Harmonic Analysis (PMS-43), Volume 43: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. (PMS-43) 9781400883929

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Harmonic Analysis Real- Variable Methods, Orthogonality, and Oscillatory Integrals

Monographs in Harmonic Analysis

I. Introduction to Fourier Analysis on Euclidean Spaces, by E. M. Stein and G. Weiss

II. Singular Integrals and Differentiability Properties of Functions, by E. M. Stein

III. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Os­ cillatory Integrals, by E. M. Stein

Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals

ELIAS M. STEIN with the assistance of Timothy S. Murphy

Princeton University Press Princeton, New Jersey 1993

Copyright © 1993 by Princeton University Press Second printing, with corrections and additions, 1995 Published by: Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Chichester, West Sussex All Rights Reserved

Library of Congress Cataloging-in-Publication Data Stein, Elias M., 1931Harmonic analysis : real-variable methods, orthogonality, and oscillatory integrals / Elias M. Stein, with the assistance of Timothy S. Murphy. p. cm. — (Princeton mathematical series ; 43) Includes bibliographical references and index. ISBN 0-691-03216-5 1. Harmonic analysis. I. Murphy, Timothy S. II. Title. III. Series. QA403.3.S74 1993 515/.785-dc20 92-44035

This book has been composed in Computer Modern. The publisher would like to acknowledge Timothy Murphy for providing the camera-ready copy from which this book was printed. The production of this book made extensive use of free computer software, most notably the TfeX typesetting system, X Windows, and GNU Emacs. Northfield Trading L. P. provided computer equipment. Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources. Printed in the United States of America 10 9 ISBN-13: 978-0-691-03216-0 ISBN-10: 0-691-03216-5

To my students and collaborators

Contents

P reface G u id e

to the

xi R eader

xiii

P rologue

3

I. R e a l -V a r ia b l e T h e o r y

7

1 . Basic assumptions 2 . Examples

9

3. 4. 5. 6. 7. 8.

8

Covering lemmas and the maximal function Generalization of the Calderon-Zygmund decomposition Singular integrals Examples of the general theory Appendix: Truncation of singular integrals Further results

II. M o r e a b o u t M a x im a l F u n c t io n s 1. Vector-valued maximal functions 2 . Nontangential behavior and Carleson measures 3. Two applications 4. Singular approximations of the identity 5. Further results III. H a r d y S p a c e s 1. Maximal characterization of H p 2 . Atomic decomposition for H p 3. Singular integrals 4. Appendix: Relation with harmonic functions 5. Further results IV. 1. 2. 3. 4. 5. 6.

H 1 AND BMO The space of functions of bounded mean oscillation The sharp function An elementary approach and a dyadic version Further properties of BMO An interpolation theorem Further results v ii

12

16 18 23 30 37 49 50 56 65 71 75 87

88

101 113 118 127 139 140 146 149 155 173 177

v iii

CONTENTS

V. W e ig h t e d I n e q u a l it ie s 1 . The class A p 2. Two further characterizations of Ap 3. The main theorem about A p 4. Weighted inequalities for singular integrals 5. Further properties of A p weights 6. Further results V I. P s e u d o - D if f e r e n t ia l a n d S in g u l a r I n t e g r a l O p e r a t o r s : F o u r ie r T r a n s f o r m

1. Pseudo-differential operators 2 . An L 2 theorem

3. The symbolic calculus 4. Singular integral realization of pseudo-differential operators 5. Estimates in Lp, Sobolev, and Lipschitz spaces 6. Appendix: Compound symbols 7. Further results V II. P s e u d o - D if f e r e n t ia l a n d S in g u l a r I n t e g r a l O p e r a t o r s : A l m o st O r t h o g o n a l it y

1 . Exotic and forbidden symbols 2 . Almost orthogonality

3. L 2 theory of operators with Calderon-Zygmund kernels 4. Appendix: The Cauchy integral 5. Further results V III. O sc il l a t o r y I n t e g r a l s

of the

F ir s t K in d

1 . Oscillatory integrals of the first kind, one variable 2 . Oscillatory integrals of the first kind, severalvariables

193 194 198 201 204 212

218

228

230 234 237 241 250 258 261 269 270 278 289 310 317 329

3. Fourier transforms of measures supported on surfaces 4. Restriction of the Fourier transform 5. Further results

330 341 347 352 355

IX. O sc il l a t o r y I n t e g r a l s o f t h e S e c o n d K in d 1 . Oscillatory integrals related to the Fourier transform 2. Restriction theorems and Bochner-Riesz summability 3. Fourier integral operators: L 2 estimates 4. Fourier integral operators: Lp estimates 5. Appendix: Restriction theorems in two dimensions 6 . Further results

375 376 386 394 402 412 414

X. 1. 2. 3.

M a x im a l O p e r a t o r s : S o m e E x a m p l e s

The Besicovitch set Maximal functions and counterexamples Further results

433 434 440 454

CONTENTS

XI. 1. 2. 3. 4.

M a x im a l A v e r a g e s

and

O s c il l a t o r y I n t e g r a l s

Maximal averages and square functions Averages over a fc-dimensional submanifold of finite type Averages on variable hypersurfaces Further results

XII. I n t r o d u c t io n t o t h e H e is e n b e r g G r o u p 1 . Geometry of the complex ball and the Heisenberg group 2 . The Cauchy-Szego integral 3. Formalism of quantum mechanics and the Heisenberg group 4. Weyl correspondence and pseudo-differential operators 5. Twisted convolution and singular integrals on H n 6 . Appendix: Representations of the Heisenberg group 7. Further results X III. M o r e

about the

H e is e n b e r g G r o u p

1 . The Cauchy-Riemann complex and its boundary analogue

2. 3. 4. 5. 6. 7.

IX

467 469 476 493 511 527

528 532 547 553 557 568 574 587

The operators 8 and □*, on the Heisenberg group Applications of the fundamental solution The Lewy operator Homogeneous groups Appendix: The 0, as well as the more sophisticated variant

I

f°°

7T J - o o

f(x -

y)

,

y 2 + t2

t > 0, which is the Poisson integral of / . For these, the limiting behavior as t —►0 is the main interest, and its deeper study is subsumed in the properties of the corresponding maximal functions. Singular integrals. A basic object in the classical theory is the Hilbert p .v .i f f ( x — y) — . Its indispensable role there is ^ J —oo y partly explained by the fact that it stands squarely at the crossroads linking real variables and complex function theory. transform /

Oscillatory integrals. Here the primordial example is the Fourier transform /

/

e 2 n i x f ( x ) dx. Of course, when thinking of it, we

should also have in mind its n-dimensional form, as well as the oscillatory integrals arising from this by symmetry considerations, such as Bessel functions. Now it was already understood early that these three concepts were, to a substantial degree, intertwined. Thus the fundamental L 2 estimate for the Hilbert transform was seen as a simple consequence of the use of the Fourier transform, and the weak-type (1,1) estimate was originally proved by using properties of the Poisson integral mentioned above. 3

4

PROLOGUE

W hat could not be guessed then, and could only be revealed with the passage of time, were the wider and deeper interconnections inherent in these examples and their successive generalizations and refinements. The insights that this yielded provide the foundations of a theory of vast scope and utility that has developed over the last thirty years, spurred by its application to such parts of analysis as partial differential equations, several complex variables, and harmonic analysis related to semisimple Lie groups and symmetric spaces. While the theory encompassing these ideas does not admit a brief summary, we do wish to touch on some of its main themes. (i) The underlying real-variable structure. A central role in the anal­ ysis of maximal functions and singular integrals is played by the covering lemmas of Vitali and Whitney types. While this was first understood in the context of R n (with its usual translation and dilation structure), significant parts of these results can be extended to much more general settings, where the analogues of these lemmas continue to hold. More­ over, as it turned out, more refined versions of the older results could be proved by examining further the techniques based on these covering arguments. (ii) Hardy space theory. We comment first on the ubiquitous nature of the Lp spaces, 1 < p < oo. First, the pervasiveness of L 2 estimates is a basic fact of analysis, given the essential part played by the Fourier transform and other devices involving orthogonality. Second, while it might have been simpler to limit considerations to L 1 and L°° estimates, long experience has shown that deep and interesting assertions of this kind rarely hold. Thus the function of Lp is twofold: as a compromise of the possible; but more importantly, that the analysis it requires often reveals fundamental properties of the operators in question. Now it is exactly with the failure of L 1 and L°° that Hardy space theory may be thought to begin. Originally developed in the context of one complex variable with a different emphasis in mind, in its modern in­ carnation this topic represents a happy culmination of the study of max­ imal functions and singular integrals by real variable methods. Not only does it yield a rich H 1 theory, making up for many of the shortcomings of L 1, but it also gives us a fruitful H p theory in the case p < 1, where Lp was entirely barren. That H p would seem destined to be of further interest in the future can be guessed from the fact that the most com­ mon “singularities” in analysis, such as those given by rational functions, or carried on analytic subvarieties, or representable by Fourier integral ( “Lagrangian”) distributions, are all locally in H p, for some p < 1. (iii) More extended singular integrals. The singular integrals alluded to so far have all been of the form ('T f ) ( x ) = J K( x , y ) f (y) dy,

PROLOGUE

5

where the singularity of the kernel K ( x , y) is concentrated in y near x. A significant departure of the current theory is that it can begin to come to grips with the situation that arises when the singularity is now “spread out” , say for y in some variety When an analysis in this context is possible, orthogonality again plays a key role, sometimes via the Fourier transform, but more often using other oscillatory integrals. An important observation is that, at bottom, what makes this possible is some sort of “curvature” property of the family {1^}. In this setting, analogues of maximal functions arise by taking averages over (proper) submanifolds of R n. Again, curvature properties play a decisive role in their study. (iv) Oscillatory integrals. As indicated above, oscillatory integrals provide a necessary tool in exploiting the geometric properties related to curvature and orthogonality in the more extended maximal opera­ tors and singular integrals that have arisen. However, these oscillatory integrals, and others of interest, are not easily classified and come in a multiplicity of forms: variants of the Fourier transform, convolution op­ erators (such as Bochner-Riesz means), and Fourier integral operators are among these forms. What is clear is that this part of the theory is in its infancy, and much more remains to be understood. (v) Heisenberg group. The study of the Heisenberg group illustrates a number of essential ideas treated in this book. In particular, it gives an excellent example of the real-variable structure mentioned above; con­ nected with this is the Cauchy-Szego projection operator, which is a nat­ urally occurring instance of a singular integral in this general context. In addition, we might point out that inherent in its structure is the notion of “twisted convolution”; it accounts for the composition formula for pseudo-differential operators (in their symmetric form) and also yields important examples of oscillatory singular integrals. But beyond these didactic uses the significance of the Heisenberg group resides in what it has allowed us to do, namely, to explore the way into the broader applications of our subject to such interesting areas as several complex variables and (subelliptic) partial differential equations.

CHAPTER I

Real-Variable Theory

We begin by setting down some of the fundamental real-variable ideas behind the theory of the maximal operator and the boundedness of singular integrals. To proceed here requires that our underlying space be endowed with a certain kind of metric structure. The model for this is R n, equipped with its usual family of Euclidean balls, which is the set­ ting appropriate for the standard translation-invariant theory.* In fact, by abstracting some simple and basic features of this case (connected with the covering lemmas of Vitali and W hitney), a number of key points of the earlier development can be carried out in a much broader context. The following additional comments may be helpful in placing the subject of this chapter in its proper perspective. (i) We prove here the weak-type (1,1) and Lp inequalities for the maximal operator in the generality alluded to above. We also deal with the corresponding facts for singular integrals. However, for the latter our results are of a conditional nature, since they depend on an addi­ tional assertion (essentially the L 2 boundedness) that must be treated separately. In the translation-invariant case, this is exactly where the Fourier transform is decisive. In our general context, other notions must also come into play, but consideration of these aspects is postponed until they are systematically taken up in chapters 6 and 7. (ii) W hat will be even clearer (in later chapters) is that maximal operators and singular integrals can be thought of as part of a threefold unity, in that these two operators are intimately tied to another con­ struct, namely that of square functions. One way to realize this unity is to consider all three as singular integrals, but now as vector-valued versions taking their values in differing Banach spaces. (iii) When we continue beyond this chapter we shall not feel con­ strained by the requirement to present matters in the generality used here. Instead, for simplicity of exposition, we shall usually content our­ selves with the standard setting of R n, and invoke the general theory only when needed in particular circumstances. Instances where the gen­ eral point of view plays an important role are the weighted inequalit As developed in, e.g., Singular Integrals, chapters 1 and 2.

7

8

I: R E A L -V A R IA B L E T H E O R Y

ties arising in Chapter 5, the maximal functions and singular integrals associated with lower dimensional varieties treated in Chapter 1 1 , the extension of the theory to the Heisenberg group and other nilpotent groups dealt with in chapters 12 and 13, and several further applications sketched in §8 below. 1. B asic assu m ption s The basic metric notions we shall be interested in have to do with the possibility of measuring the order of magnitude of “size” (or distance), and the order of magnitude of “volume”. The situations we envisage will be general enough so that these two quantities will have to be taken, to a degree, independent of each other. As a reflection of this, we shall quantify these notions in terms of different objects: size in terms of a family of “balls” , and volume in terms of a Borel measure. 1 . 1 Our considerations will always take place in the coordinate space R n .* We shall assume we are given, for each x G R n, a collec­ tion {B(x, 8{x)/Ac\. If not, taking e < l/2c i(< 1), we have

Since 28(xk) < 6(x)/2ci, by the engulfing property B ( x k,26(xk))

C

which gives a contradiction since B(xk,28(xk)) meets F = cO , while B ( x , 8(x)/ 2) C O. Using 4cie^(xfc) > e6(x) and the engulfing property again gives x e B ( x k,ci • 4cie6(xk)). We take B ( x k,ci • Ac\e6{xk)) — B k = B( xk, 8( xk) / 2); i.e., c* — 4cf, e = 1/2c* = l / 8cf, c** = 4c* = 16cf, finishing the proof.

4 . G eneralization o f the Calderon- Zygm und decom position The Calderon-Zygmund decomposition is a key step in the realvariable analysis of singular integrals. The idea behind this decompo­ sition is that it is often useful to split an arbitrary integrable function into its “small” and “large” parts, and then use different techniques to analyze each part. The scheme is roughly as follows. Given a function / and an alti­ tude a, we write / =

/

£

17

L 1 and a positive

\ f \ duJ Then there exists a decomposi-

^(Rn) Jnn

tion of f , f = g + b, with b =

and a sequence of balls {B £}, so

k

that (i) \g(x)\ < cot, for a.e. x.

(ii) Each bk is supported in B

J

\bk(x)\dfi(x) < cap(Bl),

and

j

bk(x) dfi(x) = 0.

(iii) X>(£fc) < ~ / \f(x)\dfi(x). k ol J 4.1 P ro o f. Let E a = {x : M f ( x ) > o}, where M is the uncentered maximal function defined in §3. E a is an open set, and we consider first the case when its complement is nonempty. We can apply the lemma of §3.2 (and the remarks that follow it) to O =■ E a . Thus we obtain collections of balls {Bk}, {B £}, and “cubes” {Qk}, so that B k c Qk C BI,

with ( J Qk = E a, k

(4)

and where the Qk are mutually disjoint. It follows immediately that £ > ( £ * ) < fJ’(Ec)fc

(5)

Now define g(x) = f ( x) for x £ E a , and g{x) =

f (y) dn{y),

if x e Qk.

Hence / = g + ^ bk, where bk(x) = Xq with XQk denoting the characteristic function of

(6) Qk-

^ Compare with the classical version on pp. 17-20 and p. 31 of Singular Integrals. See also §2.1 of Chapter 3 and §3.1 of Chapter 4 below. ^ Of course, this assum ption is vacuous if p.(Ftn ) = oo.

18

I: REAL-VARIABLE THEORY

Because of the corollary in §1.5 (the differentiation theorem), we have |/(x )| < a for a.e. x £ C( J Qk — {% : M f ( x ) < a}. So |^(x)| < a for x £ c\JQk- Next, we observe that (7)

because the ball intersects cE a. So from (7), and the fact that B k C Q k C B £*, it follows that \g(x)\ < ca, whenever x £ Thus (i) is proved. That bk is supported in is a consequence of the inclusion Qk C B£. Also [ \bk(x)\dfj,(x) < 2 1 J

\f(x)\dn(x) < can{B*k)

JQk

by (7) and the doubling property. Moreover the assertion f bk{x) dfj,(x) = 0 is obvious from (6); thus conclusion (ii) is proved. Again by the dou­ bling property, ^ cm ( { ^ / > a }) because of (5), and the quan­ tity on the right is majorized by (c/a) f |/ | d/4, as we see if we invoke the maximal theorem of §3.1. W ith this the proof of Theorem 2 is concluded, under the assumption {x : M f ( x ) < a} ^ 0. If we now consider the special situation where {x : M f ( x ) > a} = R n (which can happen only when /4(Rn) < oo), then we see by the maximal theorem that M R n) < J /

a J l/l m. Therefore, A(Xm) forms a coherent set of functions; that is, A(Xmi) = A(Xm2) in Om i, if mi < ra2. Hence there exists a function a(x), so that a = A(Xm) in Om and, as a result, A (fm) = a - fm, whenever / m is supported in Om. Finally, A (/) = a f for all / G L9, and a(x) is bounded since A is, proving (27).

7.3 So far we have seen how Lp and weak-type L 1 results for the oper­ ator T are related to corresponding estimates for the truncated operators Te. For some purposes—e.g., when one tries to make more precise the sense in which T f = lim Tef + a • / for certain T —it is essential to consider the associ£— ►0

ated maximal operator. We therefore define T*/ —sup |Te/ |, and our intention

£>0

is to prove for T* the same kind of estimates that hold for T. Experience shows that it is possible to do this only when the kernel K satisfies an additional hypothesis: estimates of the type ( 10 ) or (18), but with the roles of x and y reversed. More precisely, we assume there exists a Dini modulus rj with \ K ( x , y ) ~ K(x, y)\ < ri^P ^ _ X ^

[V{x,y)}~1

(29)

whenever p(x,t/) > cp(x, x). P r o p o s i t i o n 2. Suppose T and its associated kernel K satisfy (23), (10), and (29). Then

T*f < A { M ( T f ) + M (/)} ,

(2 1 ),

(22), (30)

and, more generally, for any r > 0 , T»f < Ar { M { \ T f \ r)1/r + M ( f ) } . Here M is the maximal operator of §3.

(30')

35

§7. T R U N C A T IO N O F S IN G U L A R IN T E G R A L S

Proof. As a preliminary matter we remark that for each e > 0, Tef(x) is actually continuous in x , and hence T*f(x) is semicontinuous and measurable.* Let us now fix an x G R n and an e > 0. Write / = / i + / 2 , where f i = f on B(x,e), and f 2 = f o n °B(x,e). Thus Tef ( x ) = T / 2 (x), by the definition of K e. The first point to keep in mind is that IT f 2(x) —T f 2( x) | < A ' Mf ( x )

whenever p( x, x) < e/c.

(31)

In fact, the difference in (31) is bounded by [

\ K( x , y ) - K( x , y ) \ ■\f(y)\dfi (y)

J p(x, y)>e = E

[

\K(x,y)-K(x,y)\-\f(y)\dn(y)

k = 0 J 2 fe+ 1 e > p ( x , y ) > 2 fee

If we invoke (29), then the right side is majorized by E n [ ^ j K B ( x, £2k) y 1

J

\ f ( y ) \ dy{ y ) < c

B( x, e 2k+ 1)

verifying (31). Therefore |Te/(x )| < \ Tf ( x) \ + \Tf i ( x) \ + A ' ( Mf ) ( x )

whenever x G B( x, e / c ) .

(32)

Inequality (32) provides us with a substantial set of x to exploit; it is just a matter of choosing x G B( x , £/ c ) so that neither T f ( x ) nor T f i ( x ) is too large. Now p { x G B(x, e/c) : \Tf(x)\ > a } < oTr f

\Tf(x)\r dp(x)

J B(x,e/c)

< a - ry ( B ( x , e / c ) ) M ( \ T f \ r )(x)

for any r > 0. Thus if a > 4 1/ r[M (|T /|r)(x)]1/ r, then p { x G B( x , e / c ) : \Tf(x)\ > a } < ^ p ( B( x , e/c)).

Also, by Theorem 3, p { x G B( x, e/ c) : \ Tfi (x)\ > a } < ^ = -

J \fi\dp f l/l d/i < - y ( B { x , £ ) ) Mf ( x ) - ,

< * J B(x,e)

a

so if a > 4 A ' Mf ( x ) , then fx{x £ B{x,e/c) : |T/i(a:)| > a } < | yt{B(x,e/c)). Therefore if a > max{4 1/ r[M (|T /|r)(x)] 1/ r,4 A M /(x )}, then there exists an x G B( x , e / c ) so that \ Tf(x)\ < a and \ Tfi(x)\ < a. Substituting this in (32) yields Tt f ( x ) < A { [ M( \ T f \ r )(x)}1/r + M f ( x ) } ,

which gives (30') and, in particular, (30). * This follows from (29) and the L p control of V -1 given by §8.12 below.

36

I: R E A L -V A R IA B L E T H E O R Y C o r o l l a r y 2.

Under the assumptions of Proposition 2, II^"*/|Ip 5: Ap||/||p,

1< p < q

and p{ x : T*/(x) > a } < —||/||i, for all a > 0. a The first conclusion follows directly from (30), the corollary in §5.2, and the Lp boundedness of the maximal function (Theorem 1 in §3). The proof of the second conclusion is in the same spirit but is a little more complicated. We need two observations. First, F satisfies a weak-type L 1 inequality, p{ x e R n : |F (x)| > a} < ~

for all a > 0 ,

(33)

exactly when \F\r belongs to the Lorentz space L1/r,°°, if 0 < r < 1 . Moreover, if we choose the smallest A occurring in (33), then Ar is equivalent to the L l/r,oo norm of \F\r . We then apply this observation successively to F = T*(f) and F = T (/), once we note that, by the general form of the Marcinkiewicz interpolation theorem, the mapping / i-> M ( f ) is bounded from L1/r,0° to itself, if 0 < r < 1 .*

7.4 Three concluding remarks. (i) One has T f i x ) = lim Tef ix ) -\-a(x)f(x) for almost every x, whenever e—»0

/ G L P, l < p < q , if one can prove the convergence for / lying in a dense subspace of Lp. This follows the usual pattern of proving the existence of limits almost everywhere as a consequence of the corresponding maximal inequality. * (ii) An immediate consequence of (27) and the definition of T* is the inequality

|T /(* )|< |T ./(:r)| + c|/(:r)|.

(34)

(iii) Under all the assumptions we have made (namely, (21), (22), (23), (10), and (29)), we can also conclude that T and T* are bounded on Lp for every p, 1 < p < oo, and not just for 1 < p < q. To see this, let T* be the dual of T, which is the bounded operator from Lq to itself (l/< / + l/oo; hence the doubling condition in §1.1 fails for large S. 8 .2 Suppose M is a smooth compact manifold of dimension n. We shall describe a construction of a family of “nonisotropic” balls on M. To do this, assume we have a smooth mapping © : M x M —* Rn so that 0 (x , x) = 0 for all x € M and, for each fixed x G M, the mapping y G(x, y) is a diffeomorphism of a neighborhood of x £ M to a neighborhood of 0 G Rn. Suppose also that we are given an n-tuple a\ , . . . , an of strictly positive numbers. We define a norm function p on Rn in terms of these numbers by

p{x) = p{x 1, . . . , x „ ) = max |xfc |1/ak. k

With ©, p as above, we define B{x,6) = {y : p{Q(x,y)) < 2 min ak, it is not necessarily true that the resulting balls satisfy the crucial engulfing property ( 1 ) in §1 . 1 . An example is given in Nagel and Stein [1979]. 8.3 Suppose M arises as the boundary of a smooth bounded domain Q, in C N; i.e., M = bfi, and n = 2N —1. For each boundary point x , let vx denote the unit (outward) normal to M at x. The directions orthogonal to C • vx are the “complex tangential” directions at x. Define the “polydisc” P (x, 6) C C N to be the product of: ( 1 ) A one-dimensional complex disc in the direction of i/x, with radius M with 7 (0 ) = x, 7 (7-) = y, and for which |(7 (t),0 l < 1 f°r aU * £ [0>T] and £ € T*(t) with Q 7 (t)(£) < 1. We make the key assumption that for some € > 0 , we have p(x,y) < cd(x,y)£, where d is a Riemannian distance. With dp as above, all the properties in §1 are then satisfied. See C. Fefferman and Phong [1983], Sanchez-Calle [1984], C. Fefferman and Sanchez-Calle [1986]. This extepds the results alluded to in §8.4(iii) to more general second-order operators; here Qx is not necessarily the sum of squares of linear forms. (b) For the treatment of other hypoelliptic operators that are polynomials in vector fields, the following extension is needed. We now assume that we are given a double-indexed family of vector fields {X j}, with 1 < i < r, where X j will be thought of as having degree i. We suppose that these vector fields and their commutators span the tangent space at each point. Let us define B(x,S) to consist of all y that can be joined to a; by a piecewise smooth path 7 : [0 , 1] —►M, 7 (0 ) = x, 7 ( 1) = y, with 7 (f) =

and i,j

£ > " 2> ’ (t )]2 < 1 . i,j

Then these balls, together with dp as above, satisfy all the properties in §1 . There is also a formula for the volume p(B(x,6)) analogous to that in §8.4; see Nagel, Stein, and Wainger [1985]. 8 .6 In the next four sections we shall consider R n with its usual balls B(x,6) = {y € R n : \x —y\ < £}. In this section, dp is any measure that is doubling with respect to these balls, and we note two elementary facts.

(a) R n) = oo, unless p ( R n) — 0. To see this, observe that there is a c > 1 so that p ( B( 0 ,26)) > cp{B{0, (5)) for all 6 > 0; iterating this inequality shows that //(R n) = oo.

40

I: REAL-VARIABLE THEORY

(b) If 5 C R n is a smooth submanifold of dimension k < n, then /jl(S) = 0. In particular, /i({x}) = 0 for all x G R n. Indeed, let x G 5, and let v be a unit normal to S at x. Then for small 6, B(x + (6u/2), 6/4) fl S = 0, and therefore /i(5 fl B{x, 6)) < c/i(B(x, —1/d. This is essentially in Ricci and Stein [1987]; related results are in Chapter 5, §6.5. (b) A variant of this result is as follows. Suppose M is a smooth compact manifold, and / is a smooth function on M that does not vanish to infinite order at any point. Let d/i = |/ |a dcr, where da is the measure on M induced by some Riemannian metric. Then there is a positive e so that d/i is a doubling measure when a > —e. More precisely, let k be the smallest integer so that, for each x G M, there is an a with |a| < k so that, in some coordinate system, d£f(x) ^ 0; then we can take e — 1/k. 8.8 (a) There exist doubling measures (with respect to the usual balls in R n) that are totally singular. Indeed, on R 1, the Riesz product oo dfi

= n i> +

a

where —1 < a < 1,

cos(3fe • 27rx)] cfcr,

fc=i is such a measure. For the proof that d/i is totally singular, see Zygmund [1959], Chapter 5. To verify the doubling property, it suffices to check that d/i(/) « d/i(J), where I = [(£ —1)/3J,^/3J], J = [£/Sj ,(£ 4- l)/3-7] are two adjacent intervals of length 3- J . Now oo

dfi = Pj(x)

+ acos(3fc • 2nx)] dx, k=j

and it is easily seen that Pj(x) & Pj(x), if \x —x\ < c3 K (b) There exist doubling measures dfi — f d x that are absolutely contin­ uous, but where / vanishes on a set of positive measure. To see this, partition R 1 using the measure d/i above, so that R 1 = AUB with /i(A) = 0, \A\ > 0, and /j.(B) > 0, |R| = 0. Now let F(x) =

f

(dn + dx).

Jo Since fi has no atoms, F is an increasing homeomorphism of the line, mapping intervals to intervals, and converts d/i + dx to dx. Then E\ = F(A) and

41

§8. F U R T H E R R E SU LT S

E 2 = F(B) are disjoint sets of positive Lebesgue measure whose union is R 1, while both Xex dx and Xe2 dx are doubling measures. Indeed, let I and J be adjacent intervals of the same length. Since dp is doubling, \ i n E l \ = \ F - 1( i ) \ * \ F - l (J)\ = \ J n E 1l from which it follows that

(Jn e 2 t = MF-1(/)) *

K F ~ \ J ) ) = \J

n e 2\.

Journe [1989].

8.9 Doubling measures arise in a natural way in various problems in analysis. (a) Suppose that dp is a doubling measure on R 1, and let F(x) = d/i. Then F : R 1 —* R 1 extends to a quasi-conformal homeomorphism of the closed upper half-plane R+ to itself. Conversely, every such mapping, provided that it preserves orientation, gives rise to a doubling measure as above when restricted to the boundary. See Beurling and Ahlfors [1956]; this paper also contains further examples of singular doubling measures. (b) Suppose Q is a bounded domain in R n with smooth boundary M — bFl and that { a}|.

(i) We have A(a) < c*- 1 , for all a > 0. (ii) If we make the additional assumption that p(B(x,8)) is continuous in 0.

(iii) When the assumption in (ii) is satisfied, we also have that [

J

V ( x , y ) - p dfi(x) = ( p - l r ' M B ) 1-* - R ( K nn

whenever B is a ball centered at y, and p > 1 . When p — 1 the integral diverges, unless p ( Hn) < oo, in which case it equals log(/i(Rn)/p(B)). To prove (i) and (ii), observe that {x : V ( x , y) < a ~ 1} = (J B ( y , 8), where the union is taken over all balls B{y,8) with p(B{y,8)) < a - 1 . 8.13 (a) The proof of | | M / | | < A ||/|| given in §2.1 shows that Ap = 0(\ p — I]-1 ), as p —> 1 . To see that this bound is best possible, let B\ = B(y, 8), B = B ( y , c8), where c is a large constant. Set / = r ( B i ) ~ 1^pXb1, then | | / | | = 1. However, l p

p

l p

l p

( Mf)(x) > c'p{B1)l - 1/p[ V ( x , y ) } - \

if x 6 CB.

Thus if we apply §8 . 12 , and let 8 —►0, we get that Av > c(p —l ) -1 as p —> 1. One can prove similarly that M is not bounded on L 1. Note that this argument uses the assumption that p(B(x,8)) is continuous in 8; this premise can be dropped (see §8.14 below). (b) The proof that | | T / | | lp < for the singular integral oper­ ators T given in §5.1 shows that Ap = 0([p — l] -1 ), as p —►1. In general, this bound is best possible. Indeed, assume that the kernel K of T satis­ fies \K(x,y)\ > c[V(x,y)]~1, as well as the regularity property (18'). Then for x £ jB, we have that {Tf){x) = » { B { ] - 1+xl» K ( x , V) + [

JBi

[K(x,y)-K{x,y)]f(9)dKV),

where / and B\ are as above. The estimate (from below) of the first term is again a consequence of §8.12, while the second term provides an inessential contribution, as the argument in §6.5 shows. A similar proof shows that T is not bounded on L1. If we make the further regularity assumption (29) (so that duality applies), then we can see that the estimate Ap = 0(p) as p —►oo holds, is best possible, and moreover that T does not extend to a bounded operator from L°° to itself.

43

§8. F U R T H E R R E SU LT S

8.14 The crucial weak-type inequality for the maximal function may be reversed. In fact, for appropriate constants c, c, we have p { x : (M f ) ( x ) > ca} > — /

|/| dx.

a J\f\>oc

To prove this, let E = {x : (M f ) ( x ) > a }, and decompose E as ( J Qk, according to §4.1. Since B intersects the complement of E , we have

L and thus J*

\f\dx
1, with cp > c/ (p—1), so that if / E Lp(R n) then ||M /||lp > cp||/||lp - Moreover, M is not bounded on L 1. See also Chapter 1 , §5.2 of Singular Integrals. 8.15 (a) If / G L 1 (R n,d/x) then M f € Lq(E)1 whenever 0 < q < 1 and E C R n has finite //-measure. In fact, for / E L 1 we have

L

n*

(b) The weak-type inequality for the maximal function goes through when the L 1 function / , more precisely the measure f (x) dp(x), is replaced by a measure dm (possibly singular with respect to dp), if dm is supposed to have finite total mass. Indeed, with such a measure dm, let M(dm)(x) = sup L— / |dm|. 6>o p{B{x,8)) J B(x 6) Then p { x : M(dm)(x) > a } < coT1 f Rn |dm|. Also, the L q inequality in (a) holds, with ||/ ||Li replaced by f Rn |dm|. To prove (a), let A(a) = p({x : M f ( x ) > a } HE). Then

n

poo

I ( M f ) q d/j, = q I

JE

JO

pA a q- 1\ { a ) d a = I

Jo

poo +

.

JA

Now choose A = ||/ ||l i//x(jE7) and use the fact that X(a) < p{E) in the first integral, while A(a) < ca -1 ||/||Li in the second. That the maximal inequalities for L l functions extend to finite measures as asserted is evident from the proofs given in §3.

44

I: R E A L -V A R IA B L E T H E O R Y

8.16 The corollary in §3.1 has the following extension. Whenever / is locally integrable, then H(B{x, 6)) JIB(x0 u(B{x,S)) J B(X>S) where B ( x, S) = {y : \x —y\ < 6 }. Then v { x : (M vf ) { x ) > a } < ^

j

\f(y)\dfx(y).

In particular, if we take v = n we see that the maximal theorem in §3.1 holds, for the standard Euclidean balls, without requiring that dp be a doubling measure. Note however that the assertion is made for the centered maximal operator M, and not for the uncentered version M. Similar conclusions also hold for certain families besides the centered Euclidean balls. See Besicovitch [1945], Morse [1947], de Guzman [1981]; the case of rectangles is described in Fourier Analysis, Chapter 2. 8.18 We summarize briefly the theory of singular integrals as presented in Calderon and Zygmund [1952]. Suppose we are given a function Ko(x) that is homogeneous of degree —n on R n, with |Xo(^)| < ^4|x|- n , so that \K0(x —y) — K q(x )\ < Ar)(\y\) when \x\ = 1 , as y —►0; here (as in §6.5) rj is a Dini modulus of continuity. We assume also the cancellation condition Ko(x ) dcr(x) = 0. Let us define the principal-value distribution K = p.v. K 0 by K ( f ) = lim [

f(x)Ko(x)dxi

£^ ° J \ x \ > e

for / E «S. The following assertions then hold.

45

§8. F U R T H E R R E SU LT S

(a) K is a bounded function on R n, and hence the convolution operator T f = / * K extends to a bounded operator on L2 (R n). (b) The operator T satisfies the assumptions (10), (21)-(23), and (29) (in the setting of R n with the usual balls, with dp = dx and q = 2). Thus all the conclusions stated in Theorem 3 and the Appendix §7 apply to T. The proof of the boundedness of K can be given by adapting the argument in §4.5 of Chapter 6 , where somewhat more regularity of K is required.

8.19 The distributions K that arise in §8.18 are homogeneous distribu­ tions (having degree —n). If we assume further regularity, such distributions can be characterized by the following four equivalent properties. ( 1 ) The distribution K is of the form c6 + p.v.Ko, where 6 is the Dirac delta function, and K q is a Calderon-Zygmund kernel (of the type specified in §8.18) with the additional property that Ko is C°° away from the origin. (2) K is homogeneous of degree —n and, away from the origin, agrees with a C°° function. The first statement means that K(t ) = t~nK((p) for alH > 0 and G (x/t). (3) K is a function that is homogeneous of degree 0 and is C°° away from the origin. (4) K can be written as

lim e — ►OjiV— >oo

f

f N $ t d t / t for some

G S with

£

dx = 0 .

The equivalence of (1), (2), and (3) may be proved by using arguments of the kind that appear in Chapter 6 , §4.4; see also §7.5 of that chapter, as well as Singular Integrals, Chapter 3. The assertion (3) follows directly from (4), via the formula K(£) = J0°° $(££) dt/t. Conversely, if K is given, one may obtain a representation (4) by taking 4>(£) = 77(|^|)K(^), where 77 G Co°([l,2]) and Si v(t) dt/t = 1. Some related results for homogeneous distributions of degree d ^ —n can be found in Chapter 6 , §7.5.

8.20 The results in §8.18 and §8.19 refer to homogeneity in the setting of isotropic dilations. There are closely parallel analogues that hold in the context of the nonisotropic dilations described in §2.3. Some further details may be found in B. Jones [1964], Fabes and Riviere [1966], Kree [1965]. 8.21 The Hardy-Littlewood-Sobolev inequality for fractional integration extends to the general context treated in §1. Indeed, if (/„ /)(* )= /

[V(x,y)}-1+a f(y)dn(y),

Jnn

then \\Ia f\\q < ^p, y) f [^(*>3/)]-1 f(y)dfj,(y).

Similar conclusions then hold with a = /?7 , if p(x,y) < c[V(x,y)]^. In par­ ticular, the operator J2 gives a majorant for the fundamental solution of the n

sub-Laplacian

X f arising in §8.4.

j =1 8 .2 2 Let T be the fractional integration operator, having imaginary order, that is defined (via the Fourier transform) by

T ?(0 = K r 0,

as ei, e 2 -> 0.

See Muckenhoupt [I960]; also Singular Integrals, Chapter 2, §6.12.

C.

V ector-valued singular integrals

8.23 Whenever 4> E (—x), then

and so (/, g) = / 0°° ( / * $*, g * ^ t) dt/t. Thus

l(/,0>l = < M /W ( s ) ) < (**(/)> M ^)); and our assertion follows from the direct inequality (a). The corresponding result for S& is proved in the same way, if one uses the integral identities found in Chapter 3, §4.4.3. For the condition of nondegeneracy imposed on , see also §6.19 of Chap­ ter 4. Further information about the g-function and area integral can be found in Singular Integrals, Chapter 4. 8.24 Let T : Lp{R n) —> Lp(R n) be a bounded linear mapping of scalar­ valued functions, for some p, 1 < p < oo. Let B be a Banach space and consider (as in §6.4) the space LP B of strongly measurable* B-valued functions / , for which \ f \ s € Lp(R n). We define the extension T b of T to B -valued functions by T b (/ v) = (T f ) ® v, when v E B, and / is scalar valued. The question we address is: Given T, for what B is Tb bounded from LP B to itself? If B is a Hilbert space and T : Lp —* Lp is an arbitrary bounded linear transformation, then T b : LP B —> LP B is also bounded, with the same norm. This is essentially proved in Chapter 10, §2.5.1, and goes back to Marcinkiewicz and Zygmund [1939a]. 8.25 (a) Continuing the discussion in §8.24, we suppose that T is a singular integral operator that satisfies conditions (10), (21)-(23), and (29), M is a general measure space, and B = Lr ( M), where 1 < r < oo. Then Tb : L pb -> L pb is bounded for 1 < p < oo. In fact, when p = r, the result for T b is immediate from that for T; one then uses Theorem 3 (in the vector-valued form indicated in §6.4) with q = r. See Benedek, Calderon, and Panzone [1962]. * A definition of strong measurability may be found in, e.g., Journe [1983].

48

I: R E A L -V A R IA B L E T H E O R Y

(b) Let B be a “noncommutative” analogue of an Lr space. An example is the trace-class space Cr : it consists of all bounded operators A on a fixed Hilbert space for which P | | Cr = (tr(^*A)r/2)1/r < oo. In this setting, results like those in (a) are valid when B = Cr , 1 < r < oo. See J. A. Guiterrez [1982], Bourgain [1986b]; also Gohberg and Krein [1970], E. Davies [1988]. 8.26 In connection with §8.24, the following condition ( “C-convexity”) on a Banach space B is decisive: There exists a function £ : B x B —>R that is convex in each variable separately, so that £(x, y) < \x + y\ whenever x , y € B with |rc| = |y| = 1, and with C(0>0) > 0(i) If T is one of the singular integral operators considered in §8.25 and T b '■L pb —> L pb is bounded for some p, then B is ^-convex. (ii) Conversely, when B is C-convex, then Tb : LP B —* LP B is bounded for all p, 1 < p < oo, for a large class of such operators T. The proofs require consideration of probabilistic analogues of the opera­ tors T, given as multipliers involving martingale differences. Burkholder [1981] and [1983], McConnell [1984], Bourgain [1983].

Notes The model for the real variable methods described in §1—§5 is the the­ ory of maximal functions and singular integrals in the standard translationinvariant setting of R n, as may be found in the first two chapters of Singular Integrals. The key source of that theory is the paper of Calderon and Zygmund [1952], together with some earlier work in R 1 by Besicovitch, Titchmarsh, and Marcinkiewicz. The general point of view set out here has many roots in past work. Among these are: an ergodic theorem in Calderon [1953]; a paper of Smith [1956] on maximal functions for Poisson integrals in domains in R n; the devel­ opment of the nonisotropic (translation-invariant) theory of singular integrals of B. Jones [1964], Fabes and Riviere [1966], Sadosky [1966]; the maximal func­ tions on homogeneous groups in Stein [1968]; its application to Fatou’s theorem for the complex ball in Koranyi [1969]; generalizations to boundary behavior of holomorphic functions on domains in Cn by Stein [1972]; and the singular inte­ grals that appear as intertwining operators in Knapp and Stein [1971]. Several more systematic approaches were then developed in Koranyi and Vagi [1971], Coifman and G. Weiss [1971]. It is the latter we have followed more closely in this chapter. Further details concerning the examples and topics mentioned in §6—such as the Riesz transforms, translation-invariant singular integrals, their vector valued versions and square functions, and multiplier theorems—may be found in Singular Integrals, chapters 2-4.

CHAPTER II

More about Maximal Functions

The basic properties of the maximal function, which were the sub­ ject of Theorem 1 in the previous chapter, were obtained as a direct consequence of the real-variable structure described there. It turns out that significant extensions and refinements of these properties can be de­ rived from the same circle of ideas. These deeper results are interesting in their own right, but they also foreshadow some later developments of importance. Gur presentation of this material will be organized along three main lines. (1) Vector-valued inequalities. The passage to a Hilbert space val­ ued version of a (scalar valued) operator is a useful device that arises in many situations. When the operator in question is linear, this technical step is subsumed under a general theorem of Marcinkiewicz and Zygmund (see Chapter 1, §8.24) However, the maximal operator M cannot be treated by this method; this is because it is not linear or, put an­ other way, although it can be reformulated as a linear operator, it then takes its values in L°° (and not in a Hilbert space). Thus, the maximal operator requires its own further analysis. This analysis is based in part on a weighted inequality that anticipates some of the ideas treated in Chapter 5, and particularly the role of the class A \. (2) The tent space J\f. The importance of nontangential behavior is highlighted by the definition of a certain function space AT. A key point here is that the dual of this space consists of the Carleson measures or, equivalently, that functions in AT have an atomic decomposition of a sim­ ple nature. These facts anticipate fundamental theorems taken up later, such as the duality between H 1 and BMO and the atomic decomposition in H p. In addition, the space AT is useful in a variety of applications. One that we describe allows us to characterize those collections B of (stan­ dard) balls in R n for which maximal operators fashioned from B satisfy analogues of the usual LP and weak-type (1,1) inequalities. (3) Singular approximations to the identity. These do not admit a pointwise majorization by the standard maximal function but neverthe­ less do arise in a variety of situations, most interestingly for Poisson integrals on symmetric spaces. The relevant weak-type and LP inequali­ ties still hold, but the proof requires that we use the Calderon-Zygmund 49

50

II: M O R E A B O U T M A X IM A L F U N C T IO N S

decomposition (Chapter 1, §4) and, in effect, that we think of the corre­ sponding maximal operator as being made up of vector valued singular integrals. Two remarks about our exposition here are in order. First, on a minor note, and as was mentioned above, our presentation is limited to the classical real variable setting of R n with the usual Euclidean balls. However, given the ideas presented in Chapter 1, the extension of this material to the general situation treated there is a routine exercise for many of the results in question. Second, and more importantly, the re­ sults of the present chapter (together with the weighted inequalities of Chapter 5) may well represent the limit of what can be understood by using only the real-variable theory centered around covering lemmas. Im­ portant further developments in the theory of maximal functions, which involve the use of orthogonality (via the Fourier transform and oscilla­ tory integrals), are treated in chapters 10 and 11.

1. Vector-valued m axim al fu n ction s 1.1 As we have said above, we shall present the theory in this chapter in the usual setting of R n. In the present context, the maximal function discussed in the previous chapter becomes M f ( x ) = sup ^

f

\ f(x — y)\dy.

r >o r n J\y\ 0,^ |{x : M f ( x ) > a } \ < - [ \f(y)\dy. Ot J R"

(2)

(c) If f G Lp, 1 < p < oo, then M f G IP and

\\Mf\\p < MfWp-

(3)

Before we come to the proof, three clarifying comments may be in order. (i) As opposed to the case when one deals with a linear operator, the vector-valued inequalities are not necessarily consequences of the scalar­ valued case. (For the case of a linear operator, see §8.24 in Chapter 1.) (ii) The untoward effect of the nonlinearity of M is highlighted when we consider the L°° case. For this purpose take R 1, and set fj = J = 2 ,.... Then |/ | = X(ij(x>) G L°°, but since M fj{ x) > 1/8 if |x| < 2J’, we get that (M f ) 2(x) > 1/64; hence M f ( x ) = ooev2i>\x\

ery where. The unboundedness of M on L°° is also reflected in the fact that the bound Ap appearing in (3) is of the order p 1/ 2 as p —>oo. For further discussion and a substitute result for bounded functions with compact support, see §5.2 below. (iii) While the “trivial” case p = oo fails, and so cannot be used in proving the inequalities for M, the starting point in the present situation is the case p = 2 of the inequality (3), which follows from Theorem 1 of Chapter 1 because

iiM/in =e 3

wfiWi 0, for all j and x. For any fixed a > 0, the aforementioned construction gives us a collection {Qk} of disjoint “cubes”* so that \f(x)\ < a on C| J Qk and — [ \ f(x)\ dx < A a , for all k. \Qk\ JQk Now take / = g+b, where g — f on C(J Qk and b = f on (J Qk- Thus \g(x)\ < min{a, |/(a;)|} and J \g\2 < a f |/ |. Combining this with (4) gives |{Mg > a / 2}| < ^ \ \ M 9 \\t
a/2 } | < ^H /llia

(6)

We prove (6) by deducing it from a simpler variant, to wit, the one obtained by replacing the function b (supported on the cubes Qk) by its average value on each cube. Thus we set b°(x) = \Qk\~1 Jgk f (y) dy if x G Qk, and b°(x) = 0 if x £ (J QkNow |6°(x)| < |