the unity in

K

subordinated to the covering. See

Rudin [ l ) ) Let

m.

=

sup B(Xj9~)

v

and consider the continuous function

I. CLASSICAL HARDY SPACES

28

We can do this for a decreasing sequence $(x) = 1 mj bj(x). cl > c2J> -+ 0, obtaining corresponding functions JI1, J12, . Then let u 1 = u 2 = min(a2,ul) . . . , u . = m i n ( @ . , ~ ~ - ~ ) , . In J J and this this w a y , we obtain continuous functions u 1 2 u 2 2 sequence can be easily seen to converge to v by using the upper semicontinuity of v. [I

...

...,

...

...

Of course, any real-valued harmonic function is subharmonic, since it is continuous and has the mean value property ( 1 . 2 3 ) , whcih is stronger than ( 2 . 2 ) . However, subharmonicity is all that is needed for the maximum principle. We can state T H € O R & f l 2.5. Then

v

R

Be u n u B h u n m o n i c t u n c t i o n i n u d o m u i n

it i n

cunnot u t t u i n u maximum v u l u e unlenn

V

C

Bn-

conntunt.

Proof: The proof of corollary 1 . 2 5 applies to this case with minor changes.

n

The maximum principle can also be given in this form

COROLLARY 2 . 6 . V : R 4 [-m,m) Then

R

Re u Bounded d o m a i n i n

Be u p p e n n e m i c o n t i n u o u n i n

-

R

h u n u maximum i n

V

t h e Boundanzy i f

v

an

-

R

it u t

und u t t u i n n

und l e t

und 4 u B h u n m o n i c i n

R.

t h e Roundunzy ( o n &

at

,L -

v

Proof: The fact that

has a maximum in

3

follows simply

from

the upper semicontinuity (exactly in the same way that the fact that Then, we just need to apply theorem 2 . 5 . 0

it is bounded above).

The maximum principle can be used to establish the following characterization of subharmonic functions, which is the best justification for the name "subharmonic". TH€Ol?&fl 2 . 7 .

R.

open n e t

: Then,

R

a)

i 4

b)

Uheneven

i 4

CL

hunmonic i n

G,

G

and

--+

4uLhunmonic i n

u

Be u p p e n n e m i c o n t i n u o u n i n t h e

[-m,m)

t h e t o l l o w i n g c o n d i t i o n n une e q u i v a l e n t :

nutinfie4

R

neul-uulued

continuoun t u n c t i o n i n

k e i n q u n o p e n und Bounded n e t w i t h V(X)

s

U(X)

/on

euenx

x e aG,

G, G c R,

then

29

1.2. SUBHARMONIC FUNCTIONS

v(x)

x e G.

/ o & eueny

u(x)

S

Proof: Suppose

a) holds and

G

We can assume that

satisfies the assumption made in

u

is connected. Then the function

-

G , subharmonic in

semicontinuous in

G

and

v-u

0

5

that

Conversely, assuming that

b) holds, let u s prove that

harmonic. Let

xo

holds. Since 2.4

and

f

B(xo,r)

c

also in

n.

u . 5 3

aB(xo,r).

. ..

continuous in

the function continuous in u. J

in

Dirichlet problem in

a) implies x e B(xo,r). 5

aB(xo,r),

v(x)

B(xo,r)

aB(xo,r), B(xo,r)

5

uj(x)

u1

converging to

2

v

u2

5

b) yield

u.(x) J We can write:

lim u.(xo) j+m J

=

and harmonic in

B(xo,r) u.. J

with boundary function uj+l

5

u . J

which

j

It follows

x e B(xo,r).

and every

and the already proved fact that

u J+l(x) .

1 lim --

...

in

that i s , the solution o f the

for every

Also, the subharmonicity of

v(xo)

is sub-

v

Abusing the notation a little, let u s denote also b y

b) that

It

is upper semicontinuous, we know from Proposition

v

coincides with from

aG.

We shall prove that ( 2 . 2 )

that there is a decreasing sequence of functions

... 5

in

b)

is upper

G.

follows from corollary 2 . 6

v-u 2 0

v-u

for every

u .(xO+ra)

J

and every

j

da

=

and ( 2 . 2 ) is proved. The interchanging of the limit and the integral sign is justified by the monotone convergence theorem. Observe that in the course of proving theorem 2 . 7 that if

v

is subharmonic in

u

we have also proved

R , then ( 2 . 2 ) holds whenever

B(xo,r) c R. This is stronger than what we assumed in the definition of subhafmonicity, namely, that ( 2 . 2 ) holds for small enough radii, r < r(xo)

depending on the point

Proposition 2 . 4 .

x o e R.

provides a very useful technique for dealing with

subharmonic functions. Two more examples of its use follow

PROPOSITION 2 . 8 ,

v

k e u 4ukhuxmonic kunction i n u domuin

I. CLASSICAL HARDY SPACES

30

Proof: Let

c R , and let

B(xo,r)

u1

u 2 >

2

...

Z U . ~ . . . bea

J

sequence o f continuous functions in aB(xo,r) converging to v in aB(xo,r). As before, consider extended a s a harmonic function in B(xo,r) , continuous in B(xo,r). Then, for every x e lRn with

ui

1x1 < 1

is:

v(xo+rx)

5

lim u.(x

=

T&-r

I,

(~n-l(

j+m

P(x,s)

lim u.(x J

j+m

n-1

I,

1

+rx) = lim

J

j+m

O

trs) ds

P(x,s)

u.(x +rs) ds J

n-1

=

T k T

I,

=

o

P(x,s)v(xo+rs)ds, n- 1

where w e have used the monotone convergence theorem (note that P(x,s) 2 0). Now, if v(x 0 tro) da =

I,

-m,

n-1

then also

because

P(x,s)

follows that

is positive

v(x)

=

and bounded as a function of

for every

-03

s. It

x e B(xo,r).

What we have shown above implies that the set A = Ix e R :

v(x+rO)

for some

do =

n-1 is open. But its complement B = R \ A xo e B and B(xo,r) C R , no point of because this would imply that

1 for some

v(x 0 +r'U) do

In-1 r' < r ,

since

subset of the sphere Therefore, either

A =

r > 0

with

B(x,r)

c 0)

is also open. Indeed, if A can belong to B(xo,r)

= -a

v

would be equal to

-a

in a whole open

E(xo,r').

0

or

A

=

0 , and in this latter case

v

is

1.2. SUBHARMONIC FUNCTIONS

identically

-m.

i n an i n c ~ e u ~ i ntunction y i n the intenual r l < r 2 < R. Let

Proof: Let

of functions continuous in F o r each

in

and harmonic in

aB(0,r2).

aB(0,r2)

... L

...

u . 2 J

converging t o

be a sequence in

v

aB(0,r2).

u . the function continuous in J

B(0,r2)

u.(r20)do

u .

coinciding with our original

1

--+

m(r,)

v(r

___

J

2

U)dO

m(r

=

2

)

n-1

n-1 Thus

[O,R).

Then

1

-~

u1 2 u 2 2

j , we also denote by

B(0,r2)

31

and o u r statement is p r o v e d . 0

m(r2)

It has t o be observed that in the definition 2.1 of a subharmonic function, condition

FOR eueay

ii)'

xo

E

ii) can be replaced by:

fi

and euenq

r > 0

nuch t h u t

B(xo,r)

c

fi

this for every r > 0 with fi, but just for those satisfying 0 < r < r(xo) for some

We do not even need to require

B(xg,r) r(xo)

c

with

B(xo,r(xo))

c fi.

Indeed, if we have (2.2).with

s

in place of

s

r , for every

(O,r],

we get condition ii)' just by integrating both sides of Tl-1 between o and r , exactly a s we did in the (2.2) against s E

remark following theorem 1.22. Conversely if the restriction

r < r(xo),

2 . 5 o r corollary 1.25, that

v

once this is done, we see that 2.7

a n d , consequently, v

ii)'

holds even with

then it can be seen, just as in theorem satisfies the maximum principle; and v

satisfies property

is subharmonic.

b) in theorem

I. CLASSICAL HARDY SPACES

32

v e C

For

2

(a), the

method of proof of theorem 1.22 gives a necessary

and sufficient condition for

to be subharmonic namely, we must have Av(x) 2 0 for every x e R . necessity u s e s also proposition 2.9. The criterion to arbitrary functions by interpreting A v in the sense ( s e e Stein-Weiss [ 2 ] , Ch. 11), but we shall

in terms of A v , The proof o f the can b e extended distributions not make u s e of

v

this. We shall however need the following

PROPOSI7ION 2 . 1 0 . Suppone t h a t { u n c t i o n on a n o p e n n e t

R o = {x e R: v(x)

open n e t

v

i n a non-negative

v

v

nuch t h a t

> 01

und n a t i n L i e n

i n nukhanmonic i n t h e w h o l e n e t

Proof: We already know that

v

i n o!,

continuoun

C2

clunn

Av

on t h e

Ro.

2 0

7hell,

R. Ro,

is subharmonic in

s o that we

only need to consider points xo f f? such that v(xo) = 0. Let xO be one of such points, and suppose B(xo,r) C R . W e must show that

Let

u

b e harmonic in

B(xo,r)

coinc ding with

of B(xo,r). It is enough to show that Suppose that v(x) - u(x) x e B(xo,r). and define then max

(v(x)

- u(x))

=

v(x

> o

v

in the boundary 5 u(x) for every for some x e B(xo,r),

6

x e B(xo,r)

x E A , w e have v(x) = u(x) t 6 >= 6 > 0, s o that A c f i g . On the other hand, A is B(xo,r). It turns out that A is also a non-empty closed subset o f c B(xo,r) a o . Then, open. Indeed, let x e A , and let B(x,r') since v-u is subharmonic in f i g , we have

n

6

=

v(x)

- u(x)

(v(xtr'U)-u(x+r'U))

5

dU 5 6

n-1 It follows that

v-u

6

all over B(x,r'). Since A is both open and closed, it must be A = B(xo,r), but this is impossible because v(x ) - u(x ) = -u(x ) 5 0. 0

0

=

0

0

TH€OREM 2.11. v k e a n u k h u n m o n i c L u n c t i o n i n a, and 4Upp04e i n a j ! u n c t i o n i n c n e a n i n q and c o n v e x i n IR. T h e n t h e c o m p o 4 i t e

@

1.2. SUBHARMONIC FUNCTIONS

{unction

i n u l n a nukhanmonic (&Line

@OV

. 7hat

kecomen continuoun at-m

33

do t h a t

@(-m)

@

the componition will alwayn

way,

make nenne) Proof: First we have to s e e that

@ov

is upper semicontinuous, i.e.

that

(@ov)-l ([-m,t)) is always open. But (@ov)-'( l-m,t)) = = v-'(@-'( [-m,t)) a n d , since @ is increasing and continuous, @-'([-m,t))

=

continuous

-1 v (

unless it i s empty. Since

~ - m , s )

is open. Thus,

[-m,s))

open. Next, let

v

i s upper semi-

[-m,t))

(@OV)-'(

is always

c $2. Then

B(xo,r)

the last inequality being a consequence of convexity (Jensen's inequality).

0

We shall give now our main example of a subharmonic function, namely, log IF(z)I for F holomorphic not identically 0 in a plane domain. The subharmonicity o f this function will be one of our main

F

tools. If

log

0, then

is never

IF(z)I=

Re(1og

F(z))

,

where

l o g F(z)

can be defined locally a s a holomorphic function. It follows that for F holomorphic without zeroes l o g IF(z)I i s actually a harmonic function. In the general case, some work will be needed to get rid of the z e r o e s .

I"

LErnrnA 2 . 1 2 ,

log

I1-eitl dt

=

0

-71

Proof: Since that

1-z

l o g 11-21

is holomorphic in

D

and has nozeroes, w e s e e

is a harmonic function, and the mean value property

yields: (2.13)

,:j

I l-reitl

log

Now we can let

r

tend to

theorem. Indeed, for

l l o g 11-re

for any

it

0<

I c1

=

dt = 0

log

1

1 l-reitl

< 1 ; and if

r e [0,1)

every

and use Lebesgue dominated convergence

It1 < n / 3 ,

1

for

we have:

- log -2

1

5 log

J l + r -2r cos t

It1 2 n / 3 ,

then

(J3/2) 5

1

5

Jsin tJ

I l-reitl

5

L

I. CLASSICAL HARDY SPACES

34

and

\ l o g \l-reitl

1

6

l o g 2. T h u s , the integrands in (2.13) are

bounded in absolute value by the same integrable function.

0

(Zennen'4 ,!omnula) Let F Le h o l o m o n p h i c i n D(0,R) und n u p p o n e t h u t F ( 0 ) f 0 . Let 0 < r < R und c u l l z l , . z 2 , ..., z t h e z e n o e n O F F & D(O,r) l i h t e d uccoizdinq t o t h e i n multiplicitie4. Then: 7 H E O R f R 2.14.

...

~

n

l o g IF(O)I

Proof: Suppose Define

1 j=l

t

zl,

...,

z

TI

log

- 2.r

e D(O,r) 3

G

I

IZj

l o g IG(z)I

log

it

)I

dt.

...

=

lznl =

IF(re

-TI

and

IzmtlI =

D(0,rt~)

will b e harmonic in

for some

D(0,rtE)

have :

But

Since

1

I l-ei(t-t,J ) I

-

we finally obtain: 11

log

IF(O)J

t

1

j=l

log

- -

Pjl - 2n

by the previous lemma.

0

I

-

is holomorphic and nowhere zero in

therefore

1

I"

l o g IF(re

it

)I

dt t

E

> 0,

and we shall

1.2. SUBHARMONIC FUNCTIONS

35

COROLLARY 2.75. F l e h o l o m o n p h i c , n o t i d e n t i c a l l y 0 on a n R c C . hen, t h e ~ u n c t i o n n log IF(^) 1 , logt IF(^) I = open 4 e t = max(1og IF(~)I, 0 ) I F ( z ) ~,/on ~ a n 4 o < a < m, a n e

R.

nulhanmonic i n

Proof: First of all. let u s s e e that the function

log IF(z)I

subharmonic. It is a continuous function with values in c Q , w e have: Desides, if

'v

& 1-n TI

log

(F(zo)( 5

log

(F(zo+reit

)I

is

[-m,co).

dt

This is clear i f F(zo) = 0. If this is not the case, it follows F(zotz) which is from theorem 2 . 1 4 applied to the function z holomorphic in D(0,rt~) for some E > 0 and does not vanish at 0. As for the functions

IF(z)I

log'

log IF(z)I

from composing

and

a > 0, they result

IF(z)la,

with the functions

$(t)

=

or

max(t,O)

$(t) = eat respectively, which are increasing and convex. The subharmonicity follows from theorem 2.11.

0

THEOREM 2 . 7 6 .

and

Fan F

0 s r < 1, w e

F

-/hen, t o n e a c h

e

H(D)

(that

i 4 :

F

0

5 p 2

i 4

holomonphic i n

e

incneaniny Lunction

H(D) oL

and e a c h

r

&

mr

mp(F,r)

i 4

an

[O,1).

Proof: This is just a consequence of corollary 2 . 1 5 2.9.

D),

detine:

and proposition

0

Theorem 2.16. (due to Hardy [l]) theory of Hardy spaces DEFINITION nimply

2.17.

ly

Hp

o

(Hp) < p

is the starting point o f the introduced by F. Riesz in 111

5 m,

when n o c o n , l u 4 i o n

we 4 h a ~ d i eline

i n

HP(D)

-

(a140 denoted

L i k e l y t o n n i n e ] t o le t h e

I. CLASSICAL HARDY SPACES

36

{ o l l o w i n q c l u 4 ~o# b u n c t i o n n :

F O R p = 0,

we huue t h e N e u u n l i n n u

N =

IF

0 < p < q <

e

H(D):

&&

PROPOSITION 2 . 1 8 .

F e N

huue

NI d e k i n e d Lg: < -1

sup m0(F,r) O S r < l

we c l e a & y

mI

C ~ U A A

Ha c Hq c

k e nuch t h u t

Hp

c N

F(0)

f 0.

i

log'

. Then

Proof:

-a

< log IF(O)I

1

T;; 1 T I

5

log JF(re it )Idt

-TI

=

TI

IF(reit)Idt

-n

Thus

and, consequently TI

&I

&1

TI

llog IF(re it ) I [ d t = -TI

-TI

n +

log2TI

IF(reit)l

dt S

log'

1"

IF(reit))dt log'

IF(reit)l

+ d t - log IF(0)l

-TI

-.'TI

From this inequality, o u r claim follows immediately. Next, we shall

Jensen's formula to derive a basic fact, namely:

use

F

that the zeroes of

e

OK

3

N

cannot be too far from the boundary.

F e N

7HCOi?€fl 2 . 1 9 . Suppone

z . Le t h e ze.zoen

F

& F

1 j

(1 -

i 4

not i d e n t i c a l l y

0.

t i n t e d uccoadiny t o t h e i n m u l t i p l i c i t i e n .

Then: (2.20)

u

lZjl)

<

-

-

1.2. SUBHARMONIC FUNCTIONS

Izll

Proof:We may assume

1z21

5

...

s

and

Applying Jensen’s formula, we get, for every

M

with

independent of

r. Given

I zn I

close enough to 1 , to have

..., n.

j = 1, 2 ,

is true for all Letting

r

Thus, for

r

n

1

i1 log $J

2

1

1 log

c

0.

0 < r < 1

a n d , consequently

r < 1

I zJ.I

6 r

hr

fixed

r(n)

which depends on

n.

we obtain

I

M - log I F ( O )

Since this is true for every m

F(0)

n , we just need to take r

5

bigger than certain

tend to

37

s M

-

n , we have

<

log IF(O)I

m .

1 1

Now, observing that

-

lzjl

-,

log

Izj

I

we get

( 2 . 2 0 ) . ~

The importance of condition ( 2 . 2 0 ) comes to light in the following result. Ffl&Of?&h 2.27.

< 1,

Let

(zj)

be a Aeguence

-

( 2 . 2 0 1 h u l d ~ .L e t

ouch t h a t

k

UL

0 < J z j J<

complex numbea4

be a n u n - n e p a t c v e

cntegee.

Then -

t h e ”Blaochke pnuduct”. 2

B ( z ) = zk

j

- 2

-

j=ll-z z j

2.

J

cunvengeo uncLo/zmly u n e a c h compact o u 6 4 e t u f Lunctcon u/zde/z

B e Hm 0

k

luhooe senueo a/ze p n e c c o e l y

&

t h e unct dc4k, the

k > 0.

z’.s 3

plu4

t o a

-

a Seeu u c

Proof: A l l we need to prove is that the series m

j=l

converges uniformly on compact subsets o f we have the estimate:

D. But, for

(zI

5

r < 1,

I. CLASSICAL HARDY SPACES

38

( 2 . 2 0 ) is sufficient for

that

so

B

Since

our purposes.

HOD, Fatou's theorem implies that

f

B

0 has non-tangential B(eit) =

limits at almost every boundary point. We shall write N.T., eit = lim B(z) (as z 1.

F

In general, if for some function dary value of F(eit).

F

If

e

F

D , the non-tangential boun-

in

i s known t o exist at

Hp with

p 2 1 , we know that

eit, we shall denote it by

F(eit)

exists a.e.,

since

F

is a Poisson (or Poisson-Stieltjes) integral according to theorems 1.3 and 1 . 8 . In the next section we shall extend this result to any p > 0.

B

7HEOREM 2.22.

7hen

2.21.

IB(eit)l

=

& I-n

Be the Beanchke pnoduct 1 con a.e. t &

uppewzing i n theonem

II

lim r+ 1

l o g IB(reit)l

dt = 0 .

Proof: We know that the limit exists because the integral is an increasing function of r. Also lB(z)l 5 1 , s o that 2 0 and Fatou's lemma can be used to obtain:

J

1"

7l

log(l/lB(reit)l)

dt

5

-71

lim r+l

log(l/lB(reit)l)

log(l/lB(reit)l)z

dt

-'TI

or equivalently:

I

71

lim r+l

log lB(reit)l

-n

dt

5

1'

log

B(eit)l

dt

5

0.

-71

I f we prove that this limit is also

20

w e shall be done. Let u s

call

Note that

IBn(e

it

)I

= 1

a.e. and also

IBn(reit)l

+

1

uniformly

I.3. F , RIESZ FACTORIZATION

as

r

+

Bn

1, since

T h u s , for every

C

log

12.1 6 J

is holomorphic in a neighbourhood of

-

D. Then:

n:

1"

m

n+l

39

lim ~ i ; log IB(reit)l r+1 -71

dt.

Now we realize that m

for a

C

m

depending on how small is the smallest

z

j'

It follows that

m

1

log

(2.1

n+l

J

-+

o

as

n

-+

and, consequently:

3.

F. RIESZ FACTORIZATION THEOREM F

For

e

H(D)

and

0 < p <

41,

we have denoted by

I)FllHp

the

quantity

with the obvious modification for

Hp

the space If

1

4 p 5 m

m (F+G,r)

P

get

6

consisting of those and

F,G

p =

F

f

m.

Then, we have introduced for which llFllHp < a.

H(D)

H(D), Minkowski's inequality gives: for 0 5 r < 1, s o , letting r + 1, we mp(F,r)+mp(G,r) f

I. CLASSICAL HARDY SPACES

40

Since, clearly, )I IIHp is homogeneous, and IIFIIHP = O implies F: it follows that, for 1 d p d m , Hp is a linear space and 11 11

0.

HP

is a norm on it.

llHP

If 0 < p < 1 , I( no longer satisfies the triangle inequality. Instead, in this case, we have, for F,G e H(D):

that

so

11

(F,G)e

[lip

-

IIF

Hp. Of course

is an invariant metric on

Hp

is p-homogeneous. It follows that space and 11 IIKp i s a p-norm on it. *

is still a linear

In this section we shall look a t properties of the individual func-

Hp. In later sections we shall study

tions in

Hp

as

a metric

linear space. Among all the H p A , H2 is the simplest to characterize. Indeed if e H(D) has the power series expansion F(z) = 1 a.zj, the j20 J function tt+ F(reit) will have the Fourier series 1 a rJ e i J t and by Plancherel's theorem

F

jbo

J m

so

that m

IIFllHz 2 Thus

F

e

H2

= lim

m2(F,r)'

r+ 1

1

=

laj( 2

j =O

if and only if

1

laj/' <

But, if this is the

m.

case, the Riesz-Fisher theorem J r o function

f e L2([-n,T])

the Poisson integral of

tells us that there is some with Fourier series 1 a eijt. Then, jzpjt J f will be 1 a j rJ e = F(reit). Thus, jro

1

the functions in tions in

L2

HL

whose

are precisely the Poisson integrals of funcFourier coefficients

It follows from theorem

j20

are

0

for each

J

j < 0.

a t least for

a<

1.3. that this characterization remains trle Indeed, if F e H p , 1 < p d and

1 < p 2 a. z J , w e know that

F

=

P(f)

with

f(t)

= F(eit)

41

1.3. F. RIESZ FACTORIZATION

L p , and also

belonging to

In

- F(eit)l

IF(reit)

dt

0,

+

as

r

(1 < P < "1

1

+

-n F(reit) * F(eit)

and

Lm

*-weakly in

r

as

1. This implies that

+

m

the Fourier series of

f(t)

F(eit)

=

is

1 a,eijt.

j=o

J

All we can say at this point about boundary bahaviour and representation in terms of the boundary functiol is collected in the following

Let

THfORffl 3.1.

a)

F

e

1 < p

Hp

S m.

Then:

t, t h e l i m i t

F o n almo4t eveng

N.T. F(eit)

lim F(z)

=

ewi4b. the {unction

F

(as

f(t)

F(e

=

eit)

z

it

flelongh t o

)

Lp(

[-.,IT]) &

= P(f)

(71

I-,

p =

I

F(re m,

it

)

-

F(reit)

F(eit)lP

*

* 0

dt

as

r * 1

i n t h e weak-*

F(eit)

Lm

t o p o l o g y o{

g

r * 1.

c)

F

08

i n t h e Cnuchy i n t e g n a l

i t 4

floundany i u n c t i o n ; t h a t

F

P(f)

Proof: Theorem 1.3 implies that Then, corollary 1.21. gives ce in

b)

F(e

it

follows from corollary

theorem 1.18. (a) Fatou's lemma gives from theorem 1.11.

for

p =

Ilfllp

4

=

for some

) = f(t)

1.17. (a)

f

f

i 4 :

Lp(Cn,.)).

for a.e.t. 'The convergenfor

p <

m

and from

m.

IIFllp

and the converse inequality follows

I. CLASSICAL HARDY SPACES

42

As f o r

c ) , we

r < 1, Cauchy's formula holds:

know t h a t f o r

F(rz)

=

1 -

F ( r 5 ) dC

5

2ni

- 2

T h e n a l l we h a v e t o d o i s t o l e t 7ttE0J?cfi

3,2.

Let

l i m sup r + l HencQ,

ii

F(eit)

F

1

and use

N,

\ l o g IF(reit)ll

dt <

1"

b ) . n 0 . Then:

not identically

m

-TI

= liiii F ( r e i t )

TI

nnd,

-+

&e u i u n c t i o n i n

huue p n o u e d t h i n o n l i * j o n

I

r

[ l o g IF(eit)l

e r i n t n u.e.

HP,

F

f

I

dt <

( I L u l w u g n d o e n , flu2 we

p 2 11, t h e n

m

-TI

F(eit)

connequentlg,

c u n u a n i n h o n l y on u n e t o!,

meunu~e

0.

P r o o f : A l l we n e e d t o p r o v e i s t h e f i n i t e n e s s o f t h e l i m s u p , b e c a u s e then t h e i n t e g r a b i l i t y of

lemma a n d , s i n c e

F(e

it

lF(eit))

log

) = 0

f o l l o w s by u s i n g F a t o u ' s

p r e c i s e l y where

and t h i s can only happen on a set of measure

[log IF(e

it

)I I

= m,

0, everything w i l l be

proved. Suppose f i r s t t h a t

log-

F ( 0 ) f 0. Then

IF(reit)l

dt.

Thus,

<

m

and, consequently

n

TI

llog I F ( r e i t ) l l

2IT

-77

dt I 2 sup r
log' -TI

IF(reit))

dt-log

IF(O)I =

1.3. F. RIESZ FACTORIZATION

43

which completes the proof in this case. If -m

F(0)

=

F(z)

0, let

< log IH(0)l

5

5

=

In

zk H(z)

H(0) f 0. Then

with

log IH(re

it

)I

l

dt =

n it log IF(re )Idt-n

-71

- k log r

and we get, exactly a s before,

-

log IH(0)l - k l o g r r > 1/2,

The right hand side is uniformly bounded for ends the proof.

s a y , and this

0

Now we wish to study the boundary behaviour of an F e H p with 0 < p 5 1. For p = I w e know the existence o f thz boundary function f . But w e do not know whether F has to be necessarily the Poisson integral of f. We shall s e e later that, indeed, F = P ( f ) . This i s very interesting because i f we merely knew that F is harmonic, the functions

tw F(reit)

F

could say is that

necessarily a function. For F(eit)

being still uniformly in

L',

all we

is the Poisson integral o f some measure, not p < 1 , we do not even know whether

exists.

Suppose that F f H p , 0 < p < m , and F does not have any zeroes in D. Then F(z) = G(z)' for some G f H(D). Since 1G(z)12p= ~F(Z)(~), it follows that G e H2' G(e

it

2

with IIGlk2p = IIFILp. If

1/2

5

(as L Z p , because

lim F(z)

also exists a.e. and belongs to Then we can g o to

1/4

5

p < 1/2

(as

z

--

z

eit)

2 p 2 1. It follows that

N.T. =

-

N.T. ) = lim G(z)

exists a.e. and belongs to F(eit)

p < 1, we know that

it +

e

) = G(e

it)2

Lp.

and

so

on. In other words, we can

I. CLASSICAL HARDY SPACES

44

prove by induction, that if F f €Ip, 0 < p I -, and F is nowhere 0 in D , then F has a non-tangential boundary function F(eit) belonging to Lp. Of course we know that the hypothesis "F nowhere

0

D" can be removed when

in

p

e Hp, 0 < p <

F

W e shall prove next that, if

0

1. We shall eventually s e e that

Z

0 1

We know that this holds for does with

p. Indeed, let

before, F(z) = G ( z ) ~

F

with

f

F

(even for an

Now we shall s e e that if it holds with

Hp

2p

with zeroes).

in place of

p, s o it

D. Write, as

have no z e r o e s in

IIG162p = IIFlLp. Then:

n - F(eit)lP

lF(reit)

dt

-n

dt =

-n

5

1" (1

5

C llG11i2p ( /

=

- G(eit)'IP

1G(reit):!

=

IG(reit) t G(eit)IYIG(eit)

- G(reit)lP

dt 2

-n n (G(reit) t G(eit)12'

-n

IG(reit) - G(eit)12'

dt)'/2(\n

dt)1/2 ,<

-TI

n IG(reit) - G(eit)12P

dt)ll2

0

+

as

r

-+

1.

-71

Again, by

induction, we prove the convergence for any

Observe that, for

p

=

p > 0.

1 , this convergence implies the Poisson and

Cauchy representations in terms of the boundary function f o r an

F

e

HI

without zeroes.

Now the problem is to see what can be done for functions with zeroes.

F. Riesz made the fundamental observation that the z e r o e s do not matter because we can factor them out. Indeed, if F e H p , 0 < p 5 m ( o r , even more generally, if f e N), and is not identically 0, we that the z e r o e s

have seen (theorem 2.19).

{ z . )

J

of

F , listed

according to their multiplicities, satisfy m

l

(1 -

lZjl)

<

m,

j=l

and this is precisely the condition that guarantees the convergence of the Blaschke product B formed with the sequence ( 2 . ) (theorem J 2.21). Then we have a function B e Hm with exactly the same zeroes

1.3. F. RIESZ FACTORIZATION

45

F and with IB(eit)l = 1 a.e.. !Je can write H e H(D) does not have any zeroes. as

F = B - H , where

Here is a more precise statement:

F e N

7 H € O R € M 3.3.

0 , and P e t B Le t h e ok F. Then F(z) = B(z)H(z) IIHIIN = llFllN and i# F e H p ,

Le n o t i d e n t i c a l l y

B l a 4 c h k e p n o d u c t Lonrned w i t h t h e zenoed

H e N t h e n a140

u a n i n h i n q nowhene.

H e H p & llHllHp

=

Behiden

IIF(IHp.

Proof: From the simple inequality

log+(a-b) 2

log's +

b

log'

we

obtain :

I"

log+lH(re

it

-n

)I

1).

r

log'lF(1-e

it

)I

+

dt

In

1

log

-n

IB(reit)

tends

Bn(z)

I

dt

The second integral in the right hand side 1. (theorem 2 . 2 2 ) .

to

F e Hp. Exactly a s in the proof of theorem

Suppose now that

Hn(z)

In

5

-n

5

(note IB(reit)l tends to 0 a s

denote by

dt

the finite partial products of

B(z)

and

2.22,

let

F(z)/Bn(z).

=

For each

n , lBn(reit)l

+

1

uniformly in

t , as

r

+

1. Then, if

P < m

IIHnllip

If

5

=

lim r+ 1

&

I

n IF(reit)/Bn(reit)lP

dt =

-TI

p = m

lim sup (IF(re it )/Bn(reit)l r+l

But

- IF(reit)l)

IF(z)I

+

lim mm(F,r)

=

IIFIIHm

r+l

t 5

I(Hn(IHp= IIFIIHp

IH,(z)I, for all

so

that

IIHnllHm

n. Now, f o r

r

=

IIF1lHm. Thus, for any p: fixed and

n

+

I. CLASSICAL HARDY SPACES

46

+

lHn(reit)l

IH(re

it

)I.

p <

If

m ,

we apply Lebesgue's monotone

convergence theorem, to get:

1/2 l/F11IH2p= llF211u2~ = IIFIIHp Proof: Let

F(z) = B(z)

3 . 3 . Since

H

with and

l[Glli$p F2 = G.

H(z)

be the factorization given b y theorem 2 has no zeroes, it can be written a s H(z) = G(z)

IIHIIip

=

=

IIFIIEp. T h e n , it i s enough to write

F1

=

BG

0

Proof: Write, as above

F

=

B

if we let Since

F.

J'

F1

lB(z)l

j = 1, 2,

- ((1-B)/2)H

H = ((ltB)/2)H

*

= (1tB)/2,

F2

=

F1

-

F2

(l-B)/2.

=

z e D , it i s clear that the functions do not vanish anywhere and satisfy llF4lHp 6 ~ ~ F ~

< 1

for every

Now we can extend the results on boundary behaviour, obtained for non-vanishing

THEOREM 3 . 6 . a)

Hp

functions, to arbitrary

& F

e Hp

with

0 < p <

a.

Hp Then:

7he n o n - t a n q e n t i a l l i m i t

N.T. F(eit)

=

lim F(z)

as

it

z-+e

functions.

~

F

p

1.3. F. R I E S Z FACTORIZATION

F

Proof: S u p p o s e

IIHIIHp

According to theorem 3 . 3 :

F. Write

B

Since both

H

and

=

B

0. Let

i s not identically

product formed with the zeroes of

IIFIIHp

47

be the Blaschke

F(z) = B(z)

H

and

H(z).

does not vanish.

have non-tanzential limits at a.e. boundary

F ; that i s , for

point, the same holds for

a.e. t ,

N.T.

F(eit)

B(eit)

=

z

as

H(eit ) = lim F(z)

it

e

Besides, since IB(eit)l = 1 a.e., it follows that IF(eit)l = = IH(eit)l a.e., and consequently F(e it ) belongs to Lp( [-T,T]). This finishes the proof of T o prove

J"

b),

observe that - ~

IF(reit)

a).

(

-n

)Ip

I

n

e dt ~=

~ IB(reit)

H(reit) - B(eit)

H(eit)lPdt

-TI

5 C(j"

lB(reit)

H(reit)

-

B(reit)

H(eit)lP

+

dt

-TI

+

IB(reit) H(e

it

) - B(e

it

) H(eit)lP

dt) 5

-n 5

C(i"

IH(reit)

-

H(eit)lP

dt

+

0

+

- B(eit)lplH(eit)lpdt)+

IB(reit)

-n

-TI

as

r

+

1.

Indeed, the first integral in the sum tends to

0 because

not vanish and the second integral tends to

simply by dominated

0

H

does

convergence. Finally

c)

follows from

b):

and the opposite inequality for

Fatou's lemma gives

p > 1

was already obtained from

I. CLASSICAL HARDY SPACES

48

Poisson representation. I f

1"

1

p 2 1, we have:

1"

n

dt s

IF(reit)lP

- F(eit)lP

IF(reit)

dt t

-n

-n

Letting

r

1

+

IF(eit)lP

dt

-TI

and using

b)

we g e t :

This completes the proof.

Proof:

Hp

q 2 p,

If

c Hq.

Let,

F

because

n o t h i n g h a s t o be p r o v e d , s i n c e , i n t h i s case p < q. If

therefore

1 < p,

is t h e Poisson i n t e g r a l of

is supposed t o belong t o

Lq.

For the case

F

We w r i t e

f a c t o r i z a t i o n theorem.

B

=

1 , we u s e

p

H

*

where

formed w i t h t h e zeroes o f

Blaschke product

F e Hq,

it is clear t h a t

i t s b o u n d a r y f u n c t i o n a n d this

F

the

is t h e

B

H

and

i s a non-

vanishing holomorphic function such t h a t take an integer

]HIP

=

[GInP,

n

G e Hnp

it follows that

The boundary f u n c t i o n s a t i s f i e s The f a c t t h a t t o

F(eit)

G

Lnq. T h e r e f o r e

Proof:

F e H1

Let

z = reie

Hnq

and take

=

1 T;;

*

-n letting

s

+

that the functions as

Lq

=

IIGII:Rp

F

e

I F Hei;, ( I

=

G(eit)

implies that

and, consequently

llHllPp - I I F I I i p .

=

IH(elt)l

0 < s < 1 . We k n o w t h a t ,

belongs

0

Hq.

for

.s D

F(sreie) Now,

with

IG(eit)ln

belongs t o f

l l I l l l H p = IIFllr,p. T h e n we a n d write H = Gn. S i n c e

1 < np

such that

s + 1 , we g e t :

1

1 - r ltr2-2r

2

F(seit)

dt

cOs(8-t)

and using t h e f a c t , contained i n theorem 3.6., F(seit)

tend i n

L1

t o the function

F(eit)

1.3. F. RIESZ FACTORIZATION

49

which is the Poisson representation. The Cauchy representation is obtained in a similar way starting from:

COROLLARY 3 . 9 .

Proof:

IJ

F

e H

1

,

t h e n Lon eueag

We just need to observe that, if it

e i z ) -

Re(

elt-z

=

4 it

le

=

D

z e

z

re

=

i€l

, then

Pr(e-t)

-21

and therefore, G(z) =

-

Re F(eit)

dt

is a holomorphic function, whose real part coincides with that of Hence F ( z ) - G ( z ) is constant. Since that F ( z ) = G ( z ) t i Im F(0).

G ( 0 ) = Re F(O),

F.

it follows

0

As a consequence of the Poisson representation for

F. and

we can prove a very famous theorem due t o

Let

7HEOREM 3.10.

p

T

k e a B o n e 1 m e a ~ u n eo n

H 1 functions, M. Riesz.

whone F o u a i e n

c o e , ! & i c i e n t n u u n i n h Loa a & ? n e g u t i u e t a e y u e n c i e n , t h a t i4:

7hen i.e.:

p

ii

a k ~ o 1 u f e 1 yc o n t i n u o u 4 w i t h a e n p e c t 1 dp(t) = f(t) dt { o n d o m e f e L

.

F

Proof: Let F(re

where

t o LeLenyue m e u ~ u n e ,

i8

l

=

T;;

u

be the Poisson integral of

J -n

l

r

2 1 - r l+r2-2r cos(8-t)

a,

dp(t)

=

1 --m

a,rlJ1eiJe J

I. CLASSICAL HARDY SPACES

50

where

Since

a. J

0

=

j < 0 , we have:

for each

m

F(z)

a . z J , a holomorphic funct on J

=

O

On the other hand, (1.15) gives:

F

-n

HI. Call

f(t)

=

quently, the measures

f(t)

dt

so

that

e

coefficients. Hence

COi?OLLA/?Y F(eit)

3. 1 1 .

-

du(t)

F(z)

Then

D & F(eit)

F(e

in

cun

and, conse-

as was to be proved.

l e extendedto

absolutely continuous function

a & u n c t i o n c o n i n u o u 4 on

F(eit)

h(t),

coincides a.e. with an

then, since

F(z)

is the

h , corollary 1.17 (b) will yield the continuity

up to the boundary.

We are assuming that

F(e

it

) = f(t)

for a.e.t, where

function of bounded variation. We can write g(t)

P(f)

an u l 4 o l u t e ~ yc o n t i n u o u d , ! u n c t i o n .

Poisson integral of

F

=

have the same Fourier

du(t)

dt

Proof: If we are able to see that

of

F

) . Then

and

f(t)

=

it

F e H 1 und 4uppone- t h u t t h e l o u n d u n y , ! u n c t i o n u h o d euenzywhene w i t h a j u n c t i o n O & k o u n d e d

coinciden

vanintion.

0 < r < 1

for every

(t)

' -n

=

I

ijt

f(t)

is a

= c t g(t)

with

t

du(s)

for a.e.t,u

being a certain Bore1 measure. ?or

-ll

j = 1, 2,

e

f(t)

...,

du(t) =

=

we get, b y integrating by parts: -ij J n

-n

lim ( -ij r + l

e(t)eijtdt

I"

=

-ij

I",

F(eit) eijtdt

=

. .

F(reit)

elJtdt)

=

0.

-TI

S o , by the F. and M. Riesz theorem: dp(t) = k ( t ) d t with k(t) integrable a n d , consequently, g is an absolutely continuous function.

0 7 H l O R l M 3.12.

F e H(D).

F

i5

coti,nuouj u p t o

the Coiintli~~y

1.3. F. RIESZ FACTORIZATION

w i t h un UkAO-!Uteeg c o n f i n u o u n Q o u n d u n g , ! u n c t i o n F' e H 1 , B e ~ i d e h , when F ' .e H 1, w e h u u e :

d

(F(e

it

Proof: Suppose that

f(t)

N.T.

u4

F e H(D)

its boundary function

F

ik

und o n l g

iL

) ) = ieitF'(eit)

F'(eit) = lim F'(z)

of course,

51

z

is continuous up to the boundary and F(eit)

=

is absolutely continuous. Then,

is the Poisson integral of

I

it e

f:

TI

F(reie)

=

2n

Pr(e-t)

f(t)

dt

-TI

Differentiating with respect to

Thus, the holomorphic function

0

and integrating by parts we get:

izF'(z)

is the Poisson integral of

f', an integrable function. It follows that consequently, F' e H ~ . To prove the converse, let izF'(z)

H1

is also in

F e H(D)

izF'(z)

be such that

is in

F'

H1.

f

H1

and

Then,

a n d , of course, it will be the Poisson

integral of its boundary function: ireieF'(re i 8 )

=

I

T;; l T I Pr(6-t) ieitF'(eit) dt -TI

ie

it

F'(eit)

dt. T h e n , integrating by parts:

P(g)(re i8)

We have seen this way that the functions

F(re iB)

have the same derivative with respect to

8 . It follows that their

difference depends only on

and

r. But, since this difference is harmonic

and continuous at the origin, it has to be a constant finally: F = P(g+c), function. Besides:

c.

Thus,

the Poisson integral of an absolutely c o n t i n n u s

I. CLASSICAL HARDY SPACES

52

zd( F ( e it ) )

b y the definition of

CoRoLLAY j r , 1 1 . ~ equivalence Laom

~d ( g ( t ) + c)

=

ie it F'(eit)

=

[7

g.

e tr

ee u Zoadun cuave and let F ee a conLo/rmni onto the inte/Limdomuin limited fly r . & r 1 in aectitiuele it and only ik F' e H

D

.

proof) says that for any Jordan curve can be extended continuously to D

F

r,

r.

If

r

for its

the conformal equivalence

and the extension is a homeo-

morphism between the two closures. In particular, parametrization of

111

Caratheodory (see Koosis

Proof: A classical result of

F(eit)

, F(eit)

is rectifiable

is a

will be of

bounded variation a n d , by corollary 3 . 1 1 , absolutely continuous. It follows from theorem 3 . 1 2 that F' e H1. Theorem 3 . 1 2 gives also the is given by the measure

converse and implies that arc length on IF'(eit)ldt.

0

Note that, from the fact that

is absolutely continuous,

t P F(eit)

0

it follows that it carries sets of measure measure (arc-length) 0

T

in

to sets o f

r.

in

We also have the existence a.e. of the derivative d (F(e it ) ) = ieitFt(eit) f 0 (theorem 3 . 2 ) , which provides a tangent to

=

almost every point. (with respect to arc-length).

4 . SOME CLASSICAL INEQUALITIES

with u conhiant

C

independent 08

F. m

Proof: We shall see that one can find IIGIIH1

$

IIFI(H1 and

la

.I

J

G(z)

=

1

AjzJ

.i=O 2

A

j

for every

j = 0 , 1,

Once this is done, if the inequality holds for

such that

. ..

G , which has non-

at

1.4. CLASSICAL INEQUALITIES

negative coefficients, it will also hold for

53

F

with the same

constant. Thus, we shall be reduced

to proving Hardy’s inequality 0 5 a . for every j. Let u s see J G. We apply corollary 3 . 4 and write F = F1 * F2

with the additional hypothesis that how to obtain with

llFLllH2 = llF21L2

11F11Ai2. Let

=

Then we consider m

W

H 2 tells

and the Plancherel theorem characterization of

us

that

llGlllH2 = llFlllH2 and l l G 2 1 1 H 2 = i l F A l H 2 . Now, call G = G 1 * G 2 . Then II G1 IIH2 * II G2II H2 = II FllHl and m IIGIIHl G(z) = 1 A . zJ with j=o J

s o that, clearly,

la.1 J

5

A,. J

G

Thus, the function

requirements and the claim that we can assume

0

Let u s suppose, therefore, that

5

a. 3

for

0 j

=

5

satisfies o u r a. J

is justified.

0, 1 ,

...

Let

< 1 , where we have chosen the principal branch of the logarithm. The function u is harmonic in D and

u(z)

=

Im log(1-z)

for

< u(z) < n/Z.

satisfies

-n/2

log(1 - z )

we get

and can write, for every

and, since

a. 2 0

Making

1

r

+

IzI

J

yields

From the Taylor expansion of

r < 1:

I. CLASSICAL HARDY SPACES

54

m

TI

Taking into account that

a .

F(e

it

) dt, we finally arrive

-TI

at:

C

This proves Hardy's inequality with

=

1 t

71.

0

m

COROLLARY 4 . 2 ,

& D aondun

F(z)

=

onto the i n t e n i o n

fle a conkonmad e y u i u u l e n c e

a ,zJ J

d o m a i n l i m i t e d fly u

nectikiuLle

m

r . -/hen

cunue

1

j=o

1

laj) <

m.

j =O

Proof: From the fact that

r

is rectifiable , it follows that

m

F'(z)

=

1

( j t 1) a .

j=O

ZJ

is in

H 1 (corollary 3.13).

J+1 m

F'

Then, Hardy's inequality applied to

z

gives

j=O

lbjtlI

<

m . 0

The corollary we have just proved is an strengthening, for rectifiable

of the fact that

F

extends to the boundary of

D

m

with continuity. In general, f o r

F(z)

=

1

j=o

a.zJ

in

H(D),

the

J

following implications hold:

1

=+

lajl <

j=O

F

-

extends to a continuous function on

D.

However, examples can be given to show that the opposite implications do not hold. There is another way to state Hardy's inequality. We have seen that the mapping

F(z)

+ +

f ( t ) = F(eit)

is a linear isometry. By means o f this isometry we may identify with a certain subspace o f

L1

which we may also call

ambiguity disappears with the habit we have

H1. The

H1

1.4. CLASSICAL INEQUALITIES

55

developed, of using capital letters for holomorphic functions on and small type letters for functions defined on same, on

T

D

o r , what is the

I-.,.].

If

F(z)

1 a . zJ is in H , the Fourier coefficients of f(t) jrol are f(j) = a . if j = 0, 1 , 2 , and f(j) = 0 if

=

1

=

...

= F(eit) J j < 0. Thus, we can write Hardy's inequality as:

We can also formulate Hardy's inequality in terms of the space Re H1 formed by those functions on the boundary which are the real parts of the functions in H1. After all, any f 6 H1 is determined by its real part except for an additive purely imaginary constant. If we assign to every g 6 Re H1 the function F(z) in H1 such that Re F(eit) = g(t) and F(0) i s real, w e have a one to one correspondence' and we can define

llgllRe H1 = 11~11~1. If

1 a . z J , then, i(j) = -L2 a j for j > 0 . i(j) = i a-j for jr0 J j < 0 and i ( 0 ) = Im ao. Therefore Hardy'sinequality can be viewed a s an inequality for the Fourier coefficients of any f 6 Re H', F(z)

=

namely:

where the prime is used to indicate that the sum is taken over those j f 0.

The function m

f(t)

= jf2

cos( it1 log j

is in Re L1 (see Zygmund 1.1) Ch.V, 1.) and for i t , the sum in 4 . 3 is a. This shows that Re H 1 is a proper subspace of Re L1. We shall

1 < p < m , Re H p = Re Lp. After this, Hardy's inequality may be considered an extension to p = 1 of Paley's inequality which says that for f e L p with 1 < p 6 2 .

see later that, for

I. CLASSICAL HARDY SPACES

56

Of c o u r s e , f o r theorem.

p

2 , t h i s i s j u s t a c o n s e q u e n c e of P l a n c h e r e l

=

I n t h e n e x t t w o c h a p t e r s we s h a l l s e e h o w P a l e y ' s i n e q u a l i t y

c a n be o b t a i n e d b y i n t e r p o l a t i o n b e t w e e n t h e c a s e

p

=

2

and e i t h e r

(f-(j)I s

Hardy's i n e q u a l i t y or,more s i m p l y ,

the estimate

s h a l l a l s o s e e , i n t h e c o n t e x t of

l R n , how t o e x t e n d t h i s i n e q u a l i t y

to

Hp

Hp, f o r

We

p < 1 . I t w i l l t h e n become c l e a r t h a t t h e Hardy s p a c e s

for

p s 1 , a r e n a t u r a l s u b s t i t u t e s of t h e Lebesgue s p a c e s

which a r e o n l y " r e a s o n a b l e " f o r

Lp

p > 1.

The name o f P a l e y i s a l s o a s s o c i a t e d t o t h e f o l l o w i n g r e s u l t :

m

1

Ia2jI

2

j=O

Proof: Write

and, s u m m i n g i n

F(z)

=

$cIIFII21-* H F1(z)

j

a s we w a n t e d t o s h o w .

0

-

F2(z)

with

l l F l l l H 2 = IIF211H2 = l l F l l1H/l 2 *

57

1.4. CLASSICAL INEQUALITIES Finally we shall present the s o called Fbjer-Riesz inequality:

THEOREI‘I 4 . 5 .

& F

E

0 < p <

H’,

Proof: Suppose first that

-

a. T h e n

p = 2 . We take -0 < r < 1

and apply

Cauchy’s theorem to the function F(z) F(g), which is holomorphic in followed by the semiD , and the path formed b y the segment [-r,r) circle

(reie:

o s e s

dx t ir

lF(x)12

nj.

We get:

F(reit)F(re

-it )eitdt = 0

-r and, from that

Letting

r + 1, we get the desired inequality f o r

we write

F(z)

=

B(z) H(z)

with the zeroes of function in

B

where

F , and

H

D. We know that

(lHIIIIp= IIFI(Hp. It will b e

exp A(z)

for some holomorphic function

=

exp((p/2)

A(z))

COROLLARY 4 . 6 .

& F Le

i n t e n i o n d o m a i n Lounded imuye

ofl e u c h d i a m e t e n

Lu i 4

=

p f 2,

2. If

is a non-vanishing holomorphic

=

we have

p

is the Blaschke product formed

(G(z)l

*

=

A.

IH(z)l ’,

Writing so

=

=

that

u conflonmul e q u i u a l e n c e [nom u nectifiuLle

H(z) G(z)

%ondun cunue

D

onto

r . 7hen,

u n e c t i f l i u k l e c u n u e who4e l e n y t h

i 4

ut

the

I. CLASSICAL HARDY SPACES

58

r.

m o ~ ht a l t t h e l e n q t h o &

F', which

Proof: After a rotation, apply Fejer-Kiesz inequality t o

H~

is an

function.

0

Observe that 1/2 is the best possible constant in 4 . 6 .

F

map

D

Indeed, let

{ x t iy: 1x1 < 1 ,

conformally onto the rectangle

IyI < E ) . We may assume that F ( t ) = t for -1 < t < 1. Since the perimeter of the rectangle is 4 + 4 ~ the , constant n 4 . 6 . has t o

+

be at least

2/(4

constant is

>1/2.

4 ~ ) Letting .

E: +

0, we conclude that the

H~

N e x t , w e shall use the integrability along radii of

functions

t o continue the study of the boundary behaviour of c nformal map-

pings.

COROLLARY 4. 7 .

F

Le u4 i n c o n o l l u n y 4 . 6 .

nuch t h u t t h e n o n - t u n . q e n t i u l L o u n d u n y ua1ue

I t L O ~ ~ O tWh uA t Proof:

F

i 4

und l e t

F' (eitO)

F(e it 0) - F(z) =

F'(<)

Thus, if we remain in the region

0

&

e r i 4 t 4 . 7hen

d<

larg(z) - to

F(eitO) - F(z) - Fl(eitO) = it I:ito(F'(C.) eit0 - z e 0 - 2 2-e IJaT.

it

a 1 4 0 contonrnul u t a l m o 4 i e u e a ~& o u n d u n g p o i n t

where the integral is taken over the segnent joining

as

e

I

z

and

e i t 0.

c C ( 1 - I z I ) , then

-

F'(eitO))

d<

+

0

itO

Now, suppose that

eitO

is such that

Then, we can speak about the tangent t o

r

at the point

F(eitO).

We know that this happens for a.e. boundary point e i t 0. Let y it be a curve in D which ends at e 0 and meets the boundary non-

1.5. THE CONJUGATE FUNCTION

59

tangentially, that is: arg(z-eitO) has a limit a s z + e i t O along and this limit i s not to + ( T / Z ) . Consider the curve F(Y). We

Y

F(Y)

are going to see that

and

Y

angle as the one with which

meet at and

aD

F(eitO) with the same i t 0 meet at e We have to

.

see that lim

it

arg(z-e

0) - to

a - -

= lim

z+eitO

- F(eitO))

arg(F(z)

-

z+eitO

Y

Y

o r , what i s the s a m e , that:

arg(z-e it 0) = lim

lim z+eitO

- F(e it 0)) - arg F'(eitO)

arg(F(z)

z+eitO

Y

Y

B u t this is clearly a consequence of 4 . 8 .

5. THE CONJUGATE FUNCTION

D€FINI7ION 5 , 7 .

on

Let

~ e tu(re

[-.TI,T].

f (t ) & 2 n - p e n i o d i c f u n c t i o n , i n f e q n a e l e it e e i t 4 ~ o i 4 4 o ni n t e g a a l and l e t v(re it)

L e t h e hanrnonic c o n j u g a t e o f

tion

v(0)

=

u

u n i q u e l y detenrnined k 4 t h e c o n d i -

0 . Then we d e f i n e t h e c o n j u g a f e f u n c t i o n o f

f

t o Le

the function: 2.

f(t)

= lim v(reit)

r+ 1 For this definition to make s e n s e , we need to show that the limit exists for a.e.t.

Of course, by writing

f

our attention to

f 2 0, which implies also

=

f+ - f-, we m a y r e s t r i c t u 2 0. Then the existen

c e of the limit follows from TNcCORER 5 . 2 , z

e D.

F e H(D)

Let

& F

k e 4uch t h a t

ha4 n o n - t a n g e n t i a l

Re F(z) b 0

{on

eueay

l i m i t 4 a t u e r n o n t euen.4 e o u n d a n y

point,

Proof: Let Il+F(z)l

b

G(z)

=

Re(l+F(z))

l/(l+F(z)). 2

Then

1 , s o that

G e Hm(D) IG(z)I

5

since

1. It follows from

I. CLASSICAL HARDY SPACES

60

Fatou's

theorem t h a t

and is d i f f e r e n t from = l i m F(z)

F(eit)

it

G(e

0

as

) = l i m G(z)

f o r a.e.t.

N'T*+

z

eit

But

as

z

N T

e

it

F(z) = G(z)-l

e x i s t s a.e.

,

exists

-1.

Hence

and equals

- 1.

G(eit)-l

We h a v e e s t a b l i s h e d t h e e x i s t e n c e o f F. We e v e n k n o w t h a t f ( t ) = N.T A e i t . N e x t , we s h a l l i n v e s t i g a t e t h e s i z e of z as = l i m v(z)

? . F i r s t o f a l l , we s h a l l p r o v e t h e t h e o r e m o f M a r c e l R i e s z s a y i n g that the conjugate function operator f +f i s bounded i n Lp 1 < p <

for

THfOREM 5. 3 .

m.

Here

FOR e v e q

4uch t h a t { o n e v e e x

with

p

F(z)

=

1 < p 5 2 , t h e n e i4 a c o n ~ t a n t C

U ( Z ) t i v ( z ) , holornonphic i n

L

D,

TI

dt 5 C

Iv(reit)IP

h o l d 4 { o n evenq

lu(reit)lP

dt

-n 0 < r < 1.

We a r e g o i n g t o p r o v e t h a t we c a n c h o o s e p o s i t i v e c o n s t a n t s

Dp

f o r which t h e i n e q u a l i t y

6 > 0

be s m a l l enough t o have

( n / 2 ) - 6 < 181 < n / Z , that S

=

-

I f we t a k e

6 < 181 < n / 2 .

cos((n/2)

have

1

5

P

and

n / 2 < p ( ( ~ / 2 )- 6 ) . T h e n i f

n / 2 < p((n/Z) - 6 ) < lp8l < 0 < -cos(p((n/2)

we have:

c o s ( p 8 ) 5 C O S ( P ( ( T I / ~ )- 6 ) ) < 0 . T h u s

-cos(p0).

TI/^)

C

101 < n / 2 ,

holds for

Let

P' w ith -

~ ( 0 )= 0 , t h e i n e q u a l i t y

D , v n e a l v a l u e d and

> 0

U(Z)

is t h e key t o t h i s boundedness:

D

C (sin P

P B u t if

6)'

- Dp.

181 6 ( n / 2 )

- 6 ) )5

b i g e n o u g h , we c e r t a i n l y h a v e (5.4) f o r

101 s ( n / 2 ) - 6 , we h a v e

- 6 ) 5 c o s 8 . We j u s t n e e d t o c h o o s e

holds also for

IT, S O

C

0 < sin 6 =

enough t o P O n c e t h i s i s d o n e , we a r e s u r e t h a t (5.4)

- 6.

big

1.5. THE CONJUGATE FUNCTION

61

Now we use ( 5 . 4 ) to prove the theorem:

c

=

p

JI.

cp

=

dt - D

lu(reit)lP

Re(F(reit)P)

1

=

TI

J"

dt - 2n D

lu(reit)lP

Re(F(0)P)

C

6

P

-TI

p

lu(reit)IP

dt.

-TI

We are ready to present the Marcel Riesz inequality

CO/?OLLA/?4 5 . 5 . F o n eueng 1 < p < 4 u c h t h a t , ,!!on each f e Lp(T):

'1

I?(t)lP

dt

B

S

p

-TI

m,

1'

theae e x L h

u condtunt

B

P

dt

If(t)lP

-TI

Proof: First, let 1 < p 6 2. The linearity allows us to assume f t 0, not identically 0. Let u be the Poisson integral o f f and

v

v(0)

=

= u(z)

1"

+

iv(z).

I'f(t)lP

Since

r * l

C P

lim inf r * 1

This proves

J"

=

?(t)

d t 2 lim inf

-TI

5

u

the harmonic conjugate o f

determined b y the condition

0. Then, we can apply theorem 5.3 to the function

I"

v(reit),

lim r* 1

Iv(reit)lP

dt 6

-71

lu(reit)lP

dt 6

c p

-TI

1"

If(t)lP

dt.

-TI

M. Riesz's inequality for

Suppose now 2 < p < that 1 < p' < 2 . For

F(z) = Fatou's lemma yields:

1 < p 2 2.

and let p' be the conjugate exponent, s o f e Lp(T), we denote b y u the Poisson

integral of f and by v the harmonic conjugate to v(0) = 0. W e can write, for every r < 1:

(1

TI -TI

Iv(reit)lP

dt)'"

=

v(re

sup --TI

it

) g ( t ) dt( :

u

such that

g with

I. CLASSICAL HARDY SPACES

62

But

'1

(5.6)

v(re it ) g(t)

dt =

1

TI

-

Indeed, let

h(z)

I"

g(t)

be the Poisson integral of

harmonic conjugate to

(5.7)

u(reit)

dt

-n

-TI

h

with

and

g

= 0. Then, for

w(0)

w(z)

0 <

s

the

< 1:

dt = 0, since

(v(rse it )h(seit)tu(rseit)w(seit))

-TI

v(rz)h(z)tu(rz)w(z)

is a haraonic

= Im((u(rz)tiv(rz)).(h(z)tiw(z))

function vanishing ar z = 0. Now 1 < p' < 2 e Lp'(T). We can apply theorem 5.3, with p '

g

the holomorphic function

h t iw

and h = P ( g ) with instead o f p , to

and conclude that it belongs to

w = Pig). Therefore h(seit) + g(t) in Lp' and w(seit) * "pt) in Lp a s s + 1 , s o that the integrand in (5.7) converges to v(re it g(t) t u(reit) g(t) in L~ a s s * 1 and (5.6) follows. Now, using (5.6) and observing that:

Hp'. I t follows that

From this we get, by using Fatou's lemma:

I

n

JTI

-n

t I I m F(O)I

J'i(t)JP

d t s BPp;'

)f(t)lP

d t . 0

-71

.

Proof: Let u = Re F and let v determined b y the condition v(0)

be the harmonic conjugate o f u 0. Then I m F(z) = v(z) t ImF(0).

=

1.5. THE CONJUGATE FUNCTION

63

it) is the conjugate funcNow one just needs to realize that v(re tion of u(reit) and apply corollary 5 . 5 . 0 We can easily s e e that corollary 5.5. does nos extend to p = 1 . 2 2 For example, if f(t) = Pr(t) = (1-r )/(l-r +2r c o s t), then Z(t)

2r sin t 2 l+r -2r cos t

=

the s o called conjugate Poisson kernel, which we shall denote by Qr(t).

Indeed, F)

Pr(t)+i

Qr(t)

l-rL+i 2r sen t 11 - re it12

=

n is holomorphic and

d t = 0. Now

I-nQr(t)

lQr(t)l

dt

=

l + r 4 log 1 - r +

as

m

r + l

n

IPr(t)l

whereas -n

dt = 2n

independently of

r.

By duality, that i s , b y applying 5 . 6 , we see that the conjugate function operator is not bounded in However, for

p

= 1,

La.

we have the following substitute result d u e to

Kolmogorov: THEOREM

5.9.

7 h e c o n . j u q u t e J u n c t i o n o p e n u t o n i 4 o,l

w h i c h meunn t h u t t h e n e e x i h t 4 u c o n 4 t a n t

f E~

e =

L ~ (c - n , ~ ; 3 { t e [-n, n ]:

x

and e v e n y

I?( t ) I

C,

weuk

type

( 1 . I),

~ u c ht h u t L o n e v e n y

> 0 , the 4et

> A)

ha4 L e L e h g u e meunune

IE ~ I

4uti~#yin(ic

Proof: Because of the linearity of the conjugate function operator, we may assume that Fix

f 2 0

> 0 . The function

and z

& jynf(t)

z -

ih

dt = 1 .

clearly maps the half-plane

Rez > 0

into the half-plane Im z > 0. In this half-plane Im z > 0 we choose the determination of the argument such that -51 < arg z < < 0 . With this choice, we define for

Rez > 0 :

I. CLASSICAL HARDY SPACES

64

1 z - iX h (z) = 1 t - arg X z t iX ~

0 < hX(z)

is a harmonic function such that

Then, hX

< 1.

hX(z) = k a r e , obviously, arcs o f circles passing through iX and -iA. In particular, the level line hX(z) = 1 / 2 is the semi-circle (heie: -71/2 < 0 < n / 2 } and we s e e that 1.1 >= h implies hX(z) >= 1 / 2 . It will also be convenient to observe that:

The level lines

1 n

h X ( l ) = 1 t - arg

_- -TI2 7(71

2 n2

- arc tg A )

Let

u = P(f),

v(0)

=

and let

t iv(z)),

hA(u(z)

hX(u(re u(0)

=

-

it

2X 2 --

2

1 --arc

tgh =

- A

1

x

v be the harmonic conjugate of for every

z

e

D

u

with

and we may consider

which is a positive harmonic function in )

2J 271

1

1 - - arc t g

1

u(z) > 0

0. Then

since

1 - iX = 1 t iX

+

iv(re

it

) ) d t = hX(u(0))

=

hX(l)

5

2 n

-

D , and

1 h

n f(t)

dt

1.

=

-71

On the otrher hand, if (v(reit)( > A , then also (u(reit) t t iv(reit)l > A a n d , consequently, hA(u(reit) t iv(reit)) > 1/2. Thus

s o that (It e

[-IT,TI]:

Take

r. f J

(v(reit)( > 1 . Since

T(t)

?,}I =

5

8/X

for all

lim v(re it ) : r+ 1

0 < r < 1.

1.5. THE CONJUGATE FUNCTION

65

f 2 0

In this way, we see that Kolmogorov's inequality is valid for with

C

Clearly As

4 / 7 1 . T o pass to an arbitrary

=

C

=

","

will suffice.

f

is straightforward.

0

a consequence, we can show that the conjugate function operatoris

bounded from

in the

C

whene

L1

Lp

to

0 0. Fubini's theorem implies that

We use the estimate obtained in theorem 5.9 when

For

X <Xo

we use the better estimate A0

71

I?(t)lPdt -71

S

2np

I,

XP-'dh+pC

(1"

h

I

-71

f e L1([-n,n))

v

lLe t h e

v

=

P(?).

5

271. We get:

(1

m

If(t)ldt)

and this gives us the desired inequality.

COROLLA/?Y 5 . 1 1 . Sppone Let u = P(f) and l e t u with v(0) = 0.

IE

Xp-2dX)

=

xO

0 in n u c h that

e L'([-n,n]).

c o n j u g a t e hanmonic {unction

o{

I. CLASSICAL HARDY SPACES

66

Proof: Applying corollary 5.10 to the functions u(reit) and its it v(re ) , we s e e that u t iv e H p for all 0 < p < 1. 1 Besides, its boundary function f(t) t i?(t) belongs to L It follows from corollary 3 . 7 . that u t iv e H 1 and, consequently conjugate

.

v = P(L

0

We can identify H p with a subspace of Lp(T) b y means of the isometry sending the H p function F ( z ) to its boundary function F(eit). There is no harm in denoting this space of boundary func-

tions also by

Hp. We shall also consider

ReHP, the space

by those functions in the boundary which are real parts of tions.

Hp

We can give a description of

for

1

p

5

S

OD

formed func

Hp

in terms of the

conjugate function. We shall look at the H p of the boundary, but note that, in this case, the Poisson integral sends us back to Hp(D). Suppose F(eit) is in H P , 1 5 p S a . Let u(z) = Re F ( z ) . Then

Im F ( z ) = v(z) t c , where

v

is the harmonic conjugate of

u

having v(0) = 0 and c = Im F ( 0 ) . If we call f(t) = u(eit), we CL it have: F ( e ) = f(t) t if(t) t ic. Thus, when 1 S p 5 m , every func tion in H p (of the boundary) is of the form f t i? t ic for some f e Re L p such that ? e L p , and some c e lR. Conversely, let f e Re L p , 1 5 p S m , be such that ? e L p (when 1 < p < m , the M. Riesz inequality guarantees that ? e Lp as soon as f e L p , but this is no longer true for

p

1

=

or

p

= a).

Let

u(z)

=

P(f)(z)

be the conjugate harmonic function of u with v(0) Therefore W e know from corollary 5.11. that u t iv = P(fti?). u t iv e Hp(D) and consequently, its boundary function f t i? and let

Hp. Thus. for

in

Hp

v

=

1 < p <

m,

we can write:

{f t i? t ic: f e Re L p , c e lR1

Re H p = Re Lp.

and

However

H1 = { f

t i? t ic: f

Re H 1

[f

e Re L 1 ,

?

e L1, c e

and =

e

Re L1:

?

e

L1ls Re L 1

.

lR1

=

0.

is

67

1 . 5 . THE CONJUGATE FUNCTION

That the inclusion is proper follows either from the remarks we made after Hardy's inequality o r from the fact that the conjugate func ion operator is not bounded in L1. The space R e H1 with th norm /If H1 = Ilf [I1 + ~ ~ ? ~which ~ l , is an equivalent copy of the subspace IF e H1: F(0) e lR} of H 1 , is a space different from 1 Similarly Re L

lae

.

Hm = (f + i? Re Ha

if

+

?

ic: f e R e La,

e La, c e lR}

and

Re LOD.

f +?, let us see how we can define it without getting inside the disk. We where : have defined ?(e) = lim v(reie) r+l

T o finish our discussion of the conjugate function operator

TI TI

But

r sin t (f(e-t)-f(e+t)) 0 l+r 2 -2r cos t

r sin t l+r 2 -2r c o s t

-+

1 2 tg(t/2)

as

r

-

dt

1

and

2

r sin t

the Lebesgue dominated convergence theorem gives

Condition (5.12) holds, for instance, if f'

is continuous in an open interval

f'(8)

exists. Besides, if

(a,b), the convergence is

uniform on every closed subinterval of (a,b). This implies that extends continuously to the arc (eit: a < t < b}. v(z) If (5.12) does not hold, the integral in (5.13) is no longer

I. CLASSICAL HARDY SPACES

68

absolutely convergent, it becomes a singular integral. However, (5.13) is still valid for almost every 8, provided that the integral is interpreted in the sense of Cauchy‘s principal value,

1L

f(8)

1

lim

f(8-t)

i n t h e Le&engue

8

{ o n eueag

=

A e t

f

o#

dt

and, conneyuently,

804

u l m o n t e u e n l 8. Thin limit i A culled t h e paincipul uulue of the ningulun inteqnul and i A cuntornunilq denoted iLy t h e initiuln p . u . pnecediny t h e inteqnul. 8

Proof: Let

be a Lebesgue point for

f. We shall show that the

quantity

tends to

r

-

0

as

r

tends to 2r sin t

2n

1. This will prove the theorem. But

f(8-t)

dt t

Itl
+ -1

( -

and we can always substitute

f(8-t)

-

f(8)

for

) f(0-t)

f(8-t)

dt

because

our kernels are odd. The first integral i s bounded in absolute value by

which tends to As

for

0

as

r

tends to

1 , since

8

is a Lebesgue point.

for the second integral, it is bounded in absolute value by:

r

close to

1 , say

r > 1/2. We just need to study, for exanple,

1.5. THE CONJUGATE FUNCTION

the integral over the interval

1-r < t <

IT.

69

Integration by parts

gives :

The first two termsin this sum clearly tend to The third term will b e split a s chosen later, I1 < s < 6

s 0, 6 can be chosen in such a way that, for

and

6 < <

I1 + I2

6

6,

With this choice of

and

which can also be made

< E by choosing

r

close to

1.

0

Let u s finish this section with a brief reference to the counterpart of the conjugate function in the setting of the upper falf plane.

H.

This is the s o called Hilbert transform, which we shall denote by The definition is a follows. Given we consider the Poisson integral of defined in

lR:

f e Lp(lR)

for some

1

5

f , which is the function

p <

m,

u(x,t)

as m

u(x,t)

=

P *f(x) t

t

= 1-m

2 f(Y)

dY

t2+(x-y)

The study of Poisson integrals and harmonic functions in half-spaces will b e carried out in detail in chapter 11, section 4 . There, the

I. CLASSICAL HARDY SPACES

70

reader will find complete justification for the claims we are about to make. u(x,t) is a harmonic function a n d , since lR+2 is simply connected, u will have a harmonic conjugate v(x,t), which we shall choose i n such a way that v(x,t) + 0 a s t + uniformly in x. This is indeed possible, and uniquely determines set

v. We just have to

Since

where

z = x

+

F(z) = u(z) t iv(z)

it, we see clearly that

is the

analytic function

2

By using Fatou's theorem for lR+ (corollary 4.19 can establish the existence for a.e. x e lR of Hf(x)

= lim v(x,t)

= lim

t+o

To do this, we assume

t+O f L 0

in chapter 11), we

+IW

x-v 2 -mt t(x-y) 2

f(Y)

dY.

and either extend theorem 5 . 2 . to

(which requires to make sure that the function G(z) = 1/(1+F(z)) cannot have boundary values vanishing on a set of positive measure) or else look at the bounded analytic function As

t

+ 0,

the kernel

fi

e -F(z)

tends to the singular kernel

we can show that

-

TY'

t +Y

for

a.e.x

The key to this i s to show that

v(x.t)

-

1

n

I,,,,,

-y1

f(x-y)

dy

+

0

as

t

+

0

for

a.e.x.

and

1.6. Hp AS A LINEAR SPACE

71

The proof is entirely similar to that of theorem 5.14. Actually, the computations are simpler in this case. Estimates for a rather general class of singular integrals will be obtained in section 5 of chapter 11.

6 . Hp AS A LINEAR SPACE

Now, we are going to look at have seen that, for

p

b

1, F

Hp

a s a topological vector space. We

**

llFllHp

is a norm on

Hp

while if

0 < p < 1 , this i s no longer true and we have, instead, a quasi-norm

F I--+

11 FlEp

and, cmsqently, (F,G) FllF-GIlip is an invariant distancecompatible 2 1 , Np(F) = IIFIIHp, and for 0 < p < 1 , Np(F) = lIFll{p and we shall see H p , 0 < p L m , a s a metric linear space with the distance (F,G)I+ N (F-G). We have also P shown that the mapping

with the vector structure. We shall write, for p

sending each H p function F(e it ) , is an isometry

F(z)

into its boundary function

f(t)

=

=

Proof: If it were locally convex, the ball would contain some convex neighbourhood of in its turn, contain

B,

=

IF

e

B 1 = [F e Hp: Np(F) < 11 0, say V , which would,

HP: N~(F) < € 1

for some

E

> 0 . Let

...,

us divide ( - T , T ) into n intervals of equal length 1 1 , In and construct continuous functions f l , fn such that each f .

...,

J

is zero outside o f I.,' takes a value a > 0 (to be specified later) J at the center of I . , and i s piecewise linear J

Thus, if w e take ap = (pt1)n

a

a

(it will depend on

then each of the functions Ifj(t)lP

dt < E . Consider now

n),

in such a way that

fl,

...,

fn

will

I. CLASSICAL HARDY SPACES

72 n

gn(t) =

n

1 X .f .(t) j=1 J J

A. 2 0

with

J

1 X J,

such that

j=1

= 1.

Then

n =

1

A!

j=1

A . = j

If we make

J

/

-'Ip

Thus, we can choose

n

prove the same fact for f . J

fixed. Each i.e.: given

c1

€12.

we shall have

'1

in such a way that

Lp(T)

This proves that

j -'Ip,

* J

2rr is not locally convex fo?

Hp

requires some m o r e

Ign(t)lP

dt > 1 .

0 0, we can find for each j m.

h.(t)

J

=

1

k=-m.

J

a . e ikt Jk

J

,that Ihj(t) l m .t = e 3 h j(t!. so

- f .(t)l

.

J

Then

<

f o r every -TI 6 t 6 7 1 . Let In.t - e 3 fj(t)J < c1 f o r

c1

Iqj(t)

qj(t) -71

=

6 t 5

71.

Observe that

and im .t

1

fj(t)lP Let m. J

Qj(z)

1

=

ajkZ

ktm. J

k=-m. J

so

that

Qj(e

it )

=

qj(t)

and

1 dt = T;;

I:, I

Xjfj(t)lPdt j=l

> 1

1.6. Hp AS A LINEAR SPACE

73

if

CI

is small.

if

a

is small. We have arrived at the following contradiction: On

...,

Qn e BE c V s o that, since V is convex, the one hand Q,, n n 1 XjQj e V c B1 o r , in other words N ( 1 XjQj) < 1. On the j=l n P j=1 other hand N ( 1 X.Q.) > 1. This proves the theorem. P j=1 J J

0

Of course, the first non locally convex spaces to be examined were 0 < p < 1. M.M. Day ((11) observed that, for 0 < p < 1 , the topological dual of L p ( [ O , l ] ) is zero. This is very easy to s e e . One just needs to realize that the only convex neighLp

the spaces

for

is the whole space. I deed, let

0

bourhood of neighbourhood E > 0. If

into

0, V

of

subintervals 1 1 , 1 1 Ig(x)lpdx = Ig(x)lp n

jo

j ) Ing(x)

I. foj

n

Ip

..., I n ,

lo

big, the functions

the convex combination

V

=

for

g =

Hp

(g(x)lp

ngxI

Lp.

However, the spaces

dx

1n j=l

for some

E)

we can divide

[O,l]

in such a way that j = 1, 2,

dx

ngxI

...,

if

<E.

n

V

are all in

il

be a convex

dx <

f (x)]

Lp(@,g),

1

dx = n-('-')

have sen that

ip I

{ f e Lp:

3

is any function in

g

V

n. But then is big. T h u s ,

a n d , consequently,

will be in

V

too. We

j

0 < p < 1 , in spite ob being non

for

locally convex, have a great deal of continuous linear functionals In general, if

F e Hp

z e r o , we factor it a s

ing Blaschke product and Then, let

A(z)

and write

G(z)

0 < p <

with F(z)

H

=

m

B(z) H(z)

F

and

is not identically

B

where

is the correspond

is nonvanishing with

llHllHp = lIFI(Hp.

H(z) = exp(A(z)), H(z)l and,

be an analytic function such that =

exp(pA(z)),

consequently, IIGIIH1

=

Cauchy representation

s o that

I G(z)l

IIHllip = IIFllip. Since

=

I

G e H 1 , it will have the

I. CLASSICAL HARDY SPACES

74

Therefore:

We can state: 7 f l f O l ? f f l 6.3.

To4

F e HP, 0 <

COROLLARY 6.4. 7 0 4

0 < p S

m,

If e Lp: i(j) = 0

clearly,

D

z e

m

Hp

i n u complete npuce

p 2 1 , the image of

Proof: For

nnd

p 2

Hp

( 6 . 1 ) is,

under the isometry j < 0)

for each

, which i s , of course,

Hp follows immediately. By using theorem 6.3, we can easily derive the completeness of H p for any p . Indeed, theorem 6.3. implies that if F e H p and K is a

closed. The completeness of

compact subset of

SUP

zeK

D , then:

IF(z)I

5

(SUP

(1-121)

zeK

-'Ip)

-

IIF1lH~

F is a Cauchy sequence in H p , it will converge uniform j ly on compact subsets of D to a certain holomorphic function F(z). Besides

Hence, if

TI

1

TI

d t = lim -

'1 2 1-n

(Fj(re it )-F(reit)lP

if

j

i s big. Thus, P e H p

So,

we can s e e H p a s a closed subspace of Lp(T). If 0 < p < m , is the minimal closed subspace which contains ( e ijt. j = 0 , 1 ,

Hp

n

Fj

and

2n

+

F

IPj(re it )-Fk(reit)lPdt

-n

in

Hp

as

j +

a.

5

0

.

... 1 as

k+m

r

03

(If +

1

F(z) =

1

j=o and, for r

a

ZJ

is in

Hp,

F(reit)

+

F(eit)

in

Lp

J

fixed, O

+ a).

a . r J e i J t + F(reit) J

uniformly a s

For 1 S p < m , this point of view can be used to describe the dual space (Hp)*. We know that Lp(T)* = L p l ( T ) with ( l / p ) t ( 1 ' ~ ' ) = =

1. T o each

f e Lp'(T),

we can associate the functional

1.6. Hp AS A LINEAR SPACE

Af

e

obtained by restriction to

(Hp)*

75

H P , that i s , given by

It follows from Hglder's inequality that

so

that we have a continuous linear mapping

The Hahn-Banach theorem tells us that every p, e (Hp)* = Af f o r some f (6.5) is, clearly,

A

( f e LP'(T):

with

fi 27T

eiJtf(t)

F(-j) =

IlAIl.

d

PI

is of the form

The kernel of the mapping

= 0,

j = 0, 1,

...I

=

7T

= {f

e Hp:

dt = O } .

f(eit) -TI

We shall denote this space by (6.5),

Hpl(0). Passing to the quotient in

we obtain an isometry 1

1

Lp /Hp ( 0 )

-

(Hp)*

+ I

Af

+ Hpl(0), F

> =

f+HP'(0)

Thus, we can write

with the pairing < f

27T

1"

F(eit)

f(t)

dt

-n

In the same way we can prove that: I

(6.7)

(Hp(0))* = L p / H p

t

,

l j p < m

I. CLASSICAL HARDY SPACES

76

p'

If we also exclude

m(i.e.

=

p ='l),

.

Lp

has a topological complement in

Hp'(0)

then the subspace

This is a consequence of the

boundedness of the conjugate function operator in Lpl(T) for 1 0, sgn j

-1

sgn 0 = 0, is the conjugate harmonic function of

u

sgn j

J

and

1

=

(sgn j) a . r l j l e i J t , where

j < 0

with

J

-m

1 -m

v(re it ) =

a.rTy'&jt. The function

0, since

u(reit)

+

iv(re

it

=

for m

1 a.(reit)J 1-

) = a0+2

is

1

holomorphic. B u t corollary 5.11rntells u s that ly

has the Fourier series

1 Tm

A(f)

=

Lp'(T)

1

with Fourier series

1

1

jso

F

a.e

i jt

ijt a .e J t

e Hp

sending the function into the function

aTziJ2. Thus

j>o J Hpl(0). The function

onto

P(f).

Consequent

j) a . eiJt. We see that J by setting

( 1 / 2 ) (f+ii-?(O))

is a bounded operatormin Fourier series

=

v

I

(-i)(sgn

f e Lp (T)

the operator defined o n

1

m

1 a e j=o -j

f-A(f)

A

f e Lp'(T)

A(f)

having

Lp'(T)

is the projection of

will have Fosrier series

-ijt. If we write

F(z)

=

1

a-j z j , we have

j=O

and f(t)

=

Af(t)

+

F(eWit) rn

And if we write G ( z ) = 1 F(e-it) = G ( e it ) , s o that?='

f(t)

=

Af(t)

+

-

G

a V j z J , we also have

e Hpt

and

G(eit)

By using the notations H p ' = {h(t):h e Hp') {h(-t): h e H p l ) , we can conclude that Lp'

=

and

(Hpl)-

topological direct sum in the two following ways:

Lpl

=

Hp'(0) I8 H p '

Lp'

=

Hp'(0) 0 (Hp')-

=

can be written a s a

Of course, we could have also writen topological direct sum

77

1.6. Hp AS A LINEAR SPACE

decompositions

Lpl

Hp' 4 Hp'(0)

=

Naturally, the projection on B(f)

=

Hp'

is the operator

(1/2)(f+iT+z(O)) m

f e Lpl(T)

sending each function

1

with Fourier series

-a. m

into its "analytic part"

B(f)

having Fourier series

1

j=o

ijt a.e J

a.eijt J

The topological direct sum decompositions give rise to topological isomorphisms

In each case two isomorphisms arise from the two different decomposi tions. In the first case we can either send the class into the function of (f-Af)(t)

+

Hp'(0)

t:

(l/Z)(f+i?+z(O))(t)

=

=

Bf(t)

o r else send it into this other function of

(f-Af)(-t)

=

(1/2)(f-i;+:(O))(-t)

t: =

Bf(-t)

I n the second case w e can either send the class function of

f

f+HP'

into the

t:

(f-Bf)(t)

(l/Z)(f+if-i(O))(t)

=

=

Af(t)

or else into (f-Bf)(-t)

=

(1/2)(f-i;-i(O))(-t)

= Af(-t)

In view of these isomorphisms, (6.6) and (6.7) can be restated for

I. CLASSICAL HARDY SPACES

78

1 =

2' -'F(eit)

< G,F > =

-

G(e-it)

F e H p , G e Hp'

dt,

or

(6.9)

2a

1'-'

F(eit) G(eit)

(Hp(0))*

=

H"(0)

We know that

F e HP, G

dt,

Hp'(0)

e I~P'

l < p < m

is not a complemented subspace of

L", since

this would clearly imply the boundedness of the conjugate function in L" and, by duality, in L1. S o , we still have a problem to which we shall turn i n section 9: To describe (H1)* as a space o f functions. For the time being, all we can say is that

( H ~ ) * = L"/H"(o),

7. Let

CANONICAL FACTORIZATION THEOREM

F

e

Hp, 0 c p 6

m.

We have used many times the factorization

F = B - H , where B is the Blaschke product formed with the zeroes o f F , H is never z e r o and llHllHp = IIFIIHp. Starting with this factorization, we shall get a finer one. We apply theorem 3 . 2 to the function H and conclude that, for any sequence r . .f 1, both (log t IH(rjeit)l) and (log-[H(r.eit)l) are J

J

bounded sequences in the Banach space L1 c M. They will have s u b sequences converging in the weak-* topology of measures. Thus, there exists log'

r. J

.f

1

IH(r.eit)l J

and positive measures

-

dUl(t)

and

log-

u2

pl,

IH(r.eit)l

such that --+

du2(t)

J

both in the weak-* topology. But it turns out that dul(t) = t = log lH(e it ) I dt. Indeed, we just need to observe that log' IY(rjeit)l --+ log' IH(eit)l in the L1 norm. T o see this,

1.7. CANONICAL

write, if

Ilog'

p

79

FACTORIZATION

1

b

IH(eit)l-log+lH(r.eit)l( 3

5

p-l( IH(eit)IP-IH(rjeit)IPI 5

5

J.ei.t)lP

p-'1H(eit)-H(r

Now, by corollary 3 . 9 :

J

I"

= i c + -

2"

Letting

j+

eit-2 w,

eit+z log IH(r.e it ) I dt =

+

= i arg H(0)

log H(r.2)

J

IH(r .eit) I dt -

log'

3

2a

I"-" eit+2

- log-( H(rjeit)Jh eit-z

we get:

dD2(t) = g(t)dt + da(t) where g is integrable is a non-negative singular measure. Thus

But

log H(z)

=

ic

1 + 2"

1"

it k(t)

-re

-2

k(t)

=

log + IH(eit)l

log IH(z)(

=

P(k(t)

with =

k(t).

dt

dt -

- g(t),

-

da(t))

-

I

'IT

eit+z

-n e

it

20

and

IJ

da(t)

-2

integrable. But, then, and this implies that

We have proved the following result, known a s

loglH(eit)l = canonical f a 2

torization theorem THEOREM 7 , 1 . E u e n y

F e Hp, 0 < p

6 a,

n o t i d e n t i c a l l g zeno,

hun u

canonical t a c t o n i z a t i o n

whene

B

i n t h e dlanchke

i n a neal conntunt,

and

IJ

pnoduct tonmed w i t h t h e zeaae4

UL

F, c

i n u n o n - n e g u t i u e n i n q u l n n meanune.

I. CLASSICAL HARDY SPACES

80

We shall

use

the notation

IF

and we shall say that

Ip

IEF(z

=

IEF(eit)l

F(eit)I

=

are, respectively, the inner and

'))

exp P(log(IF(eit)

F e H p , then also

that if

EF

F.

outer factors of Since

and

EF e H p

5

and

P(lF(eit)lP), it is clear IIEFIhp = IlFlkp because

for a.e.t.

F e H p is an i n n e n function if and only if We shall say that EF Z 1 ; and we shall say that F is an o u t e n function if and only if

IF

is constant.

C O / ? O f f A R y 7 . 2 , 7 h e i n n e n kunCk4OnA a n e p n e c i d e l y t h o d e f u n c t i o n n F e Hm { o n w h i c h I F(e it ) I = 1 almont euenywhene Proof: Since

EF

IEF(z)l

lEFl

1

5 1

=

exp P(log

@log

IF(e it )

IF(eit)l

=

I),

0 a.e.

we realize that JF(eit)l

=

1 a.en

Proof: It follows from theorem 7.1 that

l T I

it

exp ( log I F(e ) I d t ) and equality holds if and only if I B ( O ) I = 1 (wlich implies B z 1) and da = 0 (which implies

5

u

=

0).

0

We shall present some applications of the criterion just given for a function to be outer

1.7. CANONICAL

THEOREM 7 . 5 . 0 < p

<=

F h u t F e H p g&

Suppo4e

F

m.&

81

FACTORIZATION

F-l

f

Hp

home

{on

i n un o u t e n { u n c t i o n .

Proof: We know that

But also

Therefore, equality holds in TH€Oil€fl 7 . 6 .

Let

F

I

Suppohe

that

IF1(z)

Fj(z) & F

F(z)

uni&onmly

+

44

ke outen

Hp

e

,

J

L

IF2(z)

1

outer.

F

j

{ u n c k i o n ~& o n

...

2

is

{on

z

eueny

0

f

und

D

j +

i n an o u t e n & u n c t i o n .

j:

Now

1

TI

2TI

log'

(Fj(e it )(dt

-t

-TI

2n

log'

JF(eit)J dt

-71

as

by dominated convergence, and

by monotone convergence. Letting log

(F(O)l

a n d , consequently,

=

F

1

l T I ?;;

j

-t

log IF(e

m,

it

we get:

)I

dt

-71

is an outer function.

a

j

+

. ..

1, 2,

=

D,

o u e a c o m p u c t ~ u k 4 e k 4o &

n o t i d e n t i c a l l y zeno,

Proof: For every

F

( 7 . 4 ) and

m

m.

Then, -

82

Proof:

I . CLASSICAL HARDY SPACES

Re F ( z ) 2

I f we h a d

Il/F(z)l 5

1 / ~ s, o t h a t

would b e o u t e r . define

Fj(z)

E

F-l

> 0

t (l/j).

For

j

...,

F . e HP, and J Besides, clearly

Re F . ( z ) 2 l / j > 0 , s o t h a t e a c h F . i s o u t e r . J J I F 1 ( z ) I b IF2(z)l z and Fj(z) + F(z) uniformly i n

...

Applying theorem 7.6.,

F

Re F ( z ) 2 0 . L e t u s

1, 2,

=

then

A c c o r d i n g t o t h e o r e m 7.5.,

H o w e v e r , w e o n l y know t h a t

= F(z)

z e D,

f o r every

e Hm c H p .

we conclude t h a t

F

is outer.

D.

0

Here is a n i n t e r e s t i n g e x t e n s i o n of c o r o l l a r y 3.7.

7H€O/I€fl 7 . 8 L e t

that

4uppohe

K(z)

t w o ouiea {unctionn,

ee t h e q u o t i e n t o,!

fLelong4 t o

K(eit)

Lp

home

{O/L

0 < P 5

m

e

und

Then

K e Hp.

Then, i f

p <

a,

IK(z)

Ip

t h a t , c l e a r l y , K e Hp.

= e x p P ( l o g ( 1K(e

If

p

=

CO,

it

)

1'))

P( IK(eit) lP) use 1 i n s t e a d of p , and t h e 6

so

0

r e s u l t is obvious.

The i n n e r f a c t o r p l a y s a n i m p o r t a n t r81e i n some a p p r o x i m a t i o n problems.

Proof: a ) Let

Here i s t h e b a s i c r e s u l t

F

Suppose

=

E

F'

a n d u s e d u a l i t y . To see t h a t t o show t h a t i f

k(eit)

as a c l o s e d s u b s p a c e of

EF-/3, i s d e n s e i n

Lp ( T )

is i n

EF(ei t ) ei j tk ( e it ) d t = 0

(7.10) then

Hp

p 2 1 . T h e n we c a n s e e

I"

-TI

G(eit)k(eit)dt

=

0

Hp,

one j u s t needs

and for

for each

j = 0,

G e Hp.

Lp(T)

1, 2,

...,

1.7. CANONICAL FACTORIZATION

But ( 7 . 1 0 )

implies that

EF(eit)k(eit)

=

eitH(eit)

83

H e H

for some

1

.

We can write: it ) = e it IH(eit)EH(eit)

EF(eit)k(e

I

I

EH(eit)

= Ik(eit)l, which belongs to Lp'(T). Theorem 7 . 8 EF(eit) implies that EH/EF e I T p ' . Denote this f u n c t i m by S. Then

Thus

k(eit)

In

=

e it IH(eit)S(eit)

=

-n

since

G(eit)eitIH(eit)S(eit)dt

=

0

-TI

GIHS e H1. This finishes the proof for

b) Let now so

1

TI

G(eit)k(eit)dt

p 2 1.

0 0, we want to find

E

P e T

such that

E -

EF = K 2

where

K e HZp

and is outer, in fact

K

=

Now, given H , there is R e T such that IIH - R l l i p < ~ / 3 Since . R e HZp and EK is an outer function in HZp, there will be Q e such that I I R - EK-Q11i2p < E / 3 . Of course, this implies that

IIR

-

E ~ ~ Q I I ~ P<

€13

IIQ - EK*PIIH2p < 6

.

Finally, there will exist

for 6

EK '

P

P e P such that

to be specified later. But then

'~ ~< E~/ 3~ 2if *we choose an appropriate IIEK-Q-EFmPllgp5 ~ ~ E F 6 6. Finally Thus

I. CLASSICAL HARDY SPACES

a4

This ends the proof.

I & F e Hp, 0 < p <

COiiOLLARy 7 . I ? .

F-P &

t h e 4puce

0 m,

t h e n t h e clohune i n H p

&

IF'HP.

Proof: We just need to realize that the mapping

IF'HP is closed. If P e P , F - P = IF*EF'P e IF'HP. Thus F - P c IF'HP a n d , since IF.HP is closed, the closure F./7 will also be contained in IF*Hp. On the other hand = H p implies F - P = IF'EF'P = I *= = IF'HP. F F F in an isometry. From this, it follows that

0 8. THE HELSON-SZEGO THEOREM The Helson-Szego theorem is a characterization o f those positive (Borel) measures

IJ

i-v,~]

on

L (U).

operator is bounded in

for which the conjugate function

-f(t)

n

=

-i

1

1

sgn j

=

1

for

J

ijt (sgn j) a . e J

-n where

7 the ijt a.e and

More specifically, denote by

space formed by the trigonometric polynomials f(t) = for f E 7 a s above,we consider its conjugate functio:n

j > 0 , sgn j = -1

for

j < 0

and

sgn 0 = 0.

The Helson-Szego theorem provides an answer to the problem o f finding those positive measures on such that, for every

1

(-Tr,n]

l?(t)I2

-n

C

1

n

71

(8.1)

for which there i s a constant

f e 7 , the following inequality holds: d!J(t)

5

C

lf(t)12

dIJ(t)

-n

We shall call such measures Helson-Szego measures. By now, we have met just one of them, namely Lebesgue measure

di.i(t)

=

dt, for which,

an inequality like ( 8 . 1 ) , is a particular instance of the M. Riesz theorem, and follows immediately from Plancherel's theorem.

85

1.8. HELSON-SZEGO THEOREM

Before presenting the Helson-Szego theorem, we shall deal with another problem about measures, which has an interesting answer that will play a part in o u r treatment of the main question. The problem

u

is this: given a finite positive measure

P e

nomials

on

find the

[-TI,+

Lp(p), 1 5 p < m, from the constant function 1 to the formed by (the restrictions to T of) those poly-

distan.ce in space p(0)

/3

such that

P(0)

=

0. In other words, we want to find:

First of all we have:

u

7H€OR&M 8 . 2 . I #

u [ i n i t e p o d k i v e memcme ruhcch

i 4

n e h p e c t t o L e k e n q u e meunune),

inf

J"

+J

4 i n ~ u L u nIwiih

then:

ll-p(eit)lP

du(t);

P

e

P ( o ) ~= o

-TI

i.e.,

the point

Proof: Let every

1

i 4

f e Lp'(o)

j = 1 , 2 , 3,

P(0)

udhenent t o

...

be such that

& Lp(0) f(t)

-TI

If we can show that

e i J t do(t)

1"

f(t)

=

do(t)

0 =

for

0 , we

-7T

shall have the theorem proved. But what we are assuming is that

du(t) is a measure whose Fourier coefficients vanish for all negative frequencies. The F. and M. Riesz theorem (theorem 3.10) f(t)

says that respect to

f(t)

do(t)

is in this case absolutely continuous with

dt. On the other hand, it is clearly singular. Therefore,

it is identically zero and the proof is finished.

0

We shall present now a result, due to Kolmogorov, which shows that we can limit our study to absolutely continuous measures. W e need a previous lemma

Proof: E is the complement in it A . = { e : a . < t < 8 j } . Since J

J

T

E

the union of a sequence of arcs has Lebesgue measure z e r o ,

of

I. CLASSICAL HARDY SPACES

86

J

(Bj -

ct.)

=

5: m j ( B j

-

ct.)

J yj =

J

(Bj

<

J

a

< t < B

j

The function

f

*

j'

f(t)

m

be such that a

j

j

-+

m

( B j - aj)/2

=

21~-periodicfunction

while and f(t)

such that:

2 2 -1/2 = m a (a -(t-yj) ) j j j

= m.

IBJ

[-IT,,) since

i s integrable in

f(t)

dt = n m . a . J J

a.

E

IR to

J

Then, setting

m.

eit e E: f(t)

b) If

m. > 0

t a j ) / 2 , we define a

a) For

and

Z I T . Let

has measure zero. Actually, f

Indeed if a sequence of points not in E approaches a it is easy to see that the corresponding values of f

f3.m).

point in tend to

is a contindous mapping from

E

We just need to consider two situations: when the points

m.

belong to just two of the arcs A and approach a common endpoint j and when the points range over infinitely many of the A.'s. In the 1 m . In the second case first case the formula for f gives f(t) -+

we just need to observe that on

A.

is

J

f(t)

2

m. J

and

m

-+

j

=.

Now, since f is integrable, we may consider its Poisson integral u(re it ) = (Pr*f)(t). It is quite simple to see that if we set u(e it ) = f(t), u becomes a continuous mapping from D into (0 ,m]. Since

u

D , we are able to consider

is a harmonic function in

the conjugate harmonic function of that f is smooth on the arcs A . observation we made after ( 5 . 1 3 ) ,

v,

such that v(0) = 0. Observe s o that, according t o the

u

J'

the function

v

extends

continuously up to each of those arcs, with finite limits. It is now easy to see that the function

satisfies our requirements. Since holomorphic function in the arcs

A. J

since

both

u P 0,

F

is a well defined

D. It clearly extends continuously up to u and v do. Now if z + e i8 e E ,

< 1 , we know that u(z) -+ m and this implies that F(z) -+ 1 no matter how v behaves. As for the points eit in the boundary such it that e d E, we have IzI

I.8. HELSON-SZECO THEOREM

87

and again, if those points approach some point in

E, the fact that

f(t) -+ 00 guarantees by itself that F(eit) -+ 1 . We have shown that is a holomorphic function in D extending continuously up to the boundary of D in such a way that F(eit) = 1 whenever eit e E . Also, the two formulas above for F(z) IzI < 1 and F(eit), it e 4 E, clearly yield IF(z)l < 1 if z e B \ E.

F

0

Observe that lemma 8 . 3 . provides a direct proof of the theorem of F. and M. Riesz (theorem 3 . 1 0 ) independent of the theory of the Hardy spaces. This was actually the original proof given by

F. and M.

Riesz. It is a s follows: Let p be a Bore1 measure with Fourier coefficients vanishing for all negative frequencies, that is:

1"

eiJtdu(t) = 0

for

j = 1, 2 , 3 ,

-77

We have to show that

p

...

is absolutely continuous with respect to

Lebesgue measure. For that purpose let E c [-n,n] have IEl = 0. We have to show that p(E) = 0. We may assume that E is closed and consider the function F corresponding to E whose existence is granted by lemma 8 . 3 . Then for each (~(2))" is in 4

I

n = 1, 2, 3,

..., since

n (F(eit))"

dp(t)

= 0.

-TI

Now a s

Thus

n -+

m,

p(E) = 0

the dominated convergence theorem implies that

a s was to be proved.

Now, we can state and prove Kolmogorov's

inf PeF(0)

1"

-n

I l-P(eit)IP

result

I

n

dp(t)

=

inf PeP(0)

Proof: Let us denote respectively by

A

Il-P(eit)lPw

(t) d t

71

and

B

the infimum in the

right and the infimum in the left of the identity to be proved. Clearly A d B, and what we have to see is that also B 6 A .

I. CLASSICAL HARDY SPACES

88

Let

E

P e P(0)

> 0 . Then, there is

I

TI

I1-P(eit)

Ip

such that:

dt < A t

w(t)

E

-TI

On the other hand, by theorem 8.2.,

PI

there is

e

P(0)

such that

It will be convenient to write this as:

J"

(8.5)

it

II-P(eit)-P2(e

)Ip

du(t)

< E

with

p2 = p 1

- P

e/7(0)

-71

E

Take a closed set

contained in the support of

u , such that:

is singular, IEl = 0. Then, the preceding lemma gives us

Since

0

F e A

such that

for every

< 1

IF(z)I

for every

E (we identify

z e

E

z e

D\E

and

T

with a subset of

F(z)

1

=

in the natural

way). F(z)

Since

=

1

E , (8.5) impliesthat

on

Il-P(e it )-F(e it ) n P2(eit)IP

for every positive integer

On the other hand, if n

-+

n +

e

m

a n d , consequently

m.

Since

1-P(e

it

)-F(e

E

it

da(t) <

E

n. This, together with (8.6) yields:

is not in E , one has it it n 1-P(e )-F(e ) P2(eit)

F(eit)n -+

-+ 0

l-P(eit)

as

as

0 , we can say that a s n -+ m for almost every

has Lebesgue measure

it n ) P2(eit) -+ 1-P(eit)

t.

It follows that

J"

1 l-P(eit)-F(eit)nP2(eit)J

+

w(t)dt

-TI

as

J"

1 l-p(eit)l

P w(t)dt

-TI

n

-+ m ,

Thus, for

1" ' - T I

Il-P(e

it

n )-F(e

sufficiently large it n

) P2(e

it

))

w(t)

dt < A t

E

< At€

I.8. HELSON-SZEGO THEOREM

89

Adding up the corresponding estimate obtained f o r the singular part, we get: TI

1

it ) n P2(e it

Il-P(eit)-F(e

)Ip

< A

du(t)

+

3~

for large

n.

-TI

Now fix

n

large enough s o that the last inequality holds. Since N 6 > 0, there will exist a polynomial R(z) = a.zJ 3 < 6 uniformly in t. Then such that \R(eit)-F(eit)nl it n it ) P2(e ) . Observe that RP2 e P ( 0 ) R(eit)P2(eit) is close to F(e

Fn e A , given

and

6

Let

P + RP2

can be chosen in such a way that

Q e P ( 0 ) . We have seen that

=

1"

I1-Q(eit)lP

< A

du(t)

+

B < A

3 ~ .Thus

+

3 ~ .

B

2 A. which is

-7T

Since this is true for every

E

> 0 , we finally get

what w e wanted. The next result, due to Szegg, gives a precise formula for the distance we are looking for.

1" J

+

l o g w(t)

dt

5

-TI

1" J

T h u s , the integrability of Since we have assumed

1"

w(t)

dt <

a.

-7T

log w

log-w(t)

is equivalent to that of d t < m , we have

log-

W.

-TI

l o g w(t)

dt) = K , a positive number.

-71

Dividing w

by

K

in the formula to be proved, we see that there

is no l o s s of generality in assuming that same, that

:1

l o g ~ ( t )d t

=

K

=

1 , o r , what is the

0. Then we have to see that the left

hand side in the formula o f the statement i s in this case equal to 1.

I. CLASSICAL HARDY SPACES

90

Let us consider the function

Then, F e H(D), since

F(0)

0, and the function

=

exp(F(z)/p)

is

Hp

in

which is a harmonic fumtion whose restrictions t o circles of radius r < 1

have uniformly bounded

short, which is uniformly in

L p norms o r , a s we shall say for Lp. Besides, exp(F(z)/p) i s , by

definition, an outer function, and

P e p ( 0 ) . We set

Let

I

G(z)

=

(1-P(z))

G(reit)dt

5

T;;

lr

1 = G(0)

=

for every

2n

1"

-TI

exp(F(z)/p)

IG(re it )Idt

=w(t) U P . .Then:

5

(&I

-TI

0 < r < 1. Letting

G e HP.

since

lexp (F(e it )/p)l

r

-+

1

TI

IG(reit)(Pdt)P

-n

1 , we get:

NOW

s o that we have obtained:

w h i c h is half of what we wanted to prove, the easy part. The converse

inequality is deeper. We shall prove that equality holds by using the density theorem 7.9. Since

is a n outer function,

exp(F(z)/p)

there will be a sequence of polynomials Q . e p

Qj(z) exp(F(z)/p) as

j

L/lr

27l

as

+.

m,

that

so

Qj(z)/Qj(0)

But

-'IT

I (1-Pj(e

j

-+ m ,

-+

1

in

Hp

(Qj(z)/Qj(0))

= 1-P.(z)

J

it ))exp(F(eit)/p)l

as

j -+

J

m.

exp(F(z)/p)

for some Pdt =

such that

In particular,. Qj(0) -+

1

in

Hp

+.

j +

as

P . e p ( 0 ) . Therefore: J

lr

2nl -TI

I l-P.(eit)) Pw(t) J

dt

-+

1

and we conclude that the infimum above is actually equal

1 m.

THEOREM

I. 8 . HELSON-SZEGO

to

91

1. The theorem is proved with the restriction that

is

log w

integrable. b) In order to complete the proof o f the theorem we need to consider

,I,

log w(t)

the case when w.(t) J

j +

m

=

max (l/j,w(t)). 2 t every point.

dt =

Then

w1 L w2 L

r, -1

j = 1, 2,

S e t , for

-m.

... t

log w.(t) d t > J

w

w. + w

and

as

J

for every

-m

..., and

j

77

l o g w.(t) J

l o g w(t)

dt

dt

j +

as

m

-TI

Now

inf PfP(0) =

kj'

I1-P(eit)lPw(t)dt

1 exp ( T;;

inf

5

PfP(0)

-TI

T

l o g w .(t) d t ) -+ 0 J

-TI

as

kITII1-P(eit)lP

w.(t) J

dt =

-77

j +

Thus, even in this case the formula holds.

m.

0

We are now ready t o initiate our search for a characterization of the Helson-Szego measures.'rhe next two results will narrow the set of candidates. 7HEORLM 8.8. fveay HePnon-Sregz meunuae i4 uk4oPutely c o n i i n u o u ~

(with aedpeci

u

t o

Lekenque rnea4uae)

E be a closed subset 0. We have to show that p(E) = 0. Lemma 8.3. tells u s that with E we can associate a f u n 2 tion F e 4,. such that F ( z ) = 1 if z e E and IF(z)I c 1 if z e D\ E . Consider Fj(z) = F ( z ) J - F ( O ) j for j = 1 , 2 , Proof: Let of

T

be a Helson-Szego measure. Let IEl

with Lebesgue measure

=

...

Since

F. e A J

Gk e p(0)

and

such that

sequence depends on Re G k -+ Re F . J

and

F . ( O ) = 0 , there is a sequence of polynomials J Gk + F . uniformly a s k -+ m . Of course, the J

j. We have:

I m Gk + I m F . J

uniformly a s

k

Looking at the boundary functions Re G k , I m G k e 7 Re G k = -(Im G k ) % , s o that

G k e p ( O ) , we have

-+

CO.

, since

I. CLASSICAL HARDY SPACES

92

-TI

and from t h i s

I", e

it

e E , Re F . ( e k E , t h e n Re F . ( : i t )

But i f it

e

it

J

Im F.(eit)

+. 0 as J i n e q u a l i t y , we g e t

Proof: were

+. 1

whereas f o r every

j +

as

m,

j +

Thus, l e t t i n g

+. m.

as

m

j +.

jTIl o g

W(t) d t =

-TI

and i f t,

i n the last

such t h a t

W ( t )

+.

0

m.

as

.

If

it

a c c o r d i n g t o Szego's

P. e P(0)

J

log w

f . ( t ) = 1-P.(eit), J J f . e 7; t h e n f . ( t ) = i P . ( e i t ) and a l s o ? . ( t ) = P . ( e i t ) . S i n c e J J J J J d u ( t ) = u ( t ) d t i s a Helson-Szego measure, t h e f o l l o w i n g i n e q u a l i Il-P.(eit)12

dt

we should have,

-m,

a sequence

m,

0

p ( E ) = 0.

A l l we h a v e t o p r o v e i s t h e i n t e g r a b i l i t y o f

theorem c1.7.,

J",

j

) = l-Re(F(0)J)

+ 0

j +.

Let

%

t i e s hold:

S o , w e h a v e , a t t h e same time TI

IPj(e

it

)I2

w(t) dt + 0

and

-TI

as j tion.

jTI -TI

+. m .

0

It follows t h a t

F r o m now o n , we a s s u m e

1 l - P .J( e i t ) 1 2

l'w(t)

W(t) d t + 0

d t = 0 , which i s a c o n t r a d i c -

-TI

w 2 0 , w e L 1 (T)

O u r p r o b l e m i s t o f i n d when

w(t) dt

a s we s h a l l s a y e q u i v a l e n t l y , w h e n i s

and

also

1 log w e L (T).

is a HelsonSzego measure o r , w

a Helson-Szego weight, For

o u r f i n a l a t t a c k on t h e p r o b l e m w e s h a l l b e w o r k i n g i n t h e H i l b e r t space

2

L ( w ) . We s h a l l d e n o t e t h e i n n e r p r o d u c t i n t h i s s p a c e by

I.8 . HELSON-SZEGO THEOREM

93

and the norm by

It will be convenient to use the operator A appearing towards the end of section 6 . A sends a trigonometric polynomial

A

We already know that the operator

is very closely related to the

conjugate function operator. Actually these two identities are very easy to check for

-

'L

f = -i(Af-AF)

f e 7

% ' L

and

Af

(-f+if)/2

=

They prove the following lemma

LEMMA 8.70. tun

in

w

u Helnon-Szeg;

w e i g h t i+? und o n l y if A in (on ~~(w).

e e u n i q u e l y e x t e n d e d t o ) u eounded o p e f i u t o f i i n

This can easily b e translated into the following

7ff&OR&M 8 . 1 1 . exi4in

~1

< 1

Proof: Every f o r some

Af(t)

=

w

in

u Helnon-Szeyo

nuch t h u i ,

f e 7

P,Q e P ( 0 ) it P(e ) , and

weight

eon e u e q

if und o n l y it f h e f i e

P , Q e P(0)

is of the form

f(t)

=

P(eit)

+

eitQ(eit)

a n d , given this representation,

111112, = (P(e it )

+

e it Q(e it )

I

P(eit)

+ C~~Q(C"))~

If the inequality in the statement holds with some

=

< 1 , then

I. CLASSICAL HARDY SPACES

94

and, consequently, w Conversely, let

is a Helson-Szego weight. be a Helson-Szego weight. We want to get an ine-

w

quality like the one in the statement of the theorem. Of course, we just need to look at P.Q e R O ) with IIPIIw; llQ1lw= 1. If we call 2 f(t) = P(eit) t eitQ(eit), we know that IIPII, s CIIflIw where, of 2 course, we may take C > 1. Thus IJflIw2 1 / c . Now, using again the 2 formula for Ilfll, given at the beginning of the proof, we get TI

P(eit)

2(1tRe

e-itQ(eit )v(t)

dt) L 1 / C

-71

or, what is the same,

1

n

Re

P(eit)e-itQ(eit)w(t)dt

2

-(1-(1/2C))

-TI

Since this is also valid for

I

Re

,I,

P(eit)e-itQ(eit)w(t)

I zI

T H E O R E f l 8. ? Z , DeLine t o n

We know

{on

U.L.

thut

t.

-P, we conclude that

O e H2

Let

w(t)@(e

in

4

1-(1/2C)

=

(Y

< 1.

0

< I

I @ (eit ) I

u n o u t e n & u n c t i o n and

i t ) -2 - ei@(t).

w e i g h t i& and only i,! t h e d i n t u n c e i n nfnictly Amullen t h u n

dt

w

L"

e/zom

i4

ei@

=

w (t)

a HeeAon-Szcgb to

i5

1.

Proof: According to theorem 8.11, w and only if there is ct < 1

is a Helson-Szego weight if

such that for every

P,Q e P ( 0 ) :

I. 8 . HELSON-SZEGO THEOREM

This is equivalent to saying that, for every

(just consider

eiTP

P

instead of

95

P,Q e / 7 ( 0 ) :

for an appropriate real number

It follows from theorem 7 . 9 that @ . J ) ( O ) is dense in H2(0) -it and also e @(eit)*p(0) is dense in H2, s o that (8.13) is equivalent to

T).

2

G e H (0) and every

for every

.We know that every G

e

H

H2(0),

finally, that some

a < 1

e

H2 w

F

e H'(0)

and

H

f

H2

.

can be factored a s

lIGllH2 = llHllH2

=

llFll:{2

G*H

with

s o , we can s a y ,

such that

I

Lm/Hm

=

is a Helson-Szego weight if and only if there is

I

This is the same a s saying that the norm of =

F

is bounded by

a <

1

eim(t)

in

H1(0)*

=

o r , in other terms, that

a s we wanted to s h o w . 0 We are now ready to prove the Helson-Szego theorem, which reads a s follows.

Proof: First of a l l , let u s s e e that the condition is sufficient. u

Since from

0

= e v"(t)

is bounded, e U and with

m .

v

will be a positive function bounded away

Therefore, we just need to see that if real and

llvll,<

.rr/2,

then

w(t)

=

w is a Helson-Szego

I. CLASSICAL HARDY SPACES

96

weight. In this case

a n d , consequently, Then

T .

O(z)’

w(t)Q(eit)-’

=

= exp(P(?-iv))*eiT for some real number -i-r iv(t) e *e The constant factor i s irrele-

.

vant and we are reduced to seeing that is quite easy. We know that such that =

1

Iv(t)l

t sin2 E

since

-

2 c o s v(t)

c o s v(t)

IIvIIm< n / 2 ,

( n / 2 ) - ~ for a.e.t.

2

2 sin

sin

E

s o that there is

> 0

E

€1’

- sin

Then

-

2 1 t sin2 E - 2 s i n 2 € = 1

, We have shown that

E

< 1. But this

distLm(eiv,Hm)

distLm(eiv,Hm)

=

sin

2 E

< 1

a n d , consequently, w, i s a Helson-Szego weight. Next we shall prove the necessity. We start with a Helson-Szego weight

w

and consider the functions

0

and

@

associated to

w

in the statement of theorem 8.12. This theorem tells u s that

H e Hm

There will exist some

such that

Iw(t)Q(eit)-’-H(eit)I Since

IQ(eit)12

= w(t),

Note also that

> 0

w(t)

=

I ei@(t)-H(eit)l

< 1

2 a

for a.e.t.

it follows that

,

for a.e.t

We conclude that the boundary values

I

71

since

l o g w(t)

it 2 -“it) Q(e ) H(e

dt >

of t h e

-m.

H1

?

O‘H

function larg IzI

ZI

a r e , for almost every

< 1

t , in the sector

a < n / 2 . Since the values

6 arc sin

for Q(z)~*H(z) are obtained from the boundary values b y means of the

Poisson integral, they will also belong to the same sector. In particular

Re(Q

2

H)

2 0

outer function. Thus

Q 2H

and theorem 7.7. implies that Q 2 H is an i.r UtiV = e e where T is a real number, U

is a real harmonic function and

IV(e Set

v(t)

with

c

it

)t

TI

= -V(eit)-.r.

6

V

its conjugate. But

arc sin a and (V(eit)+T)% Then

Iv(t)l

a real constant. Thus

< n/2

and

= -U(eit)

U(eit)

= ;(t)

t U(0) t c

,

1.9. THE DUAL OF

le i@(t) From 5 l/IH(eit))

- H(eit)l 5

IIvIlrn5 arc sin

l/(l-a) <

f

e

0 < l/(lta)

We can set eC/IH(eit)I it 2 = I@(e ) I = e u(t)+v"(t)

5

with

=

u

with

0

H1.

9. THE DUAL OF Every

m.

w(t)

< n/2.

c1

a < 1 , it follows that

5

real and bounded. Then

97

H1

gives rise to a functional Af e ( H 1 ) * 1 F e H , the complex number

Lm(T)

obtained

by assigning to each

Af(F)

=

I"

F(eit)f(t)dt

-TI

Conversely, every

A e

(HI)*

is of the form

A =

Af

for some

f e Lrn(T). The kernel of the mapping Lm

is

Hm(0),

-

(H 1 ) *

s o that we can write

(H~)*

=

L~/H~(o)

Similarly

The purpose of this section is to characterize these duals a s spaces of functions. We shall look at H ' ( 0 ) . Every continuous E-linear functional over A(F) where

L

=

ReA(F)

+

is a continuous

iRe/\(-iF) = L(F)

H'(0)

is of the form

t iL(-iF)

IR-linear functional.

Conversely, every continuous

IR-linear functional

L

is of the

I. CLASSICAL HARDY SPACES

98

form L(F) = ReA(F) for some continuous @-linear functional Therefore, for some real functions @ , $ e Lm

I

71

L(F)

=

ReA(F)

1

=

Re

F(e

it

)(@(t)+i$(t))

A.

dt =

-71

71

lim Re r+l

=

F(reit)(@(t)ti$(t))

71

lim r+l

=

dt =

-71

- ImF(re it )Q(t))

(ReF(reit)@(t)

dt =

-71

I

71

lim r+l

=

ReF(reit)(@(t)t$(t))

since, observing that

I

dt

-51

F(0)

=

0, we have

71

(ReF(reit)$(t)

t ImF(reit)$(t))

dt =

-71

I

71

=

Im(F(reit)($(t)ti~(t)))

dt = 0

-71

The correspondence F ++ ReF is a one to one mapping from H'(0) 1 onto ReH ( 0 ) . We consider ReH1(0) as a real Banach space with the norm

IR-linear functionals on H'(0) correspond to the 1 ReH (0). S o , o u r problem i s really to

The continuous

continuous functionals on determine the real space

(ReHl(O))*.

We already know that the mapping + (ReH'(O))*

ReLm t (ReLm)% sending the function

1

0 t

6

into the functional

= lim

r+l

given by:

TI

71

L(ReF)

L

ReF(reit)(@(t)t$(t))dt

=

Re

F(eit)(@(t)ti$(t))dt

-TI

-71

is onto. Let us s e e what is the kernel of this mapping. If lim r+l for every

'I

ReF(reit)(@(t)t$(t))

dt

=

0

-71

F e H 1 (0), then taking

F(z)

=

z n , n = 1 , 2 , 3,

..., we

1.9. THE DUAL OF

H1

99

obtain

1

1T

(cos n t)(@(t)

+

ij(t))

dt = 0

-n F(z)

and, taking

1"

izn, n = 1 , 2 , 3 ,

=

(sin n t)(@(t)

t &(t>)

..., we

obtain

dt = 0

-1T

It follows that

(ReHl(0)) the norm in

+

Q(t)

*

=

$(t)

is actually a constant. We can write

(ReLm+(ReLm)%)/IR.

ReLm t (ReLm)%

being

where the infimum is taken over all representations o f the function as

6

+ 6

6 , @ e Lm.

with

What we have done is to transform the problem of characterizing the dual space, into the problem of characterizing those functions g which can be written in the form

6 , (I E LOD. These functions will turn out to be the functions o f bounded mean oscillation (B.M.O.).

with

DEFIMI7ION 9 . 1 ,

Let

g

Le u locully inteqnuLle

2n-peniodic

{unction on IR. We n h a l l AULJ t h u t g i A 08 Lounded meun o ~ c i l l u t i o n i8 and o n l y if thene i n u conhtunt C n u c h f h u t / o n euenv Lounded intenuul

I c IR:

7he npuce tonmed L4 u l l 2r-peniodic tunctionh 08 Lounded meun oncillution w i l l Be denoted LLJ B.M.O. ( T ) o n , n i m p l y , Lx B.M.O.

I. CLASSICAL HARDY SPACES

100

Given

g e B.M.O.,

the infimum o f all the constants

f o r which

C

(9.2)

holds will be denoted by l l g l l * . The mapping g w l l g l ) * is a seminorm on B.M.O., vanishing only f o r the constants. Thus, in o r d e r to get a normed space, w e have to consider the quotient o f B.M.O. m o d u l o constants. We shall call this space also B.M.O., hoping that this ambiguity does not cause any problem.

Proof: Suppose

I

is an interval with

) I \ > 2 n . Let

n

be an

integer such that n.2T 2 111 < (ntl) 2 n . Take an interval J 3 I with 1J1 = (ntl).2n. Then

Therefore:

6

6C.

0

Proof: We can assume 6 < 111 2 2 ~ r . Then

6 < 2n. Let

I

be an interval with

1.9.

THE DUAL OF

H1

101

Proof:

7HEOilEfl 9 . 6 . t i o n n . Then

with Proof:

c

Let

g = @

g e B.M.O.

+

5 &

@

& JI

2n-peniodic

Lw-&-

Actually

an n h o l u t e c o n n t a n t .

Lm c B . M . O . a n d 1 1 @ ) 1 * 5 21(@IL.S o , t h e p a r t of t h e theorem is t h a t t h e conjugate function

It is obvious t h a t

only non-trivial

Lm

operator sends

boundedly i n t o

B.M.O.

We s h a l l s e e i n t h e

5th s e c t i o n o f c h a p t e r I1 t h a t t h i s i s a s t a n d a r d f e a t u r e o f s i n g u lar integrals.

The proof g i v e n t h e r e c a n e a s i l y be a d a p t e d t o o u r

p r e s e n t s i t u a t i o n . We s h a l l n o t g i v e t h e d e t a i l s h e r e , s h a l l f i n d another proof of t h i s r e s u l t s h o r t l y .

Proof:

since we

0

It is enough t o l o o k a t t h e r i g h t hand s i d e a f t e r expanding

( @ ( t ) - @ ( s ) ) 2 = @ ( t ) 2 + @ ( s ) 2- 2 @ ( t ) @ ( s ) .

DEFIh'I7ION 9 . 8 .

Foe

I(@) =

sup

Izl
0

a4 i n t h e lemma, we d e e i n e (P(tJ2)(Z) - (P(@)(Z))

2 112

1

The f o l l o w i n g p r o p e r t i e s are immediately v e r i f i e d

I. CLASSICAL HARDY SPACES

102

b)

I(@+$) 5

I(@) t

~ ( u ) (use

the lemma and Minkowski's

inequality). c)

I(@ = ) 0

It follows that it is finite.

if and only if N

@

is a constant.

is a seminorm on the space of functions on which

We can s e e rather easily that f o r g to b e o f t h e form

I ( g ) < m

g = @ t

Proof: Lemma 9.7. implies that

6

with

is a necessary condition @ and @ bounded. Indeed:

I(@ 5 0 )

]I@],.

On the other hand,

t is a holomorphic function belonging to

Hp

for every

p <

m.

Thus, it will be the Poisson integral of its boundary function, and the same will happen to its real part. Therefore

so

that

and, consequently:

Next, we shall s e e that the finiteness of g e

B.M.O.

N(g)

implies that

1.9. THE DUAL OF

7H€Oil€M 9.10.

with

C

Ton

@

+ &

We may assume

2n-peniodic

I

=

I

0 < a 5

with

[-a,.)

TI

5

I tl

If

2 a , we have:

1 - r 2

2 2

t

(1-r) t 4 r sin (a12) r = l-sin(a/2), t

2

-5a

.

2 1 - 1 -2 1 - r (@(t)-ys))l+r 2 -21-cos s l+r 2-2r costdsdt

r < 1. This follows from lemma 9 . 7 .

2 2 1 - r 1 - r 2 (1-r) +Zr(l-cos I+r 2-21- c o s t

L

Then, we use the formula

TI.

2

-71

0

103

un u e n o e u t e c o n h i a n t

Proof: F o r any interval

for

H1

we have:

1 - r2 2 2 ( 1 - r ) +4r sin (t/2)

t)

1 - r 2 2 (1-r) + 4 sin (a/2) 1 - r

2

2 l + r -2r c o s t

Thus, for our choice o f

r

=

1

-

t

. In

t

particular, choosing

sin(a/2) 1 2 5 sin (aI2) 5 sin(a12)

sin(a12):

1

I. CLASSICAL HARDY SPACES

104

111

We conclude that, for

As in lemma 9 . 3 , if

5

TI:

111 > 27r:

Combining the last two theorems, we obtain the promised alternative proof of theorem 9 . 6 . :

The difficult part of the duality theorem is to show that every f with with N(f) < m can indeed b e written in the form f = @ t iJ @ , $ e L m , s o that f gives rise to a continuous linear functional over

ReH 1 (0). To reduce the proof of this fact to its essential

core we shall need several simple preliminary lemmas.

1"

W(eit)

11,

dt =

-a

c2(D)

Proof: W e

-

means that

( l o g -) 1

AW( z ) d x dy

I Z I

W

is

c2

on some open set containing

D. F o r small

DE

= ( z e

- (A =

I"

E

e:

l o g -)

1

I ZI

W(eit)dt

> 0 , we apply Green's theorem on the domain I z I 

dxdy

t €(log

=

E )

-71

We have denoted by normal.

a an

the derivative in the direction of the outer

1.9. THE DUAL O F

Now, since make

E

W(0)

0

-+

REMAilK 9 . 7 2 .

0, we have

=

to get what we wanted.

0

Oflnenve t h a t t h e l e m m a

i 4

I ~ w1(z) I

~ ( z )=

with w1

LEMMA 9 . 1 3 . L e t U & V and l e t U(0) = 0 . &

I

U(eit)

V(eit)

dt

IID(

2

=

-T



e

105

and we only have to

= O(E)

v a l i d , w i t h t h e 4ame p n o 0 8 ~

c2(5)

fle h u n m o n i c ~ u n c i i o n 4on

T

whene

IW(Eeit)l

H1

log

1

R > 1

D(O,R),

V U I V V > dxdy

) <

i n u4ed t o d e n o t e t h e 4calan p n o d u c t and

t o denote

t h e gnadient

W = U V , and observe

Proof: We just need to apply lemma 9 . 1 1 with that

LEflflA 9 . 1 4 . z = 0.

F

Let

IF(.)/

&

&

Cm

= (U(z)

2

+

V(z)

) 1'2

4uch t h a t

0 1

H(D(O,R)),

E

IF(.)/

U

with

LEMflA 9 . 7 5 .

R > 1 , and

F e H(D(O,R)),

Let

vuninh except a t

z = 0, wheae i t ha4

use lemma 9 . 1 4 .

f

=

0

on1.y , / o n

real, we see

for

z

+

0.

TO

0 4Uppo4e t h a t

zeno.

with

W(z)

F

doe4 n o t

7hen:

=

F(z)

and

0

Now we are ready to prove that the condition for

cm

a dimple

Proof: We just need to apply remark 9 . 1 2 .

V

and

is clearly

compute the Laplacian is a simple exercise.

F(z)

&

to b e of the form

f

=

@ t $

with

prove this, it is enough to see that, if 1 F e H (0) there exists

<

iV(f) @,@

N(f)

e

<

m

is sufficient

L". In order

a,

to

then, for every

I. CLASSICAL HARDY SPACES

106

1

IT

(9.16)

lim r+l

ReF(reit)

f(t)

dt

-IT

and provides a continuous linear functional on H 1 (0). Indeed, w e know that this implies the existence of @,I) e Lm such that, for every F e ~'(0) IT

lim r+l

I

ReF(reit)

- @(t)

(f(t)

- $(t))

dt

=

0

-TI

,..

In particular, taking F(z) = z n , n = 1 , 2 , we arrive at f(t) - @(t) - $(t) n = 1, 2 ,

...,

=

and F(z) = izn constant.

On the other hand, in order to s e e that the limit

(9.16) exists 1 and provides a continuous linear functional on H (0), it is enough to show that there is a constant C such that for every F e H'(0) and every 0 < r < 1 :

Indeed, suppose we know that (9.17) holds independently o f r. T h e n , 1 given F e H ( 0 ) and given E > 0, there will exist r o < 1 such that for every r , ro < r < 1

'1

(F(re it ) - F(eit)l

dt <

E .

-IT

If

rl, r2

satisfies

r o < r1 < r 2 < 1 , then

are such that ro < r < 1

belonging to

H'(0)

and with

G(z) IIGII

=

F(z)

<

E .

-

Therefore

H ReF(r2eit)f(t)dt 5

-

1"

-TI

F(rz)

r = r /r 1 2 is a function

ReF(rleit)f(t)dt

1

=

/IlITReG(r2eit)f(t)dt

C E

and we conclude that the limit in (9.16) exists and provides continuous linear functional on

a

~'(0).

note that, in order to s e e (9.17), it is enough to show that there exists a constant C such that for every F e H 1 ( O ) , every 0 < r < 1 and every 0 < s i 1: Also

5

1.9. THE DUAL O F

F

H1

107

Hl(0) has a simple zero at z = 0 and is different from 0 elsewhere. Indeed, we can write F(z) = zG(z) is the Riesz factowith G e H1 and then, if G(z) = B ( z ) H(z) rization of G , we have F ( z ) = z((B(z)-l)/Z) H(z)+z((B(z)+l)/2)H(z). 1 The two terms in this sum are functions in H ( 0 ) having just a simple zero at z = 0 and with H1 norms bounded by JJFjlH1. We can even assume that

Fix

F e H'(0)

and

s. Set

with just a simple zero at

U ( z ) = ReF(rz)

V(z)

e

=

z =

0, and fix also

r

and

P(f)(sz)

Then, after writing the integral in

(9.18) a s in lemma 9.13, we s e e

that it is enough to prove the inequality

with with

C C

independent o f =

kN(f)

and

k

r,s

or

F. We shall eventually prove 9.19 r,s or F.

independent of

As a preliminary step we use the Cauchy-Schwarz inequality, obtaining a s a bound for the left hand side of (9.19): I F ( rz) 1 1/2dxdy

5

According to lemma 9.15, the first factor equals

Thus, it will be enough to show that the second factor is bounded by CN(f) l l F l f ( 2 with C an absolute constant. We shall indeed show that

I. CLASSICAL HARDY SPACES

108

G e H 1 , C being an absolute constant, (9.20) will be a consequence of the following two results:

holds for every

THEORLM 9 . 2 1 .

( log

The m e a n u / ~ e dp(z) = I z I ~ u t i ~ t i et hne i p o l l o w i n y c o n d i t i o n :

D

Iip Lon

I,

evefig i n i e n v a l

) <

s u p (V(R(I))/III

I

the

sup.

v

on

IVV(z)12dxdy

i n

-

the net

D

having

is customarily called a Carleson measure and

m

above is called the Carleson constant of the measure.

7ff€OR&fl 9.22.

k(v).

R(1)

we d e n o t e k q

In general, a positive Bore1 measure

-rq 1 )

Let

v

l e u C u d e n o n meadune w i t h C a a l e n o n

condant

Then

b) T o n e v e f l y

G

e

H

1

:

We give first the Proof o f theorem 9.22 We just need to prove write Then

G = G 1 - G 2 with

1

Indeed, once we have a), if G e H /ze 2 G1, G 2 e H and \ l G l l l H 2 = llG2\l = \lG\l

a).

i

H

H

.

1.9. THE DUAL OF

Before presenting the proof o f

a)

109

H1

let u s say that the analog of

lRn is given a s theorem 2.18 in chapter

a) for t.he euclidean space

11. The proof that follows is based upon the same ideas.

0 < a < IT/^

For

and for

e i e e T , we shall denote by

rate)

the

Stolz region based on e i e with aperture a which i s , by definition, the interior of the minimal convex set containing e i e and the disk D(0,sin a ) . For f e L1(T) w e define

Naf(€l) is the non-tangential Poisson maximal function of aperture (y Its euclidean counterpart will be denoted Pi,a(f), but for the moment we do not want to introduce unnecessary complications,

.

The main fact is that

where

is the Hardy-Littlewood maximal function, to be systematically by studied in chapter 11. The proof of (9.23)

is essentially the same a s that of ( 4 . 1 5 ) in

chapter 11. We omit the details. We shall use the operator Let 1 < p < 1 I z I 6 7 , then

= 1/2.

m

and

N

=

g e

N

with a = n / 6 , s o that sin a Lp(T). We want to prove a). If

a

=

I. CLASSICAL HARDY SPACES

110

s o that

1,

IP(g)(z)lp

dv(z)

S

CV(D)

2 ,S 1 / 2

1"

Thus, w e can forget about the region 121

and v(D)

Ig(t)lPdt

5

Z~k(v).

-77 121

s 1/2

and concentrate on

> 1/2.

It is a simple geometrical observation that if

Given

h > 0, let

EX

= ( 8 e [-T,V):

the connected components of

Eh.

N(f)(B)

> h

IP(g)(z)l

a given z with 121 > 1 / 2 , then N(f)(B) > h if eiB an arc centered at z / ( z ( and having length 2c(l-\z\) is some geometric constant (c = 0 / 2 is enough) > A},

for

belongs to where c

and let

I . be J

It follows from the last observa-

tion that

Next we shall u s e the fact that

M

is a bounded operator in

Lp(T).

T h i s fact can be established essentially as in section 1 of chapter 11. After ( 9 . 2 3 ) ,

N is also bounded in

Lp(T),

and we can write

We come finally to the Proof of theorem 9 . 2 1 :

Observe, first o f a l l , that

1.9. THE DUAL OF

H'

111

the inequality

needs to be established only for small intervals. The reason is 0 < ro < 1 , we have:

that, given

that, once we have proved (9.24) for small intervals, say for all such that 111 4 6 for some fixed 6 > 0 , if we are given an interval J with IJI > 6 , we can write J as the union of

so

I

intervals

..., IN with ... U R(IN) U

11,

R(J) c R(I1) U < ro < 1 , we get

u(R(J)) with a bigger

2

CN(f)

2

(Ij( 5 6 D(O,ro)

and

N

5

ZIT/&.T h e n , since

provided we take

1-(6/(2~))

IJI

C.

1"

To prove (9.25) is very simple: if we write

fo(t) = f(t)-& Vo(z) = P(fO)(sz) we have VV(z) = VVo(z) since V differ only by a constant. But, for I z I 2 ro < 1 , and

f

and-'V0

by theorem 9.10. Naturally the constant in the first inequality depends o n ro tends to a a s ro 1. (9.25) follows after realizing that 1

and

+

121

log

2

1.

T o prove (9.24) we shall assume is centered at

R(1)

(11 = h < 1/2

and also that

0. Then c R h = {re

it

: ( t ( < h/2

and

and the proof will be finished if we are able

I-h

to

< r < 11 show that

I

<

I. CLASSICAL HARDY SPACES

112

u(Rh)

d CN(f)'

h

I t w i l l b e more c o n v e n i e n t t o work i n t h e u p p e r h a l f t h a t e f f e c t we map t h e d i s k

IzI < 1

onto

2

IRt

IR,.2 T o

plane

by m e a n s o f t h e

conformal mapping: Q(z) = i

On

IR,2

we shall use the coordinate

i n v e r s e of

For

1 - 2 l t z

1/2 5

0

5

=

5

t iq,

with

5,q e

IR.

The

w i l l be t h e mapping

Izl < 1

note that

and %(l-q)

1 -

)212

=

2 -

4r-

1 -

52t(1tr))2

5%

2 4q

l+rl)

so t h a t

A s i m p l e computation shows t h a t

Now, a f t e r w r i t i n g

W ( < ) = U(Q(c)),

t h e change of v a r i a b l e s formula

implies

T h i s i s m o s t e a s i l y s e e n by o b s e r v i n g t h a t t h e J a c o b i a n o f

y

is

lY'(<)l and, i f w e write V = ReG w i t h G h o l o m o r p h i c , t h e n = I ( G o Y ) ' ( < ) I 2 = lVW(<)l2 l v v ( Y ( < ) ) 1 2 p ' ( < ) 1 2 = IG'(Y(5))121Y'(<)12

1.9. THE DUAL OF

113

H1

We arrive at:

Now

2 Qh c { L e E+ : 151 < 2h). Actually, for

5 e Q h , we have

Then

Remember that set

N(f)

=

) 1’2

sup(P(f2)(z)-(P(f)(z)) then

so

that, if we

A ( z ) = P(f2)(z)-(P(f)(z))*, 0 I A(z)

5

/V(f)

2

and the integral to be estimated equals

r1>0

which, using G r e e n ’ s theorem and writing o u r domain, is seen to be equal to:

n>0

r

for the boundary o f

I. CLASSICAL HARDY SPACES

114

The inequality is due to the fact that the first of t h e two integrals 2 hd ) vanishes on r , and the second over r i s zero, since r\(l - -L 1 - 1

(q(1-% ) ) s 0 on r. Using the polar re an 1 3 )2 5 = peie, the Laplacian becomes A = ( -

one is

2 0 , since

sentation Then it is very easy to compute

o.

a0 .

dU

Finally

s

&

io /i 2h

N(f)2

sin

d e p dp

=

2

6N(f) h

as we wanted to prove (H1(0))*

We have finally identified on

T

N f) <

for which

coincides with

co.

f In order to see that this space actually as the space of functions

B.M O., some more work is needed.

For the next result we shall u s e the fact that, for

with

C

f e B.M.O.:

an absolute constant.

This is contained in corollary 3.10 of chapter I 1 and is a consequen ce of the John-Nirenberg theorem, a basic result for B . M . O .

func-

tions, which shall be studied in chapter I1 within its most natural framework. We shall also make use of the following observation: If and

I c J

are concentric intervals, then:

This is clear if

IJI

2

III

since

f e B.M.O.,

r . 9 . THE DUAL OF

T h e g e n e r a l c a s e f o l l o w s by

I c J1 c J 2 c

where

... c

H

1

115

writing

JN c J

and

We are r e a d y t o p r o v e t h e f o l l o w i n g

7HEOREfl 9.26.

with

C

Foa

/ z e a l und 2r-peaiodic

f

independent of z = r

consider

f

0

with

5

and of

z e D.

I t i s c l e a r l y enough t o

r < 1.

A c c o r d i n g t o lemma 9 . 7

w h i c h , a f t e r i n t e g r a t i n g by p a r t s a n d w r i t i n g

1-r

=

u,

is seen t o

be bounded by: m

us

Call

I

=

2

[-s,s],

2

t

ut

J =

(f(O)-f(T))’

[-t,g.

dod-r) d s d t

Then, w e have t o e s t i m a t e

I. CLASSICAL HARDY SPACES

116

Finally

,

u2s2t2 Ilfl12, (ltlon2(s/t)) 2 2 2 2 2 2 ( u ts ) ( u tt )

1:

1, I,

=

x2y2(1t2(log x)2t2(loR (1tX2)* (lty2)2

ds dt =

Y)

2

)

dx dY

Ilfll, 2

=

All the work we have done i n this section can be summarized in the following result:

7H&‘OR&fl 9.27, Let the tonun.

f

ee u n e u l

2 v p e n i o d i c &unction i n t e q n u L l e o n

7 h e n , t h e k o l l o w i n q conditiond u n e euuiualent:

1

n

F+-+

lim ri.1

Re F(reit)

f(t)

dt

-7~

i n u c o n ~ i n u o u ne i n e u n t u n c t i o n u l o n

f e B.M.O.

(C)

f i e d i d e n , t h e n u t u n a 1 noam5 unnociuted

&

~‘(0)

(e)

une u e e e y u i u u e e n t

.0

t~

f

i n

(a),

(b),

(c), (d)

1.10. THE CORONA THEOREM

117

10. THE CORONA THEOREM I n this section w e are going to view

Hm

a s a Banach algebra with

pointwise multiplication. It is easy to s e e that all the algebra homomorphisms of a weak-*

Hm

&

onto

closed subset

S

a r e , actually, continuous. They form

of the unit ball of the dual space

(Hm)*.

S , with the weak-* topology i s , consequently, a compact space which is customarily called the maximal ideal space of Hm. The reason for this name i s the one-to-one correspondence between S and the set of maximal ideals of Hm obtained by associating to an algebra homomorphism L from Hm onto &, the maximal ideal m I = If e H : L(f) = 0 ) . The corona theorem is an important step in understanding the space

S. F o r each

z e D

= { z

algebra homomorphism

e &:

bZ

IzI < 1 1 , the evaluation at z is an Hm onto & given by 6Z(f) = f(z).

of

The mapping D-S

D

is a homeomorphism between considered a s a subspace of

and a part of S , s o that D can be S. The purpose of this section is to

prove the following result, known a s the corona theorem THEOREM 70. I

D

i 4

denhe i n

S.

We shall base our proof upon three lemmas. The first one is simply a reformulation of the condition that D is dense in S. LEMMA 1 0 . 2 .

7he t o l l o w i n q 4 t a t e m e n t n ane e q u i v a l e n t :

a)

D

b)

F o n euenzy

i n denne i n

S.

fl,f2,

> 0, t h e n e e x i 4 t

...,

f n e Hm

gl,g2,

...,

nuchthat

gn e H m ,

inf maxlf.(z)l> J zeD j nuch t h a t

I . CLASSICAL HARDY SPACES

118

a), let

P r o o f : Assuming

fl,f2,

..., f n

e Hm

be such t h a t

i n f max ( f . ( z ) l = 6 > 0 . C o n s i d e r t h e i d e a l

zeD

. I

j

I = (glfl

... +

+

gnfn:

max I f . ( J ) I j

... t

gl,

..., g n

6 . T h e r e f o r e : I = Hm

2

is not contained

e Hm}. Now, I

a) t h a t

J, s i n c e i t follows from

i n any maximal i d e a l

and, consequently:

1 = glfl

+

..

J

gnfn.

Now a s s u m e , c o n v e r s e l y , t h a t

b)

+

holds.

J e S, c o n s i d e r t h e

Given

I h . ( L ) I < 6} f o r some J V = (L: (h.(L)-h.(J)I < b } . J J Now, t h e f u n c t i o n s f . ( z ) = h . ( z ) - h . ( J ) are i n J a n d , s i n c e J J J J d o e s n o t c o n t a i n 1 , i t f o l l o w s f r o m b ) t h a t i n f max I f . ( z ) l 5 0 , zeD j neighbourhood of hl,

..., h n

e Hm

J

V = J

{L:

6 > 0, t h a t is

and

from which it is clear t h a t t h e r e e x i s t s

..., n

j = 1,

be done f o r

: Ifj(z)I and

J

V

z e D

< 6 . T h i s means t h a t

such t h a t f o r every

z e V.

Since t h i s can

a) holds.

a r b i t r a r y , we c o n c l u d e t h a t

0

F o r t h e n e x t lemma we s h a l l u s e t h e d i f f e r e n t i a l o p e r a t o r s

a

i-a ) a n d ax ay i n solving the equation =

-i 2

(

a -

LEflflA 70.4. L e t g(z) Then, t h e J u n c t i o n

LAC"

-

so

SeCL

C"

# u n c t i o n w i t h cempuct n u n n o n t c'a

A o l u t i o n oJ t h e e u u u t i o n

Proof: Since IzI

-

-a = l2 ( a t i-a ) . We s h a l l b e i n t e r e s t e d ax ay

g

(10.3) & on

(null

I

I Z I

<

h a s c o m p a c t s u p p o r t a n d we a r e o n l y i n t e r e s t e d i n

< 2 , we c a n w r i t e f o r a n a p p r o p r i a t e

R > 0:

h

1.10. THE CORONA THEOREM

z = 0 , and compute

Without loss o f generality, we can take

-

af(0)

=

lim r+O

119

'I

TI

where we have used Green's theorem to pass from the two-dimensional integral to the integral over the circle.

0

The third and last lemma contains the basic idea for the proof of theorem 10.1 L€flflA 10.5. thut i n

D,

Cm

h

{unction

on

R > 1.

D(0,R).

4

u n e C u n l e h o n rneununed w i t h C u n l e n o n c o n n f u n t ~ A uely.

T h e n we c u n l i n d u L u n c t i o n

D(O,R'),

2)

Suppone

eofh

R' > 1

v(z),

which

Cm

i 4

B

ne4pecti-

o n dome

d h k

und 4 u L i h L i e 4

1 v(eit)l

< c(A~/~+B)

{on

t

eueny n e u l

whene

C

un

i h

uilnolute conhiant.

Proof: We redefine

h(z)

for

function, which we still call

(21

> (1tR)/2

h , Cm

so

in all of

a s to obtain a

@

and having

compact support. T h e n , the formula in lemma 10.4 with h in place of g , gives us a function vo such that avo(z) = h(z) for

ern

IzI < 2. The problem is that

vo

may be very large on the boundary

of the unit disk, and we have t o change it s o a s to satisfy condition

2). Now if f is holomorphic in D(O,R'), R' > 1 and we set v = v -f, we still have av(z) = h(z) for IzI < 1 , since a f 0 The idea is to choose f in such a way that 2) holds. vO(eit)

=

0.

120

I. CLASSICAL HARDY SPACES

belongs to the class

c

of

of all continuous functions on the boundary i t) A = H m n c . Let d be the L" distance of vo(e

D . Consider

from

A , that is d = inf

[I

vo-fllm = sup Ivo(e t

it

)-f(e

it

)I:

f e A).

5 d" < If d' > d , we can find f e A such that Ivo(eit)-f(eit)I < d ' for every t , and, consecuently, there is r < 1 such that r < d'. The function v = v o - ' t)-f(reit)( for every t:

satisfies 1) and also lv(eit)l < d'. S o , where f (z) = f(rz) our problem is really to compute d , o r rather, to s e e that it) in the quotient d C(A112+B). Now, d is the norm of vo(e C I A . It follows immediately from the F. and M. Riesz theorem 1 (3.10), that t e dual of C I A can be identified with H (0). Thus, according to the Hahn-Banach theorem,

space

d = sup[I/ vo(eit)F(e

dtl : F e H1(0),

it

IIFI; H1

5

11.

We can further restrict the set of F's, taking only those which are analytic on some For such an

But

A(v

0

F)

+

R > 1 , where,

D(0,R)

o f course,

R

varies with

F , we apply Green's theorem, obtaining (lemma 9.11)

=

4

4aT(voF)

ljD

F'(z)

=

4hF'

h(z)

+

log

4ahF. Thus

1

dx dy.

The first term in the sum is bounded in absolute value b y

since we are dealing with a Carleson measure, s o that theorem 9.22. b ) applies.

For the second term in the sum we use first the Cauchy-Schwarz inequality to dominate it by

F.

1.10. THE CORONA THEOREM

Observe that we just need to estimate this f o r

121

F e Hl(0)

with no

zeroes in D except f o r a simple zero at the origin (see the explg nation at the beginning of page 1 0 7 ) . For this kind o f F, we have, according to lemma 9.15:

Also, applying theorem 9.22. b) again, we get

-

1 F e H ( 0 ) , analytic on

Thus, for every

D:

wanted to prove.

+

d < C(A1/*

Adding the two estimate we conclude that

B)

a s we

0

N o w we shall prove a theorem, which, according to lemma 10.2,

implies theorem 10.1.

TH€OR€fl 70.6

b)

Thene i n

Then, 6,

& fl,

6 > 0

f2,

40

..., f n

thut

thene e x i n i n u c o n n t a n t

nuch

L e nuch t h a t

e Ha

max Ifj(z)I j

> 6

C = C(6,n),

depending o n l y

thut

P r o o f : We start by analyzing the case

n = 2.

{on

each

z e

D.

on

n

&

I. CLASSICAL HARDY SPACES

122

It is enough to prove the theorem for functions f . analytic on a J disk slightly larger than D. Indeed, the uniform bound on the 93s allows u s to use the typical convergence theorem for normal families (see Rudin

(13).

S u p p o s e we have

every

Let

z e

U(z)

fl

and

f2

with

IlflllmS 1 ,

Ilf21lm2 1

be a

Cm

U(z) 5 0 for IzI everywhere. We set

5

function, depending only on 6 1 2 , U(z) 1 for I z I 2 6

U(fi(Z)) U(f1(z))+U(f2(z))

bj(Z)

=

is a

Cm

@,(z)

+ @,(z)

for

J

=

121,

and

such that 0 5 U(z) 2 1

1*

R >

, and

We would have finished the proof if the functions

Qj/fj

Each

6 . J

a n d , for

D:

Consequently

function o n some

D(O,R),

1

+

(@,/f1)fl

(@,/f,)

f2 5 1 were

analytic in D. But they are ony Cm (Note that if f i has a zero J at z o , it is an isolated zero, and z o has a neighbourhood at which Ifj(z)l < 6 / 2 , s o that b j ( z ) = 0 in that neighbourhood.) Now we shall try to find gl

=

(@l/fl)

+

v vf,

such that and

g2 =

(O,/f,)

-

vf 1

become analytic in

IzI < 1. Once we achieve this, we shall have

and the control on

v

will give u s control over

For the analyticity of g 1 and g, we need agl Since af, = af2 = 0, we obtain the conditions:

81 =

and

ag2

=

82’

0

in

I).

1 . 1 0 . THE CORONA THEOREM

b1

But

+

E 1

@,

-

-

a@l + a @ 2

implies

123

E 0.

Thus, the two conditions are equivalent to this single one:

We shall apply lemma 10.5 with

This is clearly a

Cm

function on

R > 1

D(O,R),

and it satisfies

the estimate

A simple computation shows that:

where

C6

is a constant depending on

shall u s e the same

&(we

symbol f o r , possibly different, constants depending on Observe that theorem 9.21. implies that

.I

21

l o g - d x dy

I ZI

bounded by

From

C(llflll,

(Ifi(z)l

2

+

6

only).

lf;(z)12).

is a Carleson measure with Carleson constant

+

~ ~ f 2 ~= ~C. m )Therefore:

(10.7) we compute immediately

The second term in the right open set where either

Ifl

I

hand side vanishes identically o n the or

I f2 I

is

<6/2. In the complement

I. CLASSICAL HARDY SPACES

124

of that open set, it is bounded in absolute value by 2

+

C6( lfi(z)(2

t

(fp) 1 ).

As for the first term, it also vanishes identically wherever or If21 is < 6/2. In the complementary, it is given by A$,(z)

A(

4fl(z)fz(z)

4fl(z)f2(2)

C6(Ifi(z)12

+

)

+ U(f2(Z))

U(fl(z)) (write A = 4 3

After some computation also b y

1)

U(f2(Z

1

-

lfll

T),

this is s e e n to be bounded

1f;(Z)l2)

Using theorem 9 . 2 1 again, we conclude that:

(10.8) and (10.9) allow u s t o apply lemma 10.5 h , obtaining v, Cm in D(O,R), R > 1 , such that it and Iv(e ) I 5 C 6 for every t. Then

D

are analytic in glfl Also, since Igj(eit)I

+

and satisfy

g2f2 z 1

in

D.

I@./f . I $ 2/6 and Iv(eit)l 5 C 6 , it follows that J J C6, and therefore g . e Hm with l l g j l ( , 2 C 6 , J

This fin shes the proof for the case The case

.

n

=

2.

n > 2

is only slightly more complicated. Suppose we have . , f n e HOD such that ~ ~ f 1j for ~ ~each m ~j and fj(z)I > 6 > 0. We may further assume, as in the case

fl’f2’ inf max zeD i n = 2 that the functions function

to o u r particular a v = h in D

U

f

j

are holomorphic in

that we used for the case

-

D. We use the same

n = 2 , and write

1.11. NOTES AND FURTHER RESULTS

125

n

1

We have

bj(z)

1

D.

in

Now, we try t o find functions

vjk

j=1

such that each o f the functions n g . = (bj/fj)

c

+

Vjk

fk

k=l

J

becomes analytic. We also impose the conditions (10.10)

Vjk

=

-vkj, v . . s 0 JJ

n

1

which guarantee that

j=1

For

gj

g.f. J

1

J

to be analytic, we must have:

This may be achieved i f we have

I n order to get the

vjk

satisfying ( 1 0 . 1 0 )

find, a s in the case n = 2 , functions wjk(z), R > 1 , such that a w . = Xbk on D and Jk J k for every t.

&

Then set

Vjk

= Wjk

-

and ( l O . l l ) , we first Cm

D(O,R),

on

Iwjk(e

it

)I

5

C6

Wkj

This automatically gives us (1O.lOA and ( 1 0 . 1 1 ) . The corresponding gj are analytic on D , satisfy 1 g . f . 1 on D and also j=1 J J llgjll, 5 C 6 for every j. This completes the proof of theorem 1 0 . 6 , a n d , consequently, of the corona theorem.

0

11. N O T E S A N D F U R T H E R R E S U L T S 11.1.-

The material in sections 1 to 8 is already classical. F o r

this part, w e can give four references in b o o k form, namely:

I. CLASSICAL HARDY SPACES

126

P. Duren ll],

K. Hoffman [l],

P. Koosis cl]

and A . Zygmund .]l[

11.2.- Theorem 1.20 is due to Fatou ,]l[ However, most people call Fatou's theorem to the result asserting that every bounded holomorphic function in D has non-tangential boundary va u e s a.e. If we assume that F'(B1), is infinite in theorem 1 20., we can a s r -+ 1 but it might still conclude that U(~~~'~)-----+F'(B~) N.T. However, if in not be true that u(z)---+F'(B1) as z-eiel. addition u ( z ) 2 0 for every z e D , the non-tangential convergence holds. S e e Koosis [l] for the details. In chapter 1 1 , section 4 , we shall discuss the boundary behaviour o f harmonic functions in the upper-half-space and, give results for approximate identities in IR". will b e based upon the Hardy-Littlewood maximal be studied in the first two sections of chapter

in general, we shall The approach there function, which will 11. It is instructi-

ve to compare the two different approaches. 11.3.-

Theorem 2.16 is due to Hardy E l ] .

the

called Hardy's convexity theorem, which says that is a convex function of log r.

so

This paper contains also

l o g mp(F,r)

The proof we give of theorem 2.16 using subharmonic functions, is due to F. Riesz [l]. In this paper o n e can also find a proof of Hardy's convexity theorem based upon subharmonicity in an annulus. A.E.

Taylor ]l[

proves Hardy's theorems with Banach space methods.

A related result is Littlewood's subordination theorem, which says

with F and w holomorphic in D that if f(z) = F(w(z)) Iw(z)l 6 1 2 1 , then m (f,r) 5 m (F,r). This is contained in P P Littlewood [l].

and

11.4.- The classical source for the theory of subharmonic functions of two variables is T . Radb [l]. 11.5.- Fatou ]l[ showed that each F f Hm has a non-tangential limit F(eit) at a.e.t, and that F(eit) cannot vanish on an arc unless

F(z)

vanishes identically. F. and M. Riesz ]l[

improved

this result by extending it to H 1 functions and showing that F(z) vanishes o n a set of positive meavanishes identically if F(eit) s u r e . S z e g g Cl]

showed that if

F e H2

is not identically

0, then

1.11. NOTES AND FURTHER R E S U L T S

127

is integrable. Finally F. Riesz was able to extend log IF(eit)l this result from H2 to H p f o r any p > 0. The key for this extension is his factorization theorem (theorem 3 . 3 ) , which appears in cg. This factorization allowed him to obtain theorem 3 . 6 . ,

F. Riesz

which clarifies the boundary behaviour of an the pointwise sense and in the L p "norm".

Hp

function both in

11.6.- Theorem 2.21 was proved by Blaschke .]l[ The convergence to 0 of the integrals appearing in theorem 2 . 2 2 , characterizes Blaschk e products among all analytic functions F satisfying I F ( z ) I 5 1. This theorem is due to M. Riesz, but was first published by Frostman ( s e e Zygmund [ l ] ,

vol. I p. 281)

s h o w h o w to obtain Riesz Factorization theorem 11.7.- M. and G . Weiss ]l[ = B-H without having to worry about the behaviour of the zeroes of F toconstruct the Blaschke product B.They start by constructing H by a limiting

F

process, and then, they obtain the properties of

B , which they

show has to be essentially the Blaschke product. 11.8.- F. and R. Nevanlinna ]I[ proved that an analytic function is is the class N if and only if it is the quotient of two bounded analytic functions. This allows them to show that for F e N, the it) exists almost everywhere and nontangential limit F(e l o g IF(eit)l is integrable unless F ( z ) is identically 0. Paley and Zygmund [13

gave examples of functions which are very

close to being in the class

N

and for which the radial limit does

not exist anywhere. 11.9.-

Theorem 3.10. was proved originally by F. and M. Riesz [lJ.

The proof we give, which is possible after F. Riesz factorization theorem, is not the original one. However, the original proof of F . and M. Riesz, is also contained in this chapter ( s e e section 8, right before theorem 8.4). There are many other proofs of this r e m a k able theorem. For a proof which uses functional analysis instead of This function theory s e e Helson [l] and Helson and Lowdenslager .]l[ proof is also contained in the books of Hoffman and Koosis. A very S e e also short and elementary proof is given by Oksendal [l]. Doss [l]

.

I. CLASSICAL HARDY SPACES

128

11.10.- Theorem 3.12. and its corollary are due to Privalov (see his book .)]l[ The theorem of Caratheodory which appears in the proof of corollary 3.13. can be seen in Zygmund ,]l[ Chapter VII, Sec. 10. Koosis cl] contains a very nice treatment of this theorem and of conformal mapping in general. 11.11.- Hardy's inequality (theorem 4.1) was proved for the first and theorem 4.5. in FejCr and time in Hardy and Littlewood [ZJ, Riesz ']l[ although Hardy ]2[ anticipated it for the case p = 2. The classical book on Inequalities i s Hardy, Littlewood and Polya .]l[ The conformality of F at the boundary points where there is a tangent (corollary 4.7) was first proved by Lindelgf

0).

11.12.- Theorem 5.3.,

and consequently M. Riesz inequality (corolla-

ry 5.5) were first proved by M. Riesz [l].

A.P. Calderbn .]5[ theorem was given by P. Stein .]l[

The proof given in the

A nice proof based upon Green's By a careful modification of

text comes from

Calderbn's argument, Pichorides ]l[ found the best constants C p in the inequality 11711, 5 Cdlf / I p . These are: C = tg JL if 1 < p 5 2 TI P 2P and C = cotg if 1 5 p < m . Theorem 5.9 is due to Kolmogorov P P [Z] and corollary 5.10 is usually called Kolmogorov's inequality. 1 Since f e L1 does not guarantee that f" e L , it is interesting to mention here a result of A. Zygmund, according to which

-

Actually A. Zygmund proved also that, conversely, if and f 2 0, the integrability of f implies that of

f is in L1 f log + f , o r ,

in other words, that f e L log' L. These theorems can be found in together with many other results chapter VII of A. Zygmund ,]l[ about the conjugate function. 11.13.- The canonical factorization (theorem 7.1) is due to Smirnov (11.

It was

A. Beurling

[g

who introduced the terms "inner" and "outer" functions. His paper contains the approximation theorem 7.9 for H2. The main result of this paper is the identification of those closed subspaces E of H2 which are invariant under multiplication by the function z. They are of the form E = FH2 for F an inner function. The invariant subspace theory has been extended to

Hp, 0 < p < 1

by Gamelin ] I [

and to

Hp(p) ,the closure in

1.11. NOTES AND FURTHER RESULTS

LP(u) 7.9.

129

of the polynomials in 2 , by J.J. Guadalupe C1J. for H p appears in Srinivasan and Wang [l].

11.14.-

When we move from the unit disk

G

connected domain

natural candidates for

to a general simply

@ ,there

are three

Hp(G)

F

(1)

The space consisting of those a harmonic majorant in G.

(2)

The one formed by the functions

lFIP

averages of

D

properly contained in

Theorem

e H(G)

F

such that

e H(G)

lFIP

has

for which the

over a family of rectifiable Jordan curves

tending to the boundary, are bounded.

(3)

The closure of the polynomials in rectifiable).

Lp

of the boundary (assumed

These spaces are studied in chapter 10 of P. Duren [l]. conformal equivalence between the unit disk and @

tions on 5

@

is a

for the different types of spaces to coincide. For (1)

example,the spaces in

0 < a

If

G , there are condi-

lb'(z)l

bb

and

for every

As for the spaces in

(2) z e

coincide if and only if

D.

@'

(2) and ( 3 ) , they coincide if and only if

is an outer function. Of course, this is a property only of the domain

G , and it is independent of the particular mapping

@.The

domains enjoying this property are called Smirnov domains. The spaces Rudin [3],

Hp(G)

for

D. Sarason

G

multiply connected were studied by

111

W.

and Coifman and Weiss [2].

11.15.- The Helson-Szego theorem i s , in a sense, the starting point of the subject of weighted norm inequalities. It was proved in Helson-Szego .]l[

In that paper, the problem is presented a s one of

"Prediction Theory". Let u s give a brief justification for this. Consider a discrete, zero-mean, stationary Gaussian process, that is, (Xn), n e Z of real random variables (i.e.: measurable

a sequence

functions) in

Lp(P),

llowing conditions:

P

a probability measure, satisfiyng the fo-

I. CLASSICAL HARDY SPACES

130

1)

I

=

0

Xn Xk d?

2)

3)

Xn dP

for every

n

depends only on

n-k

Every function in the linear span of

{Xn]

is normally distri-

buted. It is readily seen that the numbers cn =

I

Xo Xn dP

form a positive

definite sequence. According to a theorem of Herglotz (see Karznelson ]l[ p . 38) the c's are the Fourier coefficients of a finite positive measure l.~ on the torus. Consequently, we get an isometry between L2(u) and the span in L2(P) of the functions Xn by int sending Xn to the function en(t) = e In this way, the ''past" and the "future" of the process (we think of n a s representing time) can be identified with the spans in L2(p) of the exponentials {en; n 5 O } and {en; n > 0} respectively. The name P r e -

.

diction theory is then applied to the study of the relation between the past and the future. For example, as theorem 8.11. shows, the boundedness of the conjugate function in L2(u) is equivalent to the fact that the "past" and the "future" are at a positive angle 2 in L ( w ) . The reader interested in this prediction-theoretic aspect S e e also Sarason Ch. 7. in advised to look at Dym-McKean .]2[

cg,

A related result is the s o called Rudin-Carleson theorem (see Carleson , ] 4 [ Rudin E4] and

Lemma 8 . 3 . was already in Fatou .]l[ Doss

[ll).

For an extension of theorem 8.4.,see Guadalupe and Rubio

de Francia (11. 11.16.- Theorem 9.27 is due to C. Fefferman [l].Section 9 is devoted to proving it along the lines set up by C. Fefferman and E.M. Stein .]2[ This basic paper studies H p for a euclidean half space. All we have done is to adapt Fefferman and Stein's proof to the context o f the disk. We shall take up the subject of Hardy spaces of several variables in chapter 111, but there, our approach will differ from the original one. The proof of the duality will be entirely b y real variable methods. There will be a third proof of the duality theorem in section 5 of chapter IV. It will be based upon a weak version o f the Helson-Szego theorem. Observe that condition (b) in theorem 9.27. provides a link between Helson-Szego weights and B.M.O. functions, namely: the logarithm of a Helson-Szego weight is a B.M.O.

1.11. NOTES AND FURTHER R E S U L T S

131

function. This relation will be further exploited in chapters I1 and

IV. The dual space of H p for 0 < p < 1 was known before the This was due to Duren, Romberg and Shields ]l[ See dual o f H'. also P. Duren [l]. The dual turns out to be a Lipschitz space. The 11.17.-

.

corresponding duality results in several variables will be obtained together with the dual of H1 in section 5 o f chapter 111. 11.18.- The B.M.O. function oscillation or to belong to

f is said to have vanishing mean V.M.O. if and only if

It can be seen that V.M.O. is a proper closed subspace of B.M.O. It clearly contains the B.M.O.closureof the space C formed by the contl m nuous functions.V.M.0.i~ to B.M.O. what C is to L . T h i s space has

[zJ

been studied by D. Sarason

(see also D. Sarason r33).

He has proved this theorem: 7he Pollowinu conditionn on t h e

B.M.O.

#unction

f , u n e eoui-

uulenf: (a)

f e V.M.O.

(c)

f

(d)

f = r$ t $

(e)

The Cunlenon meunune D unnociuted t o theonem 9.27 ~ u t i n l i e n

i n in t h e

whene

R(1)

B.M.O.

{on

centuin

clonune o/

r$,@

C.

e C.

f

UA

in

hu4 t h e 4ume meuninq u4 i n theonem

From the characterization

(e)

&

9.21.

(d) above, we can easily see that the

I. CLASSICAL HARDY SPACES

132

can be identified, in the natural way, with H'(0). dual of V . M . O . acts on V.M.O. because it acts Let u s s e e this: Of course H'(0) on B.M.O., which is a larger space. Conversely, if L is a linear functional on

V.M.O.,

we can form the composition

C x C-V.M.0.-lR

which is a bounded linear functional on will exist measures

L(@t$)

=

J

v

and

p

Q du

1

t

C

X

C. Consequently, there

such that dV.

J,

I

Q

It follows that, whenever

In particular, for

J

n = 1, 2, 3 ,

eint (dv t idu) =

Q

and

are continuous:

...

o

F. and M . Riesz theorem (3.10), for some f e H 1

s o that, according to the

d v t idu = (f t i : )

Now

L(Q t

6) =

dx

I

Q 7

t

.

I

t~ f

=

I

(Q

t

J)f"

and

f'

e

H'(0).

1 We can write ( V . M . O . ) * = H (0). The space B.M.O., strictly larger, i s not a reflexive space. cannot have the same dual. Thus, H'(0)

11.19.- The corona theorem (10.1) was conjectured by Kakutani around 1941. It was finally proved by L. Carleson El]. Hgrmander [2] gave another proof, in which he introduced the relation between the corona theorem and the solution of the

-

a-equation

(10.3).However

the

proof we have given, based upon the fundamental lemma 1 0 . 5 , i . s due to T. Wolff (circa 1979).

CHAPTER I 1 CALDERON-ZYGMUND THEORY

I n 1952, A . P . C a l d e r 6 n and A . Zygmund [l] i n v e n t e d a s i m p l e b u t p o w e r f u l method which h a s become w i d e l y known a s t h e Calder6n-Zygmund decomposition. We aim t o d e s c r i b e h e r e t h i s method t o g e t h e r w i t h some of i t s most immediate and i n t e r e s t i n g a p p l i c a t i o n s . The f i r s t two s e c t i o n s g i v e a d e s c r i p t i o n of t h e method i n c o n n e c t i $ > n w i t h t h e ( v e r y c l o s e l y r e l a t e d ) H a r d y - L i t t l e w o o d maximal o p e r a t o r . Apart from t h e u s u a l e s t i m a t e s f o r t h i s maximal f u n c t i o n , we a l s o o b t a i n some w e i g h t e d i n e q u a l i t i e s which a n t i c i p a t e t h e A t h e o r y t o P be d e v e l o p e d i n C h a p t e r IV, and we s t u d y some v a r i a n t s o f t h e HardyL i t t l e w o o d o p e r a t o r when Lebesgue measure i s r e p l a c e d by a more gene r a l measure. T h i s l e a d s u s i n a n a t u r a l way t o t h e d e f i n i t i o n and s t u d y of C a r l e s o n m e a s u r e s . T h i s i s n o t t h e o n l y maximal o p e r a t o r t o a p p e a r i n t h i s c h a p t e r . The s o - c a l l e d s h a r p maximal f u n c t i o n s h a r e s enough p r o p e r t i e s w i t h t h e H a r d y - L i t t l e w o o d o p e r a t o r , b u t b e h a v e s i n a d i f f e r e n t way i n L m , which i s somehow r e p l a c e d by o u r f r i e n d t h e s p a c e i n t e r e s t i n g r e l a t i o n between w e i g h t s and

BMO

BMO.

There i s an

which w i l l be f u r t h e r

e x p l o i t e d i n C h a p t e r IV. T h i s r e l a t i o n comes t o l i g h t a f t e r p r o v i n g t h e John-Niremberg i n e q u a l i t y f o r BMO f u n c t i o n s , which i s y e t a n o t h e r a p p l i c a t i o n of t h e Calder6n-Zygmund d e c o m p o s i t i o n . The l a s t two s e c t i o n s c o n t a i n a v e r y condensed a c c o u n t 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 and t h e i r a p p l i c a t i o n t o m u l t i p l i e r t h e o r e m s , s i n c e t h e s e o p e r a t o r s o r i g i n a l l y m o t i v a t e d t h e Calder6n-Zygmund d e c o m p o s i t i o n , and t h e y a r e s t i l l t h e most s t r i k i n g and i m p o r t a n t a p p l i c a t i o n of t h e method.

133

1I.CALDERON-ZYGMUND THEORY

134

1.

THE HARDY-LITTLEWOOD MAXIMAL FUNCTION AND THE CALDERON-ZYGMUND DECOMPOSITION

be a l o c a l l y i n t e g r a b l e f u n c t i o n i n

Let f define

where t h e supremum i s t a k e n o v e r a l l c u b e s

Rn.

Q

For

x

containing

6

Wn

we

x (cube

w i l l a l w a y s mean a compact cube w i t h s i d e s p a r a l l e l t o t h e a x e s a n d

nonempty i n t e r i o r ) , and

IQI

s t a n d s f o r t h e Lebesgue measure of

Q.

Mf w i l l be c a l l e d t h e ( H a r d y - L i t t l e w o o d ) maximal f u n c t i o n o f f , and t h e o p e r a t o r M s e n d i n g f t o Mf, 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 . Observe t h a t we o b t a i n t h e same v a l u e M f ( x ) , which c a n b e + m , if we a l l o w i n t h e d e f i n i t i o n o n l y t h o s e c u b e s Q f o r which x i s an i n t e r i o r p o i n t . I t f o l l o w s from t h i s remark t h a t t h e f u n c t i o n Mf i s lower semicontinuous, i . e . , f o r every t > 0 , t h e s e t

i s open.

I n o r d e r t o s t u d y t h e s i z e of Mf, we s h a l l l o o k a t i t s d i s t r i b u t i o n function h ( t ) = I E t l . I t w i l l be i n s t r u c t i v e t o s t a r t with t h e c a s e n = 1 , which i s p a r t i c u l a r l y s i m p l e . L e t f 6 L 1 ( R ) . The open s e t Et i s a d i s j o i n t u n i o n o f open i n t e r v a l s I . : i t s c o n n e c t e d compo3 n e n t s . L e t u s l o o k a t one of t h e I . ' s , and l e t u s c a l l i t I . 3 Take any compact s e t K c I . For e a c h x 6 K , we have some compact c o n t a i n i n g x i n i t s i n t e r i o r and s a t i s f y i n g i n t e r v a l Q,

Since

Q,

c E,,

it follows t h a t

Qx c I .

Since

K

i s c o m p a c t , we

c a n c o v e r i t w i t h t h e i n t e r i o r s of j u s t f i n i t e l y many o f t h e QI's, s a y { Q j I . We c a n even assume t h a t t h i s f i n i t e c o v e r i n g i s minimal i n t h e s e n s e t h a t n o Q j i s s u p e r f l u o u s . Then, no p o i n t i s i n more t h a n two of t h e i n t e r i o r s o f t h e Q j ' s . I t f o l l o w s t h a t :

11.1. HARDY-LITTLEWOOD MAXIMAL FUNCTION

Since this is true for every compact

+I

III

(1.1)

I

KcI, we obtain:

If(Y)IdY

This implies,in particular, that I is bounded. Then, since b c 7 and b 6 Et, we can write: 1 l-q

Finally

(1.1)

1

135

If(y)Idy

5 Mf(b1

I

Let

I

=

(a,b)

.

6 t

implies:

We have obtained the following result: L e t f 6 L'(R) . T h e n , f o r e v e r y t > 0 , t h e s e t R : Mf(x) > t } c a n b e w r i t t e n a s a d i s j o i n t u n i o n of b o u n d e d o p e n i n t e r v a l s I. s u c h t h a t , f o r e v e r y j = 1 , 2 , ... : THEOREM

Et

=

{x

1.2.

6

1'

and. a s a conseauence:

Now we seek an analogue of theorem extension is not straightforward.

1.2

in dimension

n > 1.

The

Let f 6 L1(IRn) , n > 1 , and let t > 0. Instead of looking at the maximal function Mf, we shall try to obtain directly a family of cubes { Q j } such that the average of If I over each is comparable to t in the sense that a relation like ( 1 . 3 ) holds. This is quite easy and it will be done most effectively by considering only

11. CALDERON-ZYGMUND THEORY

136

d y a d i c c u b e s . For k 6 ZL formed by t h o s e p o i n t s of

,

we c o n s i d e r t h e l a t t i c e A k = 2-kZIn IRn whose c o o r d i n a t e s a r e i n t e g r a l mul-

Let Dk b e t h e c o l l e c t i o n of t h e c u b e s d e t e r m i n e d t i p l e s of 2 - k . by Ak, t h a t i s , t h o s e c u b e s w i t h s i d e l e n g t h Z m k and v e r t i c e s 03

The c u b e s b e l o n g i n g t o D =-k Dk a r e c a l l e d d y a d i c c u b e s . in Ak. Observe t h a t i f Q , Q ' 6 D and ( Q ' ( 2 I Q I , then e i t h e r Q ' c Q o r e l s e Q and Q ' do n o t o v e r l a p (by which we mean t h a t t h e i r interiors are disjoint).

Each

Q

6

Dk

i s t h e u n i o n of

2"

non-

We s h a l l c a l l ct t h e f a m i l y overlapping cubes belonging t o D k + l . formed by t h e c u b e s Q 6 D which s a t i s f y t h e c o n d i t i o n : (1.4)

t <

1 -

IQI

If(x)Idx

Q

and a r e maximal among t h o s e which s a t i s f y i t . Every Q c D s a t i s fying ( 1 . 4 ) i s c o n t a i n e d i n some Q ' 6 C t b e c a u s e c o n d i t i o n imposes an u p p e r bound on t h e s i z e of Q ( I Q I 2 t - ' ~ ~ f ~ ~ l ) . (1.4) The c u b e s i n C t a r e , by d e f i n i t i o n , n o n o v e r l a p p i n g . Also, i f Q 6 Dk is in Ct and Q ' i s t h e only cube i n Dk-, containing Q,

we s h a l l h a v e :

but, since

I Q ' I = 2"1Q1,

we g e t :

We have a c h i e v e d o u r p u r p o s e by o b t a i n i n g a f a m i l y cubes such t h a t , f o r every

Ct

= {Qj}

of

j:

N e x t , w e s h a l l i n v e s t i g a t e t h e r e l a t i o n w i t h t h e maximal f u n c t i o n M f . Suppose x 6 IRn i s such t h a t M f (x) > t . There w i l l be some c u b e

R

containing

x

i n i t s i n t e r i o r and s a t i s f y i n g

11.1. HARDY-LITTLEWOOD MAXIMAL FUNCTION

137

We l o o k f o r a d y a d i c c u b e of c o m p a r a b l e s i z e o v e r which t h e a v e r a g e

of

If1

i s comparab y b i g .

Let

T h e r e i s a t most one p o i n t of

k

be t h e only i n t e g e r such t h a t

interior to

Ak

R.

Consequently,

of them, m e e t i n g t h e t h e r e i s some c u b e i n D ~ , and a t most 2" i n t e r i o r of R . I t f o l l o w s t h a t t h e r e i s some Q 6 D k m e e t i n g t h e i n t e r i o r of R a n d s a t i s f y i n g : '

Since

IR1 5

191

we h a v e :

< Zn1RI,

and t h e r e f o r e :

I t follows t h a t

Q c Q.

6

J

C 4-"t

f o r some

I n g e n e r a l f o r any c u b e Q and any t h e cube w i t h t h e same c e n t e r a s Q

j

I

a > 0 , we s h a l l d e n o t e by Q" but with s i d e length a times

I n o u r p a r t i c u l a r s i t u a t i o n , s i n c e R and Q meet t h a t of Q . 3 and I R I 5 I Q I , i t f o l l o w s t h a t R c Q3 c Q j . We c o n c l u d e t h a t i f and t h i s l e a d s t o the estimate t h e n c E t c U Q ? = {Qj}, 4-"t 3 1

We h a v e o b t a i n e d t h e f o l l o w i n g r e s u l t :

THEOREM

1.6.

n

Et = {x G IR of c u b e s { Q i }

Let f 6 L1(Rn). Then, f o r every t > 0 , t h e s e t : M f (x) > t } i s c o n t a i n e d i n t h e u n i o n o f a f a m i l y w h i c h r e s u l t f r o m e x p a n d i n g b y a f a c t o r of

nonoverlapping cubes

{Qj}

which s a t i s f y :

3

the

11. CALDERON-ZYGMUND THEORY

138

I t follows t h a t :

where t h e c o n s t a n t

C

d e p e n d s o n l y on t h e d i m e n s i o n

n.

0

We s h a l l d e r i v e some c o n s e q u e n c e s o f t h e b a s i c i n e q u a l i t y ( 1 . 8 ) which i l l u s t r a t e t h e r o l e p l a y e d by t h e maximal o p e r a t o r M . The importance of t h e o p e r a t o r M stems from t h e f a c t t h a t i t c o n t r o l s many o p e r a t o r s a r i s i n g n a t u r a l l y i n A n a l y s i s . A s a n e x a m p l e , we a r e going t o prove a n e x t e n s i o n of Lebesgue's d i f f e r e n t i a t i o n theorem.

.

1.9.

THEOREM

Q(x;r)

=

{y

every

x

G

G

IR":

1 L e t f G L ~ ~ ~ ( I RFX ~ ) x G IR" & a r > 0 , let n : IR ( y - x l , :m a x l y . - x . ) 5 r l . T h e n , f o r a l m o s t j J 1

We may assume

Proof:

f o r every

f

G

L1(IRn).

I t w i l l b e enough t o

show t h a t ,

t > 0, t h e set

h a s z e r o measure. m precisely U Al,j.

I n d e e d , t h e s e t where

(1.10)

does not h o l d , i s

j=1

Given

E

>O,

we c a n w r i t e

compact s u p p o r t and

Therefore

llhl <

f = g + h, E.

For

is continuous w i t h w e c l e a r l y have:

where g

g

MAXIMAL FUNCTION

11.1. HARDY-LITTLEWOOD

139

But

1

{x

6

R" : M h ( x )

11 5 C I I h l ( l / t < C

> t/2

E

/t

and

Thus

i s c o n t a i n e d i n a set of measure 5

At

b e done f o r any

we g e t

> 0,

E

lAtl

The p o i n t s x f o r which ( 1 . 1 0 ) f o r f . We c a n r e p h r a s e t h e o r e m point

x

G

COROLLARY

for

a)

f(y)dy

1

whilst If

x 3

Then, f o r e v e r y Lebesgue a.e.

point

x

G

IR":

a ) , just note that

-

f(x)

IQ(x;r)

b)

f.

f ( y ) dy

In o r d e r t o prove

\ Q ( x ; r )I

Q1

f

f(x) = l i m

Proof:

I

1

f G Lloc(Rn). and, therefore, f o r Let

il

0.

h o l d s a r e c a l l e d Lebesgue p o i n t s 1 . 9 by s a y i n g t h a t a l m o s t e v e r y

i s a Lebesgue p o i n t f o r

IRn

1.11.

x

point

=

Since t h i s can

CE/t.

1

5

IQ(x;r) I

i s a n immediate c o n s e q u e n c e o f

JQ(x;r) If(y) a).

-

f(x)Idy

'0

f and we have a sequence of cubes fl Q . = { X I , t h e n :

i s a Lebesgue p o i n t f o r Q,

3

....

with

1

3

'

f(x) = l i m flyldy Qj j+w

IQj(

Indeed i f

Q. 3

has side length

rj,

we h a v e

Qj c Q(x;rj)

and

11. CALDERON-ZYGMUND THEORY

140

rn = I Q . J J

I

J.

0,

so t h a t

r. J

-+

Therefore

0.

Ct(f) = IQ.1 be the c o l l e c t i o n 3 f o r m e d by t h o s e maximal d y a d i c c u b e s o v e r w h i c h t h e a v e r a g e o f If( i s > t ( c a l l e d Calderh-Zygmund cubes f o r f corresponding t o t ) . Let

f 6 L1(IRn)

and l e t

Ct

I

Let x $ U Q . Then t h e a v e r a g e o f If o v e r any d y a d i c c u b e w i l l be 5 t. "Let {Rk} b e a s e q u e n c e o f d y a d i c c u b e s o f d e c r e a s i n g size such t h a t Rk = { X I . F o r e a c h o f them

'

ft

I f , b e s i d e s , x i s a L e b e s g u e p o i n t f o r f ( a n d h e n c e f o r I f ( ) we g e t , by p a s s i n g t o t h e l i m i t : I f ( x 1 5 t . Thus I f ( x ) 6 t f o r

I

I

a.e. x $ U Q j . 3

IRn i n t o a s u b s e t R made up o f n o n If I i s o v e r l a p p i n g cubes Q j o v e r e a c h of which t h e a v e r a g e of between t and Z n t , a n d a c o m p l e m e n t a r y s u b s e t F where (f(x)1 5 t a.e., i s t h e f i r s t s t e p of t h e s o - c a l l e d Calder6n-Zygmund d e c o m p o s i t i o n w h i c h w i l l b e a t o o l o f c o n s t a n t u s e i n t h e s e q u e l . Let u s r e c o r d t h e f o l l o w i n g : The s p l i t t i n g o f t h e s p a c e

THEOREM

1.12.

Given

non-overlapping cubes

f c L1(Rn) Ct = C t ( f )

&

t h e r e i s a f a m i l y of

t > 0,

c o n s i s t i n g of t h o s e m a x i m a l d y a -

If

d i c c u b e s o v e r w h i c h t h e a v e r a g e of

I

> t.

T h i s famiZy

satisfies: (il

(ii)

f o r every f o r a.e.

Q

x

Besides, f o r every where

f

ct

: t <

UQ,

t > 0,

ranges over

where

1

1

IQI

I f ( x ) Idx 6 2"t

Q

Q

ranges over

ct,

is

If(x)l 6 t.

E t = { X c R n : M f (x) > t } c U Q 3

c4-nt.

0

11.1. HARDY-LITTLEWOOD MAXIMAL FUNCTION

141

Next we a r e g o i n g t o s t u d y a u s e f u l g e n e r a l i z a t i o n o f t h e maximal f u n c t i o n . Let p b e a p o s i t i v e B o r e 1 m e a s u r e on IR", f i n i t e on compact s e t s and s a t i s f y i n g t h e f o l l o w i n g "doubling" c o n d i t i o n :

f o r e v e r y c u b e Q , w i t h C > 0 i n d e p e n d e n t o f Q.We s h a l l o f t e n s a y s i m p l y t h a t . 1-1 i s a d o u b l i n g m e a s u r e . T h i s i m p l i e s , o f c o u r s e , t h a t f o r e v e r y c1 > 0 , t h e r e i s a c o n s t a n t C = C c1 > 0 , d e p e n d i n g o n l y on c1 , s u c h t h a t p(Q') S Cp(Q) f o r e v e r y cube Q. Since w e a r e i n R n , t h e f i n i t e n e s s o f p on c o m p a c t s u b s e t s i m p l i e s p ( Q ) > 0. t h a t p i s r e g u l a r . Notice t h a t f o r every cube Q , I n d e e d , i f w e h a d p(Q) = 0 f o r some c u b e Q , we w o u l d h a v e k p ( Q ) 2 Ckp(Q) = 0 , f r o m w h i c h p(lRn) = 0 , w h i c h i s e x c l u d e d a s trivial.

Now, f o r

p

as above,

f

G

1

Lloc(p)

and

x

G

IRn

,

define:

w h e r e t h e s u p . i s t a k e n o v e r a l l c u b e s Q c o n t a i n i n g x . As b e f o r e , we o b t a i n t h e same v a l u e Mu f ( X I i f we j u s t t a k e i n t h e d e f i n i t i c q those cubes Q containing x i n t h e i r i n t e r i o r . This is a consequence of t h e r e g u l a r i t y o f p .

Let f E L 1 ( p ] a n d t > 0 . We w a n t t o o b t a i n a CalderBn-Zygmund decomposition f o r f and t r e l a t i v e t o t h e measure p and, a t t h e same t i m e , w e w a n t t o e s t i m a t e t h e p - m e a s u r e o f t h e s e t E t = { x G IRn : Mu f ( x ) > t l , w h i c h i s , o f c c u r s e , o p e n . We a r e g o i n g t o a p p l y t h e same i d e a s t h a t l e d t o t h e o r e m 1 . 1 2 . We n e e d t o make two o b s e r v a t i o n s .

First, we a r e g o i n g t o see t h a t t h e r e i s a c o n s t a n t K > 1 s u c h C it follows t h a t t h a t , e v e r y t i m e we h a v e d y a d i c c u b e s Q ' + Q , p(Q) 2 Kp(Q'). To see t h i s , l e t Q" be a dyadic cube contained i n Q, contiguous t o Q' a n d w i t h t h e same d i a m e t e r . Then Q ' c Q 1 1 3 a n d c o n s e q u e n t l y , f o r C = C 3 we h a v e : i i ( Q ' ) 2 C p ( Q " ) 5 C(p(Q) - p ( Q ' ) ) . T h i s i m p l i e s ( 1 + C ) u ( Q ' ) 6 Cp(Q), which g i v e s p ( Q ) 2 Kp(Q') w i t h K = ( 1 + C ) / C > 1 , and o u r claim is j u s t i f i e d . As a c o n s e q u e n c e , i f w e h a v e a s t r i c t l y i n c r e a s i n g s e q u e n c e

11. CALDERON-ZYGMUND THEORY

142

C

o f d y a d i c c u b e s Qo Q, Q2 k p(Qk) 2 K p(Qo) m as k m. of dyadic cubes is such t h a t t h e -+

...

,

we have t h e i n e q u a l i t y

The c o n c l u s i o n i s t h a t i f a c h a i n p - m e a s u r e o f t h e c u b e s i s bounded

-+

a b o v e , t h e n t h e i r d i a m e t e r i s a l s o bounded a b o v e o r , what i s t h e same, t h e c h a i n t c r m i n a t e s a t a g i v e n c u b e c o n t a i n i n g a l l t h e o t h e r s . The o t h e r o b s e r v a t i o n w e n e e d i s t h e f o l l o w i n g :

f o r every

t h e r e i s B > 0 s u c h t h a t e v e r y time we h a v e c u b e s Q a n d which meet a n d s a t i s f y 1 Q l < A I R I , then they a l s o s a t i s f y p(Q) < B p ( R ) . To s e e t h i s , j u s t n o t e t h a t Q c R' with c1 = 2A1'n t 1, and choose B a c c o r d i n g l y . Now we g o b a c k t o o u r p r o b l e m .

D e n o t e by

t i o n f o r m e d by t h e maximal d y a d i c c u b e s

Since t h i s condition forces

Q

A > 0

R

C = Ct(f;v) the collect satisfying the condition

t o b e bounded by

p(Q)

t - ' lRn ( f ( y ) ( d p ( y ) c m , O U T f i r s t o b s e r v a t i o n i m p l i e s t h a t e v e r y d y a d i c c u b e s a t i s f y i n g o u r c o n d i t i o n i s c o n t a i n e d i n some member o f Ct. Take Q G C t . I f Q G Dk a n d Q ' i s t h e o n l y c u b e i n Dk-, containing

Q,

Thus, f o r e v e r y

we h a v e

Q

6

C t:

Let x G E t , t h a t i s : Mp f (XI > t . Then t h e r e w i l l b e some c u b e R containing x i n i t s i n t e r i o r such t h a t

A s we d i d f o r t h e c a s e when

p = Lebesgue measure,

d y a d i c cube which o v e r l a p s w i t h

R

and s a t i s f i e s

let

Q

be a

I R I 6 IQ I 2 - " t v ( R )

. A = 2

L e t B be t h e c o n s t a n t c o r r e s p o n d i n g t o s e r v a t i o n . Then

jRnQ I

' -'tv

I

f du > B-

2

143

n

i n o u r s e c o n d ob-

(Q)

and h e n c e

I t follows t h a t = I f c i-n - 1 B t

Q c Qj

,

f o r some Q j 6 3 then Et c U Q . . j J

'2 - " B - ' t

and

R c Q3 c Q 3j .

We f i n a l l y o b t a i n t h e e s t i m a t e :

This b a s i c estimate

c a n be u s e d t o e x t e n d t h e o r e m

1 . 9 and i t s

corollary, obtaining: THEOREM

With

1.14.

almost every

x

6

li

IR"

as above, l e t

f

( w i t h r e s p e c t t o 111

Then, f o r

:

u

C)

Ct(f,v) = CQ.1, I (with respect t o

In p a r t c u l a r , i f a.e.

1

Lloc(p).

f

x B U Q. j J

we have

If(x)l S t

for

v).

We c a n f i n a l l y s t a t e t h e f o l l o w i n g : THEOREM

1.15.

For

1-1

a s above,

let

t h e r e i s a f a m i l y of n o n - o v e r l a p p i n g

1 f 6 L ( p ) g& t > 0. Then, consistcubes Ct = c t ( f , p ) ,

144

11. CALDERON-ZYGMIJND THEORY

If\

i n g of t h o s e m a x i m a l d y a d i c c u b e s o v e r w h i c h t h e u v e r a g e of relative t o

il

for e v e r y

Q

for a.e. x iil -

wit

fi

p

Here

UQ

@

f

ctIc,

which s a t i s f i e s

: t

V(Q)

where

Q

IR":

M

P

jQI f l d p

5 Ct

ranges over

If ( x )

we h a v e :

pI

Et = { x

ranges over

Ct

G

respect t o

the set

> t,

f (xj>t}

I

6 t.

ct

(and

a.e.

i s

Besides, for every t >

UQ3

is c o n t a i n e e

where

1,

Q

and we h a v e a n e s t i m a t e :

r e p r e s e n t s an a b s o l u t e c o n s t a n t , p o s s i b l y d i f f e r e n t a t each

C

occurrence.

2.

NORM ESTIMATES FOR THE MAXIMAL FUNCTION

THEOREM

t

0.

2.1.

Let

b e a m e a s u r a b l e f u n c t i o n on

IR" a n d l e t

Then we h a v e t h e f o l l o w i n g e s t i m a t e s f o r t h e L e b e s g u e mea-

E t = { x c IR":

s u r e of t h e s e t

with constants

Proof:

f

Write

f,(x) = 0

C

and C '

f = f, + f2,

otherwise.

Then

M f (x) > t}:

w h i c h d o n o t d e p e n d on

f

or

t

where f , ( X I = f ( x ) i f I f ( x ) l > t / Z , a n d M f(x) 5 M f ,

( x ) + M f 2 ( x ) 6 M f l (x)+

11.2. NORM ESTIMATES FOR THE MAXIMAL FUNCTION

since

+ t/2,

which g i v e s

f2 2 t/2

implies t h a t

Mf2 2 t / 2

also.

145

Thus

(2.2).

As f o r (2.3), we may a s s u m e t h a t f 6 L 1 ( R n ) ( o t h e r w i s e we t r u r r c a t e a n d a p p l y a l i m i t i n g p r o c e s s ) . Then we u s e t h e C a l d e r B n Zygmund d e c o m p o s i t i o n f o r f a n d t . We h a v e n o n - o v e r l a p p i n g c u b e s Q j9

such t h a t

f o r every X

f

Qj

j,

and

If(x)I 5 t

implies t h a t

x d UQ.. J J w rite: we c a n

for

Mf(x) > t ,

a.e

Now, s i n c e

1% and

(2.3)

i s proved with

C'

= 2-".

c]

The n e x t r e s u l t i s p r o v e d i n e x a c t l y t h e same way.

THEOREM

Rn

2.4.

Suppose

p

i s a r e g u l a r p o s i t i v e Borel measure i n

s a t i s f y i n g a "doubling" condition l i k e

are constants,

C,C',

(1.13).

Then, t h e r e

such t h a t , f o r any Borel f u n c t i o n

f

and any

t > 0:

we c a n e a s i l y d e r i v e s e v e r a l norm e s t i m a t e s f o r t h e maximal f u n c t i o n .

From t h e o r e m

THEOREM

c

P

> 0

2.1.

p such t h a t , f o r every 2.5.

For e v e r y

with f

G

1 < p <

L p ( R n ):

m,

t h e r e is a c o n s t a n t

11. CALDERON-ZYGMUND THEORY

146

I n e x a c t l y t h e same way we o b t a i n t h e f o l l o w i n g THEOREM

Let

2.6.

1~

be a r e g u l a r p o s i t i v e Bore1 measure i n

s a t i s f y i n g a "doubZing" c o n d i t i o n l i k e with

p

1

there i s a constant

a,

(1.131.

C

f e ~p(p):

P

> 0

Then, f o r each

Rn p

such t h a t f o r every

We h a v e s e e n t h a t t h e o p e r a t o r bl i s b o u n d e d i n L p ( R n ) f o r 1 < p 5 w. H o wev er , i t i s n o t b o u n d e d i n L 1 ( I R n ) A s a matter of f a c t , f o r f 2 0 , Mf f a i l s t o b e i n L 1 u n l e s s f ( x ) = 0 f o r a . e . x , s i n c e Mf(x) 2 Clxl-" f o r l a r g e x , with C > 0 i f f

.

is not t r i v i a l .

Th e b a s i c e s t i m a t e

(1.8)

L 1- b o u n d e d n e s s , a s w i l l b e s e e n b e l o w .

a c t s a s a substitute €or As regards local integrabi-

l i t y o f t h e m a xi mal f u n c t i o n , we h a v e t h e f o l l o w i n g r e s u l t : THEOREM

Mf

I f ( x ) I l o g + l f ( x ) Idx <

(2.8) Proof:

f b e an i n t e g r a b l e f u n c t i o n s u p p o r t e d i n u b u l l i s i n t e g r a b l e o v e r B i f a n d o n l y if:

Let

2.7.

B c RP.=

If

m.

(2.8) holds, then co

Mf(x)dx = j,l{x

m

f

B : M f ( x ) > t l / d t = Z j I{x e B : Mf(x1 > Z t } / d t 5 0

11.2. NORM ESTIMATES FOR THE MAXIMAL FUNCTION

147

O b s e r v e t h a t f o r t h i s p a r t o f t h e p r o o f we do n o t n e e d t o u s e t h e f a c t t h a t f i s supported i n B. I n d e e d , t h e same p r o o f shows t h a t i f IIRn If ( x ) I l o g + l f ( x ) ldx < a, which we s h a l l i n d i c a t e by s a y i n g f E L l o g L(IRn 1 ,

that

then

Mf

is locally integrable.

Going b a c k t o t h e p r o o f of t h e t h e o r e m , s u p p o s e t h a t IBMf(x)dx < m . I f we d e n o t e by B ' the b a l l concentric with 5 b u t w i t h r a d i u s t w i c e a s b i g , we c a n e a s i l y s e e t h a t I B I M f ( x ) d x < I t is sufficient to realize that for

*

x

6

B'W,

m.

Mf(x) 5 CMf(x*),

i s t h e p o i n t s y m m e t r i c t o x w i t h r e s p e c t t o t h e bounwhere x d a r y o f B. I n t h e complement o f B ' , Mf i s b o u n d e d , s i n c e we

a r e a t l e a s t a f i x e d d i s t a n c e f a r from t h e s u p p o r t of f . I t is a l s o o b v i o u s t h a t Mf(x) 0 as 1x1 a. Thus, we c o n c l u d e t h a t , +

f o r any

+

to > 0

Mf(x)dx < jfx:Mf ( x ) > t O }

m.

to = 1 , t h i s i n t e g r a l i s b i g g e r t h a n o r e q u a l t o

For

/ ; I {X

: Mf(x) > t l l d t 2

Thus

(2.8)

( f(x) ( d x d t =

h o l d s and t h e p r o o f i s c o m p l e t e .

The t h e o r e m e x t e n d s c l e a r l y t o doubling condition.

Mp

0

f o r a measure

!J

satisfying a

T h e r e i s no n e e d t o w r i t e a new s t a t e m e n t .

Suppose now t h a t we hav.e two m e a s u r e s p a c e s w i t h r e s p e c t i v e m e a s u r e s 1-1

and

Lq(v), (2.9)

Then,

V

, and t h a t

that is:

T

i s an o p e r a t o r bounded from

LP(u)

to

11. CALDERON-ZYGMUND THEORY

148

a n d we o b t a i n v({x : ITf(x)

(2.10)

I

> t}) 5

[ c II t II LP f

When T s a t i s f i e s ( 2 . 1 0 ) we s a y t h a t t h e o p e r a t o r T i s o f weak t y p e (p,q) w i t h r e s p e c t t o t h e p a i r o f m e a s u r e s ( v , p ) . F o r example, ( 1 . 8 ) i s r e a d by s a y i n g t h a t M i s o f weak t y p e ( 1 , l ) ( w i t h r e s p e c t t o L e b e s g u e m e a s u r e ) . However, we know t h a t M f a i l s t o b e bounded i n L 1 . I n g e n e r a l , (2.10) may h o l d w h e r e a s ( 2 . 9 ) does n o t hold f o r a given operator T. I t i s c o n v e n i e n t t o see (2.10) a s a s u b s t i t u t e o r a weakening of ( 2 . 9 ) . With t h i s i n m i n d , when ( 2 . 9 ) h o l d s , w e s a y t h a t T i s o f s t r o n g t y p e (p,q) w i t h r e s p e c t t o t h e p a i r of measures ( v , p ) . Sometimes i t i s c o n v e n i e n t t o i n d i c a t e t h a t ( 2 . 1 0 ) h o l d s by s a y i n g t h a t T s e n d s L p ( p ) b o u n d e d l y i n t o L:(v). T h i s means t h a t w e a r e c o n s i d e r i n g t h e s p a c e ( c a l l e d weak-Lq(v))

L:(v)

f o r m e d by t h o s e f u n c t i o n s

g

f o r which

T h i s s p a c e w i t h t h e "norm" n ( f a i l s t o be s u b a d d i t i v e even though 9 we h a v e a s i m i l a r i n e q u a l i t y w i t h a c o n s t a n t i n t h e r i g h t h a n d s i d e ) is bigger (topologically a l s o ) than

Lq(v).

Weak type i n e q u a l i t i e s such a s (2.10) can be used t o o b t a i n s t r o n g t y p e i n e q u a l i t i e s . T h i s i s w h a t we h a v e d o n e t o p r o v e t h e o r e m 2 . 5 . We a r e g o i n g t o p r e s e n t a r e s u l t , w h i c h i s a p a r t i c u l a r c a s e o f 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 t h e o r e m a n d i s b a s e d upon t h e same i d e a a s o u r proof of 2.5. 2.11.

THEOREM

measures

S u p p o s e we h a v e t w o m e a s u r e s p a c e s w i t h r e s p e c t i v e

p

L'o(~) + ~ P 1 ( p ) pose t h a t :

ii

T

v

to

.

T

b e an o p e r a t o r s e n d i ' n g f u n c t i o n s i n

+measurable

f u n c t i o n s , 1 6 po < p 1 5

i s subadditive, that i s , f o r

fl,f2

6

m.

L PO ( p ) + L p ' ( p ) ,

%-

11.2. NORM ESTIMATES FOR THE MAXIMAL FUNCTION

ii) T

iii) T p1 < m,

i s of weak t y p e whiZe, i f

def i n it i o n :

IITf

Then, f o r every

(p,p)

(po,po),

i s of weak t y p e

~

p1 =

11 Lm (") p

149

that i s :

(pl,pl) w h i c h means t h e same a s a b o v e i f weak t y p e and s t r o n g t y p e c o i n c i d e by

m,

5 Clllfll

such t h a t

Lrn ( p 1

p,

 0 w r i t e f ( x ) = f ( x ) + f t ( x ) where f ( x ) = f ( x ) t if I f ( x ) > t a n d f ( x ) = 0 o t h e r w i s e . Clearly f t 6 LpO(u),

I

I

and s i n c e I f t ( x ) 1 t , f t 6 L p l ( p ) . S u p p o s e p 1 < m. Then, we c a n w r i t e : s i n c e ITf(x) 1 5 IT(ft) (x) 1 + I T ( f t ) (XI

I,

Thus Tf(x) 1 2 P o C o p ~ ; l t P - P " -1

I

> t1)dt 5

L:

If(x

x) =

P-Po

J '

P1-P

J

11. CALDERON-ZYGMUND THEORY

150

For t h e c a s e

where

2-1

=

CL

p, =

=

2

=

c

PO

P

we s i m p l y o b s e r v e t h a t

m y

and,consequently

C;'

co P j

1

Jf(x1

N e x t we s h a l l e s t a b l i s h a g e n e r a l i n e q u a l i t y f o r t h e maximal f u n c t i o n . This i n e q u a l i t y involves a weight f u n c t i o n

@(x).

THEOREM 2 . 1 2 . F o r e v e r y p __ with 1 < p < m y there i s a constant s u c h t h a t , f o r a n y m e a s u r a b l e f u n c t i o n s o n IRn @ 2 0 and f , -

C

P

we h a v e t h e i n e q u a Z i t y :

Proof:

E x c e p t when

trivially,

M@

M$(x) =

m

a.e.,

i n which case

is t h e d e n s i t y of a p o s i t i v e measure

(2.13)

holds

p(du(x) =

i s t h e d e n s i t y of a n o t h e r p o s i t i v e measure ( 2 . 1 3 ) means t h a t M i s a b o u n d e d o p e r a t o r f r o m

M$(x)dx) and @ w(dv(x) = $(x)dx). =

Lp(u) t o LP(v). This i s c l e a r l y t r u e f o r p = m. Indeed, i f M$(x) = 0 f o r some x , t h e n $ ( X I = 0 a . e . a n d L m ( v ) = 0 , s o t h a t nothing needs t o be proved. I f M$(x) > 0 f o r e v e r y x a n d

, we h a v e IIx : [ f ( x ) l > a l l = 0 which

Mf(x) 5

c1

a.e.

1

M$ = 0

and consequently

t If I > c d

o r , w h a t i s t h e same, Thus

I/MflI

5

If(x)I 5 a a.e., c1

from

and, f i n a l l y ,

L"(V)

Having t h e

(m,m)

result,

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

M

i s of

11.2. NORM ESTIMATES FOR THE MAXIMAL FUNCTION

151

weak t y p e ( 1 , l ) w i t h r e s p e c t t o t h e p a i r ( v , p ) , the interpolation theorem 2 . 1 1 w i l l give (2.13). T h u s , a l l we n e e d i s t o show t h a t :

We c a n o b v i o u s l y assume t h a t f t 0 . We c a n a l s o assume t h a t f G L1 (Wn) I n d e e d , we c a n f i n d i n t e g r a b l e f u n c t i o n s f . such 3' and o b s e r v e t h a t t h a t f l S f 2 5 --- + f a . e .

.

{ X : Mf(x) > t j = U j

{X

: Mf.(x) > t } .

J

f G L1(IRn) a n d f b 0 . Given t > 0 , we know (Theorem t h a t t h e r e i s a f a m i l y of n o n - o v e r l a p p i n g c u b e s { Q j } s u c h

So, l e t

1.6)

t h a t , f o r each

j:

and a l s o {X

: Mf(x) > t } c U Q3 j J

Then

3"*4"

5 t J

f ( x ) ( M @ ) ( x ) d x6

C

f(x)(M@)(x)dx

.

fl

Qj

The t h e o r e m we h a v e j u s t p r o v e d i d e n t i f i e s a whole c l a s s o f w e i g h t f u n c t i o n s @ f o r which t h e o p e r a t o r M i s bounded i n L p ( @ ) f o r every

p

E

(1

,a]

and o f weak t y p e

(1,l)

with respect t o

$,

namely, t h e c l a s s , c u s t o m a r i l y d e n o t e d by A 1 , o f t h o s e @ 2 0 s a t i s f y i n g M@(x) 2 C @ ( x ) a . e . f o r some c o n s t a n t C . For i n s t a n c e , i t i s e a s y t o c h e c k t h a t @ ( x ) = 1x1' i s an A 1 w e i g h t i n IRn provided

-n <

c1 S

0.

We s h a l l 20 back t o w e i g h t s i n c h a p t e r

IV.

11. CALDERON-ZYGMUND THEORY

152

T h e r e i s a n i n t e r e s t i n g e x t e n s i o n of t h e o r e m 2 . 1 2 whose p r o o f i s b u t a r e p e t i t i o n o f t h e a r g u m e n t s which l e d t o 1 . 6 , 2 . 5 and 2.12. I n o r d e r t o p r e s e n t t h i s r e s u l t we make s e v e r a l d e f i n i t i o n s : IR", we d e f i n e a f u n c t i o n , t 2 01 by s e t t i n g

Given a f u n c t i o n f in = { ( x , t ) : x G IRn

I

M f ( x , t ) = sup{--- 1

IQI

If(y)ldy :

Q

Given a p o s i t i v e B o r e 1 m e a s u r e ~ ( p ) in Wn by s e t t i n g

in

1-1

x

Wn+ 1

where t h e s u p i s t a k e n o v e r a l l c u b e s cube

+

Q

Q

G

,

Mf

in

and s i d e l e n g h t ( Q ) t t ) we d e f i n e a f u n c t i o n

containing

x

and f o r a

Q, 'L

Q = {(x,t)

that is,

'L

Q

f

IRn++l : x

G

Q

and

0 5 t 5 side length

(Q))

,

-i s t h e cube i n

Rn++l

having

Q

as a face.

With t h e

above d e f i n i t i o n s , we c a n s t a t e t h e f o l l o w i n g : THEOREM

C

P

2.15.

p

For e v e r y

such t h a t , f o r every

f

with

1 < p <

and e v e r y

m,

t h e r e is a c o n s t a n t

p:

B e f o r e we s t a r t t h e p r o o f , l e t u s n o t e t h a t t h i s r e s u l t i n c l u d e s t h e p r e v i o u s o n e . J u s t t a k e d p ( x , t ) = Q(x)dxrndG(t) where

Proof:

i s t h e u n i t mass c o n c e n t r a t e d a t t h e o r i g i n i n t h e t a x i s . Then since M f ( x , O ) = Mf(x) a n d , f o r o u r p a r t i c u l a r p , U p ( x ) = M Q ( x ) , (2.16) reduces t o (2.13) i n t h i s case.

6

Now we p r o v e t h e t h e o r e m . A s i n t h e p r o o f o f t h e p r e c e d i n g r e s u l t , i f we e x c l u d e t h e t r i v i a l c a s e when flp(x) = m a . e . , Up(x)dx is a measure and M i s bounded f r o m L a ( N p ( x ) d x ) t o Lm(IRn++' , p ) . A l l we n e e d t o do i s t o p r o v e a weak t y p e ( 1 , l ) & q u a l i t y and t h e n u s e i n t e r p o l a t i o n . I f we c a l l E~ = { ( x , t ) G IR"++' : M f ( X , t ) > c r ~ , we h a v e t o show t h a t t h e r e i s a c o n s t a n t C s u c h t h a t , f o r e v e r y a > o

153

11.2. NORM ESTIMATES FOR THE MAXIMAL FUNCTION

is integrable. R and s u c h t h a t

Let us s e e t h i s .

W e may assume t h a t

and s u p p o s e t h a t

(x,t)

6

Ea.

x , w i t h s i d e l e n g t h (R) t t

Let

k

f

Fix

Then, t h e r e i s a c u b e

be t h e only i n t e g e r such t h a t

< ]R1 4 2

2-(k+1)n

t h e proof t h a t l e d t o theorem 1.6, t h e r e i s some meets t h e i n t e r i o r o f R and s a t i s f i e s

Q

G

ct >

0

containing

-kn

Dk

.

As i n

which

so t h a t

3 3 I t f o l l o w s t h a t Q c Q j G C4-na f o r some j a n d x G R c Q c Q j . 3 On t h e o t h e r hand t S s i d e l e n g t h ( R ) 5 s i d e l e n g t h ( Q . ) , so t h a t "3 J ( x , t ) G Q j . Thus, we h a v e s e e n t h a t Ea c U where C4-n,= {Qj}. 1 Then

4

I n p a r t i c u l a r , i f t h e measure

f o r every cube ~ ( x S) C and

Q c IRn

(2.16)

i s such t h a t

p

with C independent of Q , then i m p l i e s t h a t f W Mf i s an o p e r a t o r boun-

f o r every p ded from Lp(IRn) t o LP(IRn++l , u ) A c t u a l l y , g i v e n any p w i t h 1 < p < m, (2.17) c i e n t but a l s o necessary f o r LP(Rn+" , P I . Indeed, s i n c e t h e b o u n d e d n e s s of

M

M M(x

with 1 < p < m. i s not only s u f f i -

t o b e bounded f r o m

Q

)(x,t) 2 1

implies t h a t

Lp(IRn) to 21 f o r every ( x , t ) G Q ,

11. CALDERON-ZYGMUND THEORY

154

P(Q plr ) 5 ja(ULXg)(x,t))Pdu(x,tl

5 ClQl

The i m p o r t a n c e of t h e o p e r a t o r M stems f r o m t h e f a c t t h a t c o n t r o l s t h e P o i s s o n i n t e g r a l o f f , P ( f ) , d e f i n e d by:

for

x

IR"

6

c

and

=

t > 0,

where

[IR"

J k

2 t <J y - x 1 s 2

Actually,

Mf

for

ktl,

1

f 2 0

'L

In p a r t i c u l a r i f f = x Q a n d ( x , t ) G Q , we g e t P(X Q ) ( x , t ) 2 a n > O , w h e r e an d e p e n d s o n l y on t h e d i m e n s i o n n . T h e n , t h e same a r g u m e n t

11.3. SHARP MAXIMAL FUNCTION AND B.M.O.

155

t h a t we u s e d f o r M shows t h a t i f t h e o p e r a t o r f W P ( f ) i s b o unded f r o m L p ( I R n ) t o LP(Rn++l,p), t h e n p s a t i s f i e s ( 2 . 1 7 ) . The m e a s u r e s p s a t i s f y i n g ( 2 . 1 7 ) a r e c a l l e d C a r l e s o n m e a s u r e s . We c a n s t a t e t h e f o l l o w i n g : THEOREM

Let

2.18.

let

Then

i f and o n l y i f

(2.17)

IRn+l and i s a b o u n d e d o p e r a t o r from L P ( R n )

b e a p o s i t i v e B o r e 2 m e a s u r e on

p

1 < p < 03. p n + l ,p) to L (IR+

f W P ( f )

i f and o n l y i f

i s a C a r l e s o n m e a s u r e , that&,

p

h o l d s for some c o n s t a n t

C.

0

O b s e r v e t h a t t h e c o n d i t i o n o b t a i n e d d o e s n o t d e p e n d on p a n d i s a l s o e q u i v a l e n t t o t h e f a c t t h a t f H P ( f ) sends L1(Rn) boundedly i n t o L ,1( I- R + ,p).

3.

THE SHARP MAXIMAL F U N C T I O N AND THE SPACE

For a r e a l l o c a l l y i n t e g r a b l e f u n c t i o n

mal f u n c t i o n

is defined a t

f#

x

G

f IRn

where t h e s u p . i s t a k e n o v e r a l l t h e c u b e s f s t a n d s f o r t h e average of f o v e r Q,

Q

f

1

Q=lQl

in

B.M.O. R",

t h e s h a r p m a xi-

by s e t t i n g

containing t h a t is:

Q

x,

a nd

f(x)dx.

Q

The s h a r p maximal o p e r a t o r

f

H f#

i s a n a n a l o g u e o f t h e H a rdy-

L i t t l e w o o d maximal o p e r a t o r M , b u t i t h a s c e r t a i n a d v a n t a g e s o v e r i t wh ich we s h a l l p r e s e n t l y s e e . O f c o u r s e , f # ( x ) 5 Z M f(x). I t is a l s o c l e a r t h a t i n t h e d e f i n i t i o n o f f#(x) one c a n t a k e o n l y t h o s e cubes

Q

containing

x

in its interior

Jtctual l y

wh e re

i s u s e d t o i n d i c a t e t h a t e a c h s i d e i s bounde d b y t h e o t h e r

11. CALDERON-ZYGMUND THEORY

156

times an a b s o l u t e c o n s t a n t . ( 3 . 1 ) i s 5f'

I t i s c l e a r t h a t t h e r i g h t ha nd s i d e o f

F o r t h e o p p o s i t e i n e q u a l i t y we see t h a t

(x).

f o r e v e r y a G IR. r i g h t hand s i d e of

I t f o l l o w s t h a t f # ( x ) i s bounde d by t w i c e t h e (3.1) We a l s o n o t e t h a t :

.

( I f l ) # ( x ) 2 2f#(x)

(3.2) Indeed

dx 5 f # ( x )

f o r each

x

G

By u s i n g

Q

we g e t

(3.11,

(3.2).

is a function I f f i s such t h a t f # i s bounded, we s a y t h a t f o f bounded mean o s c i a l l a t i o n , a n d w e d e n o t e by t h e i n i t i a l s B . M . O . t h e s p a c e formed by t h e s e f u n c t i o n s . B.M.O.

We w r i t e For

f

G

{f

=

B.M.O.(IRn) B.M.O.

G

T hus

: f #

Lloc(IRn)

G

Lwl

when we n e e d t o s p e c i f y t h e u n d e r l y i n g s p a c e .

we w r i t e x)

Of c o u r s e we g e t a f t e r

-

f

Q

Idx.

(3.1):

Thus, i n o r d e r t o be a b l e t o say t h a t make s u r e t h a t t h e r e e x i s t s

C <

w

aldx 2

IIflI.

f 6 B.M.O., and, f o r each

it s u f f i c e s t o Q, a constant

11.3. SHARP MAXIMAL FUNCTION A N D B.M.O.

(1

157

T h i s i s t h e u s u a l way t o s e e t h a t a c e r t a i n

Then [If * 5 2C. f c B.M.O.

C l e a r l y , f W l l f I I * i s a seminorm a n d IIf I I * = 0 i f a n d o n l y i f f is constant. I t i s n a t u r a l t o consider the quotient space of B . M . O . modulo c o n s t a n t s , which i s a normed s p a c e a n d , a c t u a l l y a Banach s p a c e ( t h e c o m p l e t e n e s s i s l e f t a s a n e x e r c i s e f o r t h e I t c a n b e s e e n i n Neri [l]). reader. T h i s space of e q u i v a l e n c e c l a s s e s modulo c o n s t a n t s w i l l a l s o b e c a l l e d B . M . O . The a m b i g u i t y does n o t c a u s e any problem. Of course

Lm c B . M . O . a n d IIflI. 5 211fllm. However, t h e r e a r e B.M.O. f u n c t i o n s a s we s h a l l s o o n s e e . We s h a l l g i v e

u n bounded

two r e s u l t s w h i c h p r o v i d e many e x a m p l e s o f THEOREM

a.e.,

A,

3.3.

then

w,

i.e.

log w = 4 ,

Call

1

IQI

functions.

If w i s an A1 w e i g h t , t h a t i s , i f Mw(x) i Cw(x) l o g w i s i n B.M.O. u i t h a norm d e p e n d i n g o n l y on t h e

constant f o r

Proof:

B.M.O.

e'(x)dx

the smallest

that is

w

=

C.

e'.

for

5 Ce'")

We h a v e , f o r e v e r y c u b e Q a.e. x c Q

Q

or, equivalently 1

I ,Q

eo(x)dx]

e s s . s u p (e-"')) x f Q implies

But

- 11 IQI Thus,

e'(x)dx

=

.

e s s . s ~ p . ( e - ' ( ~ ) )5 C x f Q

e x p ( - e s s . i n f . @ ( x ) ) , and J e n s e n ' s i n e q u a l i t y x f Q

2 exp($ )

Q

'Q

exp(QQ

-

ess. i n f . '(x))

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

-

e s s . i n f . '(x) x f Q

5 C

C

and, consequently,

independent of

Q,

@

satis-

the property

6 C

We e x p r e s s t h i s by s a y i n g t h a t $ i s o f bounde d l o w e r o s c i l l a t i o n , t h e c l a s s f o r m e d by a l l t h e f u n c t i o n s o f a n d d e n o t e by B . L . O .

11. CALDERON-ZYGMUND THEORY

158

Now, we s e e t h a t

bounded l o w e r o s c i l l a t i o n .

Q

Averaging over

IQI

I,

Indeed

we o b t a i n

[ $ ( X I - OQldx

and t h e i n c l u s i o n

c B.M.O.

B.L.O.

c B.M.0

B.L.O.

Observe t h a t t h e c l a s s

-

6 Z($Q

B.L.O.

ess. i n f . +I

Q

0

follows readily.

i n t r o d u c e d above f a i l s t o b e a vec-

t o r s p a c e e v e n t h o u g h i t i s s t a b l e u n d e r t h e sum a n d t h e p r o d u c t by m

.

a n o n - n e g a t i v e n u m b e r . A c t u a l l y B . L . O . n (-B.L.O .) = L Indeed, and -0 a r e i n B . L . O . , w e h a v e a t t h e same t i m e i f both

+

QQ - e s s . i n f . $

2 C

and

-$JQ

+ ess. sup. @

Q

A d d i n g u p we g e t : cube

ess.sup @

Q

-

e s s . i n f $J 6

This is only possible i f

Q.

Theorem

3.3

gives us

6 C

Q

B.M.O.

Q

independent of t h e

i s e s s e n t i a l l y bounded.

$J

f u n c t i o n s from

p r e s e n t l y see a n i c e way t o p r o d u c e L i t t l e w o o d maximal o p e r a t o r

C

A1

weights.

We s h a l l

w e i g h t s by u s i n g t h e H a r d y -

A,

M.

Let p b e a p o s i t i v e B o r e l m e a s u r e on ELn, f i n i t e on c o m p a c t s e t s , and hence r e g u l a r . I t makes s e n s e t o c o n s i d e r t h e m a xim a l f u n c t i o n Mp(x) where t h e

sup

=

sup xfQ

1 ~

dp

IQI

Q

is taken over a l l cubes containing

x.

Exactly as

i n t h e c a s e of i n t e g r a b l e f u n c t i o n s , one o b t a i n s f o r measures t h e fundamental e s t i m a t e

with

d e p e n d i n g o n l y on t h e d i m e n s i o n .

C

THEOREM Mp(x) < w(x)

=

3.4. m

Let

for a.e.

We c a n s t a t e t h e f o l l o w i n g :

be a gositive Borel measure such t h a t

p

x

( ~ p ( x ) ) y is a n

G

IR",

and l e t

0 < y < 1.

T h e n t h e function

A, w e i g h t w i t h a c o n s t a n t d e p e n d i n g o n l y

on

11.3. SHARP MAXIMAL FUNCTION AND B.M.O.

y

159

a n d t h e d i m e n s i o n n.

Let

Proof:

1

IQI

with

Q

be a f i x e d cube. w ( x ) d x 5 Cw(x)

a.e.

x

G

Q

Q

independent of

C

for

We s h a l l s e e t h a t

Q.

'Q =L Q 3 , t h e 3 - d i l a t e d of Q . We 'L P p , t h e r e s t r i c t i o n o f p t o Q. Q and, s i n c e 0 < y < 1 , a l s o

Let

write p = p, + p2 with p l = Then Mp(x) 5 Mpl(x) + MpZ(x)

( M ~ ( X ) )2 ~ ( M p l ( x ) ) ' + (Mp2(x))'. T h e r e f o r e , i t w i l l b e enough t o s e e t h a t t h e averages of (Mpl(x))' and ( M I . I , ( X ) ) ~ over Q a r e b o t h bounded by Cw(x) f o r any x c Q w i t h C d e p e n d i n g o n l y on y

and t h e d i m e n s i o n .

R

=

&[lo

We c a r r y o u t t h e s e two e s t i m a t e s s e p a r a t e l y :

m

+

I,]

b e s p l i t t h e i n t e g r a l by u s i n g an a r b i t r a r y

R > 0).

Near

0

we

u s e t h e t r i v i a l e s t i m a t e f o r t h e d i s t r i b u t i o n f u n c t i o n , which i s ,

IQI.

clearly, 5 From R t o m we u s e t h e weak t y p e e s t i m a t e , t h e d i s t r i b u t i o n f u n c t i o n i s 5 C t - 1 lip, 1 1 , where l l p l 11 = p 1 ( R n ) Thus

1-Y RIQI

f o r e v e r y x E Q . What we h a v e j u s t done i s t o r e a l i z e t h a t e v e r y in a f i n i t e measure s p a c e , a c t u a l l y o p e r a t o r of weak t y p e ( 1 , l ) t a k e s L 1 boundedly i n t o L p , i f p < 1 . T h i s f a c t i s known a s Kolmogorov's i n e q u a l i t y ( s e e a l s o C h a p t e r V , Lemma 2 . 8 . )

.

11. CALDERON-ZYGMUND THEORY

160

To d e a l w i t h

I t i s enough t o see t h a t ,

i s even s i m p l e r .

Mp2

b e c a u s e o f t h e fact. t h a t p 2 l i v e s f a r f r o m Q ( o u t s i d e a n y two p o i n t s x , y G Q ,we h a v e Mu2(x) 5 CMp2(y), w i t h i s a cube containing so t h a t

absolute constant. I n d e e d i f Q' m e e t i n g R n \ :, then Q c Q I 3 ,

&

dP2 5 3n

lQt31

'Q'

1Q

a), C

x

for an

and

1 3 d p 2 5 3"Mp2(y).

Thus

f o r every

x

For example, g i n i n Rn. x = (xl,

...

G

[I

Q.

take = 6 , t h e Dirac d e l t a o r u n i t mass a t t h e o r i Then M6(x) = Ixlmn where 1x1, = max I x j I f o r 15j5n , x n ) . Thus M6(x)

I t f o l l o w s t h a t f o r any Y w e i g h t , o r , i n o t h e r words

with 1x1'

0 5 Y < 1, Ixl-nY i s an i s a n A1 w e i g h t f o r a n y

A1 a

with

-n < a 5 0 ( o n l y f o r t h e s e a ' s a c t u a l l y , s i n c e w h a s t o b e l o c a l l y i n t e g r a b l e a n d w-' l o c a l l y b o u n d e d ) . However, o u r main c o n c e r n h e r e i s t h e f a c t t h a t l o g 1x1 i s a n e x a m p l e o f a n unboun1 is a c t u a l l y i n B.L.O. ded B.M.O. f u n c t i o n . Note t h a t l o g I n g e n e r a l we have COROLLARY

a)

3.5.

For any p o s i t i v e Bore1 measure

~ p ( x )< m f o r a . e . x G IR", l o g M ~ ( x ) w i t h norm d e p e n d i n g o n l y on t h e d i m e n s i o n . bl

log

1x1

i s in

B.M.O.

!J

such t h a t function

B.M.O.

we know t h a t f G B . M . O . implies B.M.O. is a lattice ( i f f , g G B.M.O., t h e n t h e f u n c t i o n s max ( f , g ) = ( I f - g l + f + g ) / 2 a n d min ( f , g ) = However, we may h a v e ( f + g - If-g ) / 2 w i l l a s o b e i n B . M . O . ) . For example i f : If1 G B . M 0 . w i t h o u t h a v i n g f G B . M . O . 0 f o r 1x1 > 1 if O < X < I f x) = -log X I Since ( I f l ) # ( x ) 6 2f#(x), If1 G B.M.O. Consequently

1

log

XI

i f -1 < x <

o

11.3. SHARP MAXIMAL FUNCTION AND B.M.O.

is in l-$p

it is c l e a r t h a t I f ( x ) ) = max(1og f i s n o t i n B.M.O. Since f is

average

1 2'

I

0

on e v e r y i n t e r v a l

-a

B.M.O. However, odd a n d , c o n s e q u e n t l y , h a s

[-a,a],

a If(x)Idx =

=

-

1

161

we j u s t n e e d t o o b s e r v e t h a t

log a

-+

m

for

a

+

0.

T h e r e i s an i n t i m a t e r e l a t i o n between t h e o p e r a t o r f H f # and 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 M . I t is contained i n the f o l lowing s t a t e m e n t : THEOREM

3.6.

0

a,

po <

f

If

then f o r every

J Rn(Mf (xDPdx with

C

Proof:

s

i s such t h a t

p

M f

f

such t h a t

L

PO

f o r some

po 6 p

po

with

we h a v e :

a,

2 C

independent of

f.

We may assume t h a t

f 2 0

since

Mf = M(1f

1)

and

(If[)#2

2f#.

First The p r o o f i s b a s e d upon t h e Calder6n-Zygmund d e c o m p o s i t i o n . we s e e t h a t t h i s d e c o m p o s i t i o n c a n b e c a r r i e d o u t f o r o u r f u n c t i o n f . L e t t > 0 and f o r every x 6 Q

suppose

Q

i s a cube such t h a t

fQ > t.

Then,

and t h u s tpo 2

P ( M f ( x ) ) Odx 2

1

IQI

I

Q

,

I t follows t h a t i f Q Q, c u b e s f o r e a c h of which i s 1 -

IQkI

I

...

1

IQI

P (Mf ( x ) ) Odx =

-!?

IQI

i s an i n c r e a s i n g f a m i l y o f d y a d i c

f(yldy > t,

Qk

t h e n t h e f a m i l y i s n e c e s s a r i l y f i n i t e s i n c e IQ,I i s bounded i n d e p e n d e n t l y o f k. T h u s , e a c h d y a d i c c u b e Q s a t i s f y i n g f Q > t w i l l b e c o n t a i n e d i n a maximal o n e . I f 19.1 i s t h e f a m i l y c o n s i s t i n g 3 of t h e s e maximal d y a d i c c u b e s , f o r e a c h o f them w i l l b e

11. CALDERON-ZYGMUND THEORY

162

t < --1 IQj

I

I

f ( y ) d y 5 2"t.

Qj

I n o r d e r t o i n d i c a t e t h e d e p e n d e n c e on t , we s h a l l d e n o t e t h i s For a . e . x f4 U Q t , j i s f ( x ) 2 t . f a m i l y by { Q . I . t,J 1 Observe t h a t i f

t < s,

then each

Qs,j C Qt,k and t a k e Given t > 0 w e f i x Qo = Q 2 - n - 1 two p o s s i b i l i t i e s : e i t h e r t,jo

f o r some k . A > 0. There a r e

I n t h e f i r s t case

I n t h e second case

and,taking

2"2-'-lt

i n t o account t h a t

= t/2,

we can w r i t e

Q05

dyS

Adding up i n a l l t h e p o s s i b l e

I n terms of

CI

Qo's, we g e t :

we h a v e g o t t h e f o l l o w i n g i n e q u a l i t y :

163

11.3. SHARP MAXIMAL FUNCTION AND B.M.O.

a(t) 5 [Ex Now, for

:

f#(x) > t/A}I + 2A-'a(2 -n-1 t)

N > 0, we consider

IN

=

jNpfP-la(t)dt 0 5 jNptP-lB(t)dt 0

since we are assuming

1

Mf

6

Lpo.

2

Also

I

N

N

IN s

ptp-l I{x : f#(x) > t/A) Idt + 2A-

ptP-la(2 -n-1 tldt

0

0

from which: 5 f N ptP-l I{x : f#(x)

I

>

t/A) ldt + CA-lIN

Jo

with a

depending only on

C

n

and

p.

Take

A = 2C

and obtain:

I N 4 ZINptP-ll{x : f#(x) > t/A}ldt. 0

Letting

N

-+

m,

we arrive at

and then

I

m

m

(Mf(x))Pdx

=

0

ptP-'B(t)dt

6 C 1 j0ptP-'a(t/CZ)dt

=

=

11. CALDERQN-ZYGMUND THEORY

164

We have s e e n t h a t t h e maximal f u n c t i o n s Mf and f # a r e c l o s e l y related. We have t h e t r i v i a l p o i n t w i s e e s t i m a t e f '(x) 2 ZMf(x), b u t we a l s o have a n e s t i m a t e g o i n g i n t h e o p p o s i t e d i r e c t i o n , t h i s t i m e an Lp e s t i m a t e . For f s u f f i c e s t o s e e t h a t f # c Lp

"good enough", and 0 < p < w , it t o c o n c l u d e t h a t Mf, and t h e r e f o r e

1 i p < m, saying t h a t c L p and t h i s i s a l s o e q u i v a l e n t t o saying t h a t f c Lp. The s i t u a t i o n i s d i f f e r e n t f o r p = m. S a y i n g t h a t M f c L m i s t h e same a s s a y i n g t h a t f E L m . However, t o s a y t h a t f # c L w i s e q u i v a l e n t t o s a y i n g t h a t f 6 B.M.O. The s p a c e B . M . O . i s a n a t u r a l s u b s t i t u t e f o r L m , and t h e o p e r a t o r f b f ' a l l o w s u s t o t r e a t t h e s p a c e s L p and B . M . O . i n a u n i f o r m way. We s h a l l s e e t h a t many c l a s s i c a l o p e r a t o r s c a r r y Lm t o B.M.O. b o u n d e d l y . T h i s i s what makes t h e f o l l o w i n g i n t e r p o l a t i o n theorem i n t e r e s t i n g . F o r f qfgood" and f , a r e a l s o i n Lp. Mf c Lp i s e q u i v a l e n t t o s a y i n g t h a t f #

T be a l i n e a r o p e r a t o r bounded i n some w i t h 1 < po < m. Assume a l s o t h a t T Po B.M.O. boundedZy. T h e n , f o r e v e r y p with po < p < bounded i n Lp. THEOREM

3.7.

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

Proof:

LP O

Let

m,

for T

This i s a s u b l i n e a r

f'(Tf)#.

o p e r a t o r bounded i n Lpo and a l s o i n Lw. By M a r c i n k i e w i c z ' s i n t e r p o l a t i o n t h e o r e m , i t w i l l a l s o b e bounded i n L p . L e t f G L p .n. Lpo. Then

Tf E

and

I(Tf)'p

L'O

and

M(Tf) E L p o .

S Cllf p .

On t h e o t h e r hand

(Tf)'

G

Lp

The p r e c e d i n g t h e o r e m g i v e s :

and t h i s i n e q u a l i t y e x t e n d s t o e v e r y

f E Lp.

Ll The most i m p o r t a n t r e s u l t r e g a r d i n g

B.M.O.

i s the following theo-

rem of F . John and L . N i r e n b e r g . THEOREM

Q

There e x i s t constants n , s u c h t h a t for e v e r y f and e v e r y t > 0: 3.8.

dimension

G

C,,C2 B.M.O.

d e p e n d i n g onZu on t h e = B.M.O.(IRn) , e v e r y

It

-CC,/llfll (3.9)

)Ix

E

Q : If(x)

- fQI

>

t l l 2 Cle

*

191

11.3. SHARP MAXIMAL FUNCTION AND B.M.O.

165

I t i s a g a i n an a p p l i c a t i o n of t h e Calder6n-Zygmund decompos i t i o n . O b s e r v e , f i r s t of a l l , t h a t we c a n assume IIflI* = 1 , because t h e i n e q u a l i t y ( 3 . 9 ) d o e s n o t change i f we r e p l a c e f by a c o n s t a n t t i m e s i t . We f i x Q and t a k e a > 1 . We know t h a t Proof:

We make t h e Calder6n-Zygmund d e c o m p o s i t i o n of f

-

f

relative to

Q

o b t a i n i n g cubes

a,

Q1 , j

Q ) f o r e a c h of which:

I

a < -

If(x)

Bes d e s , f o r a . e . x $ U Q ,j f o r each Q 1 , j i s I f Q , j - f Q 5 2"a. A l s o :

-

f

is

Q

Q

f o r the function ( d y a d i c s u b c u b e s of

Idx I Zna

If(x)

-

f

Q

1

I t follows t h a t

5 a.

I

On e a c h Q 1 function f of d y a d i c

I f Q2,k If(x)

-

we make t h e Calder6n-Zygmund d e c o m p o s i t i o n f o r t h e - f r e l a t i v e t o a . Thus w e o b t a i n a f a m i l y Q 2 , k Q 1 s j subcubes of Q . , f o r e a c h of which i s

f Q1,j f

I

I

<

Zna,

151 and a l s o f o r a . e .

Besides,

I a.

Q1 , j together a l l the families

ilQ2,kl

x

G

Ql,j\

(!

Q2,k)

is

Now we p u t

6 a1 l Q l , j l .

{Q2,k} corresponding t o d i f f e r e n t

and c a l l t h e r e s u l t i n g f a m i l y a l s o Q1 ,j t h e u n i o n of t h e Q 2 , k ' s we h a v e : 's

{Q2,k}.

Then, o u t s i d e

and a l s o

S u b s e q u e n t l y , we o b t a i n f o r e a c h n a t u r a l number o v e r l a p p i n g cubes union i s

If(x)

{QN,j}

- f QI

N,

a f a m i l y o f non-

i n s u c h a way t h a t o u t s i d e of t h e i r and s u c h t h a t

? I Q , , ~ 5I

a - N ~ ~ ~ .

11. CALDERON-ZYGMUND THEORY

166

If

N.2"a

S t c (N

+ 1).2"a

with

N = 1,2,

...,

then

( w i t h C 2 = 2-"-l ( l o g a ) / a ) s i n c e t < ( N + l ) Z n a 5 N2""a. On t h e o t h e r h a n d , i f t < 2"a, t h e n C 2 t < ( l o g a ) / 2 , and we u s e t h e t r i v i a 1 ma j o r i z a t i o n

Thus, we g e t

f o r every

(3.9)

C1 = / a . Finally, a v a l u e of t h e c o n s t a n t COROLLARY

3.10.

II II* f

I f

Q

by c h o o s i n g

C2

a s above and

c a n be c h o s e n i n o r d e r t o g e t a n o p t i m a l C2 (a = e). f

sup(l

,p

t

IQI

then:

B.M.O.

E

1Q

If(x)

-

fQ/Pdx]'/p

s

CpIlfl),

f , i n s u c h a way t h a t , f o r 1 < p < ff-, I l f l l * , p i s a n o r m e q u i v a l e n t t o f ) - - , l l f l I *on B . M . O .

with P

i n d e p e n d e n t of

C

i i ) For e v e r y

X

1 sup -

Q

IQl

1

0 < X < C 2 / \ \ f \ l * , where

such t h a t

same c o n s t a n t a p p e a r i n g i n

e

(3.91,

I f ( x ) -fQ I

dx <

Q

A f t e r a change of v a r i a b l e s

we h a v e : m

C2

m,

i s the

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

167

1 1 , S t i r l i n g ' s f o r m u l a T ( p ) = 6e - p p P - l I 2 c a n b e u s e d t o c o n c l u d e t h a t C 6 C - p w i t h an a b s o l u t e c o n s t a n t C . P If

II I I *

4.

ll,,p

HARMONIC AND SUBHARMONIC FUNCTIONS IN A HALF-SPACE

I n t h e f i r s t two s e c t i o n s of c h a p t e r I , we s t u d i e d h a r m o n i c and s u b h a r m o n i c f u n c t i o n s i n a d i s k o r i n a e u c l i d e a n b a l l , and s o l v e d t h e D i r i c h l e t p r o b l e m f o r s u c h d o m a i n s . Our p u r p o s e i n t h i s s e c IRn++l = t i o n w i l l be t o extend t h i s s t u d y t o t h e h a l f - s p a c e { ( x , t ) f IR"+1 : t > 01. We a l r e a d y p o s e d t h e D i r i c h l e t p r o b l e m f o r t h i s domain i n t h e f i r s t s e c t i o n o f C h a p t e r I , and s t a r t e d

=

l o o k i n g f o r a s o l u t i o n . We s h a l l c o m p l e t e t h i s t a s k i n t h e p r e s e n t s e c t i o n . The main d i f f e r e n c e between t h e s i t u a t i o n a n a l i z e d i n C h a p t e r I and o u r p r e s e n t s e t t i n g i s t h e unboundedness of t h e domain, which f o r c e s u s t o l o o k c a r e f u l l y a t t h e b e h a v i o u r o f o u r functions a t infinity. We s t a r t by showing t h a t P o i s s o n ' s k e r n e l d o e s i n d e e d s o l v e t h e A t t h e end o f s e c t i o n i n t r o d u c e d t h e P o i s s o n k e r n e l f o r iRn++'

D i r i c h l e t problem.

1

i n Chapter

I

we

11. CALDERON-ZYGMUND THEORY

168

so t h a t

a s c a n b e s e e n by i n t e g r a t i n g i n p o l a r c o o r d i n a t e s . We a l s o h a v e , f o r a n y

I

Pt(x)dx =

6 > 0 Pl(x)dx * 0

t * 0

as

1x1 > 6 / t

I X b 6

because

P1

is integrable. P,(x)

i)

x

2 0,

6

Pt(x)dx = 1

ii)

T h u s , we h a v e : IR"

,

t >

for all

o t > 0

I

J

iii)

Pt(x)dx

+

0

as

t

+

0

f o r every

6 > 0.

Ixl>6 I n o t h e r words,

Also, t h e function P (x) = t

-

i s an approximate i d e n t i t y i n

{Ptlt>O

'n a n-1 a t

i s harmonic i n

(x,t)iPt(x)

I (t2

+

IR".

IRn+l

.

Indeed

1 lx12)(n-l)/2 1

and

is Itthe" r a d i a l harmonic f u n c t i o n i n THEOREM

g r a l of

4. I.

f:

For

f

6

?? p= 1

Lp(IRn)

IRn+l\ C O l .

, we c o n s i d e r t h e P o i s s o n i n t e -

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

Then

u(x,t)

a

a)

in -

f

E

as

LP

i s harmonic i n

LP(IR") t

with

169

Rn+' and:

+

I 5 p <

-

u(*,t) * f

it f o l l o w s t h a t

m,

0.

+

If

f i s c o n t i n u o u s and bounded, it f o l l o w s t h a t u ( . , t ) * f u n i f o r m l y on c o m p a c t s u b s e t s a s t 0. c ) Lf f i s u n i f o r m l y c o n t i n u o u s and bounded, t h e c o n v e r g e n c e b)

+

as

u(-,t) * f

Lf

d)

weak-*

f

G

t *

2a

Pt*v(x)

, then

l o g y of

M(IRn)

( f i n i t e ) Borel measure i n

u

.

c i t y of

Pt(x). u(x,t)

u(.,t)

+

f

i n the

Lm.

i s harmonic and

The h a r m o n i c i t y o f

Proof:

R".

i s uniform i n

L m ( R n ) , a l l we c a n s a y i s t h a t

t o p o l o g y of

Also if

o

u(x,t)

u(.,t)

Rn * p

u(x,t)=P(v)(X,t)= i n - t h e weak-*

topo-

i s a consequence o f t h e harmoni-

Besides

-

f(x) =

(f(x

-

y)

-

f(x))Pt(y)dy.

T h u s , by u s i n g M i n k o w s k i ' s i n e q u a l i t y f o r i n t e g r a l s , w e g e t :

In c a s e s a ) and a way t h a t llf( choice of

6,

c ) , given E > 0 , 6 > 0 can be chosen i n such - y ) - flI < E / Z w h e n e v e r IyI < 6 . W i t h t h i s P t h e f i r s t term i n o u r sum o f i n t e g r a l s i s < ~ / 2

independently of

-

t.

The s e c o n d

term i n t h a t sum i s

j l y l , 6 P t ( y ) d y * 0 a s t * 0 a s we know f r o m p r o p e r t y i i i ) of t h e P o i s s o n k e r n e l . T h e r e f o r e , t h e s e c o n d term i n t h e 5 211fllp

~ / 2 by t a k i n g t c l o s e t o 0 . This proves sum c a n a l s o b e made a ) a n d c ) . The p r o o f o f b ) is e n t i r e l y analogous. P a r t d)

a n d t h e f i n a l s t a t e m e n t c o n c e r n i n g a B o r e l m e a s u r e p a r e p r o v e d by d u a l i t y e x a c t l y a s i n t h e o r e m 1 . 1 8 i n C h a p t e r I . Now M ( I R n ) is t h e d u a l o f Co(lRn) , t h e s p a c e o f c o n t i n u o u s f u n c t i o n s v a n i s h i n g at

m

w i t h t h e supremum norm, a n d t h e s e f u n c t i o n s a r e u n i f o r m l y

continuous, so t h a t p a r t COROLLARY

4.2.

Fo r

f

c)

applies.

[I

c o n t i n u o u s and bounded i n

Rn,

the func-

u(x,t) defined by u(x,t) = P(f)(x,t) 2 2 t > 0 and u ( x , O ) = f ( x ) , i s c o n t i n u o u s z' n a n d h a r m o n i c i n R n + l . I& with i s t h e r e f o r e a s o l u t i o n o f t h e D i r i c h l e t p r o b l e m i n IRn+l + -

.-t i o n

+

11. CALDERON-ZYGMUND THEORY

170

boundary f u n c t i o n

I t f o l l o w s i m m e d i a t e l y from p a r t

Proof:

rem.

f.

b)

of t h e p r e v i o u s t h e o -

0 4.3.

PROPOSITION

I(u(

W i t h t h e same h y p o t h e s i s

.,t)llpsllfllp

f o r euery

of

theorem

we

4.1,

t > 0.

Since t h e function x H u ( x , t ) i s t h e c o n v o l u t i o n of Pt and f ( u ( . , t ) = P t * f ) , t h i s i s a p a r t i c u l a r i n s t a n c e o f Y o u n g ' s i n e q u a l i t y and i t i s e a s i l y o b t a i n e d by w r i t i n g u ( x , t ) = Proof:

=

IRn P t ( r ) f ( x - y ) d y

grals.

and a p p l y i n g M i n k o w s k i ' s i n e q u a l i t y f o r i n t e -

u

Now we s h a l l o b t a i n P o i s s o n ' s r e p r e s e n t a t i o n f o r a c l a s s o f n i c e harmonic f u n c t i o n s . 4.4.

THEOREM

R'+'

monic i n

+

Let

-

u(x,t)

be a c o n t i n u o u s f u n c t i o n i n

and bounded.

Then

u

Rn+l + '

harcoincides necessarily with

x + $ u(x,O),

t h e P o i s s o n i n t e g r a l of t h e b o u n d a r y f u n c t i o n

that

i s , we h a v e t h e P o i s s o n r e p r e s e n t a t i o n :

proof:

Consider t h e f u n c t i o n I

v ( x , t ) g i v e n by:

r

if V(X,t)

=

t > O

1

T h i s f u n c t i o n i s c o n t i n u o u s i n lRn++l , b o u n d e d , h a r m o n i c i n IRn++' and v a n i s h e s a t e v e r y p o i n t of t h e b o u n d a r y . Theorem 1 . 3 3 i n chapter I t e l l s u s t h a t v is i d e n t i c a l l y 0.

[I

Theorem 4 . 4 . c a n b e r e a d by s a y i n g t h a t t h e D i r i c h l e t p r o b l e m i n Rn + 1 f o r a bounded and c o n t i n u o u s b o u n d a r y f u n c t i o n h a s a u n i q u e bounded s o l u t i o n , b u t , o f c o u r s e , we know t h a t u n i q u e n e s s d i s a p p e a r s a s s o o n a s we a l l o w unbounded s o l u t i o n s .

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

171

I n o r d e r t o e x t e n d t o o u r s i t u a t i o n Theorems 1 . 3 and 1 . 8 from C h a p t e r I , and a l s o f o r o t h e r a p p l i c a t i o n s , we s h a l l make some b a s i c e s t i m a t e s f o r harmonic and s u b h a r m o n i c f u n c t i o n s i n Rn++' . v ( x , t ) be a non-negative subharmonic f u n c t i o n IFP++w h i c h i s u n i f o r m l y i n L p ( l R n ) f o r some 1 5 p < m , &

THEOREM

in -

Let

4.5.

w h i c h we mean t h a t :

~ ( x , t ) ~ d =x

Mp <

m

.

C depending only on p & n, such ( x , t ) G Rn:.' In p a r t i c u l a r v ( x , t ) IC M t - " l P f o r every I ( x , t ) G lRn++' : t 2 i s bounded i n each p r o p e r s u b - h a l f - s p a c e v(x,t) 2 t > 01. A c t u a l l y , t h e f o l l o w i n g s t r o n g e r p r o p e r t y h o l d s : 0 v(x,t) + 0 as ( x , t ) + m i n each proper sub-half-space. Then t h e r e i s a c o n s t a n t

that

Given ( x o , t o ) G Rn++' , l e t Bo = B ( ( x o , t o ) , t o / 2 ) t h e b a l l c e n t e r e d a t (xo, to) with r a d i u s t 0 / 2 . Then

Proof:

1 v ( x 0 , t ) I-

v(x,t)dx d t i

0

I C M

denote

tin/P.

Let u s p r o v e now t h a t t o >0 . S i n c e we know t h a t g i v e n E > 0 , we c a n f i n d t l > t o v ( x , t ) IC M t - n / P , s u c h t h a t v ( x , t ) 5 E p r o v i d e d t 2 t 1 ' We j u s t n e e d t o show t h a t f o r t o It 5 t l , v ( x , t ) c a n b e made s m a l l e r t h a n E by t a k i n g x big. P r o c e e d i n g a s i n t h e f i r s t p a r t o f t h e p r o o f , we see t h a t f o r 1x1 > t l , s a y , a n d f o r t o I t It 1 T h i s p r o v e s t h e f i r s t p a r t of t h e t h e o r e m .

v(x,t)

+

0

as

(x,t)

-+

m,

t 2 t 0 > 0.

Fix

11. CALDERON-ZYGMUND THEORY

172

11

and t h e proof i s completed.

We s h a l l s e e b e l o w t h a t t h e o r e m

4.5

s t i l l holds f o r

v(x,t) =

I

w i t h u h a r m o n i c i n Illn+’ a n d a n y p > 0 . When t h e proof of theorem 4 . 5 cannot be used, s i n c e t h e second i n e q u a l i t y i n t h a t p r o o f i s J e n s e n ’ s i n e q u a l i t y which i s no

= lu(x,t) 0 < p < 1,

longer available f o r LEMMA

4.6.

Let

3. T h e n , i f -_ in

p < 1.

We s h a l l u s e , i n s t e a d , t h e f o l l o w i n g

u be harmonic i n a b a l l B c ( x o , t o ) i s t h e c e n t e r o f B,

and c o n t i n u o u s we h a v e , f o r a n y

p > 0:

where

C

P

i s a c o n s t a n t d e p e n d i n g o n l y on

p

& n.

P r o o f : O f c o u r s e , we a l r e a d y know t h a t t h e lemma i s t r u e when p 2 1 w i t h C = 1 . Let u s l o o k now a t t h e c a s e 0 < p < 1 . We P may a s s u m e t h a t B i s t h e u n i t b a l l c e n t e r e d a t t h e o r i g i n a n d a l s o t h a t j B l u ( x , t ) l P d x d t = IB . Write

r o ) lPdo do

YP

b e i n g L e b e s g u e m e a s u r e on t h e u n i t s p h e r e , a n d

mm(r) 2 1 f o r e v e r y r w i t h 0 < r < 1 , s i n c e o t h e r w i s e t h e maximum p r i n c i p l e w o u l d i m m e d i a t e l y g i v e u s t h e required inequality

We may a l s o a s s u m e t h a t

From t h e r e p r e s e n t a t i o n o f u a s a P o i s s o n i n t e g r a l o v e r t h e s p h e r e of r a d i u s r (see t h e formula r i g h t b e f o r e theorem 1.30 i n Chapter I ) , we o b t a i n , f o r 0 < s < r : m m ( s ) S 2 ( 1 t a k e s = r a w i t h a > 1 . We h a v e

- s r - l ) - nm , ( r ) .

Now

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

173

By t a k i n g l o g a r i t h m s and i n t e g r a t i n g we g e t :

l o g m ( r ) d- r P r The l a s t i n t e g r a l i s bounded by a c o n s t a n t d e p e n d i n g Dnlyon p n,

and

s i n c e o u r normalizing assumption i s e q u i v a l e n t t o

lo 1

(n

t

1)

mp(r)P

rn d r

=

1.

T h e r e f o r e , a f t e r a c h a n g e of v a r i a b l e s i n t h e i n t e g r a l on t h e l e f t hand s i d e , we c a n w r i t e :

/2

But a > 1 mm(r) 2 1,

implies

(1/2)a

< 1/2

log mm(r)

dr r

-

a n d , s i n c e we h a v e assumed

it f o l l o w s t h a t

log mm(r) 2 drr

I

r)

dr

.

Thus log mm(r) Now w e c h o o s e a 1 < a < (l-p)-'.

dr

2 Ca + ( 1

so close t o 1 t h a t Then we o b t a i n :

dr log mm(r) - 5 C

- PI l / a > 1-p,

dr ,log m m ( r ) r t h a t i s , we t a k e

P

ro > 0 ( ( / 2 ) a 5 r o 2 l ) , mm(r ) S C 0 P' a c o n s t a n t d i f f e r e n t from t h e p r e v i o u s o n e , b u t s t i l l d e p e n d i n g o n l y on p and n . Then t h e maximum p r i n c p l e g i v e s u s I 5 cP a s we wanted t o p r o v e .

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

IU(O,O)

Now we c a n g i v e t h e p r o m i s e d e x t e n s i o n o f t h e o r e m

4.5.

11. CALDERON-ZYGMUND THEORY

174

THEOREM

Let

4.7.

i s uniformly i n

u(x,t)

Lp(lRn)

Then t h e r e i s a c o n s t a n t

that

l

u(x,t)

b e a harmonic f u n c t i o n i n 0 < p <

f o r some

C

Rn+l which

+

-

that i s :

m,

p

d e p e n d i n g o n l y on

u ~ x f o r r e v e r y ( x , t ) f IR"++~ i s bounded i n e a c h p r o p e r s u b - h a l f - s p a c e

and

.

n,

such

In particular ( x , t ) G IRn+tl : t t

2 t > 01. A c t u a l l y , t h e f o l l o w i n g s t r o n g e r p r o p e r t y h o l d s 0 u(x,t) + 0 (x,t) m i n each proper sub-half-space. -+

I t i s e x a c t l y t h e same a s t h e p r o o f o f t h e o r e m ing with t h e inequality:

Proof:

with

Bo

=

4.5

start-

B ( ( x o , t o ) , t o / Z ) ) , w h i c h was e s t a b l i s h e d i n lemma 4 . 6 .

We c a n now p r e s e n t t h e a n a l o g u e s o f t h e o r e m s

1.3

and

1.8

in

Chapter I , providing c h a r a c t e r i z a t i o n s of t h e Poisson i n t e g r a l s o f LP f u n c t i o n s , p > I , o r m e a s u r e s i n IR"

.

THEOREM

4.8.

the function a) u bl u

Let

u(x,t)

1

p S

Then t h e f o l l o w i n g t u o c o n d i t i o n s on

m,

defined i n

IRnt+'

are equivalent:

i s harmonic i n and i s u n i f o r m l q i n Lp(IRn) i s t h e P o i s s o n i n t e q r a l o f some f u n c t i o n f G L p ( I R n )

and 4 . 3 , that (b) implies ( a ) i m p l i e s ( b ) . Assuming (a) holds, consider t h e f u n c t i o n s f . ( x ) = u ( x , t . ) for a 3 J s e q u e n c e t . + O . From ( a ) , (fj) i s a bounded sequence i n 1 Lp(IRn) = LP'(JRn)*. A s i n t h e proof of theorem 1.3 i n Chapter

Proof:

(a). that

We a l r e a d y know, f r o m

.

4.1

Let u s p r o v e t h a t , c o n v e r s e l y ,

( f j ) h a s a s u b s e q u e n c e c o n v e r g i n g i n t h e weak-* t o p o l o g y o f L p . I, We may a s s u m e t h a t ( f j ) i t s e l f c o n v e r g e s i n t h e weak-* t o p o l o g y t o a c e r t a i n f G Lp(lRn) , t h a t i s , f o r e v e r y g G L p ' ( l R n )

j, the function (x,t)w u(x,t t t . ) is continuous J i n IR"++l , h a r m o n i c i n IRn+l a n d , a c c o r d i n g t o t h e o r e m 4 . 7 , it i s a l s o bounded. I t f o l l o w s from theorem 4.4 t h a t t h i s f u n c t i o n

NOW, f o r each

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

175

i s t h e Poisson i n t e g r a l of i t s boundary f u n c t i o n , t h a t i s , f o r every j u(x,t + t . ) = J Letting

J IR" P t ( x

go t o

j

since the function

-

y ) u ( y , t . ) d y = jJRnPt(x J

m,

we g e t

y

H Pt(x

As in the torus,

the case p similar p r o o f , t h e following THEOREM

a) b)

in -

y)

=

1

belongs

to

y ) f .J ( y ) d y .

Lp'(IRn)

. fl

i s d i f f e r e n t , a n d we g e t , w i t h a

T h e f o l l o w i n g t w o c o n d i t i o n s on t h e f u n c t i o n

4.9.

an+'

defined i n

-

-

u(x,t)

are equivalent:

u i s h a r r n z n i c i n Rn;l and i s u n i f o r m l y i n L1(IRn) u i s t h e P o i s s o n i n t e g r a l o f some ( f i n i t e ) B o r e 1 m e a s u r e IR", that i s u(x,t) = P(u)(x,t) = P (X - Y ) d U ( y ) . JR"

B a s e d u p o n t h e same i d e a s i s t h e f o l l o w i n g r e s u l t o n h a r m o n i c m a j o r i z a t i o n , which w i l l be a v e r y u s e f u l tool i n t h e s e q u e l

1

Let

4.10.

THEOREM

v

be a n o n n e g a t i v e subharmonic f u n c t i o n i n

w h i c h is u n i f o r m Z y i n

ha: a l e a s t h a r m o n i c m a j o r a n t

Lp(JRn) f o r some 1 6 p IRn++' . M o r e o v e r , i f

iz

m. Then v p > 1 , & I@

h a r m o n i c m a j o r a n t is t h e P o i s s o n i n t e g r a l o f some f u n c t i o n i n Lp(IRn)

a n d if

Bore1 measure on Proof:

p = 1, IR" .

i t i s t h e P o i s s o n i n t e g r a l of some ( f i n i t e )

As before, consider the functions

We may a s s u m e t h a t f . sequence t . G O . J J t h e weak-* t o p o l o g y o f Lp(IRn) if p > logy of M(IRn) if p = 1. Just t o f i x Then f . J f o r every

+

f

j

E

Lp(IRn) t h e r e is

f.(x) = v(x,tj) for a J converges as j m in 1 o r i n t h e weak-* t o p o the notation, let p > 1.

i n t h e weak-* t o p o l o g y .

+

Now w e c l a i m t h a t

The way t o s e e t h i s i s t o o b s e r v e t h a t b o t h f u n c t i o n s

176

11. CALDERON-ZYGMUND THEORY

( x , t ) H v(x,t + t . )

and

1

tend t o

as

0

(x,t) R > 0

h a v e t o make

in

+ m

IRn+l

.

Thus, given

E

we j u s t

> 0,

b i g enough s o t h a t

+ t . ) - P(f.)(x,t)

V(X,t

P(f.)(x,t) 3

1

3

i n t h e boundary of t h e r e g i o n

5E

KR =

I ( x , t ) c JRn++l : I x I 2 + t 2

6

R2 1 .

Then we c a n a p p l y c o r o l l a r y 2 . 6 f r o m C h a p t e r I t o c o n c l u d e t h a t v ( x , t + t . ) - P ( f . 1 ( x , t ) 5 E f o r every ( x , t ) c KR. Since t h i s is 3 1 t r u e f o r a l l l a r g e enough R ' s , we h a v e a c t u a l l y v ( x , t + t . ) 3 - ~ ( f . ) ( x , t )5 E f o r e v e r y ( x , t ) c IR"++' . N O W , s i n c e E was 1 a r b i t r a r y , we g e t ( 4 . 1 1 ) . That t h e function ( x , t ) W v ( x , t + t . ) 3 t e n d s t o 0 a s ( x , t ) + m i n IRn+l f o l l o w s f r o m t h e o r e m 4 . 5 . As f o r t h e f u n c t i o n P ( f . ) , we j u s t nLed t o o b s e r v e t h a t f j c Co(lRn) , 3 t h e s p a c e o f c o n t i n u o u s f u n c t i o n s v a n i s h i n g a t m, a n d t h i s a l r e a d y guarantees t h a t let

h

Co(JRn).

G

By t a k i n g

R

Take

E

+

0

> 0.

as

(x,t) *

in

m

IRn++'

-+

f o r every

.

Indeed,

Then

b i g e n o u g h , t h e s e c o n d i n t e g r a l c a n b e made

absolute value. t h a t P,(x) 0 (4.11)

P(fj)(x,t)

< ~ / 2 in

F o r t h e f i r s t i n t e g r a l w e make t h e o b s e r v a t i o n as (x,t) + i n IR"++' . T h u s , we h a v e e s t a b l i s h e l j

and every

(x,t) c

.

Letting

j

+ m,

we

have:

Thus case

u ( x , t ) = P ( f ) ( x , t ) i s a harmonic majorant of p = 1 we g e t a h a r m o n i c m a j o r a n t u = P(p)

v . For t h e with G M(IRnI

That u i s indeed t h e l e a s t harmonic m a j o r a n t of v f o l l o w s from t h e f a c t t h a t P ( f i ) ( x , t ) i s t h e l e a s t harmonic m a j o r a n t of ( x , t ) l - v ( x , t + t . ) b e c a u s e , i f h i s a h a r m o n i c m a j o r a n t o f v , we 1 would h a v e h ( x , t + t . ) 2 v ( x , t + t . ) a n d , c o n s e q u e n t l y h ( x , t + t . ) t 3 1 1 2 P ( f j ) ( x , t ) . Letting j +m, we g e t : h ( x , t ) 2 u ( x , t ) . We s h a l l n e x t s t u d y t h e p o i n t w i s e c o n v e r g e n c e o f P o i s s o n i n t e g r a l s .

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

177

The a p p r o a c h w i l l be t h e same a s t h a t we h a v e a l r e a d y u s e d t o p r o v e t h e L e b e s g u e t h e o r e m ( c o r o l l a r y 1 . 1 1 ) . Given f G U L p ( I R n ) , 1 o IPt*f(x) S CMf(x,t) with end o f s e c t i o n 2 we p r o v e d t h a t +

1

M f ( x , t ) = s u p I- 1

s i d e l e n g t h (Q) 2 t )

If1 : x E Q ,

1QI

'Q

Of course, it follows t h a t

* P f(x) 5 C

Mf(x,t)

SUP

CMf(x)

=

t>O

T h u s , P * i s a bounded o p e r a t o r i n L p € o r a n y p > 1 a n d i t i s a l s o o f weak t y p e ( 1 , l ) . T h i s i s a l l we n e e d t o p r o v e t h e f o l l o w i ng

f o r some p I t w i l l b e enough t o show t h a t f o r e a c h

Proof:

Assume f i r s t t h a t

1 5 p <

w.

A~

h a s measure true that

=

{x 0.

G

f

G

LP(IRn)

Indeed, t h e s e t of p o i n t s

Pt*f(x)

+

f(x)

as

t

+

According t o theorem Consequently,

-

4.1,

f(x)

I

part

5 lirn

c),

x

G

>

s

5

0, t h e s e t

S }

IRn

is p r e c i s e l y

0,

Given E > 0 , we c a n w r i t e f = g + h , w i t h compact s u p p o r t a n d J R n l h l P < E .

l i m sup IPt*f(x) t+O

- f(x) I

: l i m sup IPt*f(x) t + O

IR"

such t h a t

where i t i s n o t m

.

U Al,j j=l

where g i s c o n t i n u o u s We h a v e , f o r e v e r y x a n d t :

Pt*g(x)

sup IPt*h(x) t + O

I

+

g(x)

+ Ih(x)

as

1

5

t

+

0.

11. CALDERON-ZYGMUND THEORY

178

I t follows that

a n d we g e t :

Since t h i s is t r u e f o r every

we a c t u a l l y have

> 0,

E

lAsl

=

0.

Thus, t h e theorem i s proved i n t h i s c a s e . The c a s e

f

s e e how.

Take

as t

f o r almost every

+

0

G

Lm(IRn) can be reduced t o t h e previous one. Let u s B = B ( O , R ) , R > 0. We s h a l l p r o v e t h a t P t * f ( x ) + f ( x ) x

G

T h i s w i l l b e enough, s i n c e

B.

R

i s a r b i t r a r y . Let B ' = B ( 0 , R + 1 ) . Write f = f l + f 2 w i t h fl(x) = f(x) if x G B' a n d f , ( x ) = 0 e l s e w h e r e . Then f,

G

L1(Rn)

and, consequently

Pt*fl(x)

-+

fl(x)

a . e . x G IR". T h u s , we j u s t n e e d t o p r o v e t h a t f o r a . e . x G B. But f o r x f B :

as

t

-+

0

for

Pt * f 2 ( x ) + f 2 ( x ) = 0

and

In general, i f

@

G

L1(IRn)

t h e approximate i d e n t i t y =

t-"@(t-'x).

at,

I,n

t > 0,

$ ( x ) d x = 1 , w e may c o n s i d e r o b t a i n e d by m a k i n g

@t(x) =

T h e n , i f we now s e t

a l l p a r t s of theorem 11,

and

4.1,

with t h e exception of t h e harmonicity of

s t i l l h o l d , s i n c e t h e y depend fsnly on t h e s e t h r e e f a c t s :

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

@(xldx= 1

ii)

t

t > 0,

I@(x)

For e v e r y 6 > 0 :

iii)

as

f o r every

179

and Idx

+

0

0.

+

Theorem 4 . 1 2 , h o w e v e r , d e p e n d s upon t h e i n e q u a l i t y P " f ( x ) 6 5 CMf(x). I f we want t o e x t e n d i t t o o t h e r a p p r o x i m a t e i d e n t i t i e s * @,, we s h a l l h a v e t o s t u d y t h e maximal f u n c t i o n @ ( f ) ( x ) =

I.

sup I@,*f(x) I f we g e t a n i n e q u a l i t y @ * ( f ) ( x ) S CMf(x), t>O e x t e n d s immediately t o t h e a p p r o x i m a t e i d e n t i t y {@t}. theorem 4 . 1 2 I f @ i s r a d i a l and d e c r e a s i n g , t h e i n e q u a l i t y i s v e r y e a s y t o =

obtain. THEOREM

$(x) t 0

Let

4.13.

IRn . Then f G u LP(IR") a n d e v e r y 15 p 9 n.

dimension -

Let

Proof:

Call

be a r a d i a Z , d e c r e a s i n g , i n t e g r a b Z e

$ * ( f ) (x) 5 C l l + l l ,Mf(x) f o r e v e r y x G I R ~ , with c d e p e n d i n g o n l y on t h e

function i n

Then

B. = B(0,r.). J J

I

m IR"

@(x)dx=

and f o r e v e r y

f

G

1

a.IB.1 <

j=1

1

Lp(iRn)

1

,

1 5 p <=

By a p p l y i n g t h i s t o t h e f u n c t i o n

11

IRn f ( x

-

m

f(x

y)@(y)dyl 5 C[jRn

Now o b s e r v e t h a t , f o r

@

m,

-

a )

we obtain

@]Mf(x).

a s above and

t > 0,

@t i s a f u n c t i o n of

11. CALDERON-ZYGMUND THEORY

180

11+, 111

t h e same t y p e a n d

=

llCplll.

Therefore

i s r a d i a l , d e c r e a s i n g a n d i n t e g r a b l e , we c a n f i n d a s e q u e n c e o f f u n c t i o n s $ ( J ) , e a c h o f w h i c h i s a f i n i t e l i n e a r comb i n a t i o n of c h a r a c t e r i s t i c f u n c t i o n s of b a l l s c e n t e r e d a t 0 w i t h

Now i f

J,

x,

c o e f f i c i e n t s 2 0, such t h a t , f o r every monotonically t o $ (x) Then

.

Since, f o r each

Cp(j)(x)

increases

j

we f i n a l l y g e t :

CII

J , * ( f ) ( x ) S CII$IIIMf(x). COROLLARY

4.14.

that the function

i s called every

f

R".

with

Cp G L 1 ( l R n )

$(XI

j I R n $ ( x ) d x = 1 and s u c h i s i n t e g r a b l e ($

= s u p { ) @ ( y ) I :IyI t 1x11

t h e l e a s t r a d i a l d e c r e a s i n g m a j o r a n t of

u

LP (R") 15p5m

G

,

Cpt*f(x)

-t

f(x),

as

is i d e n t i c a l t o that of theorem

It

Proof:

Let

t

+

0,

4.12.

4).

Then, f o r

f o r almost every

The f a c t t h a t

+

i s i n t e g r a b l e a n d J,nCp = 1 guarantees t h e uniform convergence f o r f c o n t i n u o u s w i t h c o m p a c t s u p p o r t . T h e n , we j u s t h a v e t o

observe t h a t

+ * ( f ) ( x ) 2 $ * ( l f l ) ( x ) S CllJ,ll,Mf(x).

A c t u a l l y , t h e p o i n t w i s e convergence o f a P o i s s o n i n t e g r a l a t a bound a r y p o i n t h o l d s i n a s t r o n g e r f o r m t h a n t h e o n e we h a v e d i s c u s s e d . We c a n a l l o w a n y n o n - t a n g e n t i a a p p r o a c h . L e t u s g i v e p r e c i s e definitions. F o r a p o i n t x 6 IRn a n d a n y r e a l number N > 0 , we s h a l l c a l l TN(x) t o t h e o p e n v e r t i c a l c o n e w i t h v e r t e x x a n d aperture

N , t h a t is

rN(x) = { ( y , t )

G

n + l .. y IR+

XI

< Ntj.

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

181

,

Given a f u n c t i o n u d e f i n e d on JRn+l and a p o i n t x G IRn we s h a l l s a y t h a t u ( y , t ) c o n v e r g e s t o II a s ( y , t ) t e n d s t o x ( a c t u a l l y t o (x,O)) n o n t a n g e n t i a l l y , and we s h a l l w r i t e u ( y , t )

+

II

a s ( v , t ) N;T*x, i f and o n l v i f , f o r e v e r y N > 0 , u ( y , t ) c o n v e r ges t o R a s ( y , t ) tends t o x remaining always i n TN(x). This i s t h e n a t u r a l e x t e n s i o n t o o u r c o n t e x t of t h e n o t i o n of n o n - t a n g e n t i a l convergence i n t r o d u c e d f o r t h e d i s k i n Chapter I ( i n theorem 1 . 2 0 ) . To p r o v e t h e n o n - t a n g e n t i a l c o n v e r g e n c e of P o i s s o n i n t e g r a l s we s h a l l l o o k a t t h e n o n t a n g e n t i a l P o i s s o n maximal f u n c t i o n o f a p e r t u r e N > 0 , d e f i n e d f o r any f G U L P ( R n ) as: 1< p 6 m * PV,Nf(X) = sup IPt*f (Y) I (y,t i G r N ( X ) We know t h a t I(Pt*f)(y)l 5 C M f ( y , t ) w i t h C d e p e n d i n g o n l v on n . However, i f ly < Nt, we c a n e a s i l y s e e t h a t M f ( y , t ) 5 S CNMf ( x ) , where CN i s a c o n s t a n t dependinr! on N . To p r o v e t h i s i n e q u a l i t y we j u s t n e e d t o o b s e r v e t h a t i f Q i s a c u b e c o n t a i n i n g y and h a v i n g s i d e l e n g t h 2 t, then x G Q Z N + l , so t h a t w e have

XI

and, consequently.

Thus we c a n w r i t e : (4.15)

P

*

v ,Nf ( x )

T h i s i n e q u a l i t y i s a l l we n e e d t o p r o v e t h e f o l l o w i n g :

(y,t)N;T'x Proof:

f

U Lp 1
4.16.

THEOREM

f

(IR") , x

6

then

Pt*f(y)

+

f(x)

IR'.

The p r o o f i s e n t i r e l y s i m i l a r t o t h a t of t h e o r e m

Suppose f i r s t t h a t

f

G

LP(Rn)

and

1 5 p <

a.

4.12.

Then, a f t e r f i x * P

with the operator ing N > 0, we proceed a s i n 4 . 1 2 , * i n s t e a d of P I n t h i s way, we show t h a t t h e s e t

.

as

v ,N

11. CALDERON-ZYGMUND THEORY

182

h a s measure pt * f ( y )

I f we c a l l

0.

f(x)

+

as

fl

E =

(y,t)N;T.x

we s h a l l h a v e x E.

EN,

IEl = 0

and

4

N=l f o r every

A s f o r t h e c a s e f G Lw(lRn) , we r e a l i z e t h a t t h e a r g u m e n t g i v e n i n 4.12 extends t o our c a s e , s i n c e , as can be e a s i l y s e e n , Pt * f 2 ( y )

+

as

0

( y , t ) N;T.x

if

x

6

n

B.

We s h a l l b e a b l e t o g i v e a more p r e c i s e r e s u l t by u s i n g a more d i r e c t a p p r o a c h b a s e d upon t h e i n e q u a l i t y :

x

v a l i d f o r every

lRn

G

and e v e r y

6

lRnttl

satisfying

w i t h a c o n s t a n t C N t h a t d e p e n d s o n l y on N we may a s s u m e t = 1 . n . To p r o v e ( 4 . 1 7 ) ,

IyI < N t , dimension

n e e d t o show t h a t , f o r e v e r y

lxlprovided

(y,t)

XI2

+ 1

y12

IyI < N .

t

x

G

and t h e T h e n , we

IRn,

S CN

1

This is very simple, since f o r

1x1 > 2 N ,

we c a n

make

and f o r

1x1 5 2 N

we s i m p l y u s e

+

Ix

-

yI2 + 1

5

1x1' + 1 5 4 N 2 + 1 .

The more p r e c i s e v e r s i o n o f t h e n o n t a n g e n t i a l c o n v e r g e n c e i s as f o l lows.

Proof:

Let x b e a L e b e s g u e p o i n t f o r f . G i v e n F > 0 , by d e f i n i t i o n of L e b e s g u e p o i n t , t h e r e i s a 6 > 0 s u c h t h a t f o r e v e r y

O < r < 6

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

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

g

d e f i n e d by

XI

Iy < 6 and g(y) = 0 otherwise. t e e s t h a t Mg(x) 5 C E . Now, i f Iy -

-

g(y) = f ( y )

XI

183

f(x)

if

The i n e q u a l i t y a b o v e g u a r a n < Nt,

B u t , s i n c e y - z = x - z - ( x - y ) a n d Ix - y I < N t , inequality ( 4 . 1 7 ) a p p l i e s t o y i e l d P t ( y - z ) 5 CNPt(x - z ) Hence

.

The f i r s t i n t e g r a l i n t h i s sum i s Pt * l g l ( x ) 5 CMg(x) 6 C E . Now we c l a i m t h a t t h e s e c o n d i n t e g r a l i n t h e sum c a n a l s o b e made < E by m a k i n g t c l o s e e n o u g h t o 0 . I n d e e d , t h i s s e c o n d i n t e g r a l c a n b e b o u n d e d by

a n d we j u s t n e e d t o o b s e r v e t h a t f o r every

[ I I ~ I > ~ P ~ ( Z ) +~ 0~ Za s] ' t/ ~ 0 +

This is a very simple computation

q 2 1.

A s a c o n s e q u e n c e , we o b t a i n a n e x t e n s i o n t o o u r c o n t e x t o f F a t o u ' s

c l a s s i c a l theorem ( s t a t e d a f t e r c o r o l l a r y COROLLARY

IRn+l

& I

X E

IRn

Proof:

1

.

4.19.

Then __

1.21

i n Chapter

I.)

u ( x , t ) b e harmonic and bounded has non-tangential limits a t almost every p o i n t

Let the function

u

t h e b o u n d a r y of

IRn+l

According t o theorem

.

4.8,

u(x,t)

=

P, * f ( x )

f o r some

11. CALDERON-ZYGMUND THEORY

184

f G L~(R")

.

ing theorems

Then t h e c o n c l u s i o n o f t h e c o r o l l a r y f o l l o w s by a p p l y 4.16 o r 4.18.

We s h a l l p r e s e n t now a l o c a l v e r s i o n o f Fatou's t h e o r e m .

This w i l l

imply, i n p a r t i c u l a r , t h a t , f o r a harmonic f u n c t i o n u ( x , t ) in IRn+ 1 t h e e x i s t e n c e o f n o n t a n g e n t i a l l i m i t s a t a l m o s t e v e r y boun+ ' dary p o i n t is e q u i v a l e n t t o t h e n o n - t a n g e n t i a l boundedness almost F i r s t o f a l l , we make p r e c i s e t h e n o t i o n o f n o n - t a n g e n everywhere. t i a l boundedness. F o r a p o i n t x G IRn a n d r e a l numbers N > 0 , h h > 0 , we s h a l l c a l l T N ( x ) t o t h e open v e r t i c a l c o n e w i t h v e r t e x x

and a p e r t u r e

N , truncated a t height

h rN(x) = { ( y , t )

G

Rn++' : [ y

-

h,

that is

XI<

Nt

and

0 < t < h}.

Then we make t h e f o l l o w i n g : DEFINITION

non-tangentially N > O

&

h

u

A function

4-20.

bounded a t %he p o i n t

> o

such t h a t

u

JRn++l i s s a i d t o b e

defined i n

x G IRn i s bounded i n

_ .

i f t h e r e a r e numbers

Tk(x),

that i s ,

such t h a t

sup{

(Y

O b s e r v e t h a t , whi e t h e n o n - t a n g e n t i a l b o u n d e d n e s s o f u a t x h r e q u i r e s o n l y t h e e x i s t e n c e o f a t r u n c a t e d c o n e r N ( x ) on w h i c h u i s bounded, f o r n o n - t a n g e n t i a l convergence one h a s t o c o n s i d e r a l l possible

apertures.

Thus, even though t h e e x i s t e n c e of a non-tan-

g e n t i a l l i m i t of u a t x c l e a r l y i m p l i e s t h e n o n - t a n g e n t i a l b o u n d e d n e s s o f u a t x , t h e c o n v e r s e i s by n o means o b v i o u s . A c t u a l l y t h e converse i s f a l s e i f one l o o k s a t an i n d i v i d u a l p o i n t x , b u t it i s t r u e i f one l o o k s a t a whole s e t of p o i n t s and d i s r e g a r d s a s e t of measure 0 . T h i s i s t h e c o n t e n t of t h e f o l l o w i n g r e s u l t , which c a n b e v i e w e d a s a l o c a l v e r s i o n o f F a t o u ' s t h e o r e m .

Let u

Rn++l. Let E c IRn and s u p p o s e t h a t u i s n o n - t a n g e n t i a l l y b o u n d e d a t e v e r y x G E . Then u h a s n o n - t a n g e n t i a l l i m i t s a t a l m o s t e v e r y p o i n t x c E . THEOREM

Proof:

4.21.

be a harmonic f u n c t i o n i n

A l l we n e e d t o p r o v e i s t h a t i f

u

is non-tangentially

bounded a t e v e r y p o i n t o f a s e t o f p o s i t i v e m e a s u r e , t h e n t h e r e i s some s u b s e t o f p o s i t i v e m e a s u r e s u c h t h a t

u

has non-tangential

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

185

l i m i t s a t every point of t h i s subset.

Indeed, once t h i s i s proved, i f u i s n o n - t a n g e n t i a l l y bounded a t e v e r y p o i n t x G E , then the s e t { x G E : u d o e s n o t have a n o n - t a n g e n t i a l l i m i t a t XI

n e c e s s a r i l y must have m e a s u r e

0.

Assume t h a t u i s n o n - t a n g e n t i a l l y bounded a t e v e r y x 6 E w i t h [ E l > 0 and l e t u s t r y t o f i n d F c E w i t h IF1 > 0 s u c h t h a t u has non-tangential limits a t every

x

F.

G

We s t a r t by making s e v -

e r a l r e d u c t i o n s f o r which t h e h a r m o n i c i t y of u d o e s n o t p l a y any r o l e . F o r t h e s e p r e p a r a t o r y s t e p s we c o u l d assume t h a t u i s merely continuous.

F i r s t of a l l , E c a n b e t a k e n t o b e compact. A l s o , s i n c e E = U E N,h,M) where E(N,h,M) = Cx G E : l u ( y , t ) I S M f o r e v e r y h ( y , t ) c T N ( x ) l and t h e u n i o n i s t a k e n o v e r a l l N , h , M p o s i t i v e and r a t o n a l ; we may p e r f e c t l y w e l l assume t h a t E c o i n c i d e s w i t h a g i v e n E(N,h,M) o r , i n o t h e r words, t h a t t h e r e a r e N > 0 , h > 0 , M > 0, such t h a t f o r every x c E : sup I l u ( y , t ) I : ( y , t ) G r N h (x)}S 5 M. The n e x t s t e p w i l l b e t o show t h a t , f o r e v e r y p o s i t i v e i n t e g e r

k > N,

t h e r e i s a set Ek c E w i t h l E k l > 0 and such t h a t f o r every x G E k , u i s bounded o v e r Tkk ( x ) w i t h a bound Mk independent of x . T h i s p r o o f w i l l be b a s e d upon L e b e s g u e ' s d i f f e r e n t i a t i o n t h e o r e m . We know t h a t i f x G E i s a Lebesgue p o i n t f o r X E , the c h a r a c t e r i s t i c f u n c t i o n of E , then

C o n s e q u e n t l y , t h i s c o n v e r g e n c e h o l d s f o r a l m o s t e v e r y x G E . The p o i n t s x G E f o r which t h i s h o l d s a r e c a l l e d d e n s i t y p o i n t s of E . Take 0 < q < 1 . A f u r t h e r r e s t r i c t i o n on r- w i l l b e imposed b e l o w . If

x

G

E

i s a d e n s i t y p o i n t , we s h a l l h a v e

Now, i f we t a k e with

IF\ > 0

6 > 0

small enough, w e s h a l l have a s u b s e t

such t h a t , f o r every

x

G

F

and e v e r y

F c E

0 < r < 6

is

11. CALDERON-ZYGMUND THEORY

186

I B ( x , r ) n E 1 , i-. S u p p o s e we h a v e a f i x e d i n t e g e r k > N . We c l a i m IB(x,r) t h a t if q i s t a k e n c l o s e e n o u g h t o 1 , d e p e n d i n g on k , a n d i f

1

s > 0

(4.22)

i s t a k e n s m a l l e n o u g h , a l s o d e p e n d i n g on h ziEI'N(~) f o r every

Ti(x) c

x

G

k,

we have:

F.

L e t u s show t h i s . T a k e x G F a n d ( y , t ) G T z ( x ) , t h a t i s : Iy < kt, t < s . We must p r o v e t h a t t h e r e i s some z G E s u c h that Iy - z l < N t (we may a s s u m e , t o s t a r t w i t h , t h a t s < h ) . If

XI

z,

t h e r e e x i s t e d no such

we w o u l d h a v e

B(y,Nt)=

k:I z - y ) < N t j c I E n \ E

i n s u c h a way a n d , c o n s e q u e n t l y , B ( y , N t ) fl B ( x , k t ) c B ( x , k t ) \ E , t h a t t h e set B(x,kt)\E c o n t a i n s a t least a b a l l of diameter N t . Thus :

if q h a s b e e n s o c h o s e n . Now, i f s i s taken so small t h a t k s < 6 , we would h a v e a c o n t r a d i c t i o n w i t h t h e s e l e c t i o n o f 6 d e p e n d i n g on o u r

F

s,

(4.22)

q.

I t follows t h a t , with our choice of

h o l d s and hence, f o r e v e r y

x

G

F

and e v e r y

and and

F

(y,t)

G

I

TE(x), l u ( y , t ) S M. Suppose t h a t F c B ( 0 , R ) a n d t a k e Mk2 M 2 a n d a l s o M 2 s u p { l u ( y , t ) I : IyI 2 R + k , s 5 t I k l ( s i n c e u i s k c o n t i n u o u s , i t w i l l b e b o u n d e d on t h e c o m p a c t s e t E ( o , R + k 2 ) x x [s,k] c IRn++l 1 . Then f o r e v e r y x G F a n d e v e r y ( y , t ) G Tkk ( x ) , s o t h a t , b y s e t t i n g Ek = F , we h a v e t h e s e t we l u ( y , t ) I 5 Mk, were l o o k i n g f o r .

G

O b s e r v e t h a t , g i v e n k > N a n d E > 0 , t h e s e t Ek c a n b e t a k e n such t h a t IE\EkI < E. I n d e e d , o n c e i- h a s b e e n s e l e c t e d , we r e a l i z e t h a t t h e set

Eo

of d e n s i t y p o i n t s o f

E

c a n be w r i t t e n a s

m

U A. j=1 J

where

IA~I

i s an i n c r e a s i n g f a m i l y , J E = ~J E~ ~ a s j -t a. I Thus by t a k i n g 6 > 0 small e n o u g h , s a y 6 = l / j , F = A . can be 1 t a k e n w i t h m e a s u r e a s c l o s c t o /El a s w e w i s h . Since

(A.)

-f

Our f i n a l p r e p a r a t o r y s t e p w i l l b e t o r e a l i z e t h a t t h e r e i s a :>et

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

G c E with c o n e r kk ( x ) ,

187

IGI > 0 such t h a t f o r every x G G and every truncated u i s b o u n d e d on r kk ( x ) w i t h a bound Mk d e p e n d i n g

on k , b u t n o t on x G G f o r f i x e d k . A c t u a l l y , g i v e n E > 0 , G c a n b e t a k e n i n s u c h a way t h a t I E \ G I < E . A l l w e have t o do i s t o consider f o r each integer k > N , a s e t Ek c E s a t i s f y i n g , a s i n t h e p r e v i o u s p a r a g r a p h t h a t u i s bounded on U { Tkk ( x ) : x G E k I and a l s o IE\ E k l < 2 - k ~ . Then, w e t a k e G = kfNEk so t h a t

k, u

and,for every

i s b o u n d e d on

U { rkk ( x ) : x

G

GI.

Once t h e s e p r e p a r a t o r y s t e p s h a v e b e e n t a k e n , o u r t a s k h a s b e e n r e d u c e d t o t h e f o l l o w i n g : We h a v e a c o m p a c t s e t E c J R n , and t h e h r e g i o n R = u T N ( x ) f o r some N > 0 a n d h > 0 f i x e d . We know XG E lu(x,t) I S 1 t h a t u i s a harmonic f u n c t i o n i n R n t l such t h a t f o r every (x,t) G R. Then we m u s t show t h a t f o r a l m o s t e v e r y X G E , there exists l i m u(y,t) as ( y , t ) t e n d s t o (x,O) r e m a i n i n g always i n s i d e of j , define wise. For write @ j( x )

R.

Let u s show t h i s .

For every p o s i t i v e i n t e g e r

if ( x , l / j ) G R a n d @ j ( x ) = 0 o t h e r @j(x) = u(x,l/j) ( x , t ) G lRnttl we d e f i n e Q j ( x , t ) = P t * @ j ( x ) , a n d we

u(x,t t l/j) = $.(x,t) t $.(x,t). The s e q u e n c e o f f u n c t i o n s 1 1 i s bounded i n Lm(IRn) a n d , c o n s e q u e n t l y , it h a s a sub-

sequence

converging i n t h e weak-* topology of

(@j

to a

Lm

we d e f i n e @ ( x , t ) = f u n c t i o n @ G L m ( R n ) . F o r ( x , t ) G IRnt+' P t * @ ( x ) . The weak-* c o n v e r g e n c e i m p l i e s t h a t @ j l ( x , t )- + @ ( x , t )

=

u(x,t) and f o r e v e r y ( x , t ) G IRnt+' . S i n c e u ( x , t + l / j ) ~ + ~ , ( x , +t )@ ( x , t ) , i t f o l l o w s t h a t , a l s o $ . l ( x , t ) $(x,t) for 1 e v e r y ( x , t ) G IRnttl, a n d w e h a v e : u ( x , t ) = @ ( x , t ) t $ ( x , t ) . Now, -f

+

since

I$(x,t)

=

Pt

* @ ( x ) and

@

G

Lm(IRn)

,

the ordinary

Fatou

implies t h a t @ ( y , t ) has non-tangential theorem ( c o r o l l a r y 4.19) l i m i t s a t a l m o s t e v e r y p o i n t x G IR" . We s h a l l p r e s e n t l y s e e t h a t f o r almost every x f E , $ ( y , t ) converges t o 0 as ( y , t ) (x,O) remaining i n s i d e of $2. This w i l l f i n i s h t h e proof of t h e theorem. Write t h e b o u n d a r y o f R a s a R = E U B w h e r e B = I ( x , t ) f a R : t > O I . +

We s h a l l c o n s t r u c t a f u n c t i o n

a)

H

b)

H 2 0 i n IR"" H t 2 i n B , and For almost e v e r y x tangentially.

c) d)

i s harmonic i n

H(x,t)

with the following properties:

IRn+tl

6

E

is

H(y,t)

+

0

as

(y,t)

+

(x,O)

non-

11. CALDERON-ZYGMUND THEORY

188

Assuming f o r a moment t h e e x i s t e n c e o f

we s h a l l s e e t h a t l $ ( x , t ) I 5 H ( x , t ) f o r e v e r y ( x , t ) G R , f r o m which t h e c o n v e r folgence $ ( y , t ) + 0 a s ( y , t ) + (x,O) remaining i n s i d e of R , lows i m m e d i a t e l y f o r a l m o s t e v e r y x G E . Now, t o p r o v e t h a t H,

I $ ( x , t ) ( 5 H ( x , t ) f o r e v e r y ( x , t ) G R , we j u s t n e e d t o s e e t h a t , f o r a g i v e n ( x , t ) 6 R , and f o r an a r b i t r a r y E > 0, w e have Ibj(x,t)

I

I H(x,t) +

E

for

b i g enough.

j

If

l / j < h,

call

where B . = I ( x , ~ )G an. : t > 0 1 . J J

we h a v e , f o r j b i g e n o u g h , ( x , t ) G R . a n d J so t h a t l u ( x , t + l / j ) I 5 1 . Also, 1 , and hence I J l j ( x , t ) I 5 2 , and t h i s h o l d s a l s o f o r We know t h a t 2 5 H ( x , t ) f o r ( x , t ) G B . T h u s , by g i v e n E > 0 , we s h a l l have 2 S H ( x , t ) + E f o r e v e r y p r o v i d e d j i s b i g enough. Thus, f o r j b i g enough H(x,t) + E f o r every ( x , t ) c B . . Also, since

Given ( x , t ) G R , ( x , t + l / j ) G ll,

I

I ~ $ ~ ( x , t )6 (x,t) G B j* continuity, (x,t) G B. IQj(x,t)

I

1 5

J

w i t h u n i f o r m c o n v e r g e n c e on c o m p a c t s e t s , i t t u r n s o u t t h a t

I

6 E for ( x , t ) G R and t s m a l l enough. Then, t h e f a c t I$j(x,t) t h a t H 2 0 i m p l i e s I $ j ( x , t ) 6 H ( x , t ) + E. Thus, given E > 0, we know t h a t f o r j b i g e n o u g h a n d f o r t o > 0 small e n o u g h , t h e

I

inequality I q j [ x , t ) 1 5 H ( x , t ) + E h o l d s i n t h e boundary of t h e s e t R t. 0 = I ( x , t ) G R . : t > t o } . But i s a subharmonic funcJ 1 t i o n a n d H + E i s h a r m o n i c . C o n s e q u e n t l y we h a v e I q j ( x , t ) l 6 6 H ( x , t ) + E a l l through Rto and, since t h i s i s v a l i d f o r a l l j s u f f i c i e n t l y small t o > 0 , w e a c t u a l l y h a v e I q j ( x , t ) I 5 H ( x , t ) + E f o r every ( x , t ) c R . . We h a v e s u c c e e d e d i n p r o v i n g w h a t w e w a n t e d : J g i v e n ( x , t ) c Q a n d g i v e n E > 0 , we h a v e l $ j ( x , t J I 5 H ( x , t ) + E f o r b i g enough j . From t h i s w e i m m e d i a t e l y g e t I$(x,t) I 6 H(x,t) f o r e v e r y ( x , t ) G R. I t only remains t o c o n s t r u c t t h e function

H.

Let

X

be t h e charac-

189

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

t e r i s t i c f u n c t i o n o f t h e complement o f

E,

t h a t i s X = XIRn,E.

T h e n , we c l a i m t h a t , f o r a n a p p r o p r i a t e c o n s t a n t

C,

the function

H ( x , t ) = C{(Pt * X ) ( x ) + t } s a t i s f i e s t h e f o u r p r o p e r t i e s r e q u i r e d . In f a c t , a ) , b ) and d) a r e c l e a r and o n l y c ) needs t o be v e r i f i e d . For t h o s e Ch 2 2 .

(x,t)

with

B

6

The p o i n t s

(x,t)

we have

t = h, G

B

with

0 < t < h

s e t { ( x , t ) G Rn++l : d i s t ( x , E ) = N t l , 0 < t h , we h a v e B ( x , N t ) c IRn\ E

T h u s , w e j u s t n e e d t o make H(x,t) 2 2

also

f o r those

H(x,t) 5 2

i f w e make

are included i n the

so t h a t , i f (x,t) and, consequently

6

B

and

C appropriately big t o guarantee t h a t

(x,t)

with

B

6

0 < t < h.

we s e e t h a t , i f a function u(x,t) i s harmonic i n Rn+' and it i s uniformly i n then the functions ut(x) = u ( x , t ) Lp(IRn) f o r some 1 < p < m, converge t o a g i v e n f u n c t i o n f G Lp(Rn) i n t h e Lp norm a s t + 0. B e s i d e s , theorem 4 . 1 2 tells us t h a t there is pointwise c o n v e r g e n c e a l m o s t e v e r y w h e r e , i n s u c h a way t h a t , f o r a l m o s t e v e r y is f(x) = l i m u(x,t). x G R~ By c o m b i n i n g t h e o r e m

4.8

and theorem

4.l(a)

t+O

I n case u ( x , t ) with p = m o r 4.9

i s h a r m o n i c i n IRn" and i s uniformly i n p = 1 , we c a n s t i l l s e e by u s i n g t h e o r e m

and t h e n p a r t

d)

i n theorem

4.1

o r the closing sentence i n

t h i s same t h e o r e m , t h a t t h e f u n c t i o n s f

G

ut o r t o a c e r t a i n Bore1 measure

Lm(IRn)

c o r r e s p o n d i n g weak-* Thus, f o r

u

topology a s

t

harmonic, t h e f a c t t h a t

+

Lp(Rn) 4.8 or

converge t o a c e r t a i n p G M(IRn) in the

0.

u

i s uniformly i n

Lp(IRn)

11. CALDERON-ZYGMUND THEORY

190

with 1 5 p S m implies t h e convergence of t h e family ut a s t + O i n t h e s e n s e o f t e m p e r e d d i s t r i b u t i o n s ( a l l t h e k n o w l e d g e a b o u t temp e r e d d i s t r i b u t i o n s t h a t we s h a l l n e e d i s c o n t a i n e d i n S t e i n - W e i s s [Z], Chapter I , s e c t i o n 3 ) . A l s o , t h e boundary d i s t r i b u t i o n ( f o r p a s t h e c a s e may b e ) u n i q u e l y d e t e r m i n e s u . Now s u p p o s e t h a t u i s s t i l l h a r m o n i c i n IRn++l a n d u i s u n i f o r m l y i n Lp(IRn) but Then t h e o r e m 4 . 7 t e l l s u s t h a t each ut i s a bounded f u n c t i o n , h e n c e a t e m p e r e d d i s t r i b u t i o n . The q u e s t i o n a r i s e s n a t u r a l l y : Does l i m u t e x i s t i n t h e sense of tempered d i s t r i b u t i o n s ? t+O The a n s w e r i s c o n t a i n e d i n t h e f o l l o w i n g : 0 < p < 1.

THEOREM

Let u ( x , t )

4.23.

suppose t h a t

u

IR n tl

be a harmonic f u n c t i o n i n

Lp(IRn)

i s uniformly i n

f o r some

p

and

t

-

0.

l i m u t = f e x i s t s i n t h e s e n s e of t e m p e r e d d i s t r i b u t i o n s , a n d t h i s t+O l i m i t f uniquely determines u.

we assume 0 0 , we d e f i n e u 6 ( x , t ) = = u ( x , t + & ) . Each u 6 i s h a r m o n i c i n IRn++l. From t h e o r e m 4 . 7 we We c a n u s e t h i s e s t i m a t e t o o b t a i n l u ( x , t ) ) 5 CMt-n'P. know t h a t Proof:

S i n c e we know t h e t h e o r e m t o h o l d f o r

This implies,in particular,that

I t follows t h a t , f o r each

f o r each

6 >O,

u (x,t)

g r a l o f some f i n i t e B o r e 1 m e a s u r e .

6

6 > 0

1 5 p 5

a,

is

w i l l be t h e Poisson i n t e -

A c t u a l l y , s i n c e t h i s measure has

t o b e t h e weak-* l i m i t o f t h e f u n c t i o n s

x n u6(x,t)

as

t

+

0,

we

see t h a t it coincides with t h e i n t e g r a b l e f u n c t i o n u,(x) = u ( x , 6 ) . I f q > 6 > 0 , w e have u ( x ) = u ( x , q ) = u ( x , n - 6 ) = P * u,(x). rl 6 ri-6 T a k i n g F o u r i e r t r a n s f o r m s we g e t :

This allows us t o d e f i n e a function 6 > 0.

j,

$(c)

= ~ ( ~ ) e * n ~ 6c G~ I 6 R " ,,

is a well defined continuous function because t h e r i g h t

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

hand side does not depend on

6 > 0. Now, since for any

191

t > 0,

we obtain the estimate 1$(5)e-Z'151tl where we have made

N

=

i

lu(x,t) Idx 2 CMt-n((l'P)-l)= IRn n( (l/p) - 1 ) > 0. Thus 2

Ct-N

This estimate implies that $ is a slowly increasing function and, consequently, a tempered distribution. We know (see Stein-Weiss [Z]) that the Fourier transform establishes an isomorphism from the topological vector space S ' , formed by the tempered distributions, onto itself. Therefore, there is a unique f G S' such that $ = f. Now it is easy to see that ut + f in S' as t + 0. We have to see that, for every function @ in the Schwartz class s h

as t

+

0. But, by using the Fourier transform

Observe that u.

f

=

0

forces

u = 0, so that

f

uniquely determines

1_1

It is important to note that the estimate obtained for $ in the We shall above proof involves M, and it implies Il 2 C(@)M. have to appeal to this estimate in the next chapter, when we shall d e a l w i t h t h e r e a l v a r i a b l e t h e o r y o f Hardy s p a c e s .

11. CALDERON-ZYGMUND THEORY

192

5.

SINGULAR INTEGRAL OPERATORS

A l l t h r o u g h t h i s c h a p t e r we have s e e n t h e a c t i o n o f t h e Calder6n-Zyg-

mund d e c o m p o s i t i o n i n s e v e r a l s i t u a t i o n s . We s h a l l now a p p l y t h i s p o w e r f u l d e v i c e t o t h e p r o b l e m f o r w h i c h i t was o r i g i n a l l y c r e a t e d ( i n A . C a l d e r 6 n a n d A . Zygmund [ l ] ) , n a m e l y , t h e e s t i m a t i o n o f o p e r a t o r s of t h e form Tf(x) = where

P.V.

I R n Q ( y ) lyl-"f(x

-

y)dy

i s homogeneous o f d e g r e e z e r o a n d i t s i n t e g r a l o v e r t h e

Q

I t t u r n s o u t t h a t t h e same m e t h o d w o r k s f o r a unit sphere is n u l l . w i d e r c l a s s o f o p e r a t o r s w h i c h we s h a l l now d e s c r i b e : DEFINITION

Given a tempered d i s t r i b u t i o n

5.1.

K, t h e c o n v o l u t i o n

opera t o r

is c a l l e d a s i n g u l a r i n t e g r a l o p e r a t o r i f t h e f o l l o w i n g t w o c o n d i tions are satisfied:

function

(a)

i

(b)

K

K(x)

f

L"(IR") coincides i n

Eln\

{Ol

with a l o c a l l y integrabZe

s a t i s f y i n g Hbrmander ' s c o n d i t i o n :

The f i r s t c o n d i t i o n g i v e s , by P l a n c h e r e l ' s t h e o r e m , t h e i n e q u a l i t y

and t h e r e f o r e ,

T

e x t e n d s t o a bounded o p e r a t o r i n

L2(IRn)

with

h

Under t h e a s s u m p t i o n ( a ) o n l y , n o t h i n g more c a n b e norm IIKII,. s a i d a b o u t T , b u t i f we a d d ( b ) , t h e n o t h e r e s t i m a t e s c a n b e p r o v e d , a s we s h a l l s e e b e l o w . Let u s now l o o k a t t h e e f f e c t w h i c h H o r m a n d e r ' s c o n d i t i o n i s d e s i g n d t o produce: I f K G L:oc(IRn \ I O l ) and a ( x ) i s a n i n t e g r a b l e f u n c t i o n s u p p o r t e d i n a cube Q c e n t e r e d a t t h e o r i g i n and w i t h

193

1 1 . 5 . S I N G U L A R INTEGRAL

mean v a l u e z e r o , t h e n t h e c o n v o l u t i o n K * a(x)

= IQK(x =

[K(x

Q makes s e n s e f o r

8

Writing

=

%

and x 6 Q ,

a.e. x

6

y)a(y)dy =

-

y)

-

K(x1)

and i s l o c a l l y i n t e g r a b l e away from

Q

it i s obvious t h a t

QZfi,

and c o n d i t i o n

I

-

5.l(b)

1x1 > 21yl

6

Q

.

i n an a r b i t r a r y c u b e .

If

5.2.

in a c u b e

y

then implies

By t r a n s l a t i o n i n v a r i a n c e , t h e same i s t r u e i f

LEMMA

whenever

Q.

Q

K

a(x)

is supported

T h u s , we have p r o v e d

satisfies

with

S.l(b)

a(y)dy = 0,

&

a

f

L 1 ( R n ) is supported

then (with

'L Q

= QZ f i )

Before proving t h e e s t i m a t e s f o 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 , l e t u s l o o k f o r i n t e r e s t i n g e x a m p l e s t o which t h e t h e o r y c a n b e a p p l i e d . More a p p l i c a t i o n s w i l l be g i v e n i n t h e n e x t s e c t i o n . ~EXAMPLES:

( a ) We a r e g o i n g t o d e s c r i b e 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 d e f i n e d by c o n v o l u t i o n w i t h p r i n c i p a l v a l u e d i s t r i b u t i o n s .

These d i s t r i b u t i o n s K ( @ ) = p.v. where

k G Lioc(Rn growth c o n d i t i o n

K = p.v.k(x)

J

k(x)r$(x)dx = l i m

\ {O))

E+O

J

k ( x ) r$ ( x ) dx IXI>E

i s a f u n c t i o n t o which we impose t h e

( k ( x ) Idx 5 C 1

(5.3)

a c t i n t h e f o l l o w i n g manner:

(r > 0 )

(which i s n a t u r a l i n view of t h e b a s i c example m e n t i o n e d a t t h e b e g i n n i n g of t h i s s e c t i o n ) . Assuming ( 5 . 3 1 , i t i s easy t o check that

K($)

i s w e l l d e f i n e d f o r a l l Schwartz f u n c t i o n s

r$

and

K

is

11. CALDERON-ZYGMUND THEORY

194

a tempered d i s t r i b u t i o n i f

Indeed, then

if

(5.3)

and

hold,

(5.4)

with

L

=

p.v.j

k(x)dx, Ixl
+

I

k(x)@(x)dx= !L$(O) + I1 + I2 1x121

where t h e i n t e g r a l s

I,

and

I2

a r e a b s o l u t e l y c o n v e r g e n t , and

moreover

and

A l l these principal value d i s t r i b u t i o n s give r i s e t o singular i n t e -

g r a l o p e r a t o r s provided t h a t Harmander's c o n d i t i o n i s s a t i s f i e d : PROPOSITION

5.5.

and

a n d if

15.41,

Lf

k

f

1

Lloc(Rn\

K = p.v.

k(x),

T f ( x ) = I( * f ( x ) = l i m E*O

J / yI

{Ol)

s a t i s f i e s ( 5 . lib)), ( 5 . 3 1

then k(x

-

y)f(y)dy

(f

G

S(Rn))

>E

i s a singular integral operator. I n o r d e r n o t t o d e l a y t o o much t h e s t u d y o f t h e p r o p e r t i e s 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 , t h e p r o o f s of t h i s and t h e n e x t p r o p o s i t i o n

195

11.5. SINGULAR INTEGRAL OPERATORS

a r e p o s t p o n e d t o t h e end of t h e s e c t i o n . ( b ) L e t u s c o n s i d e r now t h e p a r t i c u l a r c a s e i n which i s homogeneous of d e g r e e - n , i.e.

k(x)

w i t h R homogeneous o f d e g r e e 0 ( w e s h a l l s y s t e m a t i c a l l y u s e t h e notation: x ' = x/lxl). The o p e r a t o r T o b t a i n e d by c o n v o l u t i o n w i t h k w i l l t h e n b e , wherever i t may b e d e f i n e d , d i l a t i o n i n v a r i ant: T(f6) = (TfI6 Now, c o n d i t i o n s

(5.3)

(with and

(5.4)

f 6 (x) = f ( 6 x ) ,

6 > 0)

a r e respectively equivalent t o

and R(x')da(x') = 0 where da d e n o t e s t h e u n i t sphere.

( n - 1 ) - d i m e n s i o n a l Lebesgue measure on t h e

F i n a l l y , some c o n t i n u i t y c o n d i t i o n must b e imposed on R i n o r d e r t o e n s u r e t h a t S . l ( b ) h o l d s . To t h i s e n d , we i n t r o d u c e t h e f o l l o w i n g L 1 -modulus o f c o n t i n u i t y :

PROPOSITION 5 . 6 . S u p p o s e that R i s homogeneous of d e g r e e integrable over t h e u n i t sphere w i t h i n t e g r a l equal t o zero.

L 1-Dini c o n d i t i o n

holds,

then

0 If

the

11. CALDERON-ZYGMUND THEORY

196

is a s i n g u l a r i n t e g r a l o p e r a t o r . M o r e o v e r , i f 52 is o d d , t h e n t h e e x t e n s i o n o f T 2 L2(lRn) i s g i v e n , i n terms o f t h e Fourier tranof o r m , by ( T f I 1 ( 5 ) = i ( S ) m ( S ) , w h e r e

Observe t h a t

m(S) = ( P . v . Q ( x ) l x l - " ) ' ( C )

i s homogeneous o f d e g r e e

z e r o , s o m e t h i n g w h i c h c o u l d h a v e b e e n p r e d i c t e d d u e t o t h e homog e n e i t y o f 52(x) and t h e behaviour o f t h e F o u r i e r t r a n s f o r m w i t h r e s p e c t t o d i l a t i o n s . An e x p l i c i t f o r m u l a f o r m ( S ) when R i s n o t n e c e s s a r i l y odd c a n b e f o u n d i n S t e i n [l] o r S t e i n - W e i s s [Zl. We a l s o p o i n t o u t t h a t t h e L 1 - D i n i c o n d i t i o n i s r a t h e r w e a k ; i n p a r -

XI-^

t i c u l a r , it i s v e r i f i e d i f

IR(x)

-

R(y)

I

2 C

[ X I

-

f o r some

y'Ia

a > 0. c ) I n IR1 , t h e r e i s b a s i c a l l y o n e homogeneous s i n g u l a r i n t e g r a l o p e r a t o r , c o r r e s p o n d i n g t o Q ( t ) = rr1 s i g n ( t ) ( o r k ( t ) = 1

x),

namely

This i s t h e Hilbert transform,

(Hf)-(E)

= -i

sign

which i s a l s o d e f i n e d i n

L2(IR)

by

([)f(E)

In f a c t , t h e formula of Proposition

5.6

g i v e s i n t h i s case

I n IRn , we h a v e a f a m i l y o f a n a l o g u e s o f t h e H i l b e r t t r a n s f o r m c o r r e s p o n d i n g t o R h ( x l ) = c n ( x ' - h ) , where h i s a n y f i x e d v e c t o r i n IRn , a n d t h e n o r m a l i z i n g c o n s t a n t c n i s c h o s e n ( f o r a r e a s o n w h i c h w i l l become c l e a r i n a moment) s o t h a t

f o r e v e r y u n i t v e c t o r u ( a n Advanced C a l c u l u s c o m p u t a t i o n shows t h a t c n = TI - ( n + 1 ) ' 2 r ( y ) ) . S i n c e Qh d e p e n d s l i n e a r l y on h , it s u f f i c e s t o c o n s i d e r t h e o p e r a t o r s corresponding t o t h e v e c t o r s of t h e s t a n d a r d b a s i s : h = e l , e 2 , ..., e n , n a m e l y R.f(x) = 1

P.V.

cn

J+

f ( x -y)dy

Cj = 1 , 2 ,

...,n )

11.5. SINGULAR INTEGRAL OPERATORS

which a r e c a l l e d t h e R i e s z t r a n s f o r m s . The a c t i o n of c i a l l y s i m p l e i n t e r m s of t h e F o u r i e r t r a n s f o r m :

197

R

j

i s spe-

7

(f

T h i s i s a consequence of

(5.6)

G

L'

; j = I , 2,

..., n )

and t h e f o l l o w i n g i d e n t i t y

whose p r o o f i s now, w i t h o u r c h o i c e of c n , a l m o s t t r i v i a l : We can f i x 5 with 151 = 1 , and t h e n , t h e l e f t hand s i d e i s a l i n e a r f u n c t i o n of

h,

say

such t h a t

!L(h),

lL(h)

I

5

lhl

and

L(5)

= 1;

L(h) = E - h .

t.his n e c e s s a r i l y im plies

We c a n now s t a t e t h e main r e s u l t c o n c e r n i n g s i n g u l a r i n t e g r a l o p e r a tors : THEOREM

5. 7 .

Every 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 a t i s f i e s t h e i n -

equalities

where

~

C

P'

1 2 p 2

d e p e n d s o n l y on

m,

~ & k BK ~ ~of tmh e

p,n

and on t h e c o n s t a n t s

kernel.

Thus, we can c o n s i d e r e v e r y s i n g u l a r i n t e g r a l o p e r a t o r e x t e n d e d a s a bounded o p e r a t o r from

Lp

t o i t s e l f , from

L r = I L m f u n c t i o n s w i t h compact s u p p o r t )

L 1 t o weak-L1 t o BMO.

and from

The p l a n of t h e p r o o f i s a s f o l l o w s : We o b t a i n f i r s t t h e b a s i c e s t i mate ( 5 . 9 ) . I n t e r p o l a t i n g by M a r c i n k i e w i c z ' theorem 2 . 1 1 between t h i s and t h e L 2 -boundedness o f T , the inequalities (5.8) follow f o r 1 < p 5 2 . We t h e n p r o v e a t e c h n i c a l lemma (5.11) which w i l l a l s o be u s e d l a t e r , and from such a r e s u l t , t h e Lm-BMO e s t i m a t e F i n a l l y , we u s e ( 5 . 1 0 ) and t h e i n t e r p o l a t o o b t a i n t h e remaining L P - i n e q u a l i t i e s :

w i l l e a s i l y be d e r i v e d .

t i o n theorem 2 < p<m.

3.7

11. CALDERON-ZYGMUND THEORY

198

P r o o f of

Take

f

L1

n

L2

and, given

t > 0,

l e t {Qj} b e t h e c u b e s o f t h e Calder6n-Zygmund d e c o m p o s i t i o n c o r r e s p o n d i n g t o If I a t h e i g h t t . (see 1 . 1 2 ) . T h i s decomposition of t h e underl y i n g s p a c e i n t o c u b e s a l l o w s t o d e f i n e what i s c a l l e d t h e CalderBnZygmund d e c o m p o s i t i o n , f ( x ) = g ( x ) + b ( x ) , of t h e f u n c t i o n f i n t o i t s "good p a r t " g a n d i t s "bad p a r t " b , w h i c h a r e g i v e n a s (5.9):

G

follows: (XI

IRn\ R with

fi = U Q . 3 1

(observe t h a t

-

b(x) = f ( x )

Ifi

g(x) =

I

b. J

i s supported i n b.(x) = {f(x) 3

\ I f 11 1 )

1 b.(x) j

where

6 Ct- 1

Qj

J

a n d h a s mean v a l u e z e r o ; p r e c i s e l y :

- -

To e s t i m a t e Tg we u s e t h e f a c t t h a t T i s bounded i n L' gether with the observation t h a t \g(x) \ = (f(x) \ 5 t f o r a, a n d I g ( x ) I 5 2"t f o r a l l x G R ; t h u s a.e. x

to-

4

N e x t , we o b s e r v e t h a t b j (x)

Lemma

can be a p p l i e d t o each p i e c e

5.2

o f t h e bad p a r t , b e c a u s e

4

Tb. (x) = K * b . (XI f o r a . e . x Qj 3 1 b . by t e s t f u n c t i o n s s u p p o r t e d J

( t h i s c a n b e s e e n by a p p r o x i m a t i n g 2& in Qj). Thus, if Q . = Q . J J QJ

On t h e o t h e r h a n d , s i n c e 2

b

G

L2(IRn)

c o n v e r g e i n t h e L -norm t o b lar, ITb(x)\ 5 L)Tb.(x)) a.e., J J

, t h e series

a n d Tb, and

b. 5.1 respectively.

and

LTb. J

.J

In p a r t i c u -

11.5. SINGULAR INTEGRAL OPERATORS

199

B e f o r e s t a t i n g t h e a u x i l i a r y lemma, we i n t r o d u c e a v a r i a n t of 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 which w i l l a l s o a p p e a r i n t h e sequel : NOTATION:

LEMMA

For

5.11.

1 5 p <

Let

e x i s t s f o r every

f

E

Cp

L:(Rn)

> 0.

ITf(x)

E-n

with

f

w e denote

m,

i n d e p e n d e n t of

,

M f = M(lflP)l'P,

P

so that the integral

1 < p <

Then, f o r a l l

-

E

i.e.

we h a v e

m,

IEldx 5 C M f(0) + P P

and

f

S i n c e M f i n c r e a s e s a s p i n c r e a s e s , we c a n assume t h a t P and i n t h i s c a s e , we have a l r e a d y p r o v e d t h a t T i s bounded i n L p . Set Proof:

1 < p 5 2,

f

1

= fX

tx:

IXl<E}

'

f2 =

-

1

Then 1Tf(x)

-

IEI 5 ITfl(x)I +

and w e o n l y have t o o b s e r v e t h a t , i f of

P,

[K(x p ' = -2P-1

-

y)

-

K(-y)]f(y)dy(

i s t h e d u a l exponent

11. CALDERON-ZYGMUND

200

The f i n a l s t e p t o e s t a b l i s h of

Proof

p = 2,

with all

G

is:

5.7

We o n l y n e e d t o u s e t h e p r e v i o u s lemma w i t h ( s a y ) M2f (0) 6 ( I f and

(Im

because

An

f

(5.10):

Theorem

THEORY

d e p e n d i n g o n l y on t h e d i m e n s i o n of t h e s p a c e .

,

Lr(IRn) (Tf)

#

Thus, f o r

we o b t a i n

(0) 5 A n ( C 2 + B K I

Ilfllm

and s i n c e e v e r y t h i n g i s t r a n s l a t i o n i n v a r i a n t , we a l s o have

which i s ( 5 . 1 0 ) . I f one w i s h e s t o g e t t h e same i n e q u a l i t y f o r a l l (even though t h i s i s a r a t h e r i r r e l e v a n t q u e s t i o n ) , f E L 2 fl La j u s t d e f i n e f . = fX I x : s o t h a t T f = l i m T f . ( i n L 2 ) and J j+a J

~

Il~fllBMo6 l i m

~

SUP

l

<

~

I l ~ f j l I B M o2

~

cIlfllm *

The e s t i m a t e s of Theorem 5 . 7 seem s p e c i a l l y s a t i s f a c t o r y i n L p w i t h 1 < p < m . I t i s n o t t r u e , however, t h a t 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 map L 1 ( o r L”) t o i t s e l f , and t h i s i s what makes t h e s u b s t i t u t e r e s u l t s (5.9) and ( 5 . 1 0 ) i n t e r e s t i n g . We c a n e a s i l y s e e t h i s i n t h e c a s e of t h e H i l b e r t t r a n s f o r m a p p l i e d t o t h e f u n c tion f = 6 L~ n L”:

xLalbI

I t is clear that also follows t h a t

Hf Hf

4 L”, 4

and s i n c e (but

L1

Hf

Hf(x) 6

Q

-,b-a

(weak-Ll)fl

1x1

+

m,

it

BMO).

I n o r d e r t o g e t a b e t t e r u n d e r s t a n d i n g of t h e b e h a v i o u r of o u r o p e r a t o r s i n L 1 and L m , we p o s e o u r s e l v e s two q u e s t i o n s : (Ql) Are t h e r e f u n c t i o n s f G L 1 f o r which we c a n a s s e r t

that

Tf

G

L1

f o r a l l 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?

11.5. SINGULAR INTEGRAL OPERATORS

(4.2)

201

T i s a s i n g u l a r i n t e g r a l o p e r a t o r , i s i t poss i b l e t o d e f i n e " i n a n a t u r a l way" Tf f o r a l l f G L m , s o t h a t T : Lm + BMO?.

If

Looking f o r an answer t o t h e f i r s t q u e s t i o n , we i m m e d i a t e l y f i n d a n e c e s s a r y c o n d i t i o n : I f f G L 1 i s s u c h t h a t R . f G L 1 f o r some J we must h a v e Riesz transform R . 1'

and s i n c e t h e l e f t hand s i d e i s c o n t i n u o u s , s o must be t h e r i g h t g ( 0 ) = I f = 0.

hand s i d e , which f o r c e s :

T h i s s u g g e s t s t h e p o s s i b i l i t y of u s i n g we a l s o assume t h a t

Lemma

i s supported i n a cube

f

5.2, Q.

provided t h a t In f a c t

and t h u s , a u n i f o r m e s t i m a t e w i l l be o b t a i n e d , f o r i n s t a n c e , i f llfll,

5

Let 1.

IQI

u s c o l l e c t t h e c o n d i t i o n s imposed s o f a r and g i v e

a name t o t h e f u n c t i o n s s a t i s f y i n g them: DEFINITION

cube

Q

5.12.

i s a bounded f u n c t i o n supported i n a Ilallm 5 l a ( x ) d x = 0.

An a t o m ( * )

and such t h a t

'

IQI

Now, what t h e p r e c e d i n g r e m a r k s p r o v e i s t h a t

f o r e v e r y atom C

a(x)

d e p e n d i n g o n l y on

k e r n e l of

and every s i n g u l a r i n t e g r a l o p e r a t o r n

a n d on t h e c o n s t a n t s

BK

and

T,

I)iII

with of t h e

T.

One may a r g u e t h a t

(5.13)

i s n o t a good a n s w e r t o

(Ql)

due t o

t h e v e r y s p e c i a l c o n d i t i o n s imposed t o a ( x ) . A t r i v i a l ( b u t n e v e r t h e l e s s i n t e r e s t i n g ) e x t e n s i o n c o n s i s t s i n d e f i n i n g t h e Banach

(*)

This w i l l be c a l l e d a

(1,m)-atom i n

C h a p t e r 111.

11. CALDERON-ZYGMUND THEORY

202

1 H,~(IR~I

space

f f o r some

(atomic iff

6

(Aj)

G

II'

The s e r i e s I f = 0 f o r 5e v' jearj y

H

1

I as f o l l o w s :

f(x) =

and atoms

1j A J.

a.(x> J

a j (x)

,

and

converges i n L 1 , so t h a t f G H a1 t . Now, we have

H a1t

COROLLARY 5 . 1 4 : Every s i n g u l a r i n t e g r a l o p e r a t o r H a1 t ( l R n ) boundedly i n t o L1(IRn) ,

G.

c L1,

and c l e a r l y

T

More i n s i g h t i n t o HAt w i l l b e g a i n e d i n C h a p t e r 1 1 1 . For t h e time b e i n g , l e t u s merely observe t h a t a Schwartz f u n c t i o n f 1 b e l o n g s t o Hat i f and o n l y i f I f = 0 . Turning t o

(QZ),

l e t us f i r s t observe t h a t i t i s n o t a t r i v i a l

q u e s t i o n . I n f a c t , t h e p r o o f of Theorem 5 . 7 shows t h a t we c a n a c t u a l l y d e f i n e T : Lp fl L m + BMO f o r a r b i t r a r y p my but s i n c e Lp fl Lm i s n o t d e n s e i n L m , t h i s d o e s n o t a l l o w t o e x t e n d T by c o n t i n u i t y t o a l l bounded f u n c t i o n s . F o l l o w i n g t h e argument i n t h e proof of (5.10), one would b e t e m p t e d t o d e f i n e Tf = l i m T ( f X B j ) , where B . = I x : 1x1 2 j } . Such a l i m i t , however, may n o t e x i s t i n 1

But, any r e a s o n a b l e s e n s e ( s e e t h e example below P r o p o s i t i o n 5 . 1 5 ) . s i n c e w e want t o c o n s i d e r Tf and T(fXBj) a s e l e m e n t s of BMO, we may t r y t o add a c o n s t a n t t o e a c h t e r m T(fXBj) s o a s t o make t h e c o r r e c t e d sequence converge. PROPOSITION

nel

K,

5.15.

f o r each

This, w e can achieve:

G i v e n a s i n g u l a r i n t e g r a l opera-

f s Lm

a.e.,

with ker-

L1 and a l s o p o i n t satisfies

The s e q u e n c e t o t h e r i g h t c o n v e r g e s l o c a l l y i n wise

T

we d e f i n e

and t h e e x t e n d e d o p e r a g

T

11.5. SINGULAR INTEGRAL OPERATORS

where

is t h e same c o n s t a n t of

Cm

Observe t h a t , i f

f E L,:

203

(5.10).

t h e o l d and new d e f i n i t i o n s of

Tf(x)

d i f f e r only i n a constant: I l y l > , K ( - y ) f ( y ) d y ; thus they agree a s e l e m e n t s of BMO. I n g e n e r a l , i f f E L m , we d e n o t e by Tf t h e c l a s s i n BMO r e p r e s e n t e d by t h e f u n c t i o n d e f i n e d by ( 5 . 1 6 ) . In t h i s way, we e x t e n d t h e o p e r a t o r T : L: T : Lm + BMO w i t h t h e same norm.

.+

BMO

t o an o p e r a t o r

L e t u s d e n o t e by

g . ( x ) t h e j - t h term of t h e J (5.16). Given a compact s e t F , s e q u e n c e i n t h e r i g h t hand s i d e of l e t R b e t h e f i r s t n a t u r a l number s u c h t h a t R 2 2 s u p l x l . Then, XGF f o r a l l x E F and j > II: Proof

of

(5.15):

The i n t e g r a l t o t h e r i g h t i s u n i f o r m l y bounded (when

x E F)

by

[Im,

and c o n v e r g e s when j * m t o t h e same i n t e g r a l e x t e n d e d BKIIf over a l l ( y I > R. Thus, g j ( x ) c o n v e r g e s p o i n t w i s e a n d i n L 1 ( F ) . A s a c o n s e q u e n c e , we h a v e IITflIBMOI l i m s u p j+co

where we have u s e d f e r in a constant.

I

and t h e f a c t t h a t

(5.101

u

gj

and

T(fXB.) 3

dif-

The n e c e s s i t y o f s u b s t r a c t i n g t h e c o n s t a n t t e r m s i n ( 5 . 1 6 ) can be s e e n i n t h e f o l l o w i n g i n s t r u c t i v e example: Compute t h e H i l b e r t transform of have

f(x) = sign(x).

1 g.(x) = 7 {log 1

and l e t t i n g

=

$

j *

m

1-1X -X J

l o g 1x1

?

Since

-

we f i n d :

1

log

fXB. = 3

rr2

I

+

2

-

Xc-j,ol,

we

=

+ -

X

2 .2 l o g Ix -1

Hf ( x ) =

X[o,ji

log j

l o g 1x1.

Our n e x t o b j e c t i v e i s t o g i v e more p r e c i s e e s t i m a t e s f o r some s i n g u -

11. CALDERON-ZYGMUND THEORY

204

l a r i n t e g r a l o p e r a t o r s f o r w h i c h more r e g u l a r i t y o f t h e k e r n e l i s assumed . DEFINITION

A singular integral operator

5.17.

Tf = K * f

c a l l e d r e g u l a r i f i t s k e r n e l s a t i s f i e s t h e f o l l o w i n g two c o n d i t i o n s :

Observe t h a t

(5.19)

t r i v i a l l y i m p l i e s Hbrmander's c o n d i t i o n

S.l(b),

Ix~-~-'

I

IVK(x) 6 C o n s t . and it i s a u t o m a t i c a l l y v e r i f i e d i f away f r o m t h e o r i g i n . F o r homogeneous o p e r a t o r s a s t h o s e d e s c r i b e d i n Proposition 5.6, a s u f f i c i e n t condition t o be a regular singular integral operator is:

R c C ' (IR" \ { o I ) .

The g r o w t h l i m i t a t i o n

(5.18)

enables us t o define the truncated

operators

d i r e c t l y f o r L p f u n c t i o n s f ( x ) , 1 5 p < m, a n d o n e n a t u r a l l y w i s h e s t o h a v e a more e x p l i c i t d e f i n i t i o n o f Tf a s a l i m i t o f T E f . T h u s , we a r e i n t e r e s t e d i n u n i f o r m e s t i m a t e s f o r ( T E ) E , O , a n d we a r e g o i n g t o s e e t h a t w e c a n a c t u a l l y e s t i m a t e t h e maximal

If

T

i s a regular singular integral operator and

THEOREM

5.20.

f c Lp,

1 S p <

(i)

# (Tf) (x) 5 C M f ( x ) 9 9

(ii)

T*f(x) 5 C M f ( x )

m

9

-

then the following inequalities are v e r i f i e d :

9 4

t

( 1 < 9) CM(Tf)(x)

( 1 < 9)

By t r a n s l a t i o n i n v a r i a n c e , we o n l y n e e d t o p r o v e t h e p o i n t wise e s t i m a t e s ( i ) a n d ( i i ) a t x = 0 . The b a s i c p o i n t i s t h a t Proof:

we c a n u s e Lemma

5.11

( w h i c h h o l d s now f o r e v e r y

f

G

Lp,

since

11.5. SINGULAR INTEGRAL OPERATORS

205

was o n l y imposed t o e n s u r e t h e e x i s t e n c e o f t h e a s s u m p t i o n f 6 L: w i t h a n i m p r o v e d e s t i m a t e f o r t h e l a s t term, n a m e l y IE)

Since

Mf 2 M f 9

which i m p l i e s IE = TEf(0),

f o r every

we o b t a i n

q > 1,

( i ) ( a t x = 0 ) . On t h e o t h e r h a n d , w e o b s e r v e t h a t and it a l s o follows t h a t

a n d t a k i n g t h e supremum o v e r a l l

E

> 0 ,we g e t

(ii).

which i s a consequence of t h e s t r o n g e r L e t u s now p r o v e ( 5 . 2 1 ) , In f a c t , r e g u l a r i t y p r o v i d e d by ( 5 . 1 9 ) . m

r

(

5

5

Finally,

(iii)

is a t r i v i a l consequence of

(ii) and t h e

Lp

i n e q u a l i t i e s f o r t h e o p e r a t o r s M and T , and t h e proof of (iv) i s p o s t p o n e d t o C h a p t e r V , where i t w i l l a p p e a r i n a n a t u r a l context.

0

COROLLARY

(5.4)

,

5.22.

(5.18)

Tf = ( p . v . k ( x ) ) * f 5.5)

satisfies

Let k

L~oc(IRn\.03) be a f u n c t i o n s a t i s f y i n p (5.19). Then, the singular integral operator G

( w h o s e e x i s t e n c e i s g u a r a n t e e d by P r o p o s i t i o n

11. CALDERON-ZYGMUND THEORY

206

F o r a l l f G S(IRn) , we know t h a t l i m T , f ( x ) = T f ( x ) a t e v e r y x G IR". S i n c e S(IRn) i s d e n s e inE+' L P ( R n ) , t h e r e s u l t f o l l o w s f r o m 5 . 2 0 ( i i i ) a n d ( i v ) by a s t a n d a r d t e c h n i q u e a l r e a d y

Proof:

used t o d e r i v e t h e d i f f e r e n t i a t i o n theorem f o r 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 .

1.9

from t h e e s t i m a t e s

To c o n c l u d e t h i s s e c t i o n , we remember t h a t t h e r e s u l t s n e e d e d t o e s t a b l i s h some e x a m p l e s 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 were s t a t e d w i t h o u t p r o o f s . Here we p r o v i d e t h e p r o o f s :

K = p.v.k(x).

We must o n l y show t h a t 1? G L m ( l R n ) Let u s d e f i n e t h e t r u n c a t e d k e r n e l s

Since

R kE = K

P r o o f of

15.51:

lim

,

w he re

( i n t h e s e n s e of t e m p e r e d d i s t r i b u t i o n s ) ,

E+O, R+m

i t s u f f i c e s t o p r o v e t h a t t h e F o u r i e r t r a n s f o r m s of { k:lo < < a r e u n i f o r m l y bounded. We b e g i n by o b s e r v i n g t h a t H o r m a n d e r ' s condition t r a n s f e r s t o the truncated kernels i n the following sense:

and t h e r e f o r e ,

Now, we f i x

5

G

IRn

a n d decompose

11.5. SINGULAR INTEGRAL OPERATORS

by t h e p r e c e d i n g o b s e r v a t i o n .

On t h e o t h e r h a n d ,

( t h i s t y p e of i n e q u a l i t y i s a c t u a l l y e q u i v a l e n t t o

Proof of show t h a t

(5.6):

(5.3)

IxI-~

implies

(5.3))

The f i r s t a s s e r t i o n w i l l f o l l o w f r o m

k(x) = Q(x)

it does, because

207

and thus

(5.5)

s a t i s f i e s Hormander's c o n d i t i o n .

if we But

11. CALDERON-ZYGMUND THEORY

208

m(6)

F o r t h e s e c o n d a s s e r t i o n , we h a v e t o p r o v e t h a t

ier transform of

K = p.v.(R(x)

[XI-").

Since

k

i s t h e Four-

i s odd,

The i n n e r i n t e g r a l i s a l w a y s b o u n d e d i n a b s o l u t e v a l u e by

i t s l i m i t when E + 0 , desired expression f o r

R

+

m,

is

7I Ts .i g n ( x ' . ~ ) .

n,

and

This gives the

m(S).

MULTIPLIERS

6.

I n t h e p r e c e d i n g s e c t i o n we o b t a i n e d

Lp

i n e q u a l i t i e s f o r convolu-

t i o n o p e r a t o r s whose k e r n e l s a r e n o t i n t e g r a b l e . k

G

L1(IRn)

,

then

Tf = k * f

Of c o u r s e , i f is also a singular integral operator,

since

(Bk

denoting t h e c o n s t a n t i n Hormander's condition

t h i s c a s e , Young's i n e q u a l i t y g i v e s

1 6 p 6

m,

and theorem

Lp

S.l(b)).

In

estimates for a l l

namely

5.7

seems t o be o f no

i n t e r e s t here.

T h e r e i s , how-

e v e r , o n e p o i n t i n t h a t t h e o r e m w h i c h ma ke s i t u s e f u l e v e n f o r t h i s simple s i t u a t i o n , and it i s t h e f a c t t h a t , f o r a f i x e d 1 < p <

11c11,

m,

and

p

with

t h e e s t i m a t e of I l k * f l l p d e p e n d s o n l y on t h e c o n s t a n t s Bk, w h i c h may b e c o n s i d e r a b l y s m a l l e r t h a n IIklIl.

To p u t i t i n a more p r e c i s e f o r m : I f w e a r e g i v e n a f a m i l y o f k N 6 L 1 , a n d we w s h t o o b t a i n u n i f o r m e s t i m a t e s f o r

kernels

Ilp

IlkN*f by means o f Young s i n e q u a l i t y , we a r e f o r c e d t o i m p o s e the strong condition s u p I k N I I l < m . H ow e ve r, t h e o r e m 5 . 7 h a s N

11.6. MULTIPLIERS

209

t h e f o l l o w i n g c o n s e q u e n c e w h i c h w i l l b e u s e d f o r o u r main r e s u l t b e low: LEMMA

Let

6.1.

(kd

be a sequence o f k e r n e l s i n

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

C > 0

lkN(x-y)

b)

-

kN(x) ldx S C

&

L1(IRn), N = 1,2,

such t h a t , f o r a l l

(y

...

IR")

f

.. .

Then, T f = kN*f, N = 1 , 2 , , are uniformly bounded operators from N Lp t o i t s e l f ( 1 < p < a ) , from L 1 t o w e a k - L 1 a n d f r o m Lm to

BMO

.

This suggests a d i f f e r e n t (though equivalent) approach t o 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 d e f i n e d by p r i n c i p a l v a l u e d i s t r i b u t i o n s . Indeed, under t h e hypothesis of Proposition 5.5, the kernels k = N s a t i s f y ( a ) a n d ( b ) of t h e p r e c e d i n g lemma. = k X ~ xI :/ N S ~ S N I Therefore IIkN

with limit

C

*

flip

' CplIfllp

independent of

P

Tf

=

(f

N = 1,2

l i m kN * f

,..., (in

6

LP; 1 < p <

and f o r a l l

f

G

w)

Lp,

the

LP-norm)

N+W

exists (because it obviously e x i s t s f o r Schwartz f u n c t i o n s ) d e f i n i n g 1 < p < w, and which c e r an o p e r a t o r T which i s bounded i n L p , t a i n l y a g r e e s w i t h t h e u n i q u e bounded e x t e n s i o n t o Lp of t h e ft+ (P.v. k(x))*f, i n i t i a l l y d e f i n e d i n convolution operator: S(IR").

T h i s i s n o t , h o w e v e r , o u r r e a s o n f o r s t a t i n g lemma 6 . 1 . h e r e . We aim a t a d i f f e r e n t a p p l i c a t i o n o f t h e Calder6n-Zygmund method w h i c h gives sufficient conditions f o r Fourier multipliers in DEFINITION

6.2.

Let

1 5 p <

a.

Given

i s a ( F o u r i e r ) m u l t i p l i e r f o r Lp d e f i n e d i n L2(IRn) by t h e r e l a t i o n

m

m

f

Lm(IRn)

if t h e o p e r a t o r

Lp.

, we s a y t h a t Tm,

initiaZZy

11. CALDERON-ZYGMUND THEORY

210

satisfies the inequality

(f E L2

llTmflIp 5 C l l f l l P In t h i s ease,

Tm

h a s a u n i q u e bounded e x t e n s i o n t o

n

LP)

Lp

w h i c h we

Tm.

s t i l l denote by

Some g e n e r a l f a c t s a b o u t m u l t i p l i e r s a r e c o n t a i n e d i n 7 . 1 5 a n d a t t h e

m

By P l a n c h e r e l ' s t h e o r e m , e v e r y

end of t h i s s e c t i o n .

6

Lm

is a

L 2 , a n d t h e n o r m o f T~ a s a n o p e r a t o r i n L' is The f a c t t h a t t h e R i e s z t r a n s f o r m s a r e b o u n d e d Lp, c a n b e r e s t a t e d by s a y i n g t h a t m j ( t ) = [ . / I F 1

multiplier for e q u a l t o Ilrnll,. operators in (1 = 1 , 2

,...,n )

are multipliers for

Lp

if

1 < p <

-.

1

The Horman

d e r - M i h l i n m u l t i p l i e r t h e o r e m p r o v i d e s a g e n e r a l i z a t i o n of t h i s : THEOREM

n 7 .

If

Let

6.3.

m

G

{R-"

(6.4)

+ 1

be t h e f i r s t i n t e g e r g r e a t e r than

ca ~outside the

o r i g i n and s a t i s f i e s

Ill< I x I < 2 R

for every multi-index

LP, 1

p l i e r for

[y]

a =

L - ( R ~ ) is of c l a s s

< p

c1

-.

such t h a t

10.1

5 a,

then

m

is _a m ti- ~ _u l _

F o r many a p p l i c a t i o n s , o n e m e r e l y v e r i f i e s t h e c o n d i t i o n

which i s s t r o n g e r t h a n

m([) (with

When

of c l a s s

Ca

b E IR)

b = 0

,

(6.4).

6 - n f ($1

ib

i.e.

(which i s t h e most i n t e r e s t i n g c a s e ) ,

is dilation invariant, i.e., =

T h i s i s s a t i s f i e d by e v e r y f u n c t i o n

o u t s i d e t h e o r i g i n a n d hom oge ne ous o f d e g r e e

'Tm(fg) = ( T m f ) 6 ,

the operator

where

f6(x)

=

.

We w i s h t o p r o v e t h e t h e o r e m by r e d u c i n g i t t o a n a p p l i c a t i o n O K

Lemma

6.1.

Two s t a n d a r d t e c h n i q u e s a r e n e e d e d f o r t h i s r e d u c t i o n .

m

11.6. MULTIPLIERS

211

The f i r s t o n e i s t h e smooth c u t t i n g o f t h e m u l t i p l i e r i n t o d y a d i c p i e c e s , which i s b a s e d on t h e f o l l o w i n g LEMMA

T h e r e i s a non n e g a t i v e f u n c t i o n

6.5.

15

i n the spherical shell

Take

Proof:

@ 6

@(El

such t h a t

Cm

> 0

:

4 < 151 < 2 1

supported i n when

1 6

J2

$

supported

such t h a t

4 < 151 < 2 ,

151 S J Z .

c"(Rn)

6

non n e g a t i v e a n d

Then,

i s a Cm f u n c t i o n s t r i c t l y p o s i t i v e a t e v e r y 5 f 0 a n d s u c h t h a t 00(25) = O o ( < ) . Thus, $ ( 5 ) = @ ( 5 ) / @ o ( 5 ) s a t i s f i e s t h e r e q u i r e d

0

conditions.

The s e c o n d p r e v i o u s r e s u l t i s a n e x p l i c i t f o r m u l a t i o n o f t h e w e l l known f a c t t h a t t h e r e g u l a r i t y o f t h e m u l t i p l i e r i s t r a n s l a t e d i n t o c o n t r o l of t h e s i z e of t h e k e r n e l : LEMMA

Let

6.6.

(so that

s

a = 21. I-f

6

n + 1 + 1 , and l e t s b e s u c h t h a t a = s 2 k 6 L2(Rn) i s such t h a t i s of c l a s s C a ,

];[

then, -

Since

Ixla 6 C(lxl

la+ ...

+ I x n l a ) , and s i n c e t h e F o u r i e r i s (-2niy a Da^ .k(t) ( w i t h D . = 9/85.), we u s e 1 1 1 1 Hausdorff-Young's i n e q u a l i t y t o g e t Proof:

t r a n s f o r m of

xak(x)

We c a n now p r o c e e d t o p r o v e t h e m u l t i p l i e r t h e o r e m .

Proof of function

6.3: $

of

W e u s e t h e p a r t i t i o n o f t h e u n i t y d e f i n e d by t h e 6.5

i n o r d e r t o decompose t h e m u l t i p l i e r a s

11. CALDERON-ZYGMUND THEORY

212

s u p p ( m j ) c 1 5 : Z j - ’ < l c l < Z j + l } , and t h a t we c a n c o n t r o l t h e s i z e o f t h e d e r i v a t v e s o f m. a s f o l l o w s : J

Observe t h a t

llDa m j / I s

(6.7)

6 C Zj(-I’

I n d e e d , by t h e L e i b n i t z r u l e o f d i f f e r e n t i a t i o n , i f

and i t i s now t h a t t h e h y p o t h e s i s

(6.4)

la1 5 a

is used, together with

Holder’s inequality, t o obtain

from which

(6.7)

follows.

Now

o u r p l a n i s t o a p p l y lemma = k . ( x ) , with -N 3

$

k.(x) = i . ( - x ) = 3 J Since

Llc.(S)

J

J

=

Im(S)

I

t o the kernels

6.1

kN ( x ) =

m j (SIdS B

Lm(Rn)

,

we o n l y have t o p r o v e

I f we a r e a b l e t o d o s o , t h e o p e r a t o r s

TNf ( x ) w i l l be u n i f o r m l y bounded i n

( N = 1, 2,

Lp,

1 < p <

m,

and s i n c e

1;. J

=

...) m.

1’

11.6. MULTIPLIERS

TN

i s t h e o p e r a t o r d e f i n e d by t h e m u l t i p l i e r N

N

N m (5) =

1

which c o n v e r g e s t o Tmf

+

(in

l i m inf N - t m

L2)

I

m. ( 5 ) = m ( 6 )

j=-N

TNf

213

m(5)

when

N

f o r every

IT N f ( x )

$(2-j~)

j=-N

J

-

+

f

f

L

I

=

0

Tmf(x)

By P l a n c h e r e l ' s t h e o r e m ,

w.

2

,

and i n p a r t i c u l a r a.e.

Then, F a t o u ' s lemma g i v e s , f o r e v e r y

f

Thus, m a t t e r s a r e reduced t o p r o v i n g

(6.8).

G

L 2 fl L p

For a f i x e d

y c IR",

we s h a l l s e p a r a t e t h e sum i n t o two p a r t s : W

Ik.(x-y) I

J=-

-

k.(x)\dx=

1

jGI

1

+

1

jcII

I = { j : Z1lyl 2 1 ) and I 1 = { j : 2 J 1 y l < 1 ) . For t h e t e r m s i n t h e f i r s t sum, w e u s e t h e t r i v i a l m a j o r i z a t i o n

where

(x-y)

which i s b a s e d on

-

k . ( x ) Idx 5 3

and

(6.6)

(6.7)

(with

s

such t h a t

a-fls

1 2

= -).

Therefore

For t h e terms corresponding t o k . (x J

-

y) 2.

-

k . (x) 3

j

G

must p l a y a r o l e .

k.(x) = k.(x-y) I 3

-

11,

t h e c a n c e l l a t i o n of

Define

kj(x)

Our t a s k c o n s i s t s now i n o b t a i n i n g a n e s t i m a t e f o r t h e d e r i v a t i v e s 2. of m . which i s good e n o u g h , namely 1

11. CALDERON-ZYGMUND THEORY

214

j ( 1 - a +n/s) sup \\Da%.\\ 5 C ) y \ 2 la =a I S

(6.9)

(l5s52; 2’1y)
I

5

To p r o v e t h i s , we f i r s t o b s e r v e t h a t , f o r a l l

I Dl ( e - 2 n i y .

5 -1)

( t h i s i s o b v i o u s when

cly12J(1-l~l)

y = 0

because

IDy(e-2niY‘S 1

we h a v e

IyI > 0

I

I

=

‘L

G

supp(m.)

1

(j

Iy.[I

5 2j”

f

Iy

I ( 2 . r r ~ ) ~2 I Clyl’).

11)

I,

a n d when By t h e L e i b -

nitz rule sup

lal=a

ID

c42’

m.(c) 1

1

5

c IY

and i n s e r t i n g t h e e s t i m a es b t a i n e d i n ( 6 . 7 ) f o r D’m. we h a v e 3’ proved ( 6 . 9 ) . Now, we a r e i n p o s i t i o n t o i n v o k e a g a i n ( 6 . 6 ) , which g i v e s

and t h e r e f o r e ,

Under t h e h y p o t h e s i s o f type

(l,l),

6.3,

the operator

Tm

i s a l s o o f weak

and s a t i s f i e s

I t i s n a t u r a l t o a s k whether t h e l a s t i n e q u a l i t y c a n be improved t o o b t a i n t h e same k i n d o f e s t i m a t e t h a t r e g u l a 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 s a t i s f y (see

5.20(i)).

We s h a l l now p r o v e , w i t h a l m o s t

n o a d d i t i o n a l e f f o r t , a more p r e c i s e f o r m o f h i g h e r smoothness of t h e m u l t i p l i e r of t h e o p e r a t o r

Tm.

6.3

w h i c h shows how

implies higher regularity

III

In p a r t i c u l a r , the operators described in the

l a s t theorem always s a t i s f y : ( T m f ) # 5 C M Z f , and t h e y behave e x a c t l y a s r e g u l a 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 f we a s s u m e t h e d e c a y ( 6 . 4 ) f o r a l l t h e d e r i v a t i v e s of o r d e r 5 n

THEOREM

6.10.

__ Let

a

&an

of

m(t).

i n t e g e r such t h a t

n < a 5 n,

arid

__

11.6. MULTIPLIERS

m(S)

suppose that the multiplier Then, for e v e r y

la1 2 a .

q >

n

--,

215

satisfies the operator

(6.4)

Tm

f o r all satisfies

Proof: We u s e t h e same n o t a t i o n a s i n t h e p r o o f o f 6 . 3 . f i c e s t o o b t a i n t h e p o i n t w i s e estimate f o r e v e r y o p e r a t o r

I t suf-

N

N T f ( x ) = kN * f ( x ) =

1

k. * f(x)

j=-N

J

(with C independent of N) and t h i s w i l l b e a n e a s y consequence 9 of t h e c r u c i a l i n e q u a l i t y f o r t h e k e r n e l s kN:

where denote (6.11)

E >

0

w i l l b e f i x e d below.

t . = 2 J Iy

1 that

1,

In fact, given

and f o r an a r b i t r a r y

f,

y E IR",

we

it follows from

$ 7

j=1 m

5

cly

1 t ;-E - n / q t n 'q3M

j=1

9

f ( 0 ) 5 C'Mqf(0)

This i s a s u b s t i t u t e , f o r our p r e s e n t problem of t h e i n e q u a l i t y (5.21),

a n d t h e r e s t of t h e p r o o f i s a s i n t h e o r e m

5.2O(i).

Now, i n o r d e r t o e s t a b l i s h ( 6 . 1 1 1 , g i v e n q , we t a k e s 5 2 s u c h n that s c q . We c a n a l s o a s s u m e t h a t E = a - < 1 , and . t h i s S i s o u r c h o i c e of E . The v a r i a n t o f lemma 6 . 6 w h i c h w e n e e d h e r e is as follows: I f k G L2 ( I R n ) and i(5) i s of c l a s s C a , then

11. CALDERON-ZYGMUND THEORY

216

The p r o o f i s e n t i r e l y s i m i l a r , b u t H o l d e r ' s i n e q u a l i t y i s a p p l i e d i n

t o order

\Ir

5 11. 1 1 - ( I s l , w i t h -r = -s - -q ' remain t o h o l d f o r t h e d e r i v a t i v e s of m

ll-l\ql

t h e form: and (6.9)

a,

if

t 2

IyI

, we

j

have

T h e r e f o r e , t h e l e f t hand s i d e of

Since (6.7) 'L and m . up 1

(6.11)

i s m a j o r i z e d by

ii

and t h e proof is ended.

To c o n c l u d e t h i s s e c t i o n ,we l i s t some e l e m e n t a r y p r o p e r t i e s o f multipliers:

1 < p <

m

is a muZtipZier f o r

Lp

if and

only if it is a multiplier f o r L p ' , and t h e norms of o p e r a t o x L p and on L p ' are identical.

Tm

as an

Let

(6.22).

m.

T h i s f o l l o w s from t h e i d e n t i t y jTmf(xl (f

6

L2

n

Lp,

n i t i o n of (6.13).

then

m

Tm

If

g(x) dx

=

1

-_.

f ( x ) T m g ( x ) dx

w h i c h i s o b t a i n e d by u s i n g t h e d e f i g f L2 n L p ' ) , and P l a n c h e r e l ' s theorem.

m

is a muZtiplier for

is a multiplier=

The c o n d i t i o n on

q

Lp,

and if

I-1

~ 9 .

means t h a t , e i t h e r

-

9

p 6 q 5 p'

- 11

6

2

or

I-1

- - 11

P

2

p' 5 q S p.

The r e s u l t i s a c o n s e q u e n c e o f t h e p r e c e d i n g s t a t e m e n t a n d

11.6. MULTIPLIERS

217

Marcinkiewicz' i n t e r p o l a t i o n theorem,

2 , l l . . I f one u s e s i n s t e a d t h e (see 7 . 5 b e l o w ) , i t f o l l o w s t h a t i s n o t g r e a t e r t h a n t h e norm i n L p . In

Riesz-Thorin i n t e r p o l a t i o n theorem t h e norm o f

Tm

in

particular, taking

Lq

q = 2,

we a l w a y s h a v e

The m u l t i p l i e r s f o r

(6.14).

Lp

Ilmll,.

IITmllp,p 2

Lm(Rn)

f o r m a s u b a l g e b r a of

w h i c h i s i n v a r i a n t u n d e r t r a n s l a t i o n s , r o t a t i o n s and d i l a t i o n s .

f = T

( T f ) . The ml m2 i n v a r i a n c e p r o p e r t i e s f o l l o w from t h e b e h a v i o u r of t h e F o u r i e r t r a n s form w i t h r e s p e c t t o a f f i n e t r a n s f o r m a t i o n s : A(x) = h + L ( x )

The f i r s t a s s e r t i o n i s o b v i o u s , b e c a u s e

(L l i n e a r ) ,

which i s very simple

"v

(with

L = (L*)-').

(6.15).

all 1 -

c

The f o l l o w i n g f u n c t i o n s a r e m u l t i p l i e r s f o r p c m:

m.(S) = s i g n ( S . 1 3 J

where

P

,

j = l,Z,

i s a (bounded o r unbounded) polyhedron i n

LP(IRn)

for

...,n

IRn

, h., F

i s t h e i n t e r s e c t i o n of a f i n i t e number of h a l f - s p a c e s

In f a c t , the multiplier

m, c o r r e s p o n d s t o t h e H i l b e r t t r a n s f o r m a c t i n g on t h e f i r s t v a r i a b l e : Tm f ( x 1 , x 2 , 1

... '

xn ) = i H ( f ( . , x 2 ,

which i s bounded i n L p ( I R n ) , 1 < p and F u b i n i ' s t h e o r e m . S i m i l a r l y f o r

m,

m. 1'

..., x n ) ) ( x l ) by t h e b o u n d e d n e s s of 2 5 j 5 n.

H

C o n c e r n i n g XF, i t s u f f i c e s t o p r o v e t h e r e s u l t f o r t h e c h a r a c t e r i s t i c f u n c i t o n of a h a l f - s p a c e (because t h e product of m u l t i p l i e r s is a g a i n a m u l t i p l i e r ) , a n d by t r a n s l a t i o n and d i l a t i o n i n v a r i a n c e , we

11. CALDERON-ZYGMUND THEORY

218

o n l y have t o prove it f o r

s = s eo

= { c ~ I R ": t , = ~ * e ~ s 0 1

1

Xs

But

7.

7.1

1

= z(l

+ ml)

.

NOTES AND FURTHER RESULTS

.-

The d e f i n i t i o n o f t h e maximal f u n c t i o n i n

IR"

i s i n Wie-

n e r [l]. I t i s t h e n - d i m e n s i o n a l a n a l o g u e of t h e maximal f u n c t i o n d e f i n e d by Hardy a n d L i t t l e w o o d [l] f o r f u n c t i o n s i n t h e t o r u s T . The c l a s s L l o g L was i n t r o d u c e d by Zygmund t o g i v e a s u f f i c i e n t c o n d i t i o n f o r t h e l o c a l i n t e g r a b i l i t y of i s a c t u a l l y necessary (theorem 2.7) 7.2.-

Mf.

That t h i s condition

was o b s e r v e d by S t e i n [4].

T h e r e a r e many v a r i a n t s o f t h e maximal f u n c t i o n i n

IR".

It

i s e q u i v a l e n t , f o r i n s t a n c e , t o consider b a l l s i n s t e a d of cubes. More g e n e r a l l y , g i v e n a f a m i l y o f s e t s F w h i c h c o v e r s IRn i n t h e

" n a r r o w s e n s e " ( i . e . , e v e r y x c IRn belongs t o s e t s a r b i t r a r i l y small d i a m e t e r ) , we can d e f i n e MFf(x) =

F c P

of

SUP

I f e v e r y s e t F E F c a n b e p a c k e d w i t h i n two c u b e s : Q1 c F c Q 2 C = fixed constant, i n s u c h a way t h a t d i a m (Q,) S C d i a m ( Q 2 ) , t h e n MF a n d M a r e e q u i v a l e n t , i n t h e s e n s e t h a t C-"Mf(x)

5 M,f(x)

5 CnMf(x)

This i s not the case f o r the family

i?

o f a l l bounded n - d i m e n s i o n a l

MRf i s c a l l e d t h e i n t e r v a l s : R = [ a l , b l ] x [aZ,b2] x . . . x [an,bI,]. s t r o n g m a x i m a l function a n d c o n t r o l s t h e d i f f e r e n t i a t i o n w i t h r e s p e c t t o a r b i t r a r y i n t e r v a l s i n IRn , a s w e l l a s t h e u n r e s t r i c t e d s u m m a b i l i t y o f m u l t i p l e F o u r i e r s e r i e s ( s e e Zygmund [I], Chap. X V I I ) . I t was f i r s t s t u d i e d by J e s s e n , M a r c i n k i e w i c z a n d Zygmund [l], who proved t h a t M i s a bounded o p e r a t o r irom Lp(IRn) t o i t s e l f , K 1 < p < -, f r o m L ( 1 o g L ) " t o L' a n d f r o m L ( l o g L ) " - ' t o weak L 1 . A s a c o r o l l a r y of t h e l a s t r e s u l t :

11.7.

1i m XER diam (R)+O f o r every

IRI I R

NOTES AND FURTHER RESULTS

f(y)dy = f(x)

219

a.e.

w h i c h i s l o c a l l y i n L(1og L ) ~ - ' ( R ~ )The . estimates f o r

f

M R a r e b a s e d on i t s m a j o r i z a t i o n by t h e n - f o l d a p p l i c a t i o n o f o n e d i m e n s i o n a l m a xi mal f u n c t i o n s ( s e e s e c t i o n I V . 6 ) . The b o u n d s o b t a i n e d i n t h i s c h a p t e r f o r t h e m a xim a l f u n c t i o n

7.3.-

in

Lp(IRn)

large

n

, for a fixed [with

a > 1).

p > 1,

operator over centered b a l l s i n M(")f(x)

=

sup O
B

(with

=

are of t h e o r d e r of an f o r M[n) d e n o t e s t h e m a xim a l

Ho wev er , i f

IBlrn

rB

,

IRn

namely

If ( x + y ) Idy

I x E IRn : 1x1 < 1 1 1 ,

then, the following inequalities

hold

with

C

Stein

$1, [a].

independent of

n.

T h i s s t r i k i n g r e s u l t is due t o

A s a n a p p l i c a t i o n , o n e c a n o b t a i n p o i n t w i s e summa-

b i l i t y o f F o u r i e r s e r i e s o f i n f i n i t e l y many v a r i a b l e s . We m e n t i o n two o p e n q u e s t i o n s i n t h i s d i r e c t i o n (Ql)

I s t h e r e a weak t y p e a n a l o g u e o f

(Q2)

I s ( * ) t r u e i f we r e p l a c e t r i c , o p e n s e t i n IR"?

Concerning

(Ql),

B

for

(*)

p = l?

by a c o n v e x , c e n t r a l l y symme-

a weak t y p e e s t i m a t e w i t h a f a c t o r

o b t a i n e d by S t e i n a n d S t r o m b e r g [l].

As for

(Q2),

n

h a s been

B o u r g a i n [3]

h a s r e c e n t l y g i v e n a n a f f i r m a t i v e a n s w e r f o r p 2 2 when n B = { x E IR : IIxII 5 1 1 , b e i n g a n u n c o n d i t i o n a l norm : T h i s i n c l u d e s t h e maxi= ( x , , x2,. ,xn) ( 2x, , 2 x 2 , . ,+ x n )

11

..

11

1 1 . 11 11

..

11.

mal f u n c t i o n o v e r c e n t e r e d c u b e s .

7.4.-

C a l d e r h a n d Zygmund [l].

i n t r o d u c e d t h e d e c o m p o s i t i o n which

b e a r s t h e i r name i n o r d e r t o s t u d y t h e e x i s t e n c e a n d b o u n d e d n e s s o f singular integrals.

The a p p r o a c h f o l l o w e d h e r e t o d e r i v e t h e e s t i -

mates f o r t h e m ax i mal f u n c t i o n f r o m s u c h a d e c o m p o s i t i o n i s a l s o

11. CALDERON-ZYGMUND THEORY

220

s u g g e s t e d i n t h e same p a p e r . The more u s u a l a p p r o a c h t o s u c h e s t i mates i s v i a c o v e r i n g lemmas, a p a r t i c u l a r example of which i s B e s i c o v i t c h C o v e r i n g Lemma:

each

*en

A c IR”,

a set

suppose t h a t f o r

Q(x) c e n t e r e d & x . Then, f r o m t h e , f a m i Z y { Q ( x )l x G A i t i s p o s s i b l e t o s e l e c t a s u b f a m i l y of c u b e s { Q j } which s t i l l c o v e r A and such t h a t i x Q j ( x ) 5 4 n ( i . e . , no more t h a n 4” c u b e s Q j c a n o v e r l a p ) . x

G

A

we h a v e a c u b e

For t h i s , s i m i l a r r e s u l t s and t h e i r a p p l i c a t i o n t o e s t i m a t e s of maximal f u n c t i o n s , we r e f e r t o t h e monographs b y d e G u z m h [l], The B e s i c o v i t c h lemma c a n b e u s e d S t e i n [lJ, S t e i n a n d Weiss [2]. t o g i v e an improved v e r s i o n of theorems

2.4

and

2.6:

“Let p a r e g u l a r p o s i t i v e B o r e 1 m e a s u r e i n EXn, and l e t b e t h e m a x i m a l o p e r a t o r d e f i n e s in s e c t i o n I , b u t w i t h c e n t e r e d

cubes

only.

Then

G,,

bounded i n

Lp(p)

a n d of weak t y p e

(l,l)I!

Observe t h a t no d o u b l i n g c o n d i t i o n i s imposed. For t h e (non-cent e r e d ) o p e r a t o r M p , t h e r e s u l t i s s t i l l t r u e i n R1 , b u t f a l s e in

IR”

for

n > 1

( S j o g r e n [I]).

C o v e r i n g lemmas f o r f a m i l i e s F s u c h a s t h o s e c o n s i d e r e d i n 7 . 2 a r e s o m e t i m e s much h a r d e r t o o b t a i n . One s u c h lemma f o r t h e f a m i l y R h a s b e e n o b t a i n e d by A . CBrdoba a n d R . F e f f e r m a n yielding a d i f f e r e n t ( g e o m e t r i c ) p r o o f o f t h e Jessen-Marcinkiewicz-Zygmund t h e @ rem.

131,

7.5.-

The 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 t h e o r e m

(2.11)

can be

s t a t e d more g e n e r a l l y f o r o p e r a t o r s o f weak t y p e ( p o , q o ) and ( p l , q l ) w i t h po 5 q o , p 1 2 4,. A companion o f t h i s i n t e r p o l a t i o n t h e o r e m whose p r o o f ( b a s e d on t h e maximum m o d u l u s p r i n c i p l e f o r a n a l y t i c functions) is completely d i f f e r e n t i n s p i r i t , is: Riesz-Thorin -(p,,qo) and

tMPe

Theorem:

(Po,q0)9

a n d we h a v e

(pl,ql),

wherv

If T -

with

i s a l i n e a r o p e r a t o r of s t r o n g t y p e s

1 ,< p i , q i

5

a,

then

T

i s of s t r o n g

NOTES AND FURTHER RESULTS

11.7.

221

The M a r c i n k i e w i c z a n d R i e s z - T h o r i n t h e o r e m s a r e t h e p r o t o t y p e s o f t h e r e a l and complex methods of i n t e r p o l a t i o n , r e s p e c t i v e l y .

These

a r e d e s c r i b e d i n B e r g h a n d L o f s t r o m [l]. Thorin theorem due t o S t e i n with t h e exponents

If TZ a r e l i n e a r o p e r a t o r s depending 0 5 Re(z) 5 1 , the hypothesis

p,q:

z,

a n a l y t i c a l l y on

An e x t e n s i o n o f t h e R i e s z allows t h e o p e r a t o r T t o change

[S]

provided t h a t

7:;

a m o d e r a t e g r o a t h when

-t

and

Ml(t)

have

An a p p l i c a t i o n o f t h i s t h e o r e m w i l l

m.

be i n d i c a t e d i n C h a p t e r V , 5.19

Mo(t)

and

7.7.

Further generalizations

h a v e b e e n o b t a i n e d by C o i f m a n , C w i c k e l , R o c h b e r g , S a g h e r a n d Weiss

C1I The i n e q u a l i t y

7.6.-

i s d u e t o C . F e f f e r m a n a n d S t e i n El].

(2.13)

The i d e a o f r e l a t i n g i n e q u a l i t i e s o f t h i s t y p e w i t h C a r l e s o n measu r e s i s t a k e n f r o m t h e same p a p e r . The s p a c e

BMO

was i n t r o d u c e d i n J o h n a n d N i r e m b e r g [l],

p r o v e d t h e t h e o r e m b e a r i n g t h e i r name, S t e i n [Z] equivalence BMO

(theorems

7.7.-

lip

3.6

The c o n d i t i o n

Zygmund

C. Fefferman and

(3.8).

d e f i n e d t h e s h a r p maximal f u n c t i o n

IIMf

c11,

%

llf#llp and

where t h e y

f#

and obtained t h e

Lp

and t h e i n t e r p o l a t i o n between

and

3.7).

S.l(b)

was a l r e a d y i m p l i c i t i n C a l d e r B n a n d

b u t i t was Hormander

[l]

who f i r s t f o r m u l a t e d i t e x p l i -

c i t l y i n t h i s form and p o i n t e d o u t t h a t o t h e r o p e r a t o r s a p a r t from t h e homogeneous s i n g u l a r i n t e g r a l s c o u l d b e d e a l t w i t h b y t h e same method. 6.3,

He u s e d i t , i n p a r t i c u l a r , t o p r o v e t h e m u l t i p l i e r t h e o r e m

.

i m p r o v i n g a p r e v i o u s r e s u l t o f M i h l i n [l]

One o f t h e e a r l y

r e a s o n s t o d e v e l o p t h e t h e o r y o f s i n g u l a r i n t e g r a l s was i t s a p p l i c a b i l i t y t o problems i n l i n e a r p a r t i a l d i f f e r e n t i a l equations.

Just

t o show t h i s c o n n e c t i o n , we w r i t e a s i m p l e consequence o f theorem 6 . 3 :

Let

P(D)

of degree

b e a n e l l i p t i c c o n s t a n t c o e f f i c i e n t o p e r a t o r homogeneous k.

IIDaflIp<

Then, t h e a p r i o r i i n e q u a l i t i e s

CP 11 P ( D ) f l / p

(la1

=

k, 1 < p <

m)

11. CALDERON-ZYGMUND THEORY

222

hold f o r every =

T (P(D)f) m

f

Ck

E

w i t h compact s u p p o r t ( o b s e r v e t h a t

with the multiplier

m(E,) = E,'/P(t)

DQf =

satisfying

(6.4l)).

The t h e o r y 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 h a s b e e n g e n e r a l i z e d a l o n g many d i f f e r e n t l i n e s . five notes.

o r i n p r o d u c t domains) 7.8.-

by

J u s t a f e w o f t h em a r e d e s c r i b e d i n t h e n e x t

Others (singular i n t e g r a l s f o r vector valued functions w i l l be considered i n Chapters

One c a n c o n s i d e r " n o n - i s o t r o p i c " A x H 6(x) = 6 x

,

dilations in

V

and

,

IRn

IV.

defined

6 > 0

i s a p o s i t i v e - d e f i n i t e n x n m a t r i x (When A = i d e n t i t y , If d = t r a c e (A) is we g e t t h e u s u a l " i s o t r o p i c " dilations). where

A

t h e homogeneous d i m e n s i o n o f

IRn with respect t o these d i l a t i o n s , is true f o r kernels k(x) then t h e analogue of Proposition 5.6 S e e d e Guzman w i t h t h e r i g h t homogeneity: k(6(x)) = 6-dk(x).

p],

Chapter 1 1 , and t h e r e f e r e n c e s c i t e d t h e r e . T h i s n o t i o n o f d i l a t i o n c a n b e c a r r i e d o v e r t o t h e more g e n e r a l s e t t i n g o f l o c a l l y c o m p a c t g r o u p s ( R i v i e r e Cl]).

These i d e a s have been

s p e c i a l l y f r u i t f u l when a p p l i e d t o some n i l p o t e n t Lie g r o u p s ( l i k e t h e Heisenberg group) i n c o n n e c t i o n w i t h s o l v a b i l i t y and r e g u l a r i t y p r o p e r t i e s of n o n - e l l i p t i c p a r t i a l d i f f e r e n t i a l o p e r a t o r s ( s e e F o l l a n d a n d S t e i n [l]). The H i l b e r t t r a n s f o r m a l o n g a f i x e d d i r e c t i o n 7.9.IuI = 1 , i s d e f i n e d a s f o l l o w s f

f

u

G

IR" ,

.

L2(lRn)

T h i s i s t h e s i m p l e s t e x a m p l e o f a s i n g u l a r i n t e g r a l whose k e r n e l i s supported in a submanifold of

the line

IRn :

Itu : t

G

R1.

In

t h i s c a s e , t h e r o t a t i o n i n v a r i a n c e and F u b i n i ' s theorem immediately

HU i s b o u n d ed i n L p ( I R n ) a n d o f weak t y p e ( 1 , l ) . A e x t e n s i o n c o n s i s t s i n r e p l a c i n g t h e l i n e by a c u r v e l i k e

prove t h a t non-trivial y(t)

=

$1,

ta2

,...)t a n )

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

y

, is

tsIR

1 1 . 7 . NOTES AND FURTHER RESULTS

I t i s known t h a t i n e q u a l i t y for

1 < p < m, IIHyfllp 5 CplIflIp, p = 1 i s s t i l l an open q u e s t i o n .

a n d W a i n g e r Cl]

b u t t h e weak t y p e We r e f e r t o S t e i n

f o r t h i s and r e l a t e d r e s u l t s .

The f a c t t h a t

7.10.-

223

IIHUflIp 6 C p I l f l l p

s u g g e s t s t h a t t h e same

i n e q u a l i t y w i l l b e t r u e f o r any "convex combination'! o f d i r e c t i o n a l Hilbert transforms:

HUf ( x ) R ( u ) d o ( u )

Tnf(x) = lul=l bith

R

L

G

1

R

When

i s odd, i n t e g r a t i o n i n p o l a r coordi-

n a t e s shows t h a t

which a r e t h e o p e r a t o r s c o n s i d e r e d i n P r o p o s i t i o n l a r i t y b e i n g now a s s u m e d o n

R.

Thus, f o r

R

5.6,

no regu-

odd

C = Norm i n L p ( R ) of t h e H i l b e r t transform. T h i s simple P a n d e l e g a n t m e t h o d , c a l l e d t h e m e t h o d of r o t a t i o n s , i s a l s o d u e t o

with

[z],

C a l d e r 6 n a n d Zygmund

r e s u l t is obtained f o r

i t s m a j o r d r a w b a c k b e i n g t h a t n o weak t y p e

.

p = 1

R

w o r k s p r o v i d e d t h a t we a s s u m e I f we a p p l y

t o the Riesz transforms

(*)

turns out t h a t

IIRjlll

independent of P following:

C

for a l l

n

G

N

When R i s n o t o d d , t h e m e t h o d s t i l l G L log L(Cn-l).

and

n.

=

1,

and t h e r e f o r e ,

R . j = 1 , 2 ,...,n , it 7' I I R j f l / P 5 Cpllfllp with

T h i s h a s been improved by S t e i n

1 < p <

m.

[8]

t o the

F o r a p r o o f o f t h i s r e s u l t b a s e d on

t h e method o f r o t a t i o n s , see Duoandikoetxea and Rubio d e F r a n c i a 7.11.-

[l].

The C a l d er 6 n - Z y g mu n d m e t h o d c a n a l s o b e a p p l i e d t o o p e r a t o r s

which a r e n o t t r a n s l a t i o n i n v a r i a n t . C o if m a n a n d Weiss [l]

A g e n e r a l s e t t i n g p r o v i d e d by

is the following:

11. CALDERON-ZYGMUND THEORY

224

We c o n s i d e r

a s p a c e of h o m o g e n e o u s t y p e X , w h i c h means t h a t X i s endowed w i t h a p s e u d o - m e t r i c p and a "doubling" measure dm(x), and we t r y t o s t u d y o p e r a t o r s o f t h e form

lx

Tf(x) =

K(x,y)f(y)dm(y)

Now, H o r m a n d e r ' s c o n d i t i o n s p l i t s i n t o two p a r t s :

I

(HI

f o r some f i x e d

k, C > 0

is the condition i)

LP,

IK(x,y)

-

K(x,yJIdm(x)

5 C

P(x,Yo)>kP(Y ,Yo)

(H)

and f o r a l l

y,yo

for the kernel:

G

X,

11

llTf ( 1 6 C I I f p l u s (H) imply t h a t 1 < p < 2 , a n d o f weak t y p e ( 1 , l )

i i ) IITf

Lp,

)I2 2

2 < p <

Cllf m,

)I2

plus and from

( H I )

Lm

to

and

imply t h a t BMO.

which

(HI)

K'(x,y) = K(y,x).

Then

i s bounded i n

T T

i s bounded i n

Examples o f o p e r a t o r s i n c l u d e d i n t h i s g e n e r a l framework a r e t h o s e considered i n

7.8

a n d s i n g u l a r i n t e g r a l s i n some c o m p a c t g r o u p s o r

i n homogeneous s p a c e s u n d e r t h e a c t i o n o f s u c h g r o u p s , l i k e t h e u n i t sphere

of

IR",

t h e u n i t b a l l Bn

of

En,

etc.

When X = IRn , t h e c o n d i t i o n s a n a l o g o u s t o t h o s e o f r e g u l a 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 (see 5.17) are

i m p l i e s (H) a n d ( H I ) ) . Operators bounded i n L 2 ( I R n ) a n d d e f i n e d b y a k e r n e l s a t i s f y i n g (CZ.l) and ( C Z . 2 ) are sometimes c a l l e d Calderdn-Zygmund operators. See C o i f (it i s obvious t h a t

man a n d Meyer

(CZ.2)

111, J o u r n 6

[l].

is not a convolution operator, the Fourier transf o r m i s no l o n g e r a n e f f e c t i v e t o o l f o r p r o v i n g L 2 - b o u n d e d n e s s , a n d t h i s i s u s u a l l y t h e d i f f i c u l t p a r t when a n a l y s i n g w h e t h e r T i s a Calder6n-Zygmund o p e r a t o r . A n i c e necessary and s u f f i c i e n t condit i o n h a s b e e n f o u n d by D a v i d a n d Journt? El]. For o p e r a t o r s w i t h 7.12.-

When

T

225

1 1 . 7 . NOTES AND FURTHER RESULTS

antisymmetric kernels, t h i s c r i t e r i o n takes t h e following s p e c i a l l y s i m p l e form:

-

If

Theorem":

T

i s g i v e n by a kerneZ s a t i s f y i n g

K(x,y) = - K ( y , x ) , T ( 1 ) E BMO.

(CZ.2) and onZy i f

then

T

is b o u n d e d i n

A s a n a p p l i c a t i o n , we c o n s i d e r t h e commutator of o r d e r

(CZ.l),

if

L2(rd)

k:

where A i s a L i p s c h i t z f u n c t i o n , i . e . A ' G Lm ( s u c h c o m m u t a t o r s a p p e a r n a t u r a l l y when t r y i n g t o c o n s t r u c t a n a l g e b r a o f p s e u d o Now, s i n c e To i s d i f f e r e n t i a l o p e r a t o r s , see A.P. C a l d e r 6 n [4]). t h e 'IT1 t h e o r e m " p l u s t h e H i l b e r t transform and Tk(l) = T k - l ( A ' ) , induction gives

This i s t h e theorem of

A.P. C a l d e r B n [3],

which c a n a l s o b e formu-

l a t e d i n terms o f a n o t h e r i m p o r t a n t o p e r a t o r : along a Lipschitz curve

where

z(t)

11,

< 1,

=

r

The Cauchy i n t e g r a l

i n t h e complex p l a n e , d e f i n e d by

t + iA(t) i s t h e p a r a m e t r i z a t i o n o f r . Now, i f we c a n e x p a n d t h e i n t e g r a n d i n t o a g e o m e t r i c s e r i e s

and, w r i t i n g

The e s t i m a t e s

E

= 1/C,

(*)

(*)

is equivalent t o

have been improved by Coifman, McIntosh a n d

a l l o w i n g t o remove t h e r e s t r i c t i o n llA'\I, < E i n ( * * ) . Meyer [l], We r e f e r t o Y.hleyer T h i s h a s a l s o b e e n o b t a i n e d by D a v i d [l]. [Z. f o r a n e x c e l l e n t e x p o s i t i o n o f t h e s e a n d r e l a t e d r e s u l t s .

D],

7.13.-

Lemma

4.6

i s d u e t o H a r d y a n d L i t t l e w o o d , b u t we h a v e p r e -

11. CALDERON-ZYGMUND THEORY

226

The a r g u m e n t u s e d s e n t e d h e r e t h e p r o o f o f F e f f e r m a n a n d S t e i n [2]. t o p r o v e 4 . 1 4 comes f r o m C a l d e r d n a n d Zygmund [l]. By u s i n g S t e i n ' s s p h e r i c a l m e a n s , which w e s h a l l d i s c u s s i n C h a p t e r V , 7 . 7 , t h i s r e s u l t c a n b e p a r t i a l l y i m p r o v e d : F o r J, r a d i a l a n d i n t e g r a b l e i n Rn , t h e o p e r a t o r f ( x ) t - + ; y g l $ t * f ( x ) i s b o u n d e d i n Lp(IRn) , p > L A d i f f e r e n t r e s u l t f o r approximations of t h e i d e n t i t y i s n-1' Zo's theorem (see s e c t i o n V . 4 ) .

1

7 . 1 4 . - Theorem 4 . 2 1 i s d u e t o A.P. C a l d e r d n [l] who s u c c e e d e d i n e x t e n d i n g t o h a r m o n i c f u n c t i o n s i n IRn++' a p r e v i o u s theorem of P r i v a l o v [I] v a l i d f o r h a r m o n i c f u n c t i o n s i n t h e u n i t d i s k D . Priv a l o v ' s t h e o r e m was p r o v e d o r i g i n a l l y v i a c o n f o r m a l m a p p i n g . We s h a l l g i v e a n o u t l i n e o f t h i s p r o o f b e l o w . The method i s n o l o n g e r a v a i l a b l e i n h i g h e r d i m e n s i o n s . T h i s i s w h a t makes C a l d e r 6 n ' s c o n t r i b u t i o n a b a s i c one. In o r d e r t o given a n account of P r i v a l o v ' s p r o o f , l e t as assume, w i t h o u t l o s s of g e n e r a l i t y , t h a t u i s a harmonic f u n c t i o n i n D such t h a t f o r e v e r y t f o r which ei t 6 E ,

a s u b s e t of T = boundary of D w i t h I E l > 0 , (u(z)I is bounded by a f i x e d number n on t h e S t o l z domain r a ( t ) , where c1 is also fixed. (The S t o l z d o m a i n s a r e d e f i n e d i n p a g e 1 0 9 ) . We may a l s o a s s u m e t h a t E i s c o m p a c t . We c o n s i d e r t h e s e t U = U T a ( t ) where e i t r a n g e s o v e r E . Then U i s a n open s e t bounded by a r e c t i f i a b l e Jordan curve.

We know t h a t

I

lu(z)

2 n

on

U.

Now t h e

U and look a t t h e idea is t o use a conformal equivalence $ : D f u n c t i o n u ( $ ( z ) ) w h i c h i s h o l o m o r p h i c a n d bounded i n D . We c a n -+

a p p l y F a t o u ' s theorem t o c o n c l u d e t h a t u ( $ ( z ) ) c o n v e r g e s nont a n g e n t i a l l y a t a l m o s t e v e r y p o i n t i n t h e b o u n d a r y o f D . We know (corollary 4.7 i n c h a p t e r I ) t h a t , e x c e p t f o r t h e p o i n t s where t h e boundary of

U

h a s n o t a n g e n t , which f o r m a s e t o f a r c l e n g t h

t h e c o n f o r m a l mapping r e m a i n s c o n f o r m a l a t t h e b o u n d a r y p o i n t s . We a l s o know ( s e e a f t e r c o r o l l a r y 3 . 1 3 i n c h a p t e r I ) t h a t t h e 0,

homeomorphism i n d u c e d b e t w e e n t h e b o u n d a r i e s o f

sets of length

0

t o s e t s of Lebesgue measure

U 0.

and

D

carries

T h u s , we h a v e

t h a t f o r almost every t and f o r every 6 , $ ( r g ( t ) i s a r e g i o n non-tangential a t $(eit) w i t h r e s p e c t t o U, a n d we f i n a l l y g e t t h e e x i s t e n c e of n o n - t a n g e n t i a l limits f o r u a t a most e v e r y p o i n t o f E , w h i c h i s p a r t o f t h e b o u n d a r y of U. Note t h a t i n E a r c l e n g t h c o i n c i d e s w i t h Lebesgue measure. The method works e v e n f o r a f u n c t i o n meromorphic i n D . a f t e r u n i f o r m i z i n g t h e s i t u a t i o n , so t h a t w e c a n assume

Indeed, c1 and n

1 1 . 7 . NOTES AND FURTHER RESULTS

227

I

f i x e d , we r e a l i z e t h a t t h e f a c t t h a t lu(z) I n f o r each z s ra(t) it with e G E, i m p l i e s t h a t t h e s i n g u l a r i t i e s of u i n U a r e a l l i n s i d e a d i s k D(o,r) with r < 1 and, consequently, t h e r e is o n l y a f i n i t e number of them. M u l t i p l y i n g by a n a p p r o p r i a t e p o l y n o m i a l , we g e t a h o l o m o r p h i c f u n c t i o n on U, t i a l b o u n d a r y v a l u e s e x i s t a t a . e . p o i n t of

f o r which n o n - t a n g e n E. Obviously, t h e

same t h i n g h a p p e n s f o r t h e o r i g i n a l f u n c t i o n

u.

Note t h a t t h e same t e c h n i q u e o f c h a n g i n g t h e domain c o n f o r m a l l y , c a n b e u s e d t o p r o v e t h e f o l l o w i n g t h e o r e m , a l s o due t o P r i v a l o v [l]: u(z)

If a f u n c t i o n

meromorphic i n

T

a t e a c h p o i n t of a s u b s e t of identically

D

has non-tangential

limit

of p o s i t i v e m e a s u r e , t h e n

0

u

D.

0

By u s i n g t h e two m e n t i o n e d t h e o r e m s , o n e c a n e a s i l y o b t a i n t h e f o l lowing result, due t o Plessmer

Let u(z)

111 ( s e e

b e an a n a l y t i c f u n c t i o n i n

p o i n t s of L e b e s g u e m e a s u r e

0,

a l s o Zygmund D.

to

eit

o r e l s e , t h e image

u(r,(t))

XIV).

Then, e x c e p t f o r a s e t of

a t every point

e i t h e r the function has a f i n i t e l i m i t a s

c ] , Chap.

z

e it

of t h e b o u n d a r y ,

tends non-tangentially

i s dense i n t h e plane f o r

a.

every

A c t u a l l y A . P . CalderBn [l] e x t e n d s P r i v a l o v ' s theorem t o p l u r i h a r monic f u n c t i o n s i n a C a r t e s i a n p r o d u c t o f e u c l i d e a n h a l f - s p a c e s and o b t a i n s non-tangential convergence a t a . e . p o i n t of t h e d i s t i n guished boundary.

By u s i n g t h i s r e s u l t , h e i s a b l e t o e x t e n d

P l e s s n e r ' s t h e o r e m t o a n a l y t i c f u n c t i o n s o f s e v e r a l complex v a r i a b l e s . T h e r e i s a c o n v e n i e n t way t o r e f l e c t a n a l y t i c a l l y t h e f a c t t h a t a harmonic f u n c t i o n u h a s n o n - t a n g e n t i a l l i m i t s a t a . e . p o i n t of a s u b s e t E of t h e boundary w i t h I E l > 0 . T h i s i s a c h i e v e d t h r o u g h the area f u n c t i o n

For

u

harmonic i n

I f we j u s t f i x

A(u) D,

i n t r o d u c e d by

Lusin.

we d e f i n e

a, we c a n s i m p l y w r i t e

r(t)

and

A(u)(t).

11. CALDERON-ZYGMUND THEORY

228

F = u + iv

Observe t h a t i f

is analytic i n

D,

then

is the t h e J a c o b i a n o f t h e mapping F . T h u s , we s e e t h a t A , ( u ) ( t ) a r e a ( p o i n t s c o u n t e d a c c o r d i n g t o t h e i r m u l t i p l i c i t y ) of F ( T , ( t ) ) . T h i s e x p l a i n s t h e name " a r e a f u n c t i o n " . The f o l l o w i n g r e s u l t h o l d s :

Let u

b e harmonic i n

D with a.e. x

6

f o r a.e.

D and l e t E b e a s u b s e t o f t h e boundary o f [ E l > 0. Then, f o r u t o have a non-tangential l i m i t a t E, i t i s n e c e s s a r y and s u f f i c i e n t t h a t A(u)(x) i s f i n i t e x

6

E.

The n e c e s s i t y was p r o v e d by M a r c i n k i e w i c z a n d Zygmund [2] s u f f i c i e n c y by S p e n c e r [IJ Note t h a t i f

F = u + iV

.

is analytic in

by t h e Cauchy-Riemann e q u a t i o n s . u

has a non-tangential

v

same i s t r u e f o r

For a f u n c t i o n

A(u)

l i m i t f o r a.e.

then:

=

A(v)

point of

a n d we g e t : E,

then the

and c o n v e r s e l y .

u(x,t)

harmonic i n t h e upper h a l f - s p a c e

(generalized) area function

A.P. C a l d e r B n [7]

Thus

D

and t h e

A(u)

Rn+'

,

the

is defined a s follows:

e x t e n d e d t h e t h e o r e m o f M a r c i n k i e w i c a a n d Zygmund

t o t h i s higher dimensional s e t t i n g and t h e extension of Spencer's The g e n e r a l i z e d a r e a f u n c r e s u l t was c a r r i e d o u t by E.M. S t e i n t i o n was p r e v i o u s l y c o n s i d e r e d i n c o n n e c t i o n w i t h o t h e r r e l a t e d

@I.

11.7.

o p e r a t o r s i n E.M. S t e i n In s e c t i o n

GO].

of c h a p t e r

8

229

N O T E S AND FURTHER R E S U L T S

I11

we s h a l l i n d i c a t e some r e s u l t s on t h e

a r e a f u n c t i o n which a r e c o n n e c t e d w i t h t h e t h e o r y of Hardy s p a c e s . L 2 , the Fourier multipliers f o r Lp a r e c o m p l e t e l y c h a r a c t e r i z e d o n l y when p = 1 : t h e c l a s s of L’ m u l t i p l i e r s c o i n c i d e s w i t h t h e F o u r i e r - S t i e l t j e s t r a n s f o r m s of f i n i t e Bore1 measures i n lRn. Sufficient conditions f o r m(S) to be an Lp m u l t i p l i e r f o r a l l 1 < p < = a r e g i v e n by t h e HBrmander7.15.-

A p a r t from t h e t r i v i a l c a s e of

M i h l i n theorem, 6 . 3 , a n d by a n o t h e r c l a s s i c a l r e s u l t : t h e Marcink i e w i c z m u l t i p l i e r theorem, whose 1 - d i m e n s i o n a l v e r s i o n w i l l be There i s an e x t e n s i v e l i t e r a t u r e c o n proved i n c h a p t e r V , 5 . 1 3 . c e r n i n g more s p e c i f i c m u l t i p l i e r s f o r d i f f e r e n t r a n g e s of p ’ s . We s h a l l b r i e f l y d i s c u s s h e r e t h o s e o n e s r e l a t e d t o s u m m a b i l i t y of Fourier series o r i n t e g r a l s . Given

a 20,

f o r every f t h e r a n g e of

i s it t r u e t h a t

G

Lp(Rn) ? The q u e s t i o n i s e q u i v a l e n t t o t h e s t u d y of p ’ s corresponding t o t h e Bochner-Riesz multipliers:

Our p r e s e n t knowledge a b o u t t h i s c h a l l e n g i n g problem i s t h e f o l l o w i n g :

i) kernel

( c r i t i c a l index).

Ka

Then

ma(S)

= i?,(C)

which c a n b e e x p l i c i t l y computed and b e l o n g s t o

Thus, m_U Weiss [ZJ.

i s an

Lp

multiplier for a l l

1 5 p < =.

with a L1(Rn)

.

See S t e i n and

ii) a = 0 . For n = 1 ’ mo = X[-1,1] i s an LP m u l t i p l i e r 1 < p < m ( t h i s i s e q u i v a l e n t t o M. R i e s z t h e o r e m ) , w h i l e f o r i s n o t an LP m u l t i p l i e r f o r any p 2 n 2 2 , m 0 - X[]l’l51’ ( C Fef f erman n- 1 iii) 0 < a S The c o n j e c t u r e i s t h a t ma 1 s an Lp multi2 p l i e r i f and o n l y i f

for

+

.

-.

Pa

=

2n n + I + 2a < P < n

-

2n 1

-

2a = P A

11. CALDERON-ZYGMUND THEORY

230

The n e c e s s i t y o f t h i s c o n d i t i o n i s e a s i l y s e e n , b e c a u s e Ka $ L p a For n = 2 , t h e c o n j e c t u r e i s t r u e , and i t was f i r s t p r o v e d by

n];

.

d i f f e r e n t p r o o f s h a v e b e e n g i v e n by C . F e f C a r l e s o n and S j a l i n For n > 2 , t h e c o n j e c t u r e h a s been ferman [4] and C6rdoba [lJ. n- 1 p r o v e d when a T h i s f o l l o w s f r o m r e s u l t s o f S t e i n a n d Tomas on r e s t r i c t i o n o f F o u r i e r t r a n s f o r m s ( s e e P. Tomas [l]); it can a l s o

=.

be p r o v e d by u s i n g t h e a p p r o a c h i n C6rdoba [l].

CHAPTER

I11

REAL VARIABLE THEORY OF HARDY SPACES

In t h i s chapter we s h a l l develop an analogue of the Hp theory studied in chapter I , where the torus 'Il'w i l l be replaced by the euclidean space IRn

.

In the new context, we can no longer r e l y upon the powerful methods of Complex Analysis. Our basic tool w i l l be harmonic majorization (theorem 4 . 1 0 in chapter 11) , together with the real-variable techniques centering around the "Calder6nZ y w d theory", which were also developed in chapter 11. Instead of dealing d i r e c t l y with the case of Rn f o r a r b i t r a r y n , we have chosen t o s t a r t with the one-dimensional s i t u a t i o n , t o which the f i r s t three sections are devoted. Beginning with a definition of Hp(IRf, a s a space of analytic functions, 2 can be which p a r a l l e l s t h a t of Hp(D) f o r the disk D , we soon r e a l i z e t h a t Hp(R+)

identified with a r e a l space of tempered distributions on IR denoted by ReHp(IR) or simply Hp(IR). The main achievement i s the identification of the basic building blocks, called "atoms", i n t o which any Hp d i s t r i b u t i o n can be decomposed. This e give two proofs of the atanic atomic decomposition is carried out in section 3. W decomposition. The f i r s t i s based on a smooth version of the Calder6n-Zygmund de-

composition (theorem 3 . 6 ) . The second, given in 3.11, is obtained working directly with the harmonic extension, and it c a r r i e s over t o the n-dimensional case with only minor modifications. Two proofs a r e a l s o given f o r the Burkholder-Gundy-Silverstein theorem asserting t h a t a d i s t r i b u t i o n is in Hp(IR) i f and only i f it i s the boundary value of a harmonic function in IR; whose non-tangential maximal function belongs t o Lp(IR). The f i r s t proof follows from the work done in a l l three sections, but we a l s o give a d i r e c t proof due t o P.Koosis [Z] Function Theory.

based upon Cauchy's integral theorem of Complex

The characterization of Hp(iR) given by the Burkholder-Gundy-Silverstein theorem provides a natural way t o define Hp(IRn) f o r n > l . Section 4 i s devoted t o the study of these spaces f o r higher dimensions. The r e s t of the chapter can be seen a s a test f o r the power of the atomic description of the Hardy spaces. Section 5 contains a thorough investigation of the duals of the spaces €1P (IRn ) 231

111. REAL HARDY SPACES

232

Ocpsl. After the atomic description, the identification of the duals can be achieved by simple real-variable techniques. In particular, section 5 contains the 1 n-dimensional version of the duality H -B.M.O. which was established f o r the toms in section 10 of chapter I (C.Fefferman’s theorem). Section 6 contains some interpolation theorems f o r operators sending Hp t o Lp (or weak-LP)

.

Finally, section 7 is the culmination of our work. I t shows how t o use atoms t o obtain estimates for different kinds of operators acting on Hp. I t contains res u l t s f o r singular integrals and m l t i p l i e r s , extensions t o IF? of the inequalit i e s of Hardy and Fejer-Riesz already encountered in section 4 of chapter I ; and also the molecular description of Hp which allows us t o obtain some Hp-Hp results. 1 . H p SPACES FOR THE UPPER HALF PLANE Let F ( z ) be an a n a l y t i c f u n c t i o n of z = x

IIFII

Hp

I n t h e upper

R + = { z = K + i t ; t > 0 ) . D e f i n e , f o r any p > O 1 F ( y + i t )I . ~ u p i j - ~ / F (+ x i t ) P d x l and c a l l m F ( x ) = s u p t>O 03 I x-Y I < t

half plane =

+ it

2

‘’9

Then, w e have THEOREM 1 . 1 .

equzvalent:

IlFi

a)

<

HP

mF

bj

F

FOP

d

2

B+ a n d

analytic i n

-

LP(IR)

LP

2

M o r e o v e r , we h a v e t h e i n e q u a l i t i e s : Proof:

Clearly

p>O, t h e f o l l o w i n g a r e

b)

implies

a)

IIFII

< llmFIIp Hp -

because- I F ( x

+

5

cI(F//

HP .

i t ) / 5 mF(x)

for

’

t 0 alld, c o n s e q u e n t l y ~ ~ F ( l m~F I I ~p . HT hP a t ~a ) i m p l i e s c a n be s e e n i n t h e f o l l o w i n g way: c o n s i d e r s ( z ) = I F ( z ; i E w i t h some E s a t i s f y i n g 0 < E 1 . I t f o l l o w s from theorem 4.10 i n Chapter I1 that s ( x -# i t ) 5 ( s o i n Chapter 11)

+ +

pt)(X)

f o r some

s

0

e Lq.

Thus ( s e e ( 4 . 1 5 )

and

so

b)

holds.

/ / m F ( l p ( CilFllHp , o b s e r v e t h a t mF e LP a . e . a n d , by theorem 4 . 2 1 i n C h a p t e r I 1 has non-tangential l i m i t s a . e . In particular,

To prove t h e i n e q u a l i t y i m p l i e s t h a t mF(x) < m t h i s implies that

F

111.1. Hp FOR THE UPPER HALF PLANE

F ( x ) = $48 F ( x + i t ) is o b t a i n e d as t h e weak-* l i m i t i n

there exists

f o r some s e q u e n c e

tn+O.

Lq

f u n c t i o n , namely

tends t o for

IF(x)l'

a.e.x

R e c a l l t h a t so

f o r a.e.x.

Lq of t h e f u n c t i o n s x b s ( x + i t n ) s ( x + it ) + IF(x)l' f o r a.e.x n s ( x t i t n ) a r e bounded by t h e

But

and, s i n c e a l l t h e f u n c t i o n s same

233

+ i tn ) so(x) = IF(x)l'

i t follows t h a t

mi,

i n t9e

Lq

Hp(lFt:),

o r simply

norm.

s(x

Therefore

and

We s h a l l d e n o t e by

when no c o n f u s i o n

Hp

i s l i k e l y t o a r i s e , t h e c l a s s formed by t h o s e f u n c t i o n s

B 2 , f o r which IIFIIHp < m . and any x e @ :

analytic i n analytic i n

F(z)

C l e a r l y , f o r any

F,G

Hp i s a l i n e a r s p a c e , F k I ( F I I H P i s a norm 1 5 p < m E n d F k IIFII$ i s a p-norm i n Hp if ( a p-norm i s a n o n - n e g a t i v e f u n c t i o n which i s 0

It follows t h a t in

if

Hp

0 < p < 1

only at

s a t i s f i e s t h e t r i a n g l e i n e q u a l i t y and i s

0,

p-homogeneous

I t g i v e s r i s e t o a m e t r i c compatible with t h e l i n e a r s t r u c t u r e ) . s h a l l view being if

Hp

We

as a m e t r i c l i n e a r s p a c e , t h e d i s t a n c e f u n c t i o n

( F , G ) h [IF - GIIHP

1 2 p <

if

m

and

( F , G ) H IIF

0 < p < 1.

L e t u s c o l l e c t t h e main f a c t s r e g a r d i n g t h e boundary b e h a v i o u r o f a n Hp

function,

They f o l l o w v e r y s i m p l y from theorem

1.1

and

its proof. COROLLARY

al

l i m F(x t+O

1.2.

Let

+ i t ) exists

F ( x ) = li

t+'6

F(x

t a n g e n t i a l l y at bl

F e Hp.

lim LlF(x t+O

Then: -

for

a.e. x e R a n d t h e f u n c t i o n

+ i t ) is i n

Lp.

Actually

a . e . x e IR.

+ it) -

F(x)lPdx = 0

and

F

c o n v e r g e s non-

234

111. REAL HARDY SPACES

P r o o f : E v e r y t h i n g i s a c o n s e q u e n c e of t h e f a c t t h a t m F e Lp. a) f o l l o w s from theorem 4.21 i n C h a p t e r 11. For b ) we j u s t n e e d t o a p p l y t h e Lebesgue d o m i n a t e d c o n v e r g e n c e t h e o r e m , and p r o v e d i n theorem

We j u s t n e e d t o a p p l y theorem

s ( x , ~ )= ( F ( X , t ) l E ,

0 <

E

.t

CIIFlI

1.3.

Proof:

h a s been

n

1.1.

For e v e r y F e Hp: IF(x,t)l 2 with a constant c which does not depend o n F.

LEMMA

c)

-l/P

HP

4 . 5 i n C h a p t e r I1 t o

r_l

< p.

An i m p o r t a n t consequence o f lemma 1 . 3 i s t h a t t h e imbedding o f 2 i n t o t h e space H ( I R + ) of a l l t h e f u n c t i o n s holomorphic i n t h e

Hp

u p p e r h a l f p l a n e , endowed w i t h t h e t o p o l o g y o f u n i f o r m c o n v e r g e n c e on compact s u b s e t s , i s c o n t i n u o u s .

T h i s c a n be u s e d t o p r o v e t h e

following. LEMMA proof:

Let

that

1.3

For e v e r y p > 0 , Hp

1.4.

(F,)

be a Cauchy s e q u e n c e i n

I m I F n ( x + i t ) - F(x

+

F.

Now, g i v e n

it)lPdx 5 l i m inf

m+ m

-m

n

Suppose that

1.5.

belongs to

Proof;

,

Lp2

f o r some

F e Hpl

p2 > p l .

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

0 < E < pl. LP 1l’E

where

I t f o l l o w s from

lFn(x

E

m:

> 0

+ i t ) - Fm(x+it)lPdxz

n

i s b i g enough.

LEMMA

Hp.

c o n v e r g e s u n i f o r m l y on e a c h compact s u b s e t o f

Fn

t o a certain analytic function

if

is a c o m p l e t e s p a c e .

and its boundary function

Then

IF(x + i t )

F(x)

F e Hp2. f o r some

c

such t h a t

T h i s i s a subharnionic f u n c t i o n which i s u n i f o r m l y i n p , / ~ > 1.

It follows t h a t

111.1. Hp FOR THE UPPER HALF PLANE

mF(x) 5 C . ( M ( l F I E ) ( ~ ) ) l ' E . But

so that p2/0

m

1, a n d , c o n s e q u e n t l y ,

5

IF(x)l'

e Lp2.

F

235

is i n

L

P2/

E

,

'.

F e HP

We c o n c l u d e t h a t

1 5 p m. I t f o l l o w s from p a r t b ) o f c o r o l i s t h e P o i s s o n i n t e g r a l o f i t s boundary f u n c t i o n , which i s a n Lp f u n c t i o n F ( x ) = f ( x ) + i g ( x ) , w i t h f and g real. Indeed, f o r each t > 0 , t h e f u n c t i o n F t ( z ) = F ( z + i t ) i s

F e Hp

Suppose lary

and

that

1.2

F

t h e P o i s s o n i n t e g r a l of i t s b o u n d a r y f u n c t i o n the functions

Ft(x)

converge t o

* (f + ig)(x).

F ( x + i t ) = P,

Thus

F(x)

+ it)

Ft(x) = F(x

i n the

as

norm

Lp

and

+

t

0.

On t h e o t h e r h a n d , i f

" 1 x t h e f u n c t i o n G(x + i t ) = Q,(x) = H ( P t ) ( x ) = t 2 t x2 ' is a n a l y t i c and h a s t h e same r e a l p a r t as = P t * f ( x ) + iQt * f ( x )

.rr

F. 0

Then

t

for

F

-

+

m ,

* f(x) +

= Pt

h a s t o be a c o n s t a n t and, s i n c e

G

F - G

i Q t* f ( x )

llfllLP + f + iHf

IIF\lHp t h e form

*

= Pt

function

Lp

(f

=

real,

f

Pt

responding i n e q u a l i t y f o r previous chapter.

p

=

1

f o r any

t h e r a n g e o f t h e mapping

s t r i c t l y smaller t h a n

ReL

1

.

Hp

H f e L1.

In

ReHl

p > 1, F(x

F k f

Hp

and

becomes a

from

H1

1

ReH

or, more s i m p l y , 1

,

t o s e e H1

from t h e isomorphism

ReLl, w h e r e a s f o r

when viewed as a s p a c e o f r e a l f u n c t i o n s . Hp

F

to

is

k f When

ReL'

is ReH

.

1

f o r which

The c o m p l e t e n e s s H

is c l o s e d i n

F C , f between H1 and 1 ReH What we h a v e done i s

.

as a s p a c e o f r e a l f u n c t i o n s i n

is precisely t o study

it) =

llflL1 + IIHfIIL1.

Banach s p a c e .

which i s t h e r e a s o n f o r t h e name

q u i t e d i f f e r e n t from

+

ReLP.

f e ReLl

f o l l o w s from t h e f a c t t h a t t h e H i l b e r t t r a n s f o r m L

For

f e ReLP

any

We s h a l l d e n o t e t h i s r a n g e by

w e s h a l l c o n s i d e r t h e norm

ReH'

p = 1.

whereas t h e cor-

and t h e mapping

I t i s t h e s p a c e formed by t h o s e f u n c t i o n s

With t h i s norm

and

=

There i s

Hp.

LP ' d o e s n o t h o l d , as we s a w i n t h e

a n isomorphism between t h e Banach s p a c e s p = 1,

+ it)

is i n

p > 1

and

function of

L'

(CpIlfll

T h i s i m p l i e s t h a t , when

* ( f + i H f ) ( x ) belonging t o

= Pt

We

g = Hf.

f e ReLP

F(x

and c o n s e q u e n t l y i t g i v e s r i s e t o

h a s H f e Lp

tend t o

t h e Poisson i n t e g r a l

f o r some

iHf

the function

IIHflILP

=

we f i n d

0,

.I

* f(x) + iQt * f(x)

we have a n i n e q u a l i t y

G

t

a n i m p o r t a n t d i f f e r e n c e between t h e c a s e s p > 1

and

+ it)

F(x

Conversely,

IIHfllLP.

+ iHf)(x)

+

f

F

We g e t

-, i s

1 5 p <

o f t h e form with

0.

and, l e t t i n g

F e Hp,

have s e e n t h a t e v e r y o f an

is actually

IR.

A s s u c h , i t is

p > 1 Hp

is simply

ReLP

The a i m o f t h i s c h a p t e r

from t h i s r e a l v a r i a b l e p o i n t o f view.

236

111. REAL HARDY SPACES

0 < p < 1.

Now l e t

W i l l i t b e p o s s i b l e i n t h i s c a s e t o view

as

Hp

a s p a c e of r e a l f u n c t i o n s on I R ? . A f i r s t d i f f i c u l t y we encounter i s t h a t t h e mapping s e n d i n g t h e f u n c t i o n F ( z ) t o t h e f u n c t i o n -i R e F ( x ) i s n o t one t o one. For example, l e t F ( z ) = z ( z - 1) 1 and t h e r i g h t hand s i d e o f t h i s i n e q u a l i t y S i n c e I F ( z ) l 2 I x II x i s i n Lp i f and o n l y i! 1 / 2 < p < 1 , i t f o l l o w s t h a t F e Hp f o r 1 / 2 < p < 1 and, a c t u a l l y , f o r no o t h e r p s i n c e mF(x) ?r (1x1 Ix - l I ) - l . But R e F ( x ) = 0. The way t o a v o i d t h i s d i f f i c u l t y is t o consider t h a n as a f u n c t i o n ,

a s a tempered d i s t r i b u t i o n r a t h e r

ReF(x)

Indeed,

which i s u n i f o r m l y i n

ReF(x

+ i t ) i s a harmonic f u n c t i o n

We know t h a t t h e f u n c t i o n s

Lp.

have a l i m i t i n t h e s e n s e of tempered d i s t r i b u t i o n s a s t h a t t h i s tempered d i s t r i b u t i o n and, consequently

F(z)

ReF(x)

( s e e theorem

xwReF(x+it)

t t 0

and

uniquely determines

4.23

ReF(z)

i n C h a p t e r 11).

In our

c o n c r e t e example

(z -

11 ) ) z-

ReF(z) = Re ( 1 1

t - x2

= -Im(; 1 -

l ) z - 1

=

t

+ t2

(x - 1)2

+ t2

which c o n v e r g e s i n t h e s e n s e of tempered d i s t r i b u t i o n s as t h e measure

6o

- 61,

where

da

a.

t h a t i s , t h e u n i t mass c o n c e n t r a t e d a t d e n o t e by ReF(x)

ReHP

4.23

Indeed, i f J,

e

0 < p < 1,

0

to

a,

we s h a l l

f e ReHP

F e Hp,

w i t h t h e p-norm

i n C h a p t e r I1 i m p l i e s t h a t t h e s p a c e

i s c o n t i n u o u s l y imbedded i n t h e s p a c e

corresponding t o every

F e Hp,

corresponding t o the functions

butions.

For

+

t h e s p a c e formed by t h e boundary d i s t r i b u t i o n s

Observe t h a t theorem ReHP

t

denotes t h e Dirac d e l t a a t

s’

of tempered d i s t r i -

i s t h e boundary d i s t r i b u t i o n

t h e theorem mentioned above shows t h a t f o r

s ReF(x , t ) $ ( x ) dx HP

where, o f c o u r s e , p r o o f of theorem

C

4.23

depends on in

y

Chapter

.

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

11.

ReHP f o r 0 < p < 1 i s t o c o n s i d e r a d e n s e subspace whose members a r e f u n c t i o n s . Even f o r p = 1 i t w i l l b e u s e f u l t o l o o k f o r a d e n s e c l a s s i n ReH’ formed by n i c e A c o n v e n i e n t way t o work w i t h

111.1. Hp FOR THE UPPER HALF PLANE

functions (smooth, vanishing rapidly at typical density results. THEOREM

those

Then H; -

N > 0, l e t HE

For e v e r y

1.6.

F e Hp

etc.).

m

237

Let us give some

be t h e space formed b y

such t h a t :

i s dense i n

Hp.

Since every F(z) in Hp is the limit in Hp of the functions z W F ( z + it) as t * 0, we may start by assuming that F For such an F in Hp, we shall exhiis C m and bounded in bit a sequence of functions in H; converging to F in Hp. Let Proof:

q.

-

-

Since each Gn is C m in IR: formly in t , it follows that in Hp as n * Indeed

-.

IIGnF

- FIIP

=

HP because

COROLLARY

1.7.

a)

Ci

formed by t h o s e

+

I,I

(---- ni

0

boundedly as

FOP

x

+ ni

any

N

)

-

p,q > 0

P

1 ) IF(x)lPdx * 0

n

-r

Hp

n

(n *

-1

-. Hq

i s dense i n

Hp

(ReHP) n Lq is d e n s e i n ReHP. i s d e n s e i n ReHP, where C i s t a n d s f o r t h e s p a c e m C functions i n lR w h i c h v a n i s h a t m a s f a s t a s

and, consequently.

bl (ReHP)n

and also O( I x I - ~ ) as 1x1 * uniGnF e HE for each n. Also GnF F

for

q > 1

WN. a) Let N > l/q. Then HE c Hp n Hq since for F e HE, the boundary function belongs to C; c Lq and so, according to 1.5, F e Hq. It follows that H P n Hq is dense in Hp. As f o r the second assertion in a), just note that the boundary distributions corresponding to the functions F in Hp n H q , for q > 1 , are precisely the functions in (ReHP)n Lq. b) follows if we observe that the boundary distributions corresponding to the functions in Hi are the functions in (ReHP) n C i . 0

Proof:

238

For

111. REAL HARDY S P A C E S

p = 1 we can e a s i l y o b t a i n a much n i c e r dense c l a s s .

T H E O R E M 1.8. T h e s p a c e c o n s i s t i n g of t h o s e r e a l f u n c t i o n s f u h o s e F o u r i e r t r a n s f o r m 2. 2 Cm and h a s c o m p a c t s u p p o r t n o t c o n ReH

taining the origin, is dense in

.

1

F i r s t of a l l , we o b s e r v e t h a t e v e r y

Proof:

f e ReH'

has

I f ( x ) d x - = 0 . I n d e e d , w e know t h a t Hf e L1 a n d , consequently, ( H f ) ( x ) = - i ( s g n x ) ?. ( x ) must be a c o n t i n u o u s f u n c t i o n . ?(O)

=

But, s i n c e sgn x unless ?.(o) = 0.

h a s a jump a t

c a n n o t be c o n t i n u o u s a t

Hf

0,

As a second s t e p , w e s h a l l show t h a t i f

if for

$

e L'

L'

in

f * * t + O

t > 0 as t

and

has

f e L'

we c o n s i d e r

f(0) = 0

qt(x) = t-l$(t-'x),

0

and then

Indeed

m.

-1

S i n c e f o r t -* m , I l $ ( - - t y ) - $ I l l + 0 f o r e v e r y y and -1 I [ $ ( . - t Y ) - $ 1 ) 5 2 1 ) ~ 1 \ ( ~an, a p p e a l t o t h e Lebesgue dominated convergence theorem y i e l d s \ I f * q t l l l + 0 as t + m . I f we choose

1x1 <

g t = f - f * $t, i n L1 as t +

provided

$(XI

such t h a t

$

say f o r

0,

Q,

f o r every

= 1

x

i n a neighbourhood of

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

>O,

i t f o l l o w s from what w e j u s t proved t h a t

-.

B e s i d e s , w e have now

1x1 < n / t .

every function

Q

f

it(x)

=

gt + f f(x)(l - $(tx)) = 0

Thus, w e have been a b l e t o a p p r o x i m a t e i n

having

?(O)

=

0

L1

by means of f u n c t i o n s each o f

which h a s a F o u r i e r t r a n s f o r m v a n i s h i n g i n a whole neighbourhood o f 0. Suppose n e x t t h a t

.

is

J,

Cm

$(XI ,. = 1 f o r e v e r y

x

t

Indeed,

w i t h compact s u p p o r t , and s t i l l

such t h a t

1x1 <

Q.

For

f e L1

f ( 0 ) = 0 , t > 0 , E > O , c o n s i d e r now t h e f u n c t i o n s g t r E = ( f - f * $,) 1 uE. We c a n s e e t h a t g t t E + f +

m

and

E

+

0.

in

with L'

as

111.1. Hp FOR THE UPPER HALF PLANE

IIf

- gt,Jll

5

IIf -

IIf -

=

* VElll

f

* IE +

f

IIf *

+

It

*

f

!Jt

* vEll1 5

Ilf

* IElll 2 t

Now

( g t , €) - - ( X I = 3 ( x ) ( 1 -

does not

f e L1

t h a t every

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

and h a v i n g

- f * Slelll

IIf

+

*Itlll

+o

0)

+

Ix

if

0

.. =

E

m ,

+

G(td)?(EX)

belong t o t h e s upport of

239

or

q / t

if

EX

a compact s e t .

J,

We have shown c a n be a p p r o x i m a t e d i n L'

F(0) = 0

e a c h o f which h a s a F o u r i e r t r a n s f o r m w i t h 0. A c t u a l l y , i f f e ReH 1 , we have

gt,€,

compact s u p p o r t n o t c o n t a i n i n g gt,

+

E +

gt,E because

f

in

f

in

as

ReHl 1

t

E

0; t h a t is: not only 1

+

H(gt,, ) * H(f)

and

the functions i n not containing is t o take

ing

.

L

This i s so

whose F o u r i e r t r a n s f o r m h a s compact s u p p o r t 1 i s dense i n ReH A l l t h a t r e m a i n s t o be done

.

0,

f e ReHl

has compact s u p p o r t n o t c o n t a i n -

c a n be approximated i n

f

m

?

s h a l l need i s t h a t E

Since

define

> 0

I

is

as

E

Cm.

0 , by t a k i n g

+

$

Then

A l s o , s i n c e t h e s u p p o r t of

Let u s s e e , f i n a l l y , t h a t

ReHl

If(x) - fE(x)ldx

=

I:

fE

+

,.

211111mlf(x)I ,

as

E

0.

+

Now

as

0

as

E

+

0

E

+

i t is

IE shrinks

for

E + O

and i s

a.e.x, Thus

fE

( H ( f E ) ) - ( x )= - i ( s g n x ) ( f E ) ^ ( x ) . For 0 <

E

2

E

0.

0.

as

0

+

an i n t e g r a b l e f u n c t i o n .

s u f f i c i e n t l y small, say f o r

+

E

f

in

>O

the supports of a l l t h e

0'

( f E ) ^a r e c o n t a i n e d i n t h e s i n g l e s e t

functions

IE(x).

s),

does n o t c o n t a i n

I f ( x ) l l l - $(--Ex)ldx

because t h e i n t e g r a n d tends t o bounded by

in

f

3 *

s u f f i c i e n t l y s m a l l , we s h a l l

> 0

E

= 1).

;(O)

(fE)-(x)=

i s i n t h e Schwartz c l a s s

fE

-m

be as above ( a l l we

!J

g u a r a n t e e t h a t t h e compact s u p p o r t o f

m

by f u n c t i o n s

w i t h compact s u p p o r t and

Cm

(because

Cm

Let

f E ( x ) = f(x)$(-Ex).

( f E ) ^ is

clear that 0

is

ReH'

and has compact s u p p o r t n o t c o n t a i n -

C

T h i s can be done v e r y e a s i l y .

i n g 0. For

3

such t h a t

and show t h a t

0,

i s an

Hf

ReH'

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

L1

in

L1 function H ( g t , € ) = ( H f - H f * I t ) * !JE ( H f ) " ( O ) = 0. So f a r we have proved t h a t t h e s p a c e formed by

with

to

and

+

, but also

L

Ix:O < a < 1x1 < B }

m

f o r appropriate

a

and

p a c t s u p p o r t such t h a t a

<

0 <

1x1 < B . E

5

E

0'

8.

Let

m(x)

be a

m(x) = - i ( s g n x )

C

f u n c t i o n w i t h com-

f o r every

x

satisfying

Then ( H ( f E ) ) - ( x )= m ( x ) ( f ) - ( x ) f o r a l l x p r o v i d e d . But m = K f o r some K e L Therefore H ( f E ) = K * f E

E.

240

111. REAL HARDY SPACES

and consequently the convergence K e L 1 , imply that H(fE) + H(f)

f

+

ii

f

in

L1

L1

as

E

and the fact that +

0.

This ends the

proof.

2.

MAXIMAL FUNCTION CHARACTERIZATIONS OF

For

p > 1,

the space

ReHP

H

has been completely

identified with

ReLP. From now on, we shall concentrate our attention on ReHP for p 5 1. Our goal will be to understand ReHP, which is a space of distributions in IR, without going out of IR. Of course this has already been done for p = 1 , since we know that ReHl = = If e ReL' / Hf e L1l with the norm fWIIflI1 + IIHfII1. However, even for tion.

p

On

we consider the gauge:

ReL2

=

1,

we shall get a clearer real variable characteriza-

(2.1) The space of functions for which i t is finite becomes, once completed, the space

ReHP.

In this section, we introduce several

gauges which will turn out to be equivalent to (2.1) (in the obvious sense of dominating each other up to multiplicative constants), thus providing new ways to look at

ReHP.

The starting

point will be the gauge: (2.2) which is, of course, dominated by

(2.1).

The gauge

(2.2)

is

given by the non-tangential maximal operator P; associated to the Now we shall consider other approximations to Poisson kernel Pt the identity.

.

+

is such that I$(x)l 2 (const) (1 + I X ~ ) - ~ with In general if 2 > 1, we can associate with each f e ReL , a function defined on 2 IR+ by f(x,t) = ( f t +,)(x) where, as usual, +,(x) = t-'$(t-'x). Then we can use the function f(x,t) to define the following maximal (X

functions:

111.2. MAXIMAL FUNCTIONS

(i)

The non-tangential maximal function

ii)

In general, for N 2 1 , tion of amplitude N

* *V9N (f)(x) For

(iii)

M

1,

=

241

the non-tangential maximal func-

sup If(Y,t) I 1 x-y I
the tangential maximal function with exponent

M

Let us study the relations among these different maximal operators.

Proof:

For any

A

> 0, consider the open set

Ix e IR : *;(f)(x) > A} disjoint open intervals.

IX e IR 11x e

: t~;,~(f)(x) >

IR

**

:

V1N

EX =

.

Then E - U I . where the 1 , ' s are A - j, J J It is geometrically obvious that N A 1 c U Ij , and, consequently

=

J

(f)(x) > x 1 1 5 N

I j

1 1 . 1= N I E I . J A

Thus :

LEMMA

2.4.

where

C

Proof:

If

Mp > 1 ,

depends o n l y on

Fix

x e IR.

then: -

M

p.

Then breaking up IR: as the union of the sets k { (y,t) : 2 t LIx-yI < 2k+1t 1 , k = 0,1,2,..;

I(y,t) : Ix - yI < t),

242

111. REAL HARDY SPACES

we o b t a i n :

Thus, by t h e p r e v i o u s lemma

CI

1 - Mp < 0

Since

As a consequence o f

dominated by t h e gauge Mp > 1.

provided

**

we s e e t h a t t h e gauge f k I ( P M (f)11; is P f)-+ I I P ; ( f ) l l p m u l t i p l i e d by some c o n s t a n t ,

2.4,

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

Now we s h a l l u s e t h e P o i s s o n k e r n e l t o c o n s t r u c t a f u n c t i o n

in

0 U

the class

o f S c h w a r t z , s u c h t h a t t h e maximal f u n c t i o n

S

is

o,(f)

*U

dominated by a c o n s t a n t t i m e s that Hp.

We s h a l l e v e n t u a l l y p r o v e

PM ( f ) .

c a n be u s e d i n p l a c e of t h e P o i s s o n k e r n e l t o c h a r a c t e r i z e

o

1.1

e s with f 0. a s a b r i d g e t o p a s s from t h e P o i s s o n k e r n e l P t o

I n d e e d , t h e same can be s a i d a b o u t any

We s h a l l u s e

o

a n a r b i t r a r y smooth k e r n e l lows:

Let

t~

11.

The c o n s t r u c t i o n o f

which i s r a p i d l y d e c r e a s i n g a t m and s u c h t h a t j y s k t J ( s ) d s = 0 f o r k = 1,2,.,. ( a c o n c r e t e q p r o p e r t i e s i s g i v e n i n S t e i n [l) O( X )

=

j;

o

be a r e a l c o n t i n u o u s f u n c t i o n d e f i n e d on

y(s)Ps(x)dx.

p.

182).

i s as f o l [l,m)

jy$(s)ds = 1

and

with the required

Then d e f i n e

I t c a n be s e e n t h a t

o

e s

.

Indeed,

(I

i s c l e a r l y i n t e g r a b l e and i t f o l l o w s e a s i l y from t h e e x p r e s s i o n

.. O(X)

=

1"' q ( s ) e-2115 I x I ds, t h a t

A

0

e

s

and, consequently:

L

Other p r o p e r t i e s o f i) ii)

G

t h a t we s h a l l be u s i n g a r e :

G

i s even

o

h a s v a n i s h i n g moments of e v e r y o r d e r 2 1.

Indeed, f o r

k

=

1,2,

...,

we h a v e :

0

e

s

.

111.2. MAXIMAL FUNCTIONS

o

243

gives rise to the approximate identity

a t , t > 0,

where

The corresponding maximal function satisfies

LEMMA

2.5.

a",f)(x)

**

5 c * p M (f)(x)

depending o n l y upon

with C

Thus, if

M.

Mp > 1 , as we shall assume from now on, we have the fol-

lowing chain of gauges:

Now we pass from a t to a very general kind of approximate identity. We do it in several steps.

LEMMA and

2.6.

0 < s(l.

Proof:

Let

Let

JI

e S

and s u p p o s e that

=

5

*

Then:

(y,t) e .1:

with

Ix - yI < t.

Then:

qS

with

ce

s

111. REAL HARDY SPACES

244

El

as w e wanted t o p r o v e .

LEMMA I-

Let

2.7.

be -

Cm

where

b e t h e f u n c t i o n a p p e a r i n g i n lemma

o

with support contained i n

depends on

C

or

either

n

Proof:

I- = I-

2.5.

Let

Then:

[-1,1].

( a p o s i t i v e i n t e g e r ) b u t d o e s n o t depend on

M

f. m

*

o

*

o +

1

* 02-k-l

( I - * 0 2-k-l

k=O

-'*

a 2-k

*

a 2-k )

I n d e e d , t h e p a r t i a l sum o f t h e r i g h t hand s i d e i s

which c o n v e r g e s u n i f o r m l y t o

The above e x p a n s i o n f o r

where with

ci-

0

I-

n

as

k

-t

m ,

since

can a l s o be w r i t t e n as

= - u and G = ando2-'o+ i n place'of

'2-1(I

+

G . Now w e a p p l y lemma and, observing t h a t

2.6

111.2.

MAXIMAL FUNCTIONS

245

we g e t :

where

with

C

C

d e p e n d s o n l y on

d e p e n d i n g o n l y on

Now, u s i n g t h e f a c t t h a t

M.

M.

o-

i s e v e n and h a s a l l moments e q u a l t o

z e r o , we c a n w r i t e :

Theref o r e

But, t a k i n g i n t o account t h a t and

with

0 5 r 5 1,

C

n

has support contained i n

the following estimate holds:

d e p e n d i n g o n l y on

M,

so that:

r-l,l]

111. R E A L HARDY SPACES

246

-m

s e [O,l],

f o r every

with

C

d e p e n d i n g o n l y on

t h e e s t i m a t e w e have j u s t o b t a i n e d w i t h

2.8

M.

Now we u s e i n

.

s = 2-k,

k = 0,1,2,...

We g e t

n

w a n t e d t o show. LEMMA

With

2.9.

tained in

[-A,A]

n*

5

V,A

(f)(x)

where

us

0

f o r some

C-(

c

I

n(u))du

d e p e n d s on

C

is a

A

+

rl

&

m

w i t h s u p p o r t con-

C

2 1.

AM+'

**

I I I ( ~ + ' ) ( ~l )d u ) o M ( f ) ( x )

b u t d o e s n o t depend on e i t h e r

M

q:,Lf)(x)L

J u s t observe t h a t

Proof: = Aq(Ax)

-c

above, l e t

(

nlIA)i(f)

(x) where

function with support contained i n

Cm

we j u s t n e e d t o a p p l y

2.7

with

n

1/A

i n place of

q,

op

n,f

A

ql/A(x) = Now

[-1,1].

and r e a l i z e

that

m

Now, l e t

be any

6

C

f u n c t i o n w i t h compact s u p p o r t .

t h e smallest closed i n t e r v a l containing t h e support of d e n o t e by that

x6

the centre of

Let

and

C,

I+. Suppose t h a t t h e p o i n t

I,+ be

x

is such

d i s t ( x , I g ) < I I y [ . Then

Observe t h a t

Ix - x

the function

n(y)

=

6

I

< (3/2)11,,,1.

$(xC- y).

Let

A = (3/2)11&.

The s u p p o r t o f

Q

Consider

i s c l e a r l y con-

111.3. ATOMIC DECOMPOSITIONS

tained i n

[-A,A).

247

Since

f ( t ) y ( t ) d t / 5 ni(f)(x)

m

we c a n u s e lemma

[!If

2.9

t o conclude t h a t

( t) * ( t

L e t u s now d e f i n e t h e maximal o p e r a t o r :

where t h e supremum i s t a k e n o v e r a l l t h e p a c t s u p p o r t and s u c h t h a t

J,

the following THEOREM

Taking

2.10.

M >l/p,

*

**

SM(f)(x) 5 C*oM ( f ) ( x )

functions

Cm

d i s t ( x , I ) < 11

*

with

I.

$ w i t h com-

Then w e h a v e p r o v e d

C

d e p e n d i n g o n l y on M.

t h i s r e s u l t a d d s a new l i n k t o o u r c h a i n o f g a u g e s

which now l o o k s l i k e t h i s :

A c t u a l l y , a l l t h e s e g a u g e s a r e e q u i v a l e n t ; b u t t h i s f a c t w i l l be t h e c o n s e q u e n c e of t h e a t o m i c d e c o m p o s i t i o n t o be p r o v e d i n t h e n e x t s e c tion.

3.

A T O M I C DECOMPOSITIONS

We a r e g o i n g t o i n t r o d u c e c e r t a i n f u n c t i o n s c a l l e d a t o m s which w i l l t u r n o u t t o be t h e b a s i c " b u i l d i n g b l o c k s " i n t o which any

Hp f u n c -

t i o n or d i s t r i b u t i o n c a n be decomposed. 0 < p 5 1 and r s u c h t h a t p < r and 1 5 r , a ( p , r ) - a t o m i s g o i n g t o be a r e a l - v a l u e d f u n c t i o n a , w i t h s u p p o r t c o n t a i n e d i n

For

111. REAL HARDY SPACES

248

an i n t e r v a l i)

and s a t i s f y i n g :

I

a " s i z e condition":

I

(

111

Ilallm 2 III - 1 ' ~ i f

or ii)

k

p =

l a ( x ) lrdx )

'Ir 2

. ., [l/p]

- 1,

r <

if

-

-. k a ( x ) x dx f o r every i n t e g e r 1 : [l/p) stands f o r the biggest integer

a "cancellation condition":

= 0,1,.

III-l/p

1

where

= 0

< l/P.

L

Note t h a t i f c o n d t i i o n

a,

t h e s u p p o r t of such t h a t (l/r)

-

J

i ) h o l d s f o r some i n t e r v a l

I

containing

t h e n i t a l s o h o l d s f o r any o t h e r i n t e r v a l

a.

c o n t a i n s t h e s u p p o r t of

( l / p ) < 0.

J c I

The r e a s o n i s t h a t

In p a r t i c u l a r , condition

holds f o r the

i)

s m a l l e s t c l o s e d i n t e r v a l c o n t a i n i n g t h e s u p p o r t of

a.

F o r t h e remainder o f

t h e s e c t i o n we s h a l l w r i t e No = [ l / p ] . A is always a ( p , r ) - a t o m f o r any o t h e r r. Thus, when p r o v i n g a g e n e r a l r e s u l t a b o u t a ( p , r ) - a t o m , w e do n o t need t o (p,-)-atom

r

w r i t e a s e p a r a t e proof f o r t h e c a s e

=

-.

I n the next f i v e

r e s u l t s we s t u d y t h e a c t i o n of o u r b a s i c o p e r a t o r s on a s i n g l e atom. The n o t a t i o n w i l l b e as f o l l o w s :

a

w i l l be a

(p,r)-atom,

smallest closed i n t e r v a l containing the suppport of c e n t r e of

I

and, as u s u a l ,

and c o n s e q u e n t l y t h e image of

a

I2

=

xI + 2 ( I

-

x,).

by l o o k i n g a t t h e a c t i o n of t h e H i l b e r t t r a n s f o r m 3. I .

with

2 ( H a ( x ) ( P d x2 C ,

proof;

We j u s t use t h e f a c t t h a t

(since

r 2 l ) , and a l s o t h a t

L< ( f o r any

the

llallp 5 1. We s h a l l s e e t h a t , f o r many o p e r a t o r s , s a t i s f i e s a similar i n e q u a l i t y with a p o s s i b l y big-

We s t a r t

g e r c o n s t a n t which d o e s n o t depend on t h e p a r t i c u l a r atom.

LEMMA

I

a, x the I Observe t h a t

R > 0)

5

H

p < r.

C

H.

i n d e p e n d e n t of

i s o f weak t y p e We h a v e :

a.

(r,r)

111.3. ATOMIC DECOMPOSITIOlJS

LEMMA 3 . 2 .

x

2

I :

0 < 8

where

Ix-x

For

249

-6

I

Y

< 1. But (y-xII < Ix-x11/2, and consequently Y (y-x ) I 2 Ix-x11/2, so that:

1

1

2

Combining the estimates obtained in the last two lemmas we get:

THEOREM 3 . 3 .

and r -

but

rm

1- IHa(x)lP dx 5 C not-on t h e p a r t i c u l a r

with

d e p e n d i n g o n Z y on

C

p

a.

Proof:

C < - , since p(NO+l) > 1 . This estimate, together with the one n obtained in 3.1 yields what we wanted to p r 0 v e . =

a e ReHp and llallReHp ‘C , r, b u t n o t o n t h e p a r t i c u l a r a t o m .

c o n s t a n t t h a t depends

THEOREM 3 . 4 . p

and

Proof: Consider the function

A(z)

of

z

=

x + it, t > 0

GI:

given tJy

250

111. REAL HARDY SPACES

We s h a l l show t h a t

t

i

lA(x

+

i t ) lPdx 5 C

with

C

independent o f

2 IFt * a ( x ) l P + I Q , * a ( x ) I P ,

S i n c e - " ( A ( x + it)/'

a.

and

0

i t w i l l be enough t o p r o v e t h e e s t i m a t e s :

But t h e s e e s t i m a t e s c a n be o b t a i n e d v e r y much i n t h e same way as t h e

Ha.

estimate f o r

Here a r e t h e d e t a i l s : IPt

By o b s e r v i n g t h a t 11,

a ( x ) l jC . M ( a ) ( x )

x.

instead of

H,

a s we showed i n C h a p t e r

t h e proof of

and u s i n g t h e same argument of

3.1

with

M

we g e t :

n

Now l e t

with

x

0 5

4

Then

1'.

<

Y -

1.

Using t h e e s t i m a t e

-N- 1,

I P ( N ) ( s ) l zCN(l+lsl)

we g e t IP

No) x-xI-8 (y-x,) Y ( --

No+ 1 I

t and s i n c e

Ix-x

I

-8

Y

(y-XI)I

-N

2 1x-x11/2 III

l/r-I/p

we f i n a l l y h a v e : NO

II

No+ 1

I X-X1 I No+l x-XI

I

)

Y

1111Ir'

=

-1

111.3. ATOMIC D E C O M P O S I T I O N S

T h i s i s t h e same e s t i m a t e as i n

251

We p r o c e e d as i n

3.2.

3.3

to

I2

to

obtain

.

2 1 P t * a ( x ) I P dx I C

T h i s e s t i m a t e combines w i t h t h a t p r e v i o u s l y o b t a i n e d f o r yield

1).

Next we s h a l l t r y t o g e t

F i r s t of a l l ,

2).

To e s t i m a t e t h i s i n t e g r a l we p r o c e e d a s i n t h e p r o o f o f

r 2 - 1,

observe t h a t , s i n c e

T h i s i s enough t o c o n c l u d e

* a ( x )I p .

IPt

Q ( s )= s/(1

+ s2)

P ( s ) = 1/(1

+

x

4

I*

i s c a r r i e d o u t e x a c t l y as

We j u s t need t o r e a l i z e t h a t t h e k e r n e l

s a t i s f i e s t h e same e s t i m a t e s t h a t we u s e d f o r

2

s ) , namely,

I Q ( N ) ( s ) I ._ < CN(l

.

-N- 1

+ Is[)

This ends

El

the proof. 3.5.

COROLLARY

IIPi(a + i H a )

l i pP

<

-

c,

i n d e p e n d e n t of

a.

With t h e n o t a t i o n of t h e l a s t p r o o f , P i ( . + i H a ) proof: o u r c o r o l l a r y f o l l o w s from theorem 1 . 1 and i t s c o r o l l a r y We have s e e n t h a t tions i n

and

that

The e s t i m a t e o f t h e i n t e g r a l f o r t h e one f o r

3.1.

w e have:

ReHP.

=

mA.

Thus,

1.2.

( p , r ) - a t o m s p r o v i d e v e r y simple examples of funcThe f u n d a m e n t a l r e s u l t w i l l be t h a t e v e r y f u n c t i o n

or d i s t r i b u t i o n i n

ReHP

c a n be decomposed i n t o

(p,~)-atoms. In

o r d e r t o a c h i e v e s u c h a d e c o m p o s i t i o n , we s h a l l u s e t h e f o l l o w i n g r e s u l t , which c o n t a i n s a r e f i n e m e n t of t h e c l a s s i c a l Calder6n-Zygmund decomposi t i o n . THEOREM

by

9

>

3.6.

1).

Let Let A

f

be a f u n c t i o n i n b e a r e a l number

ReL2 >

0

could be replaced

(2

&

N

& M

integers

252

a.e.

uaZ

111. REAL HARDY SPACES

2 a n d i n L -norm, w i t h Ii = I i ( h )

s u p p o r t e d i n an i n t e r k such t h a t

j g h ( x ) I z C h , bi h

for every integer

= 0

0 2k N. B e s i d e s , t h e i n t e r u a 2 . s Ii = I i ( A) a r e g o i n g t o b e p r e c i s e Z y t h e c o n n e c t e d c o m p o n e n t s of t h e o p e n s e t : {x e I R : S i ( f ) ( x ) > h j .

Proof:

For e v e r y

a Whitney t y p e d e c o m p o s i t i o n

Ii,

I i , j},

{

j

r a n g i n g o v e r a l l t h e i n t e g e r 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:

first divide

Ii

We

i n t o t h r e e i n t e r v a l s of e q u a l l e n g t h , of which,

t h e one i n t h e m i d d l e w i l l be c a l l e d w i l l be s p l i t i n h a l f . t o the r i g h t w i l l be

I.

190

and t h e remaining ones

O f t h e two i n t e r v a l s n e x t t o

I.

I i , O , t h e one

and t h e one t o t h e l e f t w i l l b e

191

Ii,-l.

The r e m a i n i n g i n t e r v a l s a r e s p l i t a g a i n and t h e p r o c e s s c o n t i n u e s indefinitely.

With t h i s d e c o m p o s i t i o n we a s s o c i a t e a p a r t i t i o n o f

u n i t y as f o l l o w s : [-I-&,

Let

f o r some

1+6)

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

0

be a n e v e n 6 < 1/2,

from

to

1

C m function with support i n

i d e n t i c a l l y equal t o 1

+

6.

1

center of

i

that is, terval

I i , j,

“j Ii

..

9J

let

is the r e s u l t of “adapting” the function Our c h o i c e o f

6

on

1

x.

Then i f

, 7

J

,

[-1,1]

is the

t o the in-

guarantees t h a t every point i n

Ii

h a s a n e i g h b o u r h o o d t h a t m e e t s a t most two o f t h e s u p p o r t s o f t h e i r i . ( s . Thus Cri!(t) i s C m i n I . and i t s a t i s f i e s t h e i n e q u a l i J .J J for e v e r y t e I ~ .L e t t i e s 1 2 ? + ( t ) ‘2 J J

J

J

The f u n c t i o n i Z $ . = X

J

J

Ii

.

i

0. is J

Cm

and h a s t h e same s u p p o r t a s

Next we s h a l l u s e t h e f a m l y Let

i ($ . ) . . J 1,J

J

xk $’.(x)dx = 0 J Then,

if

E

d e n o t e s t h e complement o f

J

Clearly

t o decompose our f u n c t i o n

be t h e u n i q u e p o l y n o m i a l o f d e g r e e (N

P!(x)

rif.

(k

?Ii,

=

such t h a t

0,1,

...,

we c a n w r i t ?

N)

f.

111.3. ATOMIC DECOMPOSITIONS

253

Define g,(x)

=

1F

+

f(X)XE(X)

i

and f o r e v e r y

f(x)

=

!(f(x) J

= gA(x)+

- P Ji ( x ) ) @ \ ( x )

a.e..

Eb:(x) C

where

I P i ( x ) $ \ ( x ) l 2 Ch, d o e s n o t depend on P?(x) J

i

i:

bi(x) Then

P\(X)+

j

The i m p o r t a n t f a c t i s t h a t

i s an absolute c o n s t a n t , t h a t i s , i t

i , j , X or

If

Let u s prove t h i s .

f.

N = 0,

is j u s t the constant

I t f o l l o w s from t h e d e f i n i t i o n o f t h e e x t r e m i t y of

Ii

closest to

Si(f)(x) I . ., 1,J

t h a t , taking

x

t o be

M+l)

u ) I du

w e have:

m

.,!I B u t , by t h e d e f i n i t i o n o f t h e I i ' s ,

S i ( f ) ( x ) 5 A.

Also

f o r the

k i n d o f f u n c t i o n s t h a t we a r e u s i n g :

( i n d e e d , j u s t from t h e f a c t t h a t e a c h follows t h a t , f o r every

k,

4:

is a d a p t e d t o

~ - ~ l ( $ ~ ) ( k ) ( u )_ li C d IuI i

.

I.

.

1,J'

it

Be-

YJ I. . i t follows t h a t 1,J' J - $ ~% IIi,jl). Consequently lmll 5 C h . F o r t h e c a s e N > 0 , we ? see t h a t I P i ( x ) @!(x) I 5 C . 1rn31. I n d e e d , t h i s i n e q u a l i t y i s J i n v a r i a n t u n d e r t r a n s l a t i o n s and d i l a t i o n s so t h a t we j u s t n e e d t o sides, i

i s e s s e n t i a l l y c o n s t a n t on

since

prove i t f o r a f i x e d above.

can be t h e

considered

in

f o r example, $

i s an orthonormal b a s i s f o r t h e s p a n o f

{l,x,.,.,xN}

Then N

where

$,

(5,)

.

m

L 2 ( $ ( x ) d x ) , s a y , t h e one o b t a i n e d by t h e Gramm-Schmidt p r o c e s s .

Clearly

111. REAL HARDY SPACES

254

= CJml

i i I P j ( x ) +. ( x ) I Ch, it follows that lgA(x)I Z C A since, of J course, If(x)l s cs$(f)(x) a.e. (just write f(x) = l i m f * ~ , ( x )

From

E+c

with

a

J,

f u n c t i o n w i t h compact s u p p o r t , i n t e g r a l

Cm

and i d e n -

1

t i c a l l y e q u a l t o 1 i n some n e i g h b o u r h o o d o f 0 ) . I t a l s o f o l l o w s i ( i t is dominated that I b A ( x ) I C * S i ( f ) ( x ) . Then, s i n c e S L ( f ) e L2 by t h e H a r d y - L i t t l e w o o d maximal f u n c t i o n ) , Lebesgue dominated

convergence theorem t o conclude t h a t t h e s e r i e s

f ( x ) = gA(x) + fbk(x)

for

k

. . . ,N

0,1,

=

THEOREM 3 . 7 . teger 2 0

Let

&

is a l s o convergent i n

A sequence

(ai)

b)

A sequence

(Ai)

.

Finally

[7

S k ( f ) e Lp, Then t h e r e e x i s t :

where

M

i s an in-

( P , ~ )- a t o m s , and o f r e a l numbers s a t i s f y i n g

o f

such t h a t

f(x)

ges t o

f

i n the space

Proof;

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

fAiai(x)

gk(x) +

a.e.

for

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

x,

ReHP. let

k, m

=

2

with

f e ReL2

0 < p 2 1.

=

L

by t h e d e f i n i t i o n o f t h e P ' . ' s . J

a)

f(x)

we c a n a p p e a l t o t h e

.

1 bk(x)

i=l

b e t h e d e c o m p o s i t i o n o b t a i n e d i n theorem fixed As

k

N +

2 [l/p] -

--,

1.

gk(x)

-f

f ( x ) - gk(x) in

L2

C.Si(f)(x),

as

l i v e s i n the s e t {x : S i ( f ) ( x )

as w e l l ,

2k

=

and some

.

have convergence i n L2 because Igk(x)l 2 L , On t h e o t h e r hand g k ( x ) + f ( x ) a . e . t o a s e t o f measure

with

3.6

Now we p u t a l l t h e s e d e c o m p o s i t i o n s t o g e t h e r . k 0 uniformly, since Ig,(x)l C-2 We even

0

since

as

k

-t

W.

z

k 2k]

-f

m

which i s i n

because

which d e c r e a s e s

A s b e f o r e , t h e r e is convergence

I f ( x ) - g,(x)I

5 CSh(f)(x).

A l l these f a c t s

1 1 1 . 3 . ATOMIC CECOMPOSITIONS

imply t h a t and i n

L

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

f

2

,

255

a.e.

namely m

f(x) =

- gk(x)) =

T(g,+,(x) - m

1 ( 1 b ik ( x ) k

i

b;+,(x)

)

=

j

where Bik ( x ) = b ik ( x ) -

E

bi+l(x) {j:I.(k+l)cIi(k)}

.

J

O f c o u r s e t h i s c a n be done b e c a u s e e a c h

unique

Ii(k).

constant.

=

Therefore

0.

1

Bk(x)l 2 C ‘ 2k ,

With t h e same c h o i c e o f

:A

=

C,

i s a ( p , m ) - a t o m and C 2 k l I i ( k ~ ~ 1 ’ p . The c o e f f i c i e n t s

a.e.

and i n

E 7i

k

t o our f u n c t i o n

L2

f(x) = hi

i i

f Xkak(x)

T h u s , we h a v e found a d o u b l e s e r i e s f

and f o r a l l

w i t h some a b s o l u t e

l e t us define

i

Then, c l e a r l y , e a c h where

i s contained i n a

x e I i ( k ) , Bk(x) = g k + , ( x ) - g k ( x ) ,

If

Bik ( x )

other x,

I .(k+l) J

i

$

i i

$hkak(x),

satisfy:

which c o n v e r g e s

and i s s u c h t h a t

IX:

Observe t h a t f o r f i x e d c o n t a m s a t most

x

and

k,

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

in

i

ne non z e r o e l e m e n t and we c a n f i n d a n e n u m e r a t i o n

o f t h e d o u b l e i n d i c e s which a l l o w s u s t o view o u r d o u b l e s e r i e s a s an ordinary s e r i e s

J

a.(x)

J J

converging t o

f(x)

a.e.

However

t h e summation i n t h e

L2

norm h a s t o be p e r f o r m e d f i r s t i n t h e

index

k.

Still this

f o r us.

i

and t h e n i n t h e

L2

convergence i s u s e f u l

L e t us consider the following a n a l y t i c functions of

111. REAL HARDY SPACES

256

+ it, t

z = x

and f o r e a c h

> O : F ( z ) = Pt

(f

x

+ iHf)(x),

i,k, = Pt

A’(z)

* ( aik + i H a i ) ( x ) .

k Then t h e

convergence i m p l i e s t h a t

L2

double s e r i e s c o n v e r g e s i n s u b s e t s . Now theorem 3 . 4 in

Hp.

F(z) =

hiAi(z)

Thus i n o u r s e r i e s

~

~

A 2 ~ C ~,

~some H ap b s o l u t e c o n s t a n t .

z ~ . a converges i n J j f.

naturally, our function For

f e ReL N

2

P,r

ReHP

Ailp a l l o w s

I

f

T h i s f a c t t o g e t h e r w i t h t h e f i n i t e n e s s of t h e sum

u s t o conclude t h a t t h e s e r i e s converges normally i n t h a t t h e simple s e r i e s

where t h e

H2 a n d , t h e r e f o r e , u n i f o r m l y on compact t e l l s us t h a t (p,m)-atoms a r e uniformly

i t follows

Hp.

and t h e l i m i t i s ,

, let ( f ) = inf I (

llAiIp)l’p

: f(x) =

i

1 Aiai(x)

a . e . and i n

i

ReHP ). the ails being a r b i t r a r y (p,r)-atoms. Since f o r rl a ( p , r )-atom i s always a ( p , r l ) - a t o m , w e c l e a r l y h a v e :

<

r N

2

2’ P9r

(f)z

( f ) . We have j u s t proved i n theorem 3 . 7 t h a t ( N p , _(f)!P I P9r2 P T h i s a l l o w s u s t o e x t e n d o u r c h a i n o f gauges o v e r < C. I l S i ( f ) I l p . - 2 ReL , o b t a i n i n g , i f M > l / p : N

A c t u a l l y a l l t h e s e gauges a r e e q u i v a l e n t as we s h a l l s e e now. L e t f e ReL2 be t h e sum of t h e s e r i e s Z L a . b o t h i n t h e a . e . s e n s e 1 1 1

and i n

ReHP,

with

$1

Ailp

<

m,

and t h e

Then, c o n s i d e r i n g t h e a n a l y t i c f u n c t i o n s o f F ( z ) = P t * (f + i H f ) ( x ) , =

( P t + i Q t )* a i ( x )

ing i n

Hp

and f o r e a c h i n d e x

we have

F(z) =

5

being

ails

z = x

(p,r)-atoms.

+ it, t

’ 0 ,

i , Ai(z) =

AiAi(z),

t h e s e r i e s converg-

a n d , c o n s e q u e n t l y , u n i f o r m l y on compact s u b s e t s of

It follows that

Rf.

111.3. ATOMIC DECOMPOSITIONS

257

Since t h i s t h e l a s t i n e q u a l i t y b e i n g a c o n s e q u e n c e o f theorem 3 . 4 . i s t r u e f o r any s u c h r e p r e s e n t a t i o n o f f , we f i n a l l y a r r i v e a t :

which c l o s e s t h e c h a i n o f g a u g e s and shows t h a t a l l o f them a r e e q u i valent.

The c o m p l e t i o n o f t h e s p a c e d e t e r m i n e d on

ReL2

t h e e q u i v a l e n t g a u g e s a b o v e , i s a n e q u i v a l e n t copy o f we h a v e s e v e r a l r e a l - v a r i a b l e

c h a r a c t e r i z a t i o n s of

by any o f

ReHP.

Thus,

L e t us

ReHP.

s t a t e s e p a r a t e l y t h e c o r r e s p o n d i n g theorems.

THEOREM

atoms < C

-

3.8.

(aj)

I I f l ( R e H l,

verging t o b)

1

f

C o n v e r s e l y if

f

J J J

Proof:

<

a.e.,

t h e s e r i e s con-

i s a r e a l m e a s u r a b l e f u n c t i o n s u c h tJgt_

a.e.,

with

( 1 , r i - a t o m s f o r some f i x e d =

SIXj[

such t h a t f ( x ) = ZX.a.(x) J J J a l s o i n ReH 1 ,

f ( x ) = ZA.a.(x)

IIflIReH~

.

a ) Let f e ReH Then t h e r e i s a s e q u e n c e of ( 1 , m I and a s e q u e n c e of r e a l numbers ( x j ) with 2

r,

cIXjl

J

1

<

-, and t h e

<

r 5 -

then

a;

a . ’ s be,ng

J f e ReH’

&

C*Slh.l. J J

a)

We c a n f i n d a s e q u e n c e

ffn}

of functions i n

fn * f

ReHln

as n + We c a n a l w a y s assume, by t a k i n g a s u b s e q u e n c e i f n e c e s s a r y , t h a t such t h a t

IIfnIIReH1I_ IIflIReHl and

fn(x) + f ( x )

IIfn

in

ReHl

L2 m.

a . e . and

-

fn+lllReH1 < 2-nllfllReH1

Then, m

with

f o ( x ) = 0.

According t o theorem

3.7,

w e can w r i t e , f o r each

n 20 fn+,(x) with

flXnil

5C2

-n

fn(x) =

f

hniani

X)

(IflIReH1, and t..e

an i. I s

being

A l l t h e r e p r e s e n t a t i o n s considered a r e convergent i n

(1,m)-atoms. ReH 1 , Then

111. REAL HARDY SPACES

258

and s i n c e FlAlil + 1

I;

1 IA,~I

IIflIReH1

'C

n i

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

in

f

ReH' ,

b)

If

being

f(x) = Li.a.(x) J

J J

a.e.

with

J" I A . 1 J

<

and t h e

m

a 's

( 1 , r ) - a t o m s , then t h e s e r i e s converges n e c e s s a r i l y t o

f

in

R e H l and IlflIReHl2 C * z l x j l . Indeed, t h e e s t i m a t e i n 3 . 4 i m p l i e s t h a t t h e r e i s c o n v e r g e n c e i n ReHl and t h e r e s t i s s t r a i g h t f o r w a r d .

ci There i s a c o r r e s p o n d i n g r e s u l t f o r

THEOREM

a) Let (p,ml-atoms

0 < p < 1.

f e ReHP, 0

c p < 1. Then, there is a seand a sequence of real numbers ( h j ) with JL ( hJ. I P ( C ( ( f ( ( E e H p such that f = I~ x .J ~ J. -'t h e series converging to f $2 ReHP and, consequently, a l s o -in t h e sense of teapered distributions. bi Conversely if f is a tempered distribution such that f = CX.a J J j in the sense of tempered distributions with $ 1 ~ ~ 1 ' < m, a n d the a . ' s being (p,rl-atoms for some fixed r , 1 5 - r 5 then J P f e ReHP g& IlfllReHp 2 C ljlXjl P

3.9.

querzce of

(aj)

.

Proof:

I t is v e r y much l i k e t h e p r o o f o f t h e p r e v i o u s theorem w i t h

conveIsgence i n t h e s e n s e o f tempered d i s t r i b u t i o n s i n s t e a d o f c o n v e r gence

a.e.

The equival'ence between t h e g a u g e s

f+

llP;(f)

P

IIp

and

f w IIP",f + iHfIIP g i v e s u s t h e 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 P which i s known as t h e Burkholder-Gundy-Silverstein t h e o r e m .

ReHP,

THEOREM 3.10. A tempered distribution f is in ReHP, 0 < p 1, if andonly if it is the boundary distribution corresponding to a real harmonic function u ( x , t ) R: for uhich the maximal function ~

mU(x) =

belongs to Proof:

Lp.

SUP

I Y-x I < t Besides

lu(y,t)l

I/mUIIp

IIflIReHp ,

O f c o u r s e we know t h a t e v e r y

f e ReHP,

0 < p 5 1,

i s of

111.3. ATOMIC DECOMPOSITIONS

t h e form f = l i m u ( - , t ) w i t h u harmokig i n IR: Conversely l e t f = l i m u(*,t)

such t h a t

4.23

morphic i n

i n chapter

with

IR+,

IlmUIIp

Then

+

vt = H ( U t )

Let

w e have

=

v

IIflIReHP

and

~

~

t

P

c I I ~ ~ I
s u p ~ , " (tu + i v t ) ( x ) = l i m P",u, t J.0 t >o

Then, F a t o u ' s lemma g i v e s

and

i s holo-

P*(u ) ( x ) &

It follows t h a t

mF( x )

Then

Write

it) = u(x,t) + iv(x,t)

u ( x , t + s ) = Ps * u t ( x ) ,

Since

IRf.

F(x

-.

<

that there exists

I1

P P IIp,"(ut + i v t ) l l P =< c l I ~ + V ( u ~P) l-5 I

F e Hp

LCllflIReHp.

ut e L 2 ( I R n ) .

C l e a r l y each

v(x,t) = vt(x).

< mU(x).

IlmUIIp

i n t h e s e n s e o f tempered d i s t r i b u t i o n s .

t+O (x) = u(x,t).

t define

i n t h e s e n s e o f tempered d i s t r i b u t i o n s ,

be r e a l , harmonic i n

u

w e know from theorem

259

+

ivt)(x)

~ i n ~s u c h~ a way t h a t

C m l l m U FI I F ,

IIFIIHP 2 I I m F I l p 2 C l l m U I I p .

=

Now w e s h a l l make s e v e r a l o b s e r v a t i o n s a b o u t t h e r e a l v a r i a b l e characterizations obtained f o r

ReHP.

F i r s t of a l l w e n o t i c e t h a t , even though a

p-atom n e e d s t o have by

d e f i n i t i o n v a n i s h i n g moments o n l y up t o o r d e r atoms a p p e a r i n g i n theorem theorems

3.8

and

3.9,

moments as we w i s h . N

3.7

[l/p]-1,

( p , -)of

a)

c a n be c h o s e n t o have as many v a n i s h i n g

A l l we have t o do i s t o u s e theorem

a s b i g a s we w i s h .

the

and, consequently i n p a r t

3.6

with

T h i s o b s e r v a t i o n w i l l be i m p o r t a n t f o r t h e

i n t e r p o l a t i o n theorems between Hardy s p a c e s . Looking back a t t h e p r o c e s s by which w e have o b t a i n e d an a t o m i c decomposition f o r

f e L

2

fl ReHP,

l o n g and somehow a r t i f i c i a l .

we have t o a d m i t t h a t i t i s r a t h e r

The q u e s t i o n n a t u r a l l y a r i s e s o f

w h e t h e r or n o t i t i s p o s s i b l e t o o b t a i n a n a t o m i c d e c o m p o s i t i o n d i r e c t l y from t h e P o i s s o n i n t e g r a l of

f.

answer i s y e s i f one c o n t e n t s h l m s e l f w i t h

We s h a l l s e e t h a t t h e (p,2)-atoms.

This poses

t h e a d d i t i o n a l problem o f b e i n g a b l e t o decompose d i r e c t l y a atom i n t o

(p,-)-atoms,

(p,2)-

a problem which h a s a n i n d e p e n d e n t i n t e r e s t ,

and t h a t we s h a l l a l s o d e a l w i t h .

111. REAL HARDY SPACES

260

3.11.

Different Approach to the Atomic Decomposition: First, let us try to decompose f e L2 n ReHP, 0 < p '1, by using u(x,t) = Pt 16 f(x). Our basic maximal function will be * (f)(x) (the 2 will be important!). Of course IIP;,2(f)llp 2 pv, 2 < CIIflIReHP with C an absolute constant. For each integer k we shall consider the open set Ek

=

{x e IR

;

F*

v $ 2(f)(x)

> 2k } = U I.(k)

j

J

where the I .(k)Is are the connected components of J interval I, we associate the lqtentll

Ek.

To each

and let

n = U Ij(k)

J

Also, for each ure 111. 3.1).

E > 0,

A ;

Ti

call

=

Tk(€) j

h

Ij(k)\ =

Ek+'

{(y,t) e !T

: t-

(see Fig-

Figure 1 1 1 . 3 . 1 We shall use a function $ e Cm(IRj, real, even, supported in with moments vanishing up to order [l/pJ-l and satisfying:

(-1,1],

The existence of such a J, is a very easy matter. First we can obtain a function in L2 , even, supported in [-1/2,1/2] say, with the appropriate number of vanishing moments. Then we can make it smooth by convolving it with a smooth function, also even and supported in [-1/2,1/2]. Then we just have to normalize to get the condition on $ , With the aid of $ we shall construct our atoms. n

F o r each

k,j, E,

we define the function:

III.3. ATOMIC DECOMPOSITIONS

261

Since the support of J, is contained in (-1,1], the function (y,t)H Jlt(x-y), for x fixed, can be different from 0 only for points of the cone r(x) = {(y,t):ly-xl < tl. Thus gk(E)(x; caa.be J different from 0 only for points x e I.(k). Now we estimate J

llg:(E)(12.

For any

J, e

S, we have:

J

J

Then we use Cauchy-Schwarz's inequality to get:

But

On the other hand, since u is harmonic, by Green's theorem, and denoting by a the derivative in the direction of the outer normal we have for points of the boundary aTj k(E) of T!(€), tlvul 2dydt

=

71

t8(u2)dydt = 1 .

E)

J

at av

at av

262

111. REAL HARDY SPACES

with

an a b s o l u t e c o n s t a n t .

C

The r e a s o n s f o r t h i s l a s t i n e q u a l i t y

a r e t h e following:

a)

lul

tlel

tlEI

and

a r e bounded by

C2k

on

k( aT.

Among t h e s e t h r e e p r o p e r t i e s , o n l y t h e p a r t of needs f u r t h e r j u s t i f i c a t i o n ,

E)

J

a)

that refers to

We s h a l l s e e t h a t

t ( V u 1( C . 2

k

k ( e ) w i t h C an a b s o l u t e c o n s t a n t . L e t u s w r i t e t h e e s t i on aT. mate for t aaut , t h e one f o r b e i n g e s s e n t i a l l y t h e same. We au use t h e mean v a l u e p r o p e r t y o f harmonic f u n c t i o n s t o w r i t e s ( x , t ) k(E) as t h e a v e r a g e of over a b a l l B cenfor ( x , t ) e aT. at x,t J t e r e d a t ( x , t ) w i t h r a d i u s t / c where c i s a g e o m e t r i c c o n s t a n t is contained i n ( s a y c = 3 ) , b i g enough t o g u a r a n t e e t h a t B x,t r 2 ( z ) = { ( y , s ) : l y - z l < 2 s } f o r some z 4 E k + l a n d , c o n s e q u e n t l y , k+ 1 Then we u s e G r e e n ' s t h e o that IuI 5 P i , 2 ( f ) ( z ) 5 2 On Bx,t' rem t o w r i t e :

tg

au

udx

.

aBx, t We g e t : au I tz l

Figure 111.3.2

It w a s essential that aperture aperture Thus

1

k(E) e L 2 , 'j

2

w a s used t o d e f i n e

t o define the "tents'l ( s e e Figure

i s supported i n

Ij(k)

111.

and o n l y

Ek

3.2).

and

A c t u a l l y , a l l t h a t we d i d , c a n be done a l s o f o r

E

= 0,

the only

263

111.3. ATOMIC DECOMPOSITIONS

difference being that this time we cannot define

J

because the integral may not converge absolutely. Instead, we debeing fine as a distribution, its action on a function 9 e S gj

given by the formula:

1 .

J

J

Then we proceed very much like before, obtaining that

gk

is an

L2

Ij(k)

with IIgkI( 5 C2k11 .(k)1%: A similar J 2 2 J k(E) -t gk in computation shows that L as E -t 0. Observe gj j also that it follows from the cancellatim properties of $, that function supported in

the gj k(E) and gJ have vanishing moments up to order [l/p]-l and, consequently, they belong to ReHP. Now for a function h e L2 , supported in an interval I and having vanishing moments up to is a order [l/p]-l, the function a(x) = 111 (1’2)-(1’p) I/hll;’h(x) (p,2)-atom. Thus, it follows from theorem

with

C

k(E) gj

-t

an absolute constant. : g

in

ReHP

as

We shall show below that

E

f =

-t

3.4.

that

By using this inequality we obtain 0.

E . gk k,J j

in

S’.

Once this is done, if

we set a.(x) k = C-12-k lIj(k)yl’pgk(x) J

J

then, each

ak J

is a

(p,2)-atom, and we would then write

This would be the desired decomposition, since

264

111. REAL HARDY SPACES

= 1CP2kP

k

IIIj(k)l j

=

Cp

12kPIEkl k

=

P

2 CIIflIReHP

*

A l l that remalns is to show that f = 1. gk in 8'. We shall show FiJ J first that f -- lirn Z gJ(') in S Indeed, for $ e S E - 0 k,j

.

=(by Parseval's formula) =

m

since

(-2ns)e-2ns$ ( s ) ,d s +l

boundedly as

E

+O.

The interchanging of the order of integration is legitimate. Indeed, the first integral containing the Fourier transform signs is absolutely convergent as one verifies by taking absolute values and then interchanging the order of integration exactly as we did above. After all I: e-2ns I $ ( s ) Ids < m. Thus, we have shown that

111.3. ATOMIC DECOMPOSITIONS

I n o t h e r words: n o t j u s t make

E

f = lim E'U

in

= 0

k,j

265

J

t h e above c o m p u t a t i o n b e c a u s e w e c a n n o t

g u a r a n t e e t h e convergence o f t h e s e r i e s

E . since the intek, J g r a l t o i t s r i g h t i s n o t n e c e s s a r i l y a b s o l u t e l y c o n v e r g e n t . But

t h i s i s o n l y a minor c o m p l i c a t i o n . E . gk j

converges normally i n

k,J

It is c l e a r t h a t t h e s e r i e s

Let u s w r i t e

C . g k. = g i n k,J J convprges normally i n ReHP.

ReHP.

we j u s t need t o o b s e r v e t h a t

g ( € ) -+

as

E

-+

0

because IIgj k

-

k(E)

-f

gj

gkj ( E ) IIReHp

g ( € ) -+ g

gence i n

S',

consequently

in

ReHP.

Say

g!

1 . gk

k,J

J

k f j g;(€) g

in

k,j.

in

ReHP

=

g(€)

as

ReHP

(as

Thus w e

have:

S i n c e convergence i n

it follows that f =

k,Ej g kj ( € 1

By t h e same t o k e n

E

+

in E

ReHP.

* 0.

O),

Now

Indeed

and

< 2C2kPIIj(k) I

which h a s a f i n i t e sum i n and

Indeed

ReHP.

in

f = g So

g(')+ ReHP

in

f

SI

i m p l i e s conver-

a s tempered d i s t r i b u t i o n s a n d ,

a s we wanted t o show.

The proof we have j u s t g i v e n i s n o t w h a t one would c a l l a " r e a l v a r i a b l e " p r o o f , s i n c e i t r e l i e s upon t h e h a r m o n i c i t y o f

u(x,t).

S t i l l , i t i s a n i c e d i r e c t proof of t h e i n e q u a l i t y N (f)' << P,2 w i t h o u t g o i n g t h r o u g h t h e l l g r a n d l l maximal function < < 1) P:( f ) 1) , S i ( f ) . T h i s i n e q u a l i t y i s enough f o r many p u r p o s e s . F o r example, i t l e a d s immediately t o t h e e q u i v a l e n c e between t h e g a u g e s 1) P; ( f ) 1) and

IlP",f

+

iHf)llF

and, c o n s e q u e n t l y , t o t h e c h a r a c t e r i z a t i o n of

ReHP c o n t a i n e d i n theorem

3.10.

However, i n o r d e r t o g i v e a com-

p l e t e a l t e r n a t i v e a p p r o a c h t o a t o m i c d e c o m p o s i t i o n s , we s h a l l a l s o g i v e a d i r e c t proof of t h e i n e q u a l i t y

N

P,"

(f)'<<

N P , 2( f ) P .

~ 1 we 1

111. REAL HARDY SPACES

266

have t o do i s t o show t h a t e v e r y

1 X.a. J J

a

(p,2)-atom

can be w r i t t e n as

ReHP w i t h t h e a ’ s being ( p , -)-atoms and j an a b s o l u t e c o n s t a n t . T h i s c a n be done q u i t e s i m p l y by XI A j l p 2 C , u s i n g t h e o r d i n a r y Calderbn-Zygrnund decomposition ( n o t t h e smooth

a =

in

v e r s i o n o f theorem 3.12

Here a r e t h e d e t a i l s :

3.6).

Decomposition of

(p,2)-atoms i n t o (pp)-atoms

To make t h i n g s as s i m p l e as p o s s i b l e , we s t a r t by assuming t h a t 1/2 < p

so t h a t

1,

( p , 2 ) - a t o m s and

(p,-)-atoms

need

o have j u s t

one v a n i s h i n g moment: t h e a v e r a g e . Let

a

be a

(p,2)-atom,

We s h a l l c o n s i d e r t h e f u n c t i o n Ilbll:-G,lII.

M( l f l ‘ ) ” .

For

I the smallest closed ins o t h a t IlalI2 5 11 ( 1 / 2 ) - ( 1 / P ) b ( x ) = l1l1”a(x), which h a s

and d e n o t e by

t e r v a l c o n t a i n i n g t h e s u p p o r t of

a,

Our b a s i c o p e r a t o r w i l l be M 2 d e f i n e d as M2f = a > 0 , which w i l l have t o be t a k e n a p p r o p r i a t e l y

b i g a t t h e end o f t h e p r o o f , w e l o o k a t t h e open s e t

U a = [x e IR : M2b(x)

> a} =

( x e IR : M ( b 2 ) ( x )

>

I t h a s t o be

a2}.

a b i g enough ( s a y , a > & ? i s U a c I*. I n d e e d , 2 f o r x G$ I * i s M ( b 2 ) ( x ) 5 - llbl12 / d i s t ( x , I ) ( 2 . Now, l e t Ua = U I be t h e decomposition o f U a i n t o i t s c o n n e c t e d components.

noted t h a t f o r

J j We know t h a t , for e a c h

J

J

We w r i t e

is:

j,

b(x) = go(x)

+

Xh.(x),

where:

j J

P O W

=

/IJI

1

b(y)dy Ij

-

if

x e I

I

h . ( x ) = b(x) b(y)dy J II,I I< J J where. C l e a r l y I g o ( x ) l 5 a and and

since

h.(x) = (b(x) J

I f we c a l l

-

j

if

x e I

j

and

h.(x) J

=

0

else-

go(x)) X,,(x). J

(2a)-lhj(x) = b . ( x ) , J

we have a f u n c t i o n

b.

J

living in

111.3. ATOMIC DECOMPOSITIONS

267

2

with Ilbjl12L [Ij[. The idea will be now to do for each b the J J same kind of decomposition that we have performed for b (with the same a ! ) and to build an induction process which will eventually I.

lead to the decomposition of a into (p,m)-atoms. We shall use multi-indices for the successive decompositions, in the following way :

h. . lives in an interval I . J,, J, I * * * , J n . jo,...,Jn-l fixed, the intervals I . J o * * * tJn-19Jn are the connected components of the set

The function

We have

.

. For * Jn with varying

.*

*

jn,

111. REAL HARDY SPACES

268

provided

wh e r e

a >

2C.

Thus, when

w e can w r i t e :

a > 2C

1im1I i n d i c a t e s convergence i n

L

1

.

We s h a l l p r e s e n t l y s e e

t h a t t h i s s e r i e s p r o v i d e s t h e d e s i r e a atomic d e c o m p o s i t i o n f o r

a(x).

Observe t h a t I g o ( x ) I 2 a and go l i v e s i n I 2 and h a s a v e r a g e i n such a way t h a t a-1 (2111)-1’pgo(x) = a o ( x ) is a (p,m)-atom.

lives i n jo, s i n c e l g j ( x ) l 5 a and lgJ and h a s a v e r a g e 0 , t h e f u n c t i g n a ( x ) = a- ( 2 7 1 j l ) - l ’ p . ~ ( ~ ) j w i l l be a (p,m)-atom. I n g e n e r a ? , f o r e v e r y no and e v e r y

Likewise, f o r every I2

Jo

0,

TII.3. ATOMIC DECOMPOSITIONS

269

jo,...,jn-l, the function a. J,

is a

9

...

(x)

(p,-)-atom.

+

=

a-l(211

Jn-1

9

I I-%

..

jo....jn-l (XI

jo,. , Jn-1

We can write:

1

(2a)"

I Ij

I "Pa

o''", Jn-1

jo.... 9Jn-l

1

We have already seen that j

I1J o ,

,...,J~-~

Therefore, the sum of the p-?h

(XI

Jo, . . . ,Jn-1

... ,jn-lI

5

+

... 1 .

2 n (Ca- 1 111.

powers of the coefficients in the

above decomposition will be:

provided namely a

Actually the previous requirement on c1 , > (2'C) 1'(2-p). 2C, is stronger than this one. This finishes the proof

a >

of the inequality In case

0

functions

<

p

NP,"(f)'

<<

NP * 2(f)'

for the case

1/2 < p z 1 .

1/2, we need to change the definition of the

go,gj C

,.. . Igj0,j,, . . . ,jn,... .

With the same notation

used at the beginning of the proof, we have to define, for

x

I. J go(x) = PI (b)(x), where PIj(b)(x) is the unique polynomial of degree 5' [l/p]-1, such that for every k = O,l, l/p] -1 is:

...,[

e

111. REAL HARDY SPACES

270

( b ) ( x ) l $ Ca f o r e v e r y x e I w i t h C a n a b s o l u t e conI j j s t a n t ( t h e proof of t h i s f a c t i s c o n t a i n e d i n t h e p r o o f of theorem

Since

IP

we s t i l l have w i t h o u r new d e f i n i t i o n I g o ( x ) I 2 Ca. B e s i d e s , have v a n i s h i n g moments up go, We p r o c e e d i n t h e same f a s h i o n w i t h t h e d e f i n i t o order (l/p]-l. . T h a t way we c a n g u a r a n t e e t h a t t h e t i o n of each g j o , . ,Jn-i corresponding a I s a r e indeed ( p , = ) - a t o m s , because t h e y jo9 ,Jo-l have t h e r i g h t c a n c e l l a t i o n . The r e s t of t h e proof r e m a i n s w i t h o n l y

3.6),

now t h e h j l s , and c o n s e q u e n t l y a l s o

.. ...

minor c h a n g e s i n t h e c o n s t a n t s .

[1

N a t u r a l l y t h i s p r o o f c a n be s u i t a b l y m o d i f i e d t o o b t a i n a d i r e c t proof o f t h e i n e q u a l i t y

1 2 r

and

N

P,=

(f)’

<< N

P,r

(f)’

f o r any

r

such t h a t

p < r.

3.13. D i r e c t Proof of Buritholder -Gundy - S i l v e r s t e i n Theorem: We have s e e n how theorem

f o l l o w s immediately from t h e i n e q u a 2 l l P ~ ( f ) l l E f o r , say, f e L There i s a 3.10

.

l i t y IIP;(f + i H f ) l l p < < P b e a u t i f u l d i r e c t p r o o f of t h i s i n e q u a l i t y due t o P . K o o s i s .

Even

though i t i s a c o m p l e x - v a r i a b l e s argument, i t i s worth p r e s e n t i n g an By s e t t i n g u ( x + i t ) = P t * f ( x ) and * ( H f ) ( x ) , we s e e t h a t what we need t o prove i s t h a t

a c c o u n t of i t h e r e . v(x

+

i t ) = Pt

Ilm 11’ 2 C l l m U I I F f o r C an a b s o l u t e c o n s t a n t . We s h a l l prove t h i s v P inequality for O 0 . I f we a r e a b l e t o p r o v e t h e i n e q u a lity

with

C

Calling

i n d e p e n d e n t of

t > 0,

vt(z) = v(z + i t ) ,

we c a n a p p e a l t o theorem

ut(z)

=

u(z + it)

1.1

to

and u s i n g t h e same

names for t h e c o r r e s p o n d i n g boundary f u n c t i o n s , we s h a l l s e e t h a t dx

271

111.3. ATOMIC DECOMPOSITIONS

To prove (3.16) for This is clearly enough, since mut(x) 2 mU(x). a fixed t > 0, we shalltry to compare the distribution functions of F o r each h > 0, we consider the open set E X = vt and m Ut'

=

(x e n:m, ( x ) t

rem

4.7

> A}

.

from chapter

Observe that

EX

is bounded.

Iu (x + is)I t2 is e IR+:(x( < N

I1 implies that

complement of the triangle

TN =

{x +

Indeed, theo-

z -

sj

in the if N

is

big enough, in such a way that EX is contained in the interval (-N,N). Let E A = U I . be the decomposition of EX into its connecJ J Over each I . consider the path ted components I j. J T . = {x + i dist(x,lR\I.):x e I.} formed by the two legs of the J J J Denote by r the isosceles right triangle whose other side is I . J'

oriented path consisting of the points of IR\ EX plus the points of T = U T . with the orientation corresponding to increasing x as J J

shown in

Figure

111.3.3.

1. 3

Figure

111.3.3

We write

and try to estimate the second term in this sum. The function 1 Therefore, as was pointed out in sec(ut(z) + ivt(z))2 is in H

.

tion

1:

(ut(x) + iv (x))2 dx t

=

0

This, together with Cauchy s theorem,implies that jr(ut(z)

+

ivt(z)

which, by taking real parts,leads to:

On

T

it is

dy = dx

or

dy

=

-dx,

consequently

111. REAL HARDY SPACES

272

Carrying this inequality to the previous identity we get

from which:

v2 'B\E, since

utdx 2

+

2jTutdx 2 2

dx 2

ut(z) 2 h

2

'B\E1

for every

z e T

(mu

2

) dx + 2X lEhl

t

and JTdx = l E X l .

But

h ( m u ) 2dx = 102sl{x e IR:s < mu (x) 2 X } ) d S = lB\Ex t t

and therefore

w>.ich implies

+

i,,,

Then

since

This finishes the proof.

111.4.

Hp I N H I G H E R DIMENSIONS

273

Hp SPACES I N H I G H E R DIMENSIONS

4. For

n > 1,

we t u r n t h e o r e m f

DEFINITION

4.1.

0 < p 2 1.

T h e n w e say t h a t

Let

is t h e b o u n d a r y

f

sup

=

IR"+'

f or which the maximal function

lu(y,t)lbelongs+to Lp(IRn).

uniquely determines

f

s p a c e w i t h p-norm g i v e n by

For

f

+

Clearly

I1

i n chapter

4.23

i s a p-normed

HP(IRn)

f ~ + I I f I IPH P ( I R n ) * ReHP

i n t h e pre-

From now o n , we s h a l l abandon t h i s n o t a t i o n , u s i n g

ceding sections.

IRnt1

u.

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

n = 1

i n s t e a d , unless there is a r i s k of confusion.

Hp(IR)

e HP(IRn)

and

i s t h e u n i q u e harmonic f u n c t s o n i n

u(x,t)

whose b o u n d a r y d i s t r i b u t i o n i s

theorem

f,

4.7

of chapter

implies that:

with f-

=

Hp( IRn)= 'lnluII p *

that

I1

mU(x)

In t h a t c a s e , w e d e f i n e

T h i s makes s e n s e s i n c e we know from t h e o r e m

If

a n d let

if a n d only if

HP(Rn)

belongs to

IY-XI

II f l I

lRn

be a t e m p e r e d d i s t r i b u t i o n in f

HP(IRn).

distribution corresponding to a r e a l harmonic

u(x,t)

function

i n t o a d e f i n i t i o n of

3.10

an absolute constant.

C

u

I t f o l l o w s from t h i s i n e q u a l i t y t h a t

i s a c o n t i n u o u s mapping from

Harm(lR:+l)

HP(IRn)

i n t o the space

formed by t h e f u n c t i o n s harmonic i n

IRnfl

with the

t o p o l o g y o f u n i f o r m c o n v e r g e n c e o v e r e a c h compact s i b s e t o f That Let

Hp(IRn)

i s a c o m p l e t e s p a c e c a n be s e e n i n t h e f o l l o w i n g way:

be a sequence of d i s t r i b u t i o n s i n

(f.)

J [If .-fklIHP(IRn)+ 0 J

as

t i o n corresponding t o that the functions En+'

u.

J u,

t o a certain

j,k f.. J

Denote by

-t

Hp(JRn)

fixed

u (x,t) s

>

0

=

J Then, t h e i n e q u a l i t y above g u a r a n t e e s

c o n v e r g e u n i f o r m l y o v e r compact s u b s e t s o f which i s , o f c o u r s e , n e c e s s a r i l y h a r m o n i c .

u(x,t+s;

and

c

such t h a t

t h e harmonic func-

u.

A c t u a l l y , t h e i n e q u a l i t y t e l l s u s t h a t u . --t u ntl J proper sub-hall-space t ( x , t ! e IF. :t 2 t o 1, call

IRn+'.

> 0,

and

u .

JYS

we h a v e

(x,t)

uniformly o v e r each

to

>

0. F o r

s

0

u . ( x , t i - s ) . Then, f o r J I u ( y , t + s ) - u . ( y , t + s ) i-1 6: for =

J

111. REAL HARDY SPACES

274

every

(y,t) e

provided

j 2 certain

s.

I t f o l l o w s , i n particular, t h a t

x,

provided

Imus(x)

I S

( x )I 5

and

E

f o r every

E

m U j t s ( x ) + mu s ( x )

I n o t h e r words:

j )jo.

d e p e n d i n g on

jo

- muj

as

j

+

m.

By u s i n g F a t o u ' s lemma we o b t a i n :

=

Ilfjll

SUP

P <

m (x) = l i m m

and, s i n c e

U

s+o

lemma y i e l d s

Thus,

u

m

HP(IR")

j

(x),

another application of Fatou's

us

g i v e s r i s e t o a boundary d i s t r i b u t i o n

t h e p r o o f we must show t h a t

f . J

+

as

mu.J , s - u s ( x )

provided

j )jo

k

+

m

for

b i g enough.

Hp(IRn)

F o r each

We p r o c e e d v e r y much a s b e f o r e . +

in

f

j

fixed.

f e HP(IRn).

as

s > 0,

j

+

-.

To end

But

rnuj,s-~k,s(x)

+

Thus

Since t h i s is t r u e f o r every

s

> 0,

we f i n a l l y g e t :

The o b s e r v a t i o n r i g h t a f t e r t h e p r o o f o f theorem implies t h a t , f o r

where

C($)

f

e Hp(IRn)

and

$

e S,

4.23

i n c h a p t e r I1

is

i s one o f t h e semi-norms d e f i n i n g t h e t o p o l o g y o f

T h i s h a s as a consequence t h a t t h e i n c l u s i o n

Hp(IRn)C-$

c o n t i n u o u s or, i n o t h e r w o r d s , t h a t c o n v e r g e n c e i n convergence i n t h e space

S'(IR")

s .

s'(lRn) i s

HP(IRn)

of tempered d i s t r i b u t i o n s .

implies

111.4. Hp IN HIGHER DIMENSIONS

275

We shall give a characterization of HP(IRn) in terms of atoms. This characterization will imply, in particular, that H 1 (IR") coincides with the space Hit(lRn) already introduced in chapter 11, with equivalence of norms. First, let us extend the definition of atom to the case DEFINITION

n

1. 0

For

4.2.

a ( p , r ) - a t o m in

IRn

<

and

p & 1

r

p

s u c h that

<

r

is g o i n g t o b e a r e a 2 v a l u e d f u n c t i o n

s u p p o r t c o n t a i n e d in a c u b e

Q c IRn

& lzr, a , with

and satisfying:

a "sizc condition":

il

i i ) a " c a n c e l l a t i o n condition":

a(x)xadx = 0

such that L

x

..

a = (al,. ,an) -1 l a 1 = al+...+ an = < n(p -I),

f o p every m u l t i - i n d e x a

formed b y natural numbers where

x a means

. .x;n.

x 1I.,$..

A s we noted after the definition of atom for

holds f o r every

Q

n = l , here also i) being a minimal cube containing the support of a.

The first thing will be to check that, for are uniformly in

HP(IR~). Since every

r

fixed,

(p,r)-atom a

(p,r)-atoms in

IR"

is

the boundary distribution corresponding to the harmonic function a(x,t) = Pt * a(x) for which ma(x) = Pi(a)(x), what we need to prove is the following:

THEOREM

Let

4.3.

llP~(a)ll~ 5 C, Proof;

Let

a

(p,r)-atorn in

a constant i n d e p e n d e n t of

Q

IR".

a

a. Q and set N = [n(p-'-l)), the The result will be a consequence of

two facts:

For every

n

a.

be a minimal cube containing the support of

Denote b y xo the center of biggest integer n(p-'-1).

2)

a. J

x

such that

Ix-xoI

>

diam (Q)

is

276

111. REAL HARDY SPACES

Indeed,

implies t h a t

2)

I C(diam ( Q ) ) since

or,

( N + l + n ) p - n + l > 1,

everything reduces t o proving

equivalently and

1)

i s p r o v e d e x a c t l y as lemma

1)

r- ( N + l + n ) p + n - l d r

(N+l+n)p-n.

N + l > n(p-'-1).

=

<

Thus,

2).

by u s i n g t h e f a c t t h a t

3.1

P i ( a )( x ) L CM(a) ( x ) .

In o r d e r t o p r o v e and f o r e v e r y

with

C

we s h a l l s e e t h a t f o r e v e r y

2)

(x,t)

such t h a t

Jy-x

0

I

s o t h a t , once (y,t)

To p r o v e

> diam

(Q)

a(x).

(y,t)

implies > Ix-xol-t,

yields

Ix-xo(

>

diam (Q)

and

2).

which allows u s t o s u b t r a c t

N

around

x-x

from

Pt(x-y)

its

i n t h e e x p r e s s i o n of

The c o m p u t a t i o n s a r e v e r y s i m i l a r t o t h o s e c a r r i e d o u t

i n t h e proof of theorem

Pt

t

ly-XI < t :

T a y l o r p o l y n o m i a l of d e g r e e Q

R

we have:

we t a k e a d v a n t a g e of t h e c a n c e l l a t i o n c o n d i t i o n

a,

f o r t h e atom

<

i s p r o v e d , we g e t , f o r

(4.4)

such t h a t

(4.4)

ly-XI

+y-XI 2 Ix-xol-ly-xl

Ix-x

=

Taking t h e s u p . i n

Ft

2 R-t,

an a b s o l u t e c o n s t a n t .

T h i s w i l l b e enough s i n c e

any

Ix-xoI

++

a(x)

=

3.4.

I,

Here a r e t h e d e t a i l s :

Pt(x-y)a(y)dy

=

ii)

111.4. Hp IN HIGHER DIMENSIONS

where, for a multi-index a = (al,...,a n ) , dard notation a ! = (a l!)* .(an!) and

we have used the stan-

. ..

a D"

Now, since

=

al ax,

Pt(x)

la

I and where

. .. axn'n

02

e

~ " ( ~ ~ ) ( x - x ~.(Y-x,)) -e Y

=

1.

<

Y =

we have for

t-"P(x/t),

=

277

N+1

=

t-n-N-1Da P(

x-x

-8

o

y t

4Y-Xo) )

From the f o r m u l a

we easily obtain:

Therefore, f o r

la1 = N+1,

we get

I D ~ ( P ~ )O( - a ~Y ~- (~Y - X ~ ) 5 ) It-n-N-1a'.

=

I -n-N-1 -<

C(t+lx-x - 0 ;(Y-X,)

I

()il-Wl I+jx-xO-~y~(~-x o) t

CR-n-N 1

since

-

t+ I x-x0- e (Y-X,) Y because

y e Q

I

2

implies that

t+lX-X

ly-xol 5 d i m (4)/2

Going back to the expression f o r IPt * a(x)I

CR

1-ly-x0 I 2 R-ly-x I 2 R / 2

-n-N-1

Pt * a(x), lu-x0l

N+ 1

<

R/2.

we conclude that:

la(y) Idy 5

270

111. REAL HARDY SPACES

CR-n-N-l

<

(N+l)n-'+l-p

-1

101

-

as we wanted to show. COROLLARY

E v e r y (p,r)-=

4.5.

a

IR"

IlallEp(IRn) 2 C ,

a c o n s t a n t d e p e n d i n g o n l y on b u t n o t on t h e p a r t i c u l a r a t o m . 1-1

Next we obtain atomic decompositions (with tions in TUEOREM

r

HP(IR")

is in

=

p,r

2)

&

with

n,

for nice func-

HP(IR"). ~ e t f e L~(IR") n HP(IR")

4.6.

f o r some

p,

o

<

p 51.

Then, t h e r e are: a)

A sequence

(aj)

Q

b)

A sequence

(Aj)

Q r e a l numbers w i t h

c

IR",

and

b e i n g an a b s o l u t e cazstant; such t h a t

f the

(p,2)-atorns i n

=

7x.a.

j J J '

c o n v e r g e n c e of t h e s e r i e s h o l d i n g i n t h e s p a c e

S'(lRn)

;2f

tempered d i s t r i b u t i o n s .

Proof:

For

n

=

1

we have seen two ways to obtain atomic decomposi-

tions. First we gave a decomposition based upon the grand maximal function and then we presented a more direct approach based upon a non-tangential Poisson maximal function.

Here we shall see how this

second approach extends with non essential changes to the case n > 1. We shall consider the function u(x,t) = P t ++ fix). This time our ii

P (f)(x), where the number N > 1 V,N will have to be chosen appropriately depending on the dimension n. In any case basic maximal function will be

1 1 1 . 4 . Hp I N H I G H E R DIMENSIONS

C=C(n,N)is a

where on

c o n s t a n t t h a t d e p e n d s on

n

279

and

N,

but not

The p r o o f o f t h i s i n e q u a l i t y i s e s s e n t i a l l y t h e same a s t h a t

f.

of l e m m a

The o n l y d i f f e r e n c e i s t h a t , i n d i m e n s i o n

2.3.

n

> 1,

w e h a v e t o g i v e a n e x p l i c i t way t o decompose t h e open s e t E~ = { x e IR" : P : ( f ) ( x ) > A ) i n t o n o n - o v e r l a p p i n g c u b e s . We s h a l l u s e W h i t n e y ' s d e c o m p o s i t i o n lemma ( s e e 7 . i n c h a p t e r 11) which t e l l s u s t h a t any open s e t where

(Qj)

$

R

c a n be w r i t t e n as

IF?

i s a sequence of non-overlapping

=

y

Qj

cubes s a t i s f y i n g

z

Doing t h i s decomposiEln\ R) ( 4 diam ( Q j ) . dist (Q diam ( Q . ) j' 'L J n+l tion for R = E w e o b s e r v e t h a t , i f w e call Q . = { ( x , t ) e IR+ A' J n+l x e Q j and 0 < t 5 6 diam ( Q . ) } t h e n { ( x , t ) e IR+ : 'L J % Indeed, i f (x,t) 4 Q j , we h a v e e i t h e r : u ( x , t ) > A} c Qj.

y

y

x

4

or e l s e

x e Q . and

t

> 6 diam

J f i r s t c a s e , it i s obvious t h a t

(ai)

f o r some

u ( x , t ) (P:(f)(x)

j.

In the

In the

A.

t h e r e w i l l be d i s t ( Q . , l R n \ E h ) 5 4 diam ( Q j ) , J some p o i n t y e En\ E such t h a t d i s t ( Q j , y ) < 5 diam ( Q . ) . It J follows that Ix-yl 5 6 diam ( Q . ) < t . Thus ( x , t ) e I ' ( y ) , a n d , J Next we c l a i m t h a t , f o r a n consequently u ( x , t ) 5 P;(f)(y) 5 A. second c a s e , s i n c e

M

appropriate constant

= M(n,Nj

we h a v e

We j u s t n e e d t o show t h a t , by t a k i n g

M

b i g enough, w e c a n g u a r a n -

(j;.

t e e t h a t whenever x 4 QMj , t h e cone r N ( x ) d o e s n o t meet T h i s i s q u i t e e a s y : we j u s t n e e d t o t a k e M > 1 2 N h - + 1. Once we h a v e p r o v e d ( 4 . 8 ) , t h e p r o o f o f ( 4 . 7 ) g o e s on e x a c t l y as t h e p r o o f o f lemma With

N

2.3.

Ek

=

= {x e IRn:P* V, N ( f ) ( x ) > 2k I Use W h i t n e y ' s lemma a g a i n t o w r i t e

s t i l l t o be d e t e r m i n e d , l e t Ek

f o r each i n t e g e r U Q.(k) J J

k.

where t h e

Q j ( k ) ' s a r e non-overlapping

cubes such

that:

To e a c h

n-dimensional

cube

6=

Also,

f o r each

cube

( ( x , t ) : x e Q,

E

>

0

call

Q, 0 5 - t

associate the (n+l)-dimensional and l e t 2 IQI1'")

Tk(E) = ( ( y , t ) e Tk

J

J

: t

2

E).

..

111. REAL HARDY SPACES

280

We s h a l l u s e a f u n c t i o n

1,

{x e IRn : 1x1 1'

w i t h moments v a n i s h i n g up t o o r d e r

and s a t i s f y i n g , i f

i (5)

For each

e,

Since k(E)

gJ

k,j

and

real, r a d i a l , supported i n

J, e C " ( # ) ,

= $(

[n((l/p)-lU

I 5 1) :

we d e f i n e t h e f u n c t i o n :

$ ( x ) = 0 whenever 1x1 2 1, we s e e t h a t t h e s u p p o r t of 3 i s contained i n Qj(k) Now t o o b t a i n t h e e s t i m a t e

.

we p r o c e e d e x a c t l y as i n t h e one-dimensional c a s e .

We j u s t need t o

get t h e estimate 2

tl V u ( y , t ) I d y d t 5 C 2 2 k l Q j ( k )I

jTk( E )

J and t h i s w i l l f o l l o w once we a r e a b l e t o show t h a t , by c h o o s i n g a p p r o p r i a t e l y , we c a n g u a r a n t e e t h a t e v e r y p o i n t

(x,t)

N

belonging

a T k ( € ) o f T k. ( E ) i s t h e c e n t e r o f a b a l l of j J which i s t o t a l l y c o n t a i n e d i n t h e cone TN(z) f o r some The p o i n t s ( x , t ) e a T k( . E, a r e of two t y p e s : J i n which c a s e t h e r e q u i r e d p r o p e r t y o b v i o u s l y x e Ek\ E k + l ,

t o t h e boundary radius t / C z e IRn\Ek+'. either

holds with

z = x,

or else

x e Qi(k+l) for

some

i.

In t h i s

l a t t e r c a s e , we c l a i m t h a t t i s b i g g e r or e q u a l t h a n one f i f t h t h e s i d e l e n g t h of Q i ( k + l ) . The r e a s o n i s t h a t Q i ( k + l ) w i l l be surrounded by o t h e r c u b e s b e l o n g i n g t o t h e Whitney decomposition o f E k +1

, a l l of them h a v i n g d i a m e t e r a t l e a s t as b i g as one f i f t h t h e

d i a m e t e r of

Qi(k+l).

Indeed, i f

Q,

and

Q2

a r e two c u b e s touch-

i n g each o t h e r , b o t h o f them members o f a Whitney d e c o m p o s i t i o n of an open s e t

w e must have:

s1,

Let u s s e e how

N

can be chosen t o g u a r a n t e e t h a t whenever

1 1 1 . 4 . Hp I N H I G H E R DIMENSIONS

281

e Q i ( k + l ) and t 2 7 s i d e ( Q i ( k + l ) ) , t h e n ( x , t ) e r N ( z ) f o r z e IRn\ Ek+'. S i n c e d i s t ( Q i ( k + l ) , R n \ Ek+')

x

some

< 4 diam ( Q i ( k + l ) ) , dist(z,Qi(k+l)) shall have

Iz-xI

(x,t) e

(N/5) s i d e ( Q i ( k + l ) ) .

taking

N

even b i g g e r , s a y

(x,t)

N = 30n,

(x,t)

we

a s above we

6 diarn(Qi(k+l)) <

N

By

>306.

we make s u r e t h a t t h e r e

with radius

which i s t o t a l l y

t/C,

TN(z).

Thus, w i t h t h e c h o i c e supported i n

x e Qi(k+l),

T h i s i s a c h i e v e d by making

e x i s t s a b a l l centered i n contained i n

r N ( z ) provided

such t h a t

Ek+l

diam ( Q i ( k + l ) ) . Thus, f o r

'6

s h a l l a l w a y s have <

z e IRn\

we c a n s u r e l y f i n d

5 diam ( Q i ( k + l ) ) . Then, f o r any

Qj(k)3

IIgjk ( € 1

[I2

w e have shown t h a t

N = 30n,

e L2,

gJ(')

is

and s a t i s f i e s ZkIQj(k)I'

'C

Now, e x a c t l y as i n t h e o n e - d i m e n s i o n a l c a s e , w e d e f i n e t r i b u t i o n , i t s a c t i o n on a f u n c t i o n

e S(Rn)

$

as a d i s -

gj

b e i n g g i v e n by t h e

f ormul a :

J

i s a n L~ Then we s e e , j u s t l i k e b e f o r e , t h a t gj t e d i n Q j ( k ) 3 and s a t i s f y i n g t h e same e s t i m a t e k(

gj

€1,

The f u n c t i o n s Q.

k(

E)

h e L2(lRn),

+gS

HP(IRn)

cn( ( l / p ) - l ) ) ,

( 1 ~ 2 ) - ( 1 ~ p ) ~ ~ ihs~ a~ ~ (1ph, Z ) - a t o m .

corollary

4.5

in

L~

as

by c o r o l l a r y

s u p p o r t e d i n a cube

v a n i s h i n g moments up t o o r d e r

IQI

k(E) gj

Q

+

0.

'(€1

gj

+

g;

in

4.5.

Now,

and h a v i n g

the function T h u s , i t f o l l o w s from

that

That way w e s e e t h a t

E

i n h e r i t the cancellation properties

and g j gj T h e r e f o r e , t h e y belong t o

for a function a =

each

as

namely:

A s i m i l a r c o m p u t a t i o n shows t h a t

of

f u n c t i o n suppor-

H P ( I R ~ ) as

E

+o.

111. REAL HARDY SPACES

282

If we a r e a b l e t o show t h a t f = C gk i n S'(IRn), then, s e t t i n g ak = C-12-k3n/21Qj(k)31-1/pgk we ks h , Ja l Jl h a v e IIajI12 k LIQj(k)q (1/2) - ( 1/ P, ) J J s o t h a t ak i s a ( p , 2 ) - a t o m . We would t h e n w r i t e J

and t h i s would be t h e d e s i r e d d e c o m p o s i t i o n ,

since

A l l t h a t r e m a i n s i s t o show t h a t

in

f

C.gk

=

k J J

S'(IRn).

F i r s t one

f = l i m k c . gk. ( E ) i n S'(IR'). The p r o o f i s f o r m a l l y t h e IJ J E ? o c k ( € ) + z k as E + o same as i n d i m e n s i o n 1. Then we have k, j g j k ,j g J i n H P ( I R n ) . B u t , as w a s p o i n t e d o u t a t t h e b e g i n n i n g o f t h e s e c t i o n ,

shows t h a t

convergence i n f

= kFjgj

COROLLARY

S' ( R " ) .

i m p l i e s convergence i n

a

and t h e p r o o f i s f i n i s h e d .

Let f e Hp(IRn), (p,~)-atoms ( a j )

Thus

< p 2 1.

0

IR"

T h e n , t h e r e is a a n d a s e q u e n c e of r e a l num-

with

bers (Aj)

C

S'(lRn)

4.9.

s e q u e n c e of

for

H p ( IR")

in

an a b s o l u t e c o n s t a n t , such t h a t

f

=

3 A.a. J J

i n t h e s e n s e of

tempered d i s t r i b u t i o n s .

Proof:

If

f

rnonic f u n c t i o n f o r each

t

i s t h e boundary d i s t r i b u t i o n c o r r e s p o n d i n g t o t h e h a r mn+ 1 w i t h mu e L p ( I R n ) , we know t h a t u(x,t) in +

0,

>

the function

x W u(x,t) = ut(x)

L p ( I R n ) fl L m ( I R n ) c L 2 ( I R n ) , and also t o H p ( I R n ) .

a sequence

f

j

of f u n c t i o n s i n

HP(IRn)

P

'1

fj-f j + l 'lHp(

2-J Ilf

II

n

belongs t o

Thus, we c a n p r o d u c e

L2(IRn)

such t h a t

P HP( R n )

and

f . J

+

f

in

St(IRn).

1 1 1 . 4 . H p I N H I G H E R DIMENSIONS

We w r i t e

+

f = fl

the functions theorem 3 . 8

J,=Yl ( fJ.+ l - f j )

283

and u s e theorem

f o r each of

4.6

fl' f j + 1 - f j . The p r o o f i s f i n i s h e d e x a c t l y as t h a t o f

c!

a).

part

We a c t u a l l y h a v e t h e f o l l o w i n g g e n e r a l r e s u l t : THEOREM

Let

4.10.

0

<

f e HP(IRn)

p < r,

p 21,

r 2

'1

For e v e r y

in -

IRn

and a s e q u e n c e of r e a l numbers

for

C

an a b s o l u t e c o n s t a n t , such t h a t

X

j

-

Conversely i f

f e

j J J

i s such t h a t

S'(lRn)

;I

tempered d i s t r i b u t i o n s , w i t h ( p , r ) - a t o m s , t h e n f e HP(IRn)

of

for c

- c TIXjlP I HP(IR") j an absolute constant.

Proof:

L e t u s prove

hjlP

J

&

Then (p,r)-atoms

with

f = 1X.a.

tempered d i s t r i b u t i o n s . bl

a.

t h e r e i s a s e q u e n c e of

a)

a. J

i n t h e s e n s e of

f : h . a . in t h e s e n s e J J and t d e a . ' s b e i n g J f

=

IIfllP

b)

first.

We h a v e

a

(p,r)-atoms

and

A such t h a t ;I h j I p < m. By c o r o l l a r y 4 . 5 , t h e s e r e a l numbers j J c o n d i t i o n s g u a r a n t e e t h a t t h e series C h a . converges normally i n j j J HP(IRn), say t o g e HP(IRn). On t h e o t h e r h a n d , s i n c e f = C A , a . j J J in S ' ( I R n ) and we know t h a t c o n v e r g e n c e i n H P ( I R n ) i m p l i e s convergence i n

S'(IRn);

i t follows t h a t

f = g e HP(IRn)

and

a ) w i l l f o l l o w f o r e v e r y r once we p r o v e i t f o r r = m . up t o We j u s t n e e d t o s e e t h a t e v e r y (p,2)now we o n l y have i t f o r r = 2 . atom

a

in

IRn

c a n be w r i t t e n as

a

C La. i n s9(lRn)w i t h t h e J J J C, some a b s o l u t e c o n s t a n t .

=

a . ' s b e i n g ( p , m ) - a t o m s and z l " I p 2 J J T h i s h a s b e e n done a l r e a d y f o r n = 1 i n t h e p r e v i o u s s e c t i o n .

But

t h e p r o o f g i v e n t h e r e c a n e a s i l y be a d a p t e d t o h i g h e r d i m e n s i o n s . We decompose t h e f u n c t i o n

b(x)

cube c o n t a i n i n g t h e s u p p o r t o f cubes

Q . for J

b2

at height

IQI1"a(x),

=

a, a

2

.

where

Q

i s a minimal

by u s i n g t h e Calder6n-Zygmund

For e a c h of them we s h a l l h a v e :

284

111. REAL HARDY SPACES

go(x)

P

(b)

b(x)

I

=

if

4

x

U Qj

J

(b)(x)

P

x e Q.

if

J

Qj

being the unique polynomial of degree I[n((l/p)-l)l

such

Qj

that

. ..

B = ( B ~ , ,Bn)

for every multi-index

-

h.(x) = b(x) J

J

Igo(x)l 2 C a

Then

We call

=

J

b.

J

b (x),

so that

J

The two basic facts are: i) Each Qj is a subcube of ii)

On each

Qj

$QjI

Sl 2 n((l/P)-l);

h.(x) = 0 J

and

elsewhere.

is

2 llb.11 2 J 2

lQjl

and go on by

a2.

with the same

J

and

and

(Ca)-'h.(x)

decomposing

I

with

x e Q.

P~.(b)(x) if

Q , since a is at least 1 1 2 , so that

M(b2)(x)

> a

5 I {x e lRn : M(b

2

)(XI

> a

2

)I I

J

After this, the proof is practically identical to the one given f o r dimension

1.

We know that

1

H (IR)

=

1

{f e L ( R ) : Hf e L1(R) 1.

extend this characterization to role of the Hilbert transform

H

IRn

with

n

>

H1(IRn)

=

1 € e L1(IRn)

1.

Of course the

will be played now by the system

formed by the n Riesz transforms R1,..., Rn, section 5. We shall eventually prove that: (4.11)

It is time to

: Rjf e L1(IRn)

defined in chapter 11,

f o r every j=l,2,..,n}

285

111.4. Hp IN HIGHER DIMENSIONS

j

...,n

1,2,

=

e L1(IRn)

f e H1(IRn)*f

That 5.14

and

R.f e L1(IRn) J

for every

follows from the atomic decomposition and corollary

in chapter

11.

To establish the converse will require some

work. R.f e L1(IRn) for every J we associate the following n+l harmonic = ' {(x,t) : x e IR", t > 01: functions in I R:+ u1(x,t) = Pt * Rlf(x), un(x,t) = Pt * Rnf(x), un+,(x,t) = u(x,t) Suppose

j = 1,2,

f e L1(IRn)

...,n.

is such that

With

f

...,

Pt + f(x). au

a)

These auk

A =ax.' axk

n+l

=

harmonic functions satisfy the conditions:

j,k e 11

,...,n

j f k

1,

J

au.

b)

E a x --a 2 t ,

j = l,...,n

j

n

au.

j=l a x . J

at

as one easily sees by using Fourier transforms, taking into account that (Rjf)^(c) = -i(~j/l~l)?(c). If one writes x t , the equations a) b) and c) can be written as: au

(4.12)

2 auk axk = ax.

;

j,k e {l,

...,n+l 1 ,

~

+instead ~ of

j # k

J

n+l au. (4.13)

1

j=l

ax.J -- 0 J

or, in terms of the vector field curl

F = 0; div

F

=

(~~,...,u~,u~+~):

P = 0.

These are the so called generalized Cauchy-Riemann equations for the 2 vector field F. A C vector field F = ( U ~ , . . . , U ~ + ~satisfying ) (4.12) and (4.13) is called a conjugate system of harmonic functions. Observe that the harmonicity of each uk follows from (4.12) and (4.13) since

The terminology comes from the case

n = 1.

In that case equations

(4.12) and (4.13) for F(x,y) = (v(x,y),u(x,y)) reduce to v = u Y X and vx = -u which are the Cauchy-Riemann equations, necessary and Y sufficient for u + iv to be an analytic function of x + iy.

111. R E A L HARDY S P A C E S

286

In generaL for a C' vector field F = ( u ~ , . . . , u ~ +to ~ )satisfy (4.12) and (4.13) it is necessary and sufficient that F is the gradient of some harmonic function h. Indeed (4.12) implies that F is the gradient of the function

Then

(4.13)

implies that

h

is harmonic.

The crucial fact is the following: LEMMA

4.14.

F

Let

=

harmonic functions in

( u ~ , u ~ , . . . , u ~b+e ~a) system IF?" and call

j=1 E z(n-l)/n, J

Then for e v e r y Proof:

of conjugate

n+1 . is a subharmonic function in IR+

IFIE

We just need to examine the case

(n-l)/n

z z 1. E

It will

be enough to prove that A ( 1 F I E ) 2 0 in the set where F(x,t) f 0 (see the observation immediately before theorem 2.11 in chapter I).

We shall use the notation Then, writing

IFI'

=



'"

for the scalar product in we get

and

+ E ( E - ~ ) I F I € - <~ F \ -aaxF . ' J

Thus, adding in

j

=

We need to see that

1,

...,n+l:

2

.

IRn+'.

1 1 1 . 4 . Hp I N H I G H E R DIMENSIONS

But

is e q u i v a l e n t t o

(n-l)/n

287

1 / ( 2 - ~ )2 n / ( n + l ) ,

so

(4.15)

w i l l be e s t a b l i s h e d as s o o n as w e s e e t h a t

T h i s w i l l be s e e n by o b s e r v i n g t h a t f o r any tor

v

w i t h e u c l i d e a n norm

IvI

( n + l ) - d i m e n s i o n a l vec-

1 c

To p r o v e

(4.17)

d e n o t e by

A

( n 1) x ( n + l )

the

m a t r i x w i t h en-

(4.17) i s simply J lAvl , t h e s q u a r e o f t h e e u c l i d e a n norm o f t h e v e c t o r Av; and t h e 2 2 sum i n t h e r i g h t hand s i d e e q u a l s j f k l a j , k l = IllAlll , the square

tries

zj,k= a u k / a x . .

Then t h e l e f t hand s i d e i n

2

of t h e H i l b e r t - S c h m i d t

norm

lllAlll

of the matrix

A.

T h u s , i f we

write ( I A ( I = sup { [ A v l ; I v ( 5 1 1, t h e norm o f A as a l i n e a r o p e r a t o r i n t h e e u c l i d e a n s p a c e IRn+' , what we n e e d t o p r o v e i s :

Note t h a t

(4.12)

and

(4.13)

t r i c and h a s t r a c e e q u a l t o any m a t r i x

A

imply t h a t o u r m a t r i x

0.

w i t h t h e s e two p r o p e r t i e s s a t i s f i e s

t h e two norms a p p e a r i n g i n

(4.18)

Denote by

A1,A2,...,hn+l

= max

J

I x. l 2 Jo

By a d d i n g

nlh j o

n+ 1 C X j=1

whereas

/ A j [ ,

=

I2

(n+l)I x . JO

(4.18).

Since

i s a diagonal t h e e i g e n v a l u e s o f A. By t h e

i n v a r i a n c e o f t h e t r a c e , we must h a v e IlAll

i s symme-

are i n v a r i a n t under m u l t i p l i c a -

t i o n by a n o r t h o g o n a l m a t r i x , we may assume t h a t matrix.

A

We s h a l l c o n c l u d e by showing t h a t

t o b o t h s i d e s we g e t

=

A

0.

Now, c l e a r l y

111. REAL HARDY SPACES

288

which r e a d i l y g i v e s

(4.18).

L e t u s g o back t o o u r problem. R . f e L1(IRn)

f o r every

J

f e H1(IRn).

IR"":

= u ( x , t ) = Pt

F = (u,

in

...,n

We have a s s o c i a t e d w i t h

u , ( x , t ) = Pt

*

as

t

f

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

* R n f ( x ) , ~ ~ + ~ ( x= , t )

= Pt

These f u n c t i o n s form a c o n j u g a t e s y s t e m

,... , u ~ + ~ )S i. n c e

L1(IRn)

such t h a t

and we want t o p r o v e t h a t

,...,u n ( x , t )

Rlf(x)

* f(x).

f e L1(IRn)

We have

j = 1,2,

the functions

xwu(x,t)

f e H1(IRn)

t o see t h a t

-+ 0 ,

converge t o

f

i t w i l l be enough

l u ( y , t ) I belongs t o mU(x) = sup ).y-x I < t By u s i n g t h e f a c t , e s t a b l i s h e d i n l e m m a 4 . 1 4 , that lFIE

t o show t h a t t h e f u n c t i o n L1(IRn).

i s subharmonic f o r some actually, the function Since

mU(x) z m F ( x ) ,

f e H1(IRn).

(n-l)/n 5

0

<

mF(x) = s u p I F ( y , t ) l b e l o n g s t o L1(IRn). 1Y-x I < t t h i s w i l be t h e end o f t h e p r o o f t h a t

mF

Now t o see t h a t E

subharmonic.

< 1.

we s h a l l be a b l e t o show t h a t ,

< 1,

E

is i n t e g r a b l e , f i x

The key p o i n t i s t h a t

Since each

J

C

function i n

my+'

v(x,t) = IF(x,t)lE

i s a n o n - n e g a t i v e subharmonic Lp(IRn) w i t h p = 1 / >~ 1.

which i s u n i f o r m l y i n

Thus, a c c o r d i n g t o theorem h e Lp(IRn)

such t h a t

but s t i l l

1

t.

a c o n s t a n t independent of

This implies t h a t

<

t > 0

g r a b l e f u n c t i o n , we h a v e , f o r e v e r y

with

E

\FIE is i s t h e P o i s s o n i n t e g r a l o f an i n t e -

u.

E

such t h a t

4.10

i n chapter

11, there is

v ( x , t ) 5 P t * h ( x ) * o r , i n o t h e r words

Then mF(x) 2 ( P v ( h )( x ) )' l F ( x , t ) I 2 ( P t * h ( x ) 1'. C(M(h) ( x ) )' and t h i s i s a n i n t e g r a b l e f u n c t i o n s i n c e M(h) e L p ( I R n ) . We c a n e x t e n d t h i s t r e a t m e n t o f Hardy s p a c e s t o

i s a s y s t e m o f c o n j u g a t e harmonic f u n c t i o n s i n (4.19)

sup t>O

f o r some

p

1

IF(x,t)lPdx

p

<

" : R I

1.

Suppose

F

such t h a t

<

IRn

with

(n-l)/n

 1.

P r o c e e d i n g as

b e f o r e we c o n c l u d e t h a t mF e L p ( I R n ) . Thus u = u ~ + o~r ,any other conponent of F, gives r i s e t o a distribution f e I?(#). Conversely, i f f e we c a n a s s o c i a t e t o f a c o n j u g a t e s y s t e m F o f harmonic f u n c t i o n s

@(fl).

111.5. DUAL SPACES

289

in IR"+' such that (4.19) holds and f is the boundary distribu, last component of f. This takes tion corresponding to u ~ + ~the us back to function theory. Actually this was the way in which the space HP(IRn) was first introduced by Stein and Weiss [ I ] as a natural extension of the case n = 1. This approach works only for (n-l)/n c p 21. When p gets smaller one needs to consider more and more complicated notions of conjugate systems (higher order gradients).

5.

DUAL SPACES

In general if E is a p-normed space ( 0 < p 5 N - 1) with p-norm and L is a linear functional on E, it is clear that the continuity of L is equivalent to the existence of a constant C such that for every x e E IL(x)l 2 C ( N ( x ) )'Ip. The linear space formed by all the continuous linear functionals on E is called the dual space of E and shall be denoted by E * . F o r L e E X , we shall call IILII to the infimum of all the constants C in the above inequality o r , equivalently IILII

=

SUP I L(x)

: x e E

and

*

N(x) 2 l}.

.

is a norm in E The mapping L+IILII Besides, it is easy to see that E* is also comp ete, i.e., a Banach space. In this section our aim is to characterize the dual spaces n 21. The atomic decomposition for HP(IRn) (Hp(IRn))*, 0 < p '1, will allow us to settle this question in a rather simple way. Suppose

L e (Hp(IRn))*.

Then, for every

f e HP(IRn),

IL(f)l

In particular, if 12 r z m and p < r , we or every (p,r)-atom a: IL(a)I ( C l l L l l with C p,r,n but not on the particular atom. F o r N = = 0 , 1 , 2 , . .. and Q a cube, Li(Q) will be the s p c e consisting of those Lr functions supported in Q and having vanishing moments up to order N. We shall write Np(n) = cn((l/p) - l)], the biggest integer 5 n((l/p) 1). When the dimension is kept fixed, we shall drop the n and write simply N Given f e Lr (Q), it is clear P' NP

c IlLII llfllHPhlRn).

shall have, depending on

111. REAL HARDY SPACES

290

that the function

i s a ( p , r ) atom.

Therefore

L e (Hp(IRn))*

We have shown t h a t o u r f u n c t i o n a l on

provides a continuous l i n e a r

w i t h norm bounded by

L;p(Q)

lQl(l'p)-(l'r).

CIILII

Now t h e Hahn-Banach theorem a l l o w s u s t o e x t e n d t h i s f u n c t i o n a l t o t h e whole s p a c e

w i t h o u t i n c r e a s i n g t h e norm.

Lr(Q)

tarily, that

r <

functiaialon

Lr(Q)

m.

Suppose, momen-

Then we know t h a t e v e r y c o n t i n u o u s l i n e a r

i s g i v e n by a f u n c t i o n b e l o n g i n g t o L r I ( Q )

norm dominated by t h e norm o f t h e f u n c t i o n a l . c l u d e t h a t f o r e v e r y cube

Q c IR",

with

I n o u r c a s e , we cong e Lr'(Q)

there is a function

with:

L e ( H p ( I R n ) ) * i s r e p r e s e n t e d on f u n c t i o n s

s u c h t h a t our

by t h e i n t e g r a l :

f e Lg ( 9 ) P

Naturally, the function to

g

every

g

is not uniquely determined.

If we add

a polynomial o f d e g r e e

SN (5.2) continues t o hold f o r P' s i n c e s u c h f u n c t i o n s f have v a n i s h i n g moments

f e Lkp(Q),

.

up t o o r d e r C o n v e r s e l y , i f g1 and g 2 a r e s u c h t h a t (5.2) NP h o l d s f o r e a c h o f them i n p l a c e o f g , and e v e r y f e L i p ( Q ) , t h e n nec e s s a r i l y

gl

-

g2

i s a polynomial o f d e g r e e

SN

Indeed, t h e

P'

g - g 2 s a t i s f i e s janf(X)h(X)dX = 0 f o r every I n g e n e r a l , f o r f e L1(Q) and N a p o s i t i v e i n t e g e r , N 5 N we s h a l l d e n o t e by P (f) t h e u n i q u e p o l y n o m i a l o f d e g r e e function h f e LGp(Q).

having o v e r

=

Q

c a s e w e need t o t a k e Then, i f

f

Q

t h e same moments a s

e Lr(Q),

N = N

up t o o r d e r

f

N.

In our

f i x e d and we s h a l l s i m p l y w r i t e

P we s h a l l have

f

-

P Q ( f ) X Qe Li ( Q ) . P

PQ(f).

Therefore

111.5. DUAL SPACES

= 1~(f(x)-PO(i)(x)).(h(x)-P Q (h)(x))dx

Since this is true for every f PQ(h)(x) for every x e Q.

=

291

f(x)(h(x)-PQ(h)(x))dx.

e Lr(Q),

it follows that

h(x)

=

=

By writing lRn = LJ Qj where ( Q j ) is an increasing sequence of J rl cubes, choosing g . e L (Qj) which represents L over Qj and J such that each gj,l is an extension of gj; we can find r' g e Lloc(lRn) such that (5.2) holds for every f belonging to the class 0 = 0 s = (f e Lr(lRn) : f is compactly supported and has P vanishing moments up to order N p ] . Notice that, even though we have excluded the case r = m to be able to use the duality (Lr(Q))* = the conclusion we have obtained holds trivially also for = Lr'(Q), the case r = =, since Lm(Q) c Lq(Q) c L1(Q) for 1 < q < m, and since (5.1) continues to hold when r increases as one sees most easily by writing (5.1) in the form

In what follows we allow also cube in lRn. Then:

r

= m,

i.e.

r'

=

1.

Let

Q

be a

But

We have obtained the following estimate :

where,

for

r'

= m,

the left hand side has to be understood simply

111. REAL HARDY SPACES

292

DEFINITION

5.4.

shall denote by g e Lyoc(lRn) Q c IR",

1

In general, for

S

q 2

m

and -

0

2 a

<

m,

we -

L:(IR")

the space consisting of those functions f o r which there is a constant C such that for every

cube

where the left hand side is understood to be 11(g - P t ' ( g ) ) X q l l , in case q = a. For g e L;(IR") , we shall denote by IlglI C which make

the infimum of all the constants every

Q.

The mapping

g WII g II L$( R n )

(5.5)

valk!>?:)

is a semi-norm in

L:(IR")

and

IlglIL8(Rn) = 0 if and only if g coincides almost everywhere with a polynomial of degree sea]. Therefore, if we form the q u o -

tient space of ~ z ( l R " ) modulo the space of functions agreeing a.e. with poZynomials of degree <[or], we shall have a normed space

/y

Observe t h a t

= B.M.O.(IR

L;(IR")

t a n t t o n o t e t h a t on

n

)

provided

t h e semi-norm

~q,(lR")

q < =.

I t i s impor-

llg11L4,(IRn) i s e q u i v a -

l e n t t o t h i s o t h e r semi-norm: 1

sup lQl-aln.inf{(-j Q cube IQI

: p polynomial of d e g r e e Indeed, f o r e v e r y p o l y n o m i a l

5 (

I

IQI

C(-

IQI

1

Ig(x)

-

l/q P(x)lqdx) :

Q 5

[a]]

of d e g r e e

Sea]:

llq + suplP(x) P(x)lqdx) xeQ

-

Q

1 6

Ig(x)

P

lg(x)

-

llq P(x)lqdx)

Q

a n d , c o n s e q u e n t l y , for e v e r y

x e Q is:

since

- PQc a l ( g ) ( x ) l

5

1 1 1 . 5 . DUAL SPACES

f o r some a b s o l u t e c o n s t a n t

293

C.

T h e r e f o r e , i n o r d e r t o see t h a t a c e r t a i n

i s enough t o f i n d a c o n s t a n t i s a polynomial of degree

g

belongs t o

s u c h t h a t f o r e v e r y cube

C

sea], s a y

it

L2(Rn)

Q, there

P Q ( x ) , s a t i s f y i n g the ine-

qua1 i t y :

T h i s i s where w e s t a n d on o u r way t o i d e n t i f y i n g ( H p ( R n ) ) * : 0 e QIR

L e (H~(IR"))*

f e

every

]:fO

I t g II

and f o r e v e r y C

5

IILII

g

8

=

n((l/p)-l).

;1.

Fix

T h e n for e v e r y such t h a t f o r

, (5.2) holds. Besides

w i t h an a b s o l u t e c o n s t a n t

C.

L p n )

I n o t h e r w o r d s , w e have found a c o n t i n u o u s l i n e a r mapping

Notice t h a t

O l i = 0;'

i s dense i n HP(Rn) s i n c e i t c o n t a i n s a l l t h e f i n i t e l i n e a r combqnations o f ( p , q ' )-atoms. T h i s i m p l i e s t h a t t h e mapping ( 5 . 7 ) i s one t o one. We s h a l l e v e n t u a l l y s e e t h a t t h i s

mapping i s a l s o o n t o , s o t h a t i t p r o v i d e s an e q u i v a l e n c e o f Banach s p a c e s . Once t h i s i s done, we s h a l l b e a b l e t o w r i t e : ( H P ( R n ) ) * = rc/

= L(:

(En). Actually t h i s i d e n t i f i c a t i o n implies t h a t the

(l/@-l)

spaces

L:

f o r a fixed

(R")

c1

and v a r y i n g

q ' s , are a l l t h e same,

and t h e c o r r e s p o n d i n g norms a r e e q u i v a l e n t . So we c o u l d s i m p l y rJ

write

L ( R " ) . Remember, however, t h a t f o r c1 = 0 ( t h a t i s : p = l), a t h e i n d l q = m is n o t allowed. T h i s is a n a t u r a l r e s t r i c t i o n ,

since

/y

W

L~ = L m , a h e r e a s , L:

of t h e s p a c e s

Lsf

= B.M.O.

f o r any

for different finite

q's

q < m. The e q u i v a l e n c e was already established

i n c h a p t e r I1 as a consequence of t h e John and N i r e n b e r g theorem. Now i t w i l l be o b t a i n e d f o r f r e e t o g e t h e r w i t h t h e c o r r e s p o n d i n g results for

a > 0 , once w e prove t h a t t h e mapping ( 5 . 7 ) i s an

111. REAL HARDY SPACES

294

0 < p 6 1. C a l l

equivalence. This is our n e x t g o a l . Let n((l/p)-l)

=

p < 1

if

5 co

and f i x and

q

1 6 q <

admissible f o r our m

L e (Hp(IRn))"

t o construct

h o l d s . O f c o u r s e , f o r such

if

p = 1. L e t

p , t h a t is: 1 6 q g e L:(lR").

such t h a t f o r e v e r y functions

f e Oq'

1 .t '

5

We want (5.2)

we can always d e f i n e L ( f ) i s dense i n by means of formula ( 5 . 2 ) . Now, s i n c e t h e s p a c e 0 1 : HP(IRn), we s h a l l be a b l e t o e x t e n d o u r f u n c t i o n a l on

f

L

t o a continuous l i n e a r

i f we a r e a b l e t o prove t h a t , f o r e v e r y

HP(IRn)

f e 0°C: IL(f)l 6

(5.8)

with

c

II f I1

IlglI L:(lRn)

HP(lRn)

an a b s o l u t e c o n s t a n t .

C

T h i s w i l l a l s o prove t h a t t h e e x t e n d e d

L

satisfies

IILII s c IlglI , t h a t way c o m p l e t i n g t h e p r o o f t h a t ( 5 . 7 ) i s an equivalence. L p n )

The f i r s t s t e p towards ( 5 . 8 ) i s t h e f o l l o w i n g : If

LEMMA 5 . 9 .

Let

Proof

I jp8

Q

(p,q')-atom, then

be a minimal cube s u p p o r t i n g

a . Then

x ) g ( x ) dx

( p , q ' )-atoms, t h e

lXjl

w e can w r i t e f = 1 A . a . where t h e . J J are real numbersJsatisfying

f e 0''

Now, g i v e n

&

a

a

c1. X!s'

c IlflIP

J

a'.s J

are

w i t h C an a b s o l u t e c o n s t a n t ; and t h e HP(IR") convergence i s i n t h e s e n s e of tempered d i s t r i b u t i o n s . If we a r e a b l e P

4

.1

t o show t h a t

L(f) =

1 j

AjL(aj),

we s h a l l have:

111.5. DUAL SPACES

295

That way we would

rJ

Therefore, the i d e n t i f i c a t i o n

(Hp(lRn))*

= L(:

have been t o t a l l y j u s t i f i e d once we prove t h a t

O f c o u r s e t h i s i d e n t i t y h o l d s w i t h any Schwartz f u n c t i o n i n p l a c e o f

g

since

f o r every -+ m

A . a . i n t h e s e n s e o f tempered d i s t r i b u t i o n s . I f J J f i n d a sequence gk o f Schwartz f u n c t i o n s such t h a t

td

we were a b l e

k

1

f =

f e 0"

DJ

and a l s o

Inn

is

f (x)gk(x)dx

-

f ( x ) g ( x ) d x as

s C , we would have f i n i s h e d s i n c e i t

((gkl(

"

dominated convergence. So, w e have reduced e v e r y t h i n g t o a problem of approximation. To approximate

g e ~z(lR")

by

Cm

f u n c t i o n s w i t h u n i f o r m l y bounded

Lq-norms i s n o t d i f f i c u l t . We s i m p l y need t o t a k e t h e f u n c t i o n s

I

c1

g * @ , where clear that

1

is

t$

g*t$€ + g

as

in

E + 0

L ~ o c ( l R n ) , so t h a t

f ( x ) g ( x ) d x as

nn f(X)g*b, ( x ) d x

f e 0%].

t$ = 1. I t i s

w i t h compact s u p p o r t and h a s

Cm

We j u s t have t o s e e t h a t

We know t h a t f o r each cube

Q,

E

-+

0

f o r every

5 IIg*$EII L9,(R")

c IlglI LZ(1R")

t h e r e i s a polynomial

such t h a t (

1

Given a cube

By w r i t i n g

jQ Ig(')-'Q

Q

P

Q-Y

the coefficients

and

E

> 0

(g)(x-y) =

a

B

consider the function

1

I B l 5ca

a B ( y ) ( x - y l B and o b s e r v i n g t h a t

a r e c o n t i n u o u s , we r e a l i z e t h a t

PQ,€

is a

296

111. REAL HARDY SPACES

polinomial of degree

1 = (

lQ-y

Sc.3.

Ig(z)-pQ-y(g)

By using Minkowski's inequality for

I dz I1/q

(2)

5

Ilgll L(;

IQl" IR")

4 c Ilgll Lq(IRn) Lq(IR7 We still have the problem of approximating a function f e CmflL2(?Rn) The idea by Schwartz functions with uniformly bounded L:-norms. should be to use a cut-off function $ radial, Cm, supported in B(0,l) such that $(x) = 1 for every x e B(0,1/2) and 0 6 $(x)6 d 1 for all x. The function g(x)Q(E:x) coincides with g(x) for 1x1 < 1 / ( 2 ~ ) , is Cm, and is supported in B(O,l/E). The functions g$(E.) approximate g as E -+ 0 uniformly on compact subsets. It would be very nice if we could get a uniform estimate for the -:L norm of these functions. We shall be able to do this very simply for the case 0 5 a < 1 , which corresponds to n/(n+l) < p d 1. To deal with the general case, we shall need to get a better understanding of the spaces L;(IR").

We have obtained the desired inequality

IIg3, ( 1

Suppose that 0 d a < 1. Then [aJ = 0, and this implies that the space ~z(lR") is a lattice, Indeed if g e L:(lRn), we have, for every cube Q:

5

I Q I a'nIIglI L,q(lR")

Thus Igl e L;(IR"). It follows easily that f o r fl,f2 e Lz(IRn), the functions max(fl,f2) and min(fl,f2) are both in L:(IR~), with norms bounded by C rnax(llflll ((f21[ q ) . We already pointed La L :'

111.5. DUAL SPACES

out t h a t t h i s holds f o r

B.M.O.

297

i n c h a p t e r 11. The a d v a n t a g e i s t h a t

now w e c a n approximate a f u n c t i o n o f L z by means o f bounded funct i o n s w i t h u n i f o r m l y bounded Lz-noms. For g e ~ 2 ( l R " ) and k > 0

we simply t a k e

gk(x) =

k if g(x) if

-k

if

g(x) L k 6 g(x) 2 k

-k

g(x)

and

-k

4

Then i f we r e g u l a r i z e by c o n v o l u t i o n each of t h e w i t h bounded

Cm

g i s , we end up f u n c t i o n s . T h e r e f o r e , we a r e reduced t o a p p l y i n g

our cut-off process t o a function

g e C"n

L m n L:(Rn).

F o r such a

f u n c t i o n we g e t a simple e s t i m a t e

where

C

depends o n l y on

mate we w r i t e f o r a cube

Q , the cut-off function.lb g e t t h i s estiQ

with center

xo:

11@11, [Irn llgll,.

I n d e e d , t h e f i r s t term i n t h e sum i s bounded by The second term i n t h e sum i s bounded by e s t i m a t e i s good f o r b i g c u b e s , s a y f o r

IQI

T h i s Lq( CI IR") > 1. For small c u b e s ,

2lld

IQI

IQI"'

IlglI

2 1 , we u s e t h i s o t h e r e s t i m a t e f o r t h e second term:

I Q I a / n . T h i s e s t i m a t e i s o b t a i n e d by c ~ ~ V d ~ ~ , l Ql 1 ~l / n g ~ ~ ClIglI, , u s i n g t h e mean v a l u e theorem f o r Q . We s h a l l a l s o need t h e f o l l o w i n g e s t i m a t e f o r t h e d i l a t e d o f an

L:

function

xwh(Ex)

h.

(5.11) Indeed f o r a cube 1

(m 4

llgl

Q

y ) Iqdy)l/q 5 11)

follows.

298

111. REAL HARDY SPACES

arq =

a + C E llgll,. I t i s c l e a r t h a t , f o r E + 0 t h i s norms L,q(lR") u n i f o r m l y bounded. T h i s e n d s t h e p r o o f of t h e d u a l i t y ( H p ( I R n ) ) * =

IlglI

5 c

Lq (1~") n((l/P)-l)

The c a s e c1

for

n/(n+l) < p

1.

d e s e r v e s s p e c i a l m e n t i o n . I n t h i s c a%s e

p = 1

= n((l/p)-1) = 0

and t h e d u a l s p a c e o b t a i n e d i s

This

L z = B.M.O.

i s i n agreement w i t h t h e d u a l i t y theorem proved i n c h a p t e r I f o r t h e of t h e d i s k . Let u s r e c o r d t h e r e s u l t o b t a i n e d f o r t h i s s p a c e H1 special case. THEOREM 5 . 1 2 .

For every

defined b y ( 5 . 2 )

e B.M.O.

t h e linear f u n c t i o n a l

Over H ~ ( I R " )

L

extends t o a c o n t i n u o u s linear f u n c t i o n a l

C

L

over bounded functions with compact support,

with

being an a b s o l u t e constant.

This, i n conjunction with the case

p = 1

of theorem 5 . 6 . ,

c a n be

s t a t e d as f o l l o w s . THEOREM 5 . 1 3 .

(H1(lRn))"

= B.M.O.

In general, with a and q w e have a l r e a d y proved i s THEOREM 5 . 1 4 .

For

related to

p

as i n theorem 5 . 6 . ,

what

n / ( n + l ) 0 . They t u r n L:(R") o u t t o be L i p s c h i t z s p a c e s i n d i s g u i s e . L e t u s d e f i n e t h e L i p s c h i t z n s p a c e s A ~ ( I R) . D E F I N I T I O N 5.15.

a ) Let

a > 0 , c1

not an integer.

Call c.7

=

N

&

111.5. DUAL SPACES

a = N

let

+

a ' , where, of course, 0 < a' < 1. I\

space consisting of those functions the derivative

f o r every

D8g

a

g e CN(lR")

such that f o r every multi-index

C

a constant

299

(XI")

will be the

f o r which there is

8

with

=

g e

x , h e lR".=

~,(m")

we denote by

IlglI A

the smallest constant

C

(m")

which makes (5.16) valid for alla p,x,h.

Equivalently, using the notation

I I I1

N,

satisfies the Lipschitz condition

is a semi-norm on

we write:

Ahf(x) = f(x+h)-f(x)

na(lRn)

which vanishes precisely

Aa(lR")

over the space formed by all the polynomials of degree S N

. ((5.16j

'L

C = 0 implies that D'~ is a constant). Denote b y A,(l~y the corresponding quotient space. It will be a normed space. g e i

b)

Let

a > 0

be a n integer, say

a = N . Then

na(lRn)

to be the space consisting of those functions which there is a constant with

18 I

C

g e CN-l(IR") such that for every multi-index

N-1, the derivative

=

is going

DBg

for 8

satisfies the Lipschitz condi-

t ion -

f o r every

x , h e 1R".

=

Observe that the expression whose absolute

value appears in the left hand side of (5.17) =

A hA h DBg(x).

est constant

C

a h2 D8 g(x) = the small((glI A a ORn) which makes (5.17) valid for all $,x,h. Equivag e AJlR"),

is

we denote by

lent l y

I ' ' ' A ( E n ) is a semi-norm o n A a ( 1 R " ) wich vanishes precisely over %he space formed by all the polynomials of degree g N ((5.17) with C = 0 implies that g'D is a n affine-linear function). In the normed space this case we shall also denote b y A:(#) obtained by taking the quotient.

111. REAL HARDY SPACES

300

Aa and

We s h a l l s e e t h a t t h e s p a c e s

Lz

c o i n c i d e as sets and t h e

c o r r e s p o n d i n g norms are e q u i v a l e n t . The f i r s t s t e p towards t h i s equivalence i s the following. LEMMA 5 . 1 8 .

g e Aa(IRn).

Let

e x i s t s a poZynornia1

B

index

a

Ilfg(x)-D P ( x ) I

6

of degree

P

I BI

w i t h

B(xo,r), there

Then f o r e v e r y b a l l 5

[a]

such t h a t f o r every rnulti-

a:

CrCGl

" IlglI

f o r every

x e B(xo,r)

A a (El")

with

a

depending o n l y upon

C

P r o o f : a) Suppose t h a t

n.

a i s n o t an i n t e g e r . We s h a l l s e e t h a t t h e

C.7

T a y l o r polynomial of d e g r e e

at

xo

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

i s good

g

enough. Indeed, l e t

P(x) = OSI

for

c

YI

1 5[a]

D Y g ( x o ) ( x - x0 )Y

Y!

x e B.

The e s t i m a t e f o r

I DB g ( x ) - D BP ( x ) I

with

e x a c t l y t h e same way by o b s e r v i n g t h a t nomial a t

xo

of d e g r e e

b ) Suppose now t h a t

a

[a]-[

181 s DaP(x)

C

i s an i n t e g e r , a = 1 , 2 ,

function supported i n

is obtained i n

i s t h e Taylor poly-

f o r the function

rn

even

[d

B(0,l)

with

DBg.

... . L e t

@

be an

4 = 1. C o n s i d e r

t h e f u n c t i o n h = g++ar and l e t P1 be t h e T a y l o r polynomial o f h a t xo of d e g r e e a . For 1'1 = a-1, DaP1 w i l l be t h e T a y l o r p o l y nomial of d e g r e e 1 a t xo f o r t h e f u n c t i o n Dah = DBg++ar.F o r

IYI

so

= 2, D

Y ( @ r )= r - 2 ( D Y @ ) r

i s an even f u n c t i o n w i t h i n t e g r a l

0.

1 1 1 . 5 . DUAL SPACES

301

the estimate

We a l s o have

I 2DB(h-g) ( x ) I

@ r ( y ) ( D B g ( x - y ) + BDg ( x + y ) - 2 D B g ( x )1 dyl 4

I

5

lRn

IDB g ( x ) - D B P l ( x ) l 4 C Ilg(l,, - r for x e B ( x o , r ) and l B l = a - 1 . Now l e t P be a p o l y n o m i a l h aav i n g t h e same terms o f o r d e r a and a-1 as t h o s e o f P1 b u t s u c h t h a t DB P ( x ) = 5

C IlgllAa.r.

= DBg(xo)

Thus

f o r every

B

0

IBI

with

Then w e j u s t n e e d t o i n t e -

5 a-2.

If3 = a

g r a t e t h e i n e q u a l i t i e s o b t a i n e d a l r e a d y for obtain

-

i n o r d e r to

1

t h e r e q u i r e d i n e q u a l i t i e s for d e r i v a t i v e s o f l o w e r o r d e r .

0 C O R O L L A R Y 5.19. every 1 9 < Proof: Let ter

A J ~ R ~c ) L;CL?)

f o r every

>

c1

o

and

Besides the inclusions are continuous.

m.

g e Aa(lRn).

2 r , consider

x e Q

c L:(#)

If

B(xo,r)

is a cube w i t h c e n t e r

Q

and a p p l y lemma 5.18. w i t h

xo B

and diame= 0.

For

is

The o t h e r i n c l u s i o n i s o b v i o u s .

[I

I n o r d e r t o p r o v e t h e c o n v e r s e we n e e d t h e f o l l o w i n g

4

Cm

L E M M A 5.20.

Let

Let a > 0

and 0 a .

~is a I$,

B

f u n c t i o n w i t h c o m p a c t s u p p o r t in lRn such that

c1 =

n((l/p)-l). S u p p o s e n

is a m u l t i - i n d e x in

.

k

variables, such

Then

( p , m ) - a t o m for every

t,

4

C

is a c o n s t a n t

a , n a n d t h e o r d e r of d i f f e r e n t i a t i o n , but not on

t

111. REAL HARDY S P A C E S

302

Proof:

-T a k d3,(Qt(x))

t-n-161-k o(x/t) support. Then the function

with corn$ct

=

c"

for some function c p ,

is bounded in absolute value by Ct-"" where C = I l c p l l m does not depend on t. But, if 0 is supported in a cube Q centered at 0 , then t-"" @(x/t) is supported in tQ. Since ltQl = Ctn for some constant independent of t , we conclude that

ak

DE(Ot(x))

=

Cta-k-161 at(x)

where

at(x)

lives in

tQ

and

atk

satisfies lat(x)l 5 ltQl ' / and shall see that at(x) is actually at(x) with

C does not depend on t. Now we a(p,m)-atom by proving that

L a ] . F o r a multi-index

has vanishing moments up to order

I yI

6

[d ,

y

consider the moment

In case 6 5 y, which means that 6 6 a j for each by parts shows that (5.21) equals a constant times

j, integration

In case it is not f3 5 y, there will be y j c a j f o r some gration by parts in x . gives immediately that (5.21) is J

j . Inte-

0.

0

Now we are ready to prove the THEOREM 5 . 2 2 . For e v e r y

a > 0

and e v e r y

1 2 9

5m

~ ( X P c) p(mn) and t h e inclusion i s c o n t i n u o u s . a

a

g e L:(Etn). W e shall show that g coincides almost whose I\ -norm we shall everywhere with a function in A (IR")

P r o o f : Let

a

estimate.

We shall use a function @ , radial, that

0 2

$(XI

d

1

for all

x

and

a

C",

supported in

having

B(0,l)

such

@(x)dx = 1. If f o r

111.5.

DUAL SPACES

303

t > 0 is @,(x) = t-"@(x/t) the corresponding approximate identity, *n+ 1 we shall consider the function u(x,t) = (g+@,)(x) in Lemma 5.20. together with lemma 5.9. guarantee that for any integer k 2 0 and any multi-index B in n variables such that

.

+

k

+ 161

> a , we have:

I , with C

s c IlglI

a

a) We study first the case when

+ a'

a = [a]

I Bl

=

[a].

independent of

0 < a 1 < 1. Let

with

t

and

g.

is not an integer. Let

B

be any multi-index with

Estimate (5.23) gives

we have (5.24) (DBu(x,tl)-DB u(x,to) I

5

ta'-l

CIIglI

dt

5

w DBu(x,t) converge uniformly This implies that the functions x rtain (continuous) function uB(x). On the other as t + O t o a c

B

D g as t * 0 in the sense of distribuhand Dpu = (DBg) @ tions. Therefore D g = uB as distributions. Now, for x,h e lRn

B

-uB(x)I

5

a

lua(x+h)-D u(x+h,lhl)l +

(hla'. Indeed, estimate (5.24) f o r to * 0 implies L!.(lR") that the First and third terms in the sum are bounded by Ih(" ' To estimate the second term we use the mean c IlglI 6 C

Ilgll

.

L?.( En)

value thUeorem together with the estimate, obtained from (5.23)

Thus, after modifying

g

if necessary on a set of measure

0 , we

304

have

REAL HARDY SPACES

111.

g

e

Aa(lRn)

and

b) Suppose now that a is an integer. Let 6 be a multi-index such that 181 = c1 - 1. Since IB I + 2 > a , estimate (5.23) holds with k = 2. Thus I

By integrating in

t (say between

t

and

l),we get a local estimate

Here C depends on x but it can be taken to be the same for each compact subset of IR". This estimate is still good to obtain this other local estimate for 0 < to c tl.

t1

ID:u(x,tl)-Dxu(x,to) R

I

5

C

4

log(t-')dt

5

C tl(l-log tl) + 0

as t l + 0. This is enough to guarantee that the functions x++ Dxu(x,t) B converge uniformly over each compact, as t -+ 0 , to a certain (continuous) function ug(x). A s before D B g = u as B distributions and we just need to look at the second differences of uB. We write, using integration by parts

ua (x) =

=

DXB u ( x , ~ )-

Dxu(x,t)-t B

a

$u(x,s)

a Da at

ds

=

D!u(x,s)

ds.

x

2 Now we estimate the second order difference AhuB(x) by estimating the corresponding second order difference of the right hand side

C IIglI

Ihl and so L q ( lR") will be its second order difference. Next we observeathat for a smooth func ion,the second order differences A h2 are dominated by a bound for the second derivatives times (hI2. We get, by using (5.23):

with

t = Ihl. The third term is bounded by

1 1 1 . 5 . DUAL SPACES

305

2 =

c Ilgll

g

I h l . We c o n c l u d e t h a t , a f t e r m o d i f y i g Lq( IR") and s a r y on a ' n u l l s e t , g e A a ( I R n )

=

We h a v e f i n a l l y p r o v e d , t h a t for e v e r y

> 0

and e v e r y

i f n ces-

1 5 q 5

m:

i n t h e s e n s e t h a t t h e two s p a c e s c o i n c i d e as s e t s and t h e norms a r e

A -norm i s e a s i e r t o h a n d l e . T h i s i s

e q u i v a l e n t . Sometimes t h e

c1

p a r t i c u l a r l y t h e c a s e i n t h e a p p r o x i m a t i o n problem which we s t i l l have t o s o l v e i n o r d e r t o complete t h e proof of t h e d u a l i t y theorem The s o l u t i o n i s c o n t a i n e d i n t h e f o l l o w i n g LEMMA 5 . 2 6 .

Let

g e AU(IRn)

.

Lemma 5 . 1 8 g u a r a n t e e s t h a t f o r e a c h n i s a polynomial for every x

r > 0 t h e r e e x i s t s gr e Acl(IR ) s u c h t h a t 8,-g B of d e g r e e S ~ Jand I D gr(x)l 6 C r c l - l ' I IIglI

A

(B'4

and e v e r y m u l t i - i n d e x $ with < a . Then i f $ i s o u r c u t - o f f f u n c t i o n ( $ i s r a d i a l , Cm, s u p p o r t e d i n B ( 0 , 1 ) , s u c h t h a t Q ( x ) = 1 for e v e r y x e B ( O , l / 2 ) a n d 0 5 $ ( x ) 5 1 for

with

1x1 < r

a l l xi,

t h e f u n c t i o n s x ~ - - tg r ( x ) $ ( x / r ) h a v e u n i f o r m Z y b o u n d e d A - n o r m s . A c t u a l l y t h e i r A - n o r m s a r e b o u n d e d b y CIIg(l with c1 c1 Acl(IR") C an absolute constant. P r o o f : a ) Suppose t h a t with

0 < a'

< 1. L e t

a

i s n o t an i n t e g e r . Let be a m u l t i - i n d e x w i t h

t o see t h a t the f i r s t difference DB(gr(x) $ ( x / r ) )

1

=

Ah

N

=

IB I

=

[a] + a '

b] .

We have

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

CY,& D Y g r ( x ) D t $ ( x / r ) ) s a t i s f i e s t h e e s t i m a -

y+6=8 5 C lhl"'. We j u s t need t o p r o v e t h e s e e s t i m a t e s f o r the te 6 f i r s t d i f f e r e n c e s o f e a c h o f t h e t e r m s D Y g r ( x ) - D ( $ ( x / r ) ) . But t h e f i r s t d i f f e r e n c e of a product i s c l e a r l y (5.27)

A h ( F ' G ) ( x ) = AhF(x).G(x+h)+I:(x)ahG(r.).

formula with

F(x)

=

D Y g r ( x ) and

L e t us a p p l y t h i s

G ( x ) = D6 ( @ ( x / r ) ) . F i r s t o f a l l

111. REAL HARDY SPACES

306

note t h a t the function

F.G

1x1 < r

lives in

and, consequently

it s a t i s f i e s DYEr Ihl 2 r . I n t h i s c a s e

I t f o l l o w s t h a t we o n l y n e e d t h e e s t i m a t e f o r we u s e

( 5 2 7 ) . The f i r s t t e r m i n t h i s sum w i l l be bounded by

Clhl'l

if

Y = 6

and o t h e r w i s e by

-a' l h l " ' s = C(lhl/r)

by

Cra-IY!

S C

=

Cr

a'-1

Ihl =

A s f o r t h e s e c o n d t e r m , i t i s bounded

C Ihl".

Ihl

Cra-IYI-llhlr-IGI

Ihl

This concludes t h e proof f o r

'I.

c1

not an integer. b ) Let =

be an i n t e g e r

c1

6

>O. Let

be a m u l t i - i n d e x w i t h

a - 1. We s h a l l h a v e t o s e e t h a t f o r e v e r y

y

and

6

141

=

multi-

of i n d i c e s such t h a t y + 6 = 8 , t h e second o r d e r d i f f e r e n c e A the function Dygr(x)D6($(x/r)) s a t i s f i e s the estimate [ A 2h h[ 5 Clhl. We

'

o n l y need t o c o n s i d e r

bounded by

Cra-IYI

Ihl 5 r = Cr'"-laI

s i n c e the function i t s e l f is = Cr.

By i t e r a t i n g (5.27) we g e t

t h e f o l l o w i n g formula f o r t h e second o r d e r d i f f e r e n c e of a p r o d u c t :

We a p p l y i t for

F(x)

=

Dygr(x)

G ( x ) = D 6 ( $ ( x / r ) ) . The f i r s t

and

t e r m i n t h e sum i s bounded by Clhl i f y = 6 and o t h e r w i s e by -1 I 61 = C I h l . The s e c o n d term i s bounded bv Ihl r -16 Ihl = Clhl i f y = 6 and o t h e r w i s e by C r " - I y I - l Ihl r r-I6'-l Ihl F i n a l l y t h e t h i r d t e r m i s bounded by C r " - I Y I

I

=

=

This ends t h e proof. f e

Observe t h a t f o r

Ox,

and

r

b i g enough

T h u s , w e have s o l v e d o u r a p p r o x i m a t i o n p r o b l e m . Given = Can

functions

g,(x)

s u p p o r t , have

=

c1

=

g r k ( x ) $ ( x / r k ) , rk t

1 1 gk I1 L:(mn)

f o r every

g e C"l?L;(lR~

A ( l R n ) , we h a v e b e e n a b l e t o f i n d a s e q u e n c e o f

f e OEcol q ,:'

Imn

'

IIgkII A" f(x)gk(x)dx

m,

which a r e

- 1,.

s C

Cm

w i t h compact

and a r e s u c h t h a t ,

(nn) f(x)g(x)dx.

111.6. INTERPOLATION

307

This completes the proof of the converse of theorem 5.6, which can now be stated in the following way, the notation being the same as in theorem 5.6. N

THEOREM 5 . 2 9 .

by ( 5 . 2 ) tiona2

C

For e v e r y

on functions

L

f

o n Hp(En)

n < g > e L:(X ) , t h e linear functional defined e @$l, e x t e n d s to a c o n t i n u o u s l i n e a r f u n c

with

being an absoZute constant.

Altogether we can write in a single statement the fruit of all our efforts in this section, which is the following duality theorem

6. INTERPOLATION OF OPERATORS BETWEEN

The tes its can

Hp

SPACES

atomic characterization of HP(En) allows us to obtain estimafor the norm of an operator from Hp to Lp by simply checking action on an atom. It is natural to study how these estimates be interpolated.

Here is a simple variant of the Marcinkiewicz interpolation theorem THEOREM 6 . 1 .

+ Lpl(lRn)

Let

T

be an o p e r a t o r s e n d i n g functions in

to m e a s u r a b l e f u n c t i o n s i n suppose that:

il

En w h e r e

H1(Bn) +

1 < P1 s

m e

T is s u b a d d i t i v e , t h a t i s , for fl,f2 e H1(En) + Lpl(En) lT(fl+f2)(x)l 5 (Tfl(x)l+lTf2(x)l f o r a l m o s t e v e r y X.

111. REAL HARDY SPACES

308

T

ii)

1

i s of weak t y p e

c o n s t a n t Co

iii) T

( H , l ) , b y w h i c h we mean t h a t t h e r e i s a 1 t > 0: s u c h t h a t f o r e v e r y f e H (IF?) and f o r e v e r y

i s of weak t y p e

Then for e v e r y

such t h a t

p

w i t h norm d e p e n d i n g on

Co,C1,

Proof: Take a f i x e d

with

t > 0

with constant

(pl,pl)

p

1 < p < pl,

p1

&

f

i s bounded i n

T

Lp(Rn)

p.

1 < p < pl.

we s h a l l s p l i t t h e f u n c t i o n

bad p a r t

C1.

f e Lp(lRn).

Let

i n t o a good p a r t

For e a c h g t and a

v e r y much l i k e i n t h e p r o o f o f ( 5 . 9 ) i n c h a p t e r 11. The

bt d i f f e r e n c e w i l l be t h a t we s h a l l u s e now t h e Calder6n-Zygrnund

{Qj}

q

of the function

lflq

corresponding t o the value

1 < q < p . That i s : t h e

i s chosen s o t h a t

ma1 c u b e s i n t h e f a m i l y o f a l l d y a d i c c u b e s

cubes

tq where

I

Q . s w i l l be t h e maxiJ Q satisfying

These maximal c u b e s e x i s t s i n c e , by J e n s e n l s i n e q u a l i t y , e v e r y c u b e Q

satisfying (6.2) also satisfies

tP

I f ( x ) I P dx

and c o n s e q u e n t l y i t s volume i s bounded by

t-'

I I f " ~ . The c u b e s

Qj

w i l l satisfy:

We s h a l l w r i t e

Et =

u 9J. . Note

that

E t c [ x e IR": M q ( f ) ( x ) > t )

j

where M q ( f ) ( x ) = ( M ( l f l q ) ( x ) ) l ' q e Observe also t h a t almost every x e I R " \ E ~ . We decompose

f = g

and t h e bad p a r t

bt

+

b t , where t h e good p a r t

is

gt

If(x)l

5

i s g i v e n by

t

for

309

111.6. INTERPOLATION

Let us assume first that p1 < shall be treated at the end.

W .

The case

p1 =

m

is simpler and

Let us look separately at the two pieces of the decomposition a) gt e Lpl(IRn). This is clear since Igt(x)) Actually

5

2""t

for a.e.x.

b) b t e H1(IR"). Let us see why this is so and let us estimate the norm. Observe that each b{ lives in Q . , has average 0 and J satisfies:

(1,q)-atom. Then, = a.(x) is a It follows that C-lt-' lQjl-'b:(x) J lQjl aj(x), we obtain that bt e H1(lRn) with since b,t(x) = 1 Ct

1 Q .JI

=

Ct lEtl

5

Ctl-pllfll~.

a) and b) imply, for any particular defined. Besides:

t > 0, that

T(f)

is well

111. REAL HARDY S P A C E S

310

But:

P-l-P1

m

dt) [f(x)[p1 dx =

1 -

If(x)IP dx

and

P1-P J m n

1,

m

tP-l

lEtldt

5

p

tP-'

[Ix e lRn:Mq(f)(x) > tll dt =

Therefore IITflIp 5 C I I f l I p with C depending only on C o , C 1 , p1 and p . This settles the case pl( m . When p1 = m, we make a change of variables in the formula for

tP-lI[ x e En: IT(gt)(x)

+

,"

I

IlTfllF, writing with

A > 0:

> At/2] (dt+

tP-lI{x e IR": [T(bt)(x)l > At/2}I dt.

if we take A > 2C1- 2"jq, the first Now since IT(gt)(x)I 6 C12n'qt, integral in the sum will vanish. The second integral is estimated as before and the p r o o f is ended. A similar r e s u l t holds for HPo(IRn) with p o c 1 in place of H1(lRn). In general we can extend theorem 6.1. in the following way:

THEOREM 6.4. Let

T

be a n operator sending

into measurable functions in

HPo(lRn) + Lpl(lRn)

En, where 0 < po

5 1

< p1

5

m.

S u p p o s e that

il

T

is subadditive

iil

T

is of weak type

constant

iiil T

Co

(HPO , p o ) , b y which we mean that there

suck that for every

i s of weak type

(p,,p,)

f e Hro(lRn) and for every

w i t h constant

cl'

i s a

t > 0:

111.6.

Then, f o r every

p

such t h a t

w i t h norm d e p e n d i n g on

Co,

INTERPOLATION

1 < p < pl,

C 1 , p o , p1

311

T

&

i s bounded i n

Lp(lRn)

p.

P r o o f : For po = 1 , theorem 6 . 4 . c o i n c i d e s w i t h theorem 6 . 1 . which h a s b e e n p r o v e d a l r e a d y . L e t po < 1. We p r o c e e d as i n t h e p r o o f o f theorem 6 . 1 . b u t making now

and

bt(x) =

1 b:(x)

with

b i ( x ) = ( f ( x ) - P Q ( f ) ( x ) ) X Q ( X I , where

j j lj ( f ) ( x ) i s t h e unique p o l y n o m i a l of d e g r e e s[n(- l ) ] having PO Qj 1 [n(- l)] t h e same moments as f on Qj up t o o r d e r PO

P

.

Suppose

p1 <

m .

a f t e r a l l : I ~ ~ , ( f ) ( x5 ) Cl J f Q

all

x e Q. J

b ) C-lt-llQjl-l'po bt(x) =

2

Ct

It follows t h a t

a s w a s shown

n t h e p r o o f of '?heorem 3 . 6 .

b tJ ( x ) = a J. ( x )

I QJ . l l / p o

1,

a j ( x ) , we obtain that

P L P ( R " ) c H O ( l R n ) + Lpl(IR")

tP-' l E t l d t 4 C

for

i s a ( p o , q ) - a t o m . Then, s i n c e bt e HPO(lRn)

I f ( x ) I P dx

with

and, i n p a r t i c u l a r , T f

is w e l l d e f i n e d . Besides:

+ Cp

I

J

e x a c t l y as b e f o r e .

312

111. REAL HARDY SPACES

The c a s e

p1

=

i s t r e a t e d as i n t h e p r e v i o u s p r o o f .

m

A natural question a r i s e s a t t h i s point.

the hypotheses

Suppose t h a t we a r e u n d e r

0 < po < 1. Can we a l l o w

o f theorem 6 . 4 . w i t h

p

=

1

i n t h e c o n c l u s i o n ? . The R i e s z t r a n s f o r m s show t h a t t h e answer t o t h i s question i s negative ( s e e next s e c t i o n ) . T bounded i n

d o e s n o t need t o be

L'.

However we c a n e a s i l y s e e t h a t

T

h a s t o be o f weak t y p e (1,l). TO

show t h i s we p r o c e e d a s i n t h e p r o o f o f theorem 6 . 4 . p1 <

q = 1. We s t i l l h a v e , a s s u m i n g

P1-1 6 C t

b)

C-lt-l

but taking

m

IIflI1.

l Q j l -'Ipo

bj(x) = a.(x)

t

J

is a

( p ,I)-atom. This leads t o 0

P c H '(IR")

and, i n p a r t i c u l a r + Lpl(lRn) Tf i s w e l l defined. Besides, I{ x e El": \ T f ( x ) \ > t } l 6 5 I I x e IR": I T ( g t ) ( x ) I > t / 2 } 1 + I { x e IR": IT(bt)(x)I > t/2}1 C

L1(IRn)

It follows t h a t

I

C t-P1

jIRn I g t ( x H p l

is

HP(IRn).

5

ct-l

IlflIl

p 0 < p 6 1 , t h e n a t u r a l domain f o r t h e o p e r a t o r

I n any c a s e , f o r T

dx + C t -PO llbtll PO HP0( IR")

We have t h e f o l l o w i n g r e s u l t

THEOREM 6.5. Let

T

be an operator sending

into measurable functions in

IRn,

HPo(lRn)

+

Lpl(lRn)

0 < po < 1 6 p1

where

5

m

.

Suppose that

i)

T

i s sublinear, which means that

IT(Xf)(x)I

lAlITf(x)l

=

T

i s of Lleaic type

PO (H ,po)

iii) T

-is of weak-type

(pl,p,)

ii)

Then, fur cuerdy

p

sinch that

T

is subadditive and also

X e IR

f o r ezjery

po

and every

f .

y &

p 5

1

and p < p l ,

T

is bounded

111.6.

from

to

Hp(IE?)

INTERPOLATION

313

Lp(lR7.

Proof: Remember t h a t e v e r y

f e HP(lRn)

c a n be w r i t t e n as

I

1X

a . where t h e a . s a r e ( p , p l ) - a t o m s having v a n i s h i n g j J 1J , the mornefits up t o o r d e r [n(-1)] and 1. l X j l p 5 C I l f l I P PO J HP( Eln) s e r i e s converging t o f i n t h e s e n s e o f tempered d i s t r i b u t i o n s . The f =

main p a r t o f t h e p r o o f w i l l be t o show t h a t f o r e v e r y w i t h moments v a n i s h i n g up t o o r d e r

[n(-

1

a b s o l u t e c o n s t a n t . L e t u s show t h i s . Le;’ c o n t a i n i n g t h e s u p p o r t of

a)

llall

P1

5

P1

I( b)

IQI

x e IR”:

101

l-(P1/P)

ITa(x)l >

(l/P)-(l/Po)

Now we e s t i m a t e

a

is a

5

p1

a ) . It follows t h a t

io

pl<

Clt

-P1

IQI

MPJP)

b) f o r

f o r every

t > 0

T h u s , by i i ) and i ) :

t

near

0

and

a)

for

R > 0:

The c a s e

CIQI-’”

a

C, a n

be a minimal cube

Q

(po,pl)-atom.

p l a c e of

i s , even s i m p l e r . We j u s t

=

(p,pl)-atom

IITalIp 5

by i i i ) :

I l T a I l ~ by u s i n g

t b i g . We g e t , w i t h

is

a. Suppose, f o r d e f i n i t e n e s s , t h a t

. Thus, tll

- l)]

USC

I T a ( x ) l 6 ClQl

llallm 5

IQ/-’”

Thus

p-

t

Next we have to put together the estimates for the different atoms. F i r s t . we

in

m

111. REAL HARDY SPACES

314

Hp c Hpo + L p l .

have t o s e e t h a t case Write

as

f e Hp

Qj

and

J 1 = { j: j

1

A.a. e L 1 J J

JeJl is a

1 X.~j a

f =

.

6

Indeed: f o r e v e r y

1

So

l Q j l l-(l’p)

Ajlp

IlflIP

5

HP

J

X.a.

X.a

jeJO 5

1

J

Also f o r

e J1,

j

1

from which

1.

J

ing t o

X.a

1

This i m p l i e s t h a t

X.a. e L J J

jeJ

as a b o v e , By s p l i t t i n g

J J

Hpo

IhjlIIajlll

jeJl

F i n a l l y t o end t h e proof of t h e theorem, l e t f =

L1

and a n o t h e r i n

f

1IXjIp

5

j

Observe t h a t f o r

p1

5

. and w r i t e

a s b e f o r e i n t o a sum b e l o n g -

and r e a l i z i n g t h a t

T

i s at least

IT(f)(x)I 1 l h j l I T a j ( x ) l . Thus as we wanted t o show.

CIIflIP HP =

1

f e Hp

of weak t y p e (l,l), we s e e t h a t llTflIF 5 C

and

J

j

s IXjlP-’,

.

b.

hjlQjl

jeJO

e Hpo.

J

(l/Po)-(l/P)

1

=

jeJ0

1

It follows t h a t

I

J Ij:lqjl < I ijlP’ 0 co Xj a . e H and ( l / p ) - ( 1/ P o ) jeJO a. = b. j, lQjl

a j . Let

J o ) . We c l a i m t h a t :

( p , p )-atom. 0 1

IlajlIl

l i k e a t t h e b e g i n n i n g of t h e p r o o f .

’

b e a minimal cube s u p p o r t i n g

Let

2

I t w i l l be enough t o c o n s i d e r t h e p1 p1 > 1 , L1 c Hpo + L

p1 = 1, s i n c e we a l r e a d y know t h a t , f o r

a

1 , t h e c o n d i t i o n i i i ) i n theorem 6 . 5 c a n be

r e p l a c e d by I

iii)

T

1

i s of weak t y p e ( H ,1)

and s t i l l t h e c o n c l u s i o n of t h e theorem h o l d s . I n d e e d , l e t a be a (p,m)-atom w i t h moments v a n i s h i n g up t o o r d e r [ n ( - 1 - l)]. A s b e f o r e , i f Q i s a minimal cube s u p p o r t i n g a , t h e n P ( p / p ) - ( l / p o ) a i s a ( p o , m ) - a t o m and i t f o l l o w s t h a t

IQI

I{x e R n : I T a ( x ) I >

tl I

5

Ct

-P

IQ1l-(pO/p).

But a l s o now

l Q [ ( l ’ p ) - l a i s a ( l , m ) - a t o m , and t h i s i m p l i e s t h a t I { x e lRn: I T a ( x ) l > t}I 5 C t - I p r o v e , e x a c t l y as b e f o r e , t h a t need t o make s u r e t h a t what w e d i d t o show t h a t

lQll-(l’p).

From t h e s e two f a c t s , we

IITallP 5 C.

Hp c Hpo

+

Hp c Hpo

Hpl.

+

To end t h e p r o o f we a l s o

The p r o o f i s v e r y s i m i l a r t o L1.

We w r i t e

f e Hp

as

OPERATORS IN

111.7.

1

f =

h.a J j

J.

a.s

with the

i n g up t o o r d e r [n(E sum as b e f o r e , having"

(I/P)-l

a

I Qjl

j

-

1

l ) ] and .a. e H J J

6

lAjlp n

JO i s a ( ,m)-atom.

= b .

J

315

b e i n g ( p , m ) - a t o m s h a v i n g moments v a n i s h -

J

1

Hp

.

I l f ( l p We s p l i t t h e HP 1 hjaj e H Indeed J dl ~ , a =,

5 C

.

and a l s o

so

1

I t i s c l e a r t h a t , i n g e n e r a l , we c a n p r o v e t h e f o l l o w i n g r e s u l t . HPo(lRn)

be an operator sending

T

THEOREM 6.6. Let

into measurable functions in

lRn, where

0 < po < PI

+ HP1(lRn) 6

1.

Suppose

that ii

T

is sublinear

iil

T

is of weak type

(H

iiii T

is of weak type

(Hpl,pl)

Then, for every Hp(lRn)

g

PO

that

p

&

,po),

po < p < p l ,

T

is bounded f r o m

Lp(lRn).

7. ESTIMATES FOR OPERATORS ACTING ON

Hp

SPACES

We s h a l l s t a r t by c o n s i d e r i n g s i n g u l a r i n t e g r a l s . We a l r e a d y know ( c h a p t e r 11, c o r o l l a r y 5 . 1 4 ) t h a t any s i n g u l a r i n t e g r a l o p e r a t o r m a p s 1 b o u n d e d l y i n t o L ( E n ) . We s h a l l p r e s e n t l y s e e t h a t any H1(IRn) r e g u l a 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 o f p r i n c i p a l v a l u e t y p e i s bounded from

HP(IRn)

to

Lp(IRn)

n / ( n + l ) < p 5 1 , and t h a t

provided

c a n be pushed f u r t h e r down t o

0

by

p

r e q u i r i n g more and more

r e g u l a r i t y of t h e kernel.

LEMMA 7.1. Let (that means that

T

be a regular singular integral operator in

T

has a kernel

K

IR"

satisfying (5.18) and ( 5 . 1 9 )

fromchapter 11). Let a be a ( p , m ) - a t o m with Then I I T a l I p 2 C , an absoZute constant.

n / ( n + l ) < p 2 1.

111. REAL HARDY SPACES

316

P r o o f : Let

center of

Q Q

be a minimal cube supporting

;j"

and call

This implies that Actually

Ta

i

xo

the

2.

=

Q2&.

is in

m

Imn,,,lTa(x)lPdx

a. Denote by x e IRn\Q,

Then, for

Lp

on

n%

IR \Q, since

p(n+l) > n.

-(n+l)p+n-l drlQl P+(P/n)-l =

c.

C IC I QI l/nr

Since, also

1%

(Ta(x)lPdx

Q

n l T a ( x ) 1 2 d x ) P / 2 . 1 ~ 1 1 - ( p / 2 ) 2 C Ilall~lQ11-(p/2)~ C

5 (

IIR

Note that the result also holds f o r a Indeed, we st 11 have the estimate (7.2)

(p,r)-atom even if

r <

m.

ITa(x)

which is now obtained by using the appropriate version of Hijlder's %

inequality, and on Q, we can use the fact that T is bounded in Lr if r > 1 or the weak type (1,l) in case r = 1. This is entirely similar to what we did at the beginning of section 3 for the Hilbert transform

H

in

1R.

From the lemma and the atomic description of THEOREM 7 . 3 . L e t

T

b e a r e g u l a 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 of

principal value t y p e i n such t h a t

n/(n+l)

ip

w i t h k e r n e l K . T h e n , for e v e r y p T can beextended t o a bounded o p e r a t o r Lp(lRn). I n d e e d , i f f = x j aj, with I h j I P 6 C IIflIHp(lRrl) P i s an a t o m i c decornposi-

1R"

5 1,

b e t w e e n HP(IRn) & ( p , m ) - a t u r n s and 1 J t i l o n f c r , f e Hp(Rn) then

a.

HP(Rn), we get:

5

Tf

=

1 xj J

ProriS: We mayassume a l s o

p

i

Ta. J

1. Start with

in -

f

Lp(lRn). in a n i c e dense c l a s s

111.7.

of f u n c t i o n s , s a y

OPERATORS I N

n ~ ~ ( $ 1L e t

e HP(IR")

f

Hp

317

A. a. J J

f = J,

atomic decomposition f o r

f

1

with

Tf(x)

5

CIIflI

T E f ( x ) where

lim

=

.

rP

We know HP(nn) . t h a t , - f o r a l m o s t e v e r y x e R"

AjlP

J

( c h a p t e r 11, c o r o l l a r y 5 . 2 2 . )

be a n

E +O I

We s h a l l n e e d a smooth t r u n c a t i o n o f t h e k e r n e l .

For t h a t p u r p o s e ,

we s h a l l u s e a r a d i a l

such t h a t :

@ ( x )= 0

for

1x1

5

Cm

1/2,

function

@(XI

= 1

@

for

in

IRn

1x1 t 1

and

0

5 @

(x) 2 1

e v e r y w h e r e . Then:

(7.4)

J

K(y)@(y/E)f(x-y)dy* 0

as

E

* 0

f o r a.e.

x e Rn.

E/2
Now

( 7 . 4 ) f o l l o w s once t h e s e two f a c t s a r e e s t a b l i s h e d :

e S(IRn)

a)

( 7 . 4 ) holds with

b)

The i n t e g r a l i n ( 7 . 4 ) i s bounded i n a b s o l u t e v a l u e by

$

i n p l a c e of

f.

The way t o d e r i v e ( 7 . 4 ) from a ) and b ) i s a l r e a d y f a m i l i a r t o u s ( s e e , for i n s t a n c e , t h e p r o o f o f theorem 1 . 9 . o b t a i n b ) we j u s t u s e t h e e s t i m a t e

IK(x)I

5

i n c h a p t e r 11) To ClxI-".

w i l l f o l l o w once we r e a l i z e t h a t

(7.5)

Indeed i f

J

K ( y ) @ ( y / E ) d y* 0 E/

as

E

*0

2<,Yl<E

( 7 . 5 ) h o l d s , we s h a l l j u s t n e e d t o s e e t h a t

A s for

a ) , it

111. REAL HARDY SPACES

318

dy

K(Y)@(Y/E)(J~(x-Y)-J~(x))

Jl

But t h i s i s obvious s i n c e t h e smoothness of i n t e g r a n d i s a b s o l u t e l y i n t e g r a b l e around

0

+

as

E +

0

guarantees t h a t the

0.

Now, ( 7 . 5 ) f o l l o w s e a s i l y from t h e f a c t t h a t

K

s a t i s f i e s (5.4) i n

c h a p t e r 11, a f t e r i n t e g r a t i n g by p a r t s . W r i t i n g

@ ( x ) = @ ( l x l ) and

F(r) =

I

K ( r y ' ) d y ' rn-',

we g e t

'n-1 E

F ( r ) @ ( r / E ) d r=

K ( y ) @ ( y / E ) d y= IE/2

jE

Now from t h e f a c t t h a t F(r E + 0 , we c l e a r l y o b t a i n p 7 . 5 ) . 2,

Let u s look a t t h e k e r n e l s

KE.

-n

I

d r = lim

6+0

as

x f 0 , and

C l Y l x I -n-l

( u n i f o r m means, of c o u r s e , t h a t

K(y)dy + 0

<E

They s a t i s f y un form e s t i m a t e s

f o r every 5

L
C

for

0 < 2 ( y l < 1x1

does n o t depend on

E

> 0 ) . The

f i r s t e s t i m a t e i s o b v i o u s and t h e second c a n be d e r i v e d i n t h e f o l l o w i n g way: L e t

0 < 21yl

< 1x1

The f i r s t term i n t h i s sum i s bounded by

Cly(

where

C

is

.

the constant f o r K The second term i s bounded by however, t h a t t h i s second term v a n i s h e s u n l e s s C I x I -n I y/ E I Note,

.

~ / 3< 1x1 < 2 ~ s, o t h a t t h e e s t i m a t e c a n be r e w r i t t e n as

Clyl 1x1-"-!

2,

The f a c t t h a t t h e k e r n e l s

KE

s a t i s f y t h e s e uniform e s t i m a t e s

i m p l i e s , of c o u r s e , t h a t for e v e r y

( p , m ) - a t o m , w i t h our r e s t r i c t i o n

2,

on

p,

I(TEaIIp

5

C

with

C

i n d e p e n d e n t of

E.

111.7. OPERATORS IN

Hp

2.

319 2.

Observe that each K E is a bounded function with Also we have an inequality

IK,(x)l

5

CE-n.

with CE depending on E , valid for every x,y e IR". Indeed, since Kc is bounded, (7.6) needs to be proved only for IyI small, say for Iy( < ~ / 8 We . can also assume 1x1 > ~ / 4 ,since otherwise, the left hand side of (7.6) vanishes. Then we have 21yl < 1x1 and we can use the inequality proved above, to obtain:

2.

2.

2.

IKE(x-y) z KE

Now, since

- K E (X)I

4

CIXI

-n-1

-n-1 IYI s CE IY

is bounded, (7.6) implies that

IyI < 1). In other for all 0 < CI I 1. F o r n/(n+l) < p < 1, we words: KE e ACI(IRn) % have 0 < a = n((l/p)-l) < 1. Thus KE e2.An((l,p)-l) (IR"). AS we saw in section 5 , this guarantees that KE can be integrated against Hp functions, and provides a linear functional on HP(IRn). This implies that 0 < CI 2 1 (We only need to consider

for every 2.

a n d , consequently:

2.

Now, from

Tf(x)

=

Trf(x)

lim E+O

c.

a.e., we can get, by using Fatou's

.

lemma: IlTfIl~ 2 C((f(lP After this, the operator T HP(IR") extended by continuity to the whole space HP(IRn).

can be

Denote its extension also by T. Then, if f e HP(IRn) has an atomic lhjlP I c I(flIP ; since decomposition f = 1 A.a. with J J ni j J HP(IR") I"

f

lim

=

N+m

Tf

=

lim N+m

1

X.a. in

Hp(IRn), we have

,j=1 N

1

j=1

AjTaj

AjTaj

=

J

in

Lp(Rn), as we wanted to prove.

320

111. REAL HARDY SPACES

Observe that, even though the statement was made for (p,m)-atoms, the conclusion holds also for a decomposition into (p,r)-atoms with any admissible r. Now we shall impose more regularity to the kernel obtaining Hp, Lp boundedness for p ' s closer to 0. We get the following extension of theorem 7.3. T

T H E O R E M 7.7. L e t v a l u e t y p e in

Eln

b e a s i n g u l a r i n t e g r a l o p e r a t o r of p r i n c i p a l with a kernel

...,

o r i g i n , k = 0 , 1, 2 ,

161

suck thut

5

k + 1,

f1ii(j

K

of c l a s s

Ck+l

outside t h e

a n d s a t i s f y i n g , for e v e r y m u l t i - i n d e x every

6

x # 0:

n/(n+k+l) < p 5 1, T e x t e n d s t o a b o u n d e d o p e r a t o r b e t w e e n Hp(IRn) & Lp(lRn). T h i s e x t e n d e d o p e r a t o r s e n d s t h e d i s t r i b u t i o n f e HP(IR") having an atomic d e c o m p o s i t i o n f = 1 X.a. i n HP(H") , i n t o t h e f u n c t i o n J J Tf = 1 h.Ta.. T h e n , for e v e r y

J

p

such that

J

p < 1. The first thing we do is to extend lemma 7.1. to our general situation. We have a fixed p with n/(n+k+l) < p < 1. We take a (p,m)-atom a and we have to make sure that ))Ta))p5 C , an absolute constant. The main difference with the proof of lemma 7.1. is that now we estimate ITa(x)l for % x e lRn\Q by substracting from K(x-y) the Taylor polynomial of K at x-xo of degree N = [n(- 1 - l)]. By doing this we get P Proof: We may assume

ITa(x)l

5

Clx-x I -n-l-NIQll-(l/p)+((N+l)/n) 0

A l l this is possible since n/(n+k+l) n. Actually

Combining this with the inequality obtained exactly as before, we

ITa(x) I p dx 5 C , which is at IITalIp 5 C.

111.7. OPERATORS IN

Hp

321

To end the proof we proceed as in the proof of theorem 7.3. Take f e HP(IRn) n L 2 (IR") with an atomic decomposition f = 1 X.a. J J'

i

J

We consider the smoothly truncated kernels

kE

defined exactiy a s in the proof of theorem 7.3. and the corres-

ponding operators

?Eg =

Since T is a regular singular integral operator of principal value type, 'L we still have Tf(x) = lim TEf(x) for a.e. x e IR". Now the 'L kernels KE satisfy: € 4

ID B% KJx)I 6 Clxl - n - I B 1 , 161 5 k + 1 , and every

1)

that E

for every multi-index x f 0 , with

C

such

independent of

> 0. 'L

2)

n

with

K E e I\ (IR )

c1

=

1

n(-

0:

P

-

l),

Indeed, the derivative appearing in 1 ) is a linear combination of terms ofthe form

E -I0'. If U = 0 , But each of these terms is bounded by C IX~-~-''' this is the required estimate. If 0 f 0 , the corresponding term may be f 0 only for 1/2 < Ix/El < 1. Thus, we finally obtain the

estimate

CI~I-~-"'. 'L

2) follows from the fact that each KE is bounded and has bounded derivatives up to order k + 1 and [a] = cn(,1 -1)] 5 k.

t)

and 2), the proof is completed as follows: % KE, and consequently KE(x--), can be integrated

With the aid of 2) implies that against

Hp

functions and provides a linear functional on

HP(IRn).

Thus, we get

'L

Then, 1) implies that we have uniform estimates IIT a fore,

IITEflIp % P

5

1I j

'L

X.1'

llTEajIIF

6

J

using Fatou's lemma we conclude that

C

1 I A.1' j

5

J

IITflIp P

IIp

€ J P

6 C

6

C. There-

CIIflIP HP(IRn)' Ilfll:p(,n).

By The

322

111. REAL HARDY SPACES

proof finishes as that of theorem 7.3. Observe that any singular integral operator T actually maps H 1 (lRn) boundedly into itself. In order to prove this we just need , n , as soon as to see that R.(Tf) e L1(lRn) for every j = 1 , J f e H1(Rn). But R (Tf) = T(R.f) and it follows from the character1 j J ization of H (IR") by means of systems of conjugate harmonic functions, that each R f e H1(lRn). Therefore T(R.f) e L1(IRn). j J

.. .

It is natural to seek an extension of this result to p < 1 for nice singular integrals. As usual we start by analizing the action of the operator on an atom. T H E O R E M 7.8.

Let

(p,m)-atorn. T h e n

II Tall

T a n d P be a s in t h e o r e m 7.7. and let Ta e HP(Rn) and S

C, i n d e p e n d e n t of

a

be a ~

a.

HP(1R")

Since T commutes with translations, we may assume that a is supported in a ball B = B ( 0 , r ) and la(x)l s CIBI-l'p for every x, with C a geometric constant. We shall produce an atomic decomposition for Ta by decomposing the kernel K in a smooth way. Take funcLions aj(t), j = 0, 1 , 2, C m in ( O , m ) , satisfying @j b 0, 1 @ (t) = 1 f o r every t e ( 0 , m ) , Q 0 is supported in j j=o @j is supported in [2j-'r,2j+'r] for j b 1 and co, 2 r l I 14:")(t) I S Ckt-k for every t > 0, every k = 0, 1, 2, , , and every j, with Ck depending only on k and being independent of r (This partition of unity can be obtained as in chapter 11, lemma Proof:

...

.

6.5.).

...

Now we define for each j = 0, 1 , 2, : Kj(x) = K(x)Qj(lxl), and I observe that all the K.s satisfy the same estimates as K with a J uniform constant. We have a decomposition:

valid a.e. and also in the sense of distributions, since it is actually a locally finite sum. Each K.*a has vanishing moments up J to order N = [n(- 1 - 1 1 1 , since convolution preserves this property P of a. Now we just need to study the relation between the size and

111.7. OPERATORS IN

j

KO

IIKo*a112

323

K *a.

the support of each Convolution with

Hp

d

is an operator bounded in

Cllal12

5

L2. Thus:

CIBI 1/2-1/~= c(rn)(1/2)-(l/~)

Since Ko*a has support contained in hOaO where . a is a (p,2)-atom.

B(0,3r), we can write

KO*a =

=

e

To estimate K *a for j L 1 we use the fact that ID Kj(x)I 5 C(2jr) - n - l e l for J every I B I 6 k + 1, in particular for I B I = N + 1 L k + 1. We write

x

where P is the Taylor polynomial function K We get

6

for the

j'

Since K +a is supported in B(0,2J+2r), we can write K.*a(x) = j where a is a (p,m)-atom and A . = C2j (n( ( 1b)-1) -N-l) = hjaj(x) J

j

1 Note that - 1)- N - 1 < 0, and, consequently fixed finite number.

1 I hJIp

6

C, a

m

Thus, we have shown that Ta = 1 h a in the distribution sense, j j where the als are (p,2)-atomsJg8d 1 IX,lp d C. This proves j=o

J

that

Ta e HP(IR")

with

((Ta(( 6 HP(lR")

J

c.J-I

We can finally give T

T H E O R E M 7.9 Let

and

to a b o u n d e d o p e r a t o r in

Proof: Let

p

f e L2 fl Hp(IRn)

f =

1h.a

j

J

J

be as in t h e o r e m 7.7.

HP(Rn). and suppose

Then

T

extends

324

111. REAL HARDY SPACES

is a decomposition of c c IlfIlP HP(1R")'

f

into (p,m)-atoms aj

1 lAjlP

with

6

j

If we are able to show that (7.10)

Tf

1 A.

=

Ta

j

j~

as tempered distributions; then, since theorem 7.8. implies that the series converges in HP(lRn), we shall have: Tf e Hp(IRn) and

c

6

IlfllP HP( 23") 'L

(7.10) can be proved by using the smoothly truncated kernels K 'L E and the corresponding convolution operators T appearing in E theorems 7.3. and 7.7. Let us see how. First of all, a simple modification of the argument given in theorem 7.3. implies that 'L

TEf

-

Tf

in

L2(IRn)

as

E

+ 0.

This also follows from the fact (established in chapter V just before corollary 4.ll.)that the maximal operator of the smooth truncations is bounded in L2. Now 'L

T f E

=

1 A. j

'L

TEaj.

J

This is a pointwise identity and also an identity between tempered distributions. If we are able to show that 'L

T a. E J

-

Ta. in J

HP(1Rn)

as

E

-t

0

we shall have

5: J

'L

A j TEaj -+

1 A.

j~

Ta

3

in

HP(IRn)

as

E

-f

0

from which 7.10 follows immediately, that way completing the proof of the theorem.

1 1 1 . 7 . OPERATORS I N

Hp

325

To s e e t h a t , f o r a n atom a %

T a --+

in

Ta

E

HP(lRn)

as

+ 0

E

is q u i t e simple. A s i n t h e p r o o f of theorem 7 . 8 . ,

Thus, c l e a r l y , f o r each

K

E,

we decompose

j

as

L(

.*a + ~ . * a i n J J

E

* o

a n d , s i n c e b o t h f u n c t i o n s h a v e t h e same s u p p o r t : %

.*a+

K € 9

J

HP(IFP)

~ . + a in J

A c t u a l l y , as t h e p r o o f o f theorem 7 . 8 .

as

E

+

o

shows,

%

K with

b. J ,E:

.*a = p b J j j,E

E,

b. J

in

and

as

HP(IRn)

and u n i f o r m l y bounded norms, and %

TEa-

Ta

in

Hp(lRn)

as

E

K.*a = p J j bj E

+ 0

1 I uj 1'

+ 0;with

<

a.

This c l e a r l y gives

u n i f o r m l y bounded n 0 r m s . m

We have s e e n t h a t atoms a r e v e r y c o n v e n i e n t f o r s t u d y i n g t h e behav i o u r of c e r t a i n operators prove a n atom

Hp, Lp

T , l i k e s i n g u l a r i n t e g r a l s , on

e s t i m a t e s we j u s t n e e d t o s t u d y t h e s i z e o f

a . We have a l s o s e e n t h a t t h e s e o p e r a t o r s map

Hp

Hp.

To

Ta

for

bounded-

l y i n t o i t s e l f . There is an u n s a t i s f a c t o r y a s p e c t i n t h i s second k i n d o f r e s u l t s , n a m e l y , t h a t f o r a g e n e r a l atom

a, T a

i s not an

atom i t s e l f b u t h a s t o be decomposed i n t o atoms. I n d e e d , i n g e n e r a l Ta

w i l l n o t h a v e compact s u p p o r t . We are g o i n g t o f i n d a c l a s s o f

f u n c t i o n s more g e n e r a l t h a n a t o m s , which s t i l l g e n e r a t e

Hp

and

111. REAL HARDY SPACES

326

include the functions Ta for every atom a. These functions will naturally decompose into atoms, and will be called molecules. To motivate the definition of a molecule, let us look back at the estimates obtained for Ta in lemma 7 . 1 . We assume for simplicity n = 1 and p = 1. Let a be an atom supported in an interval I centered at 0. Then we have the estimates:

and

This second estimate can be rewritten as

which implies

we get m

I, I x

ITa(x

I 2 dx

d

ClIl.

This, combined with the first estimate, leads to

I

.m

1x1 2 ]Ta(x)l 2dx

m

I

C.

-m

This is a very nice condition as the following lemmas show LEMMA 7 . 1 1 . f(x)

both

Let

and

f b e a f u n c t i o n o n t h e r e a l l i n e IR s u c k t h a t x-f(x) b e l o n g t o L 2 ( l R ) . Then f e L 1 ( I R ) .

A c t u a l 1~ m

(j-mlf(~) ldxl2

5

im

If(x)12dx)1/2(lm1x121f(x) 12dx)1’2

8-(

-m

-m

111.7. OPERATORS IN

Hp

327

Proof:

By taking

1 r = IlflIs IIxf(x)112

we get

whose square is the inequality in the statement. L E M M A 7.12. S u p p o s e f is a s in t h e p r e v i o u s lemma a n d f(x)dx = 0 . Then f e H1(IR) &

Proof: We already know that f e L1 (IR). We shall prove that, also, Hf e L1 (IR). This is quite easy. First of all, since f e L2(IR) and the Hilbert transform H is bounded in L2 ( R ) , we have Hf e L 2 ( I R ) . Secondly

since f has vanishing integral. Again, by the L2 boundedness of 2 H, we get that x-Hf(x) belong to L (IR). Then, lemma 7.11. applied to Hf implies that Hf e L1(lR) as we wanted to show. Now, it seems natural to call molecule (centered at xo e IR) to any real valued function M on the line satisfying the two conditions:

J-

m

m

ii)

M(x) dx = 0

J

-m

328

111. REAL HARDY SPACES

The left hand side of i) will be called "molecular norm" of M and will be denoted by N(M). It is clear that every (1,2)-atom a is a molecule with molecular norm bounded by a constant independent of the atom. Indeed, if I is the minimal interval containing the support of a , and xo is the center of I:

so that

N(a)

1.

5

We have obtained the following characterization of

H1:

Q3

A function

f

H1

is in

if a n d only if

f(x)

1

=

M'.s J

with t h e

M.(x)

j=1

m

f o r a.ex,

J

being m o l e c u l e s a n d

Next, we face the task of extending the molecular characterization to HP(lRn) for arbitrary p and n. It will also be convenient to consider Lq norms for arbitrary q > 1 rather than just L2 norms. 0 l/p

-

l/q.

Then, 2

x0 e lRn is a r e a l - v a l u e d f u n c t i o n

M

lRn a n d satisfying:

0 = (l/p-l/q)/b, so t h a t

",1

ii)

m

xPM(x)dx

[n((l/p)-l)]

=

= 0

0 < 8 < 1

f o r every m u l t i - i n d e x

P

such that

N

The left hand side of i) will be called the "molecular normll of and it will be denoted by N(M). Observe that condition i) implies that all the integrals in absolutely convergent. Indeed, we just need to see that

and this follows from Holder's inequality in this way:

M

ii) are

111.7. OPERATORS IN

329

Hp

Note that the molecules we used to motivate the formal definition are, in our present notation (1,2,1)-molecules in

LEMMA 7.14. A (p,q,b)-molecule is also a any admissible b 0, as one seed by looking separately at the cases

for every lx-x 0 I < R

and

Taking

( 1 1 l.-xoI nbMilq / 11M11,)

R =

Ix-xol > R.

and,consequently, writing

0

=

we get:

(l/p-l/q)/L

=

0 b/6:

LEMMA 7.15. Let 0 < p 5 1 < 9 5 a. Then, a (p,q)-atom is always a (p,q,b)-molecule for any admissible b , uith molecular n o r m bounded by a constant independent of the atom. Proof: Let a be a

a, and let

xo

llallq

II

(p,q)-atom. Let Q be a minimal cube supporting be the center of Q. Suppose b > l/p - l/q. Then:

6

I Q I '/q-'/P

I .-xoI nballq

and

s Clqlb+l/q-l/P

111. REAL HARDY SPACES

330

(l-e)(l/q-l/p) + e(b+l/q-l/p) = l/q

since

It is clear that

C

- l/p + be

=

0.

is a geometric constant independent of

a.

The main result is the following: THEOREM 7 . 1 6 . Let M e HP(IR") and

with C

M

(p,q,b)-molecule in

33".

Then -

independent of the molecule.

Proof: We may assume, without loss of generality, th t M i , centered at the origin, and also that N ( M ) = 1. We define 0

by

setting

and consider the sets:

n

E~ = {x e IR : 1x1

We shall denote by Mk(x)

=

M(x)X

Mk

6 U)

and

the restriction of

M

to

Ek, that is:

(XI.

Ek We shall also consider, for each k , the unique polynomial of degree N = [n((l/p)-l)] whose restriction to Ek, denoted by k' ' satisfies

for every multi-index

a

with

I a1

6

N.

The proof will consist of two parts 1) To show that each

Mk-Pk is a multiple sequence of coefficients in tP.

of a

(p,q)-atom with a

111.7. OPERATORS IN

2)

Hp

331

To show, using summation by parts, that 1 Pk can be written as k an infinite linear combination of (p,m)-atoms, with a sequence of coefficients in .'2

Since

M

=

1 M~ k

=

1 (Mk-Pk) k

+

1 k' k

and the series in 1) and 2) will be seen to converge locally in Lq, the theorem will be proved. Let us start with part 1). It is clear that Mk-Pk has the right cancellation properties, so we just need to worry about the relation between the size and the support. B(0,2 k a). Thus, a minimal cube Qk Mk-Pk is supported in the ball containing the support of Mk-Pk has measure equal to C(2k0)n, with C a geometric constant. As for the size, we have

Qk 1)

=

CllM -P

(1

d

Lq(dx/lEkl)

, since Pk

is bounded by a constant

Ekl

times the average of

lMkl

over

Ek (see below).

The definition of 0 together with the fact that M is a molecule centered at the origin with N(M) = 1 , imply that

(p,q,b)-

where fj = (l/p-l/q)/b as in definition 7.13. It will be convenient to write this estimate as

Let us go back to the estimates for

Mk-Pk

111. REAL HARDY SPACES

332

4

lIMolI

I I M I I ~ ~ -1/q E ~ ~

=

co -n/p

Lq(dx/lEol) IIMkI\

5

C(2ko)-n/q(lM-(

5

Lq(dx/lEkl) 5

C(2ko)-n/q-nbona = c(2ko)-n/P(2"a)-k

k = 0, 1 , 2,

It follows that, for C(2 is a

na k

) (Mk-Pk) = Ak

(p,q)-atom, C

is a Ak

1 x;

=

k=O

being an absolute constant. In other words:

A k Ak

Mk-Pk

where m

...

(p,q)-atom and

-

m

1

cp Znap)-k

k=O

Xk

=

C(2na)-k. Observe that

as we wanted to show. m

Z ) , where we have to deal with

Next, we come to part

1 k=O

k'

...

For k = 0, 1 , 2, and c1 a multi-index such that l c l l 5 N , let k @,(x) be the function on Ek (actually the restriction to Ek of a polynomial of degree 4 N ) uniquely determined by the conditions

f o r every multi-index and 0 otherwise.

B

with

16 I

Then

mk c1

1

= -

lEkl

I

Mk(x)xadx.

It is clear, by homogeneity, that:

with

C

an absolute constant.

5

N , where

6a,B = 1

if

D

= c1

111.7. OPERATORS IN

Hp

Note in passing that the above expansion for immediately yield: for the

333

Pk

and the estimates

@a

estimate which we have already used in part 1. Now

“ k 1 m IE k=O. a. N, we can write.

M(x)xadx = 0

After observing that with

5

m

a

for every

m

= L ( k=O

c

j=k+

where

We have obtained

It is clear that each

Qi

has the right cancellation properties to

be a multiple of a p-atom. It is supported in a cube of measure k k equal to C(2k0)n. Now we just have to estimate the size of “a Qa m

INa( k 5

1

j=k+l

1

m

IMj(x)I 1x1 “ldx

5

C

1

j=k+l

IIMjll

(2jo)n+Ia Lq(dx/lEjl 1

I

111. REAL HARDY SPACES

334

k k It follows that P & U J a ( x ) = na -k" atom and 0 5 5 C(2 ) k,a

.

Ak,,(x)

where

A

k,a

is a

(p,m)-

Then : m

m

and

as we wanted to show. After this, we can give the following molecular characterization of HP. T H E O R E M 7.17. Let

0 l/P

lRn, b e l o n g s t o

~

if, as t e m p e r e d d i s t r i b u t i o n s

where each

M

is a

j -

(p,q,b)-molecule,and

3 N(Mj)P A c t u a l l y , if t h e a b o v e d e c o m p o s i t i o n h o l d s , t h e n

IlflIP s C 1 fl(Mj)' HP(IR") j with an absolute constant

C. A l s o , if

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

f e HP(Rn), t h e a b o v e

111.7. OPERATORS IN

C

being again an absolute constant.

Hp

335

0

We may reformulate theorem 7.8. in the following way:

THEOREM 7.18. Let T be a 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 a t i s f y i n g t h e h y p o t h e s e s of t h e o r e m 7.7. Let n/(n+k+l) l/p - l/q. T h e n , for every (p,q)-= a, Ta is a (p,q,b)-Elecule w i t h N(Ta) 5 C , i n d e p e n d e n t of a. Proof: Let center of

Q be a minimal cube supporting Q. Then

a

and let

xo

be the

and

The first term in this sum is bounded by CIQlb+l/q-l/p. Then, as in 'L the p r o o f of theorem 7 . 7 . , we have, for x e lRn\Q:

This estimate implies that the second term in the sum above is a l s o bounded by

Cl Q

lb+l/q-l/P.

Consequently, with

0 = (l/p-l/q)/b, we

have

as we wanted to prove.

REMARKS 7.19. a) Observe that, if M is a function in the Schwartz class S , then M is a (p,q,b)-molecule f o r any admissible (p,q,b) if and only if M satisfies ii) in definition 7.13., that i s , if the right amount of moments of M vanish. This is clear since, for M e s, i) in 7.13 holds automatically. b) The usefulness of molecules is enhanced in case

q

=

2

and

111. REAL HARDY SPACES

336

b = k / n , with

a p o s i t i v e i n t e g e r . With t h e s e v a l u e s of t h e p a r a -

k

i s a molecule c e n t e r e d at

M

m e t e r s , if

0 , we s h a l l h a v e

This is equivalent t o saying t h a t t h e r e is a constant f o r every multi-index

(0.1

with

O.

=:

C

such t h a t ,

k:

e

I I M I I ~ - ~ I I ~ 5~ cM. I I ~

since

lxlk

%

The f a c t t h a t

10.

F

lxcll =k

is i n

x'M(x)

Fourier transform

of

M

(see Stein-Weiss [ 2 ] ,

sense

L2

is equivalent t o saying t h a t t h e

has a derivative 1.1.9.).

Dah

i n the

L2

Indeed

By u s i n g P l a n c h e r e l ' s t h e o r e m , we a r r i v e a t t h e f o l l o w i n g c o n d i t i o n for

M:

Now we s h a l l s e e t h a t c o n d i t i o n i i ) c a n a l s o be i n t e r p r e t e d a s a O b s e r v e t h a t i) i m p l i e s t h a t , f o r e v e r y m u l t i 2 such t h a t I 5 k , D% h a s L - d e r i v a t i v e s up t o o r d e r

r e q u i r e m e n t on

4

index k -

If,

space

[I],

M.

or, as i t i s s a i d e q u i v a l e n t l y , D p G

belongs t o t h e Sobolev

L ~ - I p l ( l R n ) . We know, a f t e r S o b o l e v ' s lemma ( s e e S t e i n

p . 1 2 4 ) t h a t t h e Sobolev space

L:(IRn)

i s made up o f c o n t i n u o u s

1

f u n c t i o n s p r o v i d e d s > n / 2 . Now f o r Ip I 6 N 5 n ( - - l), s i n c e - > -I - we h a v e : k - \ p \ > n (1- - -1) - n ( - 1 - 1 ) = n P Thus ; 'D is n P P 2 P 2 ( e q u a l a.e. t o ) a c o n t i n u o u s f u n c t i o n . Note, f i n a l l y , t h a t c o n d i t i o n

:,

-.

i i ) i n d e f i n i t i o n 7.13.

IpI

5 N =

[n((l/p) -

is equivalent t o saying t h a t , f o r

111:

where t h e e v a l u a t i o n o f

D%

at

0

i s meaningful s i n c e i t i s a

continuous function. We h a v e r e a c h e d t h e f o l l o w i n g c o n c l u s i o n :

HP

111.7. OPERATORS I N

The f u n c t i o n

337

is the Fourier transform of a

F

c e n t e r e d a t z e r o i f and o n l y i f

(p,2,k/n)-molecule

:

i )I

with

a, )a1

independent of

C

DDF ( 0 ) = 0

ii)

f o r every

e

= k, where

I@ I

0 5

5

= n(l/p-l/2)/k

and

N

These two c o n d i t i o n s w i l l be t h e key t o o u r s t u d y o f t h e m u l t i p l i e r operator f-Tf

=

w h e r e , as u s u a l

*

(mf)'

denotes

t h e i n v e r s e F o u r i e r t r a n s f o r m . We s h a l l

p r e s e n t below a v e r s i o n o f H o r m a n d e r ' s m u l t i p l i e r theorem f o r

Hp

s p a c e s . The r e s u l t w i l l c o n s i s t i n showing t h a t , i f we impose t h e u s u a l c o n d i t i o n s on t h e d e r i v a t i v e s o f h i g h e r o r d e r ( s e e theorem 6 . 3 . (p,2)-atom a (centered at

m

up t o

a

i n c h a p t e r 11), then

sufficiently T

carries a

O), h a v i n g a c e r t a i n ( l a r g e ) number o f

v a n i s h i n g moments, i n t o a m o l e c u l e .

To s e e t h i s we h a v e t o show t h a t

n

(Ta)

=

:m

s a t i s f i e s t h e same c o n d i t i o n s i ) ' and

i i ) l

as

F

above.

S i n c e t h e n a t u r a l h y p o t h e s e s o f H o r m a n d e r ' s theorem a r e e s t i m a t e s f o r t h e d e r i v a t i v e s of

m , it i s reasonable t o expect t h a t success w i l l

depend on our a b i l i t y t o e s t i m a t e 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

a

f o r a a n atom. We s h a l l g i v e below s e v e r a l e s t i m a t e s o f t h i s t y p e , which h a v e a l s o i n d e p e n d e n t i n t e r e s t and e x t e n d r e s u l t s o b t a i n e d i n chapter I. c ) Observe t h a t i t f o l l o w s from t h e p r e v i o u s remark t h a t , for t o be t h e F o u r i e r t r a n s f o r m o f a

i t i s n e c e s s a r y and s u f f i c i e n t t h a t 0 5

101

5

F e S

(p,2,k/n)-molecule (centered a t 0 ) DBF(0) = 0

f o r every

N.

I n t h e n e x t theorem we g a t h e r i n f o r m a t i o n a b o u t t h e F o u r i e r t r a n s f o r m a

o f an atom

a.

111. REAL HARDY SPACES

338

a

centered at zero). Suppose that

-

k t N = [n(l/p

%

a

atom b y saying that d = l/(l/p-l/2)

= 2p/(2-p),

with -

5 CIC1

a

n

Here, a s usual, r ' C

constants

2

=

where

1,

P(x)

the function

0 2 la1 6 k :

-A(k,n)

1 > 0

and every

1 5 r 5

m:

,r,n)

+ 1) r -

2

stands for the exponent conjugate to

a) pnd

Proof: a) Da2(C)

Then, if we set

llal12

cllal12- B ( a

B(a,r,n) = d ( w

with -

1

+ 21 ) -

b ) For every multi-index 2

such that

k+l-la

k+l A ( k , n ) = d(n

IIIDaal A 2 I\,,

(p,2,k)-=).

we have the following estimates: a

a ) For every muZti-index

IDai(C)l

has vanishing moments up to order

( W e shall attjreviate this restriction on the

l)].

1,

=

do not depend on

b)

a ( ~ ) ( - 2 n i x )e ~ -2nix'Edx a

a(x)(-2nix) (e

-2nix.C

- 2 n i x -5

.

The

at

0

=

- P ( x ) ) dx

i s t h e Taylor p o l y n o m i a l o f d e g r e e xl->e

r.

a.

k - (a(

for

The t y p i c a l e s t i m a t e for t h e r e m a i n d e r

i n T a y l o r ' s formula f o r t h i s f u n c t i o n y i e l d s :

ID';(C)

I

5 CI

51 k + l - l a I

l a ( x ) I Ixlk+ldx 5

t o saying t h a t Observe t h a t , s i n c e

k 2 N

=

or, equivalently, k+l 1 1 1 1 n + 7 > P- - - = -

1 [n(P

-

l)],

so that:

we s h a l l h a v e

A ( k , n ) > 0.

IQI

5

llal12. -d

k+l > n(l-1) P

111.7. OPERATORS IN

b ) Here we just use the fact that

D"^a

Hp

339

is the Fourier transform of

r

the function a(x) (-27rix): First we deal with the cases r = m , and then we use interpolation. If

r = 1,r'

=

m,

L1,

we have, by using the

=

1

and

L m boundedness of the

Fourier transform: 1

+ 7

II all

5

lalQ 1 l-d(+ -)2 n 5 cllal12 After raising to the second power, we obtain b) for If

r

=

m,

r = 1.

r' = 1 , we use Plancherel's theorem to obtain:

which is what we wanted. Now, for

1 < r <

m,

we write

and observe that

COROLLARY 7 . 2 1 .

form

Let

f e Hp(yln), 0 < p

$

1. T h e n t h e F o u r i e r t r a n s -

A

f

f , w h i c h a l w a y s m a k e s s e n s e as a t e m p e r e d d i s t r i b u t i o n ,

is actually a c o n t i n u o u s f u n c t i o n satisfying t h e estimate:

with C Proof:

i n d e p e n d e n t of

f.

We use the preceding theorem with

la1 = 0

and

k = N = [n((l/p)-l)]. Given a (p,2)-atom a centered at -A N+l of 7.20. implies la(<)[ 6 CIS( Ilal12 , where

0 , part

a)

111. REAL HARDY SPACES

340

N+1 1 A = A(N,n) = d (- n + ? ) Also, part b) with

-

1

r = 1 , yields: Ia(c)I

5

1-d/2 Cllal12

Observe that, in the first estimate, the exponent of

Ilal12 is negative, whereas in the second estimate, the exponent of Ilal12 positive, because d 5 2 .

is

Note also that the estimates are valid for every (p,2)-atom a, even if it is not centered at zero. Indeed, when a function is translated, the modulus of its Fourier transform does not change. Now, combining both estimates we obtain a better one. We use the first one where 151N+111all;A 5 Ilal12 1-d/2 or, equivalently, n d 151 5 Ilal12. In the rest of IR", we use the second estimate.

where

Ia 5 ) since

For

N+1-An/d

n/d-n/2

=

d

5 1 " > I I ~ I we I ~ get ,

I;(C)I

1-d/2 CIIaI12

5

CIS

5

Thus, (p,2)-atoms satisfy uniformly the estimate: la(s)I Now if

C l s l n( I/P)-l)

5

f e HP(IRn), we have a decomposition f

where the

=

1 A.

j

J

a . s are J

a. J

(p,2)-atoms and

Since the series converges in

5"

and the Fourier transform is

111.7. OPERATORS IN

continuous in

Hp

341

S ' , we shall have

* I

The uniform estimate for the a . s , which are,of course, continuous J functions, together with the fact that

Xj aj

imply that the series

converges uniformly over compact

subsets to a given conthuous function. The distribution must coincide with it. In this sense f is a continuous function and

The estimate we have just obtained, can be written as: I 3 X ) Iplxln(p-2)

so that, for

5

CIIflIP Ixl-n HP(IR")

f e HP(IRn), the function

~?(x)~p~x~n(p'is 2 )in

weak L1. We shall presently see that this function is actually integrable. We first prove it for an atom. THEOREM 7.22. Let

a

(p,2)-atom in

be a

a constant independent of

IR", 0 0. F o r the first integral we use the first estimate of a as i t appears in the p r o o

of corollary

7.21

i,

Ix

p-2)dx

5

C

xlcR

where

A

=

1 1 d (-N+ n + 7 )-l=

N+1 1 n - P

(--

342

111. REAL HARDY SPACES

A simple computation yields:

I XI< R

I (: x) I I x I n(p-2 )dx

2

(N+l+n)p-n > 0, since

Note that

(N+l+n)p-n

C( R - 11 a11

N+l > n(l/p - 1).

F o r the second integral we start by using Holder's inequality with

exponents

2/p

and its conjugate

2/(2-p). Then we use Plancherel's

theorem, obtaining:

R

If we take

=

[lall:'n,

0 < p 5 1

For

COROLLARY 7 . 2 3 .

the proof is finished.

1?(x) Iplxln(p-2)dx 5 CIIf l i p

HP( IR") does n o t depend on

where

C

proof

Let

f e H'(IR"). f = l x j

with

a . being

J

j

f e HP(Rn).

Write

a. J

(p,2)-atoms and

1

P

5

CIIflIP HP( IR")

j

As we have seen in the proof of corollary 7.21., this implies that

with the series converging uniformly on compact subsets. Then

so that, applying theorem 7.22. to each

a . we get: J'

111.7. OPERATORS IN

Hp

343

as we wanted to prove. For

p

=

1 , the inequality just proved is:

dx 5 C llfll

(7.24)

H1 (IR”) This is an extension to our context of Hardy’s inequality, established in chapter I (theorem 4.1). As a preparation for the multiplier theorem, we need the following: LEMMA 7.25.

k

Let

mn

function on

>n/2. S u p p o s e t h a t

be a n i n t e g e r

m

is a

satisfuinq:

1

(7.26) (

2

jR
such that 0 6 I B I 6 k and e v e r z R > 0 , R B T h e n t h e r e is a c o n s t a n t C or n 2 n/r > 2(IBI-k) + n , i n d e p e n d e n t of rn, s u c h hat if r = 1 t h e f o l l o w i n g i n e q u a l i t y holds:

for every m u l t i - i n d e x with A

B

.

i n d e p e n d e n t of

1 -

(7.27) (

In c a s e

I#m(x) 12rdx)2r 6 CAR-’’’

1

R”

jR
2(181-k) + n < 0, then

continuous on

IF?\

IxII”[D’m(x)I

D a rn

CA

(01.

Proof: Take a non-negative, radial, C (x e I?”: 1/2 < 1x1 < 4 ) Given

5

and equal to

R > 0 and (3 such that

f(x) = R”’rl(x/R)D’rn(x),

m

and

0

g(x)

5 =

I PI

function rl 1 5

for

S

1 , supported on

1 5 1x1 5 2.

k , let

f(Rx) = R”’I~(x)DB rn(Rx). Call

s = k - I B I . Then, if v is a multi-index such that 0 S I v I S s , V condition 7.26. implies that the function D gix) = RIV’Dvf(Rx) is

in

L2(lRn) and satisfies IIDvgl12 I CA, with C independent of and R. In other words: g has derivatives in L2 up to order

rn,v,s

s. This is usually expressed by saying that Sobolev space

g

belongs to the

2

L_(IRn). This is a norrned space with norm given by zl

lllDVg112. We appeal to Sobolev’s imbedding theorem (see Stein [l] OSlVl<S

p . 1 2 4 ) , which says:

111. REAL HARDY SPACES

344

i)

For s > n/2, the functions of L:(lRn), belong to Co(IRn), the spae of hctiomcontinuous on E? and vanishing at m , and the inclusion L;(IR") c c0(mn) is continuous.

ii) There is a continuous inclusion s > n/2 - n/q.

LE(IRn) c Lq(lRn)

provided

Since s = k - 16 I , if we write 2r = q , the condition s > n/2 is equivalent to n/r > 2(1@1 -k) + n. If this holds, then:

Observe that this is valid with the usual interpretation for

r

- n/q

=

m.

The key to the multiplier theorem will be the following result. THEOREM 7.28. Suppose

m satisfies the hypothesis of lemma 7.25. (p,2,k-l)-atomcentered at the origin with 0 n(l/p-1/2) >

> n(l/p-l).

We shall prove that ma is the Fourier transform of a (p,2,k/n)molecule centered at the origin and having molecular norm bounded by C.A. In order to do this, it will be enough (see remark 7.19. b)) to show that:

for

I v I = k ; and

111.7. OPERATORS IN

Hp

345

Let us start by proving i). It will be convenient to write d = l/(l/p-1/2) as we did already in theorem 7.20. We can write i) also in this way:

k/n > l/d, the exponent

Observe that, since we have assumed

is negative. Note also that we saw (lemma 7.25) that 2 C.A. Therefore (7.29) will follow if we are able to see

1-(kd/n) Im(x)l that:

To establish this inequality will require to prove that whenever and

B

are multi-indices satisfying

la1

+ IB(

=

a

k , we have:

The case B = 0, la I = k , is quite easy. We use the estimate b ) in theorem 7.20., with r = m , together with the fact that l l m l l m 5 C.A. We get:

Next we consider the case

=

S1 + S2, where

S1

0 <

I BI

5 k, 0 5

is the sum over

la1 < k. We write:

j 5 j,,

and

jo

is an integer

to be determined later. Using theorem 7.20. a) to estimate

DO";

and condition ( 7 . 2 6 ) on

we get:

5 cllal12 -2A(k-l,n)22j(k-lal)2nj(2-nj

5 CA

2 llall -2A(k-l,n)2nj , where

'+1lDBm(x)I 2dx) 2

A(k-1,n) = d(k/n+l/2)-1

m,

111. REAL HARDY SPACES

346

These e&irnatescan S1 Since 2nJ0

s

1

2 -2A(k-l,n) CA Ilal12

j 0 2nj

=

CA211a11;2A(k-1,n)2nj0

- m

-2A(k-l,n) %

0, o b t a i n i n g :

j 5 j

be added f o r

= 2(1-(kd/n)-(d/2)),

i f we choose

jo

such t h a t

Ilallg, we s h a l l h a v e :

which i s t h e e s t i m a t e we a r e l o o k i n g f o r . Complete s u c c e s s w i l l be a c h i e v e d i f we a r e a b l e t o g e t t h e same estimate f o r

S2

jo

with

f i x e d as above.

We s t a r t by u s i n g H s l d e r ' s i n e q u a l i t y t o g e t :

Then we u s e theorem 7 . 2 0 .

a way t h a t ( 7 . 2 7 ) If

IBI

r = 1. If

> n / 2 , we t a k e

- ( n / 2 ) , we t a k e

b ) and ( 7 . 2 7 ) , a f t e r c h o o s i n g

r

=

m.

0 <

IBI

5 n/2

I t o n l y remains t h e c a s e

5 n / 2 . O f c o u r s e t h i s c a s e o n l y a r i s e s when we choose

1 < r <

Observe t h a t lemma 7 . 2 5 . , S2 5

i n such

and a l s o

so that

B

0 < 21 BI

-

IBI

< k -

k - (n/Z) 5

5

and

0 <

k 5 n. I n t h i s s i t u a t i o n ( n / r ) < 2k - n .

h a s been s e l e c t e d i n s u c h a way t h a t and

r

s a t i s f y t h e h y p o t h e s i s of

( 7 . 2 7 ) h o l d s . We g e t :

-B

cllal12

where

2"Jo

such t h a t

i n all cases, r

2lBl - ( n / r ) > 0

r

c a n be a p p l i e d . L e t u s s e e how t h i s c a n be d o n e .

%

B ( a , r , n ) = d ( 2 1 a l / n + l / r ) - 2 . Then, t a k i n g i n t o a c c o u n t t h a t d IlalI2, we a r r i v e a t : 2 2(l-(kd/n)) S2 5 C A [la11 2

111.7.

OPERATORS I N

Hp

347

i ) i s completely proved.

and

Now w e h a v e t o p r o v e i i ) . We a l r e a d y p o i n t e d o u t ( r e m a r k 7 . 1 9 . V

t h a t i ) implies the existence of 0 6

"

D ( m a ) ( O ) f o r every

5 N. We j u s t n e e d t o see t h a t t h e s e d e r i v a t i v e s a c t u a l l y

( V (

For

vanish.

V

it suffices to realize that

= 0,

a ) , so t h a t

5 C l x l k (theorem 7.20.

I;(x)l

m

i s bounded a n d

(m;)(O)

=

lim X

= 0.

b)

satisfying

V

For higher o r d e r

m(x)a"(x) =

4

w e proceed i n a similar f a s h i o n , w r i t i n g

V's,

V

D ( m ^ a ) ( O ) = l i m t - I V 'A:(mG)(o) t+0 V

v

.. .

1

V

A t = At e l A , the difference operator ten v 1 t i m e s , ..., Aten, V n t i m e s ; t h e o b t a i n e d by a p p l y i n g A t

where, f o r

real,

tel

9

b e i n g t h e s t a n d a r d u n i t v e c t o r s of

IA:(ma)(o)I

4 Cltlk

and

IR".

Since, c l e a r l y

5 N < k , we o b t a i n

IV(

e:s J

V

"

D (ma)(O)

0 , as

=

we wanted t o p r o v e . We s h a l l s a y t h a t t h e m e a s u r a b l e f u n c t i o n p l i e r on

m

on

i s a multi-

IRn

f e HP(IRn),

Hp(IRn), 0 < p 6 1 , i f and o n l y i f f o r e v e r y

the function

m%

belonging t o

HP(IRn)

i s t h e F o u r i e r t r a n s f o r m o f some d i s t r i b u t i o n and t h e r e i s a c o n s t a n t

i n d e p e n d e n t of

C

f

such t h a t

Theorem 7 . 2 8 .

immediately y i e l d s t h e

Hp

m u l t i p l i e r theorem we were

l o o k i n g f o r . I t c a n be s t a t e d i n t h e f o l l o w i n g way: T H E O R E M 7.30.

function on

j'(

Let

k

A

1/2 lD'rn(x) I 2 d x ) Rn

i n d e p e n d e n t of C

5 AR - l e l

R
l / ( k / n + 1 / 2 ) < p 6 1, m constant

is a

1 ~ " satisfying:

f o r every multi-index

with

m

be a positive integer. Suppose

B

such that R

& 6.

0 6 161 6 k

T h e n , for e v e r y

is a multiplier on

i n d e p e n d e n t of

m

&

and every

f

HP(IRn),

such that:

p

R > 0,

such that

a n d t h e r e is a

111. REAL HARDY SPACES

348

P r o o f : Fix

such that l/(k/n+l/2) p 2 1. This is equivalent to saying that k/n > l/p - 1/2, and this implies that k-1 2 N = 1 as we pointed out in the proof of theorem 7.28. = [n(E-l)] p

Let f e HP(IRn). Write f = atom and the h i s are real Then

1 J

A. a where each J J numbers such that

a

is a

J

1 I hjIP

(p,2,k-lb CllfllP HP *

J

J

in the sense of tempered distributions. It follows from theorem 7.28. and the translation-invariance of the operator g w(mg)', that each ( r n ^ a . ) is a (p,2,k/n)-molecule with Iv((ma.)') 5 CA. Then -J J so that 1 fl(hj(maj)v)P 5 CAP 1 I hjlP 5 CAPIlflIP J J HP(IR") A

A

as we wanted to prove. We shall see next that a multiplier on HP(lRn), for some 0 < p 5 1 , has to be necessarily a bounded function. Actually, the following is true : THEOREM 7 . 3 1 .

Suppose

w i t h norm

0 < p 5 1

a22 t h e c o n s t a n t s

C

m i s a m u l t i p l i e r on HP(IRn), w h e r e A ( b y norm we me an, of c o u r s e , t h e i n f i m u m

of

w h i c h make t h e i n e q u a l i t y

5

-

Il(mf)'ll HP(IR")

v a l i d for a l l

6 CIIflI

f e Hp(IRn)).

Then

m

i s a continuous

HP(IR") f u n c t i o n on such t h a t

lRn\{Ol and t h e r e i s a c o n s t a n t C Im(x)I s C.A f o r a 2 2 x e ~R"\IOI.

P r o o f : Given

p , associate with each function t > 0 , defined as follows

ft(x)

=

f

i n d e p e n d e n t of

m,

its dilation by

t-"/Pf(x/t)

Extend this definition to distributions in the usual way, that is:

111.7. OPERATORS IN

Then, it is clear that, for every

Hp

349

t > 0, and every

f e HP(lRn):

To see this we just need to observe that if a is a (p,m)-atom, then at is also a (p,m)-atom. Indeed if Q is a minimal cube supporting a, then at is supported in tQ and

Observe that

Then, applying get: Im(x and, after choosing

t

=

.

1x1 -1.

To conclude, we just need to realize that we can find

f e HP(IRn) A

5 C (a geometric constant) and such that

IlflI

f(x) = 1 HP(1R") for every x having 1x1 = 1. Indeed, if we take F e Cm(IRn) such that F is supported on the set { x e IR": 1/4 < 1x1 < 41 and is

with

identically

on

1

{x e

lRn: 1/2 < 1x1 < 2 1, it is clear (see rema-ks F = f where f is a molecule with I(f) C. A

c) that The continuity of m follows from the continuity of the Fourier 7.19.

b

and

transform of an ry 7.21.

Hp

distribution, which was established in corolla-

0.

The fact that m is a bounded function implies, of course, that m is a multiplier on L2. There are interpolation theorems which allow us to conclude that m is a multiplier on Hq (see, for example; Calder6n and Torchinsky [ 2 ] ) on Lq for for 2 5 q <

1 < q 5 2. Then, by duality, m and on B.M.O.

m

for every p < q 5 1 and also a multiplier

is a multiplier on

Lq

111. REAL HARDY SPACES

350

Hp.

We h a v e n o t g i v e n i n t e r p o l a t i o n r e s u l t s f o r o p e r a t o r s b o u n d e d o n We s h a l l c o n t e n t o u r s e l v e s w i t h a s i m p l e a p p l i c a t i o n o f m o l e c u l e s .

T H E O R E M 7.32. L e t

0 < po < 1 < q

Lq(lRn)

bounded in

Suppose

which sends each

(po,q,b)-molecule in

lRn

T

is a l i n e a r o p e r a t o r

(po,q,k)-atom in

Eln

into a

w i t h m o l e c u l a r n o r m b o u n d e d by a c o n s t a n t

T

i n d e p e n d e n t of t h e a t o m . T h e n

HP(Rn)

is b o u n d e d o n

for every

Po 1 / p 0 k 2 [n((l/po)-l] 2 [n((l/p)-l] t h a t t h e p a r a m e t e r s are s t i l l a d m i s s i b l e .

-

l/q

i

l / P - l / q , so

L e t a be a ( p , q , k ) - a t o m . L e t Q b e a minimal cube s u p p o r t i n g a . a is a (po,q,k)-atom, so t h a t Then b = I Q I l / p - l / p O Tb = Q I l/p-l/po T a w i l l b e a ( p o , q , b ) - m o l e c u l e c e n t e r e d a t a

certa n

where

x

eo

e R".

0

We s h a l l h a v e :

(l/po-l/q)/b.

=

We s h a l l a l s o w r i t e

8

=

(l/p-l/q)/b

and

d = I/(l/P-l/q).

We know t h a t

i s bounded i n

since

llallq 5

IQll/q-l/p

Lq(lRn),

lQl-l'd.

Using t h a t f a c t t h a t

T

l / p o - l / p > 0 . Now, c a r r y i n g t h i s i n e q u a l i t y i n t o o u r

previous inequality f o r

Ta, we get:

Observe t h a t =

=

we can w r i t e :

(eo/e) -

d(l/po-l/p) = ((l/po-l/q)-(l/P-l/q) )/(l/P-I/q) = 1 , s o t h a t 1-80+d(l/p0-l/p) = ( 1 - 8 ) 8,/8. Then our l a s t

i n e q u a l i t y c a n b e w r i t t e n as:

III.7. OPERATORS IN Hp

and this implies that

Ta

is a

351

(p,q,b)-molecule as we wanted to

show. We shall shift our attention from translation-invariant operators to dilation invariant operators. We shall start working on the real line. As a point of departure, we shall consider two basic inequalities due to Hardy

lo

.x

where

F(x) =

f(y)dy and 1 < p <

m

This second inequality was contained in corollary 7.23. We want to point out that these two well-known inequalities involve operatolsof the same type.

1 with K(x,y) = x+(x-y), denoting by tion of the half-line l R : @ , m)

x,

the characteristic func-

The left hand side of (7.34) is

with

K(x,y) =

1 X

e-

2~iylx

Both kernels satisfy the condition (7.35)

K(Xx,Xy)

=

(A ' 0 )

A-'K(x,y).

This condition guarantegthat the operator

T

given by

352

111. REAL HARDY SPACES

commutes with dilations (it sends the function

It is well known .that if the kernel

K

f(X.)

into

Tfo..)).

satisfies (7.35) and

m

IK(l,y)l IyI-l’pdy = C <

(7.37)

m

-03

where 1 I p < m , then the operator from L p ( 1R) to Lp( 1R+) Indeed:

.

T

given by

(7.36) is bounded

so that, Minkowski’s integral inequality yields:

(7.33) can be obtained in this way since

Obviously (7.33) breaks down for to

f e HI

L E M M A 7.38. [a,b]

p = 1. However, it can be extended

as we are about to see Let

(l,m)-atom h a v i n g i n t e r v a l - s u p p o r t

f

c 1R+. T h e n : m

5 log 2 dx 5 (log 2) b-a b+a

Proof: Since f has integral zero, F a 5 x S b , we have:

lives also on

[a,b].

For

Thus : 1

IF(x)l I b-a min (x-a,b-x) and, consequently, setting

c = (a+b)/2

and 0 =

b-a b+a’

we get:

111.7. OPERATORS IN

Hp

353

as we wanted to prove. Let us point out that, for the two inequalities contained in the lemma, the constant log 2 is sharp. Indeed, for for 0 6 x 6 1 1 5 x 5 2 for otherwise

1/2 -1/2 0 it will be

x/2 for 0 s x 6 1 1-(x/2) for 1 6 x 5 2 0 otherwise

so that

If we want to pay attention to the constants appearing in the inequalities, we should specify which, of the many different equivalent norms,are we using on H1 Sometimes we shall work with the atomic norm

.

IlflI

=

Hat

inf{

1 I xjl:

f

j

=

1 XJ. aJ. ,with j

a. J (1,m)-atoms and

lXjl < -1. J

while other times we shall take the norm

L E M M A 7.39. S u p p o s e f e H1 a n d let fe b e i t s e v e n p a r t , t h a t i s : fe(x) = (f(x)+f(-x))/2. T h e n b o t h fe and X+fe b e l o n g t o H1.

111. REAL HARDY SPACES

354

A c t u a 11 y

f(x) =

P r o o f : If m

and

1I hjI

<

m,

1

A . a . ( x ) w i t h each 3

j=l

a.

being a

J

J

(l,m)-atorn

w e have:

j=1 m

then both a and a , ( - . ) a r e (I,-)j J l i v e s i n (-m,o]. If we a r e n o t i n atoms. The same happens i f aj e i t h e r of t h e s e c a s e s , t h a t i s , i f 0 i s i n t e r i o r t o t h e i n t e r v a l If

a

j

lives in

m

[O,m),

s u p p o r t o f a * t h e n A ( a . ( x ) + a . ( - x ) ) i s a ( l , m ) - a t o m . We r e a c h j' 4 J 1 J the conclusion t h a t f e e H and \Ifell 2 211fll Hat

Hat

Now, l e t us look at m

If

a. J

lives in

m

a j = ajy,+

[O,m),

( l , m ) - a t o m and

is a

ajx+ = 0 and is interior t o the 1 -(a.(x)+a.(-x))x+(x) is a (1,m)-

then a , ( - x ) x ( x ) = 0 . If a . l i v e s i n (-m,O], J J a . ( - x ) x + ( x ) i s a (1,m)-atom. F i n a l l y , when 0 J

i n t e r v a l - s u p p o r t of atom. Thus

a . then e H1 J ' a n d

feX+

2

J

3

llfe&ll

5

Hat

Hat LEMMA 7.40. For a f u n c t i o n

f

s u p p o r t e d on

1R+,

the following

properties are equivalent: i)

f e HI

ii)

f(x) =

m

[o,m)

1

j=1

A a . ( x ) , where each

&

Jm

a.

J

1

I "I

<

J -

is a

( I , m ) - a t o m living in

m.

j=l i i i ) The e v e n extension

f

g

belongs t o

H1

.

m

Besides,

I l f l I 1,

IIgIIH1

inf

.I

J=1

}

, whei-e

t h e inf is

t a k e n o v e r ol,? d e c o m p o s i t i o n s a l l o u e d in i i ) : a r e e a u i v a l e n t norms. m

P r o o f : Assume i). L e t

tion for

f . Then, a l s o

1

A. a . ( x ) be an atomic decomposij=1 J J m f ( x ) = f ( x ) X + ( x ) = 1 x. a . ( x ) x + ( x ) .

f(x) =

j=1

J

J

111.7. OPERATORS IN

Hp

355

NOW each a,i(x)x+(x) is an atom living in [O,m) obtained iij with an infimum bounded by IIf 1 1

.

and we have

Hat Assuming ii), we can write m

g(x)

=

m

1 A. j=1

a.(x) +

J

Both

a.(x) and J

a.(-x) J

J

1

j=l

A. a J

j

(-XI.

are atoms, so that

S 211fll and iii) holds. Hat Hat Finally if we assume iii), since g = gp

g e H1

with

llgll

Here is our extension of 7.33. to

THEOREM 7.41. L e t

f e H1

and

f = gxI, lemma 7.39.

H1:

b e s u p p o r t e d in

[O,m).

Then:

dx 2 (log 2 ) Proof: Just write each atom.

Given

1

f

Hat as in lemma 7.40. ii) and apply lema 7.38. to

f e H , consider for

x > 0

Applying theorem 7.41. to the function X+fe, we obtain:

1 ; that is: with

k0(u)

=

X+(l-(u() = X [-1,l-J ( u )*

111. REAL HARDY SPACES

356

DEFINITION 7 . 4 3 . K(x,y), (x > 0, y e R ) , w i l l be said to be a Hardy kernel for H1 if and only if K satisfies condition 7.35 and to also the operator T given b y 7.36. is bounded f r o m H1(lR) 1

L (R+). One of o u r basic examples of a Hardy kernel will be KO(x,y) = 1 = X+(x-ly(). It is indeed a Hardy kernel, as the estimate 7.42. shows. We shall denote by To the corresponding operator bounded from H'(IR) to L ' ( R + ) .

x

Using To and integration by parts we shall find a sufficient 1 condition for K(x,y) to be a Hardy kernel for H

.

THEOREM 7.44. Suppose K(x,y), (x > 0, y e lR), satisfies 7.35and also assume that k(y) = K(1,y) is an even function of bounded variation with total variation s B . Then K is a Hardy kernel for H1 and the operator T defined b y 7.36. satisfies: m

j

s (log

ITf(x)ldx

2 ) Bllfll

0

Hat

Proof:

except for a denumerable set of points. Let

f

(1,m)-atom. hen :

be a

11; 14, m

Tf(x)

=

k(u)f

m

=

-

f(xu

the last identity being a consequence of the dilation-invariance of TO'

Observe that

6

ITf(x)Idx

tf(t') S

liji

is also

a(l,m)-atom. Therefore:

IT,(tf(t.))(X)ldxldk(t)l

2

111. 7. OPERATORS IN

5 2

iom

(log 2 )

result follows, a atoms.

TI-

Idk(t)l

5

Hp

357

(log 2 ) B.

usu 1, after decomposing any

f e H1

into

0

Let

k

be the even extension of a function monotone and bounded on

IR+. If we take the atom

f(x) =

I

if 0 5 x if 1 2 x otherwise

1/2 -1/2 0

5

1

2

2

then all the inequalities in the proof of theorem 7.44. are, actually, equalities. This shows that the constant in the theorem is the best possible. It also allows us to give the following necessary condition

TH&OR&R 7 . 4 5 . bide or 0.

Let

k(y)

a Handy k e n n e l [ o n

H1

IR,

be a n e v e n c u n c t i u n u n

Then, t h e connebpunding k e n n e l

i[

and only

ic

k

K(x,y) id

=

monutone u n each

(l/x)k(y/x)

9

Dounded.

To simplify statements as the one just given, we shall say sometimes that k(y) is a Hardy kernel for H1 to indicate, of course, that 1 K(x,y) = (l/x)k(y/x) is a Hardy kernel for H

.

To give a simple application of theorem 7 . 4 4 . ,

consider the operator:

We claim that:

Indeed

I

m

and

(l/x)Lf(l/x)

=

(l/x)e-IY1/Xf(y) dy

-m

has a kernel K(x,y) = (l/x)e-’yI/x, so that which has total variation B = 2.

k(u)

=

K(1,u)

=

e-IuI ,

111. REAL HARDY SPACES

358

Instead of using the atomic characterization of H1, we can use the Hilbert transform H. We obtain the following result: TH&OR&M 7 . 4 7 . L e t ad i n

7.33.:

cuheae

C

Pmof:

Let

p 2

1

&

LA independent of

f

p

f

e

Lp(lR+).

&

Then,

uith

F

defined

f.

be a linear combination of atoms living in

[O,m).

Then :

I-, m

=

X[-1,1]

(Y) Hfe(xy)dy.

A simple computation yields:

Then, since both Hilbert transforms are odd, we can write:

with

k(y)

=

log

l=i

Since .m

we immediately obtain:

For

p > 1 , this inequality can be combined with 7.33. so that we

111.7. OPERATORS IN

359

Hp

have :

Then, invoking the simple inequality a c min (-b'd-)

5

a+c b+d,

a, b, c , d > 0

we get:

Since the linear combinations of atoms are dense in

Lp(IR+), we

obtain the result. p = 1, we can give another

By examining more closely the case version of

7.42. We write, as before:

-X

Hfe(xy)dy. Then we observe that:

dy (see Selby

[l]

p . 467). and

='TI

2/2

consequently dx

In the proof obove we can take

5 71

f

IIHflI

to be an atom or an element of

any other dense class for which it is legitimate to write:

If we use, for example, the dense class appearing in theorem 1.8., we obtain a proof of (7.48.) independent of the atomic decomposition. Theorem 7.44. was inspired by inequality (7.33.). Now we shall

360

111. REAL HARDY SPACES

present another basic sufficient condition, which is inspired by inequality (7.34). TH&OY?&A 7 . 4 9 .

T

Let

be an o p e n a t o n buunded Lnom

w i t h u p e n a t o n nonm bounded by

A.

d i l a t i u n o a n d , a t Lea42 t u n atorno,

k(y) = K(1,y)

Then

T can L1(lR+) w i t h

nati4Lyinp

Q

T

Suppone t h a t

2

L2(lR)

cornmuten

L2(R+)

with

i o g i v e n bg (7.36)

Lipnchita condition

be e x t e n d e d t o an o e n a t o n bounded faom

, C

o p e n a t o n nonm -nuob

1

Hat(lR)

& J

being a n

aboolute conotant.

Let f be an atom with interval-support I = [a,b]. Then 2 llflI2 5 lI1-'j2, so that, using the L -boundedness of T , we can obtain, for any R > 0: $noof:

I

t

To get an estimate for x > R , we use the function F(t) and integrating by parts, write:

=

f(s)ds,

-m

This leads to

J a

J

-.m

But

Ij

t

F(t)I

=

-m

Setting

=

f(y)dyl

c = (a+b)/2, we get:

-m

so that:

I/ t m

f(y)dyl

Ja

J C

s min(t-a,b-t)/lII.

111.7. OPERATORS IN

36 1

Hp

and, consequently:

Putting together both estimates, we arrive at: IITflll

S

-

min (A(R/tIl)1’2 + (B/4)(R/IIl)-’) R>O

CY r1l2 + f3 r-l attains By elementary calculus, the function r its minimum over r > 0 for r3l2 = 2@/cr. In our case, the best estimate will correspond to R/III = (B/(Z?A))~/~. With this choice for R , we get: (ITflIl S CA2/3B1/3

Splittinga general

f

e H1

into atoms, we obtain

IITflIl 5, CA2/3B1/311fllH1 at as we wanted to show. The best example of application of this second criterion is to theoperator 1 1 Tf(x) = - f ( - ) 1 X X

It is clearly bounded from

x > o

,

L 2 (IR) to

L2 (IR+)

since:

We a l s o have:

with

K~(x,Y)=

This kernel K1 dilations. Also

f.

e

-2ai(y/x)

obviously satisfies (7.35), sothat T1 commutes with kl(y) = Kl(l,y) = e-2aiy is clearly Lipschitz with

t

IlkllIm = 2 n . Then, theorem 7.49. implies that

K1

is a Hardy kernel for

H1

.

We can give another necessary and sufficient condition, this time f o r odd functions.

362

111. REAL HARDY SPACES

TH&OW&M 7 . 5 0 . L e t con4.tan.t

4ign

.to

k(y)

b e an a d d f u n c t i o n i o n m e a 4 u n e l w i t h

each 4 i d e a f

0. @

k

i 4

a Handy k e n n e l f o n

H1

i f and only i f :

$nuof:

O f c o u r s e , t h e c o n d i t i o n is s u f f i c i e n t , because i t is j u s t

( 7 . 3 7 ) w i t h p = 1 and we know t h a t t h i s i m p l i e s t h a t t h e o p e r a t o r i s bounded from L 1 ( l R ) t o L1(IR+). To s e e t h a t i t i s a l s o n e c e s s a r y , c o n s i d e r t h e atom:

f(x) =

i

-1/2

if

-1 d x < 0

1/2

if

0 < x 5 1

rl

so t h a t only if

]Tf(x)( = 2

1," 1;"

otherwise

0

J

/x '

( k ( y ) ( d y , which b e l o n g s t o

1;

0

I k ( y ) Idy dx =

Ik(y) I

$

H1

(7.37

with

i f and

<m.

Apart from t h e t h r e e b a s i c s u f f i c i e n t c o n d i t i o n s f o r Hardy k e r n e l f o r

L1

k

t o be a

p = 1 , 7 . 4 4 . and 7 . 4 9 1 , we c a n

d e r i v e o t h e r s by u s i n g mappings known t o p r e s e r v e

H1.

We s h a l l g i v e

two examples b a s e d upon de H i l b e r t t r a n s f o r m and t h e F o u r i e r transform. The i d e n t i t y

allows us t o w r i t e t h a t :

TH&OW&M 7 . 5 7 . T h e H i l h e n t k,

i 4

tnanofonm

a l 4 o a Handg k e n n e l f u n

1 H

Hk

of

. C1

a Handy k e n n e l

fon

HI,

T h i s s t a t e m e n t i s n o t v e r y p r e c i s e b e c a u s e i t d o e s n o t s a y what do

w e mean by

Hk

f o r a g e n e r a l Hardy k e r n e l

k . T h i s d i f f i c u l t y can

be overcome by u s i n g t h e t h e o r y o f d i s t r i b u t i o n s . We s h a l l l i m i t o u r s e l v e s t o u s i n g theorem 7.51.

just f o r nice

k's.

111.7. OPERATORS IN

For example, let Hardy kernel for

k(y) = y X co H 1 , since

9

11(y).

Hp

Of course this

363

k(y)

is a

will also be a Hardy kernel for H1. However it-does not satisfy any of the three basic conditions. First of all IHk(y)l dY = 0 1 . m Actually the integral is m even if we add a constant to Hk since Hk(y) + 0 as y + m whereas Hk(y) + - l / v as y + 0. The two other conditions fail since Hk(y) + m as y + 1.

n

Next we shall use the Fourier transform, for which we have the formula:

This formula cannot ve applied to every Hardy kernel g because T1 1 does not preserve H Still, for g integrable or, more generally, for a finite measure u instead of g, we can take advantage of the fact that T1 sends H1 into L1. We get the following result:

.

TH&OR&M 7 . 5 2 .

The F o u n i e n t n a n 4 f u n m

Handy k e n n e l r u n

H

1

.

o r a L i n i t e mea~ufleu

, A4 a

and consequently:

For example, if k is an even function of bounded variation, then k' is a finite measure and, consequently (k#)-(y) = hyC(y) is a 1 Hardy kernel for H Thus, for such k , both k(y) and y$(y) are 1 Hardy kernels for H .

.

111. REAL HARDY S P A C E S

364

We shall end the section by presenting some extensions of the FejerRiesz inequality which was studied already, for the torus, in chaptc

I (theorem 4.5.). First we deal with the one-dimensional case and later we shall move on to higher dimensions.

where

K(t,y) = Pt(y) =

1 t 7T

Now

K

y2+t2

obviously satisfies (7.35) and

k(y) = K(1,y) =

1 1 ; 7

is

an even function of bounded variation with total ~ariation~+~2/11.

It follows from theorem 7.44. that

The more accurate estimate given by the first inequality in lemma 7.38. can be used to show that:

TN&OW&M 7 . 5 4 . W i t h t h e dame h g p o t h e 4 i A and n o t a t i o n

ole

theomm 7.53.,

lue h a v e

1,

lu(x,t)Idt

0

as

1x1

-+

m

1 < m and each a is an atom j j with interval-support I., centered at c Then, proceding as in the J j' p r o o f of theorem 7.44., we get, for a fixed x: ?noo[:

Let

f =

1 A jaj'

*

where

j

where we have used the notation:

111.7. OPERATORS IN

Suppose

365

Hp

x 4 Ij. Then

The atom y + aj(x-y) is supported in x - I and the atom j y ++ aj(x+y) is supported in I.-x. These two intervals x-I. J J and I.-x are symmetric and, because x d I . they are disjoint. J J' For s > 0, only one of the atoms a,(x-y) or a,(x+y) actually J J occurs in the expression for To(fj ( s ) . By using the first inequality in lemma 7 . 3 8 . , we get: lIjl -

CIIjl

5

lx-c . I J

(x-c. I + [ I .I J J

and, consequently: m

If

ImI

m

*a.(x)(dt IoIpt J x e I

j'

5

0

IsTo(fj)(st)ldt

dk(s)( 6

0

CII-I ' J '

Ix-c . I + I I .I J J

we use theorem 7.53 to write

lilP,*aj(x)Idt

Finally, since

u(x,t)

6

=

1 4

C

2

CIIjl

Ix-c.l+lI.I J J h.P +a (x), we get: J t J

In order to give an n-dimensional version of the Fejer-Riesz inequality, we shall use the following

id a -

Pnoo{:

(1,m)-atom i n

1R

time4 a geometaic condtant.

The cancellation is immediate since

111. REAL HARDY SPACES

366

1;

Im

a(r)dr =

-m

Suppose

rn-l

11,-1

i s s u p p o r t e d i n a b a l l of d i a m e t e r

a

-

x . Let

First, if

ro

I f , on t h e c o n t r a r y , we have t h e atoms

x ++

N 2 C(ro/6)n-1

Z(r)

la(x) 1 IC6-"

6 < ro, we have t o be more c l e v e r .

..., O N , s u c h t h a t t h e s u p p o r t s of ..., N , do n o t o v e r l a p , and

rotations

N

and

6 , we immediately g e t

5

There a r e

6

be s u p p o r t e d i n [ro,ro+6] w i t h ro >, 0 . To a we s h a l l d i s t i n g u i s h two c a s e s .

a estimate the s i z e of f o r every

a ( x ) d x = 0.

a(ru)dadr =

a(Bjx) with

C

-

=

for

j = 1,

a g e o m e t r i c c o n s t a n t . Then: N

i

1

a(O . r u ) d a J

j=1

from which we o b t a i n t h e e s t i m a t e :

n COROLLARY 7.56.

belongd

with

C

t o

Foa

f e H1(lRn),

and:

H1(lR)

independen2 o f

P a u u f : Decompose

f

the functiun

II 7 II

I

H1(lR)

c IlflI

7

defined

1R

&:

H1(IR")

f.

i n t o atoms and u s e theorem 7 . 5 5 .

Now we c a n g i v e t h e promised e x t e n s i o n o f t h e F e j e r - R i e s z to

on

inequality

lR".

Tff&OR&fl 7 . 5 7 . T o n

f 6 H1(IRn), (Pt

u ( x , t ) = Ptxf(x)

6e the

$UC-JAO~

111.7. OPERATORS IN

uhe/re

C

due4 n u t depend un

.,1

u(x,t)tn-l = cn

1

Hp

367

f.

tn

n+l ( t2+ IY 2, 2

f(x-y) dy

2

I

m

=

K(t,r)g(r)dr

0

where

K(t,r) = cn

t n+1 2 2

satisfies (7.35) with

k(r)

=

K(l,r)=

2 (t +r )

obviously even and of bounded variation, so that n+1 2 2 (l+r 1 theorem 7.44. can be applied; and g(y) = f(x-y) has II g I t = IIfll , so that, the function g associated to H1 (IR") H1(IR") g as in corollary 7.56, satisfies IlglI 5 c IlflI ThereH1OR") H (m) fore, the theorem follows. = cn

Exactly as in the one-dimensional case we can prove: TH&OW&M 7 . 5 8 .

U i t h t h e 4 a m e h y p o t h e 4 i n and

n o t a t i o n of

theufiem

7.57:

111. REAL HARDY SPACES

368

8.

NOTES AND FURTHER RESULTS

In chapter I we viewed the spaces Hp as classes of functions analytic in the unit disk D. We saw that these spaces are useful tools in studying many questions of Fourier Analysis on T, the boundary of D. However, our methods were based on Complex Analysis, so that we always had to be changing from T to D and then back to T. This was the usual way to work in Fourier Analysis during the first half of our century. Now in chapter I11 we build upon the basic real variable techniques discussed in chapter 11, and completely liberate the Hp theory from its dependence on Complex methods. The history of this liberation movement of the Hp spaces culminates in the decisive work of C. Fefferman and E.M. Stein 223. Here we list chronologically the main steps in this process: 8.1.-

1) E.M. Stein and G. Weiss cl] introduce the notion of conjugate system of harmonic functions as a natural generalization of the concept of holomorphic function. As we see at the end of section 4, it is simply a solution of a system of partial differential equations which generalizes to higher dimensions the Cauchy-Riemann system. They define HP(Rn:') for p > (n-l)/n as a space whose elements are conjugate systems of harmonic functions and extend to this context the basic theory of F. Riesz on boundary behaviour. We discuss these results in section 1 for n = 1 and in section 4 for the general case. Their main tool is that of harmonic majorization, possible after they proved lemma 4.14. It was observed that some singular integral operators can be extended from Lp, 1 < p < m to Hp, p 2 1 (see E.M. Stein [2], chap. VII for the early proof of one of these results). The proofs were rather unnatural because they depended on inequalities for some auxiliary functions, for example, on the Hp inequalities for the area function obtained by A.P. Calder6n c2] and C. Segovia .]l[ Section 7 contains a systematic study of these boundedness properties based upon the atomic description of the Hp spaces. 2)

A major breakthrough was the theorem of Burkholder, Gundy and Silverstein [l] (theorem 3.101, proved originally by probabilistic methods (Brownian motion). This theorem opens the way for defining Hp(lRn:') directly as a space of harmonic functions without appealing to any notion of conjugacy. This is what we do in section 4.

3)

1 1 1 . 8 . NOTES AND FURTHER RESULTS

369

4) The next significant advance was the identification of the dual as the space B.M.O. This is a space of of H1 by C. Fefferman [I] This identification confirmed that functions on the boundary, IR". it was appropriate to look at H1 as a space of real functions in IRn and also provided a direct method to establish the boundedness of more singular integrals in H 1

.

5) C. Fefferman and E.M. Stein [2J define Hp(WnI1) as a space of harmonic functions u(x,t) in IRnI1 such that the vector F(x,t), whose components are u and a number of conjugates depending on p, is uniformly in L'(IR"), i.e.:

The definition is the same as that of E.M. Stein and G. Weiss [l], except that, after Calder6n and Zygmund [3], they are able to allow any p > 0 by just taking cmonjugate systems of increasing complexity (higher order gradients). Starting with this definition, they show that the following properties are equivalent for a tempered distribution

in IRn : f = lim u(-,t) in the sense of tempered distributions, for t*O u e HP

f

a) some

b) c) IRn$(x)dx d)

SUP f * $,(x)l t>o SUP f * $,(x)I t>O

is in

Lp

for each

is in

Lp

for just one

$ e S $ e

having

f 0. The non-tangential maximal function

Pi(f)(x) = sup lu(y,t)I lY-xl
is in

Lp where

u(y,t) = Pt + f(y)

.

The equivalence between a) and d) is an extension to several variables of the theorem of Burkholder, Gundy and Silverstein. The equivalence of a) with b) or c) shows that the Poisson kernel does not play a significant role and may be replaced by any reasonable approximate identity. This clarifies the position of Hp as a space of real objects in

IR",

and helps to explain the permanence of

Hp

under many natural operators. 8.2.-

The first three sectiorsof the chapter are devoted to the

study of Hp( WE). These spaces were first studied by Krylov [1J by using conformal mapping from the disk. Our method is quite dif-

370

111. REAL HARDY SPACES

ferent. It uses the ideas of E.M. Stein and G. Weiss ]l[ and of plus a third ingredient, which will C. Fefferman and E.M. Stein [2J turn out to be the basic one f o r the whole chapter: atomic decompositions. The atomic decomposition for H1 follows immediately from the duality theorem by using the Hahn-Banach theorem. Indeed, one sees that B.M.O. is the dual of the atomic space, just from the was able to obtain the atomic decompodefinition. R. Coifman [l] sition constructively f o r ReHP(IR), 0 < p 5 1. In particular this gives a different proof of the duality theorem. We adopt Coifman's method, based upon a smooth version of the Calderbn-Zygmund decomposition (theorem 3.6), which was already used in Fefferman-RiviereSagher [l] for the purpose of interpolating. The atomic decomposition of HP(IRn) for n > 1 was achieved by The technical details of Latter's proof were simplified Latter (11. by Uchiyama (see Latter and Uchiyama [ I J) . We have chosen a completely different approach to the atomic decomposition in higher dimensions in order to avoid the technical differences between the cases n = 1 and n > 1 , and also because this approach, based upon a reproducing formula of A . P . Calderbn [ 8 ] , applies to many different situations (see J.M. \:ikon [ll). 8.3.-

It is important to note that in the definition of the atomic

"norm9' N

(immediately after the proof of theorem 3.7), it is P9r not legitimate to take the infimum only over the finite linear combinations of atoms even if we know that f admits such a finite decomposition.

The "norm" obtained in this way may be much larger,

as the following example, due to Y.Meyer, shows (see Meyer , Taibleson and Weiss

Let

U

[l]):

be an open dense subset of the interval

(0,l)

with

Lebesgue measure IUI < E . Write U as a disjoint union of open To each I . associate a (1,m)-atom a . supintervals U = UI 3' J j J' ported on I which equals l/IIjl on a half of I . and -l/lIj\ j'

m

f = j=1 . C ( I. ( a Clearly f e J H 1 and J j' Nl,m(f) 5 jJ1lI.I = ( U ( < E . But f is also an atom supported on J (0,1), since If(x)l = Xu(x) I 1. Now suppose that f = 9 b k=l k k where each bk is a (l,m)-atom supported on the interval on the other half, m

Let

J k c (0,l). Then, if we call

sk(x) = (l/lJkl)XJ

(x), k

we have

371

111.6. NOTES AND FURTHER RESULTS

The function

g

is continuous in all but finitely many points of

Since g 2 x u and U is dense in (O,l), it follows that g(x) 2 1 for all but finitely many x e (0,l). Consequently (0,l).

1 s

1 g(x)dx 0

We have that

n

=

1

.

lXkl

k=1

llfllHl S CE

while the inf of

CIAk[

over all finite

atomic decompositions is necessarily 2 1. Since E is as small as we please, this shows that the inf over the finite sums is not a 1 norm equivalent to the H -norm. The same can be done for p < 1. Observe that what makes this example work is the difference between the Lebesgue and the Riemann integrals. Of course, the finite linear combinatiomof atoms are dense in Hp; but, as we have seen, in order to compute their HP-l1normtt, it is not enough to look only at the finite decompositions. 8.4.- L. Carleson H1 and B.M.O. in of C. Fefferman and obtained by proving Let

[5J

IR",

which differs both from the original proof

from the atomic proof. Carleson's proof is the following decomposition theorem for B.M.O.:

h e S(IRn), e v e n a n d s u c h t h a t

a n d has compact support. depending on

and

has given a proof of the duality between

n,

tj(y) > 0

Ih

f 0.

a s e q u e n c e of f u n c t i o n s

b.(x), J

onty

B,

such that

such that

j=1

(x-y)b.(y)dy + bo(x) J

Conversely, any function o f t k i s type belongs to

11@11*

e B.M.O.(IRn)

Suppose

Then, there exists a number

t

Constant B.M.O.

and

S Cj~olIbjlIrn.

After this decomposition, the characterization of H1(IRn) b) in 8.1., leads immediately to the inequality:

and this proves the duality.

given by

312

111. REAL HARDY SPACES

Uchiyama [3J h a s observed t h a t , i n t h e above decomposition theorem, one can t a k e b j = 0 f o r e v e r y j 2 2 .

A.

Carleson' s paper

a l s o c o n t a i n s another important observation, to H1(IR)

(51

which h a s t o do w i t h an a l t e r n a t i v e g e n e r a l i z a t i o n o f s e v e r a l v a r i a b l e s v i a homomorphic f u n c t i o n s . S

of

C o n s i d e r t h e subspace

g e n e r a t e d by t h o s e f u n c t i o n s

L1(IRn)

l y t i c e x t e n s i o n t o some p o l y - h a l f - p l a n e .

f(x)

which admit ana-

The p r o t o t y p e of such a

f u n c t i o n i s one whose F o u r i e r t r a n s f o r m v a n i s h e s u n l e s s a l l c o o r d i n a t e s are

5 0.

The space

s

Then, t h e f o l l o w i n g h o l d s : coincides w i t h

To prove t h i s , we decompose

H'(IR~)

open convex cones w i t h v e r t e x at t h e o r i g i n such t h a t t h e union i s IRn\ {O}, and no n+l r j c o n t a i n s a s t r a i g h t l i n e . Then t a k e a p a r t i t i o n o f u n i t y = 1 homogeneous o f d e g r e e 0 by means of f u n c t i o n s , $ 4 e C m ( I R n \ {O)), IRn

in

. n

t

1

jcl@j

J

and such t h a t each 9 . J have, i f f e H1: f = f

+

1

f2

+

... +

I t f o l l o w s from theorem

frltl'

7.30

where

t h a t each

F o u r i e r transform i s supported i n to

ri

.

t h e cone c u a l t o

f . =

J

(4.F)" J

rj '

We

.

f . i s a l s o i n H I and i t s 3 The r e s u l t c l e a r l y e x t e n d s

p < 1.

For t h e t h e o r y of

Hp

s p a c e s d e f i n e d i n t e r m s of holomorphic func-

t i o n s o f s e v e r a l v a r i a b l e s s e e Chapter 8.5-

r;,

has support i n

A s we mentioned i n

Silverstein

[1J

8.1.,

I11

of S t e i n and Weiss

[2].

t h e theorem of Burkholder, Gundy and

w a s proved by p r o b a b i l i s t i c methods and w a s e x t e n S t e i n (21 t o n > 1. The d i r e c t

ded by C . Fefferman and E . M .

proof we g i v e i n 3.13 f o r n = 1 comes from P . Koosis [2]. There i s an e x t e n s i o n of t h i s proof t o h i g h e r dimensions a l s o due t o P . Koosis

[3],

P r o b a b l y , t h e b e s t way t o u n d e r s t a n d t h i s theorem i s through t h e a r e a function

A(u)

introduced i n

Burkholder and Gundy

[l]

7.14.

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

111.8. NOTES AND FURTHER RESULTS

-Let

u(x,t)

b e harmonic i n

lIRn(A(u)(x))Pdx

(*)

With C

and l e t

0 < p <

a.

Then

jEn (mU(x))'dx

u. If t h e l e f t hand s i d e of ("/ is lim u(x,t) e x i s t s and is f i n i t e and c o n s t a n t , f o r t+u is n o r m a l i z e d so t h a t t h i s l i m i t is 0 , t h e n t h e

i n d e p e n d e n t of

f i n i t e , then

.

x e IRn

5 C

IRnIfl

373

3

c o n v e r s e i n e q u a l i t y ho I d s :

(**I with

jnn(mu(x))Pdx C

5 C lEn(A(u) (x)jPdx

u.

i n d e p e n d e n t of

Actually their result is more general, because they allow a class of Lp "norms9'as a particular case.

'harms" which includes the

This theorem can be used to prove inequalities of the form

where v, normalized to vanish at m , is a conjugate of u. This leads immediately to the Burkholder-Gundy-Silverstein theorem. ) l [ also yields a chaThe above theorem of Burkholder and Gundy , racterization of Hp<7Rn:1) in terms of the area function, namely: The h a r m o n i c f u n c t i o n

u(x,t)

+

o

t

+ m

u(x,t) b e l o n g s t o HP(lRn++') and A(U) e LP(IR").

if and o n l y if

This characterization is considered by C. Fefferman and E.M. Stein c2] and, prior to them, by A.P. Calder6n [23, C. Segovia L1-J and G. Gasper (13. It was this characterization that allowed A.P. Calder6n to study his commutator operator, the first step in understanding the Cauchy operator associated to a Lipschitz curve (see note 7.12 in Chapter 11). The area function radial analogue)

A(u)

can be replaced by the

g

function (its

374

111. REAL HARDY SPACES

(21.

See F e f f e r m a n - S t e i n

The way t h a t B u r k h o l d e r and Gundy

p r o v e t h e i r theorem i s by

[1J

mu

comparing t h e d i s t r i b u t i o n f u n c t i o n s o f

a so-called

ttgood-Xtt i n e q u a l i t y .

and

They o b t a i n

I t i s e s s e n t i a l l y a I'good-X"

a p p e a r e d a l r e a d y i n t h i s book.

i n e q u a l i t y t h a t a l l o w e d u s t o p r o v e theorem The method o f

A(u).

This kind of i n e q u a l i t i e s has i n Chapter

3.6

11.

ttgood-Xtt i n e q u a l i t i e s a p p e a r e d f o r t h e f i r s t t i m e i n

B u r k h o l d e r and Gundy

a l s o i n c o n n e c t i o n w i t h p r o b a b i l i t y , more

(23,

concretely with t h e theory of martingales.

To u n d e r s t a n d t h e r e l a -

t i o n between m a r t i n g a l e s and harmonic f u n c t i o n s , t h e r e a d e r i s a d v i s e d t o look a t Burkholder "good-A"

For o t h e r a p p l i c a t i o n s o f

[4J.

i n e q u a l i t i e s s e e Coifman

and Hunt and Young

Coifman and F e f f e r m a n

[3],

"good- A" i n e q u a l i t i e s , B u r k h o l d e r and Gundy

From t h e i r

]l[

ClJ.

obtain

[l]

a l s o v e r y s i m p l e p r o o f s o f t h e t h e o r e m s o f C a l d e r b n and S t e i n on t h e l o c a l b e h a v i o u r a t t h e boundary of harmonic f u n c t i o n s ( s e e n o t e i n Chapter 8.6.-

I n h i s e x c e l l e n t e x p o s i t o r y l e c t u r e on

made t h e c o n j e c t u r e t h a t any n o n - d e g e n e r a t e singular

integrals in

...,K N

C.

Hp,

Fefferman

[5)

s y s t e m o f v e r y smooth

would c h a r a c t e r i z e

Eln

p r e c i s e terms, h i s conjecture w a s t h i s : K1,K2,

7.14

11).

Let

H1( R n ) .

I n more

f e L1(IRn)

and l e t

be a f i n i t e c o l l e c t i o n o f s i n g u l a r i n t e g r a l k e r n e l s m

.

K.(x) = R.(x')lxl-" where R . h a s mean v a l u e 0 and i s C SupJ J J pose t h a t t h e s y s t e m i s n o n - d e g e n e r a t e - f o r example we c o u l d assume t h a t it is e l l i p t i c , so t h a t t h e

{O } -

7R"\

f

is i n

.

H~(IR").

J . Garcia-Cuerva

J

h a v e no common z e r o e s on +t

f o r every

f e L1

j.

g i v e n by t h e k e r n e l s X

K1(x1,x2) =

fail t o characterize

Then

( o r a n y even o r d e r ) i s a c o u n t e r e x a m p l e t o 2 For e x a m p l e , i n lR , t h e R i e s z t r a n s f o r m s o f

2

t h e above c o n j e c t u r e . 2,

K.

found t h a t t h e s y s t e m formed by t h e R i e s z

(17

transforms of order order

kj(E)

F i n a l l y suppose t h a t

2

2

-.

ixl H

1

i2 ,

.

-2x x K2(xl,x2 )

1 2

=

~

i xi

4

Actually, there i s a r a d i a l function

111.8. NOTES AND FURTHER RESULTS

f e L1(lR2) whereas f

6

375

such that both K1 * f and K2 x f belong to L 1 ( I R 2 ) , H1(IR2). Note that K1(z) + iK2(z) = z - ~ , a very simple

kernel, whose associated multiplier is given, in polar coordinates, by ei2'. See also Gandulfo, Garcia-Cuerva and Taibleson (11 where this is analyzed together with the corresponding phenomenon in local fields. The following theorem holds: Let

ml,m2, ...,mN

be h o m o g e n e o u s f u n c t i o n s of d e g r e e 0 in IR", C" T . be t h e c o r r e s p o n d i n g m u l t i p l i e r o p e r a t o r s

a n d Let

in given b y

J

(T.f)^(c) = mj(c)?(c). Then, the system J c h a r a c t e r i z e s H1(IRn) if a n d only if: i*l

That

5 f 0 m.(c) # m.(-t) J J

For every

t h e r e is s o m e

j = l,.. .,N

{T1,T2,...,TNl

such that

.

. . ,TN} characterizes

IT1,.

H 1 (IR")

means, of course, that

f e L1(IRn) is in H1(JRn) if and only if T.f e L1(IRn) J N. Note that this is the same as saying that: j = 1,

for every

...,

1

is equivalent to the H -norm. That the condition ( * )

is necessary was proved by S.Janson

who

[l]

a l s o conjectured the sufficiency by analogy with the martingale set-

ting.

The sufficiency was finally established by A. Uchiyama

[2]

who showed that if (*a')

rank

for every

1

]

. . ., m N ( c ) 0 , . . . , m N G i;

ml(E), ml(-

5 e

=

2

then every function

pact support, can be written as

N f

=

1 j=1

T.g. J

J

As a corollary he obtains from

("*)

that

f e 8.M.0.(Rn)

with com-

111. REAL HARDY SPACES

316

is a norm equivalent to the H1 norm. rank

which is identity

[

(**)

.

1 1

Note that

... m N ( S ) ] ml(-S) ... mN(-S)

(*)

is the same as

ml(S)

for the system

g

2

...,TN}

{I,T1,

where

I

is the

The paper of Uchiyama is a great achievement, because the decomposiN tion f = j41 Tjgj is obtained constructively. For the Riesz Rn, Fefferman and Stein obtain a decomposition transforms R1, of any f e B.M.O. in Rn as f = g + R 1g1+...+ Rngn with n However, they get this decomposition llgllm + jCIIIgjlIm 6 CIIflI.. from the duality (H1)* = B.M.O. by using the Hahn-Banach theorem, that is, in an essentially non-constructive way, In case n = 1 , the Fefferman-Stein decomposition, which now takes the form

...,

f = g + Hh with IIgllm + IIhllm constructively by P.Jones [43 For this case n = 1 , we shall proof of the decomposition, and in section 5 of Chapter IV.

6 CIIflI*, had already been obtained by using complex function theory. give a very simple constructive consequently, of the duality theorem,

has been even able to extend the theorem above to A. Uchiyama [4] HP(R") for a range of P I S < 1. The Lipschitz spaces Aa have a long history. They appear in many different areas of Analysis. For example: in the convergence of Fourier series; in Approximation Theory and in Partial Differential Equations. For the classical theory developed by Hardy, Littlewood, Zygmund, etc., see Zygmund [l], Chapters I1 and 111. Taibleson c1J and [Z] made a systematic study of these spaces in higher dimensions by using the harmonic extension to a half-space. This point of view is also presented in Stein [l], Chapter V. The approach to the Lipschitz spaces via mean oscillation over cubes, which led us to the duality theorems, originates with the independent but simultaneous work of S. Campanato [ l ] and N. Meyers [l]. See also Campanato [Z] and J. Peetre LlJ. For an approach via mean oscillation to the general class of Besov spaces, see J . Dorronsoro [ ] I 8.7.-

.

111.8. NOTES AND FURTHER RESULTS

377

The realization that the Lipschitz spaces Acl are the duals of the Hardy spaces Hp is due to Duren-Romberg-Shields [IJ in the torus and to T. Walsh C1-J in higher dimensions. It casts a new light on the classical results of Calderbn-Zygmund about the preservation of the spaces Aa by singular integral operators (see Taibleson [2]). 8.8.- The classical approach to interpolation of operators between Hp spaces was via complex function theory. See, for example, the contributions of Calder6n and Zygmund [ 4 ] and G. Weiss .]l[ Our presentation, based upon the atomic decomposition, originates with Igari ]l[ and culminates in Riviere and Sagher ,]l[ and in Fefferman, Riviere and Sagher [l]. Atoms are already used in the paper of Igari [l], as early as 1963. It was natural for the atoms to appear as a result of the effort to generalize the Marcinkiewicz interpolation theorem by looking closely at the Calder6n-Zygmund decomposition.

The theorem of E.M. Stein [5] on interpolation of analytic families of operators which we mention in Chapter 11, 7.5, was extended to Hp spaces by Stein and Weiss r3] in the classical case of the torus using complex methods and by E. Hernhndez El] in several dimensions using the atomic approach. 8.9.- Coifman [Z] gives a characterization of the Fourier transform ? of a distribution f belonging to Hp in terms of functions of exponential type. What he does is to translate, via the Paley-Wiener theorem, the characterization of Hp obtained in Coifman [lJ. Then, he uses the characterization of Hp to obtain multiplier theorems. It is also in Coifman [2] that molecules appear explicitly for the first time. The molecular theory was further exploited in Coifman-Weiss [ 3 ] and fully developed in Taibleson-Weiss cl], which is the source of the multiplier theorem 7.30.

-

The results on Hardy kernels (theorems 7.44, 7.49, etc.) are due to Jodeit-Shaw [l]. So is the sharp inequality in lemma 7.38, which allows them to obtain theorem 7.54. Lemmas 7.39 and 7.40 and also theorem 7.55 come from Garcia-Cuerva [IJ. 8.10.- Coifman, Rochberg and Weiss rl] have been able to obtain a weak version of the factorization theorem (3.4 in Chapter I)

378

111. REAL HARDY SPACES

valid for H1(IRn) with n > 1. In order to understand their approach, let us start with the classical factorization theorem, F e H1(D)

according to which any

can be written as

F

=

G 1 - G 2 with

and IIGjlIH2 = IIFII % for j = 1,2. If we set % H % Z % and G . = g . + ig. we obtain f = glg2 t glg2. Thus, in J J J terms of boundary functions, a function is the imaginary (or real) part of an H1 function if and only if it is of the form g1z2 + 21g2 for g, ,g9 e L2 and, in that case, its H1 norm is bounded by

Gj e H2(D) % F = f + if

Note that the integral above can be rewritten dsj dt =

so that the estimate ( * ) is equivalent to the L2 boundedness of the commutator operator of the multiplicaCb = [Mb,%), tion operator Mbf(x) = b(x)f(x and the conjugate function operator

the operator

'L

Cbf(x) = b(x)f(x)

-

(b

Thus the factorization plus the duality imply the Cb a)

L 2 boundedness of for b e B.M.O. But there are other possibilities: From the factorization and the L 2 boundedness of Cb for

b e B.M.O., the duality follows. b) From the duality and the boundedness of the commutator, we get a weak form of the factorization. What Coifman, Rochberg and Weiss do is to prove directly that if is a regular singular integral kernel in B.M.O. (IR"),

IRn

and

b

K

is in

then the commutator operator

is bounded in L 2 (IR") with operator norm dominated by IIblI++. They give two proofs of this fact. One of them is based on the theory of A weights and will be given in section 7 of P j = 1, Actually for the Riesz transforms R . J'

Chapter IV. n , we have equiva-

...,

379

111.8. NOTES AND FURTHER RESULTS

lence between the fact that b e B.M.O. @ T b , R J is L 2-bounded.

and the fact that each com-

mutator

Then, observation a) above leads to yet another proof of the duality theorem and observation b) allows them to give the following result, which contains a weak factorization theorem: Let K -

b e a s a b o v e and l e t

K*

f = g1K * g2

gl,g2 e L2(lRn), t h e f u n c t i o n

with

H~(IFP)

be t h e adjoint kernel.

'llfllHl s cllg111211g2112.

- (K* *

Then i f

g,)g2

Conversely every

is in

f e H'(IR")

can be w r i t t e n a s

8.11.- Coifman and Weiss [3] have developed a theory of Hardy spaces where the underlying space X is a space of homogeneous type, as defined in Chapter 11, 7.11, that is, X is endowed with a quasi-distance

d,

which allows us to consider balls

B(y,r) =

{ x e X : d(x,y) < r), and a doubling measure P . If 0 < p < q, and p 5 1 S q 5 m, then a (p,q)-atom is, by definition, a real

=

valued function i)

a

If

q =

a(x),

such that:

is supported in a ball

m,

the left hand side of

B(y,r)

ii)

is interpreted as the essen-

tial supremum of a over the ball. In case function 1-1 (x)-(l/p) is counted as an atom. one can simply assume

~ ( x =)

P(X) < the constant Actually, in this case, 03,

1.

In a context as general as this, one cannot consider higher moments. Consequently, the Hp spaces to be defined, will coincide with the classical ones only for a range of p's close to 1. Still, the point of view of Coifman and Weiss allows a very convenient unification of the ever

growing family of Hardy spaces.

111. REAL HARDY S P A C E S

380

Since the Hp spaces for p < 1 will have to be spaces of "distributions", the next thing to do is to define the space of test functions. For a > 0, we consider La, the space of functions f on X , for which

where B is any ball containing both x and y and C depends only on f. This space is normed by the smallest C above, or, in case p (X) < m , by the smallest C plus IIXf(x)dp(x)l. Next we observe that (p,q)-atoms act continuously on La for a = (l/p)-l if O < p < 1. Then, for 0 < p < 1 5 q, HPsq(X) is defined as the subspace of the dual L i of La where a = (l/p)-1, consisting of those linear functionals admitting an atomic decomposition m m h = X.a where the a . ' s are (p,q)-atoms and &lhjlP < The j = O J j' J infimum of '/')' over all such decompositions is denoted by Ilhllp,q. It is clear that Ilhll!,q is a p-norm. For p = 1 , Hlrq(X) is defined as a subspace of L1(X).

-.

(61Xjl

Then, Coifman and Weiss show that HP" = HPlm whenever p < q m, and that, moreover, the two metrics are equivalent. This completes the definition of Hp. They go on to show that, for p < 1, the dual ( H p ) * of Hp, coincides with La' where = (l/p)-1. For p = 1 , the dual is identified with B.M.O., which is defined in the natural way. Actually, different B.M.O.'s are introduced. B.M.0.q is defined by using the Lq norm of f - fg over each ball B. It is shown that (H1")* = B.M.O.q', from which all the B . M . O . ' s are seen to coincide. observe that Carlesonls proof of the duality Coifman and Weiss [3] H 1-B.M.O. (see note 8 . 4 ) , can be extended to the setting of a space of homogeneous type satisfying a mild geometric condition. They use this to obtain a maximal function characterization of H1(X). The maximal function characterization of Hp(X) for 0 < p < 1 follows after Macias-Segovia ]l[ and (2) and A . Uchiyama

D].

Here are some instances of spaces of homogeneous type for which the Hardy spaces of Coifman and Weiss are interesting objects connected with classical problems (see Coifman and Weiss [3] for details). 1) En with a nonisotropic distance and a doubling measure. This setting is appropriate for studying singular integrals with the cor-

1 1 1 . 8 . NOTES AND FURTHER RESULTS

381

responding homogeneity (see note 7.8. in Chapter 11). 2) An interval of the line and a doubling measure naturally associated with the study of certain classical polynomial expansions (Jacobi series, etc.). 3) A compact Riemannian manifold with its natural distance and

For example a homogeneous space

X = G/K where K is a closed subgroup of the Lie group G. In particular the unit sphere of IF? can be viewed as = SO(n)/SO(n-l), and then 'n-1 d = euclidean distance and 1-1 = Lebesgue measure. 4) The boundary of a Lipschitz domain in IRn with euclidean distance and harmonic measure. 5) The boundary of a smooth and bounded pseudoconvex domain in $n with Lebesgue measure and the nonisotropic quasidistance associated to its complex structure. In particular for C2n-l this provides an example essentially different from that in 3). 6) The Heisenberg group and its generalizations (see Folland-Stein measure.

C2J * 7) A locally compact group with a system of open neighbourhoods of the identity and a measure satisfying certain natural properties. This was introduced as a reasonable setting for the study of singular integrals in Koranyi and Vagi 8.12.-

)l[

and N. Riviere

(11.

Even though the Coifman-Weiss theory briefly sketched in

8.11 covers many interesting spaces, still in many particular situations, a richer theory is obtained by considering atoms with an increasing number of

vanishing moments.

Here is a , necessarily

incomplete, list of these richer but more particular theories going beyond the Coifman-Weiss frame. 1)

The parabolic

chinsky

Cl]

and

Hp

c2]

spaces introduced by

A. Calder6n and Tor-

are associated to a space of homogeneous

type which falls within point 1) in 8.11., the measure being Lebesgue measure. Initially these spaces were introduced by means of a maximal function. Then, the atomic decomposition (with higher moments) was obtained by A.P. Calder6n [a) for the case when the parabolic metric is given by a diagonalizable matrix, and by Latter and Uchiyama (13 in general. 2) Two different kinds of weighted Hardy spaces were studied in

J. Garcia-Cuerva [l]. The underlying space is the line IR with the usual distance and the measure dp(x) = w(x)dx with w a weight belonging to the class Am to be studied systematically in Chapter IV. One kind of Hp(w) is characterized by the fact that

382

111. REAL HARDY SPACES

an appropriate (smooth) maximal function belongs to

Lp(w).

Equiva-

lently, atomic characterizations (involving higher moments with respect to Lebesgue measure) are obtained for 0 p 5 1. The other kind of Hp(w) is simply the one associated to the space of homogeneous type (IR,d,w(x)dx). There is a relation between these two kinds of spaces. For the homogeneous type H1(w) the following characterization is obtained H 1 (w) = If0w-l : f e H1(IR)). A l s o in Garcia-Cuerva (11 it is shown that if f is defined in [O,m], then x W f ( l x 1 ) is in H1(IRn) if and only if r H f ( ~ r ~ ) ~ r ~ n - l is in H1(IR). This characterization of the radial functions in H1( En), whose proof is based on theorem the counterexample alluded to in 8.6.

7.55,

is the source of

General weighted Hardy spaces have been studied by Stromberg and Torchinsky (see for example their paper [ll).

C. Kenig ( [ I ] and [2]) has given a theory of weighted Hardy spaces on Lipschitz domains of the plane. This extends the situation discussed in 2 ) and also the two different kinds of 3)

atoms appear. One thing that makes these spaces interesting is the fact that harmonic measure on the boundary of a Lipschitz domain in IRn

is given by an

A m weight times surface measure.

The theory of

weighted Hardy spaces has been extended to Lipschitz domains of for arbitrary

n

(see, for example

Fabes, Kenig and Neri

4) Regarding the space of homogeneous type considered in point of 8.11 over the unit sphere presented in Garnett-Latter ,]l[

x2n-l

lRn

[l]). 5)

of $", a richer theory is which also discusses the rela-

tion between the atomic space and the classical Hp space of holomorphic functions on the unit ball of $". From this relation they obtain weak factorkzation theorems for the holomorphic H p , extending the results obtained for holomorphic H1 by Coifman, Rochberg and Weiss . ] I [ 5) Atomic decompositions (with moments) for the Hardy spaces of the Heisenberg group, appear in Latter-Uchiyama

(A].

A

general theory

for a wide class of generalizations of the Heisenberg group, is presented in Folland-Stein (2:). 6) A particular case falling under point p-adic fields.

7)

in

8.11

is that of

The associated Hardy spaces are examples of Hardy

spaces of martingales, but the theory of these latter spaces is much more general. It was in this context that the atomic decomposition was first explicitly noted by Carl Herz [ 1 ] (see also [SJ).

1 1 1 . 8 . NOTES AND FURTHER RESULTS

8.13.-

383

There a r e s t i l l some s e t t i n g s where t h e n a t u r a l

i s f a r removed from t h e Coifman-Weiss

atomic t h e o r y .

t h i n k i n g o f t h e Hardy s p a c e s a s s o c i a t e d t o

Hp

theory

W e a r e mainly

"product s p a c e s " ,

whose t h e o r y i s f i n a l l y b e g i n n n i n g t o be u n d e r s t o o d , t h a n k s m a i n l y t o t h e e f f o r t s o f Chang and R . survey

p] 1.

See a l s o

Lin

Fefferman

[ I ) .

(see

[l]

,

(2)

and t h e

This Page Intentionally Left Blank

CHAPTER IV WEIGHTED NORM INEQUALITIES

The L p inequalities obtained in the previous chapters for seve ral kinds of operators remain true when Lebesgue measure dx is replaced by certain measures w(x)dx. This is not entirely new for us: In Chapter I we proved the Helson-Szeg5 theorem characterizing those w(x) for which the Hilbert transform is bounded in L2(w), while the boundedness of the Hardy-Littlewood maximal function in Lp(w) was established in Chapter I for every A1 weight w(x) and 1 < p < m .

We shall devote this chapter to a systematic study of this type of inequalities. F o r the maximal function and regular singular opera tors, it is possible to give a very precise and satisfactory answer to the question of finding those w for which either

or the corresponding weak type inequality hold. This answer is provided by the Ap theory, which has already reached a great level of perfection and has found applications in several branches of Analysis, from Complex Function Theory to Partial Differential Equations. The same problem for two weights is also considered (in sections 1 and 4), but here some unsolved questions remain which are probably far from being completely settled. Why should one be interested in inequalities like ( * ) ? We shall briefly sketch some answers 1) Conjugate functions, Hp spaces etc. can be defined in domains of the complex plane with a "reasonable" boundary a D . When estimating the Lp norms of operators appearing in this context, some of the problems that arise can be reduced, by a change of variables, to estimates for known operators on the line or on the torus, but with respect to a measure w(x)dx for a certain w. These ideas were actually used in CalderBn's original proof of his celebrated theorem on Cauchy integrals along Lipschitz Curves (see Chapter 1 1 , 7.12 for the precise statement).

385

386

IV. WEIGHTED NORM INEQUALITIES

2) Given a complete orthonormal system some interval I, does the series

{r$k}

in

L2(I),

for

converge to f(x) in the Lp norm, p # 2 ? For ordinary Fourier series, the partial sums are very closely related to the Hilbert transform, and M. Riesz theorem implies (Snf-flp +. 0, 1 < p < m . In some other interesting cases, the partial sums can be expressed for a certain in terms of the operator f(x) +. k(x)H(f k-l)(x) k(x), and then, convergence in Lp is equivalent to ( * ) with T = H, w(x) = Ik(x)lp. 3) Inequalities like ( * ) imply, when the structure of the weights satisfying them is sufficiently known, the following

for arbitrary u(x) 0, where N is (in the most desirable case) some kind of "maximal operator" which we can control. An inequality like ( * * ) was proved in Chapter 11, 2.12 for the Hardy-Littlewood operator, and a similar one for singular integrals will appear in corollary 3.8 of this chapter. Such inequalities are very easy to handle, and contain essentially all the relevant information about the boundedness properties of T (we shall come back to this point in Chapter VI).

4) There is an intimate connection between weights, BMO functions and C. Fefferman's duality theorem, which can be roughly described by saying that BMO consists of the logarithms of good weights. This connection is exploited in section 5, where we give a new proof o f the duality theorem and obtain, by real variable methods, a satisfac tory analogue in Rn of the Helson-Szeg8 theorem.

IV.1. THE CO!:DITION A

38 7

P

1 . THE CONDITION Ap

By a weight on a given measure space, we shall always mean a measurable function w with values in [O,m]. Our main problem is going to be the following PROBLEM 1 . G i v e n

p,

1 < p

<

i s of s t r o n g t y p e

M

f o r w h i c h t h e maximal o p e r a t o r

w(x)dx,

r e s p e c t to t h e m e a s u r e

determine those weights

a,

w

(p,p)

on Rn

with

t h a t i s , f o r w h i c h we h a v e an i n e -

quality:

We can also pose this more general PROBLEM 2 . G i v e n p , 1 < p < m , d e t e r m i n e t h o s e p a i r s o f w e i g h t s o n Rn (u,w), f o r w h i c h M i s of s t r o n g t y p e (p,p) with respect to t h e p a i r o f m e a s u r e s (u(x)dx, w(x)dx), t h a t i s , f o r w h i c h we h a v e an i n e q u a l i t y : ( 1 .2)

(IRn

(Mf(x))pu(x)dx)’/p

5

5 C(jRn (f(x)lPw(x)dx)’/P We can pose similar problems substituting weak type for strong type in the two problems above. For example: PROBLEM 3 . G i v e n

p,

1

2 p

<

m,

d e t e r m i n e t h o s e p a i r s of wi7:ghts

on Rn,

( u , w ) , f o r w h i c h ?I i s of weak t y p e (p,p) w i t h r e s p e c t to t h e p a i r o f m e a s u r e s (u(x)dx, w(x)dx), t h a t i s , f o r w h i c h we have t h e i n e q u a l i t y :

(1.3)

u(Cx € Rn : Mf(x) > tl) 5 t > O

IE

For a set E, u(E) stands for u(x)dx. This notation has been used in ( 1 . 3 ) and it will be used systematically.

IV. WEIGHTED NORM INEQUALITIES

388

We shall keep the usual conventions for multiplication in [O,m], namely m.t = t.m = m for 0 < t 5 m and m . 0 = 0.- = 0. Also - 1 = 0 and 0 - 1 = m when we consider w - 1 for a weight w. Let us start by analizing Problem 3. Suppose that the pair of weights (u,w) is such that (1.3) holds for a given p, 1 5 p < every function f and every t > 0 . Let f be a function LO. Let Q be a cube such that the average f = f(x)dx > 0. Q lQ\ Q Observe that f M(f.XQ)(x) for every x E Q. Then, for every

m,

1

Q -

t

with

0

t < f

Q C Et

Q'

by (1 .3): u(Q)

5 C

t-p

=

1,

{x E Rn : M(f.X

Q )(x)

> tl

so that,

f(x)pw(x)dx

It follows that

(fQ)Pu(Q)

(1.4)

5 C

f(x)pw(x)dx

JQ

We can actually write this inequality in a seemingly stronger form. If S i s a measurable subset of Q, we can replace f in (1.4) by f.XS, obtaining

'k 1s

(1 .6)

f (x)dx)Pu(Q)

2 C

f (x)pw(x)dx IS Of course (1.5) is just equivalent to (1.4), but ( 1 . 5 ) is more readily applicable sometimes. F o r f(x) E 1, (1.5) yields: (1.5)

(ISl/lQlIP u(Q) 5 C w(S)

From (1.6) we can draw some relevant information about the pair (u,w) : a) w(x) > o for a.e. x E R" (unless u(x) = o for a.e. x € Rn, trivial case which we shall exclude). b) u is locally integrable (unless w(x) = m for a.e. x E Rn, again a trivial case which we shall a l s o exclude). Let us prove a) and b). I'f w(x) = 0 on a set S with IS1 > 0, a set S which we could assume to be bounded, (1.6) would imply that u(Q) = 0 for every cube Q containing S, and consequently, u(x) = 0 for almost every x E En. If u(Q) = m for some cube Q and, consequently, f o r any cube containing 0, we would have w(S) = m f o r any set S with I S 1 > 0, and this implies w(x) = m f o r a.e. x E Rn. We are about to derive a necessary condition on the pair

(u,w)

for

IV.l. THE CONDITION A

( 1 . 3 ) to hold for every in the form:

f

and t.

If p

389

P

= 1,

(1.6)

can be written

(1.7) the inequality being valid for every cube Q and every set S C Q with I S 1 > 0. Fix Q and let a > ess.Qinf.(w), the essential infimum of w over Q, which is defined as the inf{t > 0 : ]{x 6 Q : w(x) < t}l > 0 1 . Then, Sa = {x E Q : w(x) < a} has lSal > 0, and ( 1 . 7 ) holds for S = Sa, from which we get: u(Q)/(QI S Ca. Since this is true for every a > ess inf. (w), we Q arrive finally at: (1.8)

1

1 IQI Q

u(x)dx

2 C ess. inf.(w) 2 C w(x),

Observe that the fact that ( 1 . 8 ) to :

Q

holds for every

Q

for a.e. xEQ is equivalent

Indeed, i t is clear that ( 1 . 9 ) implies ( 1 . 8 ) for every cube. Conversely if ( 1 . 8 ) holds for every Q, let us show ( 1 . 9 ) holds, that is, the set Cx E Rn : M(u) (x) > C w(x)} has measure 0. If M(u)(x) > C w(x), i t will be

for some cube Q containing x , and we can assume that Q has vertices with all coordinates rational. Then our set is contained in a denumerable union of sets o f measure 0 and, consequently has measure 0. Condition ( 1 . 9 ) is known as condition A, for the pair (u,w). When it holds, we also say that the pair (u,w) belongs to the class A,, viewing A 1 as a collection of pairs of weights (u,w). We often speak of the A 1 constant for the pair (u,w) which is the smallest C for which ( 1 . 8 ) , o r equivalently ( 1 . 9 ) , holds. We have just seen that (u,w) 6 A, is a necessary condition f o r to be of weak type ( 1 , l ) with resnect to the pair (u,w). It is very satisfactory to realize that this condition is actually sufficient. Indeed, let (u,w) € A 1 , s o that ( 1 . 9 ) holds. Then,

M

IV. WEIGHTED NORM INEQUALITIES

39 0

u s i n g i n e q u a l i t y ( 2 . 1 4 ) i n c h a p t e r 1 1 , we g e t

m. We s t a r t by l o o k i n g f o r a Now we s h a l l t r e a t t h e c a s e 1 < p n e c e s s a r y c o n d i t i o n . So f a r we know t h a t i f M i s o f weak t y p e (p,p) with r e s p e c t t o (u,w), then (1.5) holds f o r every f u n c t i o n

Q and e v e r y m e a s u r a b l e s e t S C Q . Let us f L 0 , e v e r y cube choose f such t h a t f ( x ) = f ( x ) p w ( x ) . T h i s g i v e s f ( x ) = = w(x) - l ' ( p - l ) . A p r i o r i t h i s f u n c t i o n n e e d s n o t be l o c a l l y i n t e g r a b l e . F i x a cube Q and t a k e S = S . = { x E Q : w(x) > j - ' } I f o r j = l , Z ,... On e v e r y S . o u r f i s bounded, s o t h a t 1 f < m. With o u r c h o i c e f o r f , (1.5) gives:

Isi

or, since the integrals are f i n i t e ,

...

m

U

S = [ x 6 Q : w(x) > 0 1 , whose i=l j complement i n Q h a s m e a i u r e 0 , a s was p r e v i o u s l y o b s e r v e d . T h u s , l e t t i n g j + m , we g e t f i n a l l y

Now

SIC S 2 c

and

We s h a l l s a y t h a t t h e p a i r (u,w) s a t i s f i e s t h e c o n d i t i o n A or ? t h a t it belongs t o t h e c l a s s A i f and o n l y i f t h e r e i s a c o n s P' t a n t C s u c h t h a t ( 1 . 1 0 ) h o l d s f o r e v e r y cube Q . The s m a l l e s t s u c h c o n s t a n t w i l l b e c a l l e d t h e Ap c o n s t a n t f o r t h e p a i r ( u , w ) . i s n e c e s s a r y f o r M t o be o f weak We have p r o v e d t h a t (u,w) E A P type ( p , p ) with r e s p e c t t o t h e p a i r (u,w). Observe t h a t (u,w) E A i m p l i e s t h a t b o t h u and w - " ( p - ' ) are locally inteP g r a b l e . I n d e e d i f one o f t h e i n t e g r a l s i n (1 . l o ) were m y t h e same would happen f o r any cube c o n t a i n i n g Q , and t h a t would f o r c e t h e o t h e r f a c t o r t o be 0 . T h i s would imply e i t h e r u ( x ) = 0 f o r a . e . x E Rn o r w(x) = m f o r a . e . h a v e been e x c l u d e d b e f o r e h a n d . made i s t h a t t h e c o n d i t i o n A1

x E Rn. Both t r i v i a l s i t u a t i o n s Another o b s e r v a t i o n t h a t has t o be c a n be viewed a s a l i m i t c a s e o f

I V . l . THE C O N D I T I O N A

the conditions (1.11)

A

P

for

+

p

1.

I n d e e d , ( 1 . 8 ) c a n be w r i t t e n a s

u ( x ) d x ) . e s ~ . ~ s u p . (-1w)

(- 1

IQI

39 1

P

<=

C

Q

while

as

p

+

1,

i.e.

l/(p-1)

-+

m.

Thus ( 1 . 1 1 ) i s t h e r i g h t companion f o r ( 1 . 1 0 ) when p = 1 . A l s o n o t e t h a t (u,w) E A1 i m p l i e s t h a t u i s l o c a l l y i n t e g r a b l e and w - 1 i s l o c a l l y bounded. Our t a s k w i l l b e now t o show t h a t , e x a c t l y a s i n t h e c a s e p = 1 , when 1 < p < m, (u,w) € A is not only necessary, but a l s o P s u f f i c i e n t f o r M t o be o f weak t y p e ( p , p ) w i t h r e s p e c t t o t h e p a i r ( u , w ) . We h a v e o b t a i n e d c o n d i t i o n A from ( 1 . 4 ) . The f i r s t P s t e p w i l l b e t o show t h a t , c o n v e r s e l y , i f then (1.4) (u,w) € A p , h o l d s f o r e v e r y f 2 0 and e v e r y c u b e Q . T h i s i s a c t u a l l y t r u e

f o r 1 S p < m. I f p = 1 c u b e Q a n d e v e r y f => 0 :

1

(L IQI

Q for

and

f(x)dx)u(Q) =

(u,w) 6 A 1 ,

i,

f(x)dx

we h a v e , f o r e v e r y

u(Q)S IQ I

C

i,

f(x)w(x)dx,

which i s (1 . 4 ) p = 1 . I f 1 0 , u s i n g H i i l d e r ' s i n e q u a l i t y w i t h p and i t s c o n j u g a t e exponent p ' = p / ( p - 1 ) ,

Thus

s o t h a t ( 1 . 4 ) h o l d s . We have e s t a b l i s h e d t h e e q u i v a l e n c e between ( 1 . 4 ) and A P'

I V . WEIGHTED NORM INEQUALITIES

392

Q

Now, s u p p o s e t h a t ( 1 . 4 ) h o l d s f o r e v e r y c u b e

f 2 0.

and e v e r y

We s h a l l o b t a i n ( 1 . 3 ) w i t h a p o s s i b l y b i g g e r C . O f c o u r s e , we h a v e ( 1 . 5 ) f o r e v e r y f L 0 , e v e r y c u b e Q and e v e r y s e t S C Q . Let f € Lp(w). We c a n o b v i o u s l y assume t h a t f 2 0 . O b s e r v e t h a t 1 L ~ o c ( w ) CLloc(Rn) a s f o l l o w s from ( 1 . 4 ) u s i n g Q s u c h t h a t u(Q) > 0 . Now we c a n a l s o assume t h a t f E L1 (R"). I n d e e d , we c a n a l w a y s w r i t e f = l i m f k where f k = f.X I f we have ( 1 . 3 ) Q(O,k) ' k+m f o r e v e r y f k i n p l a c e o f f , p a s s i n g t o t h e l i m i t we o b t a i n ( 1 . 3 ) f o r f . T h u s , a s s u m i n g f i n t e g r a b l e , we want t o e s t i m a t e u ( E t ) where E t = { x 6 Rn : Mf(x) > t l . We u s e t h e o r e m 1 . 6 from c h a p t e r 11, t o w r i t e E t C Q i , where t h e Q j ' s a r e n o n - o v e r -

0

'

l a p p i n g c u b e s f o r which t

t f(x)dx f Zn

Then, u(Et)

2

C u ( Q3. )

j

6

C 3"p

J

drip t - p c j

2

I

C C (- l3

j

[

lQjl

f(x)dx)-P

f(x)Pw(x)dx

2

Qj

Qj

f(x)Pw(x)dx

I

2

C t-p

Qj

where t h e s e c o n d i n e q u a l i t y r e s u l t e d from a p p l y i n g ( 1 . 5 ) w i t h

Q

= QS and S = Q We have a c o m n l e t e p r o o f o f t h e f a c t t h a t t h e j' s o l u t i o n t o Problem 3 i s p r e c i s e l y t h e c l a s s A of p a i r s of P w e i g h t s . We c a n c o l l e c t o u r f i n d i n g s i n t h e f o l l o w i n g

THEOREM 1 . 1 2 . Let

w

u

be w e i g h t s on

Rn

and l e t

1

2

p <

m.

Then, t h e f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t : a) M that i s :

M

i s o f weak t y p e

Lp(w)

takes

there i s a constant and e v e r y t > 0

f

2

0

in

C

2

( p , p ) w i t h r e s p e c t t o (u,w), L!(u) boundedly o r , i n other.words,

such t h a t f o r every f u n c t i o n

u ( { x E Rn : Mf(x) > t l )

2

b ) T h e r e is a c o n s t a n t

C

Rn

and f o r e v e r y c u b e

f(x)dx)pu(Q) 5 C

C t-P

f

6 L;oc(Rn)

jRn I f ( x ) IPW(X)dX.

such t h a t f o r every f u n c t i o n Q

IV.l. THE CONDITION A

c)

(u,w) € A

t h a t f o r e v e r y cube

and, i n case

Besides,

Q

393

P

that i s , there i s a constant

P'

we h a v e , i n c a s e

1 < p <

C

such

m,

p = 1,

the constants

C

a p p e a r i n g i n a ) , b ) and c ) a r e of t h e

same o r d e r .

COROLLARY 1.13. L e t (u,w) € A . T h e n , f o r e v e r y q with P p < q < m, t h e maximal o p e r a t o r M i s bounded f r o m Lq(w) to Lq(u), t h a t i s , there e x i s t s a constant C such t h a t f o r ever2 f € Lloc(Rn) 1 : ~

IRn

IMf(x)lqulx)dx

S C

1,"

I f (x) 1 w'

(x) dx

Proof: We already known that F! is of weak type (p,p) with respect to (u,w), that is, M takes Lp(w) boundedly to L t ( u ) . We shall see presently that M is also bounded from Lm(w) to Lm(u). This is a consequence of the chain of inequalities: IMflm,U 5 I M f l ,

S

lfl,

2

Since ~Mf~,,u

=

sup{a : u({x

€

Rn : Mf(x) >

a } ) > 0}

the first inequality follows from the fact that u(E) > 0 implies [ E l > 0. Likewise, the last inequality follows from the fact that IEl > 0 implies w(E) > 0, because w(x) = 0 only for the points in a set of measure 0. Once we know that M is bounded from Lp(w) to L:(u) and from Lm(w) to Lm(u), we use Marcinkiewicz interpolation theorem to conclude that M is bounded from Lq(w) to Lq(u) provided p < q < m. 0 A particular instance of corollary 1.13 is the inequality

iRn

IMf(x) IPu(x)dx

=< C

jRn [ f(x) IPMu ( W x

valid for 1 < p < m, which appeared in Chapter 11, (2.13). I t is contained in our corollary because (u,Mu) € A 1 .

394

IV. WEIGHTED NORM INEQUALITIES

The following theorem contains some simple basic facts about the classes A P'

THEOREM 1.14. a ) Let b)

Let

1 < p < q <

1 5 p <

m,

Then A I C A C A P 9

m,

0 < E <

1

( u , ~ )6

A P'

Then -

(uE,wE) 6 A E P + l - E .

1 < p < =. T h e n (u,w) E A P i f and o n l y i f u7'p-l)) E A ~ ' , pl being, as u s u a l , t h e exponent t h a t i s p ' = p/(p-l). c o n j u g a t e t o p, c) L e t

(w-1'(p-'),

Proof: a) We just need to observe that

6 ess .Qsup. (w- 1 ) The first inequality follows from Jensen's o r H6lder's inequality, since (q-l)(p-l)-' > 1. The second one is obvious. b) For

r

=

EP+~-E

is

r-1

=

~ ( p - 1 ) . Then

if C is the A constant for the pair (u,w). Again Jensen's P inequality has been used to deal with the integral containing u . The case p = 1 is included by interpreting the second factor properly. c ) Suppose

since (p-i)(pf-l) quality as:

=

(u,w) E A

P'

Thus:

pp'-p-pi+l = 1, we can write the previous ine-

IV. 2 . REVERSE H O L D E R ' S

and conclude t h a t that

(u,w) 6 A

39 5

A c t u a l l y we s e e

(w - l ' ( p - l ) , is equivalent t o

P

INEQUALITY

.o

(w

EXAMPLE 1 . 1 5 . We s h a l l g i v e h e r e a n e x a m p l e w h i c h shows t h a t c o r o l l a r y 1 . 1 3 c a n n o t be improved s o as t o i n c l u d e a l s o t h e case q = p . We s h a l l g i v e w e i g h t s u,w s u c h t h a t (u,w) € A a n d , however, M P i s n o t b o u n d e d f r o m Lp(w) t o L P ( u ) . I f p = 1 , we j u s t n e e d t o t a k e u = w E 1 . We a l r e a d y know t h a t i f g 2 0 i s i n t e g r a b l e a n d is not 0 a t a.e. x , t h e n Mg i s n e v e r i n t e g r a b l e ( s e e t h e r e m a r k T h i s same F a c t l e a d s t o a n f o l l o w i n g theorem 2.6 i n Chapter 11). e x a m p l e f o r p > 1 . Let g 2 0 , i n t e g r a b l e a n d non t r i v i a l , i n s u c h a way t h a t

Mg

e

L1.

Take

g

bounded s o t h a t you c a n guarantee

Mg(x) i s a l w a y s f i n i t e . Then ( g , Mg) € A I C A p l , and hence = ((Mg)'-p, g l - p ) € Ap. ( P ' ' ) , g " ( p ' l ) ) I f we t a k e ((Mg) u = ( b l g ) l - P a n d w = g l - p , we h a v e a p a i r ( u , w ) € A f o r which P the inequality that

jlMf(x) IPu(x)dx cannot hold, since f o r I j l f l p w = J g < m.

f

5

C

=

g

I

l f ( x ) J p w(x)dx we have:

Mg =

11.f

I n t h i s way, we h a v e s e e n t h a t t h e c o n d i t i o n s o l v e Problem 2 . I t i s n e c e s s a r y f o r ( 1 . 2 ) t o p l i e s ( 1 . 3 ) . However, i t i s n o t s u f f i c i e n t .

m

and

E Ap d o e s n o t s i n c e (1.2) i m

2 . THE REVERSE H ~ L D E R ' S INEQUALITY A N D THE C O N D I T I O N A ~ .

The t h e o r y d e v e l o p e d f o r t h e case situation:

u

THEOREM 2 . 1 .

Let

=

i n s e c t i o n 1 becomes p a r t i c u l a r l y i n t e r e s t i n g w. F i r s t of a l l , t h e o r e m 1 . 2 r e a d s a s follows i n t h i s

w

be a weight on

Rn,

and l e t

1

2 p

<

m.

Then,

t h e foZlowing c o n d i t i o n s are e q u i v a l e n t :

a) M i s of weak t y p e

( p , ~ ) with respect t o

w({x € R n : b l f ( x ) > t } ) 5 C t - p b ) 2 e r e i s a constant

C

iRn I

f

( X I I pw

w,

&

(x) dx

such t h a t , for every f u n c t i o n

I V . WEIGHTED NORM INEQUALITIES

396

f > 0

and f o r e v e r y c u b e

cl

(w,w) 6 A

Q,

f o r e v e r y cube constants

C

P'

Q

1 < p <

that i s , i n case

and, i n c a s e

appearing i n a ) , bl

m

p = 1, Mw(x) 5 C w ( x ) p.e. c) a r e of t h e same o r d e r .

The

&

When w s a t i s f i e s c ) , we s a y t h a t w s a t i s f i e s t h e c o n d i t i o n *P' and w r i t e w E A We a l s o s p e a k o f t h e Ap c o n s t a n t f o r w , w i t h P' t h e n a t u r a l meaning. N o t i c e t h a t t h e c l a s s A, i s t h e same w h i c h appeared i n Chapter I1 ( a f t e r theorem 2 . 1 2 and a l s o i n theorem 3 . 4 ) . We saw ( i n 1 . 1 5 ) t h a t a p a i r o f w e i g h t s

( u , w ) may b e i n A and P In contrast t o y e t M may n o t b e b o u n d e d f r o m Lp(w) t o L p ( u ) . t h i s s i t u a t i o n , f o r p > 1 , i t s u f f i c e s t h a t w i s i n Ap f o r M t o b e b o u n d e d i n Lp(w) . weights: T h i s f a c t depends on a b a s i c p r o p e r t y e n j o y e d by t h e A P t h e r e v e r s e H z l d e r ' s i n e q u a l i t y ( R . H . I . ) a p p e a r i n g i n lemma 2 . 5 b e l o w . F i r s t we p r e s e n t a c o u p l e o f s i m p l e p r o p e r t i e s o f t h e weights.

We s t a r t w i t h a n

a cube

Let w

and e v e r y

X

b e ay.

A

1

w(QX) where

Qx

of

Q.

LEMMA 2 . 2 . Q

w-measure o f t h e d i l a t e d

estimate for the

Ap

C

weight i n

P

2

C Xnp

is of t h e same o r d e r

as

Then, f o r every cube

w(Q) the

f = Xs

P r o o f : I n b) o f t h e o r e m 2 . 1 , t a k e

Rn.

A

constant f o r

P

with

S C Q,

w

and

Q

a

c u b e . Then (2.3)

Using (2.3) w i t h Q x w(Q ) 2 C X n p w ( Q ) .

( / s ~ / ~ Q ~ ) ~ w 5( Qc) W(S) in place of

0

S

and

Qx

i n p l a c e of

Q

we g e t

IV.2. REVERSE H ~ ~ L D E RINEQUALITY ~S

39 7

In particular the lemma implies that for an A weight w, the P measure 1~ given by du(x) = w(x)dx is a doubling measure. Actually, what we have shown is that property b) in theorem 2.1 implies that p is a doubling measure. Observe also that the same pro perty b) implies that 5 C1’p IMu((f(p)(x)?”p

Mf(x)

a.e.

where Mv is the operator introduced in Chapter I1 (after theorem 1.12). We showed there that, for 1~. doubling, Mu is of weak type (1,l) with respect t o u . We can rely upon this fact to prove that b) implies a) in theorem 2.1. Indeed w(Cx E R” : Mf(x) > tj)

where the last in b).

C

<=

w({x

E Rn : CMp(lflP)(x)

exists

Let

B

1

w € A

P’

Then, f o r e v e r y p o s i t i v e

depending on

r a b l e s e t contained i n a cube follows that

tP3)1

contains the doubling constant and the constant

The next lemma is a comparison between the measure besgue measure LEMMA 2.4.

>

a

such t h a t , whenever

Q

and s a t i s f g i n g .

w(x)dx

and

< 1,

there

c1

A /A1

Lg

i s a measu-

2

alQl,

w(A) 5 B w(Q).

Proof: We start from (2.3) where, of course, it is always C b 1 (set S = Q). If we u s e in (2.3) S = Q\A where \ A \ 5 a \ Q l , we get:

We shall use the previous lemma to establish o u r basic inequality LEMMA 2.5. on -

p

w E A

and o n t h e

A

P

P’

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

constant for

E

> 0 , depending o n l y

w , such t h a t , f o r every cube Q

IV. WEIGHTED NORM INEQUALITIES

398

w i t h a constant

n o t d e p e n d i n g on

C

Q.

The o p p o s i t e i n e q u a l i t y h o l d s , w i t h C = 1 , f o r e v e r y f u n c t i o n w and i s a p a r t i c u l a r c a s e o f H b l d e r ' s i n e q u a l i t y . T h i s i s why t h e lemma i s c a l l e d t h e r e v e r s e H o l d e r ' s i n e q u a l i t y ( R . H . I . ) . P r o o f : We s h a l l f i x t h e cube with

and

E

...

ho < h 1 <

Q

and we s h a l l g e t t h e i n e q u a l i t y

i n d e p e n d e n t o f Q . We t a k e an i n c r e a s i n g s e q u e n c e < w i t h A, = wQ = w(x)dx a n d , f o r

C

&

...

< Ak

JQ

we make t h e C a l d e r h - Z y g m u n d d e c o m p o s i t i o n o f Q f o r t h e each h k , f u n c t i o n w and t h e v a l u e A k ; t h a t i s , we c o n s i d e r t h o s e maximal (the d y a d i c s u b c u b e s o f Q o v e r which t h e a v e r a g e o f w i s > A k dyadic subcubes of s i d e o f Q i n 2N

.

r Q k , j j =1 , 2 , . . . 'k < wQk . <= 2"Ak, =

Dk

is

1 1

a r e t h e c u b e s r e s u l t i n g from d i v i d i n g e a c h e q u a l p a r t s , N = 0 , l , 2 , . . . ) . L e t them be I t follows t h a t , f o r each j is Q

while f o r a . e .

w(x) 2 X k .

Since

Ak+l

x

y

not belonging t o

> Xk,

i n Qk,i f o r some i , i n s u c h a way t h a t what p o r t i o n o f Q k , i c a n b e c o v e r e d by

each

Q k + l ,i

._

D k + l C Dk.

Dk+l.

Qk,j =

i s contained

Let u s s e e

We know t h a t :

w(x)dx = Qk,in

Dk+l w(x)dx

I Q k + l ,j

' 'k+l

lQk,i"

Dk+ll

IQk,il

Let us t a k e t h i s r a t i o equal t o

c1

< 1,

t h a t i s A k + l = 2 n a - 1 A k , A k = ( 2 na - 1 ) k A., I f we c o n s i d e r t h e @ a s s o c i a t e d t o a a c c o r d i n g t o t h e p r e v i o u s lemma, we s h a l l have w ( Q k , i n D k + l ) 5 6 W(Qk,i) a n d , summing o v e r i , w ( D ~ + ~I) Bw(Dk), which l e a d s t o

w(Dk) S

lDk+l1 S a\Dkl

and

kw ( D ) .

\Dkl B

m

I n

k=o

Dkl

= l i m lDkl

k+m

=

0.

Of c o u r s e , we a l s o have

R CY ID,\,

Then:

which i m p l i e s t h a t

399

IV. 2. REVERSE HOLDER'S INEQUALITY

I f we take E > 0 s m a l l enough to have (2" a will have a finite sum and we shall get: I

w(x)'+'ddx

2 C X:(w(Q\Do)

+

-') B < 1 ,

w(Do))

the series

C wE w Q) Q

=

Thus +c

'

0

Lemma 2.5 has far reaching consequences which we shall presently see w 6 A with 1 < p < THEOREM 2.6. I-f P s u c h t h a t w € Aq; that i s , for every Ap= A 4' 9
u

m,

t h e n t h e r e is some

p,

1 < p

m,

q

ue h a v e

Proof: Theorem 1.14(c) for the special case u = w tells us that On the other hand, from lemma w 6 A implies w -1/(P-1) 6 A P' . P 2.5 for the weight w -l'(p-l) we know that there exist c > 0 , C > 0 such that, for every cube Q:

But

1 1+E > p-l p - 1 implies

l+E p-l-q-l for some

Actually, since w itself satisfies following stronger result.

3

K.Il.I.,

T I E O R E M 2.7. i fw 6 A 1+E p E > 0 such t h a t w € Ap.

<

m,

1 5

11

1

c

q < p.

Then

we obtain the

then,

Ih~7i-e c x i s L s

c

p

400

I V . WEIGHTED NORM INEQUALITIES

Proof: I f enough :

p = 1

1,

it s u f f i c e s t o observe t h a t f o r

w(x)l+'dx

p > 1,

I

5 (A w ( x ) d x ) 1 + E 2 c w(x) 1 + E IQI

Q

s m al l

for a.e. x € Q

w 1+E 6 A 1 .

i n s u c h a way t h a t If

> 0

E

it suffices t o take

small enough t o h a v e , a t t h e

> 0

E

same t i m e :

and

O f c o u r s e , theorem 2 . 7 combined w i t h p a r t b) o f theorem 1 . 1 4 , g i v e s

theorem 2 . 6 . Now, w i t h t h e h e l p o f t h e o r e m 2 . 6 , w e c a n i m p r o v e t h e o r e m 2 . 1 a s anticipated, obtaining

Let w

THEOREM 2 . 8 .

be a w e i g h t on

R"

and l e t

1 < p <

m.

Then,

t h e foZZowing c o n d i t i o n s a r e e q u i v a l e n t :

a ) M i s o f weak t y p e ( p , p ) w i t h r e s p e c t t o w, t h a t is, t h e r e i s a c o n s t a n t C s u c h t h a t f o r e v e r y f u n c t i o n f t Lloc(Rn) 1 and e v e r y

t > 0

w({x € Rn : Mf(x) > t j ) 6 C t - P iRn b ) There i s a c o n s t a n t

f

>=

0

in

Rn

and e v e r y c u b e

c) w E A every cube

Q

P'

s u c h ttut f o r e v e r y

such t h a t f o r every f u n c t i o n

Q

t h a t is: t h e r e i s a c o n s t a n t

d ) ?I is b o u n d e d i n

C

C

If(x)IPw(x)dx

Lp(w),

f € Lp(w):

C

such t h a t f o r

t h a t i s , t n e r e is a c o n s t a n t

I V . 2 . REVERSE H O L D E R ' S I N E Q U A L I T Y

401

P r o o f : A l l t h a t remains t o be proved i s t h a t c ) i m p l i e s d ) . Here i s t h e p r o o f . We h a v e that

w 6 A

9

Since 1 < p < m, theorem 2.6 t e l l s us P' q < p . T h en M i s o f weak t y p e ( q , q ) w i t h

w E A

f o r some

i s a l s o bounded i n e q u a l i t y f o l l o w s from t h e f a c t t h a t 0 < w(x) < m respect t o

w

and,

since

M

Lm(w)

=

(this

La

for a.e.

x), the

Marcinkiewicz i n t e r p o l a t i o n theorem allows us t o conclude t h a t

i s bounded i n

Lp(w)

.

M

0

A l s o , t h e r e v e r s e H E l d e r ' s i n e q u a l i t y a l l o w s u s t o g i v e a more p r e c i s e v e r s i o n o f lemma 2 . 4 .

THEOREM 2 . 9 . 6 > 0,

Lf

C > 0

w E A f o r some p 6 [ l , m ) , then there e x i s t P s u c h t h a t , e v e r y t i m e we h a v e a m e a s u r a b l e s e t A

contained i n a cube

Q,

the following inequaZity holds:

(2.10)

P r o o f : The k e y f a c t i s t h a t w s a t i s f i e s a n i n e q u a l i t y l i k e t h e o n e a p p e a r i n g i n lemma 2 . 5 f o r some E > 0 ( R . H . I . ) . We s t a r t b y u s i n g 1 + ~a n d i t s c o n j u g a t e

Hijlder's i n e q u a l i t y w i t h exponents and t h e n w e a p p l y t h e R.H.I.

which i s (2.10) w i t h

(l+E)/f,

We g e t

6 = €/(I+€).

C o n d i t i o n ( 2 . 1 0 ) i s known a s

Am

s o o n . We a l s o s p e a k o f t h e c l a s s

f o r r e a s o n s which w i l l a p p e a r v e r y Am

which i s , n a t u r a l l y , t h e c l a s s

f o r m e d by t h o s e l o c a l l y i n t e g r a b l e w e i g h t s

w

satisfying the

Am

condition. For t h e n e x t r e s u l t ,

pl

and

u2

a r e going t o be doubling

measures, t h a t i s , both s a t i s f y a doubling c o n d i t i o n l i k e (1.13) i n Chapter 11. For t h e s e measures, w e g i v e t h e f o l l o w i n g d e f i n i t i o n :

I V . WEIGHTED NORM I N E Q U A L I T I E S

402

p,

i s comparabze t o

when t h e r e e x i s t

p2

a,B

t h a t , e v e r y t i m e we h a v e a m e a s u r a b l e s u b s e t

Q with

uZ(A)/p2(Q) 5 a,

i t foZZows t h a t

A

< 1

of a c u b e

p1(A)/p,(Q)5B.

W ith t h i s d e f i n i t i o n we c a n w r i t e

THEOREM 2 . 1 1 ,

rable set

The f o l Z 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

C > 0 a ) There e x i s t 6 > 0 , A contained i n a cube Q P 2 (A) p21Q)

' '(v) ~1 (A)

bl p2

i s comparable t o

el p l

is comparable t o

dl dp2(x) = w(x)dpl(x)

f o r some

E

that

a

and

6

p1 p2

with:

> 0

is c l e a r . Indeed, i f

P r o o f : a ) =*b)

I C a6.

p2(A)/p2(Q)

s u c h t h a t f o r e v e r y measu-

pl(A)/pl(Q)

6 a,

I t s u f f i c e s t o s t a r t w i t h some

Ca6 < 1 a n d we o b t a i n 6 B = Ca .

p2

comparable t o

p1

it w i l l be a > 0 such

with constants

b ) 3 c ) . T o s a y t h a t p l ( . 4 ) / v l ( Q ) <= a i m p l i e s p 2 ( A ) / p 2 ( Q ) is equivalent t o saying t h a t p2(A)/p2(Q) > implies

U ~ ( A ) / I J ~ ( Q )> a .

Then i f

u2(A)/u2(Q)

5 (1-8112 < 1 - B ,

2

f3

i t w i l l be

p2(Q\A)/p2(Q) > 6, i n s u c h a way t h a t p 1 ( Q U ) / p l ( Q ) > CI a n d , c o n i s coms e q u e n t l y p l ( A ) / p l ( Q ) < 1 - a . T h u s , we h a v e s e e n t h a t y l parable t o p2 with constants a' = ( 1 - B ) / 2 and B ' = 1 - a . I t becomes c l e a r t h a t , a c t u a l l y , b ) an d c ) a r e e q u i v a l e n t . Le t u s s e e now t h a t b )

We s t a r t f r o m t h e f a c t t h a t

p2 i s compa6 . We s e e , f i r s t o f a l l , t h a t p2 is absolutely continuous with respect t o p , , that is: U l ( E ) = 0 => p 2 ( E ) = 0 . Once t h i s i s p r o v e d , t h e Radon-Nikodym theorem g u a r a n t e e s t h a t d u 2 ( x ) = w ( x ) d p l ( x ) w i t h w l o c a l l y i n Let p l ( E ) = 0 a nd s u p p o s e t h a t tegrable with respect t o p l . v 2 ( E ) > 0 . S i n c e t h e measures a r e r e g u l a r , t h e r e w i l l b e an open s e t R s u c h t h a t R 3 E and p 2 ( R ) < B - l u 2 ( E ) . Let R = Q.

rable t o

pl

3

d).

with constants

a

and

uj

. I

HBLDER~SINEQUALITY

IV. 2. REVERSE

403

where the

Q J s are non-overlapping cubes. Since for each j is o = E ) 5 C X U ~ ( Q ~ ) ,we shall have w2(Qj fl E ) <= B U ~ ( Q ~ ) and, adding in j : u2(E) $ @p2(G), which contradicts the election of R. We have used the fact that the faces or edges of the cubes have measure p 2 (or p, for that matter) equal to 0. This follows easily from the doubling condition. Indeed, if 1~ is doubling, there is a constant k < 1 such that if Q is a cube and R is a half of Q (that is R is an interval contained in Q having one side equal to a half the corresponding side of Q and the other sides coinThe proof o f this is ciding with those of Q) then u ( R ) $ ku(Q). essentially the same that the argument we used in chapter I1 (after theorem 1.12) to guarantee that the Calderdn-Zygmund decomposition makes sense for a doubling measure. Viewing a face of Q as intersection of the R!s resulting from repeatedly dividing by 2 a side 3 of Q , we see that a face has 1-1 measure 0 for u doubling.

ul(~jn

Let dp2(x) = w(x)dul(x). I t remains to s e e that the inequality in d) holds. All we have to do is to repeat the proof of lemma 2.5. with p l in place of Lebesgue measure. Observe that in the proof of lemma 2.5 we just used these two facts: w(x)dx is comparable to Lebesgue measure and Lebesgue measure is doubling. These hypotheses and dul(x). Thus, we obtain still hold for du2(x) = w(x)dp,(x) the inequality in d). Finally we have to see that d) implies a ) . But this is done exactly 0 as in the proof of theorem 2.9. COROLLARY 2.12. T h e c o m p a r a b i Z i t y of m e a s u r e s is a n e q u i v a l e n c e r e Zation.

Proof: The equivalence between b) and c) in 2.11 tells us that comparability is a symmetric relation. 'Transitivity is proved very simply by using the characterization given by a ) in 2.11. COROLLARY 2.13. __ L e i w(x) 2 0 b e localZy i n t e g r a b l e i n following conditions arc? e q u i v a l e n t : il w € A

P

w(E) I Bw(Q)

whenever

a,R

E

is

T&

p € [1,m)

for some

iii Them, exist

Rn.

1

< (i

such t h a t

IE( 5 n\Q(

implies

m e a s u r a b l e subset of t h k c u b e

Q.

404

I V . WEIGHTED NORM INEQUALITIES

i i i l There e x i s t

and

> 0

E

C > 0

such t h a t f o r e v e r y

cube Q -

Proof: A l l t h e implications i )

3

i i ) =*

iii)

-

iv)

have a l -

r e a d y b e e n p r o v e d . Observe t h a t t h e p r o o f o f lemma 2 . 5 a c t u a l l y yields the fact that i i ) i i i ) . I t o n l y remains t o s e e t h a t i v ) d i ) . Let u s s e e i t . We know from theorem 2 . 1 1 t h a t w E A,

is

e q u i v a l e n t t o s a y i n g t h a t t h e m e a s u r e s dx and w(x)dx a r e compar a b l e a n d , t a k i n g i n t o a c c o u n t t h a t dx = w ( x ) - ' w ( x ) d x , the following R.H.I. must h o l d : T h e r e e x i s t q > 0 and C > 0 s u c h t h a t f o r e v e r y cube Q

5

&-i,

w(x)-lw(x)dX =

c

i4.7

Hence

Setting

rl =

-,1

p

that is:

Thus we have shown

A,

given t o condition (2.10).

=

=

u

l+rl - 1 > 1 , Ap,

we o b t a i n

w

E A

P'

which e x p l a i n s t h e name

0 A,

1 _-< p < m

A c t u a l l y , t h e name A, i s j u s t p e r f e c t , s i n c e , a s we s h a l l p r e s e n t l y show, Am c o i n c i d e s w i t h t h e f o r m a l l i m i t o f c o n d i t i o n AP as p tends t o

The l a s t i d e n t i t y above i s a s i m p l e e x e r c i s e i n m e a s u r e t h e o r y ( s e e , f o r example Rudin [ l ]

to

,

c h a p t e r 111, p r o b l e m 5 ) .

Thus, t h e c o n d i t i o n o b t a i n e d by p a s s i n g t o t h e l i m i t a s in condition A is: P l o g ( w ( x ) - ' ) d x ) 5- C (1w ( x ) d x ) IQI Q

m

1

p

tends

405

1v.2. REVERSE H~LDER'S INEQUALITY

or, equivalently:

I

1

(2.14)

IQI

w(x)dx

Q

5

C exp(- 1

IQI

log w(x)dx)

Q

The exponential in the right hand side of (2.14) is the geometric mean of w on Q, which is, of course, dominated by the arithmetic mean wQ (Jensen's inequality). Thus (2.14) implies that the arithmetic and the geometric means of w on every cube, are equivalent. The equivalence between this condition and Am is contained in the f01 lowing

L e t w(x) 2

THEOREM 2 . 1 5 .

0

Rn.

be l o c a l l y i n t e g r a b Z e i n

Then,

t h e foZZowing c o n d i t i o n s a r e e q u i v a l e n t :

a,B e (0,l)

i l There e x i s t I{X E

Q

: W(X)

5

such t h a t , f o r every cube

"WQ}~

Q:

5 BlQl

i i l w e Am i i i l There e x i s t s

C,

such t h a t , f o r e v e r y cube

Q:

Proof. Suppose i) holds. Let us prove ii). After the proof of theorem 2 . 1 1 , it will be enough to see that, for appropriately chosen y,6 e ( O , l ) , the following property holds: If E is a subset of a <= y , then lEl/lQl <= 6. To prove cube Q such that w(E)/w(Q) this property, assume w(E)/w(Q) 5 y , to be chosen later. Then we split E = E l u E2, where E l = Cx e E : w(x) > a w 1 and Q E2 = Ix e E : w(x) 5 a w 1 . For E2, i) gives the estimate Q lE21 5 BlQl. For E, we use Chebichev's inequality to get:

If we choose Adding up the two estimates, we have IEl 5 ( B + ): I Q I . y s o small that 6 + < 1, we get what we wanted with 6 = B + (y/a). To see that ii) implies iii) is quite easy. Indeed, if w e A m , it follows from corollary 2.13 and theorem 1.14 a), that there is a constant C such that, for all p large enough and for all cubes Q:

Letting

1

p

tend to

m

we obtain iii).

IV. WEIGHTED NORM INEQUALITIES

406

Finally, assuming iii), we are going to see that i) holds. Take a cube Q. Dividing w by an appropriate constant, we see that,

JQ

without loss o f generality, we may assume that wQ 5 C.

and, consequently, Then, with I{x e

Q

A >

: W(X)

I

<

still undetermined, we have:

0

5 All log(1

IIx

= +

E

Q

2 log

: l o g ( 1 + w(x)-’)

w(x)-l)dx

=

Q

=-

1 +w (x) log 7 dx w x

-

since, by assumption,

w

log w(x

=

1 lOg(l+X-’)

log w(x)dx

By using the simple inequality < C, we get:

log(1 + JQ

0.

=

log(l+w) 5 w

and the hypothesis

Q =

C

(Q(

6 1 ( Q ( if X

is small enough.

log(l+A-l) In particular: I{X

if

ct

e

Q

: W(X)

5

C1 Wg)(

2

[IX E

Q

: W(X)

2 C 0.11

is small enough. We have obtained i) with

6

=

(l/Z)lQ( 1/2.

0

In chapter I1 (theorem 3.4) we gave examples of A, weights, namely those functions w of the form w ( x ) = (Mp(x))Y where 1-1 is a positive Bore1 measure such that Mp(x) < m for a.e. x e R” and 0 < y < 1. We used this result to show that 1x1‘ is an A1 weight in R” if and only if - n < ct 5 0 . for W(X)

Starting with A1 weights one can easily generate Ap weights 1 < p < m . Let wo,wl E A, in Rn, and let 1 < p < a. Then = w0(x).wl (x)’-~ is and A weight. Indeed P

IV.2 , REVERSE H ~ ~ L D E RINEQUALITY ~S

407

We shall show in section 5 that every Ap weight w is actually of for some wo,wl e A1 (factorization the form w(x) = wo(x)w,(x)l~p theorem). F o r the time being, we shall content ourselves with giving examples of A weights. If - n < c1 5 0 and - n < B 2 0, (x(alx/B(l-P) is an A weight in Rn. Hence lxl’ is an P AP weight in Rn if and only if - n < c1 < n(p-1) since 1x1’ and have to be locally integrable.

(IX~~)-~’(~-~)

By using the R.H.I. we get a converse of theorem 3 . 4 in Chapter 11, giving the following characterization of A1 weights: THEOREM 2 . 1 6 . Let w(x) be LO and f i n i t e a . e . Then, w E A1 i-f a n d o n l y if w(x) = k(x)(Mf(x))’, w h e r e k(x) 0 i s such that k,k-’ E Lm, f i s l o c a l l y i n t e g r a b l e and 0 < y < 1 . Proof. Theorem 3 . 4 of chapter I1 implies that every function of the given form is an A1 weight. Conversely, let w e A 1 . We know that w satisfies a R.H.I.:

I

(L w(x)~+€ dx) IQI

o

C <- -

l/(l+E)

IQI

-

1

w(x)dx

5 C w(x)

a.e.

0

1 +E

~ h u s w(x) z ( ~ ( w )(x)~”(’+~) 5 c w(x). We can write W(X) = k(x)(M(~”~)(x))”(~+~) with C-l 5 k(x) 2 1 and we obtain the representation required with f(x) = w(x)~+€ and y = l / ( i + ~ ) . n U There is a relation between weights and B . M . O . functions. We have already seen in Chapter I 1 (theorem 3 . 3 ) that the logarithm of an A, weight is a B.M.O. function. We shall see presently that the same is true for any Am weight. Of course this follows trivially after the factorization theorem, but a simple proof can be given without appealing to that result which we have not proved yet. First of all, we give a characterization of A weights in terms of their P logarithms. THEOREM 2.17. a) Let Rn and l e t 1 < p <

@ 03.

be a r e a l l o c a l l y i n t e g r a b l e f u n c t i o n on

TheM

e@

two c o n d i t i o n s a r e s a t i s f i e d :

ii

b)

1 -

I91

1, @

(@(XI

e

-aQ) dx

as i n a),

5- C ,

e@

E Am

E

A

P

i f and o n l y i f t h e f o l l o w i x

i n d e p e n d e n t of t h e c u b e

Q

U n d o n l y if il h o l d s . N o t e

IV. WEIGHTED NORM INEQUALITIES

408

that for

p

c o n d i t i o n i i l becomes e m p t y , so t h a t b l i s j u s t an

= m,

p

e x t e n s i o n of a) t o

= m

c) I t foZZows from a) and b ) t h a t pnd w-”(p-l) a r e i n A,.

w

is in

Ap

i f and onZy i f

both w

Proof. a) It is clear that the two conditions i) and ii) imply together that e $ e A since P’

Conversely, suppose that

e$ e A

Then

P’

Also

b) Theorem 2.15 implies that

I

e@ e A,

if and only if

1

e@(x)dx 5 C e@Q, which is equivalent to condition i). Q c) I t follows from b) that, f o r w = eo, condition i) is equivalent to saying that w e Am and condition ii) is equivalent to saying that w -’/(P-’) e Am, 0 IQI

In case

-+I

p e

=

$(XI

2,

-4

conditions i) and ii) become:

Qdx 5 C

and

J-1 IQI

e

-

(4

-4Q)

IQI Q Q These two inequalities together are equivalent to

We can write:

dx 5 C.

IV. 2 . REVEP.SE H ~ L D E R I S INEQUALITY

COROLLARY 2 . 1 8 .

Rn.

Then

Let

409

b e a r e a l l o c a l l y i n t e g r a b l e f u n c t i o n on

eQ e A 2 i f and o n l y i f t h e r e i s a c o n s t a n t Q c Rn

C

such t h a t

f o r e v e r y cube

-

The r e l a t i o n b e t w e e n w e i g h t s and B . M . O . COROLLARY 2 . 1 9 .

w

E

A,

f u n c t i o n s i s now c l e a r .

l o g w e B.M.O.

P r o o f . L e t w f: A, and w r i t e w = e Q , t h a t i s : w e A 2 , w e know from c o r o l l a r y 2 . 1 8 t h a t

so t h a t

Q

=

Q

=

log w.

If

l o g w e B.M.O.

w e A, w e A f o r some p E [ l , , ) . If p P we h a v e w e A C A 2 a n d , a s w e have j u s t s e e n , l o g w E B.M.O. P I t follows t h a t p > 2 we l o o k a t w - l ’ ( p - l ) E A ,C A 2 . P 1 1 0 g ( w - ” ( ~ - ~ ) )= -logw eB.M.0. Thus, i n any case, P- 1 l o g w e B.M.O. In general

Observe t h a t , i f the

A

P If

constant for

w w.

A

E

l l o g wI*

P’

d e p e n d s o n l y on

p

2

2, If

a n d on

Q e B . M . O . , w e know t h a t e X 4 E A 2 f o r X > 0 small enough ( 0 < X < C 2 / l Q l * w i t h t h e n o t a t i o n u s e d i n C h a p t e r 11, 3 . 1 0 ) . I f we s e t e X Q = w , we g e t Q = X - ’ l o g w . Thus B.M.O.

= {a

log w : a

A c t u a l l y , t h e same i s t r u e f o r a n y

p

2

0,

with

2

w

E

A2)

1 < p

5

w e A 1 P We a l r e a d y know t h a t t h i s i s t r u e i f p 2 2 . For 1 < p < 2 , if e B . M . O . , we c a n w r i t e Q = a l o g w w i t h a 2 0 and w E A 2 . But u = wP-’ E A s i n c e 2 ( p - l ) + l - ( p - l ) = p ( s e e theorem 1 . 1 4 P p a r t b ) ) . Therefore Q = a log w = a log(u l / ( P - l ) ) = (a/(p-1))logu. B.M.O.

= {a

log w : a

0,

I n c o n t r a s t t o t h i s s i t u a t i o n , we h a v e { a log w :

a 2 0,

w

f

A1) = B . L . O .

I n f a c t , we a l r e a d y know t h a t

ci

$

B.M.O.

log w e B.L.O.

when

a

2 0,

( s e e t h e proof of theorem 3 . 3 . i n Chapter 1 1 ) . Conversely, l e t

w

E

A1

410

IV. WEIGHTED NORM INEQUALITIES

$ e B.L.O. Then, a c c o r d i n g t o C o r o l l a r y 3 . 1 0 ( i i ) i n C h a p t e r 11, we h a v e f o r E > 0 s m a l l enough, e v e r y cube Q and g i v e n C:

which i m p l i e s

IQ

1

e E @ ( X ) d x5 C e x p (E$ ) < C expIE(C + e s s . Q i n f .

Q -

< c -

ess. inf.(eE$)

e E @f A 1 .

I t follows t h a t

$11

Q

Thus,

$ =

log w

E-'

with

w = e E @E A1.

Combining t h i s w i t h theorem 2 . 1 6 , which t e l l s u s t h a t e v e r y

w E A1 c a n b e w r i t t e n a s w(x) = k ( x ) ( M f ( x ) ) y , w i t h k ( x ) 2 0 s u c h t h a t l o g k e Lm and 0 < y < 1 , we a r e l e d t o : B.L.O.

=

{ h + 8 log(Mf) : h

f

Lm,

f e Lloc, 1

6 2 0)

We f i n i s h t h i s s e c t i o n by o b s e r v i n g t h a t t h e Lp i n e q u a l i t y e s t a b l i s h e d i n c h a p t e r I1 ( t h e o r e m 3 . 6 ) b e t w e e n t h e H a r d y - L i t t l e w o o d maximal f u n c t i o n Mf and t h e s h a r p maximal f u n c t i o n f ' , also h o l d s when Lebesgue measure dx i s r e p l a c e d by t h e measure w ( x ) d x , where w i s any Am w e i g h t . The c o n c r e t e s t a t e m e n t i s a s f o l l o w s THEOREM 2 . 2 0 .

Mf E L

PO

(w)

such t h a t

L e t

w e A,

f o r some

po 5 p <

po

in

Rn and l e t

with 0

< po <

f

be such t h a t

w ,f o r

m.

every p

m

P r o o f . I t i s a r e p e t i t i o n o f t h e argument we u s e d t o p r o v e t h e o r e m 3 . 6 i n c h a p t e r 11. F i r s t of a l l , t h e Calder6n-Zygmund d e c o m p o s i t i o n can be c a r r i e d o u t f o r o u r f . I n d e e d i f QIC Q2C ... i s a n i n c r e a s i n g f a m i l y o f d y a d i c c u b e s f o r e a c h o f which 1 -

I

f(y)dy > t ,

then

w(Qk)

i s bounded i n d e p e n d e n t l y o f k a n d ,

IQk Qk since {Qk}

w(x)dx

i s a doubling measure, t h i s implies t h a t t h e sequence

is finite. We c o n s i d e r as i n t h e o r e m 3 . 6 . o f c h a p t e r I 1

cubes

{Qt,j}.

Given

t > 0

we f i x

t h e family of

Qo = Q - n - l 2

and t,jo

take

41 1

I V . 3 . WEIGHTS FOR SINGULAR INTEGRALS

A > 0.

P r o c e e d a s i n c h a p t e r 11. Then a f t e r we have o b t a i n e d t h e

estimate

I:

l Q t , j \ 5 2A-’IQol

j : Q t , j C Qo we u s e t h e f a c t t a t

c

w E Am

t o c o n c l u d e t h a t , f o r some

6 > 0

and

> 0,

Adding i n a l l t h e p o s s i b l e

Qots we o b t a i n t h e e s t i m a t e

A f t e r t h i s , t h e proof c o n t i n u e s p r a c t i c a l l y unchanged.

0

3 . WEIGHTED NORM INEOUALITIES FOR SINGULAR INTEGRALS Only f o r r e g u l a r s i n g u l a r i n t e g r a l s ( s e e 5 . 1 7 and 5 . 1 i n c h a p t e r I 1 f o r t h e d e f i n i t i o n ) , a s a t i s f a c t o r y t h e o r y can be given i n terms of t h e A c l a s s e s . We s t a r t by u s i n g i n e q u a l i t y 5.20 ( i ) from P chapter I 1 t o prove t h e following THEOREM 3 . 1 . L e t T be a r e g u l a 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 , 1 < p < m and w E A T h e n T i s bounded i n Lp(w). More p r e c i P‘ s e l y , t h e r e is a c o n s t a n t C d e p e n d i n g o n l y on T, w a n d p _ .

m

f E Lc ( t h a t is: f w i t h compact s u p p o r t ) , t h i s i n e q u a l i t y h o l d s : t h a t , f o r every function

1,.

I T f ( x ) l P w(x)dx 5 C

P r o o f . We s h a l l show below t h a t i f

i s a bounded f u n c t i o n

jRn I f ( x ) I P w ( x ) d x f e L,:

then

M(Tf) e L p ( w ) .

Once t h i s i s g r a n t e d , we c a n u s e t h e o r e m 2 . 2 0 t o w r i t e :

jRn I T f ( x ) I p

w(x)dx I

(M(Tf) ( x ) I p w(x)dx 5

q > 1 small p > 1 ( t h i s c a n h e done a c c o r d i n g t o enough f o r h a v i n g w E A P/q, a theorem 2 . 6 ) . Then t h e l a s t i n e q u a l i t y f o l l o w s r e a d i l y .

We have u s e d i n e q u a l i t y 5.20 ( i ) from c h a p t e r I 1 w i t h a

412

I V . WEIGHTED NORM INEQUALITIES

To f i n i s h t h e p r o o f we j u s t need t o c h e c k t h a t M(Tf) e L p ( w ) f o r f e L z . I t s u f f i c e s t o show t h a t Tf e Lp(w), T h i s c a n b e done i n t h e f o l l o w i n g way. L e t R > 0 b e s u c h t h a t f ( y ) = 0 f o r a l l y w i t h ( y ( 2 R . Then f o r 1x1 > 2 R , t h e f o l l o w i n g e s t i m a t e holds :

Now we w r i t e

since

'r

ITf(x) (pw(x)dx

Tf(x) (pw(x)dx

=

Rn

t

i,X,<2R

T f ( x ) pw (x) dx and s e e t h a t b o t h i n t e g r a l s i n t h e r i g h t hand s i d e a r e f i n i t e . F o r every M > 0 p (1 +1/ E)

E/

(1 + E)

j[xl<M ( \ l x l < Mw(x) l + E d x ) l / ( l + E ) The r i g h t hand s i d e o f t h i s i n e q u a l i t y i s f i n i t e i f E > 0 i s c h o s e n i n s u c h a way t h a t w s a t i s f i e s a r e v e r s e H U l d e r ' s i n e q u a l i t y w i t h exponent 1 + E ( s e e 2 . 5 ) , s i n c e we know t h a t T i s bounded in particular for i n L4(dx) f o r e v e r y q w i t h 1 < q < m , q = P(l+l/E). Now, i n view o f t h e e s t i m a t e o b t a i n e d f o r I T f ( x ) J 1x1 > 2 R , we j u s t need t o o b s e r v e t h a t i f w E A and P have : dx <

(3.2)

when p > 1,

m

JO<M<

Indeed I 2 k- M< m

5 <_ ( t a k i n g

1 < q < p

C

1

(2kM)-nP w(ZkB(0,M)) 5

k= 1 so t h a t

w

E A

9

and u s i n g lemma 2 . 2 )

we

IV.3. WEIGHTS FOR SINGULAR INTEGRALS

Since LT is dense i n t o t h e whole s p a c e Lp(w).

Lp(w),

T

413

c a n b e e x t e n d e d by c o n t i n u i t y

Theorem 3 . 1 s a y s n o t h i n g a b o u t t h e c a s e p = 1 . Of c o u r s e f o r p = 1 , t h e e s t i m a t e 5.20 ( i ) used i n t h e p r o o f , i s t o t a l l y u s e l e s s , s i n c e i t i n v o l v e s a q > 1 . However, we a r e g o i n g t o show t h a t i f w E A1, t h e n T i s o f weak t y p e ( 1 , l ) w i t h r e s p e c t t o w . The p r o o f w i l l b e b u t a r e p e t i t i o n o f t h e Calder6n-Zygmund p r o o f g i v e n f o r Lebesgue measure i n c h a p t e r 11. We s t a r t by e x t e n d i n g lemma 5 . 2 i n chapter I 1 t o our s i t u a t i o n . --____ LEMMA 3 . 3 . __ Let

a

w e i g h t and

K E

be a r e g u l a r s i n g u l a r i n t e g r a l k e r n e l ,

L 1 (w),

T h e n i f we s e t , a s u s u a l ,

with

C

Q

supported i n a cube

0=

Q2fi

K

&

d e p e n d i n g onZy o n

with

(Q

w

~n

A1 a(y)dy = O.

the following inequality holds:

w.

P r o o f . We c a n a s s u m e , f o r s i m p l i c i t y , t h a t

Q

is centered a t the

o r i g i n . Then, e x a c t l y a s we d i d i n c h a p t e r I 1 ( r i g h t b e f o r e lemma 5 . 2 ) , we c a n w r i t e

(note t h a t

a(x)

i s a c t u a l l y i n t e g r a b l e and t h e c o n v o l u t i o n makes

sense). Then we j u s t n e e d t o o b s e r v e t h a t I K ( x - y ) - K ( x ) Iw(x)dx 2 CMW(Y) 1x1 > 2 I Y I T h i s i n e q u a l i t y i s p r o v e d e x a c t l y a s ( 5 . 2 1 ) i n c h a p t e r 11, c h o o s i n g (3.4)

y instead of 0 t o evaluate

Mw.

S i n c e w E A 1 , we have Mw(y) 5 Cw(y) t y o f t h e lemma i s c o m p l e t e l y p r o v e d . 0

a.e.

and t h e i n e q u a l i -

We a r e now r e a d y t o p r o v e a s u b s t i t u t e r e s u l t o f t h e o r e m 3 . 1

for

p = 1.

THEOREM 3 . 5 .

w

f

A1.

Let T

Then

precisely,

T

b e a r e g u l a 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 , and

i s of

weak t y p e

t h e r e is a c o n s t a n t

such t h a t f o r every f u n c t i o n

C

(1,l)

with respect t o

d e p e n d d i n g o n l y on

f E L1(w)

and e v e r y

T

t > 0:

w.

&

More -

w,

41 4

I V . 3 . WEIGHTED NORM INEQUALITIES

Proof. We are assuming w E A 1 , This means that there is a constant C such that, for every cube Q, w(Q) 5 c w(x) for a.e. x e Q . Let f e L 1 (w) and t > 0. We ca ry out the corresponding Calder6n-Zygmund decomposition. If Q is a cube for which

Therefore, every dyadic cube over which the average of If1 is > t will be contained in a maximal one. These maximal cubes are the Cal der6n-Zygmund cubes { Q j ) . F o r each of them,

t
I f(x)

Idx

<=

2"t,

and outside of their union we know

that f(x) 5 t a.e. Next, w e obtain the Calder6n-Zygmund decomposi tion o f the function f. We write f = g+b where g, the "good part", is given by: g(x) = f(x) if x d R = Qj, and J

f(y)dy J

can be split as

b(x)

fQ

= =

1 j

x e Qj, while

if

j b.(x)

with

J

bj(x)

=

b,

(f(x)-f

In chapter 11, the corresponding result for w 5.9) was proved by using the following properties: a) T

is bounded in

the "bad part"

E

Qj

1

)X

Qj

(x).

(inequality

L'

d) lemma 5.2 in chapter 11.

J

I t turns out that these four properties extend to the case of a general w e A, i n the following way:

a!) Since w E A I C A for every p > 1 , theorem 3 . 1 tells us P that T is bounded in Lp(w) for every p > 1 .

IV.3. WEIGHTS FOR SINGULAR INTEGRALS

415

Now we c a n w r i t e a p r o o f e n t i r e l y a n a l o g o u s t o t h e o n e g i v e n i n t h e unweighted c a s e : w({x E R"

: [Tf(x)l > t ) )

+ w ( 6 ) + w({x

d

5

5 w({x

: ITb(x)l

: lTg(x)] > t/2))

e Rn

+

> t/Z})

We e s t i m a t e e a c h o f t h e s e t h r e e t e r m s . F o r t h e f i r s t one we u s e p r o p e r t i e s a ' ) ( w i t h some p > 1 ) t h a t [ g ( x ) l 2 2"t a . e . We g e t : w({x e Rn

'3 t 1,

: (Tg(x)l > t/Z}) 2

tp C -

Ig(x)lpw(x)dx =

C

5

and

b ' ) together with the f a c t

lRn

lg(x)[ Inn

C

ITg(x) Ipw(x)dx 5

(w)'-'

w(x)dx 2

C

5 t

lf(x)Jw(x)dx i,n The s e c o n d term o f t h e sum h a s b e e n a l r e a d y e s t i m a t e d , w i t h t h e same bound a s t h e f i r s t , i n c ' ) . As f o r t h e t h i r d , we h a v e : w({x k

lg(x)lw(x)dx

fi

: lTb(x)l > t/Z))

2 C

lRn,,

[ T b ( x ) Iw(x)dx

2

The n e x t t o l a s t i n e q u a l i t y i s a c o n s e q u e n c e o f lemma 3 . 3 and t h e l a s t o n e f o l l o w s from b ' ) s i n c e

b(x) = f(x)

-

g(x).

n

I f T is a regular singular integral operator with kernel K f E Lp(w) with w e A and 1 p < a, then t h e truncated P o p e r a t o r s T E , E > 0 , c a n b e d e f i n e d a c t i n g d i r e c t l y on f by means o f t h e f o r m u l a and

416

IV. WEIGHTED NORM INEQUALITIES

TEf(x)

K(X-Y) f(Y)dY

=

Indeed, this integral converges absolutely, since

I K(x-y) I

w (y) - p ' I p dy)

and it is clear that the right hand side is finite, since the second factor is bounded by a constant times W(X-Y) -"(p-l) Iy[-np' dy)l/pl, which is finite because (IlYl>E

W(X-Y) -"(p-l)

is, as a function of y , an A p l weight, and ( 3 . 2 ) holds for this function in place of w and p ' in place of p . Of course after theorem 3.1, Tf may be defined by density. But it is natural to ask whether a more explicit definition of Tf as limit of TEf f o r E + 0 is possihle. We already know that the answer to this question rests upon the study o f the boundedness properties of the maximal operator T*, defined as T*f(x) = s u p ITEf(x)[, with respecto to the weight w. E>O

We can use inequality 5 . 2 0 (ii), from chapter 11, obtaining the following result THEOREM 3 . 6 . Let 1 c p < m & w

T be a r e g u l a 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 ,

Then 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 a r i s i n g f r o m t h e t r u n c a t e d o p e r a t o r s TE i s bounded i n Lp(w). p r e c i s e l y , t h e r e is a c o n s t a n t C d e p e n d i n g on T & w, t h a t f o r e v e r y f e Lp(w): f

A

P'

f

iRn

(T*f(x))p

w(x)dx

5 C J R n If(x)

Proof. We observe that inequality for every f f: Lp(w) provided q just need to take q such that f c L:~~(R") as soon as w e A ~ / be granted by taking q > 1 close

Ip

5 C(

More

w(x)dx.

ii) in chapter I1 is valid is close enough to 1. Indeed, we e LToc(Rn), but I fIq E Lp'q(w) and we already know that this can ~ , enough to 1 . 5.20

Now we can write: (T*f(x))Pw(x)dx

T*

(Mqf(x))pw(x)dx

+

IV.3. WEIGHTS FOR SINGULAR INTEGRALS

by 2.8 and 3.1, if

q

has been chosen so that

The weak type inequality, p = 1, of theorem 3.6 is also V, 4.11. It is now clear that valid for f e Lp(w) with w

w

417

E A

P/q' 0 corresponding to the limit case, true. This will be proved in Chapter corollary 5.22 of Chapter I1 remains e AD, 1 5 p < m.

We have just seen that regular singular integrals satisfy the same weighted inequalities as the Hardy-Littlewood maximal function. The question arises naturally whether o r not the conditions imposed on w in theorems 3.1 and 3 . 5 are really necessary. We shall see next that the answer is positive. THEOREM 3.7. Let

w

pose t h a t each o f t h e

(p,p),

weak t y p e

b e a w e i g h t i n En a n d l e t 1 5 p < co. Supn R i e s z t r a n s f o r m a t i o n s R,, ,Rn i s of

...

with respect t o

w.

Then, necessariZy

w e A

P'

Proof: Let Q be a cube, and let f be a non-negative function living in Q, with f > 0. Denote by Q' the cube touching Q Q having the same side length as Q and such that ;j 2 yj for any x E Q', y E Q and j=1,2 , . . . ,n. Call Rf(x) = 1 R.f(x). Then, j=1 J for every x E Q' n ~ Q x.-y.) ( I 3 Ix-yl -n-l f(Y)dY 2 Rf(x) = cn . J=1

It follows that, for every

0 < t <

C f

9'

Q ' C {x E Rn : [Rf(x) I > t}. Now, since R is of weak type (p,p) with respect to w , we shall have: w(Q') 5 C t - p fp w. We get

5 C

(fQ)P w(Q')

lQ

(f(x))p

w(x)dx.

IQ

In particular, the choice

f = X Q leads to w(Q') 2 C w(Q). By the same token we have w(Q) 5 C w(Q'), from which we conclude that (fQ)p w(Q) 5

' IQ c2

(f(x))p

w(x)dx.

In other words:

w

E A

P'

n

The information contained in our previous results on weighted inequalities for singular integral operators is very often applied in the following useful formulation: COROLLARY 3 . 8 . Let f o r every

E

> 0

T

b e a r e g u Z a 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 . Then,

& 1

j/Tf(x)lP u(x)dx

< p <

w,

then inequality

5 CE,p Ilf(x)

Ip

M 1 + E u(x)dx

418

IV. WEIGHTED NORM INEQUALITIES

holds f o r arbitrary functions

f and u

0

f o r which t h e r i g h t

hand s i d e i s f i n i t e . T h e same i n e q u a l i t y i s v a l i d f o r t h e a s s o c i a t e d maximal o p e r a t o r

T".

'+'

is locally integrable, since otherProof: We assume that u(x) wise M,+E u(x) = +a at every point x. Then, by 2.16, M 1 + € u f: A , C A with A constant depending only on E , and P P theorem 3.1 gives jlTf(x)lP

5 I/Tf(x)Ip M 1 + E u(x)dx

u(x)dx

< C E,P IIf(X)

The inequality for

T"

Ip

5

MI+€ u(x)dx

follows in the same way from theorem 3.6. 0

The novelty in the inequality just proved (which should be compared with 2.12 in Chapter 11) stems from the fact that we try to estimate 11 Tf 11 for an arbitrary u(x) 2 0. In spite of that, LP(U) we are able to remove the T , the only price to pay for this being the change of u by another function, M 1 + € u , which is essentially of the same size (observe that M1+' is bounded in Lp for p > 1+~). A lot of information is contained in corollary 3.8; for example, by using it only when p = 2, all the Lp boundedness properties of T follow from Hslder's inequality plus duality. In this connection, see section V.6. We shall end this section by giving a weighted version of the Hsrmander-Mihlin multiplier theorem. THEOREM 3.9. Let a b e an i n t e g e r s u c h t h a t n/2 < a 2 n. S u p p o s e m f: Lm(Rn) i s o f c l a s s Ca o u t s i d e t h e o r i g i n and s a t i s f i e s

that

the condition:

(R-l

lDam(S)I2

dC)'l2

jR
ci

such that

5 CR-Ial, (a1

(0 < R <

m)

5 a. Then:

i l f-I n/a < p < w and w e ApaIn, m i s a m u l t i p l i e r f o r Lp(w), i n o t h e r w o r d s : t h e o p e r a t o r Tm g i v e n by (Tmf)"(t) = = m(c)?(S), i s bounded i n Lp(w).

ii) I f

1 c p < (n/a)l b o u n d e d i n Lp(w) . i i i ) ~fwnIa e A1,

W.

Tm

and w - 1 / (p-1 1 i s o f weak t y p e

f:

Ap,a/n,

(1 ,1)

Tm

i s again

w i t h r e s p e c t to

419

IV.3. WEIGHTS FOR SINGULAR INTEGRALS

Proof: i) We look for an a priori inequality for Tmf with f € S (:Bn). The notation will be as in section 6 of Chapter 11. serve that TNf(x) -+ Tmf(x) uniformly as N a. Indeed:

Oh

-+

IITmf - T

N

fl ,

=

Il((m-m

N

" v

If) [ I m 5

mN ( x )

N m , since -+ m(x) IIrnNIlm 5 C independently of N.

as

-+

II(m-m

as

N

-+

N A

)fU1

m

+

0

for every

x # 0

and

In general, we shall denote by Ilg(l the norm of the function PS W g in Lp(w). Since TNf(x) + Tmf(x) uniformly as N m, it will be sufficient to obtain an estimate N (3.10) IT flpyw 2 c Ifllp,w -f

with

C

independent of

N.

In the proof of theorem 6.10 in Chapter I1 we obtained a pointwise estimate N # (3.11) (T f ) (XI 2 Cq Mqf(x) valid for every q > n/a, with Cq independent of N. Even though this estimate was not written down explicitly, it was the key to prove the corresponding estimate for (Tmf)# appearing in the theorem. From (3.11) and theorem 2.20 we shall obtain (3.10). First we shall make sure that TNf e Lp(w), s o that theorem 2.20 can actually be applied. This is very simple because T N f is a continuous function and we only have to check that it has the right behaviour at m . For 1x1 large we have ,-

since (TNf)^ = m N f has derivatives in it is enough to observe that

L1

up to order

a.

Now

as it follows from ( 3 . 2 ) , since

w e ApaIn. we can write, using theorem Once we know that TNf E Lp(w), 2.20 and (3.11) with some n/a < q < p

(M( I f 19) (xj)p/q w(x) dx) l/p Of course p/q < pa/n. However q that w e A With this choice P/q'

can be chosen s o close to n/a

420

IV. WEIGHTED NORM INEQUALITIES

and

(3.10)

follows.

i i ) i s o b t a i n e d by d u a l i t y . W r i t i n g

u

=

that IlTmfll p,w =

IqTmf(x)

SUP

m dx

l l ~ l l p l

sup llgllpl ,cr5

=

'C

IIfllp,w

since p < t h e weight

n

($I

u

E

implies t h a t Apla/n.

p' >

n

a,

and we c a n a p p l y p a r t i ) t o

The p r o o f o f i i i ) i s v e r y s i m i l a r t o t h a t o f theorem 3 . 5 . L e t f e L 1 ( w ) . We a r e g o i n g t o s e e t h a t w({x e Rn : I T N f ( x ) I > t ) ) <= C t - ' Ilflll,w w i t h C i n d e p e n d e n t o f f , t > 0 and N . T h i s i s c l e a r l y enough. S i n c e w e A 1 , g i v e n t > 0 , we c a n p r o c e e d e x a c t l y a s i n t h e p r o o f o f theorem 3 . 5 , obt a i n i n g t h e CalderBn-Zygmund c u b e s { Q j l f o r f a t h e i g h t t . The c u b e s { Q . ) a r e d i s j o i n t and s a t i s f y : J

and (f(x)1 5 t Next we w r i t e

-

I

f = g+b,

f o r almost every where

g(x) = f ( x )

x d

U Qj.

if

j x d n =

u

Qi, j J 1 b.(x) j J

f(y)dy = fQ i f x e Q j , and b ( x ) = j IQjI Q ; J w i t h b . ( x ) = ( f ( x ) - f ) XQ (x) J Qj j Now we j u s t h a v e t o c h e c k t h a t t h e f o u r p r o p e r t i e s a t ) t o d ' ) i n t h e p r o o f o f t h e o r e m 3 . 5 c o n t i n u e t o h o l d f o r o u r w and t h e o p e r a t o r T~ g(x)

=

a ' ) T N i s bounded i n Lp(w) p r o v i d e d p > n / a . This follows from i ) s i n c e w E A I C ApaIn b l ) and c l ) a r e p r o v e d a s b e f o r e u s i n g t h e f a c t t h a t w e A 1 .

F o r d l ) we n e e d t o s e e t h a t lemma 3 . 3 h o l d s f o r o u r w and t h e k e r n e l K N o f T N . T h i s i s s e e n by showing t h a t KN s a t i f i e s (3.4).

IV.3.

WEIGHTS FOR SINGULAR INTEGRALS

421

The proof of [3.4) for KN and w such that wnla E A 1 will be be based upon inequalities (6.11) of Chapter I 1 with some q > n/a such that wq e At. We know that such a q exists after theorem 2 . 7 . With this choice of q we have:

II

XI > 2

I YI

IKN(x-y) - KN(x)

m

5 C

-

since

1

(2-')j

M w(y)

=

C M w(y)

j=1

M(w~)(Y)

5 ess inf w(x)q

lXl5lYl Once these four properties have been checked, the proof is idell tical to that o f theorem 3.5. 0 In the problem o f convergence of Fourier series and integrals in one variable, one is interested in the multipliers xI, I an interval of 8 , which give rise to the partial sum operators SI, (SIf)" = 3 xI. We know, see Chapter I1 (6.15), that S I is a multiplier in Lp(R) f o r all l < p c m (this fact is actually equi valent to M. Riesz theorem). Strictly speaking, S I is neither a singular integral operator nor a multiplier operator satisfying the conditions of theorem 3.9, due to the lack of regularity of XI at the end points of the interval I. Nevertheless, the weighted inequalities for SI are really an easy consequence o f known estimates : COROLLARY 3.12. Let

W(X)

0

be a weight i n R .

Then

a) If 1 < p < m, t h e p a r t i a l sum o p e r a t o r s SI a r e u n i f o r r n l x bounded ( f o r a l l i n t e r v a l s I ) Lp(w) i f and o n l y i f w e A

P'

b) The o p e r a t o r s SI a r e u n i f o r m l y of weak t y p e r e s p e c t t o w i f and o n l y i f w f A 1 . c) F o r a l l

f e Lp(w), Ils[_R,R]

with w

'

E

A

1 < p <

and

P - fllp,w * O ,

*

(1,l)

m,

with

IV. WEIGHTED NORM INEQUALITIES

422

P r o o f : Let

be t h e a n a l y t i c p r o j e c t i o n , d e f i n e d by ( A f ) ^ ( E ) = o r A f ( x ) = f ( x ) + i H f ( x ) , where H i s t h e H i l b e r t t r a n s f o r m . D e n o t i n g b y Ma t h e i s o m e t r y c o n s i s t i n g i n m u l t i p l y i n g e a c h f u n c t i o n f ( x ) by e 'Tiax, we have i S [a,-) = 'Z Ma AM-a =

A

t(5) X[,,,)(5),

= s[a,-) - s [b,m)

'[a,b)

i

=

7 (Ma

- Mb AM-b)

S i n c e H i s a r e g u l a 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 , t h e o r e m s 3 . 1 and 3 . 5 imply t h e s u f f i c i e n c y o f t h e c o n d i t i o n w E A i n p a r t s a ) and P b ) . F o r t h e n e c e s s i t y , we o b s e r v e t h a t Hf = i ( S ( - m ,o] f-S[o , m ) f ) and a p p l y theorem 3 . 7 ( s i n c e t h e r e i s o n l y one R i e s z t r a n s f o r m i n R', and i t i s p r e c i s e l y t h e H i l b e r t t r a n s f o r m ) . F i n a l l y , c ) f o l lows from a ) b e c a u s e t h e r e s u l t i s t r i v i a l l y t r u e f o r t h e f u n c t i o n s

f E L1(R) whose F o u r i e r t r a n s f o r m h a s compact s u p p o r t , and t h e s e 0 a r e d e n s e i n Lp(w).

4 . TWO-WEIGHT.NORM INEQUALITIES FOR MAXIMAL OPERATORS

The main g o a l o f t h i s s e c t i o n i s t o g i v e a s o l u t i o n o f p r o b l e m 2 ( p o s e d i n s e c t i o n 1 ) . T h i s s o l u t i o n w i l l b e b a s e d on i d e a s o f

E . Sawyer [ l ] and B . J a w e r t h [ l ]

.

I n o r d e r t o i n t r o d u c e t h e main i d e a , and a l s o f o r i t s own s a k e , we s h a l l p r e s e n t f i r s t a g e n e r a l r e s u l t o f B . J a w e r t h , which c o n t a i n s B . Muckenhoupt's theorem 2 . 8 w i t h a s i m p l e and e l e g a n t p r o o f . By a b a s i s i n Rn we s h a l l s i m p l y mean a c o l l e c t i o n 8 o f open s e t s i n R n . For a b a s i s 8 we s h a l l c o n s i d e r w e i g h t s w s u c h t h a t w(B) < 03 f o r e v e r y B E 8. Given B and w , w e s h a l l d e n o t e by M B S w t h e maximal o p e r a t o r s e n d i n g t h e m e a s u r a b l e f u n c t i o n f into the function defined as follows: I f x E B, Bet3 M f ( x ) = sup

u

,w

where t h e sup i s t a k e n o v e r t h o s e B E B s u c h t h a t x E B and w(B) > 0 . O t h e r w i s e MB,wf(x) = 0 . When w 2 1 , w e s i m p l y w r i t e Ma. S i n c e t h e s e t s o f B a r e o p e n , w e s e e t h a t M a f i s a l o w e r semicontinuous f u n c t i o n . We s h a l l s a y t h a t t h e w e i g h t

1 5 p

my

w

is in the class

i f and o n l y i f , t h e r e i s a c o n s t a n t

C

A

P,B' such t h a t :

423

IV.4. TWO-WEIGHT NORM INEQUALITIES

f o r every

B e B

(with t h e u s u a l i n t e r p r e t a t i o n of t h e second f a c -

t o r as

e s s . s u p (w-') when p = 1 ) . B Jawerth's r e s u l t is the following:

THEOREM 4 . 1 , Let 1 < p c m . CalZ

w a weight f o r B & T h e n , t h e foZZowing c o n d i -

B be a b a s i s i n Rn, U ( X ) = W ( X )- " ( p - ' ) .

t i o n s are equivaZent:

a ) MB

i s bounded b o t h i n

b ) w e A p,B;

in

and

LP(w)

MB,w

Lp(u)

i s bounded i n

MB,u

Lp'(u) i s bounded

~"(w).

P r o o f : We s h a l l s t a r t by showing t h a t b ) i m p l i e s a ) . A c t u a l l y w e j u s t n e e d t o s e e t h a t b ) i m p l i e s t h a t MB i s bounded i n Lp(w). i f and o n l y i f u E A The r e a s o n i s t h a t w E A P ' ,B' s o , P,B a s s u m i n g b ) , we want t o s e e t h a t MB i s bounded i n Lp(w) . Let f E Lp(w) . We may assume t h a t MBf(x) < w e x c e p t f o r a s e t o f n u l l w-measure ( t a k e , f o r i n s t a n c e , f b o u n d e d ) .

For every i n t e g e r Sk = {x

k,

we s h a l l c o n s i d e r t h e s e t k k+l Rn : 2 < MBf(x) 5 2

E

From t h e d e f i n i t i o n o f

MB,

SkC

Define

u

Bk

j

tisfies

Ek,l

=

Bk,ln

and, f o r

Sk

Ek,j

=

(Bk,j \

,j'

where

F.

B

sa-

j > 1:

u

s<j

Bk,s)

n

'k'

The s e t s Sk form a d i s j o i n t c o l l e c t i o n and e a c h for varying j . j o i n t union o f t h e s e t s E k , j

where

Bk,j

Sk

i s the d i s -

IV. WEIGHTED NORM INEQUALITIES

424

gk,j

=

(v I jBk,j

1

f(Y) I U(Y) -'~o~)dY)P

1 pk,j gk,j as an integral on a measure, space k,j ( X , p ) built over the set X = {(k,j)) by assigning to each (k,j) the measure pk,j ' For X > 0 , call

We view the sum

r(X)

=

G(X)

=

I

{(k,j)

We use this to estimate

: gk,j >

u

C

XI

Bk,j

(k,j)er(h) p(r(A))dX.

1 pk,j gk,j' k,j 0 valent to saying that there is Then

x

2 '

Observe that

w

E

A

such that, for every

P,B B e

is equi

B:

'k,j

Next we shall use the boundedness of

to estimate

p(r(X))

IV.4. TWO-WEIGHT NORM INEQUALITIES

425

= C I R n ( f ( x ) ( p w(x)dx

s i n c e u l - p -- w . plies a).

T h i s f i n i s h e s t h e proof o f t h e f a c t t h a t b) i m -

L e t u s p r o v e now t h a t a ) i m p l i e s b ) . F i s t t h e p r o o f t h a t t h e boundedness o f MB i n Lp(w) i m p l i e s w f A p , B , i s e x a c t l y a s t h e one g i v e n i n s e c t i o n 1 f o r t h e b a s i s o f c u b e s . Then, o b s e r v e t h a t , by symmetry, i t i s enough t o p r o v e t h a t M B , u i s bounded i n L P ( u ) . The p r o o f o f t h i s f a c t i s v e r y s i m i l a r t o t h e one g i v e n f o r t h e o t h e r h a l f o f t h e t h e o r e m . We t a k e f n i c e i n L P ( u ) and d e and E w i t h MB,a i n p l a c e o f MB. Then fine Sk, B k , j k,j

=

w(G(X)) 5 w({x

Finally

E

Rn : ( M B ( f a ) ( X ) ) p > A } )

IV. WEIGHTED NORM INEQUALITIES

4 26

I

m

5 C

w({x

E

Rn : (M (fo)(x))p

=

C

iRn

> h1)dX

=

B

0

5 C

IMB(fa)(x)lPw(x)dx

I

If(x)lPu(x)Pw(x)dx

=

R"

1 < p < m. Suppose t h a t t h e b a s i s 8 i s s u c h COROLLARY 4 . 2 . implies that M i s bounded i n Lp(u) that w e A 89 0 P 38 MB,w i s bounded i n Lp'(w). Then MB i s bounded i n Lp(w) Ji and o n l y i f w E A P,B'

A particular case of corollary 4 . 2 in Muckenhoupt's theorem, which was proved in section 2 (theorem 2 . 8 ) and is derived here in a completely different way. COROLLARY 4.3.

Then M

Let

1 < p <

i s bounded i n

w

Lp(w)

and l e t

w

i f and o n l y i f

be a weight i n w e A P'

Rn.

Proof: The reason i s that the basis Q formed by all the open cubes in Rn satisfies the hypothesis of corollary 4 . 2 . Indeed, if w E Ap,!! = Ap, then u e A and s o , lemma 2 . 2 guarantees that P' ' both w(x)dx and u(x)dx are doubling measures. Consequently, theorem 2 . 6 in Chapter I 1 implies that = Mu is bounded in LP(u) and M = Mw is bounded in Lp (w). 0 2,w Corollary 4 . 2 also applies to the basis D formed by the open dyadic cubes. (See Chapter I1 section 1 for the definition). Since the dyadic maximal functions will play a basic role in the sequel, it will be convenient to introduce short notations for them. We shall write N = MD and, in general Nw = M . Also, for R > 0 , we shall denote by N(R1, M(R), N F ) , M$Rypw the maximal operators obtained by taking in the corresponding definition just those cubes o f the basis whose side length is 2R. Here is a simple but useful observation about the dyadic maximal functions. LEMMA 4 . 4 . Let 1 < p 5 m and l e t w be a l o c a l l y i n t e g r a b l e weight i n Rn. Then Nw i s bounded i n Lp(w). Proof: The fact that w is locally integrable implies (as in the proof o f corollary 1 . 1 3 ) that Nw is bounded in Lm(w). Then, we

IV.4. TWO-WEIGHT NORM INEQUALITIES

427

just need to prove that Nw is of weak type ( 1 , l ) with respect to w and appeal to the Marcinkiewicz interpolation theorem to complete the proof of the lemma. Let us prove the weak type ( 1 , l ) inequality. Let f E L 1 (w) . Since Nwf(x) = lin NrSR) f(x) and NrSR) f(x) R+m increases with R, it will be enough to prove the inequality for N(R) with constant independent of R. But, after fixing R > 0 W and X > 0, observe that {x E Rn :NiR) f(x) > A } = Qj, where

u

Qj

are the maximal dyadic cubes of side length

$R

j

for which

J

These maximal dyadic cubes do exist because of the restriction on their size. Since they are disjoint, we get W({X

e Rn : N\SR)f(x)

>

XI) 6

6

w(Qj) 1

J

0

as we wanted to show.

COROLLARY 4.5. Let 1 < p < m , and l e t w b e a w e i g h t in Rn. T h e n N i s bounded i n Lp(w) i f and o n l y i f w e A that i s , P,U' w s a t i s f i e s t h e Ap c o n d i t i o n o v e r d y a d i c c u b e s . proof: Lemma 4.4 shows that corollary 4.2 applies to the basis

U . 0

Jawerth's approach to Muckenhoupt's theorem provides a proof which is independent of the reverse HElder's inequality. An even shorther proof has been given by M. Christ and R. Fefferman [ l ] . It is very similar to Jawerth's, but it uses Calder6n-Zygmund cubes. Here is an account o f it. ALTERNATIVE PROOF OF COROLLARY 4.3. p m and let w be an A weight in Rn. Take P hoose a constant a > 2". Then f o r each integer k , be the Calder6n-Zygmund cubes of f at height ak . Then, calling as usual a = w - 1 / (p-1) Qk,j \ Q,,,.

rt;lk

w(x)dx

5 ap

1 k,j

(---

1

lQk,j I

1

Qk,

f(y)dyIP

w ( E ~ , ~ )5

428

IV. WEIGHTED NORM INEQUALITIES

since

w

E

A

P’

Thus

The following general result of Jawerth [l] obtain a solution to problem 2 of section 1.

will allow us to

THEOREM 4.6. Let 8 b e a b a s i s , ( u , w ) a coupZe of w e i g h t s i n Rn and 1 < p < m . S u p p o s e t h a t a(x) = W(X)-”(~-’) i s such t h a t Mg,‘ i s bounded i n LP(a). Then t h e f o l l o w i n g two c o n d i t i o n s a r e equivalent:

a) Mg

LP(w)

i s bounded f r o m

2

Lp(u)

b) T h e r e i s a c o n s t a n t a union o f s e t s in

I,

8,

C such t h a t f o r every s e t t h i s inequaZity holds:

IMg(a xG)(x)

Ip

u(x)dx

G

which i s

<= c ‘ ( G I .

Proof: Implicit in the assumptions is the fact that o(B) < m for every B e B. That a) implies b) is almost immediate. If a) holds, we have an inequality

IRn

IMgf(x)IP u(x)dx 5 C If(x) IP In.(x)dx IRn valid for every function f. If we apply this inequality to the function f = o XG, we obtain:

jRn IMg(u XG)(x)lp

u(x)dx

5 C

o(x)p

w(x)dx

=

C o(G)

and

IG

b) follows. Next, assuming b), we shall prove a). Let f be a nice function in L p ( w ) . We proceed exactly as in the proof of theorem 4.1. Just like there we define the sets Sk, Bk,j and Ek,j and write

IV.4. TWO-WEIGHT NORM INEQUALITIES

429

where

1

The d e f i n i t i o n s o f

I f(y) 1 0

jBkYj

gkyj =

r(A)

and

G(A)

(y)

for

-'

A > 0

0 (y)dyIp

a r e e x a c t l y as i n

t h e v r o o f o f t h e o r e m 4.1. Then

5 C u({x

Rn : ( M , , , ( f / ~ ) ( x ) ) ~ > A ) )

E

"

where t h e n e x t t o l a s t i n e q u a l i t y i s t h e r e s u l t o f a p p l y i n g c o n d i t i o n b) t o the set

G(A)

=

Bk,j'

Now

( k , j )eT(A)

I

m

5

C

u ( { x E Rn

: ( M B y , ( f / a ) ( x ) ) P > A1)dA =

0

=

c

j R n I M B , ( ~f / o j ( x )

=

C

IRn

Ip

I f ( x ) l p w(x)dx.

a(x)dx 5

c

jRn [ f ( x ) I P a ( x ) l - p d x

We h a v e o b t a i n e d a ) .

=

n

Theorem 4.6 i s n o t i m m e d i a t e l y a p p l i c a b l e t o t h e b a s i s o f c u b e s b e c a u s e we c a n n o t c o n c l u d e from b ) t h a t Ma i s bounded i n L p ( o ) (a d o e s n o t h a v e t o b e d o u b l i n g ) . However f o r t h e b a s i s o f d y a d i c c u b e s D , t h e b o u n d e d n e s s o f Nu i s a u t o m a t i c , a c c o r d i n g t o lemma 4.4. I t w i l l b e t h r o u g h t h e d y a d i c c a s e t h a t we s h a l l b e a b l e condit o s o l v e problem 2 o f s e c t i o n 1 . A c t u a l l y , f o r t h e b a s i s D , t i o n b ) c a n b e w r i t t e n i n a s i m p l e r form. Here i s t h e s o l u t i o n o f t h e two w e i g h t s p r o b l e m f o r t h e d y a d i c maximal o p e r a t o r N .

Q

I V . WEIGHTED NORM INEQUALITIES

430

Let 1 < p c m and l e t ( u , w ) b e a c o u p l e o f w e i g h t s Then, t h e f o l l o w i n g two c o n d i t i o n s a r e e q u i v a l e n t :

THEOREM 4 . 7 .

in

Bn.

a) N i s bounded f r o m

Lp(w)

b) T h e r e i s a c o n s t a n t IN(o XQ) (XI I p

u

where

=

LP(u) s u c h t h a t , f o r e v e r y d y a d i c cube

C

Q:

u(x)dx 5 C a ( Q ) <

w‘Ql,(p-ll.

1,

P r o o f : F i r s t , we have t o s e e t h a t a ) i m p l i e s o ( Q ) < a f o r e v e r y Q . Indeed i f a(Q) = a we have (w(x)-lJ]rw(x)dx = m . I t follows that there exists =

1,

f(x)dx =

m.

f

E

Lp(w)

But t h e n

such t h a t Mf(x)

=

1,

w(x)-lf(x)w(x)dx

f o r every

x

E

Q,

=

and i t

f o l l o w s from a ) t h a t u(Q) = 0 . A c t u a l l y u ( R ) = m f o r e v e r y cube R 3 Q , s o t h a t u E 0 , a t r i v i a l c a s e which we b e t t e r e x c l u d e . Now, t o prove t h e theorem, w e j u s t need t o s e e t h a t c o n d i t i o n b ) e x t e n d s from d y a d i c cubes t o a r b i t r a r y open s e t s ( i . e , , u n i o n s o f d y a d i c c u b e s ) . Suppose b) h o l d s and l e t G be an open s e t . We s h a l l prove an i n e q u a l i t y P U

with

{Qk,jI length

C

x ) d x 5 C u(G)

i n d e p e n d e n t o f G and R . T h i s i s c l e a r l y s u f f i c i e n t . L e t be t h e maximal d y a d i c c u b e s c o n t a i n e d i n G and h a v i n g s i d e 2 R and s u c h t h a t 1

S i n c e a l l t h e cubes i n t h e doubly indexed f a m i l y { Q k , j I have s i d e length zR, e v e r y cube w i l l b e c o n t a i n e d i n a maximal o n e . Let {Qi}

be t h e s u b f a m i l y formed by t h e s e maximal c u b e s . Then

431

I V . 4 . TWO-WEIGHT NORM INEQUALITIES

What m a k e s i t p o s s i b l e t o e x t e n d t h e o r e m 4 . 7 t o t h e o r d i n a r y

i s t h e f o l l o w i n g e s t i m a t e by The a n a v e r a g e o f t r a n s l a t i o n s o f t h e d y a d i c m a xim a l o p e r a t o r N .

H a r d y - L i t t l e w o o d max i mal o p e r a t o r

M

i d e a goes back t o C. Fefferman and S t e i n [ l ] . LEMMA 4 . 8 . f

in

Rn

For e v e r y i n t e g e r

k,

e v e r y ZocaZZy i n t e g r a b Z e f u n c t i o n

x e Rn:

and e v e r y

-rtg(x) = g ( x - t ) .

where

Proof: ( T - t 0 N o - r t ) f ( X) =

(X+t) =

1

sup x e Q - t , QeD

1

IQI

f(z)dz Q-t

T - ~ O N * T i~s s i m p l y t h e o p e r a t o r

Thus

D-t

formed by t h e c u b e s

t r a n s l a t e s by - t

k, f

Given e x i s t a cube

Let

N('ftf)

1 sup f(y-t)dy = x+teQeQ I Q I Q

-

sis

=

j

course

j 5 k.

2j+'

with

Q

dyadic, t h a t i s , the

of t h e d y a d i c c u b e s . and

x,

by d e f i n i t i o n o f

5 2k

of side length

be an i n t e g e r such t h a t

t E Q(O,Zk+')

equal t o

R

Q-t

associated t o the ba-

such t h a t

x

E

R

2j-l < s i d e length of

Consider t h e s e t

R

I!c Q.

there w i l l

and

R 2 2j.

3f

consisting of those

f o r w h i c h t h e r e i s some and s u c h t h a t

k M(2 ) f ( x ) ,

Q e D-t For e v e r y

with side length t E R,

w e have:

Now i t i s a s i m p l e g e o m e t r i c a l o b s e r v a t i o n t h a t t h e m e a s u r e o f

R

432

IV. WEIGHTED NORM INEQUALITIES

'

L

3n+ 1

I

(T -

(Q(0,2k+2) 1

O N O T

jQ(0,zkt2)

t) f(x) dt .

0

We are finally ready to give our promised solution to the two weights problem for M. THEOREM 4.9. Let 1 < p < a , and l e t (u,w) b e a coupZe o f w e i g h t s i n Rn. T h e n , t h e f o Z l o w i n g t w o c o n d i t i o n s a r e e q u i v a Z e n t : a) M

i s bounded f r o m

LP(W)

b ) There i s a c o n s t a n t

1,

where

IM(0 XQ)(x)IP

C

LP(u)

to

such t h a t , f o r e v e r y cube

u(x)dx

2

C

Q:

o(Q) <

u = w- 1 / (p-1)

Proof: All we have to do is to show that b) implies a). But, after lemma 4.8, the boundedness of M is equivalent to the uniform boundedness of the operators T - ~ O N O T for ~ t E Rn. Indeed, once this uniform boundedness has been established, we can use Minkowski's integral inequality and the monotone convergence theorem to get the boundedness of M . Now, since

and JRn we see that the uniform boundedness of the operators T - ~ O N O T ~ between Lp W) and Lp(u), is equivalent to the fact that the couples ( T ~ u , T ~ w )satisfy condition b) in theorem 4.7 with a constant independent of t . But this fact follows quite easily from our condition b). Indeed, for any t e Rn and any dyadic cube Q, we have :

4 33

IV.5. FACTORIZATION AND EXTRAPOLATION

=

C(T~U)(Q).

0

This finishes the proof.

Condition b) in theorem 4.9 will be called condition on Sp the couple (u,w). It is obvious that implies Ap, since

s?

is stronger than . However, for u=w, But, of course, S P AP we know that S is equivalent to A Actually, it is very easy P P' to see directly that, in the case of equal weights, A implies P This is an observation of Hunt, Kurtz and Neugebauer [ l ] which Sp. we shall presently explain. It is clear that, for a function f living in a cube Q, the Hardy-Littlewood maximal function at a point x e Q coincides with the supremum of the averages of If1 over cubes contained in Q. Now, if R is a cube contained in Q and w E A we have

P'

so that IM(a XQ)(x)lp $ Clb!w(w-l XQ)(x) sequently, since w (x)dx is "doubling"

and (w,w) e S P' theorem.

I

for

x

E

Q,

and, con-

This gives yet another proof of Muckenhoupt's

5. FACTORIZATION AND EXTRAPOLATION

We have already seen (immediately before theorem 2 . 1 6 2 that if we have two A 1 weights wo and w 1 and if 1 < p < m , then w(x) = wo(x) W , ( X ) ~ - ~ is an Ap weight. Now we are going to show that, conversely, every weight w e AD can be written in this form for certain wo,wl e A 1 . This factorization theorem will have important consequences. The proof will be based on a simple lemma, which, as we shall see, provides a strikingly powerful method to deal with several problems about weights.

I V . WEIGHTED NORM INEQUALITIES

434

LEMMA 5 . 1 . Let S b e a s u b l i n e a r o p e r a t o r bounded i n L p ( u ) , w h e r e p 2 1 p i s a n a r b i t r a r y p o s i t i v e m e a s u r e o n some m e a s u r a b l e s p a c e . S u p p o s e t h a t S f 2 0 f o r e v e r y f f L p ( p ) . Then, f o r e v e r y

u 2 0

k

Lp(p)

there i s

6 v(x)

i ) u(x)

i i ) llvllp 2 2

v

2 0

@

Lp(p)

such t h a t :

x

for a . e .

IlUU,

i i i ) Sv(x) 2 C v ( x )

x

f o r a.e.

( C = 2llSII

i s enough).

m

Proof: I t s u f f i c e s t o take

v

1

=

( 2 ~ ~ S ~ ~ ) - j STj hu i.s s e r i e s c o n -

j=o v e r g e s c l e a r l y i n norm, a n d

Ivll,

5

,I ( 2 l l S l l ) - j l l s l l j l ~ u ~ l =P

211ullp,

J=O

which i s i i ) . On t h e o t h e r h a n d , u ( x ) 5 v ( x ) b e c a u s e a l l t h e p a r t i a l sums 2u as a c o n s e q u e n c e o f t h e f a c t t h a t S f 2 0 f o r e v e r y f .

are

Finally, since

i s s u b l i n e a r , we h a v e :

S

A c t u a l l y , w i t h t h e h e l p o f t h i s lemma, w e c a n g i v e a g e n e r a l f a c t o r i z a t i o n t h e o r e m w h i c h i n c l u d e s t h e o n e we were s e e k i n g f o r Ap

weights.

THEOREM 5 . 2 .

T

be a p o s i t i v e s u b l i n e a r o p e r a t o r a c t i n g on

(X,dx) ( t h i s means and a l s o t h a t If1 6 g i m p l i e s

m e a s u r a b l e f u n c t i o n s o n some m e a s u r e s p a c e

that

I T ( f + g ) I 5 l T f [ + lTgl

l T f l 2 Tg ) . For 1 < p < m, l e t us call W = {w : 0 5 w(x) < m a . e . and T i s b o u n d e d i n P = Lp(w(x)dx)). Also, we c a l l W,

=

{w: 0 5 w(x) <

independent

a . e . and

Tw(x)

5

C w(x)

a.e.

for some

C

XI

0 : '

Then, f o r every Q

m

Lp(w) =

n

1 < p <

m, we h a v e Wp W;ipC W1 . W i - p , that i s : 1 w and a l s o w / ( P - 1 ) then there e x i s t P" such t h a t w = B e s i d e s , t h e c o n s t a n t s C for

w E W

P w o , w l e W1 w 0 -and w i n t h e c l a s s W1 depend o n l y upon t h e c o n s t a n t s f o r -l'(p-l) i n Wp g& W r e s p e c t i v e l y , t h a t i s , on t h e w and w P' r e s p e c t i v e norms of T 3 L P ( ~ ) LP'(~-''(?-') 1. P r o o f : We j u s t n e e d t o c o n s i d e r t h e c a s e lows f r o m

W

P

n

WAyPc

W1.W;-p

1

p

2

2,

by r i s i n g t o t h e power

since it f o l 1-pl,

that

IV.5. FACTORIZATION AND EXTRAPOLATION

W

P'

Wi-p'~

n

W1

n

W1-p'.

1 i.e.,

5 2,

1 < p

So, l e t

435

and s u p p o s e t h a t

w e Wp WA;p, w e W and a l s o w - 1 / ( P - 1 ) w We want P P I . t o see that w = w i t h wo,wl e W1. After writing v - l = w ; - ~ , w e s e e t h a t t h i s i s e q u i v a l e n t t o f i n d i n g v such t h a t : a ) vw e W , ,

<=

t h a t i s : T(vw)

C vw,

and a l s o

1/(p-1)) 5

b) v valently (T(v Suppose t h a t f o r e v e r y so that:

u

i n some

c

v "(P-')

or

s p a c e we c a n f i n d

Lq

equi

Su

IT(u w)I IS(U)W

If the operator

S

s a t i s f i e s t h e h i p o t h e s e s o f lemma 5 . 1 . , we s h a l l

b e a b l e t o f i n d v >= 0 s u c h t h a t b e c a u s e t h e n we s h o u l d h a v e

S(v) 5 C v .

T h i s would s u f f i c e ,

T ( v w) 5 S ( V ) W 5 C vw T (V l / ( p - l ) p - l

A l l w e h a v e t o do i s t o l o o k f o r

5 S(v)

S

5 c v

and make s u r e t h a t i s s a t i s f i e s

t h e h y p o t h e s e s o f t h e lemma. The n a t u r a l c a n d i d a t e f o r operator sending t h e function

u

S ( u ) = IT(u w ) l w - l

F i r s t o f a l l we o b s e r v e t h a t

S

into +

Su

S

is the

g i v e n by

T(lu1 1I ( P - 1 ) ) P - l

is s u b l i n e a r ( f o r t h e f i r s t term i n

t h e sum, s u b l i n e a r i t y i s c l e a r , and f o r t h e s e c o n d i t f o l l o w s from t h e f a c t t h a t , b e i n g 1 < p 5 2 , we have l / ( p - 1 ) 2 1 , and i t i s a s i m p l e e x e r c i s e t o c h e c k t h a t u I-+ T( IuI a) ' l a i s s u b l i n e a r provided a 2 1). B e s i d e s , S i s bounded i n L p l ( w ) . I n d e e d :

w e W From t h e d e f i n i t i o n o f S , i t i s e v i d e n t t h a t P' f o r every u e L p ' ( w ) . Thus, S s a t s f i e s a l l t h e c o n d i t i o n s r e q u i r e d i n lemma 5 . 1 . Note t h a t C i n lemma S . l ( i i i ) d e p e n d s

because Su

2

0

o n l y on t h e norm o f S i n L p ' ( w ) , and t h i s d e p e n d s o n l y on t h e norms f o r T i n Lp(w) and i n L P ' ( w - 1 / ( p - 1 1 . T h i s f i n i s h e s t h e proof. 0

I V . WEIGHTED NORM INEQUALITIES

436

COROLLARY 5 . 3 . (P. J o n e s ' f a c t o r i z a t i o n t h e o r e m ) . For 1 < p < i f and o n l y i f t h e r e e x i s t Ap = A1.Ai-P, that i s : w e A P wo,wl f A1 s u c h t h a t w = w 0 w ; - ~ . P r o o f : I f we t a k e

m,

T = M = H a r d y - L i t t l e w o o d maximal o p e r a t o r i n

and W1 = A 1 . B e s i d e s W 1 i P = theorem 5 . 2 , we know t h a t W = A P P -l/(p-l)e A P AP' i P = AP b e c a u s e w e AP i f and o n l y i f w P' + T h e r e f o r e , theorem 5 . 2 a p p l i e s giving us A C A,.A;-p. The i n c l u P h a s b e e n a l r e a d y e s t a b l i s h e d i n s e c t i o n 2. sion A1.A;-PC A 0 P By combining t h e f a c t o r i z a t i o n t h e o r e m 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 A1 for A

w e i g h t s g i v e n by theorem 2 . 1 6 , we o b t a i n a g e n e r a l expression w e i g h t s i n t e r m s o f maximal f u n c t i o n s . T h i s i s t h e n a t u r a l

P e x t e n s i o n t o p > 1 o f t h e o r e m 2 . 1 6 . Then, by u s i n g t h e J o h n - N i r e n berg theorem, t h i s y i e l d s an e x p r e s s i o n f o r B.M.O. f u n c t i o n s i n

terms o f maximal f u n c t i o n s : COROLLARY 5 . 4 .

a)

Then, w

e A

a.e.

Let w

be a w e i g h t i n Rn s u c h t h a t i f a n d o n l y i f i t can b e w r i t t e n a s

P

w(x) = k ( x )

w(x) <

( M f ( x ) I a (Mg(x)) B(1 - P I

1 f , g e Lloc(Rn), k bounded away f r o m 0 m and 0 < a , 6 < 1 . I n t h i s r e p r e s e n t a t i o n , k c a n be t a k e n b e t w e e n two p o s i -

with

t i v e bounds which depend o n l y o n t h e

and

b ) T h e r e a r e c o n s t a n t s C1 mension n, s u c h t h a t e v e r x I$

with

f,g

f

C2

B.M.O.

E

constant f o r

~

w.

d e p e n d i n g o n l y on t h e d i Rn c a n be w r i t t e n a s :

+ ( x ) = b ( x ) + y l o g Mf(x) - n l o g Mg(x) 1 L. , y J n 2 0 & lbl, + Y + n 5 C1ll+(l*.

ConverseZy, e v e r y B.M.O.

Ap

*

II+I*2

I$

w h i c h can be. w r i t t e n a s a b o v e , b e l o n g s t o

C2(llbIlm

+

Y

+

n).

c ) we can w r i t e a s t a t e m e n t l i k e B.M.O. and n = 0 . -

with

b)

B.L.O.

i n p l a c e of

function d) As a consequence of b) & c ) , every B.M.O. can be w r i t t e n a s a d i f f e r e n c e o f t w o B . L . O . f u n c t i o n s . I n s h o r t : B.M.0.C B . L . O . - B . L . O . P r o o f : a ) The " i f 1 ' p a r t i s c l e a r , s i n c e it f o l l o w s from theorem 2 . 1 6 t h a t b o t h (Mf(x))a quently w e A P'

and

( M ~ ( X ) ) a~r e

A1

weights and, conse-

437

I V . 5 . FACTORIZATION AND EXTRAPOLATION

Conversely i f w e A corollary 5.3 implies t h a t P’ w = w wlPp with wo,wl e A1. Then we j u s t need t o a p p l y t h e o r e m 0’ 1 2.16 t o o b t a i n t h e d e s i r e d r e p r e s e n t a t i o n . Observe t h a t , i n t h e p r o o f o f t h e o r e m 2 . 1 6 t h e l o w e r bound o b t a i n e d f o r t h e f u n c t i o n k d e p e n d s o n l y upon t h e c o n s t a n t C i n t h e r e v e r s e H o l d e r ’ s i n e q u a l i t y f o r t h e A1 w e i g h t , and t h i s , i n t u r n , d e p e n d s o n l y upon i t s A1 c o n s t a n t . The u p p e r bound o b t a i n e d f o r k i n t h e p r o o f o f t h e o r e m 2.16 i s j u s t 1 . I n o u t p r e s e n t s i t u a t i o n , t h e f a c t o r i z a t i o n theorem t e l l s u s t h a t t h e A1 c o n s t a n t s f o r wo and w1 depend o n l y upon the A constant f o r w . Therefore i n our r e p r e s e n t a t i o n f o r t h e P A weight w , t h e f u n c t i o n k i s bounded away from 0 and m P w i t h bounds d e p e n d i n g o n l y upon t h e Ap c o n s t a n t f o r W .

B . M . O . w i t h norm i n d e p e n d e n t o f f . C o n s e q u e n t l y , i f $ r e p r e s e n t a t i o n e x h i b i t e d i n b ) , we have @ E B . M . O . with

11$11* 5

f o r some a b s o l u t e c o n s t a n t

n)

C2(nbllm + y +

Conversely, i f

e B.M.O.,

@

is i n has t h e

l o g Mf(x)

b ) A c c o r d i n g t o C o r o l l a r y 3 . 5 o f C h a p t e r 11,

C2.

i t f o l l o w s from c o r o l l a r y 3 . 1 0 i n

C h a p t e r I 1 and c o r o l l a r y 2.18 t h a t , t a k i n g

A = C/U$ll*

with

C

an

i n t h e n o t a t i o n of c o r o l l a r y 3.10 absolute constant (say C = C z / 2 C h a p t e r 1 1 ) t h e f u n c t i o n w(x) = e A @ ( x ) i s i n A 2 w i t h a n A 2

Q

c o n s t a n t independent of

( l o o k a t t h e end o f t h e p r o o f o f t h e

f o r e m e n t i o n e d c o r o l l a r y 3 . 1 0 and r e a l i z e t h a t , w i t h t h e c h o i c e t h e A 2 c o n s t a n t f o r w t u r n s o u t t o depend o n l y on C1 i n t h e n o t a t i o n u s e d t h e r e ) . A p p l y i n g p a r t a ) t o o u r W , we o b t a i n

A = C2/2,

log W(X) = log k(x) Since

Q = 1 - l log w ,

b = A-l

Observe t h a t t h e s i n c e A-1

we g e t t h e d e s i r e d d e c o m p o s i t i o n w i t h

log k ,

Lm

y = A

norm o f

C-’ll@l]* and

=

a l o g Mf(x) - B l o g Mg(X)

+

0

Ilbllm

log k

5 a +

Y

< 1, +

rl

-1

a,

11 = A - l

d o e s n o t depend on 0

5

B < 1,

with

is actually in B.L.O.,

Ilbllm <

m

and

y

Then,

5 C o n s t a n t IIQII*

s o t h a t any

a r e a l number

@.

we have

c ) A s we o b s e r v e d i n t h e p r o o f o f t h e o r e m 3 . 3 l o g Mf(x)

B

20,

Q

=

i n C h a p t e r 11, b

+ y

l o g Mf

w i l l a l s o belong t o

B.L.O.

For t h e c o n v e r s e , t h e p r o o f i s v e r y much l i k e t h e one i n p a r t b ) . The d i f f e r e n c e i s t h a t , a s we n o t e d i n t h e s e c o n d remark f o l l o w i n g c o r o l l a r y 2 . 1 9 , i f Q e B . L . O . , t h e w e i g h t w ( x ) = e A@(x) i s

IV. WEIGHTED NORM INEQUALITIES

438

a c t u a l l y i n A,, b e f o r e , b u t now

n o t j u s t i n A2. Then we c a n u s e p a r t a ) a s p = 1 , s o t h a t we o b t a i n t h e r e p r e s e n t a t i o n w i t h

q = 0.

F i n a l l y d ) f o l l o w s o b v i o u s l y from b ) and c ) .

Next we a r e g o i n g t o a p p l y theorem 5 . 2 t o s t u d y t h e f o l l o w i n g q u e s t i o n : Let RC Rn b e a m e a s u r a b l e s e t w i t h IRI > 0 . L e t f be a m e a s u r a b l e f u n c t i o n on 5 2 . We want t o know how f h a s t o b e i n o r d e r t h a t t h e r e e x i s t s F d e f i n e d on t h e whole s p a c e b e l o n g i n g t o B . M . O . , and s u c h t h a t F(x) = f ( x ) f o r a . e .

Rn,

x

E

R.

I n o t h e r w o r d s , we want t o c h a r a c t e r i z e t h e s p a c e c o n s i s t i n g o f t h e

R o f t h e f u n c t i o n s i n B.M.O. We s h a l l o b t a i n t h e s o l u t i o n t o t h i s problem from t h e s o l u t i o n t o t h e c o r r e s p o n d i n g r e ? t r i c t i o n problem f o r w e i g h t s .

restrictions t o

We s h a l l u s e t h e o p e r a t o r g r a b l e o v e r bounded s u b s e t s o f

defincdon functions

VR,

0 , by

where t h e s u p i s t a k e n o v e r a l l t h e c u b e s o f We s h a l l s a y t h a t

w(x) $ 0 ,

c l a s s Ap,R, 1 5 p .: m , s u c h t h a t f o r e v e r y cube

0

(only t h e cubes

inte-

g

d e f i n e d on

Rn

containing

R,

x.

belongs t o t h e

C

i f and o n l y i f t h e r e i s a c o n s t a n t Q o f Rn

intersecting

i n a s e t o f p o s i t i v e measure

R

w i l l be r e a l l y r e l e v a n t h e r e ; f o r

p = 1,

t h e second f a c t o r has t h e

u s u a l i n t e r p r e t a t i o n a s t h e e s s e n t i a l supremum o v e r w(x) - ' ) .

QnQ

of

The f o l l o w i n g theorem i s p r o v e d e x a c t l y a s theorem 1 . 1 2 .

THEOREM 5 . 5 . L e t

1

2

p <

(u,w)

m

g a t i v e measurabZe f u n c t i o n s on

R.

b e a c o u p l e of n o n - n e -

Then, t h e f o l l o w i n g t h r e e condi-

tions are equivalent:

a ) Ma

i s bounded f r o m

i s a constant

C

LP(R;w)

s u c h t h a t , for e v e r y

L;(Q,u), f

on 0 -

that is, there and e v e r 2

> 0:

I V . 5 . FACTORIZATION AND EXTRAPOLATION

u ( { x E 0 : M,f(x) b) There e x i s t s f o r e v e r y cube

5 C A-p

> A])

C

439

J,

I f ( x ) I P w(x)dx

on

2 0

f

such t h a t , f o r every

and

fi

B":

Q

c ) There e x i s t s

C

s u c h t h a t , f o r e v e r y cube

Q

0 f -

Bn:

w i t h t h e u s u a l i n t e r p r e t a t i o n f o r t h e second f a c t o r i n case

In p a r t i c u l a r ,

w,

i s o f weak t y p e

Mn

i f and o n l y i f

w

E A

>=

0

(p,p)

p = 1

with respect

to

PlQ' Here i s t h e s o l u t i o n t o t h e r e s t r i c t i o n p r o b l e m f o r w e i g h t s .

THEOREM 5 . 6 . Let

w

on

&

R

1 5 p <

Then, t h e f o l l o w i n g

m.

conditions are equivalent:

W

i ) There e x i s t s W(x) = w ( x )

for a.e.

i i ) There i s

E

> 0

E

A

P

(W

d e f i n e d on

R"),

such t h a t

x e R. s u c h that

w(x)

1 +E E

A

P,R'

Proof: I t is c l e a r t h a t i) implies i i ) . Indeed, i f w is t h e r e s f o r some then t h e r e w i l l be W1+€ e A triction to 0 of W f A P P' E > 0. On r e s t r i c t i n g t o R , we s h a l l h a v e w l + € E A P,R' Next w e w a n t t o show t h a t i i ) i m p l i e s i ) . Let p > 1 . S u p p o s e t h a t 1+E f o r some E > 0 . T h e n , a s i n 1 . 1 4 b ) , we c a n f i n d E A P,R It 6 , q with 0 < 6 < E , 1 < q < p , s u c h t h a t w ~ E+ A ~ q,R' f o l l o w s t h a t MR i s o f weak t y p e ( q , q ) in 0 with respect t o

w1+6.

Since it i s also of type

( s e t s o f measure z e r o a r e

(m,=)

e x a c t l y t h e same a s t h o s e f o r L e b e s g u e m e a s u r e ) , i t f o l l o w by i n t e r polation that

'+'

MQ

i s bounded i n

LP(w1+')

o r , i n o t h e r words,

w b e l o n g s t o t h e c l a s s Wp corresponding t o the operator we s h a l l On t h e o t h e r h a n d , s i n c e ( w ' + ~ ) - ' ' ( ~ - ' ) E .4 MR. 9,' l+6 -l/(p-l) f o r some 1 < r < p . Then MQ i s o f ) *r,Q h a v e (w weak t y p e ( r , r ) w i t h r e s p e c t t o ( w l + [ ' ) - l ' ( p - l ' a n d , by i n t e r p o 1+6 - l / ( p - l ) w Th eo r em 5 . 2 c a n b e a ppl i e d , obtaining lation, (w ) P' . 1+6 1 -p w - wo w , with wo,wl e W 1 , t h a t i s : MQ W . 5 C w i , j = 0,l. J We e x t e n d e a c h w . m a k i n g i t 0 o u t s i d e oi R a n d c o n s i d e r t h e J f u n c t i o n V . = I\l(w.) d e f i n e d o n R". Of c o u r s e , f o r a . e . x t: a , that

y'

1

w e have

1

w - ( x ) 5 M-(x) = M w . ( x ) .l J 1

= hl

K.(x) 5 C

R I

!a:.(?().

1

Therefore,

IV. WEIGHTED NORM INEQUALITIES

440

= ~ f o r a . e . x e R , Mj(x) ;w j ( x ) , s o t h a t w ( x ) ~ + = Mo(x)Ml(x) l - p k ( x ) , where k i s a p o s i t i v e f u n c t i o n d e f i n e d on

R

and bounded away from 0 and

We c a n w r i t e

m.

Observe t h a t , a c c o r d i n g t o theorem 3 . 4 i n C h a p t e r 11, t h e f u n c t i o n s d e f i n e d on R n , M]/(1+6) = M(wj) 1 / ( 1 + 6 ) a r e A1 w e i g h t s . I f we f o r x e R and K(x) = 1 f o r x L 0 , d e f i n e K(x) = k ( x )' 1 / ( 1 + 6 ) and w r i t e , f o r x e Rn W(x) = Mo(x) 1/(1+6) ( M l ( x ) 1 / ( 1 + 6 ) ) 1 - P ~ ( x ~ , we s e e t h a t W e A s i n c e K i s o b v i o u s l y bounded away from 0 and P a. C l e a r l y W(x) = w(x) f o r a . e . x e a. In the case

p = 1,

t h e p r o o f i s d i r e c t , w i t h o u t need o f f a c -

0

torization. COROLLARY 5 . 7 .

Per f

d e f i n e d on

R,

the following conditions are

equivalent:

i ) There e x i s t s F(x) = f ( x ) f o r a . e . i i ) There e x i s t a constant

CQ,

F

d e f i n e d on

Rn,

such t h a t

F

E

B.M.O.

&

Q

Rn

x e R. A > 0,

C > 0

c&,

f o r e v e r y cube

f0

such t h a t :

J-1 IQI

A

e

Qfln

P r o o f : T h a t i ) i m p l i e s i i ) f - l l o w s i m m e d i a t e l y from C h a p t e r 1 1 , c o r o l l a r y 3.10 i i ) . L e t u s s e e t h a t i i ) i m p l i e s i ) . I t f o l l o w s from i i ) , e x a c t l y a s i n c o r o l l a r y 2.18, t h a t e Af e A2,R. T a k i n g 0 < X o c A , we 'Ofll+E A have ( e f o r some E > 0 . A c c o r d i n g t o theorem 5 . 6 , 2 ,Q t h e r e w i l l e x i s t W E A 2 s u c h t h a t e x O f ( x ) = W(x) f o r a . e . x e s2.

I f we w r i t e

G e B.M.O.

Thus

F = AilG.

0

tions

W = eG

f(x) =

we h a v e , by c o r o l l a r y 2 . 1 9 , t h a t

h i iG(x)

for a.e.

x

E

R.

This is i) with

I t seems n a t u r a l t o d e n o t e by B . M . O . (a) t h e s p a c e o f f u n c f satisfying condition i i ) i n corollary 5.7.

We c a n g i v e a v e r s i o n o f t h e o r e m 5.2 v a l i d f o r some l i n e a r , n o t n e c e s s a r i l y p o s i t i v e o p e r a t o r s , provided

p

=

2.

The c a s e o f t h e

441

I V . 5 . FACTORIZATION AND EXTRAPOLATION

Hilbert transform

class

H,

is particularly interesting.

the We know from t h e o r e m s 3 . 1 and 3 . 7 t h a t , f o r 1 < p < m, Wp(H) a s s o c i a t e d t o H a s i n t h e o r e m 5 . 2 , c o i n c i d e s w i t h

the class

A

Now we g i v e t h e f o l l o w i n g :

P'

DEFINITION 5 . 8 . Me shaZZ d e n o t e b y those functions

O
tions:

a)

-m

w(x)

>=

R

0

W1(H) t h e c Z a s s f o r m e d b y s a t i s f y i n g t h e foZZowing c o n d i -

q d x < m l+x

b) T h e r e e x i s t s $ s u c h t h a t I $ ( x ) I 5 C ~ ( x ) for a . e . w i t h C i n d e p e n d e n t of x, and t h e f u n c t i o n F ( x + i t ) = = Pt*(w+i$)(x) i s h o Z o m o r p h i c i n R.+. 2

I t f o l l o w s from b ) t h a t

Pt*$(x)

x

i s a c o n j u g a t e harmonic f u n c

I t can be s e e n ( a s i n theorem 1.20 o f Chapter N T I ) t h a t P t * $ ( y ) -P $ ( x ) ( a s y+itx) f o r a . e . x , s o t h a t I$ would d e s e r v e t o be c a l l e d t h e H i l b e r t t r a n s f o r m o f w . Actually tion for

Pt*w(x).

f o r w s a t i s f y i n g a ) , Hw i s a l w a y s w e l l d e f i n e d a s a n e q u i v a l e n c e c l a s s modulo c o n s t a n t s , s i n c e Pt*w a l w a y s h a s a h a r m o n i c c o n j u g a t e and t h i s harmonic c o n j u g a t e h a s n o n - t a n g e n t i a l b o u n d a r y v a l u e s because t h e corresponding holomorphic f u n c t i o n

F

does.

( T h i s f o l l o w s from t h e f a c t t h a t Re F 2 0 . We j u s t n e e d t o c o n s i d e r t h e h o l o m o r p h i c f u n c t i o n e - F = G which s a t i s f i e s I G I 5 1 a n d u s e F a t o u ' s t h e o r e m 4 . 1 9 i n C h a p t e r 1 1 ) . We c a n s e e b ) above a s a c o n c r e t e form o f t h e s t a t e m e n t l H w l 2 C w. Here i s t h e f a c t o r i z a t i o n t h e o r e m f o r t h e H i l b e r t t r a n s f o r m .

w e A2

THEOREM 5 . 9 . E v e r y

w i t h wo

,Mi,

P r o o f : For

E W1 (H)

@

R

can be w r i t t e n a s

.

u e L'(w),

w

=

w w- 1 0 1

define

Su(x) = IH(u.w)(x)

1

w(x)-'

+

IHu(x)I

T h i s makes p e r f e c t l y good s e n s e , s i n c e we know ( t h e o r e m 3 . 1 ) t h a t H i s bounded b o t h i n L2(w) and i n L 2 ( w - l ) , and we h a v e u e L 2 (w) 2 A l s o , c l e a r l y , S i s bounded i n L 2 ( w ) . By and u.w E L ( w - ' ) . 2 a p p e a l i n g t o lemma 5 . 1 , we c o n c l u d e t h a t t h e r e e x i s t s v E L (w) x . T h i s i m p l i e s IHv(x) 1 <= such t h a t Sv(x) 5 C v(x) f o r a . e . 5 Cv(x) and a l s o IH(v.w)(x) I 5 C v ( x ) w ( x ) f o r a . e . x . Note a l s o t h a t we h a v e v e L 2 (w) and vw e L 2 ( w - l ) a n d , s i n c e w and w - 1

442

are

IV. WEIGHTED NORM INEQUALITIES

A2

against against

weights, it follows t h a t both

v

and

vw

are integrable

( 1 + x 2 ) - l . A c t u a l l y , more i s t r u e : t h e y a r e i n t e g r a b l e ( 1 + l x l a ) - ’ f o r some 0 < c1 < 1 . I n d e e d , f o r example:

i f we t a k e

c1

c l o s e enough

t o 1 (see 3 . 2 ) .

-1 I f we w r i t e w o = vw and w 1 = v , we h a v e w = w 0w 1 and we c a n s e e t h a t b o t h wo and w1 b e l o n g t o W1(H). The r e a s o n i s t h a t w e have IHwj(x)l. 2 C w j ( x ) a . e . f o r j = O , l and P t * ( w . + i H w . ) ( x ) i s a n a l y t i c a s c a n be s e e n from t h e i d e n t i t y 1 3 I n f a c t , t h i s i d e n t i t y i s obvious Pt*(w.+iHw.)(x) = (Pt+iQt)*wj(x). 1 1 f o r n i c e ( s a y , Schwartz) f u n c t i o n s , and, i n o r d e r t o extend i t t o a r b i t r a r y f E L 2 (w) ( i n p a r t i c u l a r t o f = w . ) i t s u f f i c e s t o 3 o b s e r v e t h a t , f o r f i x e d x e R and t > 0 , b o t h f k P * f j f ( x ) and f Q t * f ( x ) a r e bounded l i n e a r f u n c t i o n a l s on L (w) ,

2

because

\ \ P t ( x - . ) \ \2 , w <

m

and

IIQt(x-.)

\12,w

<

m.

CI

Next we s h a l l s e e t h a t t h e f u n c t i o n s b e l o n g i n g t o W1(H) s a t i s f y t h e c o u n t e r p a r t i n t h e l i n e o f t h e Helson-SzegG c o n d i t i o n appearing i n theorem 8.14 of Chapter I . THEOREM 5 . 1 0 . E v e r w(x) = e u(x)+H+ [VI,

w E W,(H) c a n b e w r i t t e n i n t h e form o r some u v s a t i s f y i n g IIullm <

m

and

< a/Z.

w E W1(H). A c c o r d i n g t o d e f i n i t i o n 5 . 8 t h i s i m p l i e s t h a t t h e r e i s a f u n c t i o n F ( x + i t ) , holomorphic i n t h e upper h a l f p l a n e , s u c h t h a t F ( x + i t ) = P t * ( w + i $ ) ( x ) w i t h I $ ( x ) ) <= C w(x) f o r a . e . x . Now I P t * $ ( x ) l <= C P t * w ( x ) . Observe a l s o t h a t a l w a y s Pt*w(x) > 0 , and l A r g F ( x + i t ) 1 = ( a r c t g ( P t * $ ( x ) ) ( P t * w ( x ) ) - l 1 , $ 2 a r c t g C < a / 2 . I t f o l l o w s t h a t we c a n w r i t e F ( x + i t ) = e G ( x + i t ) w i t h G h o l o m o r p h i c s u c h t h a t IIm G I 6 a r c t g C < a / 2 . T h i s c o n d i t i o n , a l l i e d t o F a t o u ‘ s t h e o r e m , g u a r a n t e e s a s u s u a l t h e existence o f boundary v a l u e s G(x) f o r a . e . x . We s h a l l h a v e , modulo c o n s On t h e t a n t s , G(x) = Hv(x) - i v ( x ) w i t h IIvII, I a r c t g C < n / 2 . o t h e r h a n d , s i n c e I F ( x ) I = Iw(x) + i $ ( x ) I , we h a v e : w(x) s I F ( x ) I 6 w(x) + I $ ( x ) I 5 ( 1 + C ) w ( x ) , s o t h a t we c a n w r i t e w(x) = ko(x) IF(x) I = k ( x ) e H v ( x ) where k and ko a r e > O and bounded away from 0 and a . A f t e r w r i t i n g u = l o g k , we g e t what we w a n t e d . Proof: Let

n

By combining theorem 5 . 9 w i t h theorem 5 . 1 0 . we g e t a weak form

443

I V . 5 . FACTORIZATION AND EXTRAPOLATION

o f t h e h a r d p a r t o f t h e Helson-Szeg8 theorem, namely: THEOREM 5 . 1 1 . E v e r x w e A 2 = eu ( x ) + H v ( x ) f o r some u

and

R v

w(x)

can be w r i t t e n as satisfying

=

jlullm <

< n.

iivn,

-1 P r o o f : A f t e r t h e o r e m 5 . 9 , we c a n w r i t e w = w 0 w 1 with T h e n , u s i n g t h e o r e m 5 . 1 0 , w e w r i t e w o , w l e W1(H). uo (XI +Hvo ( X I wo(x) = e

and

[ujll,

with

<

m

and

Ivjllm < n / 2

for

j=O,l.

I t follows t h a t

u -ul+H(v0-vl) w

=

w 0 w-1 1

=

e

= e

0

A s we s a i d , t h i s i s a weak f o r m o f t h e H e l s o n - S z e g B t h e o r e m b e c a u s e we a r e n o t a b l e t o g e t t h e bound )Ivllrn < n / 2 c o n t e n t o u r s e l v e s w i t h t h e w o r s e e s t i m a t e IIvllrn < T I . o b t a i n a s a consequence COROLLARY 5 . 1 2 . B.M.O.(R) f e B.M.O.(R)

a c o n s t a n t and

=

Lm(R) + HL"(R).

can be w r i t t e n a s

I$II,

+

IIqII,

2

and w e have t o H o w e v e r , we

More c o n c r e t e l y , e v e r y

f ( x ) = c + $ ( x ) + H$(x)

C l!fll*

where

c

with

c

i s an a b s o l u t e cons-

tant. f e B.M.O.(R).

P r o o f : Let

Then, as w e e x p l a i n e d i n t h e Droof o f

c o r o l l a r y 5 . 4 . ( b ) , t a k i n g A = C/llfll* c o n s t a n t , t h e f u n c t i o n w(x) = e X f ( x ) t a n t independent of

f.

with

C a certain absolute i s i n A 2 w i t h a n A2 c o n s -

Then, keeping t r a c k o f t h e c o n s t a n t s i n t h e

p r o o f s o f theorems 5.9 and 5 . 1 0 , we r e a l i z e t h a t w e c a n w r i t e u ( x ) + Hv(x) w(x) = K e with K z 0, gives

IIullrn

<=

C,

f(x)

=

c

an a b s o l u t e c o n s t a n t , and

+

$ ( x ) + H$[x),

llvll,

<

TI.

This

where

c = ( l o g K ) llfll*/C, $ ( X I = u ( x ) llfll*/C a n d @ ( X I = v ( x ) llfil*/C. C l e a r l y II@(l, + ~ $ l l r n 2 C Ilfll* f o r some a b s o l u t e c o n s t a n t C . We a l r e a d y know ( P r o p o s i t i o n 5 . 1 5 i n C h a p t e r 11) t h a t , c o n v e r -

L"

sely,

+ HL"CB.M.0.

w i t h t h e r i g h t e s t i m a t e f o r t h e B.M.O.norm

We h a v e e s t a b l i s h e d t h e i d e n t i t y b e t w e e n t h e s e two B a n a c h

spaces :

n

IV. WEIGHTED NORM INEQUALITIES

444

B.M.O.

=

( L ~+ H L ~ ) / R

the norm in the right hand side being the quotient norm corresponding to the seminorm inf tII9II,

+

INI,

: f = 9

+

W I

Now the space (Lm(R) + HLm(R))/R is naturally identified with the dual of H1(R), by doing exactly the same that we did for the torus at the beginning of section 9 in Chapter I. Thus, corollary 5.12 gives yet another proof (the third one!) of the duality theorem (H')* = B.M.O. Observe that the proof of the decomposition in coroi lary 5 . 1 2 is a constructive one, whereas the two previous proofs of the duality theorem (sections 1.9 and 111.5), do not give an explicit construction o f the decomposition. COROLLARY 5.13.(C. Fefferman's duality theorem). (H1(R))* = B.M.O.(R).

=

In Chapter I we have proved the Helson-Szep8 theorem only for the torus. However, the theorem also works for the real line substituting the Hilbert transform for the conjugate function. The proof is not essentially different, but some changes have to be made due to the unboundedness of the domain. By using this Helson-SzegB theorem, we can prove a converse of theorem 5 . 1 9 , namely: if IIvllm IT/^ then eHv(x) = k(x)w(x), where k > 0 i s bounded away from 0 and m and w E W,(H). Indeed, the Helson-Szeg8 theorem im plies that eHv E A Z , so that i t has the right integrability (condition a) in definition 5 . 8 ) . Besides, eHv-iv = eHV(cos v - i sin v) is the boundary function of a holomorphic function F in the upper half plane, and it can be shown that F(x+it) = Pt*(e Hv-iv1 (XI * Thus, F(x+it) = Pt*(w+i$)(x) where w(x) = eHv(x) cos v(x) and $(x) = -eHv(x) sin v(x). But, since IIvllrn < ~ / 2 , we have: 0 < cos 5 cos v(x) 5 1 , s o that if we write k(x) = l/cos v(x), k is bounded away from 0 and a . Also I@(x) I 2 eHv(x) 2 c w(x) with C = l/cos IIvII,. We have obtained eHv(x) = k(x)w(x) with k > 0 bounded away from 0 and m and w f: W1(H).

IIv~/~

What we have is a characterization of W1(H). Using the Helson-SzegG theorem once more, we can write: exp(Lm).W1(H) = = exp(Lm) .A2. In other words, the functions belonging to W 1 (H) coincide with those in A2 up to multiplication by functions bounded away from 0 and m . This can also be written as W1(H) - A2.

IV.5.

445

FACTORIZATION AND EXTRAPOLATION

A s a f i n a l a p p l i c a t i o n o f t h e f a c t o r i z a t i o n t h e o r e m we s h a l l g i v e a theorem t h a t r e l a t e s t h e f a c t t h a t w belongs t o A2 w i t h the fact that

log w

THEOREM 5 . 1 4 . @

t a n t s C,

a) e

'

e A2

@

C2,

is close to

Lm

i n t h e m e t r i c of B.M.O.

b e a f u n c t i o n on

Then, t h e r e e x i s t cons-

Rn.

d e p e n d i n g onZy o n t h e d i m e n s i o n

provided

n,

such t h a t :

W

i n f { ( l @ - g ( I * : g e L 1 5 C1

b ) inf{ll@-gll* : g e Lm} 5 C 2

e@ e A2.

whenever

P r o o f : a ) I f i n f {ll@-gllx : g e Lm) < C , we s h a l l be a b l e t o f i n d g e L m , s u c h t h a t @ = g+h w i t h llhll* < C . Now i f C i s small enough, i t f o l l o w s from t h e J o h n - N i r e n b e r g t h e o r e m ( c o r o l l a r y 3 . 1 0 i n C h a p t e r 11) p l u s t h e c h a r a c t e r i z a t i o n o f A 2 w e i g h t s i n c o r o l l a r y 2 . 1 8 , t h a t e h e A 2 . Then e' = e g e h e A 2 . I f n = 1 , p a r t b ) c a n b e d e r i v e d from t h e Helson-Szegijtheorem. The weak v e r s i o n c o n t a i n e d i n t h e o r e m 5 . 1 1 s u f f i c e s . I n d e e d , i f

e A 2 , we s h a l l b e a b l e t o w r i t e e' = e'+Hv w i t h )Iullm < IIvllm < TI. Thus I$-ul/, = llHvll* 2 C IIVI, < Ca, s o t h a t m i n f t11$-g1!, : g E L 1 6 CTI = c 2 .

e'

w

and

I n t h e g e n e r a l c a s e , p a r t b ) c a n be d e r i v e d from c o r o l l a r y 5.4. ( a ) . I n d e e d , i f e0 E A 2 we h a v e : '(x) = b ( x ) + h ( x ) , where b(x)

=

log k(x)

i s a bounded f u n c t i o n , and h ( x ) = c1 l o g Mf(x) Ihll* 6 C i n d e p e n d e n t o f

- 6 l o g MP,(X) i s a B.M.O. f u n c t i o n w i t h

@. Therefore W

inf {ll@-gll* : R e L

1 5 IIhll* 5 C

=

IJ

C2.

A n o t h e r way t o f o r m u l a t e t h e o r e m 5 . 1 4 i s t h e f o l l o w i n g : COROLLARY 5 . 1 5 .

(Garnett-Jones theorem).

A(f) = sup{h > 0 : e A f e Az}.

For

f

f

B.M.O.,

e Lw}

-

A(f)-l.

we w r i t e

Then

d i s t ( f , L m ) = inf{I(f-gl(, : g B.M.O.

P r o o f : C, and C 2 w i l l have t h e same meaning a s i n t h e o r e m 5 . 1 4 , and " d i s t " w i l l mean t h e d i s t a n c e i n B . M . O .

=

C1,

a ) L e t X = C l / d i s t ( f , L m ) . Then d i s t ( X f , L m ) = X d i s t ( f , L m ) = s o t h a t , a c c o r d i n g t o p a r t a ) o f t h e t h e o r e m , e Af e A 2 o r ,

i n the notation introduced i n the statement: -1 t h e same a s s a y i n g : A ( f ) - ' 5 C, dist(f,Lw).

X 5 A(f),

which i s

I V . W E I G H T E D NORM INEQUALITIES

446

0 < A <

b ) Let

A(f).

e Af e A 2 ,

Then

so t h a t according t o

p a r t b) o f t h e theorem: A d i s t (f,Lm) = d i s t ( h f , L m ) 5 C2.

d i s t (f,Lm) 5 C2 A ( f ) - l .

It f o l l o w s t h a t

0

We s h a l l f i n i s h t h e s e c t i o n by p r e s e n t i n g a n e x t r a p o l a t i c n

The k e y t o t h i s e x t r a p o l a t i o n t h e o r e m theoren for the classes A P' i s the following. LEMMA 5 . 1 6 . S u p p o s e w E A t o r a g i v e n 1 < p < m , and l e t

<=

0 < t

P

1.

Consider t h e operator

S ( u ) = (M(lu1 ' I t w) . w - ' ) ~ Then: -

a) S

LPtIt(w)

i s b o u n d ed i n

t h e A - c o n s t a n t for w . P b) For e v e r y f u n c t i o n

2

u

w i t h norm d e p e n d i n g o n l y u p o n

LP'lt(w),

belonging t o

0

the pair

(uw, S ( u ) w ) b e l o n g s t o t h e c l a s s A r of p a i r s of w e i g h t s for r = ( 1 - t ) p + t , w i t h an Ar-constant f o r t h e p a i r which depends exc l u s i v e l y upon t h e

A - c o n s t a n t for P

w.

Proof: a)

w'-~'

j

~ ( u ) ~ ' / t =w E A

=

A -constant for P

b ) The c a s e S ( u ) w = M(uw).

w. t

=

P

j

M(lull/tw)p'w1-~'

, with

1

A

P'

Q,

- c o n s t a n t d e p e n d i n g o n l y upon t h e

i s c l e a r , since then

Now l e t cube

5 c j l u l ~ ' / t w since

0 < t < 1. Observe t h a t we h a v e t o e s t i m a t e :

r-1

=

r

=

1

and

(1-t)(p-1).

For a

On t h e s e c o n d f a c t o r we u s e t h e f a c t t h a t

On t h e f i r s t , a f t e r w r i t i n g

w

=

w ~ w ' - ~ , we u s e I-IBlder's i n e q u a l i t y

IV.5. FACTORIZATION AND EXTRAPOLATION

with exponents

if

C

l/t

and

(l/t)l

=

l/(l-t).

447

We get:

is the

A -constant f o r w. P Combining lemma 5.16 with lemma 5.1, we obtain

LEMMA 5.17.

ant

Let W, p

b e a s i n lemma 5 . 1 6 .

t

Then, f o r every

u e LP’/~(~), t h e r e e x i s t s a n o n - n e g a t i v e

non-ne a t i v e

such t h a t :

a) u(x)

<= v(x)

for a.e.

lI~llp‘,t,w

b) IIVllpl/t,w 2 2 V.W e A,

C)

for

d e p e n d i n g o n Z y on t h e

x

r = (1-t)p+t, w i t h a n A - c o n s t a n t f o r w. P

Ar-constant

Proof: It is enough to apply lemma 5.1 to the operator S defined in lemma 5.16. Given u 2 0 belonging to LU’It(w), we have v satisfying a) and b) above and, besides, Sv(x) 5 C v(x) with C = 2 USll. We know that (vw, S(v)w) e A, with an A,-constant that depends only upon the A -constant for w. Combining this P with the fact that Sv <= Cv, with C depending only upon the A P -constant for w , we obtain v.w E A r with an A,-constant depending only on the A -constant for w. P The extrapolation theorem will follow immediately from the next lemma, which is little more than a reformulation of lemma 5.17. 1 < p < w , 1 2 r < m , r # p. D e n o t e b y s -1 -- 11 - -r1 . Let w e Ap. Then, for S P i n L’(w), there exists a v 2 o L’(w) such

LEMMA 5.18. S u p p o s e

t h e exponent g i v e n by

u

every that: -

2 o

a) u(x)

v(x)

b ) Nvlls,w

2 c lI~NS,w9 __ with C

c)

If

A,.-constants

for

~

w.

r < p,

f o r a.e.

vw e A,,

x

and i f

an a b s o l u t e c o n s t a n t p < r,

v-?

E

t h a t , i n b o t h c a s e s , d e p e n d onZy o n t h e

with

AT,

A -constant P

I V . WEIGHTED NORM INEQUALITIES

448

P r o o f : 1 ) I f r < p , t h e n r = ( 1 - t ) p + t f o r some 0 < t <= 1 , and l / s = 1 - ( r / p ) = 1 - (1 - t + ( t / p ) ) = t / p ' , so t h a t s = p ' / t . T h e r e f o r e , i n t h i s c a s e , t h e r e s u l t c o i n c i d e s w i t h lemma 5 . 1 7 . 2 ) L e t p < r . Then l / s = ( r / p ) - 1 = ( r - p ) / p , that is: s = p/(r-p). Now r ' c p ' , and i f we w r i t e l / s * = 11 ; ( r ' / p ' ) / ,

we have wimp e A we s e e t h a t s * = s ( r - 1 ) - S i n c e w e A P' P' We a r e i n a p o s i t i o n t o a p p l y t h e f i r s t p a r t o f thelemma a l r e a d y p r o v e d , w i t h p ' i n p l a c e o f p , r ' i n p l a c e o f r , and w l - p ' I f u 2 0 b e l o n g s t o Ls(w), we s h a l l have i n p l a c e of w. s/s* wp'/s* E L'*(W'-~'). Therefore, there w i l l e x i s t uo = u

-

e L'*(W'-~')

e Arl.

w i t h uo 2 v o , I f we w r i t e vo

IvoII = vs / s

5

C Iu,II

wP'/S*,

and, besides, t h a t i s , i f we

= v s * / s , - P ' / S , we s e e t h a t define 0 and b e s i d e s : v - l w = v 0- s * / s w l + ( p ' / s ) -- vo1 - r w ( r = (vow - W P - 1 ) ) 1 - r = (vow1 -P

Here i s , f i n a l l y , t h e e x t r a p o l a t i o n theorem

Let T b e a s u b Z i n e a r o p e r a t o r . L e t 1 5 r < m, 1 < p < m . S u p p o s e t h a t T i s bounded i n Lr(w) ( r e s p e c t i v e Z y of weak t y p e ( r , r ) w i t h r e s p e c t t o w) f o r e v e r y w e i g h t w e A r , w i t h a norm t h a t d e p e n d s onZy upon t h e A r - c o n s t a n t f o r w. Then, f o r every w E A T i s bounded i n Lp(w) f r e s p e c t i v e Z y , T 2 P' u t weak-type (p,p) w i t h r e s p e c t t o wl, w i t h a norm t h a t d e p e n d s o n l y upon t h e A - c o n s t a n t for w . P THEOREM 5 . 1 9 .

P r o o f : We s t a r t by d e a l i n g w i t h t h e c a s e i n which be o f s t r o n g t y p e . T h e r e a r e two p o s s i b i l i t i e s .

T

i s assumed t o

1 ) L e t r < p . Take w e A We a r e g o i n g t o a p p l y lemma 518, P' o b s e r v i n g t h a t , w i t h t h e n o t a t i o n u s e d i n t h a t lemma, l / s = 1 - ( J p ) , s o t h a t s = ( p / r ) ' , We have = j / r f ( x ) Iru(x)w(x)dx

Ls(w), s u c h t h a t u 2 0 a n d I I U ~ ~ =, 1~ . I f we t a k e a s s o c i a t e d t o u a s i n lemma 5 . 1 8 , we s h a l l h a v e : pw(x)dx) r / p 5- J I T f ( x ) l r v ( x ) w ( x ) d x 5

I V . 5 . FACTORIZATION AND EXTRAPOLATION

5

C Ilf(x)l'

v(x)w(x)dx

449

<=

5 C ( j ( l f ( x ) I r 1 p / r w ( x ) d x ) r / P ( j v ( ~ ) ( ~ / ~ ) ' w ( x ) 11 d x( )p / r ) ' =

c

IIvIIs,w

I I f l l p ,rw

I I f l l p ,rw

5 C'

Observe t h a t t h e c o n s t a n t

=

d e p e n d s o n l y on t h e

C'

A -constant f o r

P

W.

w e A

2 ) Now l e t p < r . Take from H 8 l d e r ' s i n e q u a l i t y t h a t

I f 0 < a < 1 , it follows P' (Ifla 5 (/Iflu-')"(\ ua'('-") 1 ,

and we know t h a t , a c t u a l l y , t h e l e f t hand s i d e e q u a l s t h e minimum o f t h e r i g h t hand s i d e o v e r a l l p o s s i b l e c h o i c e s o f

u,

that is: u-1

w ( x ) dx

Observe t h a t , w i t h t h e n o t a t i o n o f lemma 5.18 l / s = ( r / p ) - 1 = = (r-p)/p, s o t h a t s = p / ( r - p ) . T h u s , g i v e n f e Lp(w), w e h a v e

I ll p ,w

j l f ( x ) l r u ( x ) - ' w(x)dx

=

f

f o r some u 2 0 w i t h I ( U I I ~ , ~ = 1 . Now c o n s i d e r t h e f u n c t i o n a s s o c i a t e d t o u a s i n lemma 5 . 1 8 . We c a n w r i t e : ( j i w x ) I P ~ ( X ) ~ X )=~ II/ iPT f i r n p / r , w

5 ~ ~ T f ( x ) I r v ( x ) - l w ( x ) d x . ~ ~ v [ s2, wC 5

C I I f ( x ) Iru(x)-'w(x)dx

Observe t h a t t h e c o n s t a n t

=

I

v

2 If(x)lrv(x)-'w(x)dx

5

CIIfll'P , W A -constant for w. P i s o f weak t y p e , w e o n l y n e e d a

d e p e n d s o n l y on t h e

C

When we j u s t assume t h a t

T

minor m o d i f i c a t i o n i n t h e two a r g u m e n t s g i v e n a b o v e . For e x a m p l e , let

r A )

XEX(x)u(x)w(x)dx

f o r each

X > 0:

I V . WEIGHTED NORM INEQUALITIES

450

2

0

(x)v(x)w(x)dx

6

f o r some u E Ls(w) s u c h t h a t c o r r e s p o n d i n g v , we h a v e : W ( E , ) ~ ' 5~

1

xE

x

u

I I U I I ~ , ~=

and

1.

Taking t h e

C A - ~J l f ( x ) I r v ( x ) w ( x ) d x

Thus w(EA) 5 C A-p ~ ~ f ~ and ~ ~ ,T w i s, o f weak-type ( p , p ) r e s p e c t t o w a s we wanted t o p r o v e . The same m o d i f i c a t i o n a p p l i e s when

p

c

2

with

0

r.

W e may s t r e n g t h e n t h e o r e m 5.19 by combining i t w i t h 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 ( t h e o r e m 2 . 1 1 i n C h a p t e r 1 1 ) . We g e t t h e following COROLLARY 5 . 2 0 . Let T b e a s u b l i n e a r o p e r a t o r . Let 1 5 r < m, 1 < p < S u p p o s e t h a t T i s of w e a k - t y p e (r,r) with respect t o

-.

w f o r e v e r y w e i g h t w e A,, w i t h a norm t h a t d e p e n d s o n l y u p o n t h e T i s bounded i n A r - c o n s t a n t f o r w. T h e n , for e v e r y w e A P' Lp(w), w i t h a norm t h a t d e p e n d s o n l y u p o n t h e A - c o n s t a n t f o r w .

P

Proof: Let w e A Then we know ( t h e o r e m 2 . 6 ) t h a t w e A for P' P-E some E > 0 . The E and t h e A - c o n s t a n t f o r w depend o n l y P-E f o r e v e r y q > p . Now, upon t h e A - c o n s t a n t f o r w . A l s o w E A P 9 a p p l y i n g theorem 5 . 1 9 , we g e t t h a t T i s o f w e a k - t y p e ( P - E , P - E ) w i t h r e s p e c t t o w and a l s o o f weak t y p e ( q , q ) w i t h r e s p e c t t o w f o r e a c h q > p . The norms depend o n l y upon t h e A - c o n s t a n t P f o r w . Then, by a p p l y i n g 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 t h e o r e m , we c o n c l u d e t h a t T i s o f s t r o n g t y p e ( p , p ) w i t h r e s p e c t t o w ( t h a t i s , bounded i n L p ( w ) ) , w i t h norm d e p e n d i n g o n l y upon t h e A -constant f o r w. 0 P 6 . WEIGHTS IN PRODUCT SPACES

The o p e r a t o r s c o n s i d e r e d s o f a r i n t h i s c h a p t e r , a s w e l l a s i n p r e v i o u s c h a p t e r s , were a s s o c i a t e d t o t h e o n e - p a r a m e t e r f a m i l y o f dilations: x w t x , t > 0 , i n the sense t h a t the conditions sat i s f i e d by them w e r e n o t a f f e c t e d b y s u c h a change o f s c a l e i n B n . I n some c a s e s , t h e o p e r a t o r s t h e m s e l v e s were i n v a r i a n t u n d e r t h e s e dilations, i.e. T(ft)

=

(T f ) t

where

ft(x) = f(tx),

t > 0

( e . g . , t h e H a r d y - L i t t l e w o o d maximal f u n c t i o n o r a homogeneous s i n -

IV.6. WEIGHTS IN PRODUCT SPACES

451

gular integral operator). The classes of weights associated to such operators were, accordingly, dilation invariant: If w E A then P' also wt E A with the same Ap constant as w. P Now, we shall study operators and weights which are invariant under the n-parameter dilations in Rn, defined by X

6(x)

h

=

( 6 1 x1,62 X2,"',6,

Xn)

Here are the simplest examples of

with 6 = (61,62,...,6n) e (R,)". such operators:

i) The strong maximal function

where R denotes the family of bounded n-dimensional intervals R = [al,bl] x [a2,b21 x...x [an,bnl. ii) The multiple Hilbert transform, defined by

H f(x)

I'

1im

=

f(Y)dY

n

n or, in terms of Fourier transforms: = (-iIn s i p ( S ~ E ~ . . . ~ ," ~, (! s J .

xn-Yn) (ff f)^(S)

=

iii) The partial sum operators

sI

f(x)

=

J,~ ( S Ie21rix.S dS

where 1 is a (not necessarily bounded) n-dimensional interval. These are the analogues in Rn of the partial sum operators in R studied in 3.12. For an arbitrary function f and 6 = ( 6 1 ,62,...,6n) E ( R , ) n , we shall denote by f6 (x)

=

f6

the function

f(6(x))

=

f(6,X1,62X2 ,..., 6,Xn)

Then the following identities hold for the operators just described: M,(f6)

=

ff(f6)

(Msf)6;

=

(H f)&;

S6(Il(f 6)

=

(SIf) 6

The boundedness of these operators in Lp(Rn) follows by writing them as products o f one-dimensional operators. Given an operator T acting on functions in R, we denote by Tj, j=1,2 ,..., n, the operator defined on functions in Rn by letting T act on the j-th variable while keeping the remaining variables fixed, namely Tjf(x)

=

T(f(xl ,x2,. . . , X ~ - ~ , . , ,. X ~. .+,xn))(xj) ~

IV. WEIGHTED NORM INEQUALITIES

452

If T = Tm is a multiplier operator, with to verify that (TJf)A(S1,S2,...,Sn)

=

p(S1,S2,

m

E

Lm(R),

it is easy

...,c;,)m(Sj)

In particular, we have Hf(x) SIf(X)

H 1O H2

=

0 . .

.oHnf(x)

SI 1 0SIZ0.. 2

s;

f(x) n where H is the (ordinary) Hilbert transform and I = I 1XI 2x...xIn, with I intervals in R . Also, if M denotes the one-dimensional j Hardy-Littlewood operator, then =

. O

1

J

and a moment's thought shows that MS f(x) 5 M 1

0

M2

0 . .

.o

Mnf(x)

On the other hand, we make the following simple observation based on Fubini's theorem: If T is a bounded operator in Lp(R) with norm

pdx and similarly for T2,. ..,Tn. From the identities for ff, SI and the inequality for MS previously displayed, it is now evidentthat, f o r all 1 < p < 00 and f E Lp(Rn), we have IIMSf

(Ip

5

cp I1 f l,

cp u f Ilp llSIfllp 5 cp Ilfllp IIHfllp 5

(Cp

independent of

I)

There is no weak type (1,l) inequality for these operators. For 1 , we have example, letting f(x1,x2) = P1(x1)P2(x2) = IT (Xl+l)(x;+l) x1 x2 , which is not in weak-L1 (R 2 ) . By using Hf(x,,x2) = 2 2 77 (x,+l) (x2+1) the estimates near L (see Chapter 11, 2 . 3 )

'

for the Hardy-Littlewood maximal function one can prove that MS is bounded from

IV.6. WEIGHTS IN PRODUCT SPACES

453

L(1og L)"-l (R") to weak-L1(R") . However, we are rather interested in Lp estimates, and our next objective will be the analogues in Lp(w), for suitable weights w(x) in Rn, o f the above inequalities. Our previous experience with A weights suggets that the P condition similar t o Ap with n-dimensional intervals instead o f cubes must be necessary, while the method used in the unweighted case indicates that we can easily deal with weights w(x) such that w(x1,x2 ,..., X ~ - ~ , . ,,,..., uniformly in X ~ + xn) e A (B) P to give a precise meaning (x1,x2,. . . , X ~ - ~ , X ~ + ~ , . . . , XIn ~ ) order . to this, we establish the following: NOTATION: G i v e n

& Rn, n 2 1

v(x) '> 0

the A -constant f o r -

P

v

(which can be

we d e n o t e by

+a),

I.[

( R")

namely

The last definition is, of course, motivated by (2.15). As usual, Q stands for an arbitrary cube (interval if n = 1) in Rn. DEFINITION 6.1. Let s a y t h a t w e A* i-f (R+)n,

i.e.9

P

sup 6

we

be a w e i g h t i n Rn. For 1 5 p 5 0, w6 E A u n i f o r m l y i n 6 = (61,62,...,6n) e

W(X)

P

[w6]Ap(Rn)

<

m'

Since dilations by 6 = (1S~,6~,...,6,) e (R+)n transform cubes in arbitrary n-dimensional intervals, a change of variables proves that, for 1 < p < m, w E A* if and only if P

and similarly for Thus, A* is the P nal intervals.

p = 1,

Ap

p = m (see theorems 6.5 and 6.7 below). class naturally associated with n-dimensio-

THEOREM 6 . 2 . G i v e n a w e i g h t

W(X)

following statements are equivalent:

a) w e A* P

Rn

1 0

s u c h t h a t , for e a c h

j = 1,2,

...,n

have -

[ w ( x ~ ~ x ~ ~ ~ ~ ~ ~ x j - ~ Y ~ #< x cj + ~ ~ ~ . ~ ~ ~ ~ ~ ] ~ f o r almost every C)

MS

(x1,x2,.. . , X ~ - ~ , X ~.+.,xn) ~ , . e Rn- 1

i s a bounded o p e r a t o r i n

Lp(w)

d) The p a r t i a l sum o p e r a t o r 4 (SI) a r e u n i f o r m l y bounded i n LP(W) e) tl

is a bounded o p e r a t o r i n

Lp(w)

Proof: For notational simplicity, we shall prove the theorem f o r n = 2 , since this case contains all the essential difficulties of the general situation. The mutual implications of statements a)-e) will be established according to the following diagram

(a) implies (b): Given R we define

w(x1,x2)

Now, given another interval apply Hijlder's inequality

where R = I x J I = (xl-6, x1+6)

and and

6

+

P'

for each interval

mJ(xl)

is the A*-constant for P 0, we obtain for

a.e.

x1

over

w.

E

I

J

in

and

Letting

B

sup mJ(xl)P, and it is enough to consider J the countable family of intervals J with rational endpoints. This proves [w(xl, .)IA 5 C (a.e. x 1 e R), and similarly, [w(xl

'.)lAp(R)

A*

I, we average

5

mJ(xl)P

But

C

in

[W(*,X2)]Ap(R)

5

=

P

(a.e* x2 e R).

IV.6. WEIGHTS IN PRODUCT SPACES

455

(b) implies (c): By Muckenhoupt’s theorem 2.8 applied to the maximal operator M in R, the hypothesis implies

and

for almost every x, and x2 in R. Now, both inequalities consecutively with g = MLf, that MSf 5 M1(M2f).

f2 0

(c) implies [a): For every function > (=

1

(

f)X,.

taking into account

R e R,

and

MSf

Thus, from (c) we get

Taking f = w - l/(p-l)xR ( o r an increasing sequence of bounded fun2 tions in R converging to w -l’(p-l)) it follows that w E A* P’

(b) implies ( d ) : It is proved exactly as (b) -4 2 f), and I = I1 x 12, we know that SIf = S 1I (SI 1 2 w together with Corollary 3.12 imply that both bounded in Lp(w) with norms independent of 1 1 ,

I

(d) implies (e): Take Then Hf and therefore

=

=

-(S1f

[O,m)

x [O,a)

S-If)

+

(Sif

+

IIHfllp,w 5 C Ilfllp,w

?

and

+

s

for all

=

(c). Given the conditions on S;l

s:,

and

are

12. [O,m)

x

(-m,O].

if) f

f

LP(W).

(e) implies (a): Given R = [a,a+h] x [b, b+l], consider the rectangle fi = [a-h,a] x [b-L,b] which is symmetric to R with respect to the common vertex (a,b). Take f > 0 supported in R. Then, for every x = (xl,x,) e

Using the fact that (6.3)

H

is bounded in

(&i, flp 1-

Now, we choose first

1?

f(x)

=

w 5

c

Lp(w)

1

R

w(x) - ” ( p - ’ )

lflP w in

(6.3).

This gives

IV. WEIGHTED NORM INEQUALITIES

456

But we can also interchange the roles of R and then take f(x) = XR(X). This gives w(R) 5 C w(E) in (6.4), proves the condition A* for w(x). 0

in ( h . 3 ) , and which, inserted

P

One of the consequences of the preceding theorem is the equivalence between A; and the one-dimensional A conditions on each P variable (uniformly in the remaining variables). It is natural to ask whether such an equivalence continues to hold in the extremal cases: p = 1, p = m . The next two theorems will give affirmative answers in both cases, while yielding some other equivalent characte rizations of At and Acf, (remember that A; and were already defined in (6.1)).

At

THEOREM 6.5. The f o Z Z o w i n g c o n d i t i o n s f o r a w e i g h t

& Rn

W(X)

are equivalent:

i) w 'A;

c 2 c

ii) M~ w(x) 5 iii) ~j w(x)

f o r a.e.

x

w(x)

f o r a.e.

x e R"

iv) T h e o p e r a t o r s t o weak-Ll (w) .

Mj,

Observe that iii) means for almost every j = 1,2 ,...,n.

R"

w(x)

j

=

1 ,2,.

E

. . ,n

[w(xl,x2

and

j

a r e bounded f r o m

1

L1 (w)

,...,X~-~,.,X~+,,...,X n)] A,(W n- 1

~ + ~ xn) e R (x1,x2,..., X ~ - , , X ,...,

Proof: By a change o f variables, the definition of reformulated as

1 -

I,z,...,n

=

and

A;

can be

5 C (ess. inf. w(x))

for all R e R x e P The equivalence between this and ii) can be seen exactly as in the case of A 1 weights (see the argument after (1.8) and ( 1 . 9 ) ) . IRI

w(x)dx

R

To show that ii) and iii) are equivalent, we s i m p l y use the ma jorizations ~jw(x) 5 ~ ~ w ( x )5 hi1

0

~2

0 ,

. . o

~"w(x)

(a.e. x

E

R")

-

The second one is already known, and the first one follows by a step by step repetition o f the argument we used to prove (a)

(b)

IV.6. WEIGHTS IN PRODUCT SPACES

in theorem 6.2, where we take now

p

=

457

1.

Let us now prove the equivalence of iii) and iv). We apply the inequality (2.14) of Chapter I 1 to arbitrary functions f(x1,x2, ...,xn) and weights w(x1,x2, . . . ,xn) considered as functions of x, only, with the remaining variables fixed. By integrating then with respect to x2,x3, ...,xn, we get

and similarly for M2,M3,. . . ,Mn. This shows that iii) implies iv) . Conversely, if iv) holds, we apply the weak type inequality for M1 to a function of the form f(x) = g(xl)XE(F), with g 0 and E a measurable subset of Rn-’ (we are denoting x = (xl,x) E Rn, with x1 e R, x E: Rn-l). Then, M1f(x) = XE(?) Mg(x,), and we get

,

-

w(xl ,x)dxlI dx

eR :Mg (x ) > A 1

and

I

Since E is arbitrary, this implies that, f o r each X > 0, the inequality

g 2 0

holds f o r a.e. 2 e Rn-l. Let A be the set of points ? e Rn-l for which (6.6) holds for every function g = X I , where I is any interval with rational endpoints, and every A rational. Let B

=

{x of

=

(xl,x)

e

-

Rn : x

e

A

and

x1

is a Lebesgue point

-

w(.,x)l. -

Then, clearly, IRn\BI = 0. Now, if (xl,x) e B and x 1 is an interior point of an interval I with rational endpoints, we take intervals { I ~ I ~ = with , rational endpoints such that x 1 E I k C I and lIkl + 0. Then we apply (6.6) to gk(yl) = XI (yl). Since k > IkI 111 for all y 1 e I, we obtain -

/

1 -

III

the supremum f o r all intervals such as

I

I V . WEIGHTED NORM INEQUALITIES

458

-

-

M 1w(xl,x) 5 C w(xl,x) The same argument works for

for all

M2,M3, ...,Mn,

-

(x,,x) e B and iii) is proved.

0

Next, we shall establish several equivalent formulations o f the condition A: THEOREM 6 . 7 . The f o l l o w i n g c o n d i t i o n s f o r a w e i g h t

i n Rn -

w(x)

are equivaZent:

i ) w e c

R e R

iii) For e v e r y

and e v e r y m e a s u r a b l e s e t

f o r some f i x e d c o n s t a n t s

iv) T h e r e e z i s t a whenever we have A C R V) T h e r e e z i s t

I IX

E:

R :

C,

E

> 0

A C R

depending o n l y on

w.

and f.

R

CY

W(X)

a 5-

I RI

s a t i s f i e s a reverse Hzlder's inequality over a l l n - d i m e n s i o n a l i n t e r v a Z s R, i.e., f o r some c o n s t a n t s C,

vi) w

E

> 0

Proof: By a change o f variables, each one of the statements ii)-vi) is equivalent to saying that the weights Iw6 16e(R+)n satisfy the corresponding condition with cubes Q instead of intervals R E R , uniformly in 6. Taking into account the different characterizations of Am (see 2.13 and 2 . 1 5 ) these statements are equivalent to [w 6]Am(Rn) 5 Constant (for all 6 e (R+)")

IV.6. WEIGHTS IN PRODUCT SPACES

459

which is the definitiorsof w e A.: Thus, all the conditions i)-vi) are equivalent. On the other hand, we know that 6 < C implies [w ]Ap(Rnl 5 B for some p = p (C,n) < m [w A,(R"I and B = B(C,n). Therefore, i) implies vii), and the converse is obvious because [ 2 [ . 1 ( ~ n ) by Jensen's inequality. Finally, we prove the equivalence between vii) and viii) by observing that each of them is equivalent to the following statement:

.

C

(*) There e x i s t

p

such t h a t

i m

Indeed, vii) \-a ( * ) by theorem 6.2, while ( * ) lary 2 . 1 3 applied t o weights in R. 0

viii) by corol-

A consequerice of the characterization of :A provided by theorem 6.7.(vi) is that, if 1 0. This follows also from the correspondP ing property for A weights and definition 6.1. We shall now use P this fact in order to describe the classes A* in terms of maximal P functions. Given u(x) > 0 and v(x) > 0, we shall write: u(x) v(x) to indicate that both u ( x ) and v(x) are L" functions.

'+'

-

m

THEOREM 6.8. Let a) w e A; g(x)

function

W(X)

b) For A* P

-

W(X)

<

a.e.

m

Then

i f a n d o n l y i f t h e r e e x i s t a ZocaZZy i n t e g r a b l e and a p o s i t i v e number 6 < 1 s u c h t h a t

-

(M'g(x))'

1 < p <

- ... -

(PPg

XI l 6

we h a v e

00

= A;(A;)'-~

(Mzg(x ) 6

=

w, e A:}

{w0 Wi-p

1 locaZZy i n t e g r a b l e f u n c t i o n s

M1g(x) M'h(x)

-

g(x)

M2g(x) M2h(x)

- ... ... -

h(x) Mng(x) Mnh(x)

and some

such t h a t

460

IV. WEIGHTED NORM INEQUALITIES

P r o o f : a ) By c o n s i d e r i n g ( M j g ( x ) ) & a s a f u n c t i o n o f x o n l y and j u s i n g ( 3 . 4 ) i n C h a p t e r 11, i t f o l l o w s t h a t Mj((Mjg)')(x) 5 < C , ( M ~ ~ ( X ) a) .~e . Thus, i f w(x) (Mjg(~))~ for a l l j=1,2 n , t h e n Mjw(x) 5 C w(x) a . e . ( j = l , 2 , . . . ,n ) , which i s

-

,...,

e q u i v a l e n t t o w e A;. w l + € e A: f o r some

C o n v e r s e l y , assume t h a t w e A;. Then which means Mj(wl+')(x) - w ( x ) ~ + ~ 1+E 1 and 6 = we n . L e t t i n g g ( x ) = w(x) for a l l j=1,2 1+ E (MJg(x))' ( j = 1 , 2 , ...,n ) a s d e s i r e d . have w(x) E

> 0,

,...,

-

b) We j u s t n e e d t o p r o v e t h e f a c t o r i z a t i o n r e s u l t : A* = P A;(A;)l-P. But t h i s i s a p a r t i c u l a r c a s e o f theorem 5 . 2 , s i n c e A Yt a r e t h e c l a s s e s o f w e i g h t s a s s o c i a t e d t o t h e s t r o n g maximal o p e r a t o r , i . e . A; = Wp (MS), 1 5 p a, ( b y 6 . 2 a n d 6 . 5 ) and A* = P' =

The r e a d e r c a n e a s i l y c h e c k t h a t t h e s t a t e m e n t : W(X)

-

(M1gl(x))6

0 < 6 < 1

f o r some

-

(M2g,(x)l6

- ... -

(Mngn(x)) 6

and some ( d i f f e r e n t ) l o c a l l y i n t e g r a b l e f u n c -

which i s a p p a r e n t l y weaker t h a n t h e one i n tions gl,g2, ...,g,, However, we p a r t a ) of t h e o r e m 6 . 8 , i s a l s o e q u i v a l e n t t o w E: A;. do n o t know i f

(M,f)&

i s , f o r every

f e Lloc(Rn)

and

0 < 6 < 1 , a w e i g h t i n A: ( t h e converse i s t r u e due t o t h e r e v e r s e H i j l d e r ' s i n e q u a l i t y : w e A; i m p l i e s w(x) (MSf(x)) 6 w i t h f = w l + E , 6 = - and E > 0 ) . T h u s , t h e p r e c e d i n g theorem 1+ E d o e s n o t g i v e an e a s y method t o p r o d u c e "enough" A; weights i n

-

Lp(Rn), b u t we c a n f i n d s u c h a method by i m i t a t i n g t h e argument used f o r t h e f a c t o r i z a t i o n t h e o r e m , namely: Given

where

Mg

1 < p <

u e Lp(Rn),

denotes, as usual, the

e q u a l s t w i c e t h e norm o f in

Lp

MS

in

we t a k e

m,

q =

k-th i t e r a t e of Lq.

6

and d e f i n e

MS,

and

C

Then, t h e s e r i e s converges

and u(x)

U e A;

with

5

U(X)

A;-constant

d e p e n d i n g o n l y on

p

L i k e w i s e , t h e whole p r o c e s s t h a t l e d t o t h e e x t r a p o l a t i o n t h e o r e m

IV.6. WEIGHTS IN PRODUCT SPACES

.

applies mutatis mutandis to weights in state the final result

461

We limit ourselves to

THEOREM 6.9. S u p p o s e t h a t we r e p l a c e

statements

Ap & A*P e v e r y w h e r e i n t h e o f t h e o r e m 5 . 1 9 and c o r o l l a r y 5 . 2 0 . T h e n , t h e r e s u l t i n g

statements are a l s o true.

The fact that, not only the extrapolation theorem, but also the lemmas we used in order to establish it can be extended to the product setting, will be important for our next result. What we shall need is the particular case of Lemma 5 . 1 8 corresponding to r = 1, which we now recall: LEMMA 6 . 1 0 . Let w e A * 1 0

&

E A;

u.

The last objective in this section will be the study of the strong maximal function with respect to a regular Borel measure in Rn, which is defined by

This is the analogue of the operator Mu studied in Chapter 11, where only cubes (instead of arbitrary n-dimensional intervals) were considered. In that case, Mp was shown to be bounded in Lp(p), 1 c p 5 m, provided that p is doubling: p ( Q 2 ) 5 C p ( Q ) f o r every cube Q . In our present situation, a trivial result is that is bounded in Lp(p) f o r 1 < p 5 m when p is a product MS,!J measure: p = p l @ p 2 @ . . . P pn for some regular Borel measures p 1 , p 2 , ...,pn in R. In fact, in this case we can repeat the argument given for Lebesgue measure, since f(x) 5 MA

M 2 o...O Mn f(x) 1 p2 % and M p , is bounded in Lp(R;pj), 1 < p < m, for each j=l,Z,.?.,n. (This has been proved in Chapter 11, theorem 2.6 when the p . ' s are doubling measures, but the doubling condition is not 1 necessary for one-dimensional measures; see 7.4 in Chapter 11). MS,u

A more significant result will be obtained for measures

which are not necessarily products of measures in

R, namely:

p

IV. WEIGHTED NORM INEQUALITIES

462

*

dp(x) = w(x)dx with w e Am. This will be based on covering arguments for the family R. Since this technique is new for us, we shall establish first the general result which allows to pass from covering lemmas to boundedness properties of maximal functions: LEMMA 6 . 1 1 . G i v e n a b a s i s 8 ( i . e . , a c o l l e c t i o n of o p e n s e t s ) i n Rn, and a measure p s u c h t h a t 0 < p ( B ) < m f o r a l l B e 8 , suppose t h a t , from any f i n i t e sequence s u b s e q u e n c e {iikj s u c h t h a t

f o r some f i x e d c o n s t a n t s

C,

{BjIC 8

we c a n s e l e c t a

1 < q <

C' > 0

m.

Then, the

maximal o p e r a t o r

is of weak t y p e q' 0, set

Bn : M g Y P f(x)

belongs to some

B

e

> XI

8 such that

will b e covered by a countable family

u

EY = l<jzN B j y -

satisfying ( 6 . 1 2 )

and select from

.

IBjIlcjLN

I,

f

dv > A P ( B ) ,

{B.}.

3 JZ1*

a subsequence

Let

fi.1 3

Then

letting N -+ m, the weak type inequality is proved. The strong type (p,p) for p > q' follows from Marcinkiewicz interpolation theorem.

0 It is remarkable that the converse of this lemma is also true (it was proved by CBrdoba [ Z ] ) , but we shall have no need of this fact. For results of this kind and generalizations we refer to M. de Guzm5n [Z] , Ch. 6 .

IV.6. WEIGHTS IN PRODUCT SPACES

THEOREM 6.13. Let

dP(x)

=

for all

w e A.:

w(x)dx

maximal o p e r a t o r w i t h r e s p e c t t o

463

~.r,

MS,p,

Then, the strong

is bounded i n

Lp(p)

p 5".

1

Proof: We know that w e A: for some r < m , and it will be enough to prove the inequalities (6.12) for r 5 q < m, since this will for q ' arbitrarily close imply the weak type (q',q') o f MS,Ll to 1 . Given IRjIlcjiN in R , we take R 1 = R1 and, once ." R 1 , R 2 , * - * s R k - 1 have been selected, we choose Rk to be the first interval in the given sequence (if any) such that at least half of it (in the sense of Lebesgue measure) is not covered by the intervals already selected, i.e.

-

- -

Now, we claim that

Indeed, let x -

E

f

U Rj,

uk

be the characteristic function of

there are two possibilites: either

x

6,.

If

belongs to some

J

Rk, in which case MSf(x) = 1 , or x e R. for some R . which has 3 J been discarded in our selection process, in which case we must have 1 1 I R j n ( U 6,) 1 lRjl, and therefore biSf(x) NOW, since k we know that MS is bounded in Lr(w)

z.

which is the second inequality in (6.12). To prove the first inequa lity we use duality

o f unit norm. Since w e A ~ A:,C we apply for some u f: ~:'(w) lemma 6.10 to find v(x) u ( x ) such that 5 C 1 and vw e A; with A;-constant 5 C 2 (independent of u ) . We denote E k = B,\ (ilu . . . a,-,), so that { E L } is a disjoint scquence and lk,l i 1 l E k l . Then

\IVI~~,,~,

fi2u

< 2 C, -

C

&li

u

\ E l i [ ess.inf. (v(x)w(x))

xeRk

5 2 C2

C

li

1

v(x')w(x)dx Ek

=

I V . WEIGHTED NORM INEQUALITIES

464

and t h i s completes t h e proof.

0

The p r o o f we h a v e g i v e n o f t h e l a s t t h e o r e m i s b a s e d on t h e i n Lp(w), w e A* For an a l t e r n a b o u n d e d n e s s p r o p e r t i e s o f MS P‘ t i v e ( a c t u a l l y , t h e o r i g i n a l ) a p p r o a c h w h i c h d o e s n o t u s e A*-weights P for p < m, s e e R . F e f f e r m a n [ Z ] . We o b s e r v e t h a t , c o n v e r s e l y , t h e f a c t t h a t MS i s b o u n d e d i n Lp(w) f o r w e A* 1 < p < m, P ’ i s a consequence o f theorem 6 . 1 3 t o g e t h e r w i t h t h e i m p l i c a t i o n :

w E A* P o f Ap imply

-

w E A* f o r some q < p . I n d e e d , j u s t as i n t h e c a s e 9 w e i g h t s , t h e c o n d i t i o n w E A* a n d H E l d e r ’ s i n e q u a l i t y 9

t h a t i s , MSf(x) 5 C {M (lf]q)(x)31’q, and t h e r i g h t hand s i d e S,W i s , a c c o r d i n g t o 6 . 1 3 , a bounded o p e r a t o r i n Lp(w), p > q .

7 . NOTES A N D FURTHER RESULTS

-

Weighted i n e q u a l i t i e s f o r o p e r a t o r s of t h e w i t h w e i g h t s o f t h e f o r m 1x1‘ were o b t a i n e d i n Aljart f r o m t h a t , t h e c l o s e s t a n t e c e d e n t s o f A P p r o b a b l y Rosenblum [l] a n d t h e F e f f e r m a n - S t e i n 7.1.

t y p e considered h e r e E.M. S t e i n [3]. weights are inequality 2.12

o f c h a p t e r 11. The b o u n d e d n e s s o f t h e maximal f u n c t i o n i n Lp(w) for w e A i s d u e t o Muckenhoupt [ l ] . The c o r r e s p o n d i n g r e s u l t P f o r t h e H i l b e r t t r a n s f o r m was e s t a b l i s h e d s h o r t l y a f t e r w a r d s i n

.

f l u n t , Muchenhoupt a n d Wheeden [ l ] The more s y s t e m a t i c a p p r o a c h o f Coifman a n d C . F e f f e r m a n [ l ] a l l o w e d them t o d e a l w i t h r e g u l a 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 T i n Rn b y means o f t h e i n e q u a l i t y :

465

IV.7. NOTES AND FURTHER RESULTS

(w e Am, 1 < p < m). This approach is different from the one followed in section 3, which relies upon the sharp maximal function. More general weights for which ( * ) is true are found in E.T. Sawyer

121

*

The fact that

Am

=

u

Ap

is due independently

to

Muckenhoupt

n>1

[3] and Coifman and C. Feiferman [l].

All other characterizations of Am given in 2.13 and 2.15 are more o r less implicit in both papers except for 2.15 (iii) (the equivalence of arithmetic and geometric means over cubes) which was found by the authors while writing this monograph and, independently, by HrusEev [ l ] . The characterization of A 1 given in 2.16 is due to Coifman (see Coifman and Rochberg [ l ] where it appears in connection with B.L.O.); CBrdoba and C. Fefferman had previously proved that (Mf)& E Am if 6 < 1 , a result which they used to obtain inequalities like corollary 3 . 8 (see CBrdoba and C. Fefferman [ l ] ) . 7.2.- The aforementioned papers of Muckenhoupt and Coifman-Feffennan prompted a long series of results about weighted norm inequalities by many different authors. We cannot even attempt to give here a short account of these developments. As a sample of them, let us merely mention the following:

a) The l a s t part of section 3 (specially theorem 3.9) is taken from Kurtz and Wheeden [l], where other related results are also obtained. Weighted multiplier inequalities beyond the A classes P have been investigated in Muckenhoupt, Wheeden and W.-S. Young [l], 121 * b) Let mu(x) and Au(x) denote, respectively, the nontange; tial maximal function and the area function (defined in 7.14 of Chap ter 11) of a harmonic function u(x,t) in the upper half-space Ign+1 . The equivalence between the norms in LP(Rn) of both func+

tions, 0 < p < m , which was stated in chapter 111, 8.5, has the following extension: Let w B Am and 0 < p < a ; if Au E Lp(w) and u(x,t) is suitably normalized, then

See Gundy

and Wheeden [l].

IV. WEIGHTED NORM INEQUALITIES

466

c) The following weighted version of the Carleson-Hunt theorem on pointwise convergence of Fourier series (see Carleson [ 3 ] , Hunt [l]) was obtained in Hunt and W.-S. Young [l]: For a funtion f in R, let S*f(x) = sup ISIf(x) I ; then S* is a bounded operator on I Lp(w) if w E Ap and 1 < p < m. This is clearly a strong improvement of corollary 3.12 (a). 7.3.- Analogues of the contexts, such u s :

AP

theory have been developed in different

a) Spaces of homogeneous type, as defined in chapter 11, 7.11. F o r the maximal operator in this setting see A.P. CalderBn [6] (from which the proof we have given of the R.H.I., lemma 2.5, was taken)

.

For the Bergman projection in the unit ball of (cn and Jawerth [l] and other domains, weighted inequalities are obtained in Bekoll6 [ l ] .

b) Martingales. For regular martingales (an abstract generalization of the process of dividing each dyadic cube of R” into 2” dvadic subcubes) the R.H.I. can be obtained by essentially repeating the proof given for Rn, and a complete analogy was found by several authors (e.g. Izumisawa and Kazamaki [ l ] ) . For general martingales, the R.H.I. may fail (Uchiyama [l]), but the boundedness of the maximal operator in Lp(w) for w e A can nevertheless be proved by using P the ideas of section 4 ; see Jawerth [l]. c) Ergodic theory. See Atencia and de la Torre [l], and also Martin Reyes [l]. 7.4.- We shall state here some simple general facts about weighted inequalities for an arbitrary operator T. To simplify matters, suppose T is defined in L”(Rn), 1 < p < m , and is linear and (essentially) self-adjoint. Consider the class Wp(T) consisting of 0 such that T is a bounded operator in those weights w(x) Lp(w), and denote by Np(w) the least constant such that (/lTf(x)

Ip

for all (good enough)

w(x)dx)l’P f.

5 Np(w) (jIf(x)

Ip

w(x)dx)”P

Then:

a) Wp is a cone, and Np(.) satisfies the ultrametrical inequality: N (u+v) 5 max (N ( u ) , Np(w)). Also, N (Au) = ND(u) if P P P

A > 0.

b) Wpl(T)

=

W (T)’-’’, P

i.e.,

w

E

W

(T) P’

if and only if

IV.7.

u E W (T) P

w = u ' - ~ ' with Np

I

(w)

= Np

(u)

467

NOTES AND FURTHER RESULTS

and, i f t h i s is the case, then

*

c ) Wp(T) i s a l a t t i c e : u , v E W (T) i m p l i e s m a x ( u , v ) E W (T) P P a n d min ( u , v ) E W ( T ) . M o r e o v e r P N (max ( u , v ) ) 5 , ' I p m a x ( N p ( u ) , N p ( v ) ) P To p r o v e t h i s , u s e ( a ) t o g e t h e r w i t h u + v 5 max ( u , v ) 5 u + v . A similar i n e q u a l i t y holds f o r

Z'Ip

min ( u , v )

with

Z1/p'

instead of

(use b ) ) . d) I f

T

i s t r a n s l a t i o n i n v a r i a n t , so a r e

W

P

and

Np(.).

As

a c o n s e q u e n c e , t h e y a r e a l s o i n v a r i a n t by c o n v o l u t i o n w i t h a f u n c -

t i o n k(x) 2 0 u E W and k P < Np(U).

*

( o r a measure p 2 0) u(x) < m a . e . , then

i n the following sense: I f * u E WP a n d N p ( k * u ) 5

k

The same p r o p e r t i e s h o l d f o r p a i r s o f w e i g h t s . A s a n a p p l i c a t i o n o f p r o p e r t y ( c ) , when p r o v i n g a n i n e q u a l i t y f o r Ap w e i g h t s , i t i s e n o u g h t o c o n s i d e r w e i g h t s w w h i c h a r e b o u n d e d away f r o m 0 and m ( i . e . w and w-l belong t o L m ) , p r o v i d e d t h a t t h e c o n s t a n t i n t h e r e s u l t i n g i n e q u a l i t y depends o n l y upon t h e constant Ap for w, a n d n o t upon llwll, o r IIw- 1 \loo. T h i s o b s e r v a t i o n c a n b e used f o r i n s t a n c e i n theorem 3 . 1 . A p p l i c a t i o n s o f p r o p e r t y (d) w i l l be given below. 7.5.-

We showed i n c h a p t e r I t h a t i f t h e c o n j u g a t e f u n c t i o n o p e r a -

must b e a b s o l u t e l y t o r i s bounded i n L 2 ( p ) , t h e n t h e measure c o n t i n u o u s w i t h r e s p e c t t o Lebesgue measure. I n t h i s c h a p t e r , i n e q u a lities in = w(x)dx,

have been considered only f o r measures dp(x) = s o t h a t t h e f o l l o w i n g q u e s t i o n a r i s e s n a t u r a l l y : Let Lp(p)

R. j = 1 , 2 , . . . ,n d e n o t e t h e R i e s z t r a n s f o r m s i n 1' r e g u l a r m e a s u r e s p i n Rn do t h e i n e q u a l i t i e s

R";

f o r which

( o r t h e i r weak t y p e a n a l o g u e s ) h o l d ? The a n s w e r i s t h a t ( * ) i m p l i e s t h a t p i s a b s o l u t e l y continuous (and t h e n , of course, " AP a s we know f r o m s e c t i o n 3 ) . The a r g u m e n t s k e t c h e d b e l o w was i n d i c a t e d

=

t o u s by B . J a w e r t h : I f (*I h o l d s , we c a n a p p l y t h e a n a l o g u e o f p r o p e r t y ( d ) i n 7 . 4 t o show t h a t t h e same i n e q u a l i t y i s s a t i s f i e d b y t h e w e i g h t f u n c t i o n s

468

IV. WEIGHTED NORM INEQUALITIES

u g * p = w6,

where

u

is

2 0 c o n t i n u o u s w i t h compact s u p p o r t ,

I u = 1 , and u 6 ( x ) = 6 - " u ( t ) . By theorem 3 . 7 , w 6 E A C Am, P w i t h A m - c o n s t a n t i n d e p e n d e n t o f 6 . Now, g i v e n a compact s e t E C R " w i t h [ E l = 0 , we f i x a cube Q c o n t a i n i n g E i n i t s i n t e r i o r , and t a k e open s e t s V . s u c h t h a t l V j l < and E C V j C Q . S i n c e J w6 + p ( 6 + 0) i n t h e weak-* t o p o l o g y o f m e a s u r e s , U r y s h o n ' s lemma g i v e s

with

C,

E >

0

fixed; therefore,

p(E) = 0 .

A n o t h e r r e l a t e d q u e s t i o n i s : Can t h e H a r d y - L i t t l e w o o d maximal operator o r a s i n g u l a r i n t e g r a l operator v e r i f y a s t r o n g type ( 1 , l ) i n e q u a l i t y w i t h r e s p e c t t o a p a i r o f r e g u l a r m e a s u r e s i n Rn? F o r Lebesgue m e a s u r e , we know t h a t o n l y t h e weak t y p e ( 1 , l ) i n e q u a l i t y h o l d s , and n o t h i n g b e t t e r c a n h o l d f o r any o t h e r m e a s u r e s i f t h e o p e r a t o r b e i n g c o n s i d e r e d i s t r a n s l a t i o n and d i l a t i o n i n v a r i a n t . I n < c llfll 1 (where R i s some R i e s z t r a n s f a c t , i f II Rfll 1 L [wl L ful f o r m ) , we c a n assume t h a t d u ( x ) = u ( x ) d x and d v ( x ) = v ( x ) d x w i t h u and v c o n t i n u o u s and e v e r y w h e r e p o s i t i v e ( j u s t c o n v o l v e w i t h a p o s i t i v e i n t e g r a b l e f u n c t i o n ) . By d i l a t i o n i n v a r i a n c e , t h e same i n e q u a l i t y h o l d s f o r t h e w e i g h t s u 6 ( x ) = u ( 6 x ) and v 6 (x) = v ( 6 x ) , and l e t t i n g 6 + 0 we o b t a i n t h a t R i s a bounded o p e r a t o r i n L 1 ( R n ) , b u t we know t h a t t h i s i s a b s u r d . T h i s s o r t o f argument w i l l a p p e a r a g a i n i n c h a p t e r V I , 2 . 8 and 2 . 9 . 7 . 6 . - The a n a l o g u e o f Problem 2 ( i n s e c t i o n 1 ) f o r t h e F o u r i e r t r a n s f o r m i s p r o b a b l y o n e o f t h e most i n t e r e s t i n g and h a r d p r o b l e m s concerning w e i g h t e d i n e q u a l i t i e s . The o b s e r v a t i o n made i n t h e p r e c e d i n g n o t e a b o u t t h e R i e s z t r a n s f o r m s no l o n g e r a p p l i e s i n t h i s c a s e , and o n e l o o k s f o r i n e q u a l i t i e s of t h e form

f o r some p , q 2 1 and r e g u l a r Bore1 m e a s u r e s p,v 2 0 . The b e s t p o s s i b l e i n e q u a l i t i e s o f t h i s t y p e w i t h m e a s u r e s o f t h e form d v ( x ) = ( x 1 8 d x a r e known ( s e e S t e i n [5] a n d , f o r du(x) = (xl'dx, the p a r t i c u l a r case p = q , c o r o l l a r y 7 . 4 i n chapter VI). Extensions

469

IV.7. NOTES AND FURTHER RESULTS

t o more g e n e r a l a b s o l u t e l y c o n t i n u o u s m e a s u r e s d p ( x ) = u ( x ) d x , d v ( x ) = v ( x ) d x a r e i n Muckenhoupt [4] ( s e e a l s o t h e r e f e r e n c e s c i t e d t h e r e ) , where a s u f f i c i e n t c o n d i t i o n f o r i n the case q = p ' , i s s p e c i a l l y simple:

i s given which,

(*)

(j

o(x)dx) 5 C for a l l r > 0 u(x)dx) ( l { u ( x ) > B Yl Io(x)
r. F o r m e a s u r e s which a r e n o t a b s o l u t e l y c o n t i n u o u s , t h e c a s e more e x t e n s i v e l y s t u d i e d c o r r e s p o n d s t o d v ( x ) = dx and d p ( x ) = r o t a If (*) t i o n i n v a r i a n t m e a s u r e s u p p o r t e d i n t h e u n i t s p h e r e En-,. h o l d s i n t h i s c a s e , we s a y t h a t t h e (Lp,Lq) r e s t r i c t i o n ( t o t h e u n i t s p h e r e ) holds f o r t h e F o u r i e r t r a n s f o r m . Observe t h a t t h i s h

means, i n p a r t i c u l a r , t h a t i t makes s e n s e t o d e f i n e which i s a s e t o f a measure 0 , f o r e v e r y t h a t t h i s a c t u a l l y happens f o r some p > 1 E.M.

f

in

f t- Lp(Rn). The f a c t was f i r s t n o t i c e d by

S t e i n , and an argument o f A . Knapp ( b a s e d on t h e d i l a t i o n p r o -

p e r t i e s o f t h e F o u r i e r t r a n s f o r m ) shows t h a t n e c e s s a r y c o n d i t i o n s f o r the (Lp,Lq) r e s t r i c t i o n i n Rn a r e

The c o n j e c t u r e i s t h a t t h i s i s a c t u a l l y t h e b e s t p o s s i b l e r a n g e . I n 2 R , t h e c o n j e c t u r e was c o m p l e t e l y p r o v e d by Zygmund [3]. For n L 3 , the conjecture is t r u e i n the case q = 2 , i.e., (Lp,L2) r e s t r i c n+ 1 t i o n h o l d s i f and o n l y i f 1 5 p 5 2 (3); s e e Tomas [l] . T h i s r e s u l t i s c o n n e c t e d w i t h t h e s p h e r i c a l summation m u l t i p l i e r s d i s c u s s e d i n 7 . 1 5 o f c h a p t e r 11; s e e 7.7.-

c.

F e f f e r m a n [4].

The c h a r a c t e r i z a t i o n o f p a i r s o f w e i g h t s f o r which t h e Hardy-

(theorem 4 . 9 ) L i t t l e w o o d maximal o p e r a t o r i s o f s t r o n g t y p e ( p , p ) T h i s c a n b e use'd t o s o l v e , was f i r s t o b t a i n e d i n E . T . Sawyer [ l ] . t h e f o l l o w i n g weak v e r s i o n o f t h e two f o r t h e maximal o p e r a t o r M , w e i g h t s p r o b l e m : F i n d t h o s e u ( x ) > 0 ( r e s p . w(x) > 0 ) f o r which

Lp(w) t o L p ( u ) f o r some w(x) > 0 ( r e s p . S a v y e r [3]. The p r e c i s e s t a t e m e n t s c a n be f o u n d i n c h a n t e r V I , where a d i f f e r e n t and s y s t e m a t i c a p p r o a c h t o t h i s k i n d o f p r o b l e m s w i l l be f o l l o w e d .

M

i s bounded from

u(x) > 0 ) ;

see

E.T.

For 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 , and even f o r t h e H i l b e r t t r a n s f o r m , t h e two w e i g h t s p r o b l e m ( i . e . , Problems 2 and 3 o f s e c t i o n 1 )

I V . WEIGHTED NORM INEQUALITIES

470

r e m a i n s c o m p l e t e l y o p e n ; s e e , however C o t l a r a n d S a d o s k y [2] f o r some p a r t i a l r e s u l t s i n t h e s p i r i t o f t h e H e l s o n - S z e g t i t h e o r e m . The i d e a o f r e l a t i n g C a r l e s o n m e a s u r e s a n d

w e i g h t s , which

A1

appears i n s e c t i o n 2 o f c h a p t e r I1 ( f o l l o w i n g C . Fefferman and S t e i n [ l ] ) can be f u r t h e r e x p l o i t e d , giving a u n i f i e d approach t o Carleson or S c o n d i t i o n s f o r weights. Consider t h e measures and t h e A P P maximal o p e r a t o r M f ( x , t ) d e f i n e d i n c h a p t e r 11, r i g h t b e f o r e

t h e o r e m 2 . 1 5 , w h i c h maps f u n c t i o n s f ( x ) i n Rn i n t o f u n c t i o n s d e f i n e d i n R Y c l = Rn x [ O , m ) . We t r y t o f i n d p a i r s (w,p), where w(x)

is a weight i n

that

M

Rn

i s bounded from

Lp(R:+',

p).

(a)

j6

and

The s t r o n g t y p e IM!a

a p o s i t i v e measure i n

p

Lp(Rn; w ( x ) d x )

(p,p)

XQ)(x,t))Pdu(x,t)

w h i l e , f o r t h e weak t y p e dition is

(p,p),

to

LP(R:+',

R:", p)

such or

h o l d s i f and o n l y i f

5

C a(Q)

t h e n e c e s s a r y and s u f f i c i e n t

con-

0

w h e r e , as u s u a l , a ( x ) = w(x) - l ' ( p - l ) and denotes t h e cube 1 R n + Q x [0, l ( Q ) ] i n . When w(x) : 1 , b o t h (a) a n d ( b ) a r e e q u i +

i s supp o r t e d i n Rn x { O } a n d i s g i v e n b y d p ( x ) = u ( x ) d x , t h e n ( a ) a n d ( b ) become, r e s p e c t i v e l y , t h e S and A conditions for the pair P S e e R u i z [ l ] a n d R u i z and T o r r e a 711. (u,w).

v a l e n t t o t h e C a r l e s o n measure c o n d i t i o n f o r

7.8.-

p.

When

p

This note is intended t o i l l u s t r a t e t h e second motivation f o r

t h e study of weighted i n e q u a l i t i e s given a t t h e i n t r o d u c t i o n t o t h i s d e n o t e t h e o r t h o n o r m a l s e q u e n c e o f Legenc h a p t e r . Let IP,(x) d r e P o l y n o m i a l s . We want t o know f o r w h i c h v a l u e s o f p i s i t t r u e that

(1

N

1 -1

f

pk) pk(x)

~

* f(x)

LP

(in

norm)

for every f E Lp([-l,l]). The N - t h p a r t i a l sum, T f i s an i n t e N 'N g r a l o p e r a t o r g i v e n by t h e D i r i c h l e t k e r n e l D N ( x , y ) = Pk(x)Pk(y) w h i c h , by t h e C h r i s t o f f e l - D a r b o u x f o r m u l a , c a n b e w r i t t e n a s DN(X9Y)

N+l

=

- X-Y

('N+l(')

'N(Y)

(KN(x)LN(y)

-

- pN(x) ' N + , ( y ) )

KN(y)LN(x))

+

=

t r i v i a l e r r o r terms

IV. 7

NOTES AND FUTHER RESULTS

where IKN(t) I 5 C(1-t2)- /4 -1 5 x 5 1, and denoting by TNf(x)

=

and

H

ILN(t)I

5

471

C(l-t2)1/4.

Thus, for

the I-iilbert transform

KN(x) H(fLN(x))

- LN(x)H(fKN)(x)

+

error

and a sufficient condition to have TN uniformly bounded in Lp 4 < p < 4. is (l-t2)’P/4 e A P (in the interval [ - l , l ] ) , i.e. 3 This can easily be shown to be the exact range of p ’ s (see Pallard [ I ] , Newman and Rudin [ l ] ) . By using the same argument together with the Hunt - W.S. Young result stated in 7.2 (c), i t a l s o follows that T,f(x) = s u p ITNf(x)l is bounded in L p ( [ - l , l ] ) if

4

N> 0 3, converges a.e. to f(x).

f e LP,

For similar results concergining other orthogonal expansions we refer to Muckenhoupt and Stein [ l ] , Askey and Wainger [ l ] , Benedek, Murphy and Panzone [1] . 7.9.- The first proof of the Ganett-Jones theorem (in LJ. Garnett was long and complicated. Further elaboration of and P. Jones [ l ] ) the ideas in that proof led to a still more complicated stopping time argument, by means of which, the factorization theorem for A P weights was originally proved (in P. Jones [ l ] ) . This result had been previously conjectured by bluckenhoupt. The approach followed here, based on the reiteration method of lemma 5 . 1 , originates in Rubio de Francia [ S ] , where the extrapolation theorem was also found. That approach was later clarified in Coifman, Jones and Kubio de Francia [ l ] (which contains also the application to the weak version of the Ilelson - S z e g i j theorem and the constructive decomposition of BMO) and Garcia-Cuerva [ 2 ] . B. Jawerth [ l ] further exploited these ideas. obtaining general versions of the factorization and extrapoweights and lation theorems. The result about restrictions of .4 P B.M.O. functions to arbitrary sets (5.6 and 5.71 are due to ‘l’, Wolff [ l ] . Another application of lemma 5.1 appears i n Neugebauer [ l ] : “ G i v e n p o s i t i v e function u ( x ) , v ( x ) - R”, t l z e nc,c17sstiY’ld i 7 i l ; i sufficient c o n d i t i o r i f o r t k P e x i s t i ~ i ? o sof w t‘ A szAch Lhz:, !’ u ( x ) 5 w ( x ) 5 v(x) 7:s that ( u ’ + € , v l C E ) E. .I for. s c m c E > 0“. I’ This result is interesting for the two weights problem for maximal and singular integral operators.

472

7.10.-

IV. WEIGHTED NORM INEQUALITIES

The extension of the AD theory to the product setting her straightforward, but we can mention heretwonon-trivial If A;

1 f E Lloc(Rn) and MSf(x) for every 6 < I ? ( + )

<

m

a.e., is it true that

Does the analogue of the Fefferman-Stein inequality (2.12 in chapter 11) hold for the strong maximal function, i.e.

("1 for arbitrary

j(MSf(x)ip

w(x)dx 5 jlf(x) IpMsw(x) dx

( 1 < PI

w(x) 2 O?

The best we know concerning the second problem is that ( * ) holds for all p > 1 if w e ;:A see Lin [ l ] . Weights of product type can also be defined in Rn+m = Rn x Rm. The family of intervals to be considered in this case is p = {Q(') x Q(2) : Q(') and Q(21 cubes in Rn and Rm respectively), the classes Ap(P) are defined as expected, and the whole section 6 applies with the obvious modifications. In particular, if T(l) and T(2) are regular singular integral operators in Rn and (defined as the comuosiR ~ ,respec ively, then, T = T(l) P T(2) tion of T ( acting on the first n variables and T (2) acting for every w e Ap(P), on the last m variables) is bounded in Lp(w) l < p < m . There is, however, a more interesting extension of the theory of s ngular integrals to product domains, which was initiated in R. Fefferman and E.M. Stein [l]: One can easily write the cancellation and size conditions satisfied by the kernel o f T = T(l) 0 T(2), and taking these as the starting hypothesis, without assuming that T is of product type, the LP-boundedness of T as well as A -weighted inequalities can be proved. A far-reaching P theory of singular integrals in product spaces (which we cannot describe here) has been built from this, including Hp and BMO spaces and non translation invariant operators. We refer to the survey by A. Chang and R. Fefferman [ I ] .

The close connection between B.M.O. functions and A weights, P as well as important consequences of it, became apparent in sections 2 and 5 of this chapter. Some other results emphasizing this connection are collected in the last four notes: (+) This question has recently been answered in the negative by

F. Soria.

IV.7. NOTES AND FURTHER RESULTS

473

7.11.- Let h : R -+ R be a homeomorphism of the real line. Suppose, for instance, that h is increasing. Then, h preserves B.M.O. (i.e. f(h-l(x)) belongs to B.M.O. whenever f(x) does) if and only if h' e A,. If this is the case, then also (h-')' E A, (by theorem 2.11) and f f0h-l is actually an isomorphism of B.M.O. onto itself. See P. Jones [2]. The corresponding problem in Rn was solved by H.M. Reimann, who showed that h preserves B.M.O. (R") if and only if h is quasiconformal (Reimann [l]).

+

7.12.- Given a measurable function b(x) in B, we denote by Mb the operator of pointwise multiplication by b(x). The commutator of this operator and the Hilbert transform H is [Mb,H]f(x)

=

b(x)Hf(x)

- H(bf)(x)

=

P.V. TI

lm

b(x)-b(y) X-Y

-m

f(y)dy

The following result was proved by Coifman, Rochberg and Weiss [ l ] "FOP any

p,

and o n l y if

1 < p < m, [Mb, H] b e B.FI.O.(R)".

i s bounde d i n

Lp(R)

3

We shall sketch the proof of the "if" part in the case p = 2. We suppose that b(x) is real, and consider the power series in the complex variable z = r e ia

By Cauchy's formula, the term corresponding to

n

=

1

is

But IITzll 5 llT,ll if I z I = r (11.11 stands for the norm as an operator in L2) and, for r > 0, Tr i s bounded in L 2 if and only if e 2rb e A2, which is certainly the case for small enough r, since b e B.M.O. Therefore 11 rMb, 1 5 1 llTrll < O1

The same proof applies to higher order commutators and to regular singular integral operators in R". The boundedness of [Mb, Rj] for b e B.M.O.(Rn) where R. J are the Riesz transforms is then used by Coifman, Rochberg and Weiss to give a weak factorization theorem for H1(Rn), See note 8.10 i n

IV. WEIGHTED NORM INEQUALITIES

474

chapter 111. 7.13.- The following characterization of the closure of Lm in B.M.O. can be obtained from the Garnett-Jones theorem. Given a to indicate that w satisweight w(x), we write w e R.H.I.(q) fies a reverse Hb'lder's inequality with exponent q (i.e., lemma 2.5 holds with 1+F. = q) . Then, for a real function f E B.M.O.(Rn), setting w = e f , the following assertions are equivalent: belongs to the closure of

a) f b) w

E

R.H.I.(q)

c) w e A d) w

E

A

P

P

e) wq e Am

and

w-'

for all

p > 1

and

E

w-'

A

P and w - E~ Am

E

Lm

in

R.H.I.(q)

and

w

for all for all

E

B.M.O. for all

R.H.I.(q)

q <

m

for all

q <

m

p > 1

q <

Observe that every f E V.M.O.(Rn) (defined as in the case of the torus, see 11.18 in chapter I) satisfies a). This is used by Guadalupe and Rezola (11 to prove A,-weighted estimates for the conjugate function operators in certain curves of the plane. 7.14,- The typical example of a function in B.M.O.(Rn) which is not bounded is log 1x1. The following extension of this fact has be a poZinomiaZ of been noticed by E . M . Stein [ l l ] : "Let P(x) degree k Rn; then log \ P ( x ) ) B.M.0. & \ \ l o g l P \ \ \ * 5 < ck,n ( c o n s t a n t d e p e n d i n g onZU o n k a n d n)". Here is the simple proof of this result. It is enough to prove a uniform R.H.I., for instance

for all cubes Q and all po ynomials of degree k. By translation and dilation invariance of the polynomials, it suffices to prove i t for the unit cube Q = [0,1]". But then, both sides of ( * ) define norms on the finite-dimensional space of polynomials of degree k in Rn, and therefore, they are equivalent.

CHAPTER V VECTOR VALUED INEQUALITIES

The estimates for the operators that we have considered so far can be formulated in the context of Banach space valued functions. The first half of this chapter is devoted to describe the general methods which allow to obtain such estimates. There are at least two reasons which motivate this type of extension lapart from the search of greater generality forits own sake): a) The non-linear operators which appear in Fourier Analysis can be viewed, in almost all cases, as linear operators whose range constists of vector valued functions. This will be the case for the maximal operators and square functions which are studied in sections 4 and 5 respectively.

b) There is an intimate connection between vector valued inequalities and weighted norm inequalities. We begin to explore this connection in the last section, but further insight will be gained with the theory to be developed in Chapter VI.

1 . OPERATORS ACTING ON VECTOR VALUED FUNCTIONS: SOME BASIC FACTS

Let (X,m) be a a-finite measure space. If B is a Banach space, a function F : X B is said to be (strongly) measurable if the following two conditions hold +

i) There is a separab e subspace F ( x ) 6 Bo

for

a.e.

ii) F o r each b ' E B" = dual of R , xt+ is measurable

Bo

of

B

such that

x E X the mapping:

A very casy consequence of this definition is that the positive f u n & is measurable. As in the s c a l n r l y v a l u e d tion IF[B : x+ [F(x)llg

case, we shall identify B-valued functions which coincide m-a.e. For 0 < p < m, we define the Bochner-Lebesgue space L E ( m ) con475

476

1:. VECTOR VALUED INEQUALITIES

s i s t i n g o f a l l measurable B-valued f u n c t i o n s

F

f o r which

( w i t h t h e u s u a l m o d i f i c a t i o n when p = m ) . Then Li(m) i s a Banach ( p - B a n a c h i f 0 < p < 1 ) s p a c e . I n t h e same way, w e d e f i n e t h e s p a c e L!B(m) = weak - LE(m) which i s “normed“ by

The f o l l o w i n g f a c t s a r e v e r y s i m p l e t o v e r i f y : (1.1)

f 6 Lp

LE,

If.bl

b E B, =

lfl,

IblB.

then Lf

(f.b)(x) = f(x)b 0 < p <

LPB f o r m e d b y a l l f i n i t e l i n e a r c o m b i n a t i o n s of f u n c t i o n s

in

belongs t o

LpPB

t h e subspace

m,

f.b

i s dense

L~B.

F = I: f . . b . € L 1 P B , we c a n d e f i n e i t s i n t e g r a l (which w i l l 3 1 j be a n e l e m e n t o f B ) i n t h e o b v i o u s way: Given

I F ( x ) dm(x) = C ( j ( 1 . 2 ) For e v e r y

Fo 6 L 1 P B ,

i

f j ( x ) dm(x))b

I I P o dmlB

I

5 IFoI

j

1.

T h u s we c a n

LBn t i n u i t y . If by c o e x t e n d t h e m a p p i x FW F dm t o a l l 1 then I F dm i s t h e o n l y e i e m e n t of B w h i c h s a t i s f i e s F E LB, <]F If

dm, b ’ > =

L B1

I

b’)

( b ’ E B*)

F E L; with 1 2 p 5 m, and G 6 L,: t h e n (x) = G(x)> i s t r i v a l l y a n i n t e g r a b l e f u n c t i o n , and we have

T h i s means t h a t

LEi

i s ( w i t h t h e n a t u r a l i d e n t i f i c a t i o n ) isome-

t r i c a l l y c o n t a i n e d i n t o t h e d u a l of

When

dm(x)

B

&

LE,

i s r e f l e x i v e one a c t u a l l y h a s

b u t we s h a l l have no n e e d o f t h i s f a c t .

L:?’

=

L E l C (L!)’.

(LEI*,

1 2 p <

m,

V. 1 . VECTOR VALUED I:U?.CTIONS

477

In this chapter we shall deal with operators on spaces LE. The following is a method of producing such operators which occurs naturally in many instances and which follows the same pattern used in ( 1 . 2 ) to define the integral: Suppose that T is a linear operator which maps L p into L q ; then T can be extended to an o p e rator TB = T @ IdB on Lp @ B by

TB (C f..b.)(x)

(1.4)

If

TB

j

J

C

=

J

j

Tf.(x)b. J J

happens to be bounded, i.e. (fj E Lp,

J 6 B) b. -

then it can be uniquely extended to a bounded operator from LE B into L z which we still denote by T . In this case, we say that T has a B-valued extension, and the following identity holds: (1.5)

< TBF(x),

b'>

=

T(
.)

,b'>) (x)

(FELg, b'€B*)

The same can be done

(this is actually a characterization of TB). with an operator T mapping L p into Lj.

< r < m. Then a measurable B - v a Examples 1.6.(a). Let B = Lr, 1 lued function is just a sequence F(x) = (fj(x))jEa where each f. J is a (real or complex valued) measurable function, and c lfj(x)lr < m a.e. The space Lg LP(L~) consists of all se-

j

quences

(f.) J

such that r l/r

I,

<

Given a linear operator T of strong type (p,q), TB is defined with a finite number of nonvanishing c o p on functions F = (f.). J JEN ponents by

Thus, 1' has a B-valued extension if and only if the following ine qua 1 it y h o 1d s (1.8)

I (? 3

r l/r ITfjI 1

I,

< -

c no ifj( j

r l/r

1

I,

for a l l countable sequences of functions fj 6 Lp. If this is the case, TB is defined by ( 1 .7) over the whole space LP(Lr).

V. VECTOR VALUED INEqUALITIES

178

where (b) As a more general example, take B = Lr(Q,u), is another o-finite measure space. Then L E ( m ) can be idec tified with the mixed norm space LrYP(~xX, ulm) which is formed by all p@m-measurable functions f(w,x) such that (Q,p)

The details of this identification are left to the reader, who is If an operator T has a also referred to Benedek and Panzone [ l ] . B-valued extension TB, this is defined by the analogue of ( 1 . 7 ) , namely (1.9)

TBf(w,x)

=

T(f(w,.))(x)

(c) There are also interesting operators on vector valued If functions which do not arise a s extensions of operators in L p . A and B are Banach spaces, we denote by L ( A , B ) the space of all bounded linear operators from A into B. Given a measurable L(A,B)-valued kernel K(x,y) defined in XxX such that K(x,.) 6 Lp’ f o r a.e. x , then the operator L (A,Bl (1 .lo)

TF(~) = j~(x,y).F(y)dm(y)

is well defined on Li(m), and one may try to investigate its conis, for each tinuity properties. If A = B = Lr, and if K(x,y) (x,y), a diagonal operator with entries (Kj(x,y)) j 6 N , then T is given by

where

T. is the linear operator defined on J

Lp

by the kernel

Kj (X,Y) We shall now prove two elementary results. First of all, we present the simplest case in which the R-valued extension of an operator can be assured to exist THEOREM 1 . 1 2 . L e t 1’ b e a l i n e a r o p e r a t o r iuhicii ?’s p o s i t i v r i i . ~ . g(x) 1. 0 i m p Z i e s Tg(x) 2 0 ) a n d b o u n d e d , with n o r m ITI, as i i r i o p e r u t o r f r o m LP(m) to L q ( m ) jzpsp. ~:(m)). hen, f o r a n y B a n a c h s p a c e B, T h a s a B - v a l u e d e x t e n s i o n T B w h i c h maps ~,:(rn) in~o l,i(mj iresp. L:B(m)) w i t h t h e samc n o r m : ITB I = I l l ,

V.l. VECTOR VALUED FGNCTIONS

Proof: I t is enough to show that,

for every

n

1 f.(x)b. and let (bi)ksg J' j=1 J of the unit sphere in B* such that lslB Let

F(x)

every

ITBF(x)

I

=

sup I
= SUP

=

IT(
k

,

sup I<s,

b;(>l,

for

k

{bl,b2,. . . ,bn].

in the span of

F 6 Lp f3 B

be a countable subset

=

s

4 79

Then, we have

b;(>( =

bi>) (x) I 5 T(IFIB)

(XI

where we have used the identity ( 1 . 5 ) and, f o r the last inequality, the positivity of T. 0 In particular, for a positive linear operator Lp, we have the inequalities

T

which is bounded

in

(when

r

=

this must be understood in the usual way).

m

The second result deals with operators of the type described i n the example 1.6.(c), and it generalizes the fact that convolution with 1 1 n a kernel in L (R") produces a bounded operator in L ( R ) . THEOREM 1 . 1 4 . Let K(x) b e a m e a s u r a b l e Rn and s u c h t h a t

L(A,B)-valued k e r n e l de-

fined i n

(1.15) Then, f o r every

dx 5

IK(x).alB 1 n ), F E LA(R

IalA

the integral I

( 1 .16) exists a.e.,

TF(x)

JK(x-y) .F(y)dy

=

,

5 C lFILl A Fubini's theorem and ( 1 . 1 5 ) give

and t h e o p e r a t o r

T

verifies:

lTF1

LB

Proof:

Given

F E LA(Rn),

and both assertions of the theorem follow immediately.

0

480

\I,

VECTOR VALUED INEQUALITIES

Simple as i t is, this result will nevertheless be useful in proving some Littlewood-Paley inequalities in section 2. Remarks 1.17.

If (1.15)

is replaced by the stronger condition

p W ( A , B )

dx <

m

Li

then the operator T defined by (1 . 1 6 ) is bounded from Lf3 to for all 1 5 p 5 m . More generally, Young's inequalities for convolution hold in this context if K € L:(A,B). However, one cannot obtain from (1 . 1 5 ) Lp boundedness of T if p > 1 , as the following example shows: Given functions k. 6 L 1 (Rn) (j=1,2,. . . ) with lkjil = 1, define A with values in the kernel KA(x) =' (kj(x))jsN L(L1,a) = L", where kx. = k.X The corresponding operator J {x : Ikj(x)l 5 A}' J Tx : L'(L1) L 1 is given by TA((f.))(x) = C k?*f.(x), and i t is I j J J clear that (1.15) holds with C = 1 . Suppose f o r a moment that T A : Lp(L') +. Lp is bounded for some p > 1 uniformly f o r all A > 0. By duality (see (1.3)) and letting h m, this would imply -+

-+

lsyp I k j

* fHpI 5

c

PIpl

which is known to be false if we take for instance where {R.}. 1 ICY tangles in R

(f E: LP'(R")) k. = (R.1' 3 I

1

xR,,

is a "dense" sequence in the family of all reccontaining the origin (see de Guzmgn [2]).

J

As i t is well known, interpolation is a basic tool to deal with operators in Lp spaces, and this turns out to be also the case f o r operators acting on B-valued functions. Here we shall simply state the Marcinkiewicz and Riesz-Thorin theorems in the form that we shall need to use them. No proof will be given since both theorems are proved exactly as in the scalar case. THEOREM 1.18. (Narcinkiewicz). L e t A , B b e Banach s p a c e s , a n d 2 T be a n o p e r a t o r d e f i n e d on L F + Lpl ( 0 < po < p 1 5 m) whose A v a l u e s a r e B-valued f u n c t i o n s . I f T i s sublinear (with the natu( p o , p o ) and (pl,pl), raZ d e f i n i t i o n of t h i s ) and of weak t y p e i.e. -

V.l.

t h e n , for a22

where

Cp

p

VECTOR VALUED FUNCTIONS

with

po < p < p,,

Co,

depe nds o n l y on

C,

481

we h a v e

&

p.

(Riesz-Thorin). Let T b e a l i n e a r o p e r a t o r w h i c h and in Lpl (Lrl), d. i s b o u n d e d i n Lp0(Lro)

THEOREM 1 . 1 9 .

(i=O,1; ci IF1 Pi ri L i ('I-& e and w i t h 1 5 po,pl,ro,rl 5 m . Lf - = - + P Po P1 -

I TFI LPi(Lri)

with

0 < 8 < 1,

then

Pi

F E: L

<

T

i s boundeu i n

Lp(Lr),

1 = 1-8 r

r.

(1 '1) +

r0

6

'1'

and more

precisely

(F E LP(Lr)) Further generalization o f these results can be found in BerghIn particular, the Riesz-Thorin theorem remains true Lofstrom [ l ] . r. if T' is bounded from Lpi(Lii) into Lpi (eB1), i=O,l, where A, B are fixed Banach spaces, and

This observation is sometimes useful for the study of certain opera tors which we now describe:

T

I_ zs c a l l e d l i n e a r i z a b l e if t h e r e e x i s t s a l i n e a r o p e r a t o r U d e f i n e d i n LP(m) whose v a l u e s a r e B-va2ued f u n c t i o n s ( f o r some Banach s p a c e BI a n d DEFINITION 1.20. A n o p e r a t o r

defined i n

LP(m)

such t h a t

ITf(x)

I

=

IUf(x)AB

( 5 E. ~"(m))

I t is clear that linear operators are linearizable, but there are some other interesting examples like (1

(1 where

(Tn n6B

is a sequence of linear operators

(we take

B

=

Lm

482

\I.

VECTOR VALUED INEQII.I?I,ITIES

for the operator M and B = t 2 for G ) . Continuous analogues of the operators defined by ( 1 . 2 1 ) and ( 1 . 2 2 ) are also linearizable, and these include all kinds of maximal operators and g-functions of wide use in Fourier Analysis. COROLLARY 1 . 2 3 . Let T b e a Z i n e a r i z a b l e o p e r a t o r w h i c h i s b o u n d e d i n Lp(m) f o r some 1 p < m. Lf T is p o s i t i v e ( i n t h e s e n s e that: If(x)l 5 g(x) a . e . i m p l i e s ITf(x)i 5 Tg(x) a . e . 1 t h e n , the following i n e q u a l i t i e s hold:

Proof: Observe that ( 1 . 2 4 )

is trivially true if

r

=

p,

and since

it is also true for r = m . Now ( 1 . 2 4 ) would follow for all p 5 r 5 m if we were allowed to use interpolation. But we are, because ITf(x)I = IUf(x)lB for some linear operator U , and thus, ( 1 . 2 4 ) is equivalent to the boundedness of the linear operator:

u defined by

: LP(P)

c((fj) j 6 B ) (x)

=

-+

LP(Q

(Ufj (x)) j6N. 0

It may be interesting to remark that what we have just proved is a generalization of ( 1 . 1 3 ) (only for some values of r ) to a wider

class o f operators. I t is natural to ask whether ( 1 . 2 4 ) would be true for 1 < r < a. We shall present below some examples in which the answer is affirmative, but, for the general class of operators considered in the Corollary, ( 1 . 2 4 ) is best possible.

2. A THEOREM OF MARCINKIEWICZ AND ZYGMUND

The basic result to be proved in this section it that 12 -valued extensions of linear operators in LP spaces always exist. TO begin with, we need to describe some elementary facts from Probabi lity. DEFINITION 2 . 1 . By a G a u s s i a n s e q u e n c e we s h a l Z mean a s e q u e n c e

(

z

~

Q random v a r i a b l e s

)

~

~

~

( i . e . r e a l valued measurable f u n c t i o n s )

V . 2 . THEORE?? OF MARCI!JKIEWICC.-2YYGMUND

i n a p r o b a b i l i t y space

( R , P ) w h i c h a r e i n d e p e n d e n t and i d e n t i c a l l y

h ( x ) = e-TIx2,

distributed with density function

P ( z j E A)

(2.2)

483

=

P({w € R : z j ( w ) E A}) =

& - TIX 2 e dx

I

(j€:eJ)

A

f o r every Borel subset

A

0 f

R.

5

What ( 2 . 2 ) means i s t h a t , f o r e v e r y j N, u n d e r t h e m a p p i n g z . : R + R i s e-TIx d x . J is

t h e image m e a s u r e o f P An e q u i v a l e n t c o n d i t i o n

f o r e v e r y B o r e l f u n c t i o n $ ( x ) on R f o r which t h e i n t e g r a l t o t h e r i g h t e x i s t s . I n p a r t i c u l a r , we s e e t h a t z . € I,'(,) for all J r < m, s i n c e

What makes G a u s s i a n s e q u e n c e s u s e f u l f o r u s i s t h e f o l l o w i n g p r o p e r t y : LEMMA 2 . 4 .

(Xj)jsN r < m,

Let

(zj)jEN

(fi,P),

be a Gaussian sequence i n

6 12. T h e n , t h e sereies X.z converges i n J j and t h e f o l l o w i n g i d e n t i t i h o l d s

( 0 < r < -)

(2.5)

where

br

i s d e f i n e d by ( 2 . 3 1

( t h u s , i t d e p e n d s o n l y on r l

Proof: I t s u f f i c e s t o prove (Z.5)

N

such t h a t

and l e t

L r (0) f o r a l l

1

\hjI2 = 1.

Let

.. z

f o r a f i n i t e sequence =

c 1.z.

J J' o r t h o g o n a l t r a n s f o r m a t i o n i n RN sudh t h a t : = (1,0,. ..,0). Then, f o r e v e r y Borel s u b s e t i=l

. I

P ( z € A) =

j

N ( x ~ ) ~ = ,

a n d d e n o t e by

h(xl)h(x2)

p(h,

A

an

, x z , . . . , l N )=

of

...

p

R: h(xN)dx =

I{x€RN: C X J- X J.€A}

I =

--

dXl

= \ A

Therefore,

e ~X€RN:x,€A~

z

i s a l s o d s t r i b u t e d w t h t h e same d e n s t y f u n c t on :

V. VECTOR VALUEP INEQUALITIES

484

2

h ( x ) = e-"

,

and

1121,

=

br.

0

Remark 2 . 6 . G a u s s i a n s e q u e n c e s d o e x i s t . An e x p l i c i t c o n s t r u c t i o n may b e t h e f o l l o w i n g : T a k e R = 'R = RxR x . . . x R x . . . , a n d d e f i n e P a s t h e p r o d u c t o f c o u n t a b l y many c o p i e s o f t h e m e a s u r e d v ( x ) = =

h(x)dx;

t h e n , a Gaussian sequence

( ' j ) jEN

in

(Q,P)

i s defined

by z.(w) = J

( 0 = (wl , w 2 , . .

0 .

J

. ' W n , . . .)

Now we a r e a b l e t o p r o v e o u r m a i n r e s u l t . A s a l w a y s ,

THEOREM 2 . 7 .

( M a r c i n k i e w i c z a n d Zygmund)

,

T : Lp

m, w i t h norm b o unded l i n e a r o p e r a t o r , 0 < p , q 2 h a s a n L - v a l u e d e x t e n s i o n , and more p r e c i s e l y :

with

P 5 9 .

C

depending o n l y on

p

and

P19

Proof: Consider f i r s t t h e case sequence

(zj)jEN

in

(fi,P),

and t h e t h e o r e m i s p r o v e d w i t h C = 1. P9P For every satisfies

(X,m)

will

u - f i n i t e m e a s u r e s p a c e , a n d we s h a l l w r i t e s i m p l y LP(X,m).

be a f i x e d instead of

-

E n)

q.

IT1

Moreover,

+

.

C

Lq

Lp

be a T

Then,

P 34

= 1

9

q 5 p . I f we t a k e a G a u s s i a n Lemma 2 . 4 i m p l i e s

= b /b In particular P99 P 9' Now we c o n s i d e r t h e c a s e p < q , a n d d e n o t e s = q / p . u(x) 0 w i t h I u ~ , , 5 1 , t h e o p e r a t o r Tuf = u 1 I P . T f

C

485

V . 2 . THEOREM OF I,!.~RCINKIEWICZ-ZYCMUND

and by the case already proved, we have

4 J

2 1/2 lfjl 1 1,.

Next, we shall establish the analogue of the theorem for weak type operators. The dea is of a function as a supremum of L r -norms, by Kolmogorov condition, which can be stated as LEMMA 2.8. Let

0 < r < q <

m,

n

Marcinkiewicz-Zygmund to express the L2-norm means o f the so called follows: f(x)

and for e a c h f u n c t i o n

0,

define

Iflq*

t m(Ix

= SUD

t>o

N

q9r

(f)

=

sup

E

If

: If(x)l

> t}) 119

'Elr

(-1 =

m

s

w h e r e t h e r r ~ u p ' ir s t a k e n f o r a l l m e a s u r a b l e s e t s

0 < m(E)

<

m.

-1 r

E

- -) 1

q

with

T h e n we h a v e t h e i n e q u a l i t i e s

Observe that I .Iq* is the usual weak-Lq norm. On the other hand, if instead of XE we allow in the definition of N (f) arbitrary q,r functions @ E Ls, then we get exactly If1 (by the converse of 9 Halder's inequality). Proof: Given t > 0, for an arbitrary set of finite measure, we have

E C {x : If(x)l

> t}

and the first inequality follows. To prove the second inequality, we can assume Iflq* = 1, so that the distribution function of f satisfies

Then, for every set E of finite measure, it is clear that X (t) 5 inf(m(E),t-q), and we get fXE

V . VECTOR VALUED INEQIJALITIES

486

< {r

1,”

m(E)

= ( h r m(E) +

T a k i n g h = m(E) - l l q desired. THEOREM 2 . 9 .

Let

tr-q-l dt}’lr =

rhr-q l/r

1 q-r

(p,q),

1/r

l f X E Q r 5 (&)

we o b t a i n

0 < p,q <

r a t o r of weak t y p e

,,”

tr-’dt + r

and a s s u m e t h a t

a,

m(E)”’

T

as

i s a l i n e a r ope-

i.e.

m({x : I T f ( x ) I > t l ) l / q 5 $l l f l p

Then

T

L 2 - v a l u e d e x t e n s i o n o f weak t y p e

has an

precisely,

there exists

C’

that

P,9

d e p e n d i n g o n l y 03

m({x : (C l T f j ( x ) \ 2 ) 1 ’ 2 > t } )

-

j

P99 t

(p,q). p

and

More q

such

,

1,

P r o o f : W r i t e H = L 2 and F = ( f . ) 6 LE, so that 1 j€N T H F(x) = ( T f . ( x ) ) j c N . Take r q , and u s e t h e p r e v i o u s lemma t o 1 obtain

l T E f l r 5 (&)’Ir

(2.10) where

E

IXEls

lflp

i s an a r b i t r a r y s e t o f f i n i t e measure,

and T E f = (Tf)XE. t o (2.10) gives

( f € LP) l/s = l/r - l/q,

Now, t h e Marcinkiewicz-Zygmund t h e o r e m a p p l i e d

and t h e p r o o f i s ended by i n v o k i n g t h e f i r s t i n e q u a l i t y i n Lemma 2 . 8 . We remark t h a t t h e o r e m s 2 . 7 and 2 . 9 a r e e q u a l l y t r u e f o r a l i n e a r operator

T

from

LP(X,m)

to

Lq(Y,u)

(or

L:(Y,p)),

where

( y , ~ ) i s a n o t h e r measure s p a c e .

The f o l l o w i n g p o s s i b l e g e n e r a l i z a t i o n o f t h e o r e m 2 . 7 a r i s e s n a t u r a l l y : Given a f a m i l y

T

o f l i n e a r o p e r a t o r s u n i f o r m l y bounded i n

wip5 M vnp

(f 6 Lp,

T 6 T)

Lp(m):

487

V. 2. THEOREM OF ?IARCINh'IEWICZ-ZVC?!UND

does the

L 2-valued inequality

(2.11)

I (;J

2 1/2 lTjfjl 1 I],

2 1/2

5.

c I ( Xj l f j l

1

IP

(Tj E 7 )

automatically hold?. The answer is trivally affirmative when but only in this case, as the following examples show.

p = 2,

EXAMPLES 2.12. (a) Let us consider in Lp(R) the translation operators T.f(x) = f(x-j), j=l ,2,3,. . . . If (2.11) is supposed to hold J for some p > 2, we apply it to (j=1,2,. . . ,N) to fj = 1-1 ,1 -jl obtain

i/ c ~ N 1 p , which is impossible for large N. On the i.e., N ~ other hand, if (2. 1 ) holds for some p < 2, we take fl = f 2 = . . . = fN = x so that T.f. = X and we J J [j ,J+11 /2 [ 0 , ! 1 ' < C N which again is absurd for large N. obtain ,'Ip -

(b) A much deeper and interesting counterexample is provided by the directional Hilbert transforms in R", n > 1 . These are defined by (Huf)^(c)

= -i

sign([.u)!?(<)

for each unit vector u 6 C n - l = Ix E Rn : 1x1 = 1). If p is a rotation taking u into the vector ( l , O , . . . , 0), one easily shows that HUf(PX)

=

H(fOP(. 'X2,. . . ,xn)) (x,)

where H is the Hilbert transform in R 1 . Thus, each HU i s a bounded operator in Lp(Rn) with norm equal to the norm of H as an operator in Lp(R), 1 < p < -. However, the inequality

is false for every p # 2. This was shown by C. Fefferman [2] in order to prove that the characteristic function of the unit ball is not a multiplier in Lp(Rn). We refer to M. de Guzmrin [2] for this and related results. Here is however an example in which inequality (2.11) does hold. By an interval I in Rn we shall mean the Cartesian product of n

V. VECTOR VALUED INEQUALITIES

488

i n t e r v a l s o f R . The " p a r t i a l sum" o p e r a t o r the multiplier xI:

SI

i s t h e n d e f i n e d by

(SIf)^(S) = m x I ( s )

Let

COROLLARY 2 . 1 3 .

n = 1,

In t h e c a s e

{Ij]

be a r b i t r a r y i n t e r v a l s i n

Then

Rn.

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 a l s o h o l d s :

(2.15)

Proof: Matters a r e e a s i l y reduced t o considering i n t e r v a l s o f t h e form x...x

I ( a ) = [al,m)x[a2,m) a n d we w r i t e

Sa

instead of

(sof)"(SI

SI(a).

= m x p ( S 1) X P ( S 2 )

( a 6 R")

[an,m) Then

...

Xp(Sn)

(P =

[O,m))

a n d we c a n w r i t e S o = 2 - " ( I d + i H l ) o ( I d + i H 2 ) O . . . O (Id + iHn)y w h e r e Hk d e n o t e s t h e H i l b e r t t r a n s f o r m i n t h e d i r e c t i o n o f (k) e k = ( O , O , . . . ,O , l , O y...,O). T h e r e f o r e , So i s a bounded o p e r a t o r i n Lp(Rn), 1 < p < m. On t h e o t h e r h a n d

a n d Theorem 2 . 7 g i v e s u s 2 1/2

IP

J

When

n = 1,

the operator

So =

71 ( I d

+ iH)

i s a l s o o f weak t y p e

( l , l ) , a n d t h e same a r g u m e n t c o m b i n e d w i t h Theorem 2 . 9 g i v e s (2.15).

n

T h i s c o r o l l a r y i s v e r y u s e f u l i n L i t t l w e o o d - P a l e y t h e o r y , a s we s h a l l s e e i n s e c t i o n 4 . We c a n a l r e a d y show how i t w o r k s t o p r o d u c e some i n e q u a l i t i e s o f L i t t l e w o o d - P a l e y t y p e w h i c h a r e s i m p l e r b u t l e s s known t h a n t h e u s u a l o n e s .

V . 2 . THEOREM OF E~4L\P,CINI;IEWICZ-ZYI;MUND

489

THEOREM 2 . 1 6 . Let I b e a bounded n - d i m e n s i o n a 2 i n t e r v a l , a n d f o r e a c h l a t t i c e p o i n t k 6 2" consider the trans2ated i n t e r v a l k + I .

f 6 Lp(Rn)

Then, f o r e v e r y

with

2 5 p

t h e foZ2owing i n e -

m,

G a l i t y holds:

1';

Isk+IfI

2 1/2

8,

5 'p

iflp

@ ( x ) i n R", and c o n s i d e r t h e 2 f i n i t e - d i m e n s i o n a l H i l b e r t s p a c e B = 1 ( F ) , where F i s a f i n i t e

P r o o f : Take a Schwartz f u n c t i o n subset of

2".

maps L:(Rn) for q = 2

and f o r

The o p e r a t o r

b o u n d e d l y i n t o Lq(Rn) for all 1 5 q 5 2 . In fact, t h i s i s a consequence o f P l a n c h e r e l ' s theorem, s i n c e

q = 1, K(x)

i t f o l l o w s f r o m Theorem 1 . 1 4 , s i n c e t h e k e r n e l

(e

=

defining the operator

=

where

Q

2nik.x T

@(x))k6F

= L(B,C)

s a t i s f i e s , f o r each

b = (bk)ksF

IblB denotes t h e u n i t cube of

R",

and

C

because

By i n t e r p o l a t i o n we o b t a i n t h e r e s u l t f o r @ 6 S(Rn). The a d j o i n t o p e r a t o r = (Tkf

(where

(Tkf)^(S)

=

f(c)+(c-k))

from Lq'(Rn) i n t o Lj'(Rn) letting F increase t o a l l

1

5 q 5 2.

kEF

i s t h e r e f o r e a bounded o p e r a t o r

w i t h norm i n d e p e n d e n t o f 2" we g e t

F,

so t h a t

490

V. VECTOR VALUED INEQUALITIES

Now, assume that we take @ such that $ ( E ) = 1 for all 5 E I. Then SkcIf = Sk+I(Tkf), and we can invoke Corollary 2.13 (applied to the sequence of functions (Tkf)kEzn) together with the preceding inequality to complete the proof. 0 Remarks 2.17. (a) One can replace in Theorems 2.7 and 2.9 L" by any other (separable) Hilbert space B. In particular, we can obtain a continuous version of both theorems by choosing B = L 2 (R, w(t)dt), where w(t) 2 0. If this is applied in the proof of 2.13, one obtains the corresponding continuous version of that Corollary, namely:

Where ft(x) is a measurable function of (t,x) € RxR", and I(t) are intervals in Rn whose vertices are measurable functions of t. Of course, there i s also the corresponding continuous version of the weak type inequality (2.15)

.

(b) By taking into account the known weighted norm inequalities for the Hilbert transform and the product operators arising from i t (see Chapter IV, (6.2)) it follows immediately that inequality (2.14) holds in Lp(w) provided that w E A* and 1 1, we cannot replace the intervals { I j } in Corollary 2.13 by parallelepipeds in arbitrary directions. This fact is actually equivalent to the counterexample 2.12 (b) (see however 7.3 below). (d) The inequality of Theorem 2.16 is false if p < 2. To N -1 see this, define fk in R by 1, = X [k,k+l] and = fk 2nikx k = o (so that f^ = X[o,N~). Then, Sk+If(x) = fk(x) = e f0W, k = O,l, . . . ,N-1, and Theorem 2.16 implies:

f(x)

=

Nfo(Nx),

so that the inequality of

V.3. VECTOR VALUED SINGULAR INTEGRALS

which, when

N

+ m,

forces

49 1

2 5 p

3. VECTOR VALUED SINGULAR INTEGRALS

So far, we have proved two general results concerning the existence of B-valued extension for a linear operator T which is bounded in LP(m): The extension exists if T is positive (theorem 1 .12) 3 r if B is a Hilbert space (theorem 2.7). This is all that can be said in general. It is the study of particular cases what has to be done now (take an interesting non positive operator T appearing in Fourier Analysis, take some fixed Banach space, such as B = L r with r # 2, and try to know wheter T has a B-valued extension o r not). This is, by far, the more interesting part, and the first natural question would concern the Hilbert transform: Does the inequality

hold? An affirmative answer can be given if r = 2 and 1 < p < m by the theorem of Marcinkiewicz and Zygmund, and this can be slightly improved by interpolating with the obvious case: 1 < p = r < -. However, as early as 1939, Boas and Bochner [l] proved that (3.1) does hold for all 1 < p , r < a. They used a very clever argument involving complex function theory, but now, with the Calder6n-Zygmund machinery available, the proof of (3.1) and its n-dimensional analogues becomes fairly easy. We shall adopt a quite general viewpoint: Let A and B be Banach spaces, and consider a kernel K(x) defined in Rn whose values We are bounded linear operators from A to B : K(x) 6 L(A,B). assume that K(x) is measurable and locally integrable away from the origin, so that the integral (3.2)

is well defined for all F € Li(Rn) with compact support and for To make the Calder6n-Zygmund theory work in this all x i! supp(F). context, some HUrmander type condition must be imposed on K[x), like

V. VECTOR VALUED INEQUALITIES

492

Now, we can state the fundamental result, essentially ta.sn from Benedek, CalderBn and Panzone [ l ] : THEOREM 3.4. Let T b e a bounded l i n e a r o p e r a t o r from Li(Rn) to Li(R") f o r some f i x e d r, 1 < r < m, and a s s u m e t h a t , when F bounded and w i t h compact s u p p o r t , TF(x) i s d e f i n e d b y ( 3 . 2 ) for e v e r y x i? supp(F), g j t h a k e r n e l K(x) s a t i s f y i n g 1 3 . 3 1 . Then, T c a n be e x t e n d e d t o an operdator d e f i n e d i n

that -

ti,

1 5 p <

m,

such

LA

every

A-atom a(x)

The space BMO(B) appearing in (3.7) is defined in a natural way: it consists oE all B-valued functions F(x) for which

Observe that

F € BMO(B)

I (IFllB)IB)lo

implies

5

lFIB € BMO,

and

IFIBhfo(B)

On the other hand, A-atoms are also as expected, i.e., functions m a E: LA supported in a cube Q and such that: -1 Ia(X)8,

5

1Q1

Proof of Theorem 3.4: I t is a straightforward generalization of Theorem 5.7 in Chapter 11. One first proves (3.6) by using a Calder6n-Zygmund decomposition of IFIA, where F € Ll L;, and estimating its "good" part by means of the L*-boundedness of T (a slight modification is required here in the case r = m ) and its "bad" part by means of (3.2) and (3.3). Interpolation (use Theorem 1.18) thengives the estimates (3.5) for 1 < p 5 r.

n

T h e second step is (3.7), whose p r o o f ( a s well as that of (3.8)) is

493

V.3. V E C T O R VALUED SINGULAR INTEGRALS

e x a c t l y a s i n t h e s c a l a r c a s e . Then, t h e s u b l i n e a r o p e r a t o r SF(x)

=

ff

(ITFIB) (x)

satisfies

..

because t h i s i s t r u e f o r Theorem 1 . 1 8 . Now, s i n c e

p = r,

p

=

and we c a n a p p l y a g a i n

m

( s e e C h a p t e r 11, 3 . 6 ) t h e r e m a i n i n g i n e q u a l i t i e s i n ( 3 . 5 ) a r e a l s o proved.

n

As i n the scalar case, the constants

p,

on

M

a n d on t h e

Cp,

(Li,Li)-norm of

1 5 p

5

m,

depend o n l y

T.

T h e r e a r e c e r t a i n r e m a r k s t o be made now which w i l l p a v e t h e way t o some o f t h e a p p l i c a t i o n s o f v e c t o r v a l u e d s i n g u l a r i n t e g r a l s . The f i r s t one i s t h e s t r i k i n g f a c t t h a t t h e v e r y s t a t e m e n t o f Theorem 3 . 4 . l e a d s t o i t s immediate self-improvement i n t h e f o l l o w i n g s e n s e : THEOREM 3 . 9 . Under t h e h y p o t h e s i s of t h e p r e c e d i n g t h e o r e m , t h e

following i n e q u a l i t i e s a r e a l s o v e r i f i e d f o r I{X

: ( TJ :

I T F ~ ( x ) I ~ )l/q

>

All

<

c x -’

- 9

1 < p,q < I(1

m:

I F j ( x ) I A )l / q d x

j

1 < q < m, t h e Banach s p a c e o f a l l a . E A s u c h t h a t I(..)[= 3 J The p r e v i o u s t h e o r e m t e l l s u s t h a t

P r o o f : Denote by l q ( A ) , sequences ( a j ) j c N with =

(z

l a j l A ) ’Iq <

m.

j

T : L: -+ L i i s a bounded o p e r a t o r . I t i s t h e r e f o r e t r i v i a l t h a t t h e o p e r a t o r .T : ( F . ) . I-+ (TF.). maps ~9 boundedly i n t o J JcN J JEN lq (A) Lq , and i t i s d e f i n e d by t h e k e r n e l R(x) E L ( l q ( A ) , l q ( B ) ) eq ( B ) g i v e n by k ( x ) . ( ( a j l j c N ) = ( K ( x 1 . aJ. ) J. E N The o p e r a t o r norms o f

K(x)

and

IK(x-y) - K ( x ) l

so t h a t

k

K(x) =

c o i n c i d e , and s i m i l a r l y

IK(x-y) - K(x)[

s a t i s f i e s ( 3 . 3 ) a n d Theorem 3 . 4 a p p l i e s a g a i n p r o v i n g

V. VECTOR VALUED INEQUALITIES

494

0

the desired inequalities.

The second important fact t o be noted is that, if the Banach space A is one-dimensional: A = C , then L(A,B) = B. Moreover, in this case the L 1 -estimate (3.8) has a more interesting meaning due to the atomic decomposition of H 1 (Rn) described in Chapter 1 1 1 . We state explicitly the corresponding results as a Corollary. COROLLARY 3.10. Let

T

: Lr(Rn)

o p e r a t o r f o r some f i x e d

Tf(x) f o r some

=

r,

1

r 5

m,

(x

B-valued measurable f u n c t i o n

1 n LB(R )

K(x)

(with I . I

e z t e n d s t o a bounded o p e r a t o r f r o m

1 n H (R )

and assume t h a t

If(y)K(x-y)dy

Harmunder‘s c o n d i t i o n ( 3 . 3 1

from

b e a bounded l i n e a r

Li(Rn)

+

and f r o m

Lp(R ) 1 n LB(R )

e

supp(f))

which v e r i f i e s

.IB).

I Then T LE(R”), 1 < p < 1 n t o weak-LB(R ) . =

2

a,

1 n ) can be defined in terms of atoms for an arbiOf course, HA(R trary Banach space A , and if this is done, the estimate (3.8) can 1 1 be read as follows: T : HA LB. -+

A final remark, which should have been observed by anyone trying to adapt the proof of Theorem 5 . 7 in Cahpter I1 to the vector valued case, is the following: In the proof of the inequalities (3.6), (3.8) and (3.5) for 1 < p 5 r , one can replace (3.3) by the weaker condition

(for all a 6 A , y E R”). However, (3.7) and the remaining inequalities in (3.5) do not follow from (3.3’). We do not emphasize very much the interest of this weaker condition since, in most of the applications of vector valued singular integrals, (3.3) does hold. Turning to the problems stated at the beginning of this section, we have the first immediate applications of our general result. Let THEOREM 3.11. -

be a sequence o f tempered d i s t r i b u t i o n s

(k.)jcN 1ijim

i n Rn with s u p < m. Assume t h a t e a c h k. c o i n c i d e s a u a y 3 J f r om t h e o r i g i n w i t h a l o c a l l y i n t e g r a b l e f u n c t i o n , and

V . 3 . VECTOR VALUED SINGULAR INTEGRALS

495

T h e n , t h e f o l l o w i n g i n e q u a l i t i e s h o l d , p r o v i d e d t h e r i g h t hand s i d e

is f i n i t g : (3.12)

1

<

> A l l 5

lkj*fj(x)\r)”r

I { x : (;

cr

-1

A

r l/r

I(C Ifjl 1 J

(1 < r <

I1

m)

P r o o f : F i r s t o f a l l , we m u s t r e c a l l t h a t , u n d e r t h e h y p o t h e s i s , k.*f 1

1 1

Lp

and t h e

m,

for a l l

and c o n s i d e r a s Banach s p a c e s :

l
A = B = br.

The

operator TF(x) = ( k . * f . ( x ) ) . J 3 JCN

Li,

i s t h e n bounded i n TF(x) where

=

K(x) 6 L ( B )

(I:

(f.). 6 L;) 1 JGN

=

and i t i s c l e a r t h a t

[ K(x-y) .F(y)dy

(x

J

e

supp(F1)

i s t h e d i a g o n a l o p e r a t o r w hose e n t r i e s a r e

kj ( x ) : K(x) . b = ( k . ( x ) b . ) . 1 1 JCN

(b

=

( b . ) 6 B) 1

Since

I K x - y ) - I < ( x ) \ ~ ( =~ )s u p I k . ( x - y ) - k 1. ( x ) l , (3.3) is 1 1 v e r i f i e d , a n d T h eo r em 3 . 4 a p p l i e s g i v i n g t h e d e s i r e d v e c t o r v a l u e d i n e q u a 1i t e s .

Cl

l h e h y p o t h e s i s o f t h l s t h e o r e m are fulfillcd i n t h c f o l l o w i n g s p e cially interesting cases:

k (XI

a ) When e a c h

lip)I 5 b ) IVhen

is

J

C,

k (x) J

=

C’

o u t s i d e t h e o r i g i n and

IVkJ(x)l 5 C h(x)

f o r a11

Ix~-~-’ J ,

where

( 1 c‘ 8 )

c

6 I,*(R”)

,ind

496

V. VECTOR VALUED INEQUALITIES

The second case can also be obtained as an application of Theorem 3.9 which A = B = C (complex numbers). What this case means is that every singular integral operator has a bounded Lr-valued extension if 1 < r < m. In particular, we can take k(x) = p.v.-TI1X in R , proving (3.1) for all 1 < p , r < m . An application of this inequality is the following improvement of Corollary 2.13 (without modifying its proof) : COROLLARY 3.14. Let (2.15)

&!

(c I

.Ir+'

r

be f i x e d ,

1

r <

m.

The i n e q u a l i t i e s (2.14)

r e m a i n t r u e if we s u b s t i t u t e e v e r y w h e r e

(C

r.

1.1

2) 1/2

j

j

Not surprisingly, weighted estimates can also be obtained for vector singular integral operators, provided that we assume more regularity for the kernel K(x), namely

We invite the reader to look for a complete analogy with the results for regular singular integral operators which, in the scalar case, were described in Chapter 11, Theorem 5.20, and in section IV.3. Here, we merely state the result which we shall need to apply in the next section. THEOREM 3.16. S u p p o s e t h a t T i s a s i n T h e o r e m 3 . 4 w i t a k e r n e l K(x) s a t i s f y i n g , i n s t e a d of Hsrmander ' s c o n d i t i o n , t h e s t r o n g e r < a s s u m p t i o n ( 3 . 1 5 ) . S u p p o s e a l s o t h a t , f o r some f i x e d r, 1 < r a n d for a l l w e i g h t s w E A1, w e h a v e

w i t h C(w)

(3.17) for every

(3.18)

depending o n l y on t h e

w(Ix

: \TF(x)lB

w E A1,

> tl)

A1-constant of

5 C(w) t-l ]lF(x)II$

and a l s o , €or every

jITF(x)lgw(x)dx

w.

w € A

with

P -

T h e n , we

m,

have

w(x)dx 1 < p <

m,

5 Cp(w) IIF(x)Og w(x)dx.

The proof of (3.17) is again a n exact repetition (except for a minor change in the estimation of "good part" when r = m ) of the one given in Chapter IV, Theorem 3.5, for scalar functions (observe that the hypothesis IK(x) 1 5 Clxl-" was not really needed there).

V.4. SOME MAXIMAL INEQUALITIES

497

Once the weak type result is proved, (3.18) follows by the extrapolation theorem of Chapter IV, ( 5 . 2 0 ) . 0

4. APPLICATIONS: SOME MAXIMAL INEOUALITIES

Almost every nonlinear operator appearing in Fourier Analysis is linearizable (in the sense of Definition 1.20). It is sometimes useful to look at such operators as linear operators taking complex valued functions into B-valued functions, and if the kernel happens to satisfy the appropriate condition, the theory developed in the preceding section can he applied. In particular, we shall see that the Hardy-Littlewood maximal operator and some of its generalizations can he viewed as vector valued singular integrals. Given

6 L1(Rn),

@

M f(x) @

=

we define

sup 6O.

he maximal operator

lf*@&(x)

it is Denoting B = L m ( R + ) , which is certainly bounded in Lm(Rn). equivalent to consider the linear operator T : Lm + Li defined (as since in Corollary 3.10) by the B-valued kernel K ( x ) = ($6(x))6,0, we have: M@f(x) = ITf(x)lB. Now, HGrmander's condition for the kernel K(x) amounts to the following:

and we are led to THEOREM 4.2. L e t il M weak t y p e

@

1

n

6 L (R )

be a function satisfying ( 4 . 1 ) .

i s an o p e r a t u r bounded i n

@ (1 ,1) E d m a p p i n g

or

iii

every

f

c

u

H 1 (R")

Lp(Rn), 1 < p c L1 (R").

m,

Then: Q

into

LP(R")

1 ip<m I

iiij The f o l l o w i n g v e c t o r valued inequa2itie.s hold f o r a l l 1 < p,q <

m:

498

V. VECTOR VALUED INEQUALITIES

Proof: Since fx@&(x) depends continuously on 6 , it suffices to obtain in (i) and (iii) uniform estimates for the operators I

where

F

is an arbitrary finite subset o f KF(X)

=

(@6(x))6cF

E

R,

and

Lm(F)

Now, we are under the hypothesis of Theorem r = m, A = C , B = L m ( F ) ) , which give (i) Finally, (ii) follows in the usual way (see instance) since it is trivially verified by compact support. 0

3.4 and 3.9 (with and (iii) respectively. Chapter 11, (1 .9), for a l l continuous f with

A trivial but important observation is that, if

is such that ( $ 1 5 1 4 1 , the whole theorem, except for the H’ + L1 result, In particular, we can take a s $ holds also for the operator M$. the characteristic function of the unit cube centered at the origin (thus, M+ = Hardy-Littlewood maximal operator) and as $ a nonnegative Schwartz function such that $(x) 1 when x E Q. Then, we have $

COROLLARY 4.3. T h e H a r d y - L i t t l e w o o d m a x i m a l o p e r a t o r v e r i f i e s t h e v e c t o r v a l u e d i n e q u a l i t i e s l i i i l of t h e p r e v i o u s t h e o r e m .

Next, we observe that, since Lm(w) = Lm(Rn) for every w 6 A 1 , the operator M$ satisfies the estimate required in Theorem 3.1b with r = m , and we can therefore obtain weighted estimates for this kind of maximal operators: THEOREM 4.4. S u p p o s e t h a t I$(X-Y)

-

L1(Rn)

$ 6 $(XI

I

5

satisfies

C l y l 1x1 - n - l

I I

(1x1 > 2 Y > O )

T h e n , t h e m a x i m a l o p e r a t o r M$ i s bounded i n L P ( ~ ) 1 < p < m and w E A and i t i s a l s o of weak t y p (1,l) P’ Q w E A, r e s p e c t t o the m e a s u r e w(x)dx _ .

with -

Proof: I t suffices to apply 3.16 taking into account that the condition imposed on $ is dilation invariant, so that the same

499

V.4. SOME MAXIElAL INEQUALITIES

inequality is verified by sup l$6(x-Y) -

Q6

for all

6 > 0 , i.e.

5 c IYI

@,(XI1

6>0

(1x1 >21yl

and this is exactly the condition (3.15) for the K(x) = ($,(X)),>0-

'

0)

Lm-valued kernel

One can consider more generally maximal operators defined by a i family Q = ( @ )iEI, namely sup * f(x)l iEI where, either I is countable o r I = R+ and $ I ~ ( X ) depends continuously on i. If (4.1) is verified by the family ( @ ' ) , and if M@ happens to be bounded in Lq(Rn) for some q (this is certainly i true with q = m when s u p I @ 1 , < m ) , then, parts (i) and (iii) of Theorem 4.2 hold foriE' Mo. Mof(x)

=

In particular, we can consider approximations of the identity (G6 ) corresponding to a lacunary sequence ( 6 k ) of dilations, and we k have THEOREM 4.5. S u p p o s e t h a t (4.6)

(4.7)

@

s a t i s f i e s t h e f o l l o w i n g two e o n d f t i o r z :

/$(x)I log(2+lxl)dx

Jl

w,(tI

dt

7

<

m,

where

<

m

w,(t)

=

sup

IhlZt

T h e n , t h e c o n c l u s i o n s of T h e o r e m 4 . 2

i

I$(x+h)-@(x)ldx

h o l d for t h e d y a d i c m a x i m &

o p e r a to r

irepZacing

i n part

(iil

lirn C+O

Ky

lim). k+m

Proof: Instead of proving (4.1) for the sequence obtain the stronger inequality

(2-k)ym,

wc shall

V. VECTOR VALUED INEQUALITIES

500

Observe that the left hand side of this inequality remains unchanged 1 if we substitute y by 2y, so that we can assume: 5 IyI 5 1 . Then, we study the sums corresponding to k 2 0 and to k < 0 separately: m

m

,

1 k=o

dx

11,

=

=

< -

< 2 f due to ( 4 . 6 ) .

For the remaining sum we have to use (4.7):

dt

=

C2 <

m

and this completes the proof. Examples 4 . 9 . (a) Given a function n ( x ) in Rn which is Dositive, homogeneous of degree 0 and integrable over the unit sphere, we consider the following nonisotropic maximal function, which was suggested by E.M. Stein and studied by R. Fefferman [l]:

This is the maximal operator associated to $(x) = Q(x')x,(x), where B is the unit ball. Since @ 6 L 1 (R n ) and has compact support, (4.6) is verified. Now, as we did in Chapter I 1 for the homogeneous kernels of singular integrals, we define o,(R,t)

=

I

sup

Ihlzt

IQ(x'+h) - n ( x ' /x'I=l

and we try to see the relationship between modulus of continuity of @ , w,(@,t). If

m,(R,

t

rj

rn-1dr

w1

Ih

and the L 1 < t 5 1, we have

n,t)

V . 4 . SOME MAXIMAL INEQlJALITIES

501

and t h e r e f o r e

Thus, we can s t a t e our r e s u l t :

I n f a c t , t h i s i s t r u e f o r t h e o p e r a t o r N Q d e f i n e d by t a k i n g t h e k supremum o v e r a l l r = 2 , k 6 Z ( d u e t o 4 . 5 ) , b u t , s i n c e k @,(x) 5 Z n @ ,(x) when Z k - ' < r < 2 , we h a v e MQ f(x) 5 ZnN Rf(x) 2 follows. f o r a l l p o s i t i v e f , a n d t h e r e s u l t f o r M, (b) For every

consider the kernel

a > 0,

OC1(Xj = - max(1

r la)

Then

, 01~-

h a s compact s u p p o r t and i t s

Oa € L 1 ( R " ) ,

tinuity satisfies (4.7)

2 1x1

-

5 Cata,

wl(Qcl,t)

1 L -modulus o f con-

so t h a t t h e Dini condi ion

i s v e r i f i e d . T h e r e f o r e , t h e max i m a l o p e r a t r NNf(x)

i s bounded i n

Lp(R"j,

1

sup k€Z

=

5

1 < p

f(~-2-~)o"(y)dy, m,

a n d o f weak t y p e ( 1 , l )

An e a s y

c o m p u t a t i o n s h o ws t h a t

or'

lim

=

( i n the d i s t r i b u t i o n sense)

0

01+0

where

0

(Lebesgue measure i n t h e u n i t s p h e r e ) i s i d e n t i f i e d w i t h

a s i n g u l a r Bore1 measure i n

Rn

supported i n

: 1x1 = 1 1 .

{x € Rn

Therefore, the limiting oaerator (corresponding t o

a

=

0)

is given

by Nf(x)

=

Nof(x)

=

sup k6Z

I

f ( ~ - 2 - ~ y ' ) d o ( yI ' )

l/y'l=l

H o w e v e r , t h e c o n s t a n t s o b t a i n e d by a p p l i c a t i o n o f The ore m 4 . 5 t o

N"

b l o w u p when

operator

N

c1

+

0,

a n d n o t h i n g c a n b e s a i d a b o u t t h e m a xim a l

by t h e m e t h o d s d e v e l o p n e d s o f a r . E s t i m a t e s f o r

Nf

w i l l be o b t a i n e d i n 5 . 1 9 below.

Now, f o l l o w i n g F e f f e r m a n a n d S t e i n [ l ] ,

we s h a l l u s e t h e v e c t o r

v a l u e d i n e q u a l i t i e s f o r t h e H a r d y - L i t t l e w o o d m a xim a l o p e r a t o r i n

V. VECTOR VALUED INEQUALITIES

502

order to obtain some estimates for Marcinkiewicz integrals. Given a closed set F whose complement has finite Lebesgue measure, the Marcinkiewicz integral of order X > 0 corresponding to F is the function:

H X (F;x) where

6(y)

=

H (x)

6 (Y)

=

Ix-y(n+X + 6(X) n+X dy

JR\F

denotes the distance o f the point y from

F.

On the other hand, given a family of disjoint b a l l s (or cubes) {Bj}, the Marcinkiewicz integral of order A corresponding to this family is defined as: d?" SA(X) = c j (x-c. + dn+'

' J

1

where

c. J

denotes the center of

R. J

and

d. J

its diameter.

x Then, t h e X > 0 , we Z e t q = 1 + 6. i n te g r a Is HA & r g Sx d e f i n e d a b o v e s a t i s f y :

COROLLARY 4.10. G i v e n Marcinki ewi c z

Moreover, p = :;

we h a v e t h s c o r a r e s p o n d i n g weak t y p e i n e q u a l i t i e s f o r

l{x :

fIX(X)

I{x

SX(x) > t)I

:

> t}l

5 CX t - ' / q 5 C X t-"'

IR\F/

; lBjl 3

Proof: Let us consider first S X . If B denotes the unit b a l l centered a t the origin, i t is very easy t o see that: M ( X B ) ( x ) > > Const.(l + lxln)-'. (Ilere M stands for the Hardy-Littlewood maximal operator). By translation and dilation invnriance of M , we have dn

and there fore

SOME MAXIMAL INEQUALITIES

V.4.

503

The inequalities for Sx result now from an application of Corollary 4.3 with f . = X B . . It is clear that this argument works equally J well for a familyJ of cubes, instead of balls. In order to prove the results for H A , we first observe the dilaX tion invariance of H A in the following sense: Hx(tF;x) = Hx(F;y). This a l l o w s us to assume IRn\Fl = 1 . Now, taking into account that 6(Y) 5

where

+

Rn\F

=

Ix-YI

u

3 p. 1 6 ) ,

Stein [ l ] , and diameter

dj)

is a Whitney decomposition of

Qj

{Qjl

i.e., and

-

x !6

and, since

( R n\ F l

=

1,

I:

c. J

( x 6 Qj)

I

for some absolute constant A. Therefore, letting Marcinkiewixz integral corresponding to the family

On the other hand, for

(see

are disjoint cubes (with center

A-'d. < 6(x) < Ad. J -

Rn\F

SA

be the

{Q41

we can make the Trivial estimate

thc results for

of the corresponding inequalities for

S

A'

HA

are now a consequence

n

We conclude this section by showing how vector valued singulor i n tegrals can be used to prove the w e a k type (1 ,1) for 'i-*f(xi

=

sup I >o

11,

K (y) f ( x - y ) d y

1

y , >k

where K is the kernel of 11 regulrlr singular integral oper;itor. This fact was stated without proof in Chapter 1 1 , Theorem S . 2 0 , hut we did prove there that 'I,* is Ihouncled in 1.' for a l l 1 i p c a'.

v.

504

VECTOR VALUED INEQUALITIES

where h ( t ) i s a C1 f u n c t i o n s u c h t h a t 0 5 h ( t ) 5 1 , h ( t ) = 0 when 0 5 t < 1 a n d h ( t ) = 1 when 2 5 t < a. The f i r s t t h i n g t o notice is that IT*f(x) - T*f(x)

I 2

sup E>O

O

Thus, e s t i m a t e s f o r vation is t h a t

'E<\y\<2E

I12E

T"

or

7"

a r e e q u i v a l e n t . The s e c o n d o b s e r -

I h ( l x - y l ) - h ( l x l ) l 5 CIyI/IxI

1x1 > 2 1 ~ 1 . T h e r e f o r e , i f we h a v e

KE(x)

=

whenever

K(x)h(lxl/E)

and

~ K ' ( X - Y ) - K E ( x ) I 2 1 ~ 1 ,

+

with C ' independent of E . This implies t h e i n e q u a l i t y ( 4 . 1 ) f o r t h e f a m i l y ( K E ) E > O , a n d s i n c e T* i s b o u n d e d i n L p , t h e remark p r e c e d i n g 4 . 5 a p p l i e s a n d t h e weak t y p e ( 1 , l ) f o r M o r e o v e r , s i n c e we h a v e r e a l l y p r o v e d t h a t t h e

?*

follows.

Lm-valued kernel

(KE) E > O s a t i s f i e s ( 3 . 1 5 ) , a n d we know t h a t T* ( a n d t h e r e f o r e a l s o i s bounded i n Lp(w) i f w € A , and 1 < p < m ( s e e Chapter IV, 3 . 6 ) we c a n a p p e a l t o Theorem 3 . 1 6 a n d s t a t e t h e f o l l o w i n g

t*)

COROLLARY 4 . 1 1 . __ Let let

T"

T

b e a r e g u l a 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 , and

be t h e a s s o c i a t e d maximal o p e r a t o r d e f i n e d a s above. Then,

f o r every weight

w E A1

w({x : T * f ( x ) > t l ) 5 C w t

5 . APPLICATIONS: SOME LITTLEWOOD-PALEY THEORY

I n t h i s s e c t i o n we c o n t i n u e t h e a p p l i c a t i o n o f t h e t h e o r e m s f o r v e c t o r v a l u e d s i n g u l a r i n t e g r a l s t o some i m p o r t a n t l i n e a r i z a b l e o p e r a t o r s . T h i s p r o g r a m was c a r r i e d o u t i n s e c t i o n 4 f o r t m - v a l u e d k e r n e l s , and h e r e w e s h a l l u n d e r t a k e t h e c a s e o f H i l b e r t s p a c e valued kernels. This w i l l lead us t o various kinds of square

V.5.

LITTLEWOOD-PALEY THEORY

505

functions, one of the typical ingredients of what is called Littlewood-Paley theory. We emphasize, however, that this theory is much deeper and richer than what we can afford to present here, which must be considered only as an introduction to some of its basic aspects. Suppose we are given a sequence of kernels form the square function m

Gf(x)

(

=

1

j=-m

Ikj

*

f(x)l

kj(x)

in

Rn

and we

2 1/2

1

Then Gf(x) = ITf(x)lB, where B = 1’ and T is the linear operam tor taking f(x) into the B-valued function (kj*f(x))j7-m. By Plancherel’s theorem, G (equivalently T) is bounded in L2(Rn) if and only if (5.1)

If this is verified, Corollary 3.10 can be applied provided that we also have

Looking for some concrete examples satisfying ( 5 . 1 ) find out the following: THEOREM 5.3. Let

@(x)

&

$(O) and

an i n t e g r a b l e f u n c t i o n in

\@(x)dx

=

I@(X)l

(5.5)

/l@(x+h)

5

c 1x1

Rn

suck t h a t

0

it verifies

> 0,

a s s u m e t h a t , f o r some (5.4)

=

and ( 5 . 2 ) we

-n-a

(x 6 Rn)

and __ - @(x)ldx

5 C lhla

(h

€

Rn)

V. VECTOR VALUED INEQUALITIES

506

a r e bounded i n

LP(R"),

1 < p

a r e a l s o b o u n d e d o p e r a t o r s from

<

m,

a n d of weak t y p e

H1(Rn)

2

1

(1 ,I).

hey

n

L (R ) .

= As usual, we have denoted by @t the dilation of I$ : @,(x) Observe that A f is the continuous analogue of Gf = t - n I$(:). S (this is more evident if we make the change of variables t = 2 the definition of nf).

in

Proof of the Theorem: The results for Gf will follow from Corollary 3.10 once we show that the kernels verify (5.1) and (5.2). Similarly, for A f we must verify the conditions:

(@2j)y=_,

and

The properties ( 5 . 1 ) and (5.1') are derived from two different estimates for the Fourier transform of @ : For small 151 we have

?)

with B = inf(?;1 z 0. The estimate for large 151 is based on 2nih.S - 1 ) is the Fourier transform of the fact that @ ( < ) ( e @(x+h)-@(x), and choosing h = [ / ( 2 1 , 1 2 ) we get

2 1 ? ( 5 1 1 5 ~l@(x+h)-@(xlldx 5 C 151-' Since the left hand side of (5.1') is invariant under dilations of 5 , it can be assumed that 151 = 1 , and then

The proof of (5.1) is quite similar, and it is left to the reader. Now, we observe that (5.4) and (5.5) are respectively stronger than the hypothesis of Theorem 4.5, under which, the inequality

(which is stronger than ( 5 . 2 ) 1 was obtnined. Thus, only (5.2 I ) remains to be proved, and again, by dilation invariance, we can assume that

V . 5 . LITTLEWOOD-PALEY THEORY

507

Iy1 = 1 . Then, Schwarz i n e q u a l i t y and F u b i n i ' s theorem a r e u s e d t o m a j o r i z e t h e l e f t h a n d s i d e by

On t h e o t h e r h a n d , i f we u s e ( 5 . 5 ) a n d t h e f a c t t h a t

@(x)

is

bounded : I ( t ) 5 C [ / Q ( x-

$1 - @ ( x ) ( d x<

C t-'

and combining b o t h e s t i m a t e s , w e f i n a l l y o b t a i n t

Remarks 5 . 6 .

'

-

t2a-1

iy

dt +

0

t - l -n/2dtl

<

;".

0

( a ) Th e h y p o t h e s i s ( 5 . 4 ) a n d ( 5 . 5 ) c a n b e s l i g h t l y r e -

l a x e d , b u t t h e y a r e g e n e r a l enough t o c o v e r a l l c a s e s o f i n t e r e s t . I n p a r t i c u l a r , b o t h o f them a r e t r i v i a l l y v e r i f i e d i f llowever, t h e h y p o t h e s i s

&(O)

=

@ 6 S(Rn). is absolutely necessary.

0

(b) I t o c c u r s q u i t e o f t e n t h a t one has t h e e q u i v a l e n c e lGflp

"

lflp,

IGF],

and n o t o n l y t h e i n e q u a l i t y

f a c t , i f B i s a H i l b e r t s p a c e and t o r such t h a t

T : L2

+

5 Cp l f l , .

In

Li- i s a l i n e a r o p e r n (f E L2)

t h e n , a n i n e q u a l i t y o f t h e form

ITfl,n

5

C

LB

some p 2 1 , a u t omn t i c a 1 1y imp 1i e s IfUPl s e e t h i s , we a p p l y p o l a r i z a t i o n t o t h e a b o v e

a n d t a k e t h e supremum o v e r a l l

g

such t h a t

If!,

(f 6 L ' n

Idp)

for

V. VECTOR VALUED INEQUALITIES

508

(c) Vector valued inequalities for the operator C; are obtained by an application of Theorem 3 9. In particular, if this is done for q = 2 , we obtain

provided that

@

satisfies the hypothesis of Theorem 5.3.

Our first application of Theorem 5 . 3 will he one of the classical results obtained (in the periodic setting) by Littlewood and Paley in their fundamental series of papers. In order to state it, same notation must be introduced: The dyadic intervals in R are those of the form:

I.

=

[ 2 j-1 , 2 j ) ;

I

and they form a partition of R \ { O } . The family of dyadic intervals in Rn, denoted by A = A(Rn), consists of all n-dimensional intervals which are the Cartesian product of n 1-dimensional dyadic intervals. I t is plain that the intervals in A(Rn) are pairwise disjoint, and their union covers almost all Rn. Thus, Plancherel's theorem implies I\f(x)12dx

(5.7)

where

SI

IS

=

p ) I 2dx

stands for the partial sum operator:

THEOREM 5 . 8 . T h e r e e x i s t c o n s t a n t s

c

such that

cp

(5.9) When

n

1,

=

I I X

( f E:

lf~,

5 I(

1

I€A

C

P'

P

> 0

(SIf)^

=

(1 < p <

I,5 cPlfpp

2 1/2 I S ~1 ~ I

2 n L (R 1 )

:XI. a)

(f t:

LP(R"))

(f t:

H~ (R))

we a L s o h a v e t h e weak t y p e r e s u l t :

1

: (

I €A

l s I f ~ x l 1 2 ) 1 ~'2 A l l 5

c x-l

lflHl

Proof: By the observation made in 5.6.(b), and taking into account ( 5 . 7 1 , i t is enough to prove the second inequality in ( 5 . 9 ) . We shall consider first the case n = 1. Take @ € S(R) such that $ ( O ) = 0 and G ( 5 ) = 1 for all 5 6 I o . Then

rsI.(@2-j* f 1 1 ~ 5 )= 1

i.e.,

S

Ij

f

=

S I . ( $ ~ ~* -f)~ 1

.

? ( E ~ Z - ~ C ) X ~ . ( C =) ? ( O X I. ( 5 ) I

I

Now we use inequality (2.14) and

V.S.

LITTLEWOOD-PALEY THEORY

509

Theorem 5 . 3 t o g e t

.

a n d t h e same i n e q u a l i t y h o l d s f o r t h e i n t e r v a l s (5.9)

{(-I.)}? Thus, 3 J=-Y i s p r o v e d i n t h i s c a s e . The weak t y p e r e s u l t f o l l o w s i n t h e

same way b y u s i n g now ( 2 . 1 5 ) , i n s t e a d o f ( 2 . 1 4 ) :

(c

({x :

I S I , f ( x ) 1 2 ) 1 / 2 > All 5 C A - 1 J

J

where

Gf(x)

lGfll

5

C'

A-1

i s t h e s q u a r e f u n c t i o n d e f i n e d i n Theorem 5 . 3 .

The p r o o f f o r

Rn

w i l l b e by i n d u c t i o n on

t h a t (5.9) holds f o r

Rn-l

.

n,

s o t h a t we a s s u m e x € Rn a s

We s h a l l w r i t e a p o i n t

x = (xl,x') w i t h x 1 € R a n d x ' € R n - l . Let A a n d A ' denote r e s p e c t i v e l y t h e f a m i l i e s o f d y a d i c i n t e r v a l s i n Rn a n d i n R n - l ,

A

so that

= A+

u

A _ , where

and i t s u f f i c e s t o p r o v e t h e s e c o n d i n e q u a l i t y i n ( 5 . 9 ) f o r t h e family

Let

A+.

@ E

S(R)

b e d e f i n e d as i n t h e f i r s t p a r t o f t h e

p r o o f , a n d d e f i n e t h e o p e r a t o r T . i n Lp(Rn) a s c o n v o l u t i o n w i t h J in the first variable, letting the other variables fixed, i.e. 92 - j ( T j f l A ( C l ,<'I = ? ( C l , < ' ) ? ( 2 - j E 1 ) By F u b i n i ' s t h e o r e m , t h e r e m a r k 5 . 6 . ( c ) operators

Ti

c a n be a p p l i e d t o t h e

t o yield

and t h e induction hypothesis i m p l i e s

Combining b o t h i n e q u a l i t i e s , we g e t

one immediatly v e r i f i e s ( b y c o m p u t i n g F o u r i e r t r a n s f o r m s ) t h a t : S I f = S I o T j OSRxI f . T h e n , t h e t r u n c a t i o n a r g u m e n t ( b a s e d on C o r o l l a r y 2 . 1 3 ) a l r e a d y Finally, f o r every interval

I

=

1 . ~ 1 6' A + , J

V. VECTOR VALUED INEQUALITIES

510

used in the case o f

R

applies again, and this completes the proof.

0 One would like to have the Littlewood-Paley inequalities (5.9) for other different decompositions of Rn. For instance, we can consider the decomposition into lacunary spherical shells:

Now, the proof of Theorem 5.8 presents basically two steps. First, the result is proved for a smooth modification of the desired decom position, by using Theorem 5.3. This works in many other situations and, in particular, in the one we have in mind. Thus, we canstate: COROLLARY 5.10. Let m(t) R, = ( 0 , ~ ) s u c h t h a t

in

be a

m(t)

Cm =

1

f u n c t i o n w i t h compact s u p p o r t f o r a22

t 6 [1/2,1]

and

m

1

m ( ~ ~ t =) 2

(0 < t <

k=-m

For e a c h

f 6 L~

n

LP(R")

a)

we d e f i n e t h e s q u a r e f u n c t i o n

The second step in the theorem just proved is the truncation of the smooth decomposition by means of Corollary 2.13. This part is more specific of n-dimensional intervals (though it also holds in othcr cases where the proof is harder, see 7.3 below). In particular, we should need in o u r case the inequality

?xD),

and this is certainly false for every p # 2, since the characteristic function of a spherical1 shell is not a multiplier for I,P(R") i f n > 1 and p # 2 (c. Fefferman [?.I). ( (SDf)^ =

*

A

I n the study of multiplier transformations: (T,f) = fm, the equivalence of Lp-norms given by (5.9) can be used as a scissor theorem [we have borrowed this expression from A. C6rdoba), since i t allows to cut the multiplier into its dyadic pieces. To be precise, we have COROLLARY 5 . 1 1 . L e t

m 6 I."(Rn),

and Z e t ~-

u s d e c o m p o s e -it us

V.5.

LITTLEWOOD-PALEY THEORY

51 1

1 mI, w h e r e , f o r e a c h d y a d i c i n t e r v a l I, we d e n o t e I€A mI = mXI. G i v e n 1 < p < m, m i s a n Lp m u l t i p l i e r , i .e. i f a n d o n l y if t h e i n e q u a Z i t y ITmflp 5 Cp lfl,, m

=

hoZds for a r b i t r a r y f u n c t i o n s

fI € L 2

n

Lp(Rn).

then ( 5 . 1 2 ) follows from Proof: If Tm is bounded in Lp(Rn), the consecutive application of Corollary 2 . 1 3 and the MarcinkiewiczZygmund theorem. On the other hand, if ( 5 . 1 2 ) holds, we take an arbitrary f 6 L 2 n Lp and apply Theorem 5.8: -1

ITmflp 5 cp -1

=

cp

l'f

I

'E

2 1/2

lSITmfl 1 2 1/2

lTmlSIfl 1

I,

5.

I,

c, I

=

(I:I P I f l 2 1 1 / 2 I,

5

c; lfl,.

n

The corollary remains true if we use instead of any other decomposition of Rn for which the conclusions of Theorem 5.8 hold; = {IxR"-~ / I 6 A(Rk)}. in particular we can take We shall now see how this works to produce significant multiplier theorems. For the sake of simplicity, we shall consider only the case of R . The operators St defined by (Stf)^ = ^fX [t,-) are uniformly bounded in L p ( R ) , 1 < p < m , and so is eve y "linear combination" of them

(X € L

T

But

is the operator corresponding to the multiplier

and every bounded C 1 Function m(<) whose derivative is integrable can be written in this form. Thus, we have established a simple result: 1'1f m c L~(R) i s o p c l a s s c 1 , a n d ml c L'(R), t h e n ITmfl, 5 C, Ifl,, 1 < p < -'I. The Littlewood-Paley theory provides the following improvement of this result: (Marcinkiewicz multiplier theorem). Let m € Lm(R) C1 o u t s i d e t h e o r i g i n and s u c h t h a t Im'(C) Id6 5 B f o r e v e r y d y a d i c i n t e r v u Z I. T h e n , t h e o p e r a t o r

COROLLARY 5 . 1 3 . be

1,

Q

function of class

51 2

Tm d e f i n e d by

V. VECTOR VALUED INEQUALITIES

(Tmf)^

=

?m

i s bounded i n

P r o o f : F o r every Schwartz function

taking Fourier transforms that

Lp(R),

1 < p <

f , one easily verifies by

1,

T f(x) = m(2j-')SIf(x) + StSIf(x).m'(t)dt mI where I is the dyadic interval: I = [Zj-', Z j ) , and S t defined as above. Therefore, the hypothesis on m imply ITm f(x)I 5 C {ISIf(x)I I By Corollary 2.13, we have

+

-.

(1,

is

IstSIf(x)12~m'(t)ldt)1/2}

n

I and ft = f I , where, given t € R, we define J(t) = [t,-) if I is the dyadic interval to which t belongs. The second term can be estimated by means of the continuous version of Corollary 2.13 stated in 2.17.(a) so that it is majorized by

Thus, we have proved that (5.12) applies. El

holds, and the preceding corollary

The hypothesis of this corollary can be slightly weakened: It suffices to assume that m E L m ( R ) and that m has uniformly bounded variation on each dyadic interval. As the reader may guess, there is a corresponding version of the Marcinkiewicz multiplier theorem for Rn. F o r this we refer to Stein [ l ] , Ch. IV. We shall now turn to one of the early applications of quadratic expressions, namely, the majorization of maximal functions in order to establish results of pointwise convergence. The idea is roughly as follows: We need to obtain Lp estimates for the maximal operator sup ITkf(x) I k and we take an appropriate Schwartz function t rivi a1 major i zation T*f(x)

=

$(x)

to make the

V.5. LITTLEWOOD-PALEY THEORY

T*f(x) 5 sup /Tkf(x) k 5

(i ITkf(x)

-

I

@ -k*f(x)

2

- @ -k*f(x) I 2)1/2

+

2

I@

sup k

+

51 3

2

-k*f(x)

I

CMf(x)

where M denotes the Hardy-Littlewood maximal operator. Thus, matters are reduced to estimate the Lp norm of 2 1/2 Qf = (1 lTkf - @ - k x f ( I k 2 When p = 2 , this is easily made by means of the Fourier transform, and the method dates back to Kolmogorov [ l ] , who proved the a.e. convergence of lacunary partial sums of Fourier series for L2 periodic functions. Now, a basic feature of Littlewood-Paley theory is that it allows to extend orthogonality arguments (which are simple in L2) to Lp, 1 < p < m . Not surprisingly, therefore, it was up to Littlewood and Paley to extend Kolmogorov's result to Lp functions, 1 0.

m(0)

=

1

C1

& Im(c) I 5

a

CIcl-'

define the multiplier transfor-

we

mat i o n s

(Tkf) ( 5 ) A

=

Then, t h e maxima2 o p e r a t z 2 n 1,

(R )

^f ( 5 ) m(

2-

T*f(x)

=

kc)

(k 6 2 )

s u p ITkf(x)

@

k

I

i s bounded i n

2 n lim Tkf(x) = f(x) a.e. ( f 6 L (R 1 ) k-tm b) Trike m = X I i n p a r t ( a ) , w h e r e I i s b o u n d e d o p e n i n t e r -~ v a l i n Rn c o n t a i n i n g t h e o r i g i n . T h e n , Tkf = S f, and t h e f(x) I i s i n ' t h i s c a s e b o u n d e d m a x i m a l o p e r a t o r S*f(x) = sup I S k 2 1 i n Lp(Rn) f o r a 2 1 1 All 5

a.e.

holds f o r every

c x-l

If$

(f6

u

Lp(Rn)) 1
f 6 H1(R),

and

(f 6 H 1 ( R ) I

Proof: We shall limit ourselves to prove the maximal inequalities.

v.

514

VECTOR VALUED INEQUALITIES

The c o r r e s p o n d i n g p o i n t w i s e c o n v e r g e n c e r e s u l t s a r e i m m e d i a t e l y d e r i v e d from them, s i n c e l i m T k f ( x ) = f ( x ) t o t h e f a c t t h a t m(0) = 1 ) . a ) Take

$ € S(Rn)

such t h a t

G(0)

=

for all

1.

f € S(Rn) (due

Then

Im(S) 5 CIS1 a n d Im(S) - ? ( 5 ) 1 5 C I S 1 - E , a n d b o t h i n e q u a l i t i e s t o g e t h e r i m p l y ( a s i n t h e p r o o f o f Theorem 5 . 3 ) t h a t

I f we d e f i n e that

Qf

and t h i s p r o v e s :

6

as above, i t f o l l o w s from P l a n c h e r e l ' s theorem

lT*f12 5 Const. l f I 2 .

b ) Take $ E S ( R n ) s u c h t h a t $ ( O ) = 0 a n d $ ( E ) i n a neighbourhood o f a I . Then, ( 1 - $ ) x I 6 C:(Rn),

e x i s t s @ € S(Rn) s u c h t h a t $ = ( 1 - $ ) x I . Qf a s a b o v e , a n d o b s e r v e t h a t

With t h i s

=

1 for all and t h e r e

9,

we define

S k f - @ * f = S (+ * f ) 2 1 2-k 2% 2-k T h e n , t h e same a r g u m e n t o f Theorem 5 . 8 Theorem 5 . 3 ) p r o v e s t h a t t o be proved.

(i.e.,

l Q f l p 5 Cp l f l p ,

c ) I t s u f f i c e s t o o b s e r v e t h a t , when

i n e q u a l i t y (2.14) p l u s

w h i c h was t h e o n l y p o i n t

n = 1,

the operator

d e f i n e d i n p a r t ( b ) maps H'(R) b o u n d e d l y i n t o weak-L1(R) p r o o f i s a g a i n a s i n Theorem 5 . 8 ) .

Q

(the

We h a v e s t a t e d p a r t ( b ) b e c a u s e o f t h e s i m p l i c i t y o f i t s p r o o f . A much d e e p e r r e s u l t i s t r u e , h o w e v e r : I f then (5.17) n = 1,

1 < p <

m,

I

?(S)e2nix*Sdg = f ( x ) a.e. tI t h i s i s t h e t h e o r e m o f C a r l e s o n [3] a n d Hunt [ l ] lim

t-tm

For

f € Lp(Rn),

.

The

r e s u l t f o r Rn c a n b e e a s i l y r e d u c e d t o t h e o n e - d i m e n s i o n a l c a s e ( s e e C . F e f f e r m a n [3]). We m u s t m e n t i o n , h o w e v e r , t h a t p a r t ( c ) i s s h a r p i n two s e n s e s : F i r s t o f a l l , ( 5 . 1 7 ) , d o e s n o t h o l d f o r e v e r y f E H1(R) ( o n e c a n a c t u a l l y h a v e l i m s u p I S t I f ( x ) I = +m a . e . for such

f);

t

+

s e c o n d l y , one c a n n o t r e p l a c e

m

H1(R)

by

L1(R)

in the

V.5. LITTLEWOOD-PALEY THEORY

51 5

statement of 5.14.(c). Proofs of both facts, in the periodic setting, will be found in Zygmund [l], Ch. VIII. A n application of Theorem 5.14(a)

denotes the unit ball in

with gives:

R",

COROLLARY 5.18. For e v e r y

m(5)

=

XB([)

where

B

f E L2(Rn)

The corresponding result for Lp(Rn) is false if p < 2 and n > 1. This is a consequence of the negative result of C. Fefferman [ Z ] for the ball multiplier together with some general principles to be studied in the next chapter (see VI.2.8(c)). Concerning L2 functions, it is not known (and this is a beautiful open question) wheter dyadic partial sums can be replaced by arbitrary partial sums or not, i.e.: Does (5.17) hold for every f 6 L 2 (Rn) after replacing I by the unit ball B?. Another application of Theorem 5.14(a) is the followi.ng: Remember the kernels introduced in Example 4.9(b) Qa(x)

=

2

max(1 - \ x 1 2 , 0la-l

which we shall consider now defined €or each complex number a with Re(a) > 0. By the usual formula for computing Fourier transforms of radial functions (see Stein-Weiss [l], Ch. IV) we have

*,

where, for each 6 > - 1 Jg(t) denotes the corresponding Bessel function. The definition of ma makes sense whenever Re(a) > and for every such a we introduce the maximal operator N"f(x)

=

sup

/j~([)ma(2-k~)e2nix.FdEl

(f E

kEZ

S(Rn))

which coincides with the one defined in 4.9(b) when a 2 0 (for a = 0, we identified 0' with Lebesgue measure in the unit sphere, 0 , and it turns out that ;(<) = mO(c)]. We can now state. The maximal o p e r a t o r ( d e f i n e d a p r i o r i i n COROLLARY 5.19. -

Nf(x)

=

sup k62

f(x-Z-ky') do(y') jly' \=I

I

S(Rn))

516

V. VECTOR VALUED INEQUALITIES

Lp(Rn)

is b o u n d e d in

Proof: In 4.9(b)

Q

n 2 2

1 0)

function in F O , m ) JB(t) 5 Cgt' 1 2

(t m ) , the multiplier ma satisfies (except f o r a multiplicative constant depending on a ) the hypothesis of Theorem 5.14(a), and we -f

get lNUf12 5

ca

(f E S(Rn),

If12

Re(a)

>

'1 - n

We shall omit the technical de ails needed to conclude the proof, but the reader who is familiar with the interpolation of analytic families of operators (see Ste n-We s s [ l ] , Ch. V) should have no difficulty in interpolating both estimates for the operators N " , after a suitable linearization, to obtain lNaflp

I cayp

lfl,

(f E S(R"))

1 - n < Re(a) 5 0.

for ID 1 - TI 1 < 7 7 Re(a) and what we wanted to prove. 0 +

The case

a = 0

is

To finish this section, we shall describe some results corresponding to continuous type square functions. We begin by the so called Littlewood-Paley g-function, which was first studied by complex methods, in the periodic setting and for n = 1 , as a previous step to the inequalities of Theorem 5.8 (This approach is described in Zygmund [l] , Ch. XIV). F o r a function f(x) in Rn, we define

(1

m

g(f)(x)

=

tlVu(xyt)12dt)"2

0

where u(x,t) we have

=

Pt*f(x)

denotes the Poisson integral of

f. Then,

COROLLARY 5.20. Let 1 0 such t h a t P'

Proof: Let us first l o o k at what happens f o r

P

p = 2:

V . 5 . LITTLEWOOD-PALEY THEORY

51 7

T h u s , we a r e i n c o n d i t i o n s o f u s i n g t h e o b s e r v a t i o n 5 . 6 ( b ) , a n d i t w i l l be enough t o p r o v e

I g ( f ) l p 5 Cp b e made s e p a r a t e l y f o r t h e o p e r a t o r s

Take

a

@(x) =

Pt(x)

I t=T.

1 < p <

-.

This w i l l

A s i m p l e c o m p u t a t i o n shows t h a t

I4(x)I 5 C(1 IV@(X)I 5

ifi,,

c

+

1x1

IxI(1

+

2 -(n+1)/2

1

1x121-(n+3)'2

The f i r s t i n e q u a l i t y shows t h a t ( 5 . 4 ) h o l d s , w h i l e t h e s e c o n d e a s i l y implies (5.5).

On t h e o t h e r h a n d ,

+(c)

=

2 1 ~ 1 5 1 e - ~ ' 1 ~ 1 s, o t h a t

a n d w e c a n a p p l y Theorem 5 . 3 t o t h e o p e r a t o r G(0) = 0 , b y means o f 4 , w h i c h c o i n c i d e s w i t h t h e o p e r a t o r g o ,

ga

u(x,t) =

(3 a Pt)*f(x)

=

A

defined because

t - l a t.* f ( x )

The i n e q u a l i t i e s f o r g k ( f ) , 1 < k < n, f o l l o w i n t h e same way b y a n a p p l i c a t i o n o f Theorem 5 . 3 t o t h e f u n c t i o n ~ ( x )= a P, ( x ) . 0 axk The n e x t a p p l i c a t i o n i s c o n c e r n e d w i t h a n o p e r a t o r i n t r o d u c e d by Marcinkiewicz. For a f u n c t i o n F i n R , we d e f i n e u(F) (x) =

(1

t - 3 l F ( x + t ) + F ( x - t ) - 2 F ( x )1 2 d t ) " 2

m

0

The f u n c t i o n

p(F)

i s c l o s e l y connected with t h e d i f f e r e n t i a b i l i t y

p r o p e r t i e s o f F(x) : A t a l m o s t e v e r y p o i n t x a t which F ' ( x ) e x i s t s , one h a s p ( F ) ( x ) < ( s e e Zygmund [ l ] ) a n d t h e c o n v e r s e i s t r u e i f t h e d e r i v a t i v e i s t a k e n t o e x i s t i n t h e L 2 s e n s e . The ~0

r e s u l t t o be proved h e r e is: COROLLARY 5 . 2 1 . G i v e n integral of

f , i.e.

f E: L 1l o c ( R ) , F(x) =

/:

f.

u e d e n o t e b2

F

the indefinite

Then, t h e i n e q u a l i t i e s

V. VECTOR VALUED INEQUALITIES

518

Proof: Define

I$

=

X [-1 , o ] -

-'1

X[O,l]

Then

*

t

(It*f(x)

=

t

[f(x+y)-f(x-y)]dy

=

t-l [F(x+t)+F(x-t)

-ZF(x)]

0

Thus, with this choice of 4 , p(F) equals the function nf o f Theorem 5.3. It is clear that I$ satisfies (5.4) and ( 5 . 5 ) , and ; ( O ) = 0, so that: lp(F)pp 5 Cp lfl,, 1 < p < m. On the other hand :

I

m

I$(tC)l2

$

=

Im 1 1

1,-2

0

- cos 2ntlzt-3

=

c

<

w

(5

+

0)

0

and this proves the reverse inequality by a new appeal to 5.6(b).

0

Finally, we shall mention the continuous analogue of the dyadic maximal operators considered in Theorem 5.14. Given a multiplier m E Lm(Rn), we define the maximal operator (5.22)

sup R>o

(A iR

(Tsf(x) (2ds)1'2

0

where (TSf)^(C;) = ?(()m(sC). As we did for the dyadic case, this maximal operator is trivially majorized by m

c Eff(x)

+

(J

ITsf(x) - aS*f(x)

0

where

@

E S(Rn),

'ds) - 1/2 S

and we can state:

COROLLARY 5 . 2 3 . _The r e s u l t s of Theorem 5 . 1 4 r e m a i n t o hoZd a f t e r T*f by t h e maximal o p e r a t o r d e f i n e d b y (5.w.

repZacing

6. VECTOR VALUED INEQUALITIES AND A_ WEIGHTS r

A new application o f weighted norm inequalities will be given now:

They can be used to obtain inequalities like (1.8) for a (not necessarily linear) operator. This fact was actually at the very source o f the A theory, since the inequality (see Chapter 1 1 , P 2.12)

V . 6 . VECTOR INEQUALITIES AND An WEIGHTS

519

which i s t h e c l o s e s t a n t e c e d e n t o f

A w e i g h t s , was p r o v e d b y P F e f f e r m a n a n d S t e i n a s a t o o l t o o b t a i n t h e r e s u l t we s t a t e d a s Corollary 4.3. This approach t o v e c t o r valued i n e q u a l i t i e s i s not t h e o n e we h a v e f o l l o w e d h e r e , b u t i t i s n e v e r t h e l e s s i n t e r e s t i n g and w e a r e g o i n g t o d e s c r i b e i t .

THEOREM 6 . 1 . Let p > 0

and

s 1. 1

(Tj)

be f i x e d , and l e t

s e q u e n c e of s u b l i n e a r o p e r a t o r s i n Lp(Rn) such t h a t , t o each u c L;(R") we c a n a s s o c i a t e u c L:(R") with 1 ~ 1 , 5 I u I s and satisfying

(6.2) f o r a21

l l T j f ( x ) I p u ( x ) d x 5 Cp I I f ( x ) J p U(x)dx

f € Lp(U)

E d every

j.

(Tj)

Then, t h e o p e r a t o r s

u n i f o r m l y bounded i n

Lq(Rn), where q = p s ' , following vector valued i n e q u a l i t y holds

and m o r e o v e r ,

the

P r o o f : We s h a l l v e r i f y t h e s e c o n d a s s e r t i o n , w h i c h c o n s i s t s i n a v e r y s i m p l e a p p l i c a t i o n o f H B l d e r ' s i n e q u a l i t y . Given a s e q u e n c e (fj)

in

Lq(Rn),

there exists

u(x)

2

0

such t h a t

1.1,

= 1

and

I n t h e c a s e 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

(T. = M f o r 1 j) we c a n a p p l y t h e t h e o r e m w i t h a r b i t r a r y p > 1 a n d s > 1 by c h o o s n g U(x) = cs Mu(x), w h e r e c s d e n o t e s t h e i n v e r s e o f t h e norm o f M i n L s ( R n ) . T h u s , we o b t a i n all

w h i c h i s t h e more d i f f i c u l t h a l f o f t h e s t r o n g t y p e i n e q u a l i t i e s c o n t a i n e d i n 4 . 3 . More g e n e r a l l y , we c a n s t a t e t h e f o l l o w i n g THEOREM 6 . 4 . Let

( T j ) be a s e q u e n c e of Z i n e a r i z a b Z e o p e r a t o r s ( i n t h e s e n s e of 1 . 2 0 ) and s u p p o s e t h a t , for some f i x e d r > 1, these o p e r a t o r s a r e u n i f o r m l y b o u n d e d i n Lr(w) for e v e r y w e i g h t

V. VECTOR VALUED INEQUALITIES

520

w 6 Ar,

3 JITjf(x)Irw(x)dx

with C,(w)

5 c,(w)

!lf(x)/'w(x)dx

d e p e n d i n g onZy on t h e A,

(w E Ar)

c o n s t a n t of

v e c t o r v a l u e d i n e q u a l i t 2 ( 6 . 3 ) h o l d s f o r alZ

1 < p,

w(x). q <

Then, t h e

(with

oJ

c = cP991 . Proof: It suffices to prove (6.3) for p = r and 1 < q < m , since the extrapolation theorem for A weights proved in Chapter IV P shows that the hypothesis of the theorem are actually verified for every r > 1. If r < q and s = (q/r)I, we take, for instance, s+l o = 7 ,s o that 1 < u s, and observe that, given u E Ls(Rn), the function w(x) = {M(u5) (x) }l/u satisfies: i) u(x) 5 w(x) ii) ,1.1 5 cs IUI, iii) w 6 A I C A, with

A1

constant depending only on

Therefore, the previous theorem can be applied with and the proof of this case is complete. Now, we consider the case are linearizable, i.e.

=

Ci'w(x),

We recall that the operators T. J

r > q. ITjf(x)l

U(x)

5.

=

lujf(X)IB

which are uniformly bounded from for some linear operators U j Lr(w) t o Li(w) whenever w E Ar. Here B is a Banach space which, for notational simplicity, will be assumed to be the same for all j. Given a sequence (fj) E Lq(lr), there exists a sequence (gj) E E Lq'(tr') with unit norm and such that

=

sup

1

kgj(x).U.f.(x), J J

Gj(x)>

dx

1 m

where the supremum is taken over all G . 6 LB* with IGj(x)IB* 5 1 J a.e. (see 1 . 3 ) . Now, if U! denotes the adjoint of U (which will 1 j map B*-valued functions into complex valued functions):

V.6. VECTOR INEQUALITIES AND A

P

WEIGHTS

521

defines a and matters are reduced to proving that the sequence ( U ! ) 1 bounded operator from Lq' (Lf;:) into Lq' (lr'). But, since r ' < q', this in turn will be a consequence of the part already proved (which works equally well for vector valued functions) once the following weighted norm inequalities are obtained: IIU;F(x)l"w(x)dx

5 Crl(w) IlF(x)lii

w(x)dx

(W 6

Art)

(with Crl(w) depending only on the A r l constant of w). Given -r/r w E A r l , we denote v = w E Ar (observe that the A, constant of v is equal to the A r , constant of w raised to the power r F), and take an arbitrary F € Li:(w). There is some g 6 Lr(Rn) with lgl, = 1 such that (jlUiF(x) Ir'w(x)dx) '/" =

<

kuj (iI I f;:

(wl'r'g) (x), F(x)>

F (x)

w (x) dx) 'Ir '

IU;F(x)

=

w(x)l/''g(x)dx

v(x)l"w(x)l/rldx

(1I

=

<

T (v-llrg) (x) I rv(x) dx) ' I r

and the last factor is bounded by theorem.

n

Cr(v)

by the hypothesis of the

As an application of this theorem, we obtain again the vector valued inequalities

IxI-~

where [ c j l m5 C, Ikj(x)l 5 C which were proved in section 3.

and

lvkj(x)l

5 Clxl - n -1 ,

Remarks 6.5. We mention here some extensions of Theorem 6.4: a) The conclusion of the theorem can be strenghtened as follows: If 1 c p,q < m and w E A then the weighted vector valued ine9' quality

V . VECTOR VALUED INEQUALITIES

522

ho ds. This can be easily proved by using the relationsh p be tween weights of different A classes, as in the proof o f the extrapolaP tion theorem. A direct proof of (6.6) for the Hardy-Littlewood maximal operator and f o r singular integrals has been given by Andersen and John [ l ]

.

b) The class Ar can be replaced in the statement of the (This relaxes the hypothesis, since theorem by the class .:A In this case, the weighted vector valued inequalities of A:c A,). the previous remark still hold, with weights w E A" 9'

A final comment about Theorem 6.1 is in order. What this theorem (which can be stated for an arbitrary measure space (X,m) instead of (Rn,dx)) shows is that vector valued inequalities for a family of operators can be obtained if one has enough information about the weights associated to these operators. In 6.4, this process was efectively carried out by using our information about A weights. P What is perhaps more surprising, is that the process can be reversed, and some information about the weighted norm inequalities which an operator verifies can be obtained if one knows that certain vector valued inequalities hold (and the results in sections 1 , 2 and 3 can be used to this end). Bringing to light the general principles on which this reversed process is based is the principal aim of the next chapter.

7 . NOTES AND FURTHER RESULTS

The only linear operators which admit B-valued extensions for every Banach space B are essentially the positive operators. To be precise: Let T : Lp + Lq be a linear operator having a bounded 1 1 -valued extension; then, there exists a bounded linear operator T+ : Lp + Lq which is positive and verifies: lTf(x) 1 5 T+(lfJ) (x) for every f € Lp. The same happens if t 1 is replaced by 1". See Virot [ I ] . 7.1.-

Theorem 2 . 7 was first proved by Paley [l] in the case q = p and with a bigger constant C The version presented here P,P' follows Marcinkiewicz and Zygmund [ l ] rather closely. F o r a general operator of weak o r strong type ( p , q ) , the values of r for which 7.2.-

V.7. NOTES AND FURTHER RESULTS

523

a bounded Lr-valued extensions exists are described in Rubio de Francia and Torrea [ l ] . An interesting extension of theorem 2.7 which is based on Grothedieck's fundamental theorem is the following: Given Banach lattices A , B and a bounded linear operator T : A -+ B, then, for arbitrary fl,f2,..., fn € A:

where KG is the so called Grothendieckls universal constant, whose exact value is still unknown (however, 1 < KG < 2). See Krivine [l]. 7.3.- The following result is of interest in relation with the negative result of C. Fefferman described in 2.12(b) : Take in R' a (for instance, lacunary sequence of directions u . € E l J u. = exp(2-jni)). inequality Then, the L 2-valued .7

3

2) 1 /2

I,

5

cp I ' f

2 1/2 lfjl 1 1,

holds. A related result is that the maximal operator associated to the family of all rectangles having one of their sides parallel to 1 2, see Stromberg [ l ] and CBrdoba and R. Fefferman [2]. The general case, p > 1 , was proved by Nagel, Stein and Wainger [ l ] . 7.4. The more usual form of the Marcinkiewicz integral corresponding to the closed set F is

x

J (XI =

j s(YP

Ix-Yl

-n-X dY

(x E F)

This is the natural extensions to Rn of Marcinkiewicz' original definition for the case n = 1 , which he successfully applied to several problems of convergence and summability of Fourier series. However, JA(x) is infinite for all x E F , and the modified definition, H A , has the advantage o f being defined over all R", while HX(x) = JX(x) for every x 6 F. On the other hand, the Marcinkiewicz integral S , corresponding to a family of intervals in R has been used by Carleson [3] in proving the pointwise convergence of Fourier series. We refer to the expository paper by Zygmund [2] for more details o n this.

V . VECTOR VALUED I N E Q U A L I T I E S

524

Jx and

The i n t e g r a l s

HA

have been extended t o t h e p a r a b o l i c

s e t t i n g ( d e s c r i b e d i n 1 1 . 7 ) b y M . W a l i a s [ l ] and C. Calderdn [2] . The r e s u l t s o b t a i n e d a r e e x a c t l y a s i n 4 . 1 0 , w i t h n r e p l a c e d by t h e homogeneous d i m e n s i o n . 7 . 5 . - The s t a t e m e n t s ( i ) a n d ( i i ) o f Theorem 4 . 2 a r e d u e t o Zo [ l ] ,

w h i l e 4 . 3 was p r o v e d by C . F e f f e r m a n a n d S t e i n [ l ] b y t h e m e t h o d d e s c r i b e d i n s e c t i o n 6 . The u n i f i e d a n d s i m p l e r a p p r o a c h f o l l o w e d h e r e o r i g i n a t e s i n Rubio d e F r a n c i a , R u i z a n d T o r r e a [ l ] . Given a Banach s p a c e

B,

i f the Hilbert transform has a

bounded B-valued e x t e n s i o n

HB

t o L i ( R ) f o r some q , t h e n HB 1 < p < a, a n d o f weak t y p e ( 1 , l ) .

7.6.-

i s a l s o bounded i n a l l ,L{(R), T h i s f o l l o w s f r o m Theorem 3 . 4 . The good s p a c e s

f o r which t h i s

B

h a p p e n s a r e c h a r a c t e r i z e d by a c o n d i t i o n c a l l e d c - c o n v e x i t y : B

c - e o n v e x if t h e r e e x i s t s

is

<

: BxB + R

symmetric, convex

i n e a c h v a r i a b l e and s a t i s f y i n g :

<(o,o)

'

See Burkholder [ Z ]

c(x,y) 5 Ix+YIB

0;

5 IlYlB.

lxlB 5

when

f o r t h e s u f f i c i e n c y and Bourgain [l] f o r t h e

necessity. The R i e s z t r a n s f o r m c a n b e d e f i n e d i n 1 n < - c o n v e x , a n d t h e n , t h e s p a c e HB(R ) coincides with

1 HB

=

1

I f 6 LB

such t h a t

LE,

1,

p

when

B

d e f i n e d i n terms o f

B 1 R . f € LB, J

j=1,2,

is B-atoms

. . . ,n l

A l s o , i f B i s a <-convex Banach l a t t i c e of s e q u e n c e s , t h e n e v e r y r e g u l a 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 h a s a bounded B - v a l u e d e x t e n s i o n t o L E ( R n ) , 1 < p < m, and t h e f o l l o w i n g e x t e n s i o n o f t h e Fefferman-Stein i n e q u a l i t i e s holds: jI(Mfj(x)IjsrlBP

(1 < p <

m).

d x 5 C P ll(fj(x))jsNIi

S e e B o u r g a i n [Z]

dx

.

7 . 7 . - The o p e r a t o r N = N o d e f i n e d i n 4 . 9 ( b ) a n d 5 . 1 9 i s t h e d y a d i c v e r s i o n o f S t e i n ' s maximal s p h e r i c a l mean:

Mf(x)

=

sup

I

f(x-ry

' 1 do ( y ' 1 I

(f E S(Rn))

V.7. NOTES AND FURTFIER RESULTS

525

For Re(") > 1 - n are defines also Ma, whose dyadic version N" was considered in 5.19 and (when a > 0) in 4.9(b). Stein's theorem is the following:

and this is best possible when n 2 3 (see Stein [ Z ] ) , while the results for R 2 remain unknown. The proof of this theorem is not far avay from the techniques developped in section 5, and we can sketch it: One applies analytic interpolation to the two estimates: a) lM"flp Ic",p

lfl,

(Re(")

= 1;

1 < p < ")

When Re(") = 1, I Oa(x) I = C,XB(x), and M a is essentially the Hardy-Littlewood maximal operator. For the second estimate, one first proves the following inequality which is based on the properties of Bessel functions 1 sup (E 5 Ca-B R>o

A4"f(x) A

:1

ITEf(x)[2ds)"2

(a-B > -) 1

2

* B

(where (T'f) = fm ; see the discussion previous to 5.19), and observes that the maximal operator to the right is bounded in 1-n L2(Rn) if B > -z (by Corollary 5 . 2 3 ) . We refer to Stein and Wainger [l] for details. The fact that the dyadic operator N is bounded in all the Lp spaces (i.e., 5.19) was observed by C. CalderBn [ l ] , and also by R. Coifman and G. Weiss. 7.8.- Let Snf f 6 L'(T).

of

denote the n-th partial sum of the Fourier series If 1 < p,q < and (1 lfj]q)l'q 6 Lp, then j

This is a vector valued version of the Carleson-Hunt theorem which follows very easily from Theorem 6.4 and the weighted norm inequalities for S*f = sup lSnf] described in Chapter IV, (see Rubio de n Francia [ l ] ) . 7.9. Vector valued inequalities can be used to obtain mixed norm estimates, i.e., estimates involving the Benedek and Panzone

526

V. VECTOR VALUED INEQUALITIES

Lq2p-norms defined in 1.6(b): "Let S be a linear operator in Rn+m = RnxRm which is translation invariant and satisfies

for some fixed p , q 2 1. Then S is a bounded operator in L ~ ~ P ( R " X R ~AS ) ~ a~ .corollary, every classical singular integral o p e rator in Rn is bounded in Lp1'p2'""pn(Rn) for arbitrary p1,p2,pn > 1, and every multiplier operator in LP(Rn+m) is bounded in L2,P(RnxRm). Details and generalizations can be found in Benedek, Calder6n and Panzone [ l ] , Herz and Rivi6re [ l ] , Rubio de Here we can sketch a proof in the compact Francia and Torrea [l]. case ( T " + ~ instead of Rn+m): First of all, it is not difficult to replace lq by Lq(Tn) in the vector valued inequality, so that:

Then, given g 6 L q y p , we define fu(x,y) = g(u+x,y), and the result is proved because Lebesgue measure and the operator S are translation invariant. 7.10.- A generalization of theorem 2.16 has been recently found by J.L. Rubio de Francia (unpublished). For an arbitrary sequence { I . ) 1 of disjoint intervals in R, the inequality

holds. The p r o o f is a combination of the ideas in 2.16 and 3.8.

CHAPTER VI FACTORIZATION THEOREMS A N D WEIGHTED NORM INEQUALITIES

A chief objective of t h i s chapter consists i n establishing a general

p r i n c i p l e o f e q u i v a l e n c e between c e r t a i n t y p e s o f w e i g h t e d and v e c t o r v a l u e d i n e q u a l i t i e s . S u ch a r e s u l t a p p e a r s i n s e c t i o n 5 , a n d i t

i s e s s e n t i a l l y a t r a n s l a t i o n i n t o a l a n g u a g e more a c c e s s i b l e t o FOE r i e r a n a l y s t s o f r e s u l t s d u e t o B . Maurey w h i c h a r e o b t a i n e d i n t h e previous section. One c a n a c t u a l l y p r o v e d i r e c t l y t h e t h e o r e m s o f s e c t i o n 5 , w i t h no r e f e r e n c e t o f a c t o r i z a t i o n r e s u l t s (see Rubio de F r a n c i a [ 3 ] , where t h i s i s d o n e ) , b u t t h e i d e a s l e a d i n g t o them f o r m a l o n g r o a d w hose m i l e s t o n e s w o u l d b e p l a c e d a t Kolmogorov [l]

[2]

, XIII. 1 . 2 2 ) , Stein and

Maurey [I]

I

[Z].

[?I,

C a l d e r B n ( s e e Zygmund

B u r k h o l d e r [ 3 ] , S a w y e r [l] , N i k i s h i n [ l ] , T h u s , we f e l t more r e a s o n a b l e t o g i v e a n g

[6],

t u r a l c o n t e x t t o such theorems by s t a r t i n g from N i k i s h i n ' s theorem ( p r o v e d i n s e c t i o n Z), w i c h i s a l s o a n i m p o r t a n t g e n e r a l p r i n c i p l e i n v i e w o f i t s a p p l i c a t i o n s , a n d by g i v i n g some r e s u l t s i n i t s d o u b l e f o r m u l a t i o n : a s f a c t o r i z a t i o n o f o p e r a t o r s a n d a s w e i g h t e d norm i n e q u a l i t i e s . For a r a t h e r c l o s e

look a t t h e h i s t o r i c a l p a i h

up t o

N i k i s h i n ' s t h e o r e m we r e f e r t o de Guzm5n (21, Ch. 1 1 . lVe h a v e f o l l o w e d h k u r e y [ I ] ,

/:Z]

f o r t h e p r o o f s o f t h e main r e s u l t s

i n t h e f i r s t f o u r s e c t i o n s . The o r g a n i z a t i o n o f t h e r n a t c r i n l i n t h e s e s e c t i o n s a l s o owes v e r y much t o t h e v i e w p o i n t s o f . J . G i l h e r t a n d C . P i s i c r whi"F1 b o t h g e n e r o u s l y s h a r e d with u s . S e c t i o n s and 8 p r e s e n t j u s t obtained.

: i

v a r i e t y o f appl ic:itions

0,

o f t h e g e n e r a l principle

7

V I . F A C T O R I Z A T I O N THEOREMS

528

I . FACTORIZATION THROUGH W E A K - L P We m u s t b e g i n by e s t a b l i s h i n g some n o t a t i o n a n d b a s i c f a c t s . A l l t h r o u g h t h i s c h a p t e r , B w i l l d e n o t e a Banach ( o r r - B a n a c h ) s p a c e , a n d (X,m) a u - f i n i t e m e a s u r e s p a c e . By Lo(X,m) = Lo(m) we s h a l l i n d i c a t e t h e s p a c e o f all m e a s u r a b l e f u n c t i o n s p r o v i d e d w i t h t h e t o pology of l o c a l convergence i n measure: f.

1

f o r every

-+

0

(in

Lo(m))

and

A > 0

E C X

m(En{x : I f j ( x ) I > A})

iff with

m(E) <

m.

-+

0

The n o n i n c r e a s i n g

rearrangement of a function f i s t h e only nonincreasing function f * : R, R, which i s equimeasurable w i t h f , and i t i s d e f i n e d -+

by

The f o l l o w i n g f a c t s a r e e a s i l y v e r i f i e d , a n d t h e r e a d e r i s u r g e d t o s u p p l y a p r o o f f o r them: (1.1)

When

m(X) <

CPC Lo(m)

a s e t of f u n c t i o n s

m,

i s bounded

( S o r t h o c o n v e r g e n c e i n m e a s u r e ) i f and o n l y if t h e f u n c t i o n : C(A) = s u p m ( I x : I f ( x ) I > A ) ) f€@ l i m C(A) = 0 . E q u i v a l e n t l y , 0 i s b o u n d e d i n

siriisfies

Lo(m)

A-+m

and o n l y i f

fA(t) 0

for ever3

a sublinear operator

T : B

-t

L0(m)

i s con-

( o r b o u n d e d ) i n m e a s u r e i f and onZy i f a n y o f t h e follo-

w i n g e q u i v a l e n t c o n d i t i o n s ho Id:

b)

EGP e v e r y

t > 0,

there e x i s t s

( T f ) * ( t ) 5 K(t) l f l B I n many a p p l i c a t i o n s ,

X

is

Rn

K(t) <

such t h a t :

( f € B). a n d h a s n o t f i n i t e m e a s u r e , b u t we

can s t i l l apply t h e preceding r e s u l t s t o g e t h e r with:

VI . I . FACTORIZATION THROUGH WEAK-LP X

(1.3)

6 X.I

=

'

j.

i f and o n l y i f

<

3

is

9 C Lo(m)

Then,

m.

{fX : f 6 0 1 i s b o u n d e d i n Lo(Xj,m) j' SimilarZy, T : B + Lo(m) i s c o n t i n u o u s i n measure s o a r e t h e o p e r a t o r s T.f = (Tf)XXj for a l l j.

b o u n d e d if and o n l y i f f o r every

m(X.)

with

-

i=l J

529

@.

=

3

3

In V.l, the spaces LE were defined for every p > 0. Just in the same way, Li(m) is defined as the space of all strongly measurable B-valued functions, and F . + 0 in L;(m) simply means that J The three results we have just stated remain IFjlB + 0 in Lo(m). true if B-valued functions are considered instead of complex valued is a E m ones. On the other hand, if B is a Banach space, Li(m) p l e t e metric linear space, so that such general principles a s the Banach-Steinhauss or the closed graph theorems can be applied. In particular, we can state the so called Banach continuity principle:

PROPOSITION 1.4. ~ e tT. : B I

+

j E: B,

Lo(m),

b e a s e q u e n c e of e o n -

tinuous linear operators. I f

T*f(x)

sup

=

ITjf(x)l

<

a.e.

m

(f C

B)

j

Then T* __ i.e.,

i s a c o n t i n u o u s ( s u b l i n e a r ) o p e r a t o r from

EC X

f o r every subset

of f i n i t e m e a s u r e :

m ( { x 6 E : T*f(x) with

lim

CE(x)

=

Lo(m),

B

>

XllflBl) 5 CE(X)

(f € B)

0

A+-

Proof: Let U f = (T.f)y=l. Then U is a well defined linear opera J tor from B to Lo (m), and by the closed graph theorem, it is contibm

nuous. Since

T*f(x)

=

luf(x)ll

.ern'

T*

is continuous.

0

Remark 1.5. In the situation described by the last proposition, suppose that i) The space

B

ii) The operators

riant,

iii) B i.e.

as well

1 Lloc(Rn)

is contained in

T . are positive: I

as the operators

lTjfl 5 Tj(lf/)

T. are translation inva3

1'1. FACTORIZATION THEOREMS

530

f E B =

j 6 B

for a l l

fh E B,

implies

lfhaB = l f l B and

T.(fh) = 1

(TjfIh h 6 Rn,

and

where

fh(x)

f(x-h).

=

Then, t h e f o l

lowing a l t e r n a t i v e h o l d s : E i t h e r

to

a) T*f(x) LOCR") o r

=

sup I T j f ( x ) j

I

f E B

b) t h e r e e x i s t s

i s a c o n t i n u o u s o p e r a t o r from

such t h a t

In f a c t , i f (a) f a i l s t o occur, t h e r e a r e that

by

IEo I >O and T*f(x) = m { q k } a n d e n u m e r a t i o n o f Qn

for a l l

T*f(x) = g E B

(the points of

a.e.

m

and

x E Eo.

B

EoC Rn

such

Then, d e n o t i n ?

Rn

with rational

m

for a l l

c o o r d i n a t e s ) we d e f i n e f(x) =

f 6 B

so t h a t x E E,

But

E

and

lg(x-qk)l

E

k

T*f(x)

2 sup

2

-k

*

T g(x-qk) =

k

where

h a s p o s i t i v e measure and a d e n s e subgroup o f p e r i o d s : f o r a l l q E Qn,

q+E = E

IRn-EI

and it f o l l o w s t h a t

=

0.

B y a more c a r e f u l a r g u m e n t ( s e e K a t z n e l s o n [ l ] , 11.3 and S t e i n [h]) t h e p o s i t i v i t y a s s u m p t i o n ( i i ) c a n be e l i m i n a t e d , a n d t h e a l t e r n a tive still stands. Now, w e s h a l l i n t r o d u c e o u r ( r e s t r i c t e d ) c o n c e p t o f f a c t o r i z a t i o n o f operators:

Definition 1 . 6 . A continuoua o p e r e t h e Banach ( o r r-Rancrch) rabZe f u n c t i o n

To : B

-t

L

space

g(x) > 0

a.e.

L

c

T : B

~.O(rn)

Lo(m) factors through if t h e r e exists a measu-+

and a c o n t i n u o u s o p e r a t o r

w h i c h m a k e t h e f o l l o w i n g diagram c o m m u t e

B

T

'

Lo(m)

VI . I . FACTORIZATION THROUGH WEAK-LP

M

Where

531

M f

stands for the muZtip1ication operator:

g

We shall be interested in the cases = weak - LP(m).

L

LP(m)

=

=

,g

L

and

=

gf. Lt(m)

=

In order to state our first important result, given an operator T : B + Lo(m), we shall denote by ?. its extension mapping sequences in B into sequences in ~ ' ( r n ) :

THEOREM 1 .7. Let and l e t 0 i p < a) T

T : B

+

Lo(m)

be a c o n t i n u o u s s u b l i n e a r o p e r a t o r ,

The 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.

L[(m)

factors tFiroue

that __

L:

w(x)dm(x)

5

> 0

w(x)

b ) There e x i s t s a measurable f u n c t i o n

oflB P

a.e.

___ such

(f C B ; X > 0 )

(T)

ITf(x) / > X I

c ) ( o n l y when

CE > 0

and

E(E)

m({x d)

?

c

m(X)

X

2' E ( E )

For e v e r 2

< a).

with

rn(X-E(&))

: ITf(x)(

> A))

E

5

C E ( 7 )

is a b o u n d e d o p e r a t o r f r o m

there e s i s t

> 0,

E

such t h a t

<

UflB

Lp(B]

P

(f6B;

X>O)

Lo (m) L"

Before proving this theorem, a few remarks must be pointed out: First of all, the equivalence of (a), (b) and (c) is rather trivial, and the interest of the theorem lies in the fact that any of them is implied by (d). Observe that, when B = LC1(u dm) for some u ( x ) > 0 , condition (b) is a weighted weak type inequality. 'Thus, the difference between (this kind of) factorization results and weighted norm inequalities is just a matter of languafle. Condition (c) w a s first considered by Nikishin [ l ] , [2], and is sometimes expressed b y saying that T is almost bounded from B to Lt(ni). Fiii:illy, to clarify condition (d), we state i t more explicitly i n the c a s e in(X) < a: There exists C ( 1 ) such that l i m C ( 1 ) = 0 2nd i+cv

(1.8)

c lfjli 2 .l

1

implies

m ( { x : sup /Tfi(x)I > A } ) .I

< C(A).

532

V I . FACTORIZATION THEOREMS

P r o o f o f Theorem 1 . 7 : I n v i e w o f ( 1 . 3 ) , i t c a n b e a s s u m e d t h a t m(X) = 1 i n a l l c a s e s . Then, we s h a l l prove t h e f o l l o w i n g c h a i n o f implications:

(c)

(b): This i s very easy. I t s u f f i c e s t o take m

w(x) = j =1l k j ' E ( l / j ) ( ' ) m

where

(b)

=*

1

k. > 0

and

(a).

Given

1

j=1

k. C 1

w(x) > 0

. 111

1.

=

a.e.,

E . = { x : 1 < w(x) < J j -

1 j1)

( j = 1 ,2,3,. . . )

2jXE,, a n d l e t us J i s a bounded o p e r a t o r f r o m B t o

which a r e d i s j o i n t and c o v e r a l l prove t h a t

we d e f i n e t h e s e t s

X.

Tof(x) = g(x)-'Tf(x)

Set

g = 1 j

LE(m) : m

m(Ix : ITof(x)l > A } ) m

( 3 )

=+

(d):

=

j =1l

m(Ix € E. : ITf(x)I 1

> ZjA))

I

Now,

Tf(x) = g(x)Tof(x)

bounded o p e r a t o r f r o m we h a v e

(t

B

to

L!(m).

a n d we know t h a t Thus, given

Therefore, (1.8)

To

is a

(fj)j21 € Lp(B),

is verified.

where

C(t)

(d)

( c ) : Let C(X) b e t h e f u n c t i o n a p e a r i n g i n (1 . 8 ) a n d , g i v e n t a k e R l a r g e e n o u g h s o t h a t C ( R ) < E . We a r e g o i n g t o

E.

=+

> 0,

-+

0

+

m).

VI.2. NIKISHIN'S THEOREM

Fc X

consider the family of measurable subsets m(F)

ITf(x)

Ip

>

533

such that

(a.e. x 6 F)

RP

(1.9) for some

f 6 B

lflB

with

5 1

If there is some set F satisfying (1.9), let F be the family of all countable sequences of disjoint sets which verify it. We order F in the usual way: ( F j ) d ( G j ) when (Fj) is a subsequence of (Gj). By Zorn's lemma, there is a maximal sequence (Fj) in F , and denoting by (f.) the corresponding elements of B , and also 1 c . = m(Fj)l'p, F = F j , we have J

u j

s u p IT(c .f. ) (x) J J j

> R

(a.e.

x E F)

and

F.) J

=

m(F) 5 1

Both inequalities together imply m(F) < C ( R ) < E = E(E) = X - F verifies m(X-E) < E and

E,

so that the set

since otherwise, we could choose f with lfiB = 1 such that = En {x : ITf(x)I > A} satisfies (1 .9), thus contradicting the maximality of ( F j ) . Therefore, (c) is proved in this case. It remains to consider the possibility that no measurable set F satisfied (1.9). But, if this is the case, (1 .lo) holds with E = X by 0 the same previous argument.

E'

2.

NIKISHIN'S THEOREM

We have already obtained a necessary and sufficiente condition (1.7 (d)) for the factorization of an operator through weak-LP. However, this condition is not easy to apply on specific examples, and o u r next step will consist in showing that, when the space B has a c e x tain structure, 1.7(d) is automatically verified (no matter how good the operator T is). Moreover, it is specially simple to check whether the space B = L q has the required structure or not, and this will lead us to formulate Nikishin's theorem and obtain some applications.

VI. FACTORIZATION THEOREMS

5 34

Denote by (rj(t))jjl the sequence of Rademacher functions, which by are defined in the interval [ O , l )

where Ik. stands for the characteristic function of the interval [(k-1)2-j, k2-j). The Rademacher functions form a (not complete) o r thonormal system in L2( [ 0 , 1 ) ) having many interesting and well known properties, among them, the following two which we shall need: ( 2 . 1 ) T h e Rademacher f u n c t i o n s f o r m a s y m m e t r i c s e q u e n c e

f o l l o w i n g sense: Given a measurable f u n c t i o n and n u m b e r s

F(rl(t),r~(t),

E .

J

=

in the

F(xl ,X2,. . . )Xn)

+ 1 (j=l)Z,...,n), the functions and F[Elrl(t),E~r2(t) ,...y~nrn(t))

-

. . . ,rn(t))

Rn

are

equidistributed.

0 p < m, then there e x i s t constants t h a t K i n t c h i n e ’s i n e q u a l i t i e s h o l d : ( 2 . 2 1 I-f

f o r e v e r y f i n i t e sequence

(aj)

kp,

K

P

> 0

such

of c o m p l e x n u m b e r s .

Detailed proofs of both properties will be given in Appendix A.l for completeness. Since p 5 2 5 q implies L ~ ( [ o , I ) ) c L ~ ( [ o , I ) c 7 Lp([O,l)) and .lpC L - C Rq (with continuous inclussions), we observe that Kintchine‘s inequalities imply

and

Both (2.3 a n d (2.3’) hold only f o r the specified ranges of p ’ s and q ‘ s . For instance , t:i k i n g “1 = “2 = * . + - - aN = 1 in (2.3) ,ind u s i n g Kin tc ti i n e s i n e q u a l i t i e s , i t follows that N1/’ < kL’KpN’’p, which, ior large N , f o r c e s p 5 2. Now, we come to our main resiilt in this section: TIEOREM 2.4. 0 0

such t h a t ,

NIKISHIN'S THEOREM

(fj) C B

f o r e v e r y f i n i t e sequence

T : B

Then, e v e r y continuous s u b l i n e a r o p e r a t o r through

535

+

Lo(m)

factors

Lt(m).

Proof: It can be assumed with no loss of generality that m ( X ) = 1 . Our task is then reduced to showing that (1.8) is verified, and by a simple limit argument, it suffices to show it for a finite sequence f l , f 2,..., fN 6 B such that X If.lp < 1. For every t € [O,l], J B j define gt

= C

j

r.(t)f. '

3

B

E

Then

and, for each fixed x 6 X , both terms on the right hand side are equimeasurable as functions of t (since the second one is obtained from the first one by changing rj(t) into -rj(t) for all j # k). Therefore I{t

: /Tgt(x)I > ITfk(x1

1

111 2 7

( I . I stands for Lebesgue measure in [O, 11). Since x this holds for k=1,2, . . . , N, we can replace ITfk(x)I and thus sup lTfj (x) 1 ,

is fixed, and by

j

All 5 2 1

> A)[drn(x) IIt : ITgt(x) X Let E = It 6 [O,l] : lgtlB > and denote by C A ) the function appearing in l.2(a) which exists b y the continuity of T. Then, we apply (5) and Fubini's theorem to get

m(Ix

: sup ITfj(x)l j

>

< 21EI 2

+

IR,lB)dt

5

A)

Since ? ( A ) is independent of N and rified (1.8), and Theorem 1.7 applies.

lim X+m

0

?(A) = 0 , we h a v e v e -

536

V I . FACTORIZATION THEOREMS

According t o t h e terminology introduced i n t h i s theorem, that all

C

( 2 . 3 ) means

{complex numbers} i s o f t y p e 2 , and h e n c e , o f t y p e

=

0 2 , due t o t h e f a c t t h a t (2.3) is b e s t p o s s i b l e . E v e r y B an ach s p a c e i s o f t y p e 1 . V o r e g e n e r a l l y ,

Examples 2 . 6 ( a )

every r-Banach space is of type

r.

(b) B = Lp(p) i s o f t y p e i n f ( p , 2 ) , 0 2 , w e u s e Kintchine's inequalities:

J

More g e n e r a l l y , i f

is of type

p0'

i s a B an ach s p a c e o f t y p e

B

then

Lg(p)

inf(p,po)

Some g e n e r a l i n f o r m a t i o n a b o u t t h e t y p e o f Ba na c h s p a c e s i s c o n t a i n e d i n 9.2.

For o u r p u r p o s e s , example (b) i s s p e c i a l l y intercsting,

s i n c e i t g i v e s t h e f o l l o w i n g immediate a p l i c a t i o n : COROLLARY 2 . 7 .

Let

( N i k i s h i n ' s theorem).

m e a s u r e s p a c e , and l e t

T : Lp(p)

0 0

such

a.e.

that -

w here

q

5

I

Ig(x)

=

inf(p,2). implies

Proof: Since

Lp(p)

M o r e o v e r , if

ITf(xj1 5 lTg(x)I

i s of type

q,

i s positive

T

-l

t h e n We c a n t a k e

f r o m T h e o r e m 2 . 4 . On t h e o t h e r h a n d , i f

T

lfj,

5 1

m

(fj)j=l

j

lf.Ip < 1, J P -

we define

in

Lp(p) The n

j

m({x : s u p I T f i ( x )

I

> A})

5 m({x : I T f ( x )

I

> XI)

5

5

q = p.

i s p o s i t i v e , we s h a l l

f = (I I f . I P ) l / P . j J and sup I T f . ( x ) I 5 I T f ( x ) l , and t h e r e f o r e J C

I

the f i r s t assertion follows

v e r i f y t h e c o n d i t i o n 1 . 7 ( d ) d i r e c t l y : Given such t h a t

If(X)

(&

C(X)

VI.2. NIKISHIN'S THEOREM

where m(x)

<

537

C(X)

is as in 1.2(a) (we have assumed, as always,

m ) .

0

The weak point of Nikishin's theorem is that it says nothing about the weight w(x), but no information could really be expected under so general hypothesis. However, the more we know about the operator T and its invariance properties, the more w(x) becomes determined. The next two corollaries will be enough to illustrate this principle. COROLLARY 2.8. (Stein's theorem). Let G b e a l o c a l l y c o m p a c t g r o u p m, and l e t T : Lp(C;) + Lo(G), 0 < p < m,

w i t h l e f t Haar m e a s u r e

b e c o n t i n u o u s i n m e a s u r e , s u b l i n e a r and i n v a r i a n t u n d e r l e f t t r a n s lations:

T(fy) (where exists

i f

G

(y E G ;

(TfIy

f E Lp(G))

fy(x) = f(yx)). Then, f o r e v e r y compact s e t CK > 0 s u c h t h a t

q = inf(p,2)

with

=

lor

q = p

i s a compact group,

then

K C G,

there

T is p o s i t i v e ) . I n p a r t i c u l a r ,

i-f T

i s a n o p e r a t o r of weak t y p e

(p,q) * Proof: Starting from the weighted weak type inequality of Corollary 2.7, and using the left invariance of m and T , we obtain

1 Take h E L+(G) with lhll = 1 , and integrate both sides of the inequality with respect to h(y)dm(y) to get

Then, w*h(x) We can assume from the beginning that w E L"(G). turns out to be continuous and everywhere positive, and it suffices to take CK = (inf w*h(x))-'. x6K

n

A global weak type result can be proved in the non-compact case by

assuming further invariance of the operator

T:

.

538

VI

COROLLARY 2.9. Let + Lo(Rn)

0 < p

T : LP(Rn)

FACTORIZATION THEOREMS

5 2. E v e r y s u b l i n e a r o p e r a t o r w h i c h i s c o n t i n u o u s i n m e a s u r e and i n v a r i a n t (p,p):

u n d e r t r a n s l a t i o n s and d i l a t i o n s , i s of weak t y p e

I{x E: Rn : ITf(x)I

3

T

>

XI1 5 C(lfl,/XIp

i s positive, the result i s valid for a l l

0 < p <

m.

Proof: Arguing as in the previous corollary, we obtain

with w(x) continuous and everywhere positive. Now, we replace f(x) by f6(x) = f(6x) and use the dilation invariance of T (which means: T(f6) = (Tf)6 for every 6 > 0) to get the above inequality with w(x) replaced by w(6-l~). Letting 6 + m, Fatou's lemma gives the desired weak type inequality with

c

a

= W(O)?

The applications presented below are intended to illustrate the power of Nikishin's and Stein's theorems, but we have limited oursel ves to examples which do not require a heavy background nor a long discussion. EXAMPLES 2.10. (a) Let T = [ O , l ) denote the torus. F o r each f E L1 ( T ) , the conjugate function was defined in Ch.1 as

;(x)

lim Tr(e21Tix) = lim C - i sgn(k) rIkl?(k)e 21~ikx r +1 rfl k where the limit was proved to exist by a short argument from complex function theory. Since f -+ ir is, for each fixed r < 1 , a continuous operator in L 1 (T), we conclude from Banach's continuity prig is continuous in measure ciple that the operator: f + ?* = s y p and translation invariant. Thus, 2.8 applies giving Kolmogorov's in5 quality =

lZr1

Therefore, the basic inequality for the conjurate function is really a consequence of the mere existence of the operator at almost every point. The same can be done for the Hilbert transform in using 2.9.

L1(R)

by

VI.2. NIKISHIN’S THEOREM

539

(b) Kolmogorov’s example of a divergent Fourier series such that can be modified to produce a function f 6 H ’ ( T )

diverges a.e. (see Zygmund [ l ] , Ch. VIII) . Something more can be said in this direction: “ G i v e n a t r a n s Z a t i o n i n v a r i a n t B a n a c h Z a t tice B C L1(T) s u c h t h a t t h e t r i g o n o m e t r i c p o Z y n o m i a Z s f o r m a e x i s t s f 6 B such t h a t d e n s e s u b s p a c e of B, = t h e r e lim s u p ISN f(x) I = m a.e., t h e n , t h e same i s t r u e f o r some N - + w

f E B

m

of a n a Z y t i c t y p e :

f(x) -

1

?(k)e

21rikx,, . In other words,

0

for the problem of almost everywhere convergence of Fourier series, there is no difference between the spaces B and HB

=

{closure in

B

1 cke Znikx 1

of analytic polynomials:

0

In fact, by translation invariance, i t is easy to prove (see 1 . 5 ) the following alternative for the maximal operator S*f(X)

=

s u p ISNf(X) N

1

Either S* is continuous in measure in HB, o r S*f(x) = m a.e. for some f 6 HB. Let u s suppose that the first possibility happens. Since every Banach space is of type 1 , we must have (f 6 HB)

and translation invariance can be put into play to show, as in Corollary 2.8, that we can take w ( ~ ) = c (constant.). Now, given a trigonometric polynomial

N =

1

-N

‘k

2nikx

we form another polynomial of analytic type: and observe that

(0

5 n 5 N) .

Therefore:

h(x)

=

e

2 niNx

,e(x),

VI. FACTORIZATION THEOREMS

540

By density, the weak type inequality holds for all g E B, and then, we should obtain a.e. convergence of Fourier series for all functions in B , which we assumed to be false. (c) We try to study the almost everywhere convergence of the spherical sums

in the case

n > 1.

3

1 5 p

t h a t lim s u p ISRf(x)I

=

R-+m we should have

=

m

<

G.

2, t h e r e e x i s t s f 6 Lp(Rn) such In fact, if this were not the case,

sup ISRf(x) I < m a.e. for every R>o (by the same alternative stated in (b)), and by Corollary f E Lp(Rn) 2.9, S * would be an operator of weak type (p,p) . A fortiori, the ball multiplier operator S1 would be of weak type (p,p) , and interpolating with the obvious L2 result: S*f(x)

lS,fIq 5

cpi,

(P

< 4

5, 2)

which contradicts Fefferman's negative result (Fefferman [2]). The argument can be modified to prove the same divergence result in Lp, p 2, f o r every fixed sequence R. m. 1

-+

(Y,u)

be an arbitrary measure space, and s u p pose that T. : Lp(p) -+ Lo(m) is a sequence of continuous linear 1 operators such that the limit ( d ) Let

Tf(x)

=

lim

T.f(x)

j

J

exists a.e. for every f E L p ( p ) . Then, the 12-valued version of this property is equally true: " F o r e v e r y s e q u e n c e (fk) s u c h t h a t we h a v e ( C lfk12)1'2 E LP(p), k

To prove this, we start from Nikishin's theorem applied to which gives = s u p ITjf(x) 1 ,

T*f(x)

1

(2.11)

m({x

: T*f(x)

> A})

5 (If

> 0 a.e., and for some measure dm(x) = w(x)dm(x) with w(x q = inf(p,2). Now, we linearize T* by means of the operators

=

VI.2. NIKISHIN'S THEOREM

54 1

where v(x) is an integer-valued measurable function. It is clear that (2.11) remains true if we replace T*f(x) by ITvf(x) I , for a y bitrary v , and then, we can invoke the weak type version of the Marcinkiewicz-Zygmund theorem ( V . 2.9) to get

where

B

=

L2,

F

=

(fk)LZ1 E L g ( u )

the proof, we observe that, once so that SUP

F

and

TVF

=

is fixed,

(Tvfk)FZ1. v(x)

To finish can be chosen

lTjF(x)llB 5 21Tv~(x)lB

j m

with T . F = (Tjfk)k,l. 'Therefore (2.12) gives raise to a weak type J inequality for the maximal operator: F -t sup (T.FIB. Since j

J m

ITjF(x) - TF(x)/jB + 0 a.e. for every F = (fk)k=, with a finite number of nonvanishing components, and these ones form a dense subthe same is true for a general F € L g ( p ) . space of Lg(p), (e) The fact that, in the theorems of Nikishin and Stein, we do not obtain a bounded operator from Lp to weak-LP when p > 2, deserves a word o f erplanation. Consider a multiplier 2 n A operator n L ( T ) : Tmf(x) C f(k) mke2Tik*x, where k m = (mk)kcZn is a bounded sequence. Tm is bounded in L2(Tn), and a fortiori

Should Tm be of weak type ( p , p ) f o r some p > 2, it would be bounded in Lq for all 2 < q < p (by interpolation), but this is certainly not true for an arbitrary bounded sequence (mk). The simplest way to see this is asfollows: Pick a function f 6 L2(Tn) and consider the operasuch that, for every q > 2, f e Lq(Tn), tor T, associated to the multinlier sequence E = ( & k ) kcZn with Ek = + I . For some choice of signs E (actually, for almost every (this can be easily E), we have T,f € Lq(Tn) for all q < proved by means of Kintchine's ineoualities (2.2); see Zygmund [l] ; Ch. V). Since TE(TEf) = f, it is clear that TE cannot be a bounded operator in Lq if q > 2.

V I . FACTORIZATION THEOREMS

542

The maximal Bochner-Riesz operator in

Tn

(see Stein-Weiss [2], Ch. VII) provide another interesting example o f the same situation. For every a > 0:

I s:fi but and

Sz ct

5

can fi

5

can f t

is not an operator o f weak tyne < (n-1)/2.

(2

IP ( " 1

( q , q ) if

2n q > n-1-2% ~

3. FACTORIZATION THROUGH Lp

In this section, we try to carry out a Frogram f o r the problem of factorization though Lp similar to that o f sections 1 and 2 for the factorization though weak-LP. We begin with an auxiliar result which is interesting in itself. PROPOSITION 3.1. Assume t h a t m(X) = 1 , a n d l e t A b e a c o n v e x 0 Then, f o r s u b s e t of L+(m) = { n o n - n e g a t i v e f u n c t i o n s i n L o ( m ) } . every E > 0 , there i s a measurable subset S(E) C X such t h a t m(X - S ( E ) ) < E & 5 2 sup f€A

f(x)dm(x)

SUP

f6A

IS(€)

f*($)

To clarify the meaning of this statement, observe that, for a single one can define S ( E ) = {x : f(x) 5 f * ( E ) } , function f E Ly(m), and then : m(X

-

S(E))

i E

and

f dm <

f*(E)

The aim of the proposition is obtaining essentially the same for all the functions in a convex subset of L y ( m ) simultaneously. Just as in Theorem 1.7 a certain form of the axiom o f choice was used, here we shall need to apply the followinE lemma (whose proof nre give in Appendix A . 2 ) in an essentially nonconstructive way. MINIMAX LEMMA 3 . 2 . L e t ~

A

and ~

B

e o n v e x s u b s e t s of c e r t a i n v e c -

VI . 3 . FACTORIZATION THROUGH Lp

543

t o r spaces, a n d a s s u m e t h a t a t o p o Z o g y i s g i v e n in i s cornpact. If t h e f u n c t i o n i ) @(.,b)

: AxB

@

+

R U

A

is a c o n c a v e f u n c t i o n on

iil @(a,.)

B

is e c o n v e x f u n c t i o n o n

iiil @(a,.)

is lower s e m i c o n t i n u o u s on

B

f o r w h i c h it

satisfies:

{+m}

b E B

for each

a E A

for e a c h

B

for each

a € A

then, t h e f o l l o w i n g i d e n t i t y holds:

min bEB

sup aEA

@(a,b)

=

sup aEA

min bEB

@(a,b)

Proof of 3 . 1 : Define

B Then, B function

= {g E

Lo(m)

is a convex

o

:

5 g(x) 5 I ,

l g dm

2

*-weakly compact subset of

@(f,g)

I

=

fg dm

(f E

A,

1 -

E

2)

Lm(m),

and the

g E €3)

satisfies the conditions of the minimax lemma. In fact, to verify condition (iii) put Et(f) = Cx : f(x) 5 t J , so that

and each function Ot(f,.) is *-weakly continuous. Let us denote )I = sup f*($), and assume that PI < m , since otherwise there is f€A nothing to prove. Then,we observe the following: F o r every f € A

,

i.e.,

f 6 A min gEB

implies sup fEA

:SUP f€A

JfR

'E!,(f)

XE,,(f) dm

=

sup f€A

'

and therefore

min gEB

/fg d m

f dm 5 I,

If the minimum is attained at t x : go(x) 1. 71, s o that

=

B,

€

go E

B,

we define

S(E)

=

VI. FACTORIZATION THEOREMS

546

and m(X-S(E))

< 2

- go(x))dm(x)

(1

5

E

q.e.d.

0

Now, we can state the analogue of Theorem 1 . 7 for o u r present problem: THEOREM 3 . 3 . Let T : B + Lo(m) be a c o n t i n u o u s s u b l i n e a r o p e r a t o r , and l e t 0 0

b ) There e x i s t s a measurable f u n c t i o n

a . e . such

that

jlTf(x1 IPw(x)dm(x) m(X)

c) ( o n l y w h e n

CE > 0

&

d)

E(E)

?

c

X

5

< ml.

such t h a t

lfli

(f € B )

For e v e r y

m(X-E(E))

i s a bounded o p e r a t o r f r o m

E

< E

Lp(B)

> 0,

there e x i s t

and

5

Lo (m)

LP

As in the case of factorization through weak-LP, the interesting part of the theorem is that (d) implies the other three conditions. A more explicit formulation of (d) is: There exists C ( A ) such that lirri C ( A ) = 0 and

x

+m

(3.4)

C

j

<

J B -

1

implies

m({x

: (1 ITfj(x)lp)l'p j

> A})

5

Proof of Theorem 3 . 3 : The proof of the implications

is quite similar to (and sometimes simpler than) the corresponding implications in Theorem 1 . 7 . Thus, we shall limit ourselves to prove that (d) implies (c), We assume as always m(X) = 1 . Since (gr)*(t) = g*(t)' f o r every r > 0, the continuity of '? can be expressed as follows:

545

VI . 3 . FACTORIZATION THROUGH Lp

with

K(t) <

f o r each

( s e e (1.2) and t h e remarks a f t e r

t > 0

( 1 . 3 ) ) . Now, we a p p l y o u r P r o p o s i t i o n 3 . 1 t o t h e c o n v e x s e t

A

ITf.lp : f . E B and 1 3 j and w e f i n d a s e t S(E) C X such t h a t {C

=

f o r every

f 6 B

asserts.

0

with

lfiB 5 1.

Zlf.lP < 1) 3 B m(X-S(E)) <

and

E

But t h i s i s p r e c i s e l y what ( c )

I n view o f N i k i s h i n ' s t h e o r e m , t h e r e i s now a q u e s t i o n w h i c h a r i s e i n a n a t u r a l way: I s t h e r e a c o n d i t i o n o n t h e s p a c e

B

t h a t e v e r y o p e r a t o r deEined on through

COROLLARY 3 . 5 .

(of N i k i s h i n ' s theorem).

s p a c e of ( R a d e m a c h e r l t y p e

P+E

l i n e a r o p e r a t o r d e f i n e d on

B

through

w h i c h ensures

p # 2:

H e r e i s a l l t h a t we c a n s a y f o r

Lp?

B

and c o n t i n u o u s i n measure f a c t o r s

E

> 0,

9

B

0 < p < 2

f o r some

then, every sub-

and c o n t i n u o u s i n m e a s u r e f a c t o r s

LP.

T h i s i s e s s e n t i a l l y c o n t a i n e d i n Th eo r em 2 . 4 , s i n c e , f o r f i n i t e L!+' C Lp ( s e e V . 2 . 8 ) . I t i s o n l y i n t h e

measure s p a c e s , w e have

p

case

=

2

THEOREM 3 . 6 .

t h a t we c a n p a r t i a l l y i m p r o v e N i k i s h i n ' s t h e o r e m :

Let

B

be a Banach s p a c e of

every continuous linear operator

T : B

+

(Rademacherl t y p e

Lo(m)

2.

Then,

f a c t o r s through

L2(m). P r o o f : T h e f i r s t i d e a c o n s i s t s i n r e p l a c i n g t h e Ra de m a c he r f u n c t i o n s b y a G a u s s i a n s e q u e n c e i n t h e d e f i n i t i o n o f t y p e 2 . To b e p r e c i s e ,

we s h a l l D r o v e t h a t , i f

B

is of type 2 , then ( f . E B) 3

(3.7)

f o r some ( a n d h e n c e f o r a l l ) G a u s s i a n s e q u e n c e V.2.1)

{zj(w)}

i n t h e p r o b a b i l i t y space

(see Definition

(n,P).

Take m o m e n t a r i l y ( 3 . 7 ) f o r g r a n t e d , assume a s a l w a y s t h a t

m(X) = 1

V I . FACTORIZATION THEOREMS

546

and f i x

f l , f 2 ,... , f n E B

C

with

J

we s e t

YX

ifjli 5 1.

For e v e r y

X > 0,

{ x E X : (1 l T f j ( x ) [ 2 ) 1 ' 2 > X)

=

j a n d , a s i n t h e p r o o f o f Theorem 2 . 4 , we d e n o t e gW = C

j

z.(w)f. J

E =

6 B;

3

{W

E

R : [gwlB > X1/2}

Now, from t h e p r o o f o f V.2.4 we r e c a l l t h a t , g i v e n complex numbers a , ,a2

,.. . , a n

with

4

laj

In p a r t i c u l a r , f o r every

where C ( . ) measure o f

I

=

1:

x E Y X , we have

denotes the function expressing the continuity in T ( s e e ( 1 . 2 ) ) , and C i s t h e c o n s t a n t i n ( 3 . 7 ) . T h u s ,

we have o b t a i n e d c o n d i t i o n 3 . 1 a s d e s i r e d , and i t o n l y r e m a i n s t o in a p r o v e ( 3 . 7 ) . We s t a r t w i t h a G a u s s i a n s e q u e n c e { y j ( w ' ) ) . J EN p r o b a b i l i t y s p a c e ( R ' , P I ) , and form t h e s e q u e n c e {zj ( w ) IjEQ defined i n

R

=

R ' x [0,1]

by

The p r o b a b i l i t y we c o n s i d e r i n n i s t h e p r o d u c t o f P ' and Lebesgue m e a s u r e . S i n c e t h e Rademacher f u n c t i o n s a r e i n d e p e n d e n t random v a r i a b l e s , i t i s a t r i v i a l m a t t e r t o v e r i f y t h a t {zj'jE:w is a g a i n a G a u s s i a n s e q u e n c e . Now, s i n c e B i s o f (Rademacher) t y p e 2 :

V I . 3 . FACTORIZATION THROUGH L p

547

( f . 6 B ; w 1 6 0 ' ) . I f we i n t e g r a t e t h i s i n e q u a l i t y w i t h r e s p e c t t o J d P f ( w l ) taking i n t o account t h a t m 2 2 -71x dx = c < m (jSN) I y j ( w ' ) I dP' (wl) =

n

we o b t a i n ( 3 . 7 ) .

COROLLARY 3 . 8 .

T : Lp(p)

-+

Let

Lo(m)

be a n a r b i t r a r y m e a s u r e s p a c e , and l e t be l i n e a r and c o n t i n u o u s i n m e a s u r e (Y,u)

P r o o f : P a r t ( a ) f o l l o w s d i r e c t l y from t h e p r e v i o u s t h e o r e m . To p r o v e (b) we s h a l l v e r i f y 3 . 4 d i r e c t l y . Given ( f . ) C Lp(p) w i t h J Z I f . l p < 1 , we d e f i n e f = (1 I f . l p ) l / p , s o t h a t l f l , 5 1 . j J P j J S i n c e T i s l i n e a r and p o s i t i v e : (;

J

ITfj(x)lP)"?

sup a

 0. 1 -

There-

fore >

with

C(X)

a s in (1.2).

EXAMPLES 3 . 9 . ( a ) Let

G

XI) 5

0 b e a l o c a l l y compact g r o u p w t h l e f t Haar

m e a s u r e m , and l e t T : L P ( G ) -+ L o ( G ) be l i n e a r , c o n t i n u o u s i n m e a s u r e and i n v a r i a n t u n d e r l e f t t r a n s l a t i o n s . I f 2 5 I, m, then, f o r e v e r y compact

K C G r

(3.10) JK

there exists

-

ITf ( x ) I 'dm(x)

5 Ci

CK > 0

1";

such t h a t ( f E LP(G))

548

VI

.

FACTORIZATION THEOREMS

The proof is as in Corollary 2.8. Observe that assuming the linearity of T allows to improve the weak type inequality of that corollary to the strong type result (3.10). I

f f denote the conjugate function operator on 1 1 . [7 ) , and consider the weighted L 2 -inequality

(b) Let the torus

T

-+

z,

2

(3.11)

jlf(x)lZu(x)dx

< Ilf(x)12v(x)dx

The question, raised for the first time by Pluckenhoupt [ 2 ] , is the following: Find all v(x) (resp. all u(x)) such that (3.11) holds f o r some u(x) (resp. some v(x)). Although a more systematic study of this kind of problem will be given in section 6 , we can already give here the following simple answer due to Koosis [4]: (3.11) h o l d s f o r some non-trivial u(x) if and o n l y if v - 1 € L1(T), and it h o Z d s f o r some nontrivial v(x) if and o n l y if u € L1(T) By a simple duality argument (which we urge the reader to supply) both assertions are actually equivalent, so that we shall only prove the first one. First of all, by translation invariance, we can assume u(x) > 0 and v ( x ) > 0 for all x (argue as in Corollary 2 . 8 ) , and then

Now, if (3.11) holds and we write every x € [0,1/4)

I-,,

fl

=

IfIXc-1,4,0),

we have for

0

?,(XI

=

I f ( y ) l cotg n(x-y)dy

and therefore,

similar inequality holds for the intervals and [1/4, 1/2). Thus, the mapping

A

f

-+

\f(x)dx

=

[-l/Z,

-1/4),

[0,1/4)

If(x)v(x)-'v(x)dx

is a continuous linear functional in i.e. v - 1 t: L~(T).

L2(v),

which means

v - 1 E: ~'(v),

The converse follows from Theorem

3.6.,

because

v-'

6 L1(T)

549

VI . 4 . OPERATORS WITH VALUES IN Lq

implies t h a t

L2(v)

i s c o n t i n u o u s l y embedded i n

L 1 (T) :

and t h e r e f o r e , t h e c o n j u g a t e f u n c t i o n i s a c o n t i n u o u s l i n e a r o p e r a t o r from L 2 ( v ) t o L i ( T ) (and, a f o r t i o r i , t o Lo(T)). ( c ) We h a v e a s i m i l a r r e s u l t f o r o n e o f L i t t l e w o o d - P a l e y ' s o p e r a t o r s : Let A d e n o t e t h e f a m i l y o f d y a d i c i n t e r v a l s i n R , and l e t H and S I s t a n d f o r t h e H i l b e r t t r a n s f o r m and t h e p a r t i a l sum o p e r a t o r c o r r e s p o n d i n g t o I , r e s p e c t i v e l y . I f v - l € L1 ( R ) , then the inequality

h o l d s form some u ( x ) > 0 . The r e a d e r w i l l have no d i f f i c u l t y i n f i l l i n g i n t h e d e t a i l s ( u s e Theorem V . 5 . 8 ) . We p o i n t o u t t h a t t h e t e r m Hf i n t h e r i g h t hand s i d e c a n n o t b e o m i t t e d .

4 . FACTORIZATION OF OPERATORS WITH VALUES IN L q .

THE DUAL PROBLEM

I n t h i s s e c t i o n , we a r e g o i n g t o c o n s i d e r o p e r a t o r s T : B * Lq(X,m), f o r some q > 0 . F o r s u c h a n o p e r a t o r , f a c t o r i z a t i o n through

and

means

a r e continuous o p e r a t o r s . This c a s e , p r e s e n t s g some d i f f e r e n c e s which we a r e g o i n g t o comment:

where

To

LP(m)

M

a ) S i n c e T f a c t o r s t h r o u g h LP(m) i f and o n l y i f T maps B i n t o LP(wdm) f o r some w(x) > 0 , when 0 < p 5 q f a c t 2 r i z a t i o n t h r o u g h LP(m) a l w a y s o c c u r , b e c a u s e w e c a n t a k e w € L1 n Lm s o t h a t Lq(m) C LP(wdm). T h u s , o n l y f a c t o r i z a t i o n t h r o u g h LP(m) w i t h p > q i s i n t e r e s t i n g now. b) I n t h e c a s e

0 < q < p,

an a r b i t r a r y m e a s u r a b l e f u n c -

d o e s n o t d e f i n e a c o n t i n u o u s mapping t i o n , g(x) > 0, M * LP(m) -+ Lq(m). I t i s n e c e s s a r y t h a t g € Lr(m), where g * l/q = l/p + l / r . I n o t h e r w o r d s , i f T f a c t o r s t h r o u g h LP(m),

V I . FACTORIZATION THEOREMS

550

then the range of T =

is contained in

Lp(w dm)

with

w -l/P

=

g 6 Lr(m).

Here is the factorization theorem corresponding to this situation THEOREM 4 . 2 . Let T : B 0 < q < p < m and l/q

+

=

Lq(m) b e a s u b l i n e a r o p e r a t o r . If l/p + l/r, t h e n t h e f o l l o w i n g s t a t e m e n t s

are equivalent: (a) T

Idr2

-.1

1 and

admits the factorization ( 4 . 1 ) with

g(x) > 0 ,

POI2 c.

( b l There e x i s t s a measurable f u n c t i o n

and

1 [r/p 5 1

w(x)>O

a.e. such

(c) The f o l l o w i n g v e c t o r u a l u e d i n e q u a l i t y h o l d s

In order to emphasize the similarity with our previous factorization theorems (see 1.7 and 3 . 3 ) , we can state the equivalence between (a) and (c) as follows: A continuous sublinear operator

t h r o u g h LP(m) i f and o n l y i f from . ~ P ( B ) into ~ ' 1(m). .tp

T : B

i'

-+ Lq(m) factors i s a bounded o p e r a t o r

The theorem is somewhat more precise, since it also gives the equality of and the factorization norm of T (which is defined as the product o f the norms of the operators To and M ) . In proving g the theorem, we shall need a simple technical result: LEMMA 4.3. G i v e n a m e a s u r a b l e f u n c t i o n s > 1, the function @(b)

=

I,

a(x) L 0

a(x)b(x)-adm(x)

t

-+

t-"

a > 0,

( b 6 LS(m))

i s c o n v e x and w e a k l y l o w e r s e m i c o n t i n u o u s o n

Proof: Since the function:

and numbers

Ls(m).

is convex in

[O,m),

the first

551

VI.4. OPERATORS WITH VALUES IN Lq

assertion is immediate. Given

k 6 R,

we have to show that

is weakly closed. But A is convex, and therefore, i t suffices to see that A is closed in the norm topology of LS(m) . Now, if

bn E: A,

lbn-bls

then, for some subsequence we have Fatou's lemma implies that b 6 A.

+

0

lim bn,(x) J0 J

=

b(x)

a.e., and

Proof of Theorem 4.2. I f (a) holds, then (b) holds with the some constant C and w(x) = g ( ~ ) - ~ , and conversely. Thus, we only have to prove the equivalence between ( b ) and (c). Suppose first that (b) is satisfied, and let such that 4 I fj I 5 1 . Then J I

ITfjlp)l/pl 9

=

{I(;

f. € B, 1

j € H,

lTfjlp 1q/Pw(l/P w'q/Pdm]l/q

be

<

-

1

Now, if we assume that (c) holds, the minimax lemma 3.2 can be applied as follows: Let c1 = p / q > 1, and define

B

=

Ib E Ld(m)

: b(x)

2 0, lbl,, 5 1 )

which are both convex sets, and B is also weakly compact in L"'(m). The function O(a,b) is defined in AxB as a(x)b(x)-,dm(x)

=

where a = C ITf. l p E A and b E B. Observe that, for each b 6 B, 1 @(.,b) is a linear (therefore concave) function on A , while the previous lemma says that, for each a € A , @ ( a , . ) is convex and lower semicontinuous on B. Now, given a = Z ( T f .I p E A , the 3 inequality:

VI. FACTORIZATION THEOREMS

552

=

@(a,b) 1 l a

holds for all b 6 B, but for a certain b 6 B, it becomes equality, according to the reciprocal of Hiilder's inequality, and thus min @(a,b) b6B

= {

This being true for every

IITf(x) for all

f E B

a 6 A,

the minimax lemma gives

min sup @(a,b) 5 Cp b6B aEA b 6 B such that

i.e., there exists

Ip

b(x)-a

5 1.

with

dition (b), because

< Cp

(Z ITf.lP)q/P}P/q l j 3

lw-

IrIp

dm(x) < Cp Then, w(x)

=

IbI:;'

=

b(x)-a

satisfies cop

0

5 1.

We shall now study the dual factorization problem, which consists in giving conditions for the factorization through LP(m) of an operator T : Lq(m) + B, i .e., we try to decompose T in the form (4.4)

T : Lq(m)

Mg

-

LP(m)

TO

B

By the same considerations made at the beginning of this section, this problem is only interesting when 0 < p < q , and the condition g 6 L'(m) , with l/p = 1/q + l / r , is then required. In order to find a reasonable guess for a theorem solving this problem, we can start assuming in Theorem 4 . 2 that B and Lq(m) are Banach spaces and T is a linear operator. Then, the adjoint of T is an operator T* : Lq'(m) + B*, and one can easily translate (a), (b) and (c) into equivalent statements in terms of T*. In this way, a theorem is found which turns out to be true without any restriction on B, q or T: THEOREM 4.5. L e t t h e o p e r a t o r following sense:

T : Lq(m)

+

B

be s u b l i n e a r i n t h e

VI.4. OPERATORS WITH VALUES IN Lq

(a') T

Idr 5

and

1

with g(x)

admits the f a c t o r i z a t i o n ( 4 . 4 1

> 0,

POIi c. w(x)

l b ' l There e x i s t s a measurable f u n c t i o n such t h a t

553

lwlr/p 5 1

> 0

a.e.

%&

ITflg 5 C P Ilf(x)lPw(x)dx

(f 6

Lq(m))

I c ' ) The foZZowing v e c t o r v a l u e d i n e q u a l i t y hoZds

Proof: The equivalence of ( a ' ) and (bl) is again obvious, with w(x) = g ( ~ ) ~ , and the implication: (b') 3 (c') follows directly from Hiilder's inequality. Thus, once more, the important part of the theorem is the sufficiency of (cl), and this will be proved by a new appeal to the minimax lemma 3 . 2 . Let N = q/p > 1 , so that a ' = r/p, and define the convex sets

and the function

Q(a,b)

in

AxB:

Observe that Q is bilinear and weakly continuous with respect to b , so that we can use the identity min bEB

sup aEA

O(a,b)

Thus, there exists b € B we have the inequality

=

sup aEA

min bEB

Q(a,b)

=

such that, for every

This implies, in particular, that for a l l f such that lTflB 5 1 , true for an arbitrary f E LP(m).

sup

(-1.1

aEA a = C

j

d P )

I f .( p

E A

'

(b') holds, with w(x) = b(x), and, by homogeneity, the same is

0

The achievements of both factorization theorems can be summarized a s follows:

V I . FACTORIZATION THEOREMS

554

In the factorization theorem 4 . 2 we improve the range of the operat o r by showing that it is smaller than what was initially assumed: range (T) C Lp(w) C Lq (with p > q). On the other hand, the dual factorization theorem ( 4 . 5 ) aims at enlarging the domain of the ope rator, while mantaining its range: dom(T) = Lp(w) 3 Lq (with p < q). In fact, it follows from condition (b’) that T has a unique extension to a bounded operator from Lp(w) to B. As in the previous sections, conditions on the space B can be given ensuring that factorization through Lp always occur for ope rators T : B + Lq, but we shall not pursue this matter any further. For the applications we have in mind, a slightly stronger form o f the factorization theorems is needed, which consists in taking a f a mily T of operators and studying their simultaneous factorization through Lp (i.e., factorization with the same operator M for g all T E T ) . Since the proofs go exactly the same, we merely state the results: THEOREM 4 . 2 ’ . T : B Lq(m) +

3

T

and

i s a f a m i l y of s u b l i n e a r o p e r a t o r s l / r = l/q - 1/p > 0,

hoZds i f and o n l y i f t h e r e e z i s t s

w(x)

then, the inequality

> 0

such t h a t

and -

THEOREM 4 . 5 ‘ . T : Lq(m) B +

3 and

T

_.

i s a f a m i l y of s u b l i n e a r o p e r a t o r s

l/r

=

l/p - 1/q > 0,

hoZds if a n d onZy i f t h e r e e z i s t s

lwlr/p 5 1

w(x)

then, the fnequaZitY

0

such t h a t

and (f E Lq; T € T )

5

1

VI.5. WEIGHTED AND VECTOR INEQUALITIES

555

5. WEIGHTED NORM INEQUALITIES AND VECTOR VALUED INEQUALITIES

The norm of eP(B), which appears in the vector valued inequalities (c) and (c') of the previous section, is not easy to handle in general. However, it becomes almost trivial when B is precisely an LP-space, and then, the factorization theorems have significant consequences. In particular, we shall derive from them the promised g e neral principle of equivalence between weighted and vector valued inequalities. to L s ( R n ) , Let T be a family of sublinear operator from Lq(Rn) and consider the following vector valued inequality, which may or may not hold: (5.11

ICC

ITjfjln)'"I

j

< s -

ClCC

(fjl

P 1 W(

3

for all fj € Lq(Rn) and all T . E J. Here, the exponents p , q 1 and s are fixed and such that, either p < min(s,q) or max(s,q) < p. The general principle mentioned above can be stated as follows: THEOREM 5 . 2 .

0 < p,q,s <

a,

and d e f i n e

a

&

f3

&

i) I-f p < min(s,q), then ( 5 . 1 ) h o l d s i f and o n l y i f , u 6 Ly(Rn), we c a n a s s o c i a t e U E L!(Rn) such t h a t

PI, 2 IUl,

clnd jlTf(x) Ipu(x)dx 5 Cp i(f(x)IpU(x)dx

each luIa 5

to

(T 6 T I

ii) If max(s,q) < p, t h e n ( 5 . 1 ) h o l d s i f and o n l g i f , t o B n ), u € L+(R we c a n a s s o c i a t e U E L:(Rn) such t h a t

l u l , and /ITf(x)\PU(x)-ldx

< Cp j]f(x)Ipu(x)-'dx

(T € T )

Proof: Consider first the case p < min(s,q). It was proved in Theorem V.6.1 that the weighted norm inequality implies (5.1) (it was assumed there that q = s, but this makes no real difference). Conversely, if ( 5 . 1 ) is verified, let u E L:(Rn) be given with By lula 5 1, and consider the space B = Lp(u) = Lp(Rn; u(x)dx).

VI. FACTORIZATION THEOREMS

556

Ls(Rn) C B

Hijlder's inequality,

and

I .IB

5

I .Is,

so

that, if

Tj E T : (C [Tjfjlg)l/p = I ( Cj j

5

c I(C

IT.f.lp)l/pI J J B -<

lfjIP)'/Plq

J

Now, Theorem 4.5' applies, and there exists such that /lTf(Pudx = ITflE 5 C Suppose now that max(s,q) < p , part. Given functions fj € Lq, norm such that

I

U 6 ,!L

(fIp U dx

with

l U I B 5 1,

(T E T I

and let us prove first the easy we can find u E L!(R") of unit

(this is due to the reciprocal of HGlder's inequality, since 1 -). If the weighted norm inequality corresponding to this 9 P case is verified, then we take U(x) > 0 associated to u(x), so that l U l , 5 1 and

I , =' -+

which is the vector valued inequality (5.1). Finally, if (5.1) holds, max(s,q) 0 is given with Then, the choice of and IuI, 5 1 , we define B = LP(u-'). H6lder's inequality imply

B

c L~(R")

so that the operators of

T

1.1,

and

are defined in

Now, Theorem 4.2' can be applied with

'

I'IB

5 B

= - -

S

and verify

'

- (i.e. P

r a = -1,

P

VI.5. WEIGHTED AND VECTOR INEQUALITIES

557

and, denoting U(x) = w(x)-', this gives exactly the desired 0 weighted norm inequality. The iteration argument used in Chapter IV for the factorization and extrapolation of A weights, can be applied here in order to unify P the weights appearing on both sides of the inequalities obtained in Theorem 5.2.: COROLLARY 5.3. Let 7

b e a f a m i l y of s u b l i n e a r o p e r a t o r s s a t i s f y i n g

the vector valued inequality a =

Then, t o e v e r y

(:)I.

( 5 . 1 ) with

u E L:(R"),

s = q >

p,

and l e t

w(x)

we c a n a s s o c i a t e

such t h a t :

i ) u ( x ) 5 w(x) i i ) ,1.1

f o r eaery

x E R"

5 Zlul,

m

in La(Rn) inProof: Given u(x), we define a sequence (where U . = U. for every J ductively: uo = u , and u . J+1 J J denotes the weight associated to u . according to S.Z(i)). Then, 1

m

w(x)

=

1

2-1

Uj(X)

j=O

for every satisfies (i), and since Iuj+,la 2 lujl, 5 ... < 1., the series converges in La(Rn) and (ii) is also verified. Finally, for each T E T , the inequalities j,

J 5 2 C /\Tf(x)lP Z-ju.(x)dx can be summed up to yield (iii).

+'(x)dx

(j+')u. 1

0

There is, of course, a converse of Corollary 5.3: If its conclussion is verified, then (5.1) holds with only a trivial loss in the constant (we obtain 8"PC instead of C). The unifying weight argument does not seem to work in the case s = q < p . The same result can be obtained, however, provided that the operators T € T are linearizable (in the sense of V.1.20) and 1 < q < p < m. This follows from the case already proved by an obvious duality argument which the reader will easily supply.

VI. FACTORIZATION THEOREMS

558

A consequence of our general principle together with the theorem of Marcinkiewicz and Zygmund ( V . 2 . 7 ) is that almost all the information that one may wish concerning the boundedness properties of a linear operator, is contained in the weighted-L2 inequalities that this operator satisfies. Just as a sample of this fact, we give the following

1 = 1 1 - -12 . A l i n e a r o p e r a t o r COROLLARY 5.4. Let 1 < p < m and P T i s bounded i n Lp(Rn) if and o n l ; i f , f o r e v e r y u € Ly(Rn), we c a n f i n d w s u c h t h a t : u(x) 5 w(x), IwI, 5 2 1 ~ 1 , and T 2 0 bounded i n L (w ) ( w h e r e CJ = 1 2 5 p u = - 1 when p < 21 w i t h norm i n d e p e n d e n t of u . Proof: The "if" part is, in both cases, a trivial consequence of H61der's inequality. To prove the "only if" statement, consider first the case 2 p. Since T is bounded in Lp(Rn), we know (see V . 2 . 7 ) that

I (C

2 1/2 ITfjI 1

1

I,

<

-

IT1

I(; l f j l 21 1/2 I, 1

and we only have to apply Corollary 5.3. Now, if p < 2, we consider the adjoint operator T" : Lp'(Rn) + Lp'(Rn) and apply the 1 = 1 1 - -12 = 1 1 - 2 and previous case taking into account that c1 P ? that the inequalities IIT*f(x)I2 w(x)dx

5 C jlf(x)12

I2 0

5 C jlf(x) I

w(x)dx

and jlTf(x) are equivalent.

w(x)-ldx

w(x)-'dx

We invite the reader to search for more general formulations of our last corollary, involving, for instance, weighted inequalities in LP(lr) for the operator T.

6 . THE TWO WEIGHTS PROBLEF FOR CLASSICAL OPERATORS

Let T be a singular integral operator in Rn, i.e., Tf = Kxf, where the tempered distribution K has bounded Fourier transform and coincides away from the origin with a locally integrable func-

VI.6.

TWO WEIGTHS PROBLEM

559

tion satisfying HBrmander's condition. The two-weights problem for this operator consists in finding all pairs (u,v) of positive functions for which the inequality ~lTf(x)IPu(x)dx I Cp(u,v) jlf(x)lp

(6.1)

v(x)dx

holds true. F o r regular singular integrals operators, (f 6 Lp(v)) it was proved in Chapter I V that this inequality is verified if and 1 < p < m . The general problem, however, remains u = v 6 A P unsolved, and we shall only deal here with the following weak variant which was already studied i n 3.9(b) in a particular case: Find c o n d i t i o n s on

V(X)

i s s a t i s f i e d b y some

u(x)

u(x)/ such t h a t i'resp. v(x)).

(resp.

(6.1)

This is really a factorization problem, and i t can be easily solved by means of the results of the previous sections. We begin with a lemma which is a consequence of Kolmogorov's condition and the e s t i mates for vector valued singular integrals

LEMMA 6.2. Let

T

be a

s i x g u l a r i n t e g r a Z o p e r a t o r whose k e r n e Z

IKCXIl

5 Clxl-"

satisfies

(6.3) a n d Zet

0 < r < 1 < p.

(x 6 Rn) k

=

0,1,2 , . . .

and each, S k , k> l , is a spherical shell: -

Proof: Given k 2 0, we decompose each function as f = f ' where f ' = fX f" = fX and Bk = {x : 1x1 < 2 k + 1 1 . 'h'

f",

+

A

R"-B~

consequence of (6.3) is that IK(x-y)I 5 Clxl-" and therefore, for all x 6 S,,

when

1x1

z

2 1 ~ 1 ,

VI. FACTORIZATION THEOREMS

560

with

w(x)

=

( 1 + Ixl)-".

Thus, Elinkowski's inequality gives

and the inequality for the functions

(f'.') follows immediately. I

On the other hand, we use V . 2 . 8 and V.3.11. to obtain:

< -

cr,p

5 'r,p

ISk

Isk

(we have denoted by [ completes the proof.

.I

. This

the "norm" in weak-L1(R"))

n

THEOREM 6.4. Let 1 0

a.e.,

it

i s sufficient that

I

(6.5)

=

lRn

V(X)'-~' ( 1 + Ixl)-nP'dx <

In t h i s case, u c a n be f o u n d s u c h t h a t i n t e g r a b l e , p r o v i d e d t h a t c1 < p' - 1 .

+ Ixl)'"P'

u(x)-"(l

f i i ) I n o r d e r t h a t ( 6 . 1 ) h o l d s f o r some ~

m

v(x)

m

is -

a.e.,

it

is sufficient that ~~

I n t h i s case,

v

can be found such t h a t

tegrable, provided t h a t

u(x)a(l

i s in-

+

a < 1.

i i i i l B o t h c o n d i t i o n s ( 6 . 5 ) E d (6.6) a r e a l s o n e c e s s a r y i n order that ( 6 . 1 ) holds for the Riese transforms:

T

=

R 1 , R 2'".'Rn

Proof: Since T is essentially self-adjoint, a simple duality argument shows that the pair (u(x), v(x)) satisfies (6.1) for the e x ponent p if and only if the pair (V(X)'-~', U(X)'-~') satisfies the same inequality with exponent p ' . Thus, parts (i) and (ii) of

VI. TWO WEIGHTS PROBLEM

561

the theorem are actually equivalent statements, and we limit ourselves to prove (i) . a < p ' - 1,

Given 1

- = p-1. B

take r < 1 such that k 0, the operator

For each Tk P(X)

=

(1

T(g(y)

+

IYl)")(X)

x~

1 a = J? r -

1,

and let

(X)

k has, according to the previous lemma, a vector valued extension mapping L1(lp) into Lr(Cp) with norm < C Zknlr. Then, Theorem 5.2(ii) can 6- nr) we associate Wk E L:(Sk) be applied, and to each w E L+(R and with IWk[, 5 ,1.1

1%

I Tf (x) I c:,p

Wk(x) "dx

2knp/r

5

If(X)lP

w(x)-l(l

+

x I ) -np dx

Now, we simply take w(x) = v(x)-l(l + IXI)-~P so that [w[[ = I < a, and (6.1) is verified if we def ne m

u(x) with that

E

> 0.

and since small enough

1

=

2

k=o To check that

= E

1< P:r

is

p',

Ek

2-knp/r W k ( X ) 3

(x) 'k

u(x)

has the correct size, observe

the series converges provided that a

chosen.

Now, we shall prove (iii). Suppose that u(x) > 0 and v(x) < m are fixed and such that (6.1) is verified by each one of the Riesz transforms. Let E. 3

=

{x E R"

1

m a x ( ~ x l ~ , ~,..., x 2 ~lxnl)

=

x.1 J

(j=1,2,...,n)

so that R" = E l U (-El)U E2 U (-E2) U . . . U En U (-En). Take a bounded subset A C - E , o f positive Lebesgue measure and such that v(A) = v(x)dx < w . 'Then, for every x E E: 1'

and therefore, (6.1) implies

VI. FACTORIZATION THEOREMS

562

jE.(l

+

Ixl)-npu(x)dx

< C v(A)

<

m

J

The same can be proved for E i and for all j=1,2, ..., , by usi in each case the corresponding Riesz transform, and thus, (6.6) is verified. The necessity of (6.5) follows by the duality argument mentioned at the beginning of the p r o o f .

n

For regular singular integral operators, we always have the inequa-

lity (see IV.3.8). jlTf(x) Ipu(x)dx 5 C

P,S

IIf(x) IpMsu(x)dx

(l
s<m)

with u(x) 0 arbitrary, and Msu = M(U~)'/~. Observe that, if q > 1, the mapping: u -+ Msu is bounded in Lq(Rn) provided that we chose s < q , and this is what makes the above inequality specially useful. It is therefore important to know that this fact remains true for the most general kind o f singular integral operators, namely: THEOREM 6 . 7 . Let T Then, g i v e n q > 1

that

be an a r b i t r a r y s i n g u l a r i n t e g r a l o p e r a t o r :

u

€

L:(Rn),

U € L?(Rn)

we c a n find

PIq5 14, a n d jlTf(x)IPu(x)dx

Moreover,

i f

5 CP , 9 (T) [\f(x)IPU(x)dx

1 n) u E L+(R

q u a l i t y h o l d s f o r some

a < 1

i s given,

(1 < p <

m)

t h e n t h e same i n e -

U € Lyoc(Rn).

Proof: The argument is basically as in the preceding theorem, the details being even simpler here. The first statement follows from Theorem 5.2 applied to the vector valued inequality (see V.3.11)

with s = p q ' . The second assertion is equivalent, by duality, to finding f o r each v E L;'-l some V 2' such that loc IITf(x)IpV(x)-ldx

5 C jlf(x)

Ip

v(x)-ldx

and this is again a consequence o f Theorem 5.2 applied to the w e a k type vector valued inequality I{x : (X ITfj(x)Ip)l/p j

>

XI\ 5 C

I(;

1

Ifj(x)lp)l'Pdx.

VI.6. TWO WEIGHTS PROBLEM

563

The preceding results remain true for vector valued singular integrals. To be precise, we consider an operator T : Li(Rn) Li(Rn) which is bounded for some fixed r > 1, and is given by convolution with a kernel K(x) E L(A,B) satisfying Hgrmander's condition. Then, T can be extended to a bounded operator from L1 weak-l1 LPO) LP(B) (see V.3.4 and V.3.9), and therefore, Theorem 6,7 holds for inequalities of the form -f

-f

Likewise, Lemma 6.2 and Theorem 6.4(i) are also verified in this context, provided that I K(x) I (A,Bl < Clx(-", and since the adjoint T* : :;L +. L** r ' is an operator o f the same kind, part (ii) of Theorem 6.4 also holds. This seemingly trivial remark will be useful to deal with the twoweights problem for the Hardy-Littlewood maximal operator, M. As before, we look for inequalities of the form (6.8)

/Mf(x)P

u(x dx 5 C (u,v) P

I

If(x) IPv(x)dx

and our first step is an ana ogue of Lemma 6.2 with the space L' ( (1 + I xI ) -"dx) , replaced by the largest Banach space on which can be defined, namely:

L LEMMA 6.9.

=

3

k=0,1,2,. . . ,

M

{f E: 0 < r < 1 <

where

Sk a r e t h e s u b s e t s of

Rn

i n t r o d u c e d i n 6.2.

The proof follows the same pattern of 6.2: Given k 2 0 , we decompose f = f' + f " , and we have the trivial estimate Mf"(x) 5 C

sk)

while the vector valued inequality for the sequence from Fefferman-Stein's inequalities ( V . 4 . 3 THEOREM 6.10. S u p p o s e t h a t

1 < p <

m.

(f

i)

f o l lows

V I . FACTORIZATION THEOREMS

564

l i l I n o r d e r t h a t ( 6 . 8 ) h o l d s f o r some n e c e s s a r y and s u f f i c i e n t t h a t

(6.11)

v(x)

it i s

satisfies

R- np '

SUP

u(x)>O

v(x)l-p'dx <

m

ilxl,

R>1

( i i l I n o r d e r t h a t ( 6 . 8 ) h o l d s f o r some n e c e s s a r y and s u f f i c i e n t t h a t

u(x)

v(x) <

a,

it i s

satisfies (6.6)

( i i i )G i v e n a < 1 , u ( x ) c a n b e f o u n d i n ( i l s u c h t h a t u ( X ) " ( l - P ' ) ( i + ( x ( ) - " P ' i s i n t e g r a b l e , and v ( x ) c a n be f o u n d i n l i i l such t h a t

v(x)"(l

+

Ixl)-np

i s integrable.

P r o o f : ( i ) The n e c e s s i t y i s e a s y , b e c a u s e ( 6 . 8 ) i m p l i e s and, i n p a r t i c u l a r

sup { R - n R21 L

u ( x ) d x l {R-n

v(x)l-P'dxp-'

L

R

(u,v) E A

=

c <

P m

R

so t h a t t h e l e f t hand s i d e o f (6.11) i s m a j o r i z e d by u ) ' - ~ ' . To p r o v e t h e s u f f i c i e n c y , we o b s e r v e t h a t ( 6 . 1 1 ) means e x a c t l y t h a t

Lp(v)C L

and

Hb'lder's i n e q u a l i t y ) . Thus, t h e o p e r a t o r w e l l d e f i n e d i n Lp(v) and s a t i s f i e s

lflL

5 Clfl

(by LP(V) M k f ( x ) = Mf(x)XS ( x ) i s k

b y Lemma 6 . 9 . I n t h i s c a s e , we a p p l y d i r e c t l y t h e f a c t o r i z a t i o n t h e o r e m 4 . 2 ( t h e Banach s p a c e b e i n g B = L p ( v ) ) , a n d we f i n d u k ( x ) > 0 i n Sk s u c h t h a t

and

I

Mf(x)p uk(x)dx 5 C y

'k Now, we s i i n p l y d e f i n e

,P

Zknplr

j l f ( x ) IPv(x)dx

m

u(x) =

2 - ~ k2 - k n p / r u k ( x ) X ( x ) w i t h k=o 'k E > 0 s m a l l enough, s o t h a t (6.8) h o l d s and t h e f i r s t a s s e r t i o n of ( i i i ) i s also v e r i f i e d . ( i i ) Consider t h e v e c t o r valued s i n g u l a r i n t e g r a l

T : Lp(Rn)

+

Li(Rn),

1 < p <

m,

d e f i n e d by

VI.6. TWO WEIGHTS PROBLEM

Tf(x)

=

(ks

*

B

f(x))scq,,

565

=

.Lm

where k(x) is a positive Schwartz function such that k(x) 2 1 when 1x1 5 1 (see section V.4). Then, 6.4 (ii) applies to T , and when f '> 0, this proves the sufficiency since blf(x) 5 IITf(x)lB part as well as the second assertion in (iii). The necessity is proved essentially a s in Theorem 6.4. Take a bounded subset A C R n of positive Lebesgue measure such that v(A) < m . Then M(xA)(x) C ( l + \XI)'", and (6.8) implies: /(l

+

Ixl)-npu(x)dx

5 CFP

1

M(xA(x)pu(x)dx

Remarks 6.12. ( a ) So far, we have only used the general principle of section 5 for families consisting o f a single operator. Considering a huge family, one is allowed to have the same weights for all the operators in the family. Let us consider for instance regular sing; lar operators T f = K*f, and denote by C(T) the least constant C such that the inequalities (6.3), l i l m5 C and

hold. Then, given that

v(x)

satisfying (6.5), we can find

IlTf(x) Ipu(x)dx

< C(T)p

u(x)

such

[ If(x) Ipv(x)dx

for every regular singular integral operator T. The simi ar statement corresponding to 6.4(ii) i s a l s o true. To show this one can argue a s above taking into account that, for each sequence (Tj) with (say) C(Tj) 5 1 , the vector valued inequalities of Theorem V.3.11 apply.

(b) A comparison between Theorems 6.4 and 6.10shows that the maximal operator )I behaves only slightly better than a singular integral operator. I n fact, condition (6.5) implies (6.11), but this in turn implies that (6.5) holds for p - ~with arbitrarily small In particular, the limiting condition of both (6.5) and (6.11) when p + 1 is the same, namely:

E

> 0.

v(x)

>

C(l

+

(XI)+

V I . FACTORIZATION THEOREMS

566

and t h i s t u r n s o u t t o be n e c e s s a r y and s u f f i c i e n t f o r t h e e x i s t e n c e o f some u ( x ) > 0 s u c h t h a t t h e weak t y p e i n e q u a l i t y

holds, e i t h e r f o r Tf(x) = Mf(x), o r f o r an a r b i t r a r y s i n g u l a r i n t e g r a l o p e r a t o r T o f t h e t y p e c o n s i d e r e d i n 6 . 4 . The s u f f i c i e n c y c a n b e p r o v e d i n b o t h c a s e s by a p p e a l i n g t o N i k i s h i n ’ s t h e o r e m .

IxIa

7 . WEIGHTS OF THE FORM

F O R HOFIOGENEOUS OPERATORS

We i n t e n d t o show h e r e how t h e g e n e r a l f a c t o r i z a t i o n t h e o r e m s c a n b e u s e d i n o r d e r t o o b t a i n w e i g h t e d norm i n e q u a l i t i e s w i t h c o n c r e t e

w e i g h t s , p r o v i d e d t h a t t h e r e i s enough i n v a r i a n c e i n t h e o p e r a t o r s u n d e r c o n s i d e r a t i o n . We h a v e t o i n t r o d u c e some n o t a t i o n : Given a n o p e r a t o r Tp

tor

T

a c t i n g on function i n

d e f i n e d by a r o t a t i o n

p € SO(n)

Rn,

the rotated opera-

is

-1 T f(x) = T(f P )(Px) P a n d t h e d i l a t i o n o f T c o r r e s p o n d i n g t o some O

tor

T6

i s t h e opera

g i v e n by T6f(x)

A family

T

f o r every tor

6 > 0

=

6

),(I

x

(with

f 6 ( y ) = f(6y))

of operators is r o t a t i o n (resp. d i l a t i o n ) invariant i f , T 6 T a n d e v e r y p 6 SO(n) (resp. 6 > 0 ) , the opera-

(resp.

Tp

T(f

Ts)

belongs t o

T.

We c a n now s t a t e t h e m a i n r e s u l t o f t h i s s e c t i o n

THEOREM 7 . 1 . G i v e n p, q with 1 5 p < m and 1 < q < m, let T __ be a f a m i l y o f l i n e a r i z a b l e o p e r a t o r s ( s e e V . 1 . 2 0 ) Lq(Rn) w h i c h ~

is

r o t a t i o n and d i l a t i o n i n v a r i a n t , and s u p p o s e t h a t t h e v e c t o r

valued i n e q u a l i t y

is v e r i f i e d f o r a l l -

T. € T J

(lTf(x) Iplxl”dx where

a

=

n(p - 1 ) . Y

&

f j E Lq(Rn).

5

[ l f ( x ) lPlxladx

Cp

Then (T t: T I

VI.7. W E I G H T S FOR H O M O G E N E O U S OPERATORS

*

Only f o r t h e p r o o f o f t h i s t h e o r e m , t h e sym bol l u t i o n i n t h e m u l t i p l i c a t i v e g r o u p R+ = ( 0 , m ) :

1

m

*

h

k(s) =

w i l l d e n o t e convo-

dt t

h(p) k ( t )

567

'

(s

0)

0

a r e d e f i n e d w i t h r e s p e c t t o t h e i n v a r i a n t measuLa(R+) R+: d t We s h a l l n e e d t h e f o l l o w i n g t e c h n i c a l r e s u l t :

The s p a c e s

r e on

t'

L E M M A 7.2. G i v e n

&

c(

&

h 6 L"(R+)

with

E

k E La'(R+)

0 <

E

< 1 < a <

lkl,,

for euery

s 6

[E,

The n o r m a l i z i n g c o n s t a n t s a r e c h o s e n s o t h a t

lhl,

P r o o f : Take

N >

1

*

k(s)

2

1-E

find

&

1

=

1 / ~ ]

and d e f i n e

t h e o t h e r hand, f o r every

s € [N-',

h * k ( s ) = (2N + 21og N ) -

=

we c a n

m,

=

h

lhl,

such t h a t

+ 22N l o g N 1

(2N

=

lkl,,

On

1.

=

N] , we h a v e eN

/a(2N) - 1 /a'

la >

(the last inequality holds i f we take

-d s _

-

1-E

N

n

l a r g e enough).

A f e w w o r d s w i l l p e r h a p h s c l a r i f y w h a t t h i s lemma r e a l l y a c h i e v e s .

5 1

s, a n d m o r e o v e r , i t i s w e l l known t h a t h*k 6 C o ( R + ) (i.e., h*k i s c o n t i n u o u s and h * k ( s ) 0 when s + 0 o r s -+ m ) . Thus, nothing First of a l l , H6lder's i n e q u a l i t y imply:

h*k(s)

f o r every

-f

e s s e n t i a l l y b e t t e r t h a n w h a t h a s b e e n p r o v e d c a n b e h o p e d f o r . On t h e o t h e r h a n d , w e o b s e r v e t h a t t h e lemma i s t r i v i a l f o r rx = 1 , s i n c e we c a n t a k e

k(t) E 1

obtaining

h*k(s) = 1

f o r every

s.

C o n s i d e r f i r s t t h e c a s e p 5 q , and d e n o t e s o t h a t T h eo r em 5 . 2 g i v e s u s t h e i n e q u a l i t i e s

P r o o f o f Theorem 7 . 1 .

a

=

(:)I,

(7.3)

where

j I T f ( x ) I p u ( x ) d x 5 C p / l f ( x ) IPU(x)dx

u 6 Ly(Rn)

found such t h a t

(T C T)

c a n be chosen a r b i t r a r i l y , and t h e n ,

[Ul, 5

Iu~,.

Suppose t h a t

we

choose

U

c311 he

u(x)

V I . FACTORIZATION THEOREMS

568

radial, u(x) = uo( 1x1). By rotation invariance, we can then replace U(x) by U ( p x ) in the right hand side of (7.3)) and integrating with respect to normalized Haar measure, dp, in SO(n), we obtain (7.3) with the radial function I

instead of U . Now, we write v(r) = uo(r)rn/" and V(r) = Uo(r)rn/a, so that v, V E La(R+), v being at our disposal and V being associated to it with IVl, 5 Ivl,. Since T is also dila tion invariant, we can apply (7.3) to T6, and it becomes =

-n = a in this case. The next step consists in taking Observe tha k 6 L"'(R+) multiplying both sides of the last inequality by k(6-l) and integrating with respect to T h u s , we are led t o

2.

jlTf(x) Ipv*k(lxl) lxladx 5 CU jlf(x) IPV*k(lxl) lxladx Finally, if we choose v and k as in the Lemma with Ivl, = and we use in the right hand side the simple majoriza= Q k l a l= 1 , tion V*k(lxl) 5 1, it follows that

and we let

E

+

0.

The case q p will be obtained from the previous one by duality, and i t is to make this argument work that the hypothesis of being linearizable the operators of T was posed. In fact, associated with each T E T , there is a linear operator U : Lq(Rn) + L;(Rn) such that ITf(x)I = IUf(x)lg, and we shall denote by u the family of all these operators U . The Banach space B could also depend on T, but we shall s u p p o s e for simplicity that it is the + Lq' same for all the operators. The adjoint operators U* : ;:L satisfy the inequalities

for all

fj E ::L

and

U . E U. J

Since

p ' < q',

the previous case

VI.7.

WEIGHTS FOR HOWOGENEOUS OPERATORS

569

can be applied now (the fact that each U* acts on B*-valued functions makes no difference at all), and we obtain /lU*f(x)lP'lxlbdx 5 C p ' (U 6 U)

with

b

=

/lUf(x)l{

n ( q - 1). 9

/lf(x)lEA

Ix

dx

By duality, this is equivalent to

Ixlb(l-P)dx < Cp \\f(x)/p x I ( -PIdx

( U E U) =

and this is the end o f the proof, because n(E - 1) = a 0 9

b(1-p)

=

Our first application is a simple proof of certain well known inequalities for the Fourier transform. COROLLARY 7 . 4 . f(x) Rn, (7.5) h o l d s if

(Pitt's inequalities).

9

1 < p <

m.

For f u n c t i o n s

the inequality

ll~(~)lpl~\-nb

d6 5 C p , b /lf(x) IPlxlnadx

0 < b < 1

0 5 a

=

b+p-2.

Proof: We shall consider only the case 1 < p 5 2, since the case p 2 2 is equivalent to it by duality. Let x * x* denote the inver sion in Rn leaving the unit sphere fixed, i.e. x* = x/lxl 2 . By using polar coordinates one easily establishes the identity

for every

g(x) 2 0. Therefore, the operator Tf(x)

=

:(x*)

1x1 - n

is an isometry in L 2 (Rn ) , and since ITf(x)l 5 Iflllxl'n, it is also of weak type ( 1 , l ) . By interpolation, and by the theorem of Marcinkiewicz and Zygmund (V.2.7), T has a vector valued extension

On the other hand, T is clearly invariant under rotations and dila tions, s o that Theorem 7.1 applies and, together with (7.6), gives

570

VI. FACTORIZATION THEOREMS

where b+p-2 2-p 5 b < 1.

=

p - 1

and

q

q

can be chosen with

1 < q 5 p,

i.e.

n

It can be proved that (7.5) only holds for the specified values of a

and b. In particular, the relation a = b+p-2 is forced by the behaviour of the Fourier transform with respect to dilations: (f6)^(C) = 6-”?(6- 1 5). There are also estimates from LP(lxIadx) to Lq(lxlbdx), with p # q , which can be obtained by a suitable modification of the method shown here. We refer to Stein [5] for the precise statements. Let us now see how Theorem 7.1 works for maximal operators. COROLLARY 7.7. Let F.

20’s condition

$ 2‘

L1 (R”)

be a r a d i a l f u n c t i o n s a t i s g y i n g

(see V . 4 )

T h e n , t h e f o l l o w i n g i n e q u a l i t i e s h o l d f o r t h e maximal o p e r a t o r

M$f(x)

=

sup

l$6*f(x)

I

associated t o

$:

6>0

(7.8) (1 < p <

m;

- n < a < n(p-1)).

is dilation invaProof: By its own definition, the operator M$ riant, and since $ is radial, M$ is also rotation invariant. On the other hand, we know (see V.4.2) that the inequalities

hold. Thus, it suffices to apply Theorem 7.1.

0

F o r the Hardy-Littlewood maximal function, (7.8) was already known

(see Chapter IV), and in this cases, the range - n < a < n(p-1) is best possible. What we have just proved is that these inequalities are also true for a more general kind o f maximal operators, M 4. This, in turn, is only a particular case of operators of the form

where

p

is a positive Bore1 measure in

Rn.

VI.8. DIRECTIONAL HILBERT TRANSFORMS

COROLLARY 7.9. S u p p o s e t h a t q <

m.

Then,

i s bounded i n

lJ

Lq(Rn)

f o r some

the inequalities:

q 5 p <

and

m

I f (XI I I x I adx

5 CP,a

lMpf(x)pIxladx hold f o r

M

571

0

5 a

<

n(p - 1). 9

Proof: We apply Theorem 7.1 to the inequalities

I

(1 (M,,fj)+

j which hold because 0 v.i.23).

D

1 l/p

4

-

j

4

(9

5 P

<

m)

is a positive linearizable operator (see

Mp

An interesting example arises when LI is Lebesgue measure in the unit sphere c ~ -(which can be considered as a singular Bore1 me2 ~ sure in Rn supported in z ~ - ~ ) Then, . Mv is Stein's maximal spherical mean

11

sup f(x-ty ' )do (Y ' 1 I t>o 1y'I=1 which was considered in V.7.7. The last corollary gives in this case: Wf(x)

=

Plxladx for n 3 and 0 5 a < p(n-1)-n. be best possible f o r positive a.

This inequali y can be shown to

8. DIRECTIONAL HILBERT TRANSFORM As a final application of the equivalence between weighted and lq-valued inequalities, we shall push a little b i t further some ideas of A. CBrdoba and R. Fefferman [l] relating the study of certain multipliers to vector valued inequalities for directional Hilbert trans forms and estimates for maximal functions. We fix a countable set of unit vectors V = {v.). in R 2 , and 3 12' write H . for the Hilbert transform along vj, i.e., I I . is a 1 3 bounded operator in Lp(R2), 1 < p < m, such that ( H . f ) A ( c ) = - i sign(<.v.):(E) 3 3

The family o f all rectangles in

Rz

( f € L2f7

LP)

one of whose sides is parallel

572

VI. FACTORIZATION THEOREMS

to some

v . will be denoted by 1

RV.

Finally, a polygonal region E = E ( V ) with infinitely many sides can be defined in terms of the given set o f directions. We assume for simplicity that each v 1. points upwards: v j = (v. v.3 9 2 1 1,:’ with v > 0. Then, we set E = Ej, where E. i s contained 392 3 j- 1 j Y in the dyadic vertical strip: 5 x 1 < 2 j , and limited from below by a straight line segment perpendicular to vj, i.e.

u

E.

=

1

Ix

=

(xl ,x2) E R2 : 2j-1 5 x 1 < 2j

for certain constants

c. € R

3

and

X.V. > c.) 3 -

1

(see Figure VI.8.1).

All through this section, the notation SA will be used to represent the partial sum operator corresponding to the multiplier x A ’ where A is any measurable subset of R 2 : ( S A f ) ^ ( c ) = ?(c)XA(c).

q P -

I I

I

0)

I

I

I

I I I

I I

I I I

I

I I

I

I

I

I

I

I

I

I

I

1

2

I I

8

4 Figure VI.8.1.

THEOREM 8.1. W i t h t h e n o t a t i o n j u s t d e f i n e d , g i v e n 1 < p < la)

the following

m,

I ( Cj

( b l SE

p

_with

assertions are equivalent:

IH.f.12 ) 1 / 2

(fj E LP)

1 1

i s a bounded o p e r a t o r i n

Lp(R2)

VI.8. DIRECTIONAL HILBERT TRANSFORMS

such t h a t :

{+jR

(8.2) Moreover,

{2}

e i t h e r reduces t o

=

11 -

2

pi,

we c a n f i n d w

5 21~1, &

\,

w(x)dx){h

9

,1.1

u(x) 5 w(x),

p's

t h e s e t of

1

with

u E Ly(Rn),

Icl For e v e r y

573

5 C

w(x)-'dx}

(R E RV)

f o r w h i c h t h e s e s t a t e m e n t s a78e t r u e ,

o r i s an open i n t e r v a l :

-

po

p < . :p

(b) * (a) in the case Proof: We shall prove that (a) =+ (c) p > 2 . This is enough, because no one of the statements is changed if we replace p by p ' . (a) 3 (c). Let H? denote the Hilbert transform in the direction 1 Then, ( 8 . 2 ) is equivalent to orthogonal to j' * l{[Hjf(x)(2+lH;f(x)12}w(x)dx

< C'

j/f(x)12w(x)dx

(j

2 1)

(with C ' depending only on C and viceversa), because w 6 A; (see IV.6.2) if and only if the Hilbert transforms in the direcand e2 = ( 0 , l ) are bounded operators in L2(w), tions el = ( 1 , O ) uniformly in j, where p . and ( 8 . 2 ) simply means that wop. E A; 3 I is the rotation taking e, into v.. Now, by a trivial change of 3 variables, (a) implies that the same vector valued inequality is v e rified by ( f i x ) , and then, (c) follows immediately from Corollary I 5.3. (c)

*

I.

( b ) . Let

I J.

=

3

15

=

be the strip (C1,E2)

€

R

. 2j-1 5 E l

< 2J}

(j 2 1)

By Littlewood-Paley theory in 1 variable (see Theorem V.5.8) p l u s Fubini's theorem, it follows that

s u c h that

IuI,

for some

u(x)

Now, let

P . denote the half-plane I

0

=

1.

P. J

=

15 : 5.v. > 0 1 , I -

so that

V I . FACTORIZATION THEOREMS

574

n

I. E = (a. J J j - t h side of

n

P.) I where a . i s t h e middle p o i n t o f t h e J j' J a E ( s e e F i g u r e V I . 8 . 1 ) . Then 21~ia..x -2nia. .y s a . + p .g ( x ) = 7 e (Id+iHj)(e g(y))(x) J J and t h e r e f o r e , t h e o p e r a t o r s { S a . + p 1 . a r e u n i f o r m l y bounded i n j j JL1 L2(w) i f w s a t i s f i e s ( 8 . 2 ) . T h u s , w b e i n g t h e w e i g h t a s s o c i a t e d t o u a c c o r d i n g t o o u r p r e s e n t h y p o t h e s i s , we h a v e +

and w e a r e d o n e , b e c a u s e

Iwi,

5 2.

( b ) =, ( a ) . T h i s i s b a s e d on a w e l l kmwn i d e a d u e t o Y . Meyer w h i c h was c r u c i a l f o r t h e n e g a t i v e r e s u l t o f C . F e f f e r m a n [2] f o r t h e b a l l m u l t i p l i e r ( s e e a l s o d e Guzm5n [ 2 ] ) . By t h e t h e o r e m o f M a r c i n k i e w i c z a n d Zygmund a p p l i e d t o t h e l i n e a r o p e r a t o r S r E , (b) i m p l i e s

f o r a r b i t r a r y f u n c t i o n s f . € L F J , p o i n t s a . 6 R' and c o n s t a n t J 1 r > 0. If the points a a r e chosen a s above, t h e set r ( E - a . ) j 1 t e n d s t o P . when r + m a n d , a t l e a s t f o r S c h w a r t z f u n c t i o n s f . J 3' (8.3) y i e l d s

which i s e q u i v a l e n t t o (a) b e c a u s e

Sp, = J

I

(Id + iH.).

1

The l a s t a s s e r t i o n o f t h e t h e o r e m i s a c o n s e q u e n c e o f t h e r e v e r s e H i i l d e r ' s i n e q u a l i t y . I n f a c t , i f w(x) s a t i s f i e s ( 8 . 3 ) , t h e n w ( p j x ) b e l o n g s t o A; with constant C f o r the r o t a t i o n p .

1 r > 1 and C1 < m ( b o t h depending o n l y on C ) such t h a t w ( x ) ~ s a t i s f i e s (8.3) with c o n s t a n t C1. But t h i s means t h a t , o n c e we know t h a t ( c ) i s t r u e f o r some q < m , i t i s a l s o t r u e f o r some 91 = < 9, a n d t h i s i s e x a c t l y what we h a d t o p r o v e . 0 taking

el

into

v.

and t h e r e f o r e , t h e r e e x i s t

1:

:

I t i s n o t d i f f i c u l t t o f i g u r e o u t w h i c h i s t h e maximal f u n c t i o n r e lated t o the multiplier operator SE:

VI.8. DIRECTIONAL HILBERT TRANSFORMS

MVf(x)

=

To state the connection between weak additional hypothesis

[*]

" t h e r e e x i s t s no s e t __ and

Mv(XA) (x)

1

sup x€R€RV

=

1

If' and

A C R2

a.e.

[*]

THEOREM 8.4. S u p p o s e t h a t

FZv

575

we need to impose a

SE

such t h a t

(A1 <

m

l'

i s t r u e . T h e n , t h e s t a t e m e n t s (a),

( b ) and ( e l o f t h e p r e v i o u s t h e o r e m a r e a l s o e q u i v a l e n t t o

( d l MV where

p

&

q

i s a b o u n d e d o p e r a t o r in are related as before:

Lq(R2) I

-

9

=

11 -

I ,L

Proof: We shall show the equivalence o f (c) and (d). First, if (d) the weight holds, we can associate to each u E L ? ( R 2 ) m

w(x)

=

1

(ZC)

- k MVu(x) k

k=0 C is the norm of MV as an operator in L q ( R 2 ) and k k-1 M: = Id., MVu = MV(MV u). Then, u(x) 5 w(x), 1 . 1 < 2 1 ~ 1 , and MVw(x) 5 2 C w(x). The last inequality means that wq is an A1 weight with respect to the family of rectangles R V , and this certainly implies that (8.2) holds with constant 2 C . where

In order to establish the converse, we begin with a simple observation: If w(x) satisfies ( 8 . 2 ) and A = {x : w(x) > 1 1 , then

In fact, given

x €

R 6 RV,

we have

which can a l s o be written a s

and taking the supremum for all possible

R,

we obtain ( 8 . 5 ) .

Now, it suffices to prove that (c) implies the weak type ( q , q ) of MV, since (c) also holds for some q 1 < q and therefore, interpo-

5 76

VI. FACTORIZATION THEOREMS

lating by means of Marcinkiewicz theorem, we get the strong type ( q , q ) for MV. We argue by contradiction. Since M V is positive and invariant under translations and dilations, if it were not of weak type (q,q), there would exist some u 6 Lq(Rn) such that MVu(x) = m a.e. (this is a consequence of Nikishin's theorem; see 2 . 9 . and 1 . 5 ) , Let w(x) be the weight which, according to ( c ) , we can associate to u(x), and put A = Cx : w(x) > 1 ) . Then IAl < m, because w E: Lq, while (8.5) gives M~(x~)(X)

L

1 -

C { M ~w(x11-l > 1 - C I M ~ ~ ( ~ ) I =- '1

contradicting our hypothesis

[*I.

a.e.

0

Remarks 8.6. (a) The same results can be obtained for bounded polygonal regions. It suffices to make the same construction of E(V) < ZjI but considering instead vertical strips I . = ix : 2 j - l < x, J for all j 5 0, and then cutting each strip above and below by straight lines perpendicular to some v . € V. 1

(b) The implication: (d) * (a), ( b ) and ( c ) does not require the additional hypothesis p], as the proof itself shows. It also follows from the proof that, if (d) holds for some q < m, then it holds for an open interval: q o < q 5 a. This is actually true for every maximal operator associated to a family of rectangles invariant under translations and dyadic dilations, due to the fact that A1-weights with respect to such a family satisfy a reverse H6lder's inequality. (c) In order to get some feeling about the hypothesis p], let us explain what the failure of such a condition would mean. If [*] fails to hold, we can find a set A of finite Lebesgue measure and a familify of rectangles C R ~ in I ~R~ ~ covering ~ almost all R ~ , while each Ri has 9 9 . 9 9 9 per cent of its area inside A. However difficult this seems to accomplish, there are some wild maximal fun2 tions for which it actually happens. For instance, [*] is false for the maximal function corresponding to the family of all rectangles in R2 (we are indebted to B. L6pez-Melero for pointing this out to us). On the other hand, if a huge family of rectangles does not satisfy [*I, it is not likely that either (a), (b) or (c) are verified, so that one would expect that condition [k] could be avoided in theorem 8.2.

577

VI.9. NOTES AND FURTHER RESULTS

9 . NOTES AND FURTHER RESULTS 9 . 1 . - I n t h e f i r s t v e r s i o n o f 2 . 7 , g i v e n by N i k i s h i n [ l ] , i t i s o n l y p r o v e d t h a t T maps L p ( v ) i n t o L P - € ( w dm) i f 1 5 p 5 2 . The Nikishin's (more p r e c i s e ) weak t y p e r e s u l t a p p e a r s i n N i k i s h i n [ 2 ] . p r o o f s , h o w e v e r , a r e somewhat more c o m p l i c a t e d , a n d a p p l y t o l i n e a r i z a b l e o p e r a t o r s o n l y . The f a c t t h a t t h e r e s u l t h o l d s f o r s u b l i n e a r o p e r a t o r s seems t o b e o b s e r v e d f o r t h e f i r s t t i m e i n Maurey [2]. Theorems 3 . 3 , 3 . 6 a n d 4 . 2 f o r m p a r t o f t h e g e n e r a l t h e o r y d e v e l o p p e d by Maurey [l] . Many a p p l i c a t i o n s o f t h i s t h e o r y d i f f e r e n t f r o m t h o s e considered h e r e can be seen i n G i l b e r t E l l . 9 . 2 . - T h e r e i s a c o n d i t i o n f o r B a n a c h s p a c e s w h i c h i s i n some s e n s e d u a l of h a v i n g t y p e p : B h a s (Rademacher) c o t y p e q , where 2 5 q <

w,

iff

E v e r y Banach s p a c e h a s c o t y p e

m

( w i t h t h e obvious meaning o f t h i s )

a n d i n g e n e r a l , c o t y p e qo i m p l i e s c o t y p e q f o r a l l q 2 9,. Some i n t e r e s t i n g f a c t s a b o u t t y p e a n d c o t y p e a r e l i s t e d b e l o w :

H 1 (R")

i ) LP(p) h a s c o t y p e s u p ( p , 2 ) , 1 5 p 5 m. The s p a c e 1 n behaves i n t h i s s e n s e e x a c t l y as L ( R ) , and t h u s , i t h a s

the best possible cotype: 2. i i ) If

B

has type

p

1. 1 ,

its dual

B*

has c o t y p e

p'.

This i s immediate, b u t t h e corresponding a s s e r t i o n interchanging t h e r o l e s o f t y p e a n d c o t y p e i s f a l s e : L1 h a s c o t y p e 2 , w h i l e Lm = ( L 1 ) * h a s o n l y t h e t r i v i a l t y p e : 1 . i i i ) There i s , however, a d u a l i t y formula f o r t y p e and cotype. Let p(B) = s u p i p : B

has type

q(B) = i n f { q : B

has cotype

I f p(B) > 1 , t h e n p r o v e d by P i s i e r [ l ] .

' + & m

i v ) A Banach s p a c e

B

=

1.

pl q]

T h i s v e r y d e e p r e s u l t was

and i t s b i d u a l

B**

always have t h e

VI. FACTORIZATION THEOREMS

578

same type and cotype. This is due to the so-called "local reflexivity principle": Every finite-dimensional subspace o f B** is almost isometrically isomorphic to a subspace of B. v) A Hilbert space has type 2 and cotype 2 . The converse is also true, but not obvious (see Kwapien [l]) . 9.3.- One may ask wheter "B of type p" is the weakest possible condition on B under which theorem 2 . 4 holds. In fact, it is not: F o r p fixed, 1 5 p 5 2 , the following two conditions are equivalent

for a Banach space

through

T : B

(a) Every continuous operator Lz (m) (b) B*

has type

B

has cotype

p'

Lo(m)

+

(this is slightly weaker than

"B

p")

A similar result holds for factorization through the conditions

through

factors

(a1) Every continuous operator LP(m) and has cotype

(b') B*

q

T : B

for some

+

Lp : If

Lo(m)

1

5 p

<

2,

factors

q < p'

are equivalent. The proofs of both statements can be found in Elaurey c31.

If B has a predual Bo [i.e., B t = B ) , then condition (b) is This is due to the also equivalent to IlBo has cotype p ' " . property iv) of the preceding Note. An interesting application results by taking Bo = L I ( R n ) o r Bo = H 1 (Rn ) : COROLLARY: Let

B

=

Lm(Rn)

tinuous Z i n e a r i z a b l e o p e r a t o r

w(x) > 0

or

B = BFIO(Rn).

T : B

+

Lo(m),

Then, f o r e v e r y conthere exists

such t h a t

This means that Nikishin's theorem applies to

Lm

and BE10 with q = 2 .

9.4.- The inequality (3.10) can be improved in case the group G is amenable (a class which includes both compact and Abelian g r o u p s ) .

V I . 9 . NOTES A N D FURTHER RESULTS

579

In that case, the operator T is automatically bounded in L2(G), see Cowling [ l ] . There are, however, examples of non-amenable groups I; and operators T of weak type (2,2) and translation inva riant, which are not bounded in L 2 ( G ) ( N . Lohu6 ]l[ , R. Szwarc

C11). On the other hand, for every 1 5 p < 2 , there are examples of translation invariant linear operators in the torus Tn, which are (see Zafran [1]) of weak type ( p , p ) but not of stronp; type ( p , p ) This shows that, even for linear operators, one cannot obtain anything better than Corollary 2.8 for p < 2 , and it also proves indirectly that the statement analogous to 3.6 for spaces of type P < 2 is false. 9.5.- The approach followed to establish the results of section 6 was intended to illustrate the applicability of factorization theorems. There are, however, constructive proofs of theorems 6.2 and 6.5. These are due, for 6.2 and 6.5(i), to L. Carleson and P. Jones [ l ] , and for 6.5(ii) to Gatto and Gutisrrez [l], E.T. Sawyer (oral communication) and W.-S. Young [ l ] . I t is also possible to obtain 6.5(i) from E.T. Sawyer's characterization of the pairs of weights associated to the maximal function (see Chapter I\', 7.7). A different approach (in the spirit of the Helson-Szeg6 theorem) to

the two-weights problem for the tlilbert transform H is followed by Cotlar and Sadosky [ 2 ] . They characterize the pairs ( u ( x ) ,v(x)) 2 5 p < m. such that H is bounded from Lp(v) to L2(u), denote the Hardy-Littlewood maximal operator in R" 9.6.- Let defined by means of centered balls. A s we mentioned in Chapter 11, < p < m, with a norm indepen 7.3, M(n) is bounded in Lp(Rn), dent of n. Stein's argument in [7] can be adapted to show, moreover, that

with 1 < p , q < m and C independent of n. If we apply P,O theorem 7.1 to this inequality, it follows that, given p , n such

VI. FACTORIZATION THEOREMS

580

for every

n E N.

9.7.- The results proved in section 8 are o f a conditional nature. They simply establish the equivalence between certain statements, without giving any clue about the truth or falsity of all of them. The interesting question is: Given the vectors {IT.}, for which 3 values o f p (resp. q) do (a) and (b) (resp. (c) and (d)) hold? The answers we know so far correspond to extremal cases: i) If the closure of

Iv.1 in the unit circle has posi3 tive length, then the statements of both theorems are only true in the trivial cases: p = 2 and q = m . The same happens, for insT I m tance, if { vI . ~= {(cos L, sen -11 21 2 j j = 1 * See de Guzm5n [2], ii) If the sequence Iv.1 converges lacunarly to some vecI tor of unit lenght, then all four statements are true for the largest range: 1 < p < m and 1 < q < m. See Nagel, Stein and Wainger [ I ] .

APPEND1X

We s h a l l e s t a b l i s h h e r e s e v e r a l r e s u l t s c o n c e r n i n g Ra de m a c he r f u n c t i o n s a n d mi n i max lemmas w h i c h were s t a t e d i n s e c t i o n s 2 a n d 3 o f C h a p t e r \ / I , a n d wh o s e p r o o f s w e r e p o s t p o n e d . A.l.

RADEMACHER FUNCTIONS

were d e f i n e d i n s e c t i o n T h e f i r s t f a c t a b o u t t h em s t a t e d t h e r e was t h e f o l l o w i n g

T h e R a d e m ach er f u n c t i o n s VI.2.

(rj(t))j,l

Symmetry o f t h e R ad emach er S e q u e n c e : G i v e n a B o r e l f u n c t i o n E . = +1 ( j = 1 , 2 ,...,n ) , J &a ~ ~ r ~ .( t. , )~ ,, r ~ ( t ) )a r e e q u i d i s t r i b u t e d .

F ( x 1 , x 2,..., x n ) F(E1r, ( t )

fi

Rn,

,

the

and numbers

F(rl I t ) , r 2 ( t ) ,

functions

.. ., r n ( t ) )

.

. . ., r n ( t ) )

r ( t ) = ( r l( t ) , r 2 ( t ) ,

P r o o f : Denote

( ~ , r ~ ( t ) , ~ ~, .r. .~,E( ntr n) ( t ) ) . B o r e l m e a s u r e s i n Rn =

p(E)

p,(E)

rcct)

=

Then, w e have t o d e t e r m i n e t h e

=

I I t e [0,1) : r ( t )

=

I{t

E

and

[O,l)

Ell

E

: rE(t)

E

E}l

a n d show t h a t t h e y a r e e q u a l . T h i s i s e n o u g h b e c a u s e , f o r a n y B o r e l

of R , I{t E [ 0 , 1 ) : F ( r ( t ) ) e BII = p(F-'(B)). First o f a l l , i t i s c l e a r t h a t b o t h p and pE a r e s u p p o r t e d i n a s u b s e t c o n s i s t i n g o f 2" p o i n t s , namely o f R" subset

B

s

{ + I , - 1 1 ~=

=

Moreover, f o r each =

In r(t)

s e S,

[(k-l)2-",k2-") =

s

for all

k

intervals

IS

e R"

: IsjI = I , j = 1 , 2

, . . . ,n }

there is an interval

( f o r some

k

=

k(s) e {l,2,...,2nl)

such t h a t

t e Ik S i n c e t h e r e a r e a s many p o i n t s i n n' t h i s gives a one-to-one correspondece

{In}ltkz2n,

between b o t h s e t s . Thus p(isj)

for a l l

s

f

=

S.

( I t e [o,I)

: r(t)

=

On t h e o t h e r h a n d , 58 1

s)/ r,(t)

=

I I ~ ( ~=) 2''I n = s

i f and o n l y i f

S

as

582

APPENDIX

r ( t ) = s E = ( E ~ s , , E ~ s ~ , . . . , E ~ and s ~ ) t ,h e r e f o r e , 2-" f o r a l l s E S . T h i s shows t h a t pleting the proof. 0 = u({sEl) =

w,({s}) p = pE,

=

com-

Observe t h a t w e have a l s o shown t h a t t h e measure 1-1 i n Rn (which i s t h e j o i n t p r o b a b i l i t y d i s t r i b u t i o n o f t h e random v a -

..

r i a b l e s r l ( t ) , r 2 ( t ) , . , r n ( t ) ) i s t h e p r o d u c t o f n m easures i n 1 e a c h o f them e q u a l t o (61 + 6-1). As a consequence, w e have n F.(r.(t))dt (1.1) n F.(r.(t))dt = n J J j=1 0 J J o j=

B,

I'

1'

.

f o r a r b i t r a r y Bore1 f u n c t i o n s F1 , F 2 , , , , F n . What ( 1 . l ) means i s t h a t t h e Rademacher f u n c t i o n s a r e i n d e p e n d e n t random v a r i a b l e s . I n p a r t i c u l a r , t h e y a r o r t h o n o r m a l i n L 2 ( [O,l) , d t ) , b e c a u s e a l l o f them h a v e z e r o i n t e g r a l . Let u s now p r o v e t h e s e c o n d f a c t , s t a t e d i n V I . 2 . 2 , a b o u t Rademacher f u n c t i o n s . We d e n o t e by 11. (Ip t h e norm i n Lp( [ 0 , 1 ) , d t )

-f Kintchine's Inequalities: I kp, K > 0 such t h a t P

f o r every f i n i t e sequence

0 < p <

(aj)

of

m,

there e x i s t constants

compZex n u m b e r s .

P r o o f : For p = 2 , we have e q u a l i t y ( k 2 = K 2 = 1 ) due t o t h e o r t h o g o n a l i t y of ( r j ( t ) ) . . T h e r e f o r e , ( 1 .2) needs t o be proved 11' o n l y for p < 2 , and t h i s w i l l be a c o n s e q u e n c e o f ( . 3 ) b e c a u s e , 1 = 1 - t + t and f = c c1.r l e t t i n g t be s u c h t h a t , we have J 2 P j t

a n d , s i n c e a l l t h e norms a r e f i n i t e , ( 1 . 2 ) f o l l o w s w i h k = K-t/(l-t) P 4 Thus, i t s u f f i c e s t o prove (1.3) w i t h p an i n t e g e r (since and

11 . [ I p 7 i s ;laj\- = 3

2

an increasing function of p ) , the a . ' s r e a l 3 1 . Under t h e s e a s s u m p t i o n s , we u s e t h e i n e q u a l i -

A.2. MINIMAX LEMMA

ties

< p! (ex lxlp 5 p! elX1 -

e-X),

+

n + exp(-C a..r.(t))ldt 1 3 j

=

2p!

583

x e R,

and (1.1) to obtain

rl

ll j=l

]o

exp(a..r.(t))dt 3 1

=

in (1.3) is of the order The estimate we have obtained for Kp K(p) = O(p), p * a, but a slightly more careful proof shows that K(p) = O ( P ~ ’ ~ ) . On the other hand, the best possible value of k l in (1.2) is known to be k l = - (Szarek [l]). We mention also Jz that J.-P. Kahane has extended Kintchine’s -inequalities to the case where (cx.)n are elements of an arbitrary Banach space (see 1 3=1 Lindenstrauss-Tzafriri [l] , vol. 11). A.2. THE MINIMAX LEMMA The setting for lemma 3.2 in Chapter VI was as follows: We are given convex subsets A and B of certain vector spaces, a topology i s defined in B for which i t is a compact (Hausdorff) space, and we have a function 0 : A x B -+ R U I + = } . Then, the statement to be proved is MINIMAX LEMMA:

If @

satisfies

i) @(.,b)

is a concave function o n A

ii) @ ( a , . )

is a c o n v e x function on B

iii) @(a,.)

is l o w e r semicontinuous o n then, the foZZowing identity h o Z d s

(2.1)

min beB

sup aeA

Q(a,b)

=

sup aeA

min beB

b

f o r each

for each

B

a

f o r each

B

E

E

A a

E

A

Q(a,b)

Results of this kind are used in the Theory of Games, and have their origin in the well known paper of Von Neumann [l]. The proofs are usually based on techniques of Convex Analysis, an example o f which is the next proposition containing the most difficult step in the proof of (2.1). We have followed H. Kneser [l] and K. Fan [ l ] rather closely (the second paper contains several generalizations

APPENDIX

584

o f t h e minimax lemma). PROPOSITION 2 . 2 . gj : B

-+

Let B

b e a s a b o v e , and s u p p o s e t h a t

j = 1 , 2 , . . . ,n

B U I+ml,

a r e c o n v e x and l o w e r s e m i c o n t i n u o u s .

L f max lijzn t h e n t h e r e e x i s t n o n n e g a t i v e numbers h l g l ( b ) + A2g2(b)

+,..+

1

=

A1,X2,

{b e B : g j w

b e B

...,A n

Xngn(b) > 0

Proof: Let us consider f i r s t t h e c a s e

B.

for a l l

n = 2.

such t h a t

f o r a12

b e B

The s u b s e t s o f

5 01,

B

j=l ,2

a r e c l o s e d ( h e n c e , compact) and c a n be assumed t o b e nonempty, b e c a u s e , i f B1 = 0, we s i m p l y t a k e A 1 = 1 and X 2 = 0 . The hyp o t h e s i s of t h e p r o p o s i t i o n i m p l i e s t h a t g 2 ( b ) > 0 2 g l ( b ) f o r a l l b e B1, s o t h a t t h e f u n c t i o n - g l ( b ) / g 2 ( b ) i s w e l l d e f i n e d and u p p e r s e m i c o n t i n u o u s on

and a t t a i n s i t s maximum

B1,

S i m i l a r l y , we d e f i n e

(2.4)

p 2 = max

beB2 Now, we wish t o f i n d b e B. b e B1

u

-@4

fil(bJ

X > 0 such t h a t

This i s obviously s a t i s f i e d i f B 2 we have

gl(b)

+

g2(bI

2

-

Xgl(b) + g 2 ( b ) > 0 b L B,U

~ 2 g) l ( b )

B2,

(if

for all

while f o r

b E B2)

T h u s , i t s u f f i c e s t o f i n d X > 0 s u c h t h a t l - X p l > 0 and A-p2 > 0 . Such a e x i s t s i f and o n l y i f p 1 p 2 < 1 . L e t u s p r o v e t h a t t h i s i n e q u a l i t y h o l d s . We c a n o b v i o u s l y assume t h a t p 1 # 0 and p 2 # 0 . Then, we t a k e b l E B1 and b 2 E B 2 f o r which t h e c o r r e s p o n d i n g maximum i n ( 2 . 3 ) and ( 2 . 4 )

But

gl(bl) < 0 < gl(b2),

are attained, i.e.

s o t h a t we c a n t a k e

bt = t b l + ( l - t ) b 2

585

A.2. MINIMAX LEMMA

with

0 < t < 1

such t h a t gl(bt1 5 tgl(b1)

+

(l-tIgl(b2) = 0

By c o n s i d e r i n g t h e same convex c o m b i n a t i o n o f b o t h e q u a t i o n s i n ( 2 . 5 ) , it follows t h a t 1-11LJ2 t g 2 ( b , )

+

( 1 - t ) i?2(bz) = 0

On t h e o t h e r h a n d , t h e h y p o t h e s i s o f t h e p r o p o s i t i o n i m p l i e s t h a t g 2 ( b t ) > 0 a n d , by t h e c o n v e x i t y of g 2 t gz(b1)

Therefore,

(l-t)g2(b2) > 0

+

plp2 g2(bl) < gZ(bl),

which f o r c e s

ulp2 < 1

(because

gz(b1) > 0) To p r o v e t h e p r o p o s i t i o n f o r a r b i t r a r y n , we u s e i n d u c t i o n . We assume t h a t t h e r e s u l t h a s been p r o v e d f o r n - 1 f u n c t i o n s , and t a k e t h e s u b s e t of B Bn = I b e B : g n ( b ) 5 0 1

If Bn

=

0, we c h o o s e

= 0

X1 = X 2 =...=

and

An

=

1.

Otherwise, w e r e s t r i c t g l , g 2 , . . . , g n - 1 t o Bn (which i s compact and convex) and u s e t h e i n d u c t i o n h y p o t h e s i s t o f i n d hl,XZ,...,Xn-l 2 0 such t h a t go(b) = Algl(b)

+

AZg2(b)

+...+ln-1gn-1(b)

> 0

f o r a l l b e Bn. Then, go and gn a r e convex l o w e r s e m i c o n t i nuous f u n c t i o n s on B , and max ( g o ( b ) , g n ( b ) ) > 0 f o r a l l b E B . S i n c e t h e p r o p o s i t i o n h a s b e e n p r o v e d i n t h e c a s e o f two f u n c t i o n s , we f i n d

?.,,A,

2 0

such t h a t , f o r a l l b e B.

Proof o f t h e Minimax Lemma: We have t o e s t a b l i s h t h e i d e n t i t y ( 2 . 1 ) . The i n e q u a l i t y i s o b v i o u s , and t o p r o v e 5 w e c a n assume t h a t t h e r i g h t hand s i d e i s f i n i t e . S i n c e n o t h i n g i s c h a n g e d by s u b s t r a c t i n g a f i n i t e c o n s t a n t from

O(a,b),

we c a n a l s o assume t h a t

s u p min Q ( a , b ) = 0 aeA beB Then, by t h e h y p o t h e s i s i i i ) , t h e s u b s e t s of (2.6)

B

APPENDIX

586

Ba = {b e B : @ ( a , b ) 5 O }

( a e A)

a r e c l o s e d and nonempty, and we s h a l l s e e t h a t t h e y v e r i f y t h e f i n i t e i n t e r s e c t i o n p r o p e r t y . I n d e e d , s u p p o s e t h a t we have

f o r some j=1,2,

...,

we f i n d

n

Ban...flBa = 0 al 2 n a , , a 2 ,...,an E A . W r i t i n g g j ( b ) = O ( a j , b ) , n , we a r e u n d e r t h e c o n d i t i o n s of P r o p o s i t i o n 2 . 2 , and B

h,,h2,

..., A n

X1 @ ( a l , b )

+

2 0

such t h a t

A 2 O(a2,b) +...+

We c a n o b v i o u s l y n o r m a l i z e t h e X , + A 2 +...+ An = 1 . I f we s e t c o n c a v i t y h y p o t h e s i s i) g i v e s

An O(an,b) > 0

(for a l l

b

E

B)

X.'s

s o t h a t they s a t i s f y 3 a, = X,al + A2a2 + . . . + Anan, (for a l l

4(ao,b) > 0

b

E

the

B)

C o n t r a d i c t i n g ( 2 . 6 ) . T h e r e f o r e , {BaIaeA i s a f a m i l y o f c l o s e d s u b s e t s o f B w i t h t h e f i n i t e i n t e r s e c t i o n p r o p e r t y , and t h e compactness of @(a,bo)

B

5

implies 0

n

aeA f o r every a

min s u p beB aeA a s w e wanted t o p r o v e .

B a # 0.

Take

bo e

n

Ba.

aEA E

A,

@(a,b)

D

and

2

sup aeA

O(a,bo)

5

0

Then,

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INDEX

Cauchy integral operator, 225. Cauchy-Riemann equations, 2, 285. Chebichev's inequality, 405. Commutator operator, 225, 378. Comparable measures, 402. Conjugate function, 59, 63, 68. Conjugate Poisson kernel, 63. Conjugate system of harmonic functions, 285, 288, 322. Continuous in measure (operator), 528 c = continuous functions on T, 120. C( = continuous functions on [ - v , T ] , 5. C~(IR") = continuous functions vanishing at -, 176.

Abel Means, 12. Approximate identity, 7, 9, 11, 16, 168, 178, 243, 303. Area function, 227, 465. Atom, 201, 247, 275, 335, 337, 341, 379; A-atom, 492. Atomic Decomposition, 257, 260, 266, 278, 282, 370. Atomic H1, 202. A(f) (f+i?-f(0))/2, 76, 93 A = H" n C = disk algebra, 85, 120. A1, 151, 157, 158, 160, 389. A , 390, 396. ,!A 401. A_*, 453, 456, 522. 'A 422. ApfB' 438.

[-,,,,,I)

PrQ'

Ball multiplier, 515. Besicovitch Covering lemma, 220. Blaschke product, 37, 38, 45, 57, 13, 78. Bochner-Riesz multipliers, 229, 542. Burkholder-Gundy-Silverstein theorem, 258, 270. B(f) = (f+if+f(0))/2, 77. B.L.O. = functions of bounded 10 wer oscillation, 157, 158, 160, 409. B.M.O. = functions of bounded me an oscillation, 99, 100, 101, 102, 131, 156, 157, 158, 160, 292, 298, 307, 409; B.M.O. (B) 492. Calder6n-2ygmund decomposition, 140, 141, 145, 161, 192, 198, 251, 266, 283, 398. Calder6n-Zygmund operators, 224. Canonical Factorization theorem 79. Carleson-Hunt theorem, 514. Carleson measure, 108, 119, 123, 124, 155, 470.

Density points, 185, 186. Dilation invariant operator, 538. Dini condition, 195. Directional Hilbert transform, 481, 571. Dirichlet problem, 4, 11, 19, 21, 23, 24, 26, 29, 167, 170. Distribution function, 134, 159. Doubling measure, 141, 397. Dual factorization problem, 552. Dyadic cubes, 136. D(0,R) = { z e C : IzI < R}, 3, 119. do, = Lebesgue measure on aB(xO,S), 15. 6 = Dirac delta, 14, 160, 236. Extrapolation theorem, 448, 461, 471. EF(z) outer factor of F , 80. Factorization theorem (for Hardy spaces), 45, 46, 48, 379; (for weights), 407, 433, 436, AP441, 471. Factorization through L, 530; through L t , 531, 535; through Lp, 544, 549, 550. Fatou's theorem, 14, 26, 183, 184.

601

602

INDEX

Fejgr-Riesz inequality, 57, 58, 364, 366. f# = sharp maximal function, 155, 221. f* = nonincreasing rearrangement, of f, 528. = conjugate function, 59, 63, 68, 84. F f ( 5 ) = f^ ( 5 ) = Fourier trans*form, 25. @ (f)(X) = SUP lQt*f(x)I I 179t>O Garnett-Jones theorem, 445 Gaussian sequence, 482. g(f), 516. r ( e ) = Stolz regi6n, 109. r'J(x)= I(y,t) ~ : + l :ly-xl < N< Ntl, 180. Hardy kernel, 356, 361. Hardy's inequality, 52, 54, 55, 67, 343. 351. Hardy-Littlewood maximal function, (operator), 109, 110, 134, 161; (Vector valued inequalities), 498. Harmonic majorization, 175, 176. Helson-Szego measure (weight) , 84, 91, 92, 93, 94, 95, 96. Helson-Szego theorem, 84, 95, 443. Hilbert transform, 69, 70, 196, 200, 248, 316, 358, 362. Hilbert transform along a curve, 222. Hormander's condition, 192, 194, 208. Hormander (-Mihlin) multiplier theorem, 210, 229, 337, 347, 418. H = Hilbert transform, 69, 70, 196, 200, 248, 316, 358, 362. H(D) = functions holomorphic in D, 35. H,(F;x) = Marcinkiewicz integral, 502. Hp(D) = Hardy space, 35, 36, 38, q9, 66, 67, 71. HP ( 0 1 , 75, 76, 77, 78. H"(O), 97. H ( l R f = functions holomorphic i n 5 h e upper half plane, 234. HP(IR+) = Hardy space, 233. HP(gn) = Hardy space, 273, 284, 1328, 334, 375. H (IR"), 494. f n ) = atomic H1, 202, 275. Hat(IR Hprq(X), 380.

H f = directional Hilbert transU form, 222, 487. Inner function. R O interpolation for analytic families of operators, 221, 377, 516. IF(z) = inner factor of F, 80. Jensen's formula, 34, 36. John-Nirenberg theorem, 114, 164, 293. Jordan curve, 52. J A = Marcinkiewicz integral, 523. Kintchine's inequalities 534, 582 Kolmogorov's condition, 485. Kolmogorov theorem, 63. Lacunary partial sums, 513. Lebesgue's differentiation theorem, 14, 138, 177, 18 5. Lebesgue points, 14, 139, 182. Linearizable operator, 481. Littlewood-Paley type inequalities, 488; Littlewood-Paley theory, 505; Littlewood-Paley operator, 549. Local convergence in measure, 528. Lp - modulus of continuity, 10, 195. L210q, L(IRn), 147. L S (lR ) = Sobolev space, 336, 344. L:(v) E weak-Lq(v), 148, 392. Lp (m) = Bochner-Lebesgue space, B475. LP(lr), 477 Lo(m), LO,(m) = space of all (strongly) measurable functions, 52.8, 529. Lp@B, 476. LyB(m) = weak-Li(m), 476. LrrP(o x X, p @ m) = mixed norm space, 478. L ( A , B ) = bounded linear operators from A to B, 478. La, 380. L:(IRn), z!(Wn), 292, 293. na(Rn) = Lipschitz spaces, 298, 301, 321, 376. Marcinkiewicz integral, 502, 523. Marcinkiewicz interpolation theorem, 148, 307, with L -B.M.O., 164; vector-valued versibn, 480. Marcinkiewicz multiplier theorem, 511. Marcinkiewicz and Zygmund theorem,

603

INDEX

4 8 4 ; weak type version, 4 8 5 , 486, 541. Maximal spherical means, 5 2 4 , 525. Maximal singular integral, 2 0 4 , 416, 503. Maximum (minimum) princip le, 1 8 , 1 9 , 28, 1 7 2 . Mean value property, 1 5 , 1 6 , 1 7 , 22. Minimax lemma, 5 4 2 , 5 8 3 . Molecule, 3 2 6 , 3 2 7 , 3 7 7 . Muckenhoupt's theorem, 4 2 6 , 4 3 3 . Multiple Hilbert transform, 4 5 1 . Multiplier, 2 0 9 , 2 1 6 , 3 3 7 , 3 4 7 . Mf = Hardy-Littlewood maximal function, 1 0 9 , 110, 1 3 4 ; M p , 158.

m (x) = nontangential maximal Ffunction, 2 3 2 , mU(x), 2 7 3 , 2 8 8 , 465. = multiplication operator, '530, 5 3 1 , 5431j , 199. M f = M(lflp)

M

P

M f, M f, 4 9 7 , 4 9 9 . M:f, 438, 5 0 0 . M'h,wf, 4 2 2 M S ..U.*

461.

Mf = maximal spherical means, 5 2 4 ; Maf, 5 2 5 . Mf(x,t), 1 5 2 , 1 7 7 . M ( L-n 1 ,11 ) = space of Bore1 meas2 5. res on [ - n , n ] , Nikishin's theorem . . . 5 3 6 , 5 3 8 , 545.

Non-tangentially bounded, 1 8 4 . Non-tangential Poisson maximal function, 1 0 9 , 2 4 0 , 2 5 1 , 2 7 5 , 369.

N = Nevanlinna class, 3 6 , 4 2 . N.T. = Non-tangential convergence, 1 2 , 1 4 , 3 8 , 4 1 , 4 3 , 4 6 ,

-

180, 181, 184, 227.

N f = non-tangential Poisson amaximal function for the disk, 109.

N f, 4 9 9 . IV! = /Of = dyadic maximal spheri cal means, 5 0 1 ; Naf, 5 0 1 , 5 1 5 7 "o), 101. N(~i)(x),1 5 2 . ( 1 . I I * = B.M.0.-norm, 1 0 0 , 1 5 6 ; I1 . I1 *,pi 1 6 6 . Outer function, 80, 8 1 , 8 2 , 9 0 . 94.

Paley's inequality, 5 5 , 5 6 .

Partial sum operators, 4 2 1 . Partition of the unity, 2 7 , 3 7 2 . Pitt's inequalities, 5 6 9 . p-norm, 2 3 3 , 2 7 3 . Poisson integral, 4 , 6 , 7 , 1 9 , 2 6 , 40, 48, 51, 69, 154, 1 6 8 , 174, 288.

Poisson kernel, 3 , 6 , 8, 1 9 , 2 0 , 25, 1 6 7 .

positive operator, 4 3 4 , 4 8 2 . principal value singular integral, 68, 195.

P ( t ) = Poisson kernel for the runit disk, 3 . P(x,s) = Poisson kernel for the ball, 1 9 lRn+l P ( x ) = Poisson kernel for +

t25, 26, 167. P* (f) = non-tangential Poisson v'amaximal function, 1 0 9 , 181, 2 4 0 , 251, 2 7 5 $ * 2 7 8 , 3 6 9 . Y;(f),Yv,N(fi,+M (f), 2 4 1 .

Qr(t) = Conjugate Poisson kernel, 6 3 , Qt(x), 2 3 5 , 4 4 2 . Rademacher functions.. . 5 3 4 , 5 8 1 . Regular singular integral operator, 2 0 4 , 315, 3 7 8 , 4 1 3 . Reverse Holder's inequality (R.H.I.), 3 9 6 , 3 9 8 , 4 5 8 . Riesz Factorization theorem, 3 9 , 40, 107.

Riesz's (M.) inequality, 6 1 . Riesz-Thorin interpolation theorem. 2 1 7 , 2 2 0 ; vector valued version 4 8 1 . Riesz (F-M) theorem, 4 9 , 5 0 , 6 0 , 229, 386.

Re Hp, 6 6 , 6 7 , 2 3 6 , 2 3 7 , 2 4 0 , 2 4 9 , 257, 273.

R.f = Riesz transforms, 1 9 7 , 2 1 0 , l 2 8 4 , 312, 322, 374, 417, 467.

Sharp maximal function (operator), 155, 1 6 1 .

Singular integral operator, 1 9 2 , 193, 194, 197, 222, 320, 335.

Sobolev imbedding theorem (or Sobolev lemma), 3 3 6 , 3 4 3 . Sobolev space, 3 3 6 , 3 4 3 . S condition, 4 3 3 . $ace of (Rademacher) type p, 5 3 4 , 5 3 5 , 5 3 6 ; (Rademacher) cotype q, 5 7 7 . Spherical sums, 5 4 0 Square function, 5 0 5 Stein's theorem, 5 3 7 , 5 3 8 . Stolz region, 1 0 9 , 2 2 6 . Strong maximal function, 2 1 8 , 4 5 1 ,

604

INDEX

460, 461. Strong type (p,q), 148, 149. Subadditive operator, 148, 307. Subharmonic functions, 26, 28, 32, 35. 286. Sublinear operator, 312, 434, 519. Szego's theorem, 89. = partial sum operators, 421, "451. S = Marcinkiewicz integral, 502. Z?x0,s) = sphere of center xO an radius s, 15. = scalar product in lRn+l, 286. Translation invariant operator, 529, 530, 537. T = Torus, 4. = T P IdB, 477. I : trigonometric polynomials, 84. c - convexity, 524.

zB

V.M.O. = functions of vanishing mean oscillation, 131, 474.

Weak type (l.l), 63. Weak type (p,q), 148, 149, 308, 310, 389. weight function, 150, 151, 387. weighted vector valued inequaiity, 521. Whitney decomposition, 252, 279. wP' 434. W1(H), 441.

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