Wavelets: Calderón-Zygmund and Multilinear Operators (Cambridge Studies in Advanced Mathematics)

CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 48 EDITORIAL BOARD D.J.H. GARLING, W. FULTON, K. RIBET, T. TOM DIECK, P. WALTERS

WAVELETS

Calderon-Zygmund and multilinear operators

Wavelets Calder6n-Zygmund and multilinear operators

Yves Meyer

Professor, Ceremade, University Paris-Dauphine Ronald Coifman Yale University

Translated by David Salinger University of Leeds

CAMBRIDGE

UNIVERSITY PRESS

iv

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

The Pitt Building, Trumpington Street, Cambridge CB2 112P, United Kingdom CAMBRIDGE UNIVERSITY PRESS

The Edinburgh Building, Cambridge CB2 2RU, United Kingdom 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia

0 Herman, ed.iteurs des sciences et des arts, Paris 1990 © English edition. Cambridge University Press 1997 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without. the written permission of Cambridge University Press. First published in English 1997 Printed in France A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Meyer, Yves.

[Ondelettes et operateurs. Tome 2 3. English] Wavelets: Calderon Zygmund and multilinear operators / Yves Meyer & Ron Coifman; translated by D. H. Salingcr. p. cru. - (Cambridge studies in advanced mathematics; 18) Translation of vols. 2 3 of : Ondelettes et operateurs. - ) and index. Includes bibliographical references (p. ISBN 0 52142001 6 (hardback) 1. Wavelets (Mathematics) 2. Calderon Zygmund operator. 1_ Coifman, Ronald R. (Ronald Raphacl) 11. Title. 111. Series QA403.3.M493513 515'.2433-dc20

1996

96-13536

ISBN 0 521 42001 6 hardback

CIP

Contents

Introduction to Wavelets and Operators

x xi xiii xv

F+.

Translator's note Preface to the English edition Introduction 7 The new Calderdn-Zygmund operators 1

2

4 5 6

7 8 9

1

8 +m-i

3

Introduction Definition of Calderon-Zygmund operators corresponding to singular integrals Calder6n-Zygmund operators and U' spaces The conditions T(1) = 0 and tT(1) = 0 for a Calderon Zygmund operator Pointwise estimates for Calderdn Zygmund operators Calderon Zygmund operators and singular integrals A more detailed version of Cotlar's inequality The good ), inequalities and the Muckenhoupt weights Notes and additional remarks

13

22 24 30 34

37 41

8 David and Journe's T(1) theorem r-4

1

2

3

4 5

Introduction Statement of the T(1) theorem The wavelet proof of the T(1) theorem Schur's lemma Wavelets and Vaguelets

43 45 51 54 56

Contents

vi

6

8 9 10 12

60 64 65 71 73 76

-63

11

57

r/1

7

Pseudo-products and the rest of the proof of the T(1) theorem Cotlar and Stein's lemma and the second proof of David and Journe's theorem Other formulations of the T(1) theorem Banach algebras of Calder6n-Zygmund operators Banach spaces of Calder6n-Zygmund operators Variations on the pseudo-product Additional remarks f3.

9 Examples of Calder6n-Zygmund operators 1

2

Introduction Pseudo-differential operators and Calder6n-Zygmund

77 79

operators

3 RI,

4 5 6 7

Commutators and Calder6n's improved pseudo-differential calculus The pseudo-differential version of Leibniz's rule Higher order commutators Takafumi Murai's proof that the Cauchy kernel is L2 continuous The Calder6n-Zygmumd method of rotations

89 93 96 98 105

natl.

10 Operators corresponding to singular integrals: their continuity on Holder and Sobolev spaces

6 7

Sobolev spaces Continuity on ordinary Sobolev spaces Additional remarks

119 122 124

0--

111 112 114 117

5

Introduction Statement of the theorems Examples Continuity of T on homogeneous Holder spaces Continuity of operators in Ly on homogeneous

1

2

3 4

11 The T(b) theorem

5

6 7 8

b1>

3 4

Introduction Statement of the fundamental geometric theorem Operators and accretive forms (in the abstract situation) Construction of bases adapted to a bilinear form Tchamitchian's construction Continuity of T A special case of the T(b) theorem An application to the L2 continuity of the Cauchy kernel

126 127 128 130 132 136 138

p-,

1 2

141

Contents

r~1

11 12

r--1

C-".

The general case of the T(b) theorem The space Hb The general statement of the T(b) theorem An application to complex analysis Algebras of operators associated with the T(b) theorem Extensions to the case of vector-valued functions Replacing the complex field by a Clifford algebra Further remarks

142 145 149 150 150 152 153 155 U15

13 14 15 16

Lam.

9 10

vii

12 Generalized Hardy spaces

6 8

9 10

171 178 181 185

r-i

7

spy

'CS

'L7

157 158 163

1-- 1.-

5

C3,

4

ego

3

Introduction The Lipschitz case Hardy spaces and conformal representations The operators associated with complex analysis The "shortest" proof Statement of David's theorem Transference Calderon-Zygmund decomposition of Ahlfors regular curves The proof of David's theorem Further results "a'

2

r-i

1

189 191 194

13 Multilinear operators

8

Conclusion

Cs.

p0'

2 3

=:°x

4 5 6 7

Introduction The general theory of multilinear operators A criterion for the continuity of multilinear operators Multilinear operators defined on (BMO)k The general theory of holomorphic functionals Application to Calderon's programme McIntosh's theory of multilinear operators

1

195 197 202 207 210 215 220 226

r-'

14 Multilinear analysis of square roots of accretive operators 1

2 3

4 5 6

7 8

Introduction Square roots of operators Accretive square roots Accretive sesquilinear forms Kato's conjecture The multilinear operators of Kato's conjecture Estimates of the kernels of the operators L,(,2) The kernels of the operators Lm

227 228 232 236 238 239 245 251

Contents

Viii

9

Additional remarks

254 .l%

15 Potential theory in Lipschitz domains 1

2

3 4 5 6

7

255 Introduction 256 Statement of the results Almost everywhere existence of the double-layer potential 261 266 The single-layer potential and its gradient 270 The Jerison and Kenig identities The rest of the proof of Theorems 2 and 3 274 275 Appendix 1w1

16 Paradifferential operators 2

3

4 5 6 7

Introduction A first example of linearization of a non-linear problem A second linearization of the non-linear problem Paradifferential operators The symbolic calculus for paradifferential operators Application to non-linear partial differential equations Paraproducts and wavelets

277 278 280 285 288 292 294

References and Bibliography References and Bibliography for the English edition

298 311

C+7

Index

.x'

I

313

fir

end

,.?

"Ce a quoi Pun s'etait failli, l'autre est arrive et ce qui etait inconnu a un siecle, le siecle suivant 1'a iclairci, et les sciences et arts no se jettent pas en moule mais se forment et figurent en les maniant et polissaut a plusieurs fois [... ] Ce que ma force ne pent decouvrir, je ne laisse pas de le sonder et essaycr et, en retastant et petrissant cette nouvelle matiere, la remuant et I'eschanfant, j'ouvre a celui qui me suit quelque facilite et la Jul rends plus souple et plus maniable. Autant en fera le second an tiers qui est cause que In difficulte tie me doit pas desesperer, ni aussi peu mon impuissance... " r..

Montaigne, Les Essais, Livre 11, Chapitre XII.

"Where someone failed, another has succeeded; what was unknown in one century, the next has discovered; science and the arts do not grind themselves into uniformity, but gain shape and regularity by carving and polishing repeatedly [... ] What my own strength has not been

able to uncover, I cease not from working at and trying out and, by reshaping and solidifying this new material, in moulding and heating it, I bequeath to him who follows some facility and make it the more supple and malleable for him. The second will do the same for the third, which is why difficulty does not make me despair, nor, any the more, my own weakness..."

Translator's note

This book is a translation of Ondelettes et Operateurs, Volume 11, Operateurs de Calder6n-Zygmund, by Yves Meyer, and Volume III, Ope-

rateurs rnultilineairr s, by R.R. Coifman and Yves Meyer. The original numbering of the chapters and of the theorems has been retained, so that it is still possible to follow the forward references in Volume I. Chapter 12 is where Volume III of the French version started. The references to Wavelets and Operators (Cambridge University Press, 1992), are to the translation of Volume I, Ondelettes, by Yves Meyer. David Salinger, Leeds, June 1996.

Preface to the English Edition

There has been great progress during the few years which separate the first edition of this work from the translation.

1. In the area of multilinear operators, Pierre Louis Lions made the following conjecture. We consider two (arbitrary) vector fields E(x) _ (El (x), ... , EE(x)) and B(x) _ (Bl (x), ... , B,,(x)), satisfying the following conditions: and E5 (x) E Lr(Rn) BB (x) E LQ(lR),

where l<j 0, and where the mean of rn(e) on the unit sphere is zero. For the Riesz transforms, C?/ICI. This theory gave a unified treatment of earlier work of J. Marcinkiewicz (1938) and G. Giraud (1936). Calderon and Zygmund showed that the Marcinkiewicz multipliers associated with elliptic partial differential equations were the Fourier coefficients of the (periodified) distributions PV Sl(x)lxI-" used by Giraud. An essential tool in that unification was the notion of the Fourier transform of a tempered distribution (introduced by L. Schwartz between 1943 and 1945). By these means, Calderon and Zygmund rediscovered, in a very natural way, the rules for composing two Calderon-Zygmund operators, Tl and T2: it is enough to employ the usual product of the corresponding However, the symbol m3 (e) = ml (t )m2 (e) multipliers ml (e) and m2 does not always have zero integral on the unit sphere, so it is necessary

to consider the algebra of operators cl + T, where c is a constant and T is defined by (1.1).

7.1 Introduction

3

After doing this, Calderdn and Zygmund tried to extend the continuity properties of Calderdn-Zygmund operators to LP(Rn), 1 < p < 00. The complex variable method, introduced by Littlewood and Paley, and used by Marcinkiewicz to prove his multiplier theorem, did not work in the n -dimensional case. It was this obstacle which led Calderon and Zygmund to invent the real variable method. The essential ingredient is the Calderon-Zygmund decomposition which, for any parameter a > 0, lets us write any function f in La (R') as a sum g + h, where g E L2 (]R,) satisfies 119112 < Cvf,\ and where h is the sum of a series of oscillatory

terms, each with support in a certain cube Qj, such that the sum of the measures of the Q. does not exceed C/a. These oscillatory terms foreshadow the "atoms" of Coifman and Weiss, as well as wavelets. The L2 estimate for an operator T, together with quite a weak hypothesis on the kernel K(x, y) of T ( f x-vI>21v'-vl IK(x, y') -K(x, y)I dx < C is enough) gives the L1-weak-L' estimate which we shall describe explic-

itly in this chapter. This, together with the L2 estimate, allows us to derive the LP estimates, for 1 < p 0, such that, for every functionf f (x) in L2(IR"), we have T f (x) = m(x) f (x) + lire K(x, y) f (y) dy, 3- O x-UI?e, (x) for almost all x E R". In the case of a convolution operator, E j (x) does not depend on x; if ej can also be taken to be an arbitrary sequence tending to 0, then we are back to the classical idea of the principal value of a singular integral. The representation (1.13) explains the sense in which the operator T "corresponds to a singular integral". Besides the examples we have already mentioned (the Calderon commutators, the Cauchy integral on Lipschitz curves and the double-layer potential), third generation Calderbn-Zygmund operators arise in a completely different context. To show that a wavelet basis '0,\, A E A, constructed by the algorithms of Chapter 3 of Wavelets and Operators, is an unconditional basis of a classical space of functions or distributions B, we need to show that the operators T : L2(R") -+ L2(R") which are diagonal in our wavelet basis are also continuous on the space B. Now such operators are automatically Calderon-Zyginuud operators, whose continuity on B will be established by the real variable methods developed in this chapter and in Chapter 10. (1.13)

8

7 The new Calderdn-Zygmund operators

2 Definition of Calderon-Zygmund operators corresponding to singular integrals As we stated in the introduction, we do not want to define a CalderGnZygmund operator as the principal value of a singular integral. More precisely, we also want to use kernels K(x, y) for which the limit lim

CIO

Ix-vI>e

K(x, y) f (y) dy

might not exist in the usual sense, even when f (y) is a test function or K(x, y) is antisymmetric. Now, in the antisymmetric case, that limit exists in the sense of distributions. This leads us to base the theory

on the bilinear form J(f, g) _ (T(f ), g) = (S, g (9 f), where f and g are test functions, T is a linear operator from D into D' and S is the distribution-kernel of T. Then K is derived from S by the condition that K is the restriction of S to the complement of the diagonal. Here is an example to clarify these distinctions. Let 0(x) be an odd function in D(R) such that f o. 0(x) sin x dx = 1. Further, let the support of 0 be the union of the intervals [-4/3, -2/3] and [2/3,4/3]. Then we define the kernel K(x, y) by 00

K(x, y) = > 2k0(2k(x -

y))e42k(x+v)

_

0

We immediately get K(y, x) = -K(x, y), IK(x, y)l < Coax - yj-1 and 10k/8nI < Cl Ix - yl-2. If f is in D(R) and equals 1 on [-10,10], the existence of the limit limelo f x-yI>e K(x, y) f (y) dy, for -1 < x < 1, would imply that the series Eo e42kx converged. This series diverges everywhere (pointwise), but converges in the sense of distributions. This leads us to consider

JJ(f, 9) =

J J Ix-vt>e

K(x, y)f (y)9(x) dy dx,

where K(x, y) is an antisymmetric kernel with IK(x, y) I < CO Ix - yj-1 and where f and g are functions in D(R). Now

JE(f, 9) = 1 11 2

x-vI>e

K(x, y) (f (y)g(x) - f (x)9(y)) dy dx

and we can define J(f, g) by the absolutely converging integral

J(f, 9) = 2 Jf K(x, y)(f (y)9(x) - f (x)9(y)) dy dx . Then J(f, g) = limj0 J. (f, g), which enables us to complete the construction of T : D -+ V by the formula (T (f ), g) = J(f, g). The case of non-antisymmetric kernels is more subtle: we can no longer use a kernel to define the operator. Reversing roles, we start with

7 .2 Singular integral operators

9

an operator T : D - V. Let S E D'(IR" x R') denote the kernel of T. The existence of S is guaranteed by Schwartz's kernel theorem and S is related to T by the identity (T(f ), g) = (S, g ® f). The left-hand side involves the duality between D(IR") and D'(IR"), whereas the duality on the right-hand side is that between D(R" x IR") and D'(IR" x IR"). H

We let 11 = {(x, y) E IR" : x # y} and we consider the restriction K(x, y) of S to fl.

Definition 1. Let T : D(IR") -+ D'(R") be a continuous linear operator. We say that T is a Calderdn-Zygmund operator if there are constants Co, C1, C2, and an exponent -y E (0,1] such that the following four conditions are satisfied: (2.1) K(-c, y) is a locally integrable function on Cl and satisfies

lK(x,y)l Colx - yl-" (2.2) if (x, y) E Cl and Ix' - xl < lx - y1/2, then lK(x',y)-K(x,y)l:Ci'1lx'-xl7lx-yl-"-7;

(2.3) if (x, y) E Cl and ly' - yl < Ix - y1/2, then l K(x, y') - K(x, y)l 1 and for every u of length 1, we have

hL(Ax, u) - L(Ax', u)l < Cl Ix'- xl7, when Ix'- xl < 1/2. Letting A tend to infinity concludes the argument. Thus condition (2.2) fails when Ix - yl is large.

10

7 The new Calderdn-Zygmund operators

Ll.

'SC

A subset B C L(L2(1R"), L2(1R")) is called a bounded set of Calder6nZygmund operators if the operators T E B satisfy (2.1), (2.2), (2.3) and (2.4) with the same exponent -y and the same constants Co, C1 and C2. Let 9 denote the group of unitary isomorphisms of L2(1R") of the

E-+

NAB

form U f (x) = b-n/2 f (b-1(x - x0)) for xo E 1R" and b > 0. Then, if T is a Calderon-Zygmund operator, the collection UTU-1, U E Q, is a bounded set of Calderdn-Zygmund operators. The kernel of UTU-1 is b-"K(5 ' (x - xo)) and we note again that conditions (2.1) to (2.4) are Ian

invariant under change of scale. We shall use the following "weak compactness theorem". MHO

tea, 'i7

a''

Erg

Proposition 1. Let T3, j E N, be a bounded scqucncc of Calde.ronZygmund operators. Then there exist a Calderdn-Zygmund operator T and a subsequence Tj(m) such that (2.5)

(7's(m)(f),9) - (T(f),.9),

for f E L2(1R") and g E L2(]ll"). `-'

We shall write T3(,,,,) - T in the above situation. Since the operator norms of the Tj : L2 (1R") - L2(IR") form a bounded sequence, there is a subsequence Tj(m) which converges weakly to a linear operator T :L2(1R") L2(R").

`.7

,fl

t/]

It remains to prove that T is a Calder6n-Zygmund operator. We let S,,, and S denote the kernels of Tj(,n) and T. Then S is the limit of the Sm in the sense of convergence of distributions in D'(IR" X R"). The restrictions of the S. to each compact subset of 1 are continuous !!ten

functions satisfying Holder conditions uniformly in m. Ascoli's theorem gives the uniform convergence of Sm to S on those compact subsets.

Conditions (2.1), (2.2) and (2.3) on S are obtained by passing to the limit in the inequalities. aid

Conversely, every Calderon-Zygmund operator T can be written, as in (2.5), as the limit of a bounded sequence of Calder6n-Zygmund operators Tm, whose kernels Sm are not only distributions, but also infinitely differentiable, hounded functions. More precisely, we have

Proposition 2. Let T : L2(1R'') -' L2(1R") be a Calderdn-Zygmund operator with kernel K(x,y). Then there is a sequence Km(x, y) in cad

L°°(1R" x1R") such that,9K,n/8xj and 8Km/8yj are also in Lm(1R" xR") and the following properties hold:

(2.6) the operators Tm defined by Tm(x) = f Km(x, y) f (y) dy form a bounded sequence of Calderon-Zygmund operators; (2.7) for every function f E L2(1R"), fit;

lim IIT(f) - Tm(f)112 = 0.

7.2 Singular integral operators

11

To see this, we let 0 E D(1R") denote a radial function of mean 1, we put (m(x) = m"O(mx), and we write R,,, for the operation of convolution by cbm. Consider T,,,, = RTTR,n. Let S E D'(R" x Rn) denote the kernel of T and let Sm denote that of T,n. Then

(2.8) Sm(x, y) =

Tx'Ym/

where Tx denotes translation by x. We can suppose that the support of 4) is contained in Ix1 < 1: we shall just deal with the case where y = 1 in (2.2) and (2.3). If Ix - yJ > 4/m, the supports of and Tx4m are disjoint, so that

Sm(x, Y) = JJm(X - u)K(u, v)0. (v - y) du dv =

JfK (x - --, y - m) 0(u)&) du dv .

Properties (2.1), (2.2), and (2.3) are thus satisfied by Sm(x, y), with uniform constants, as long as Ix - yJ > 4/rn.

For Ix - yJ < 4/m, we use the continuity of T on L2(1R") to get ISm(x, y)I < Cm". Further, dx3 Sm(z, y) =

JJ(x

_-

- u)S(u, v)46. (v - y) du dv a (9x3

4)nn/

which gives

ax S. (x, y) I < Cmr}1.

These estimates give (2.1) and (2.2) for Ix - yJ < 4/m. (2.3) is done similarly. The convergence of T,(f) to T(f) in L2(1R") follows immediately and concludes the proof of Proposition 2. One last general property merits attention. Let T be a Ca.lderon-Zygmund operator. Suppose that the kernel K of T is zero. Then T is the operator of pointwise multiplication by a function m(x) E L°°(R"). To prove this, we approximate as above, but in a more rudimentary fashion. Let Q3 denote the collection of dyadic cubes 2-3k+2-3 [0,1)n, k E Z", and let V3 C L2(1R") be the subspace of functions constant on

each cube Q E Q3. Lastly, let E3 : L2(IR") - V be the corresponding conditional-expectation operator. If T is bounded on L2(IR") and local (that is, the support of T (f) is included in the support of f, for f E L2(IRn)) then the same holds for E2TE3 : L2(IR") V. As a consequence, E3T, restricted to V3, is the operator of pointwise multiplication

7 The new Calder6n-Zygmund operators

12

by a function mj(x), constant on each cube Q E Q3. Moreover, we have

Ilm;il. = IIE,TE;II < IITII fly

The conditions of compatibility between E3T and E?+1T immediately give m3 = E9 (mp+1). So the sequence of functions mj (x) is a uniformly-

`'1

bounded martingale which converges to m(x) E L°O(R"). A simple passage to the limit shows that T is pointwise multiplication by rn(x). To conclude: we could have proved Proposition 2 by replacing the regularization operators I,,, by the operators E. corresponding to a multiresolution approximation of regularity r > 1. The approximation of T by a bounded sequence of Calder6n-Zygmund operators Tj would °5.

I30

C4'

then have been given by E?TE3. By Proposition 2, the theory of Calderbn-Zygmund operators can be written using only absolutely converging integrals (in the first place) and then finishing by passing to the limit. This point of view is taken in [76] which is, to our knowledge, the first major reference to the general theory which we are presenting here. The second approximation technique for Calderbn-Zygmund operators is based on the truncation of kernels. For this we use a function 4) E D(IR") which is radial, equal to 1, when IxI _S 1, and to 0, when Ixl > 2.

We then replace K(x, y) by K6(x, y) = K(x, y)(1- 4)(6-1(x - y))). It follows that K6(x, y) = 0 when Ix - yI < 6 and the singularity of the kernel has disappeared. We let T6 be the operator defined by the kernel K6. The relationship between T and T6 is described by the following proposition.

Proposition 3. The operators T6 form a bounded set of Calder6n0 and a function Zygmund operators. There exist a sequence hj m(x) E L°° (R") such that, for every pair (f, g) of functions of L2(R"), we have (2.9)

(Tf, g) = 9l-00 (T5, f, g) + Jrn(x)f(x)(x) d--

We start by verifying the first assertion. We have 4)(u) =

(27r)-" Jetc) de

and

K6(x,y) _

(=

(1_4)Qfj!))sis,Y,

where S(x, y) is the distribution-kernel of T. We now separate the variables by representing 4)(6-1(x - y)) as a Fourier transform:

fes6-1ey-1 (27r)" J

}4.

7.3 Calderdn-Zygmund operators and LP spaces

13

This gives

Kb(x,y) = S(x,y) - R5(x,y),

where Rs (x, y) = (2 n Jelx Ca-1 S(x, y)e-s& fb-1 (

)d.

Let Rb denote the operator whose kernel is the distribution Rb(x, y) and let M(C) be the operator of pointwise multiplication by e'-{. We have Rb

(2i_L_

r)n

jM(Le)TMl()

and the operator norm of Rb : L2(R")

dC

L2(Illn) is bounded above by

(27r) -n JJTJJJJr111.

The estimates (2.1), (2.2) and (2.3) are easy to verify as long as the

cases Ix - yJ < 6, 6 < Ix - yI < 26 and Ix - yI > 26 are dealt with separately To prove the second assertion, we start by taking a subsequence b; such that (Tb, f, g) - (L f, g), for a certain Calderdn-Zygnnmd operator L. If f and g are test functions of disjoint support, then (Tb, f, g) = (T f, g), for large enough j. It follows that (T f, 9) = (L f, g). The kernel corresponding to T - L vanishes and (2.9) is verified. We shall see later that the approximation technique described by Proposition 2 is significantly better than that which we have just used. For example, if T is bounded on the Hardy space H'(IR"), or its dual BMO(lR' ), the same holds for the operators T,,, of Proposition 2, but not for the approximations of Proposition 3.

3 Calderdn-Zygmund operators and LP spaces The purpose of this section is to study the continuity of CalderonZygmund operators on LP(Rn), when 1 < p < oo, and to describe what happens in the limiting cases p = 1 and p = oo. Let K(x,y) be a function which is locally integrable on lrln x Rn. Suppose that the operator T f (x) = f K(x, y) f (y) dy is continuous on L°O(Rn). Then it necessarily follows that (3.1)

ess sup J I K(x, y) I dy < oo XER"

and IITII,,,,.,, is given by the left-hand side of (3.1). Similarly, if we assume that the operator T is continuous on L' (R'1), then ess supXER, f I K(x, y)I dx is finite and equals IITII1,1

7 The new Calderina-Zygmund operators

14

Calderon-Zygcnund operators do not satisfy these conditions because, in general, (2.1) allows the kernel to have too large a singularity. A Calderon-Zygmiind operator is thus not continuous as an operator from Lu to L' itself. However, it can be extended as a continuous linear operator from L'(]R") to weak Ll, which space we shall now define. We shall throughout use IEI to denote the Lebesgue measure of the measurable set E. 4-,

Definition 2. A measurable function f (x) on R" belongs to weak L' if I1f1Iw =supAI{x : If(x)I > A}I A>O

is finite.

The essential point is that IIf + gllw < 211f II,, + 2IIyII,,, even though IIf IIw is not a norm. The verification of the inequality is straightforward,

indeed, if If (x) + g(x) I > A, then either if (x) I > A/2 or Ig(x) I > A/2. This gives the following relationship between sets: {x : If (x) +g(x)I > A} C {x : If (x)I > A/2} U {x : Ig(x)I > A/2} finally leading to

NIA 2

1(x : If (x) +g(x)I > All < - 11f1I. + -II.4IIw -

This implies that weak L' is a complete metric vector space. But r-1

weak L' is not a Banach space. To see this, we look at dimension 1 and observe that Ix - xoI-1 belongs to weak L' with norm equal to 2. But the sequence of convex combinations 2-i r_o0, (3.2)

I{x E R" : ITf(x)I > A}I
21&-&'I}

I K(x, y') - K(x, y) I dx < CI , }{'

and give the proof under these hypotheses. Clearly, we may suppose that CI = C2 = 1 and that 11f III = 1. We now carry out a Calderdn-Zygmund decomposition off of threshold A. This gives the cubes Qj, j E J, as described by Theorem 2. Let QQ denote the "doubled" cube: the centre of Qj is that of Qj and the diameter of Qj* is twice that of Qj. Then fl* is the union of the Qj* and ago

we get I

IQ;I A is contained in E U )* and we know that ISi*I
jE J bj (x) and then let E, and E2 denote the subsets of E where, respectively, ITg(x)I > A/2 and ITb(x)I > A/2. Then E C E, UE2, which leads us to find upper bounds for the measures of El and E2.

For IEII, we simply use the continuity of T on L2(Rn). We have IIT9112 < 119112 < 2?A1/2The Bienayme-Chebyshev inequality gives 2

4 IEII < IIT91I2
jE J bj with the sum converging in 4n+lA-I.

L2 (1R') (as well as pointwise) if f E L' f1 L2, as we have assumed. Hence Tb(x) = jE J Tbj (x). For x V fZ*, we can use the representation of T by its kernel. This gives ay"

Tbj (x) =

j

K(x,y)b3 (y) dy

Q,

= f(K(L,Y) - K(x, yj))bj (y) dy , Q9

where yj is the centre of Qj. We then estimate the integral

'=1
I ° AjT.(aj) = 0-

Then '7we observe that the T,,,, form a bounded sequence of Caldcr6n-

Zygmund operators. Hence I IT,,, (a j) II 1 _< C, by the argument above.

It is easy to check that, for each j, T,,,(aj) converges in L' norm to T(ai). Finally, we get Eo AjT(aj) = 0, because Eo tends to 0 in L' norm, as m tends to infinity. This proves that Calderon-Zygmund operators are (H', L') continuous.

As a corollary, we can define the extension of Calderbn-Zygmund op-

,0.

E-+

erators to L°° by using the (L1, L°°) and (H1,BMO) dualities. More precisely, if b(x) E L°-(R'), we define T(b) as a continuous linear form on H1 by (3.11) (T(b),u) = (b,T*(u)), u E H1. The right-hand term makes sense because the transpose T* of T is again a Calderbn-Zyginund operator and, thus, T*(u) E L'. So T(b) is in BMO(R-).

F.+

It is worthwhile to make the connection between this definition and the action of T on L2(ll8"). To do that, we observe that, if b c L°°fL2 and if u is an atom, then (3.11) becomes an identity. Further, if b c L°°(lll;"),

let us define the functions bj(x) by b3(x) = b(x), when Ixi < j, and bj (x) = 0, otherwise. Then T(bj) is defined by the action of T on L2 (R'). If u is an atom of R', then (T(bj), u) = (bj,T*(u)) - (b,T*(u)) . Indeed, IIbjII= < I1bIl=, bj(x) - b(x) almost everywhere, and T*(u) E Ll(lR1), so that we can apply Lebesgue's dominated convergence theorem.

The functions T(bj) thus form a bounded sequence in BMO(lR') and this sequence converges to T(b) in the a(BMO, Hl) topology. The following lemma describes this convergence precisely.

20

7 The new Calderdn-Zygrnund operators

Lemma 2. If T is a Calderdn-Zygmund operator and if the functions b., are formed by truncating the function b E L°°(IR"), then there is a sequence c(j) of constants such that T(b,) - c(j) converges, uniformly on compacta, to a function in BMO(R") which is a representative of T(b) modulo the constant fuctions.

.Oh

Indeed, we put c(j) = fi 2d

as required. The "integral on the interior" f x-xola(A)z/ia(x) E H' and Ea(A)zj'a(x) E H1 are equivalent. To do this, we consider the unitary operator U : L2(RI) L2(R") defined by U(z/',,) = a. The

7 The new Calderon-Zygmund operators

24

..,

'ti

...

kernel of U is the distribution S(x,y) = EXEA rb (x),ia(y) and it follows immediately (by using the regularity and localization properties of wavelets) that S(x, y), restricted to y * x, satisfies conditions (2.1), (2.2) and (2.3) of Definition 1, with -y = 1. The operator U is thus a cep

,L+

fig,

Calderon-Zygmund operator.

We now use Lemma 3 to show that U(1) = 0 and U*(1) = 0. We (+,

La.

require approximations U,,,,, m E N, of U, defined as follows. Let A"t be an increasing sequence of finite subsets of A whose union

a,,

t.7

is A. We define U,,, by the distribution 5,,,(x, y) = E,EA,,, a(x)r ,\W which is its kernel. Then the S,,,,(x, y) satisfy conditions (2.1), (2.2) and (2.3), uniformly in m, while the U,,, converge strongly to U. Each °u3

'C3

U,,,.(1) = 0, because f sJia(y) dy = 0, and, similarly, Uf/z(1) = 0. It follows

t=.

`'r ...

that U(1) = 0 and U*(1) = 0. Theorem 3 gives the required result. This line of proof has many variants. For example, it shows (but we already know the result) that the wavelets rfia, A E A, form an unconditional basis of H'(R"). For every bounded sequence rn(A), A E A, we consider the operator M whose eigenfunctions are ip,\, A E A, and whose eigenvalues are m(A). Once again, M is a Calderon-Zygmund operator and M(1) = 'M(1) = 0, so M can be extended as a continuous linear operator of H'(R") into H'(R"). Thus F, a(A)V),\ (x) E H' implies that E m(A)a(A)Via(x) E H', which shows that the s/), (x) form an unconditional basis of H' (R"). This can be further generalized by replacing H'(1&") by other spaces B of functions or distributions. If the Calderon Zygmund operators T t.'

,°c

'"'

.w.

satisfying T(1) = tT(1) = 0 can be extended as a continuous linear in.

operator from B to itself and if the subspace So(R"), of those functions in S(R') whose moments all vanish, is dense in B, then the wavelets rJ,a, A E A, form an unconditional basis of B. Following the same pattern, we could show (although we already know it) that the wavelets rJ)a arising from an r-regular multiresolution approximation formed an unconditional basis of the Hardy space Hr(1[t") when n(l/p- 1) < r. The condition tM(x°i) = 0, tal < n(1/p-1) follows from the cancellation properties of wavelets (Wavelets and Operators, Chapter 3, Section 7 ). The operator M is defined by M(V)a) = m(A)g5a, with m(A) E l°°(A).

C,;

5 Pointwise estimates for Calderon-Zygmund operators One of the purposes of this section is, if possible, to define a CalderonZygmund operator as the principal value of a singular integral. That is,

7.5 Pointwise estimates for Calderdn-Zygmund operators

25

we want to know whether

Tf (x) = lim

(5.1)

K(x, y)f (y) dy

ELo f1x-8I>_E

fps

for every function f E L2(IR") and almost all x E 1R". Experience shows that (5.1) is proved by first establishing (5.1) for a

Ins

dense linear subspace V of L2(IR"). Usually, V = D(1R"), but, in the case of the Cauchy kernel (z(x) -z(y))-1, where z(x) = x+ia(x) is such

that -M < a'(x) < M, we must take V = (f (x)z'(x) : f c D(Rn)). Once we know that (5.1) holds in a particular instance, we look at the corresponding maximal operator ([219]).

To construct that maximal operator, we define the truncated kernel F's

KE(x, y), for each e > 0, by Ke(x, y) = K(x, y), when Ix - yj > e, and KE (x, y) = 0 otherwise. The operator TE is given by Tj(x) (x) = fir' C..

f KE (x, y) f (y) dy and then, for all .r E R", we define the maximal operator T.. by (T,,f)(.r) = sup,,>o ITEf(x)I -

coo

..C

The operator T*, thus defined, is sublinear. More precisely, if f is a function in L2(R"), T* f is a positive measurable function and we have T'*(Af) = IAIT*(f) and T. (f, +f2) f f (x), but it is easy to see that f*(x) < 2'f,* (x), so the two definitions are equivalent from the point of view of integration theory. We shall assume the following classical results ([126], [127]).

7 The new Calderdn-Zygmund operators

26

Theorem 4. The sublinear operator Al : f

, f* has the following

properties: (5.4)

I{x E IR" : f*(x) > A}I s

llf*llp c

C(n) p

C(n)

iff E L'(R') arid A >0,

llflll

if1 0 and each x' E R' satisfying Ix' - xI < e/2, we have

IT. Ax') -TEf(x)I < Cf*(x). We first find an upper bound for I f x_5I>E (K(x', y) - K(x, y)) f (y) dyl.

This is done by using (2.2) and (5.9). We then estimate the difference between K(x', y) f (y) dy and f y_yI>E K(x', y) f (y) dy. Since Ix' - xl E/2, the symmetric difference

`-'

of the sets W - yI > 6 and Ix - yI > E is a subset of the shell E/2 < Ix' - yI < 3e/2 (and, by symmetry, of the shell E/2 < Ix - yI < 3E/2).

Thus, in that region, I K(x', y) I may be bounded above by 2'Coa-'°, and the difference between our two integrals is less than C f * (x). .L7

We return to the proof of Cotlar's inequality. Suppose that E > 0 and let TE be the corresponding truncated operator. Let B be the ball IxI < E/2 and let b be the ball Ixl < E. We then write fr for the product of f by the characteristic function of B and put fz = f - fl. We intend to show that, for every E > 0, (5.10)

ITEf(0)l

2(Tf)*(0)+Cf*(0).

Taking the supremum over E > 0 of the left-hand side of (5.10). Will give Cotlar's inequality.

To prove (5.10), we observe that TE f (0) = Tf2(0) which we shall compare with the other values taken by Tf2 on B. Lemma 6 shows that, for each x E B, (5.11)

ITf2(x) - Tfz(0)I < C(n, 7)f*(0)

In order to estimate T f (x), we note that, for almost all x c B, T f2(x) = T f (x) - T f, (x) and (5.11) gives, for almost all x E B, (5.12)

ITf2(0)I 5 lTf(x)I +

+C(n,'Y)f*(0)

7 The new Calderdn-Zygmund operators

28

The last part of the proof consists of estimating the "weak L1 norms"

of both sides of the inequality (5.12), with respect to the uniformly distributed measure t on the ball B, that is, the measure of constant density IBI-1 on B and whose support is B. If f : B - C is a measurable function, we put N(f) = supa>0(\µ{x E B : If (x)I > A)). If If I 0 and verify that I{x E 1&" : w(f;x) > a} I = 0. Indeed, let Q > 0 be a real number, which will tend to 0, and let g c V denote a function such that Ilf - 9IIp :5,8. Then w(f; x) < w(f - g; x) + w(g; x) = w(f - g; x) almost everywhere and, thus, I {x : w(f; x) > a}I C a-pllw(f - g)IIp

c 2'a-9IIT*(f - g) IIp
e K(x, y) f (y) dy exists, for every C' function f of compact support. Indeed, the integral can be written as fEe

(z(x) - z(y))-lz (y)f (y) dy = -

Jl.c-yl?e 1''(y)f (y) dy I'^

f

D°5

= log(z(x + E) - z(x))f(x r+ E) - log(z(x - E) - z(x))f(.c - E) iii

+ JIx-vl>e log(z(y) - z(x))f'(y) dy. For almost all x E R, z(x + E) - z(x) lim E10

= z' X = lira ( )

E

F10

z(x) - z(x. - e) E

Hence, the integrated terms converge almost everywhere to -irp f (x). The integral f x_y1>e log(z(y) - z(.c)) f'(y) dy is absolutely convergent. In this last example, the kernel K(x, y) defines the operator T by the

L7'

`C1

bias of the distribution PV(z(x) - z(y))-'. The dense subspace V of L' (R) is the set of products z' f , where f is a C' function of compact 'L7

support. Of course, Cotlar's inequality can only be applied once we have shown that the operators in question are L2 continuous.

6 Calderon-Zygmund operators and singular integrals Cotlar's inequality will let us give a complete answer to the following question: if T is a Calderdn-Zygmund operator with kernel K(x,y), can we construct the operator T just using the kernel K?

7.6 Calderon-Zygmund operators and singular integrals

31

'"J

One answer has already been given. We let 0 E C°°(P") be a radial function, equal to 1 when IxI _> 2 and to 0 when IxI c 1. For each e > 0, we form the truncated kernel Ke(x, y) = 0((x - y)le)K(x, y) and denote by Tf the operator defined by this kernel. Then the Te, e > 0, form a hounded set of Calder6n-Zygmund operators. Thus, there is a

...

's3

C-'

sequence ej tending to 0, such that the operators TES converge weakly to a Calderon-Zygrnund operator L. Here T = L + M, where M is the operator of pointwise multiplication by a function rn(x) E L°°(R"). The kernels K(x, y) of T and L are identical and it is obvious that K(x, y) can give us no information about rn(x). We have to pass to a subsequence even in the simplest examples. Consider the operator My = (-A)'-Y/', ,y c R, which is defined, via the If y 0 0, the kernel K(x, y) Fourier transform, by (M.y f) (t;) = l l' f of M.y is c(n, -y)lx - y['"-'"Y. For a C' function f of compact support, limE1 to f x_yl>Ej [.e; - yI-"-'Y f (y) dy exists if and only if the limit of e3 `7 "'!

`J'

cr.

does. We see this by reducing the range of integration to E j < I x-yj < R,

for large enough R, and then writing f (y) = f (y) - f (x) + f (x). We conclude that the operators TE, converge weakly if and only if e.7 `-r

tends to a limit, that is, if (y/2rr) log ej converges modulo 1. A more elaborate example will be useful in what follows. We consider the pseudo-differential operator a(x, D) defined by the symbol a(x, ) = (1 + gt2)gck(x)/2, where a(x) = 2 + sinx and x, E R. This symbol

ban

b^°

6°d

belongs to all the Hormander classes S°,,, ([68], Chapter 2) and the corresponding operator a(x, D) is thus a Calderon-Zygmund operator. The kernel K(x, y) of a(x, D) is the sum of a principal term Ko(x, y) and an error term Kl (x, y). The principal term is given by -i-ia(x) pH,

--,

Ko (x, y) _ 'y(x) Ix - yl

where t.7

.Hi

y(x) = -2ir(1 + 2ia(x)) sinh rra(x),

fix'

r being Euler's Gamma function. The error term Kl (x, y) belongs to L°° (IR x R) and thus does not give rise to a singular integral. 0, with e(j) > 0, such that Suppose there is a sequence e(j) limj-- f K(x, y) f (y) dy exists almost everywhere, whenever f (x) is a C' function of compact support. The existence of this limit is equivalent to that of limy, a-'a(x) logy(j). However, the latter cannot exist almost everywhere, by the following lemma ([239], p.316).

r.;

Lemma 7. Let Aj be a sequence of real numbers whose absolute values tend to infinity. Then the set E of real numbers t, such that e'ta' tends to a limit, is of measure zero.

7 The new Calder6n-Zygmund operators

32

To see this, we let f (t) be the limit of the functions Eicaj on E. Then, for every compact interval [a, b], on applying Lebesgue's dominated convergence theorem, we get b

1 eiea, XE(t) dt -+

(6.1)

a

Ja

b f (t)XE(t) dt,

'L"

where XE is the characteristic function of E. Next we observe that the left-hand side of (6.1) is the Fourier transform at -Aj of the characteristic function of E f1 [a, b]. This function is in L1(R), so the Riemann-Lebesgue lemma shows that the limit in (6.1) chi

is zero. So, for every a E R and every b > a, fn f (t)XE(t) dt = 0. As a consequence, f (t)XE(t) = 0 almost everywhere. But If (t)I = 1 on E. "d"

We conclude that JET = 0. We return to the basic problem of calculating the action of a Calderon-

Zygmund operator T on a function f e L2(R) via a generalization of ..,

.C.

i-+

°+,

chi

PV f K(x, y) f (y) dy, where K is the kernel corresponding to T. Let Te he the operator given by TE f (x) = fz_vl>e K(x, y) f (y) dy, for e > 0. Theorem 5 tells us that the operators Te form a bounded subset of L(L2(R"), L2(R")). It is therefore possible to find a subsequence ej, tending to 0, such that T., converges weakly to an operator L. Then L is a Calder6n-Zygmund operator whose kernel is precisely K(.c, y). Finally, [RD

C07

2'+

Jam'

T = L + M, where M is the operator of pointwise multiplication by a function m(x) E L°C(ll?" ).

We thus have the following integral representation formula, where f and g belong to L2(R"):

`ti

K(x, y) f (y)g(x) dy dx + Jrn(x)f(x)(x) d.a. (6.2) (T f , g) = Ii m J z-vl?ej We can ask for more, namely, that ff, f (x) should converge to L f (x) almost everywhere, as e j tends to 0. We have already seen that we then need to replace the constants e, by functions e j (.c) tending to 0. These remarks lead to the following result. L.'

y°-

.-,

I.,

.4Q

Theorem 6. Let T : L2(R7) - L2(R") be a Calderon-Zygmund operator. Then there exist a sequence e, (x) of strictly positive measurable functions on R", such that limj-_.,0 ej(x) = 0, and a function ^,,

;'3

M(x) E L°O(R") such that, for every function f e L2(R"), r`'

(6.3)

y'.

moo

-~.

K(x, y)f (y) dy, T f (x) = m(x)f (x) + lim y-.ooJk-vl?ej(x) for almost all x E R". The right-hand side of (6.3) also converges to T f in L2 norm, for f belonging to L2(R").

7.6 Calderon-Zygmund operators and singular integrals

33

As we have seen, Theorem 6 is the best possible result. It is basically a reformulation of Cotlar's theorem. To prove Theorem 6, we use the sequence mj(x), j E N, of continuous functions on R", defined by mj(x) = f 1+1)-1gx-yI A}. Then (7.5)

C,,A-1

IEI
O and every ry E (0, yo), (8.2) w{x E llt : T f (x) > 2A and f * (x) < yA)

Al is thus open. Since T. f (x) = O(IxJ "), as IxI oo (due to the conditions we have imposed on f), 11 is a bounded open set. Because of this, up to a set of measure 0, H is a disjoint union of dyadic cubes Qj C 12, which are maximal with respect to the inclusion relation. We let Qj denote the "parent" of Qj, that is, the dyadic cube containing Qj with double the diameter. Then Q j is not contained in St and there exists a point a j E Q j outside ft. W e Put Ixf = sup(Ixl It ... , Ix"I), X E W", so that Qj is defined by Ix - ,Q3 I < d3 /2, where N3 is the centre of Q p . Hence, T* f (a3) < A and

Iaj-$3I 2A and f*(x) _< -yA, does not intersect Q3. Since E is a subset of f2, we have E C U Q j and, hence, w(E) = Ejej w(E (1 Qj). For each j E J, let E3 be the set of x E Q3 such that T* f (x) > 2A. We shall show that, for every j c J. (8.3)

IE3I 0. We then use Theorem 8 to get C' IEjI A} .

Then, if Cyys < 2-p, it follows that (8.12) Ilullp 5 (2-" - Coys)-1/py 'IIvIIp in.

Indeed, from (8.11) we obtain the inequality p{Iu(x)l > 2A} < p{Iv(x)I > yA} + Coyap{Iu(x)I > AI. We multiply both sides of (8.13) by A"-' and integrate with respect to A, using (8.10). This gives (8.13)

2-"Ilullp < -y-"IIvIIp +

7 The new Calderon-Zygmund operators

40

%.,

For 2-P > Co-? and lIull, < oo, the conclusion of the lemma follows. In the application to Calderon-Zygmund operators, we suppose that f is continuous, of compact support and such that T,, f (x) = O(Ixj-n) as Ixl --> oo. If 1 < p < oo, then T* f E LP(R", w dx), when w E AP. If 0 < p < 1, we suppose that f l.1,1 JxI-"Pw(x) dx < oo and that ensures that f(f *)Pw dx < oo and f xI>R(T* f )Pw dx < oo, for sufficiently large R. To deal with f xj 1, w,,(x) = inf (m,, w(x)) satisfies (8.9), with a constant C which does not depend on M-

Let us assume this result. We still suppose that f is continuous and of compact support, but we replace w by w,,,. Then f xj exp

JBI

f f (x) dx

by the convexity of the exponential function. It follows that each of the two terms forming the product in (8.14) is not less than 1. Each is thus bounded and the condition w E Ap can be written as the two simultaneous conditions -O(x)-FB dx < C' JB and

eIp1dx < C' J We can carry out two final manoeuvres with these estimates. Firstly, we 1

I

B

replace Q(x) -fib by (fi(x) -fb)+ in the first of them and by (fi(x) -Qb)-

7.9 Notes and additional remarks

41

in the second, because the only values we have to take into account, on integrating, are those greater than 1. Then, we may replace 16B (the mean of O on B) by a real constant A(B), whose relationship with ,8(x) we do not know, a priori. Indeed, if we simultaneously have C' (8.15)

4

ifB1

and

(8.16)

1

BI

fe(-(A(x)-A(B))/(p-1))+ dx < C1,

then, on adding the inequalities, we get -

f cosh(T(/3(x) - A(B))) dx < C' ,

dx < C" This allows us to work back,

where r = inf(l, (p-1)-1). It follows that TB17

and, finally, that IA(B) - fiB

C1.

....

replacing A(B) throughout by (3B, which only changes the constants. Now Lemma 13 becomes obvious. We simultaneously replace 6(x)

by fm(x) = inf(,3(x),1ogm) and A(B) by A..(B) = inf(A(B),logm). Then (fJ,,,,(x) - \,,,(B))4 < (,a(x) - A(B))+ and the same is true for (,6,n(x) - Am(B))

A final remark is necessary. Muckenhoupt's clasp A,o is the union of all the AP's. If w E Ate, then log w will belong to the space BMO of John and Nirenberg. We see this straight away by returning to the conditions (8.15) and (8.16). Conversely, if (3(x) belongs to BMO, there exists a sufficiently small E > 0 so that w(x) = cEQ(x) is a weight in Ate.

9 Notes and additional remarks The theory which we have described is only one aspect of the research in progress on singular integrals. After the pioneering work of Nagel, R.iviere and Waingcr ([195]), Stein and Phong systematically studied the operators T : L2(R") - L2(R") whose kernels have stronger singuilarities, in a certain sense, than those described by (2.2) and (2.3). Their starting point was the Hilbert transform along a parabola This is an operator T : L2 (1R') - L2 (R") defined by

Tf(x,y)=-7rPV J7 00f(X - t' Y _ t2) tt. 00f(x-t,y-t2)

In other words, T f = f * S, where S is the distribution which is the image of 7r-1 PV t-1 under the mapping t i.-+ (t, t2). The most recent results of Phong and Stein are about a generalization

7 The new Calderdn-Zygmund operators

42

9);

cad

of the "Hilbert transform along a parabola", a generalization in which the singularities of the kernel S(x, y) lie on a sub-manifold V. containing x and whose translations or whose geometric form satisfies a certain curvature condition ([203]). These extensions of the idea of CalderonZygmund operator come close to Fburier integral operators. Finally, a very active branch of research concerns the case of Banachspace valued functions (the kernel remaining scalar-valued). All of the theory of this chapter remains unchanged, but the difficulty that arises is that of proving the basic L2 estimate. Even in the case of the Hilbert transform, this estimate is not automatic. J. Bourgain put the coping stone on work of D. Burkholder's: these authors characterized those Banach spaces B for which the Hilbert transform defines a continuous operator on L2(R; B) (square-surnmable functions on R taking values in B). Those Banach spaces are the UMD spaces ([26], [30] and [33]). The operators we have studied in this chapter are not convolution operators, which suggests that we can dispense with the group structure of the underlying space IIt". This point of view has been developed systematically in [75]. The theory, extended to homogeneous spaces, applies, for example, to nilpotent Lie groups. The main difficulty, which will be resolved in the next chapter, is to replace the (non existing) Fourier transform by some other tool in order to give the L2 estimate on which the theory is based. A similar difficulty occurs in the work of Bourgain and Burkholder, where the Calderdn-Zygmund operators are convolution operators, but the functions take their values in a Banach space. If the Banach space in question is not a Hilbert space, we do not have a Plancherel theorem at our command to give the basic L2 estimate. ((DD

tip'

..,

X33

'7'

David and Journe's T(1) theorem

1 Introduction This chapter continues and completes the preceding one, correcting a weakness to be found there. The definition of Calderdn-Zygmund operators that we have given includes four conditions, of which the first three can be verified directly. These involve the restriction K(x, y) of

the distribution-kernel S(.r, y) of the operator T to the open set y # x in R' x R", and are (1.1) IK(x,y)I CoIx-yl-", I K(x', y) - K(x, y)I C, Ix' - xl7Ix - yl-'-'Y, (1.2)

if Ix' - xl < Ix-yl/2, and IK(x, y') - K(x,y)I < Clly' - ylrylx -

(1-3)

VIA

iffy' - yl 1, then, for every s > 0, there exists a constant C(s) such that (2.5) I(Tf,g)I 0 and R > 2r, the supports of f and g are contained in Ix - xoI < r and Ix - sot < R,

...

respectively. We shall exploit the fact that r is small compared with R and verify the inequality -"'

(2.6) I1(f,g)I < Cr"log RIIfIIoollglloo -I-

+Cr" E rlalllaafll. lal 0 such that the restriction of S to the open set lxi > R is a O(1x1-n-,Y) as JxJ - oo. If continuous function and such that S(x) = y > 0, then the integral fR S(x) dx = (S, 1) converges. To see this and to define the integral, we write 1 = 0o(x)+4i (x), where

¢o E D(iR") and ¢e = 1 in a neighbourhood of lxJ < R. Then (S, 1) is defined by (S, 4'o) + f S(x)(bi(x) dx. The integral converges absolutely. It follows immediately that (S, 1) is independent of the decomposition.

The transpose of T : V --+ V' is the continuous linear operator tT : V -y V given by (tTf.g) = (f,Tg), for all f,g E V. The hypotheses (2.8), (2.9) and (2.10) are all invariant under transposition, the kernel corresponding to the transpose tT being L(x, y) = K(y, x).

Now, if f E D(IR") and f f (x) dx = 0, we can define (T (1), f) by (tT f,1). Indeed, if the support of f is contained in lx) < R, then tea,

tTf(x) = J(K(y,x) - K(0,x))f(y)dy = O(IxJ "')

for lxJ > R.

As a mathematical object, T(1) is thus a continuous linear form on the subspace Do C D(IR") defined by f f (x) dx = 0. We extend T(1) to a distribution S E D'(R") as follows. Let 0 E D be a function of mean 1: then every f E D can be written uniquely as f = where A = f f (x) dx and g E Do. We make an arbitrary choice for the value of (S,¢) and put (S, f) = A(S,0) + (T(1),g). As a mathematical object T(1) is now a distribution modulo the constants. Another possible definition of T(1) is given by the following direct approach. We start with a function ¢ E D(IR") which equals I at 0 and put 0, (x) = O(ex), for every e > 0. We then consider the distribution Se = T(4E). Lemma 2. We can "correct" the distributions SE by `renormalization constants" c(e) such that limelo(SE - C(E)) exists in the sense of distributions. This limit coincides with T(1). Indeed, if f E Do, we have (T (,OE ), f) = (.0£, tT f ). We must verify that the right-hand side converges to (1, tT f ). Putting 1 = u + v, where u E D and u = 1 in a neighbourhood of the support of f. we can handle

8 The T(1) theorem

50

lim,Elo0E,uLTf), using the definition of a distribution. The Lebesgue dominated convergence theorem then shows that lim to (WE, v tTf) exists.

Let w E V be a function of mean 1 and put c(E) = (SE, w). Then, for

f EVandA= ff(x)dx, .n-.

where g = f - Aw. (SE - c(E), f) _ (SE, f) - A(SE, w) = (S, g) Then g E Do, so the required limit exists. Having made these definitions, we are in a position to state the theorem of David and Journe.

Theorem 1. Let T : V -j V' be a continuous linear operator corresponding to a singular integral as in Definition 2. Then a necessary and sufficient condition for the extension of T as a continuous linear operator on L2(Rn) is that the three following properties are all satisfied:

(a) T(l) belongs to BMO(llr); (b) tT(1) belongs to BMO(fl"); (c) T is weakly continuous on 1R". We first note that an equivalent form of condition (b) is T*(1) E BMO. Here, T*(1) is the complex conjugate of tT(1). Let us use a few simple examples to show that conditions (a), (b) and (c) are independent. If T is defined by pointwise multiplication by a function 7rt(x) E BMO,

we clearly have K(x, y) = 0 and T(1) = tT(1) = m(x) E BMO. Property (c) is only satisfied if m(x) E L' (R"). Another example in the same

direction is that of convolution by the distribution S = fp 1x1-'. The kernel K(x, y) = Ix- yl-n satisfies (2.8) to (2.10) and T(1) = tT(1) = 0, because T is a convolution operator. But, once again, (c) is not satisfied. An interesting example where (a) and (c) are satisfied, but (b) is not, is that of the pseudo-differential operators o(x, D) whose symbols o(x, l;) satisfy the "forbidden" conditions

l&oo(x,l;)I = C(a,Q)(1 + 16 )I0I-I0I It is not hard to show that the corresponding kernels K(x. y) satisfy (2.12)

IOOK(x, y)I < C'(a, [3) Ix - yi " I(VI-IAI In fact, (c) is just a consequence of the hypothesis C. In general, however, such an operator is not bounded on L2(IR"). Nevertheless, T(1) = o(x, 0) E L°°(IR"). What is missing is condition (b). We refer the reader to [96] for a more detailed discussion of such examples. We also pursue these matters further in Chapter 9. A noteworthy special case of Theorem 1 is when the kernel S(x, y) of T is PV K(x, y), where K(x, y) = -K(x, y) and K(x, y) satisfies (2.8) (2.13)

°.3

may'

8.3 The wavelet proof of the T(1) theorem

51

and (2.9). Then 'T = -T and (c) is trivially satisfied. For such antisymmetric kernels, the L2 continuity of T is equivalent to the condition T(1) E BMO. For example, consider the kernels

Km(x, y) = (A(x) - A(y))'/(x - y)"'+1, where A : R -i C is a Lipschitz function and m E N. The corresponding operators T,,, are the "Caldcrdn commutators" and a simple integration by parts gives the following remarkable identity: (2.14) T,,,(1) = Tm_1(A') form > 1. Assuming Theorem 1, we show how the continuity of the T,,, follows.

To is obviously continuous, since To = 7rH, where H is the Hilbert transform. Then we argue by induction on m > 1. If we know that T,,,_ 1

is a Calderdn-Zygmund operator, it follows that T,,,,_1 is continuous as a mapping from L°O to BMO (Wavelets and Operators, Chapter 5, section 3). But A'(x) is in L°°(R") and thus T,,,,(1) E BMO. This gives the L2 continuity of T,,, and the induction proceeds. A closer examination of this recursive proof gives the estimate (2.15)

IITm(f)II2 < irXmIIA'IImIIfII2,

where X > 1 is a constant whose value is not given by this method of proof.

Before approaching the proof of Theorem 1, we observe that (a), (b) and (c) are indeed necessary for the L2 continuity of T. As far as (a) is concerned, we have already remarked that every Calderdn-Zygmund operator is also continuous as a mapping from LO° to BMO. Of course, the constant function 1 is in L°°. The transpose of a Calderdn-Zygmund operators is still a Calderon-Zygmund operator, so (b) is satisfied. Finally, any operator which is L2 continuous is automatically weakly continuous.

3 The wavelet proof of the T(1) theorem The T(1) theorem is a statement which becomes more powerful as the exponent -y > 0 decreases. We may therefore suppose that 0 < ry < 1 in what follows. To prove Theorem 1, we must establish the L2 continuity of T, given the conditions (2.8), (2.9), (2.10), (a), (b), and (c). As we have seen, we may suppose that V is the topological vector space of C' functions of compact support. We now fix a multiresolution approximation Vj, j E Z, of L2(R) giving a function 0 and wavelets 3Ga which are Cu and have compact supports. Recall that O(x - k), k E Z", is an orthonormal basis

8 The T(1) theorem

52

ca,

of Vo and that zba(x) = 2'2/20,(2?x-k), where E E {0,1}n\{(0,...,U)} and A = 2-1k + 2_9-'e. We further let Q(A) denote the dyadic cube defined by 2?x - k E [0,1)n: the support of ipx(x) is contained in mQ(A)

where rn > 1 is a fixed integer and mQ(A) is defined by 22x - k E [-m/2 + 1/2, m/2 + 1/2)n. .-.

The functions 0,N are real-valued and the idea of the proof is to estimate the absolute values jT(A, A')j of the entries T(A, A') _ (Tz1', ,via.) =

?+.

(Via, tT*a-) of the matrix representing T with respect to the wavelet basis of the Hilbert space. We try to show that these entries become very small when the cubes Q(A) and Q(A') differ in either position or magnitude. To do this, we exploit the trivial remark that an integral f f (x)g(x) dx is very small if one of the terms of the integrand varies very slowly whilst the other is sharply localized (on a very much smaller

L10

scale than that of the variations of the former) and has mean 0. In our particular situation, we can always suppose that IQ(A)l is (much) larger than JQ(X)j, by changing T to its transpose, if necessary. If JQ(A)j is larger than IQ(A')I, the wavelet Via is 'fiat" where Vix, has zero mean. To conclude, we need to know that tTVa, also has zero mean, so we need T(1) 0. The condition tT(l) = 0 is needed to deal with the case IQ(A')I > IQ(A)I. tom,)

Vim]

These considerations suggest that the matrix M of T in the wavelet basis is almost diagonal, in the sense of Schur's lemma, when T(1) = tT(1) = 0. This is what we shall now confirm, by establishing the next result.

Proposition 1. Let T : V -} V' be a weakly continuous operator on L2(Rn) which corresponds to a singular integral, as in Definition 2. Suppose, further, that T(1) = LT(1) = 0. Let Via. A E A, be an orthonoru al Cl wavelet basis of compact support. Then there exists a constant C .II

.I.

such that, for A = 2-'k + 2-j-'E and A' = 2-j'k' + 2-3 -'e', 2-.1 + 2-A'

(3.1) IT(a,

A')1IEZ" c(l) = 0, since r_ q5(x - 1) = 1 and tT(1) = 0. Then. r(), )') = f g(x) ,\, (x) dx, where V;,\, is "flat", whereas g(x) is well localized (about )) and of mean 0. because of the properties of the coefficients c()). The estimate (3.1) then follows from the next lemma. (3.3)

Lemma 3. Let f (x) be a C' function whose support is contained in the unit ball of R" and whose partial derivatives Of ldxj, 1 < j < n, satisfy Ilaf/dx;II. < 1. Let -y E (0,1) be an exponent and let g E L'(R") be a function Ixl)-"-y and f g(x) dx = 0. Then there is a satisfying I q(x)I _< (1 + constant C = C(n, -y) such that, for all xo E R' and every real R > 1. (3.4) and

(dxl

c,1

(3.5)

lfg(x)f

< CR-y

if Jxcl < R

J lfg(x)f

(dxl
R.

To establish (3.4) and (3.5), it is enough to write

g(x) =

&I g1 (X) + ... + .9x. gn(x) ,

Ixl)-',+1-'Y and C(1 + then integrate by parts. The calculation of the matrix coefficients r((), A')) _ (T1PA, y')y) involves two separate scales (2-3 and 2-3'). It is worth remarking that

where Igi (x)l < C(1 + 1_T1)-n+1-,Y,..., I gn(x)I

8 The T(1) theorem

54

the calculation can be done using a preliminary computation involving only one of those scales (the smaller one). Writing Ojk instead of ?PA,

for ), = 2-2k + 2_,-1E, where E E {0,1}n \ (0.....0), we define three "non-standard" matrices corresponding to T by A = (T1G,k,;1),

B=(T-Oik,it), C = (T-0gk, 10?l) ,

for j E Z, k E 7Gn, and I E Zn. Their entries are denoted, respectively, by a(j, k, l), /3(j, k, l), and ry(j, k, l). The singular integral operators of Definition 2 are characterized by the simple conditions Ic(j, k, 1)1 C(1 + Ik - 1I)-n-1 ,

/(j,k,l)1 < C(1+ Ik - ll)-n-ry, and

I-f(j, k, 1) 1 < C(1 + lk -

ll)-n-ry

.

Once we have computed these three non-standard matrices, the standard matrix (T A, V),\,) may be obtained by a very simple manipulation, consisting of writing vPa. E V.,, for j' < j -1, and decomposing V' with respect to the orthogonal basis 0jk, k E Zn, of V

In [E4J, G. Beylkin, R. Coifman, and V. Rokhlin demonstrate that the above remarks have a considerable significance in numerical analysis. The reader may also refer to [E7] and [E19].

4 Schur's lemma For the moment, we shall forget the problem of the continuity of singular integral operators, and recall the statement of Schur's lemma.

Lemma 4. Let M = (m(p, q))p,gEN be an infinite matrix and suppose that w(p) > 0 is a sequence of positive real numbers. Suppose, further, that, for every p E N, (4.1)

E Im(P, q)Iw(q) < w(P) qEN

and, symmetrically, for every q E N,

E Im(p, q)lw(p) < w(q).

(4.2)

pEN `ti

Then M : 12 (ICY) , l2 (N) is bounded and the norm of M is not greater than 1.

Indeed, suppose that Eo jx(q)12 < 1 and put y(p) = Eo m(p, q)x(q).

8.4 Schur's lemma

55

We intend to show that Ta ly(p)12 < 1. To to do this, we write m(p,q)j1/2jx(q)j-

Im(p,q)Ilx(q)l = Im(p,q)j1/2w1/2(4)w 1/2(q) Applying the Cauchy-Schwarz inequality gives jm(p,

Im(p.4}lw(4)

Iy(p)I 2 0

Q)jw-1(Q)Ix(t1)12

0 0C

<w(p)

1(q)Ix(q)I2 0

We sum with respect to p, change the order of summation and (4.2) gives what we want.

The magic of Schur's lemma is that the order of the indices is of no account.

Let us return to the problem of the 12(A) continuity of the matrice. M whose coefficients satisfy (3.1).

We apply Schur's lemma with w(A) = 2-"j/2. This means we must estimate

Eit E

2-nj'/22-Ij-j'I((n/2)+ry)

k,

2-a + 2-9

2-j + 2-j' + 12-jk - 2-j'k'I

We start with the case j' > j. Putting d = j'- j, we bound the quotient above by 2(1+Ik-2-dk'l)-1. Now, > k, 2-"d(l+Ik-2-dkl)-n-ry = Ek, 2-"d(1 + 2-d9k'j)-n-ry < C(n, y), as the latter series can be interpreted as a Riemann sum. This leaves

2-ry(j'-j) 2-nj/2 = C(,y)2-nj/2 = C('Y)w(,\).

Similarly, if j' < j, we put d = j - f and the quotient is bounded +12-d k - k'I)-1. This leads to a summation over k' which is bounded above, uniformly with respect to k and d. We are left with

above by 2(1

2-n3 /22-(7-j')(n/2+ry) = 2-nj/2 r 2-dry = j' <j

C('Y)2-n3/2.

dd>>Ob

o,^

So we have proved Theorem 1 when T(1) = tT(1) = 0. Choosing w(.X) = 2-n7/2 has the following significance. If we look back at section 8 in Chapter 6 of Wavelets and Operators, we can see that the operators T and tT are continuous on the homogeneous Besov

space b"", when T(1) ='T(1) = 0. These two estimates give the L2 continuity by interpolation.

8 The T(1) theorem

56

5 Wavelets and Vaguelets Before leaving the special case of the T(1) theorem, considered above, we shall give a corollary of the proof we have just presented.

Definition 3. The continuous functions f,,k(x), j E 7G, k E Z'L, on IIBn, are called vaguelets if there are two exponents a ,fl, satisfying 0 < Q
2a(j, k)fj,k(x)II2 < C'(>EIa(j, k)12)112 .

To establish (5.4), we compute the square of the L2 norm of the series

EEc (j, k) f3,k(x) = f (x). To do this, we must compute the integrals

I(j,k;j'k') =

ffj,k(x)f9',k'(x)dx.

By symmetry, we can always reduce to the case where j' > j. To use (5.1), we first note that 2n"' C(n, a) (1+I2ix-kI)n+a(1+I2fx-k'I)n+'

- (1+29Ik'2-a' -k2-jJ)n+a

From this, we deduce that the integral we are interested in is bounded above, in modulus, by 2-2 + 2-j C'2-(n/2)1'-.j'1

2-i + 2-"' + I

k2-i

- P2-fl

The second estimate we use is given by

I J(f3,k(x) -

fj,k(2-I' k'))f j',k'(x)

< C23(n/2+0)


0, A E A, we have w(A)p(A) < fR. w(x) dx ,

(6.9) AEA

w(x) = sup W(A). Q(A)3r

In our case, if we ignore the constants, we have p(A) = Ia(A)12, W(A) _ IOA (f) 12 and thus w(x) _ (f *(x))2, where f * (x) is the Hardy-Littlewood

maximal function of f . We finish by observing that fa (f*(x))2 dx 0. let Sgt be the set of x such that w(x) > t: then Sgt can also be expressed as the union of all the cube Q(A) such that w(A) > t. By the Bienayme-Chebyshev inequality, Isntl < t-' f w(x) dx, and we may suppose that the integral is finite, because, if not, (6.9) loses all interest. Let Qk denote the maximal dyadic cubes contained in fIt. u t is the union of the Qk and, for t > 0, we get VII 'w`

ACA

X('\'t)p('\) < E p(A) Q(A)Cls

_

E p(A). k Q(A)CQk

We now use the assumption about p(A) and get Ek IQk I = I0t I as an UPper bound for the double sum. We finish by observing that f 11 I dt = fan w(x) dx.

Proposition 2 is now completely established and gives the operator R

8 The T(1) theorem

60

we have been trying to construct. To define S, we replace a(A) by 0(a), and S is the transpose of the operator R constructed using the ,6(A).

7 Cotlar and Stein's lemma and the second proof of David and Journe's theorem We state Cotlar and Stein's well-known lemma.

Lemma 6. Let H be a Hilbert space and let T9 : H -4H, j E 7G, be bounded linear operators with adjoints TT . Suppose that there exists a

sequence w(j) > 0, j E Z, such that , MA)'/2 < 00, forallj,kEZ, (7.1) IIT;TkII <w(j-k), and for all j,kE7L. IITj7k1I <w(j-k), Then, for all x E H, the series E-> T9 (x) converges in H. Putting T(x) _ ooT!(x) gives IITII < E (w(j))1/2. (7.2)

We follow [108] and first consider the finite sums S = SN = ENN T3. Of course, IISII' = IIS*SII and, since S*S is a positive self-adjoint operator, IIS*SII = II (S*S)MII1/M for every integer M > 1. Now (S*S)M EE...EETj e > 0, if E(m) does not tend to 0. In that case, we can take disjoint intervals [Mk, m'] and zk = mk 2SjTAj +F, AjTA-, .

T = E(Sj+1TSj+1 - SjTSj) = -00

-0-0

-00

-OC

8.7 Cotlar and Stein's lemma

63

The details of the proof are very simple. We give them for the convenience of the reader.

Definition 4. Let L : V --+ V' be a continuous linear operator. We say that L is regular if lim (LSSu, S3v) = 0 00

for u, v E V.

The significance of the definition is that a regular operator L is the sum of its telescoping series (SS+1LS3+1 - SSLS1). It is not hard to see that every weakly continuous operator on L2(IR") is regular. On

the other hand, it is easy to give examples of operators which are not regular.

The theorem of David and Journe then follows from

Proposition 3. If T satisfies the hypotheses of Theorem 2 and if, further, T(1) = tT(1) = 0, then the lemma of Cotlar and Stein applies to each of the three series 0c

00

Co

EA3TS3,

ES2Tt3,

and

1: t3Tt,

.

-cc

In fact, a more precise result is valid: the condition T(1) = 0 together with the weak continuity of T is enough to deal with the first series; 1T(1) = 0 is what is needed for the second; the final series requires only the weak continuity of T. We concentrate our efforts on the first series. Let r1j = 83+1- 03. The kernel T3 (x, y) of the operator o?TSS is calculated by the same rule as the coefficients of a product of three matrices. We have (7.10)

MX, y) = I:E rl3 (x - u)S(u, v)03 (v - y) dudv

where S(u, v) is the distribution-kernel of T. We put and 0,(V) (v) = 03(v - y), so that we also have (7.11)

773 (x - u)

T3 (x, y) = (T83v), 71W)

If the distance from x toy is not greater than 4-2-3, the estimates are a consequence of the weak continuity of T, whereas, if Ix - yJ > 4.2-', we can replace S(u, v) by the kernel K(u, v) corresponding to T. From here, (7.4) can be obtained by the usual calculations. The verification of (7.5) and (7.6) is simple and left to the reader. In equation (7.7), the integral is T3 (1) = A3TS9 (1) = A3T(1) = OJ (0) = 0. This formal calculation is easy to justify and we leave that task to the reader. Verifying (7.8) is even more straightforward.

8 The T(1) theorem

64

8 Other formulations of the T(1) theorem To deal with certain applications, it is convenient to have several different ways of expressing the necessary and sufficient conditions which appear in the T(1) theorem. We suppose, once and for all, that conditions (2.8), (2.9) and (2.10)

are satisfied by an operator T : V - V' and the corresponding kernel K(x, y). We shall examine, in greater depth, the relationship between T(1), tT(1) and the property of weak continuity.

Proposition 4. Suppose that T and K(.c, y) satisfy (2.8), (2.9) and (2.10) and that T is weakly continuous. Then T(1) and tT(1) belong to the homogeneous Besov space Bo-'. Conversely, if u and v are arbitrary elements of Mfo, there exists an operator T : V --+ V' satisfying properties (2.8), (2.9) and (2.10), which is weakly continuous and such that T(1) = u and tT(1) = v.

To prove the first part, we observe that B°' is the dual of B°'1. r-,

Moreover, the elements of Bo,l have an atomic decomposition using special atoms. The special atoms are constructed from basic functions ip E D(n) whose supports lie in the unit ball jxj < 1 and which satisfy f )(x) dx = 0 Then the "special atoms" are the functions of the form a-"V) (a-1(x - x0)), where a > 0 and xo E R" are arbitrary. After using the isometric invariance of the hypotheses under the action of the group ax + b, everything boils down to showing that the estimate (T(1), V)) < Co holds, where Co depends only on the constants which appear in the hypotheses made about T and on 1101011., for a certain integer q > 1. This estimate is obtained by repeating, word ran

:Yo

'f5

rya

U4'

.ti

for word, the argument used to define T(1). The property of weak continuity appears in the term (T(4a), 1), where tbp E D(IR"), 00(x) = 1,

for IxI3. ..a

To show the converse, we look at the construction of the pseudoproducts used in the proof of Theorem 2. The details are left. to the reader. In the opposite direction. it is sometimes useful to he able to deduce the weak continuity from another condition.

Proposition 5. Suppose that property (2.8) is satisfied by the kernel K(x, y) corresponding to an operator T : V V'. Then the weak Ll continuity of T follows from the continuity of T: Bo'1 coo

Here we can again reduce the problem to estimating I(Tg, f) 1, where f and g are sufficiently regular functions with supports in the unit ball

8.9 Banach algebras of Calder6n-Zygmund operators

65

fin'

Ix) < 1. Consider h(x) = f (x) - f (x - xo), where Ixo1 = 3. Then h c Bl'1 (in fact, h is a special atom) and we get (8.1)

(Tf, g) = (Th, g) + JJK(x,

y)f (y - xo)g(x) do dy.

We estimate the first term, using the continuity of T : Bi'1 - L', and the second term, using (2.8). The continuity of T : B°" -> L' is a consequence of the condition (8.2)

IIT( ,a)112 < C, eon

where ?ia, ), E A, is the orthonormal wavelet basis of compact support of Wavelets and Operators Chapter 3, section 8. Since IIV)a 112 = 1, condition (8.2) expresses a weak form of the L2 continuity of T. To check our assertion, observe first that (8.2) and (2.10) imply (Chapter 7, section 3) that (8.3)

IIT(&a)II1
0, M, is an algebra. The proof is a simple verification. We just need to prove the existence of a constant C(n, -y) such that, for each A0 E A and Al E A, (9.5)

E U-y(Ao, A)wy(A, A1) < C(n,'y)u-r(Ao, Al) . AEA

With the obvious notation, we can distinguish three partial suns in

(9.5), corresponding to j > j1 > jo, jr > j > jo and ji > jo > j (by symmetry, we may suppose that ji > jo). In the first case, we note that, if 0 < 17 < E < 1 and x, y E 1[8n, then r7kl)-n-7 (1 + Ix - Ekl)-n-~(1 + l y kEZ' jo > j can be dealt with similarly. The situation where ji > j > jo is more interesting: we then need to observe that ji)2)-1

(1 + (j - jo)2)-1(1 + (j -

< C(1 + (j' - jo)2)-1

h ?i?io

Omitting these terms would have introduced a "logarithmic" singularity. Let V)a, A E A, be a wavelet basis arising out of a multiresolution approximation of regularity r > -y. We define (9pM.y to be the set of

8 The T(1) theorem

68

operators T whose matrices, with respect to the (orthonormal) wavelet basis, belong to M.y. We need to show that this definition does not depend on the particular choice of wavelet basis. To do this, let Via be another orthonormal wavelet basis of regularity r > y. Then the matrix M = (m(A, A')) of the unitary operator defined by U(P,,) = ,\, belongs zlia.) and the verification of (9.4) is to My, because m(A, A') _ immediate (and similar to that carried out in section 5). From now on, we consider only the case where 0 < y < 1. We let .A.y C A denote the collection of Calderon-Zygmund operators satisfying (2.8), (2.9) and (2.10) with exponent -y and such that T(1) = tT(1) = 0. Then we have

Theorem 3. ff 0< y< 1 and T E OpM.r, then T E A.. Conversely, if T E A., and if 0 < y' < y, then T E OpM,, . V"!

In one direction, (3.1) gives T E OpM.y', for 0 < -y' < y. In the other, we must verify, for T E OpM,, (9.6) (9.7)

I K(x, y)I 5 Coix - yl-' , IK(x', y) - K(x. y) I 1 be an integer such that the support of the wavelet V),\ is contained in mQ(A) = mQ(j. k). In the last two series, j, k and 1 are connected by the conditions x E mQ(j,k) and y E mQ(j,1). The product 2-jlk-1I is thus of order of magnitude Iy-xI, while 2("°+'')3(1+ Ik 1I)-'x-7 is essentially constant and equal to Ix - yl-n-'r. Once this estimate has been done, we sum over j, which gets rid of the term (1 + (j' _j)2)- 1. Finally, the sum over j' gives Iy' - yIT. We have established (9.8). The estimate (9.7) follows by symmetry. We now show that T(1) = 'T(1) = 0. Let A,,, he an increasing sequence of finite subsets of A whose union is A and let Tm be the = a(A, A'), if corresponding truncated operators, defined by (A, A') E Am x Am, and am (,\, A') = 0, otherwise. Then T,,,(1) = 0 and 'T,,,,(1) = 0: for once, these equations can be interpreted naively. The

8 The T (I) theorem

70

operators T,,, form a bounded set of Calderou-Zygmund operators, to which we apply Lemma 3 of Chapter 7 (section 3). Passing to the limit,

we getT(1)=tT(1)=0. The operators T E OpM j have some noteworthy continuity properties which we would not normally expect of Calder6n Zygmund operators. For quite general y > 0 we have

every T E QpM7 is a Theorem 4. For all y > 0 and each s E bounded operator on the homogeneous Besov space B. Once again, the verification is purely numerical. We work with the wavelet basis 0a, A E A, which comes from the Littlewood-Paley multiresolution approximation (Wavelets and Operators, Chapter 3, section 2). We then use the characterization of the Besov spaces given by (10.5) of Wavelets and Operators, Chapter 6, namely, EC(A)v',\(x) E Bp,q if and only if I/P Ie(A)IP2nj(P/2-1)

(9-9)

= 2-s3e3

and

Ej E Pp.

SEA,,

still fulfils conSo it is enough to show that ry(a) = KWeA dition (9.9), when a(A, A') satisfies (9.4) and (9.9) holds for (A), A E A. To achieve this, we use the following lemma.

Lemma 10. For y > 0 and n > 1, there exists a constant C(-y, n) such that, if 0 < E < 1, 1 < p < oo, and 1/p + 1 /p' = 1, then the norm of the matrix AE = (1 + Ik - e1I)-"-" as an operator on 1P(Z") is not greater

than C(-y, fl)'/P' . In order to prove Lemma 10, we first consider the case p = 1 and p' = oc. Then the matrices AE are uniformly bounded on l1(Z'), because

>k(1 + Ik - Ell)-"-'I < C(-y, n). The other extreme case is p = oo and p' = 1. This time, we observe that >i E"(1 + Ik - Ell) _n_7 < C(-y, n), because the series is a Riemalm sum of a convergent integral. The general case of Lemma 10 is obtained by interpolation between the two extreme cases. We return to Theorem 4. Define the weighted coefficients xE(j, k) by xe(3, k) =

C())2n3(1/2-1/P)2s2

,

where A = 2_j(k+E) and E = (El/2,...e"/2), with (El,...) Ell) E {0,1}" \ ((0..... 0)}. With this notation, the condition for belonging to the Besov space becomes (Fk Ixe(j, k)IP)1/P E 1 (Z"). Similarly, we define yE(j,k), using ry(a) and, to simplify the notation, we omit the

8.10 Banach spaces of Calderon-Zygmund operators

71

subscript e. All this gives

y(j,k) _

JQ(3,3',k,k')x(j',k')

j'

k' EE(...)+E>(...)=u(j,k)+v(9,k).

j'?j k'

3' lP(Z") is uniformly bounded for 1 < e and 1 < p < oc. We conclude,

-s+n/p>0.

10 Banach spaces of Calderon-Zygmund operators Let H be a separable Hilbert space. A bounded sequence T,,, of continuous operators on H converges weakly to T : H -' H if (Tx, y) (Tx, y), for all x, y E H. It is well-known that, given a bounded sequence T,n.: H - H. we can extract a subsequence T,,,.,, which converges weakly to some operator T : H -+ H. This weak compactness theorem is a special case of the following general result.

Lemma 11. Let E he a separable Banach space. Then the unit ball B of the dual space E* of E is a topological space which is metrizable and compact for the weak topology a(E*, E).

For the case of the unit ball B of £(H, H), we let E denote the tensor product H®,TH. Then E* is exactly £(H, H), which is how the two statements are connected. In Chapter 7, we showed that, for a bounded sequence T,, of CalderdnZygmund operators (that is, T,,, satisfy (2.8), (2.9) and (2.10), with the same exponent -y > 0 and uniform constants Co and CI), the limit T is also a Calderon-Zygmund operator. Will Lemma 11 help us to interpret this remark? We shall see that the answer is "yes", by showing that the Calder6n-Zygmund operators sort themselves, in a natural way, into a family of Banach spaces L-,, 0 < -y:5 1, satisfying the following conditions:

(10.1) if (2.8), (2.9) and (2.10) are satisfied by T, then T E c.e, for

8 The T(1) theorem

72

0 < y' < ry, and the norm of T in L.y only depends on the his

constants Co, C1, on the norm of T : L2 - L2 and on -y and ry'; (10.2) if 0 < Y < ry, then L y c L,,; (10.3) if T E L.y, then (2.8), (2.9) and (2.10) are satisfied with the same

r; (10.4) L.y is the dual of a separable space E.y.

o,.

To define & and L.,, we go back to the decomposition T = R + S + N used in the proof of Theorem 1. We recall that R is the pseudo-product with the fraction T(1) E BMO, that S is the transpose of the pseudo-

product with tT(1) and that N satisfies N(1) _'N(1) = 0. We write T E L,y if (and only if) N E (9pM.y. The norm of Tin L.y is (10.5)

IIT(1)IIBMO + II'T(1)II13r4o + II NII oPM, ,

where the last norm is defined to be the infimum of the constants appearing in (9.4). As a Banach space, OpM, is isomorphic to t°°. Indeed, IINpopM., is the norm in 1' (A2) of a(,\, a')/W.y(A, A') and the bounded sequences in question are entirely arbitrary (because za is a Hilbert basis). The Banach space L.y is thus isomorphic to BMO(R") ®BMO(1Rn) e1°°(A2), which is itself the dual of H' (litre) ® H' (1Rn) ® 1' (A2).

What we have done proves the statements (10.1) to (10.4), but it moo.

F'+

might be useful to give the relevant isomorphisms explicitly. To do that, we use the following notation. If E is a Banach space whose dual is E*, if U E E and v E E*, then u®v e L(E, E) is defined by (u.®v)(x) = v(x)u,

for allxEE. "o'

['+

We also need the idea of unconditional basis of a dual space E*. Let E be a separable Banach space with an unconditional basis e3, j E N. Then every vector x E E can be written uniquely as x = aoeo+a,el +- and this series converges to x commutatively. The scalars a, are given ICI

by a3 = (e?, x), where (-, -) denotes the bilinear form which encapsulates

the duality between E* and E. Let x* E E*. We define ,3i = (x*, ej) and it follows that, for all y E E, lo(e0*,y)+...+,(1?(ej*,y)+... (10.6) (x*,y)= converges to x* in the In other words, the series ,Qoeo + . - - +,Q3e3 +

VI-

o(E*, E) topology. Further, there exists a constant C > 1, such that all the series \o(3oeo + - + AJ,O?e; + - -, with Ian I < 1, converge, in the v(E*, E) topology, to elements y* E E* satisfying II y*IIE. < CIIx*IIE.. We can then state Theorem 5. If O,, is given by (6.6), then the union of the three families '.1

b)®Ba,AEA,Ba®aPa,aEA,and ?ia®zp>,'.(A,A')EA2,forms an unconditional basis of L.,, where the norm is defined by (10.5).

8.11 Variations on the pseudo-product

73

In other words, every operator T E Gy can be written uniquely as

(10.7) T = E a(A)?LA®9A+E AEA

Q(a)6A®1JA+E E y(A,

ACA

A

Al

Even more explicitly, (10.7) means that the distribution kernel S(x, y) of T is given by (10.8)

A(y}

a(A) A(x)9A(y) +

S(x, y) =

AEA

AEA

+ E E y(,\, )7GA(x) A' (y) A

Al

Further, if A,, in E N, is an increasing sequence of finite subsets of A whose union is A, then the operators T,,,,, defined by (10.7), but with A replaced by form a bounded sequence of Calderon-Zygmund operators whose limit is T. Given this information alone, it is easy to identify the three series

which appear in (10.7). Indeed, we have T,,,(1) _ and thus, by a simple passage to the limit, T(1) _ >AEA a(a*A(x). Thus, the first series on the right-hand side of (10.7) is the pseudoproduct with T(1). The second is the transpose of the pseudo-product with 'T(l) and the last series is the operator N E OpMy. It is worth noting that the conditions on the coefficients of (10.7) apply to their absolute values (this is a general consequence of the definition of an unconditional basis). Conditions (6.3) and (6.4) apply to a(A) and 13(A) and we have condition (9.4) for y(A, X).

11 Variations on the pseudo-product The proof we have given of the T(1) theorem is based on a bilinear operation 7r : BMO x L2 - L2 such that, for every function b E BMO, the operator f --+ 7r(b, f) is a Calderdn-Zygmund operator. Explicitly, (11.1)

ir(b,f) _

(b,V,A)(f,eA) A AEA

We have called this operation a pseudo-product. This is why. If b E LP(Rn) and f E LQ(R ), 1 < p, q < oo, then 7r(b, f) belongs to L', whexe 1/r = 1/p + 1/q. Moreover, we can show that, when b and f are both in L2(Rn), then ir(b, f) belongs to the Stein and Weiss space Hl. To see this, we let g be a function in BMO and try to show that CIIbII2IIf II2II9JIBMO is an upper bound for I (Tr(b, f ), g) I. In fact, we can

see immediately that (7r(b, f ), g) = (b, 7r(g, f )) and we then apply the initial description of 7r. The meaning of the above is that the pseudo-product improves on the

8 The T(1) theorem

74

pip

fig'

ado

2.9

usual product as far as Holder's inequality is concerned: the product of two functions in L2(lR") is an Ll and not an H1 function. Similarly, the product of a BMO function by an L2 function does not, in general, belong to L2. These new algebraic operations on functions are one of the revolutionary aspects of Calderon's achievements. Let us recall how Calderon's argument goes. We let lIP, 0 < p < oo, denote the space of functions which are holomorphic in the upper half-plane and satisfy L].

00

sup

If (x + iy) I1 dx < oo.

`''

v>o f-00 If f E TIP and g E TIP, then Calderon defines a third function h(z) holomorphic on sm z > 0, by h'(z) = f (z)g' (z) and h(ioo) = 0. Since the derivative of f g is given by f g' + f'g, we can think of h(z) as one "half' of the product of f and g: the other half would he given by defining h'(z) = f'(z)g(z). In 1965, Calderon showed that his pseudo-product behaved like the usual product. In particular, his proof of the L2 continuity of the first commutator (whose distribution kernel is PV((A(x) - A(y))/(x - y)2), "54

A' E L°°(R)) relies on the inequality IIhII 15 C'IIf1I2IIgII2 This procedure

coo

'++

can be found again in the proof of the T(1) theorem, which relies on the continuity of the pseudo-product. Another operation related to the pseudo-product is J.M. Bony's wellknown paraproduct ([161) which we shall encounter once more towards the end of this volume. To define the paraproduct, we let -y(x) denote a radial function, in 1 if the Schwartz space S(R"), whose Fouriex transform satisfies cad

Iei51and y(C)=0if ICI >3/2. For all jEZ,we let .'3

the operator of convolution with 2"jy(2jx) and we put Oj = S.7+1 - S3. t-!

The decomposition 1 = E ' Oj is the Littlewood-Paley decompoIf f belongs to L2, the support of the Fourier transform of f j = /(f) is contained in the annulus 1 j defined by 2j _< ICI < 3.2j.

sition.

Bony's paraproduct is the bilinear operation defined by B(f, 9) _ E00Sj-1(.f )Oj (g) 00

The difference between the subscripts is no accident. Its effect is that the product hj = Sj _, (f )Oj (g) has a good "frequency localization". [n'

`".

sw.

To see this, we note that, if the spectrum (that is, the support of the Fourier transform) of a function u is a compact set A and if that of v is a closed set B, then that of the product uv is contained in A + B. On applying this remark to hj, we find that the support of h3 is contained

8.11 Variations on the pseudo-product

75

III

^'T

in (1/4)23 < ICI < (154)23. The frequency localization ensures that the h3 are orthogonal when j - 0 (mod 4), or ..., or j - 3 (mod 4). Thus 1/2

IIB(f,9)I12 0, and. lastly, JL(x, z)dv(z) = 0,

where da was the usual surface measure of the unit sphere Izi = 1 in R. The operator T was then defined by T f (x) = PV f L(x, x - y)f (y) dy. The latter class consists of those operators which correspond to symbols E C°°(Rn x Rn \ {0}) satisfying for It l = IOov(x, C)I 0.

It is not necessary to suppose that the mean of the symbol on Itl = 1 is zero, as long as we allow the inclusion of symbols which do not depend on t (these correspond to the operators of pointwise multiplication by bounded functions of x). The relationship between the symbol a(x, C) and the operator T is given by T f (x) = Basically, T(e'x'{) = f e a(x, ) f a(x, C)e'x-£ and this operation is the analogue of amplitude modulation in radio detection. Finally, kernel and symbol are related by (1.1)

PV / L(x, y)e-t£"' dy = a(x, t)

(here, the Fourier transform is to be understood in the sense of distributions) .

The relationship (1.1), discovered by Calderon and Zygmund in the

9 Examples of Calderon-Zygmund operators

78

1950s, opened the way to all later developments in which the pseudo-

Q..

.,,

Obi

differential operators were defined using algebras of symbols, without reference to any kernels. After the golden age just described, the two points

1111

.]'

c.,

of view diverged: Kohn and Nirenberg, for their part, and Hormander, for his, systematically favoured the definition of pseudo-differential operators by symbols. Research on kernels remained very active in the school of Calderon and Zygmund and led to what we now call the "CalderonZygmund operators." It remains to be seen whether the operators in question can be defined by symbols satisfying simple conditions of regularity and rate of growth at infinity. Unfortunately, this is not the case for the set C of the operators of Calderon's programme. However, there does exist an algebra A... C C of operators which can be described, either by their kernels, or by their symbols, or by the matrices of their coefficients in a wavelet basis. For example, suppose we start with symbols satisfying the illicit estimates (1.2) -°C

We know that such conditions do not give operators which are bounded on L2(Rn). In the light of the T(1) theorem, however, we can see that

it will be sufficient to require that the symbols of T and its adjoint T* satisfy (1.2). The set A, of these operators is a subalgebra of L(L2(1[ln), L2(R" )). One of the purposes of this chapter is to show that

.C'

'ti

the algebra A, can be characterized just as well by conditions on the distribution-kernels of the operators. We may also characterize the operators T E A. by their matrices in the orthonormal basis of Littlewood-Paley wavelets, and this property enables us to investigate the symbolic calculus. This algebra is unlike the usual algebras of pseudo-differential operators, in that there exists an operator T E A,, whose inverse does not belong to Aa, even though tip

.x°

T is an isomorphism on L2(l[$n).

r']

The operators T E A,,, arise when we use wavelets as unconditional bases in classical spaces of functions or distributions. If the orthonormal basis 4 , A E A, is an unconditional basis of B, every bounded operator on L2(lR') which is diagonal with respect to the basis rba can be extended COO

as a continuous linear operator on B. These are the operators which

belong to A. 'L7

The second group of examples that we deal with in this chapter cannot be described in the usual language of pseudo-differential operators. In the course of Chapter 13, we shall see that we can apQUA

9.2 Pseudo-differential operators and Calder6n-Zygrnund operators 79

proach these examples by extending that language to multilinear operators. What we are discussing are the commutators ri = [A1, Li], r2 = [Al, [A2, L2]], ... , rk = [Ai, [A2, -, [Ak, Lk]] ...], where the L; mss.

^J+

L1.

are classical pseudo-differential operators of order j and the A; are operators of pointwise multiplication by bounded functions aj (.e) of limited regularity. Calderon's programme consists of exploring the algebras of operators containing the usual pseudo-differential operators and the operators of multiplication by functions which are. only slightly regular. We thus look for conditions of minimal regularity on the functions aJ (x) so that the commutators r1, ... , rk, ... are bounded on L2(1R"), whatever the choice of operators L1, .... Lk,..., of order 1, ... , k,... . By choosing Lk to be a differential operator of order k with constant coefficients, we see that the as (x) must, necessarily, be Lipschitz functions. The fact that this necessary condition is also sufficient is one of the nicest applications of the T(1) theorem. The final example of a Calderon-Zygmund operator that we present in this chapter is the Cauchy kernel on a Lipschitz curve and the operators that can be constructed therefrom by the method of rotations of Calderon and Zyginund. We shall study the Cauchy kernel more systematically in Chapter 12, but here we want to show that its continuity is a consequence of the T(1) theorem and some new real-variable methods introduced by G. David and then simplified by T. Murai ([1931, [1941). G-^

'T'

,77

.t7

LL'

+-,

2 Pseudo-differential operators and Calderdn-Zygmund operators Cs'

_.I

Let E L°°(RT x R") and let a(x,D) : S(1R") - C°°(]l ) be the operator defined by (2.1)

a(x. D)f (x) =

(27r)n,

JeY(x, C)f (C) dd .

If R is the translation operator defined by (R f) (.r) = f (x - xo), then R-la(x, D)R = r(x, D), where r(x, t;) = a(x + xo, l;). If, now, &,

a > 0 is the dilation operator defined by (8n f)(x) = f (a 1.r), then 8Q ia(x, D)6. = r(x, D), where r(x, C) = a(ax, a-] 0. The hypothesis a(x, C) E LOO (1R" x Rn) is isometrically invariant under these two operations. This enables us to make the following observation.

Lemma 1. If a(x,t;) E L°O(IRn X W'), then the operator a(x, D) is `p'

weakly continuous on L2(IR").

9 Examples of Calderon-Zygmund operators

80

t'.

Indeed, let f be a CQ function, q > n/2, with compact support. We first show that E L'(R"). We suppose that q is an integer: then x° f (C) belongs to L2(Rn) for Ial < q, since &'f is continuous with compact support. So (1 + ICIQ)!(C) E L2(Rn)

and it follows that

f(C) E L'(R"), by the Cauchy-Schwarz inequality.

Thus, if f is a C4 function, q > n/2, with support in the unit ball IxJ < 1, it follows from (2.1) that Ia(x,D)f(x)I < C11 fllc4 which implies the weak continuity property. To pass to functions with support in an arbitrary ball, we use the remarks about invariance under the action of the group ax + b, a > 0, b c lR'. t1,

Lemma 2. Suppose that a#,(;) belongs to C°°(R" x lib" \ {0}) and that (2.2)

I

aa(x, )I

/t

a, J3 E Nn.

c(a, 13)ISII'3I-lal . °»,

Then the distribution-kernel S(x, y) of a(x, D) satisfies (2.3)

I8 D S(x, y) I < C'(a, 8)I x - yI-n-lal-Iltl,

a,

E N1'

To see this, we use a technique of approximation by truncated symbols. be a function in D(IRn) which equals 1. if JCJ < 1/2, and Let a equals 0, if IZ;I > 1. We put 01 = 1-¢e and replace a(x,C) by aj(x,t;) = a(x, t;)¢o(j-'t;), j > 1. If Tj = aj (x, D), it is an immediate consequence that, for every pair (f, g) of functions in S(lR'). (2.4)

him (T3 f, q) . (a(x: D) f, g) = .7-00 7-r

The principle of the proof will be to show that the distribution-kernels S3 of T3 satisfy (2.3) uniformly. Since S is the weak limit, of the S3, we obtain (2.3) by passing to the limit.

In what follows, another truncation is going to be useful. The hypothesis on a(x, t;) does not exclude a discontinuity at C = 0. If we want .>'

to avoid this problem, as we shall need to later, we replace a(x, ) by

a(x,0O1(j04o(j-' ) The essential fact is that the estimate (2.2) is not affected by these adjustments. We shall therefore forget about the subscript j in the calculations which follow. Moreover, by the previous considerations we `.1

can suppose that a(x, ) E C°°(1R' x R"), a(x, 0) = 0...., a' a(x. 0) _ 0.... and that, with respect to , the support of a(x, ) is compact.

9.2 Pseudo-differential operators and Calderon-Zygmund operators 81 With these hypotheses in mind, the relationship between symbol and kernel is, evidently, Icy

S(x,y) _

(2.5)

(27r)-

J°' dt .

To prove (2.3), we begin by supposing that Ix - yi = 1 and we first find a bound for IS(x, y)I. We use the partition 1 = oo(t;)Je'e once again by putting + (x-y)o(x,t;)5o(t;)dt;

So(x,y) _ (21)" and

S1 (X, y) =

(2)n fe'_b0o.(x,e)&(e)de.

The required estimate of ISO (x, y) I is obvious and we integrate by parts 2m times to get S1 (x, y)

(-1)'"

f

(a(x, )(b ( )) d Fro

'^^

We now exploit the hypotheses applying to the symbol and the fact that c9'01(t;) = 0, if a # 0 and ICI > 1 or 1t;1 < 1/2. From the latter considerations, we conclude that the term (f'or(x,t;))b1(t;) is the only

f-3

c$1

one for which the range of integration is unbounded, and, for 2m > n+1, the hypotheses allow us to conclude.

mar"

The estimates for I8S(x, y) I are similar, when Ix - yJ = 1. For the general case of y # x, we use the action of the group ax + b, a > 0, b E '. For xO E R' and a > 0, we observe that, if we conjugate the operator T = v(x, D) by the translation by xo and the dilation by a, we pass from the kernel S(x, y) to the kernel a'S(ax + xo, ay + yo) and from the symbol v(x, Z;) to the symbol cr(ax + xo, a-'t;). Now, the hypotheses we have made on the symbol o are not affected by these changes of variables. It is this observation that allows us deduce the general case from that in which Ix - yJ = 1. The fundamental problem is whether the operator T = a(x, D) can be extended as a continuous linear operator on L2(IR"). To this end, we apply the T(1) theorem. The process starts with the following lemma.

Lemma 3. With the hypotheses of Lemma 2, T(1) = 0.

,C3

""1

We establish this result by the method of approximating the symbol v(x, t;) of T by the truncated symbols o , (x, t;) corresponding to the operators T. Clearly we have T, (1) = 0, in the simplest sense. Moreover, the distribution-kernels S; of T, satisfy (2.3) uniformly in j and are weakly continuous, uniformly in j. The T, converge weakly to T in the sense of

9 Examples of Calderon-Zygmund operators

82

(2.4). As a consequence, Tj(1) converges to T(1) in the space B 00 , in the a(B° °°, B°-') topology. The proof of this remark is a straightforward adaptation of the proof of Lemma 3 in Chapter 7. Hence T(1) = 0. We still have to calculate a = tT(1).

We already know that a belongs to B °°. Because of this, a is a continuous linear functional on B°". In order to proceed with the calculations, we need the following lemma.

Lemma 4. The vector space V of functions f E S(R"), whose Fburier transforms are zero in a neighbourhood of 0, is dense in B°" Here is a very simple proof of this remark. We let S denote a continuous linear functional on b?" which satisfies and we choose an arbitrary (S, f) = 0 f o r all f E V. Then S E B representative of S, modulo the affine functions, which is a tempered distribution. We still use S to denote this representative.

On passing to the Fourier transform, we get (S, g) = 0, whenever g E S(IR") is zero in a neighbourhood of 0. Hence the support of the distribution S is 0 and thus S is a polynomial. Since S belongs to BLOC,

it follows that S must he a constant. Hence the class of S in B 00 is zero.

We can now proceed to the calculation of a = 1T(1).

Proposition 1. Let a(x,t;) be a symbol satisfying condition (2.2). Then the integral (27r)-" f e'T-Ca(x, t) dx definers a distribution T on R" \ {0} and 1T(1) is the generalized Fourier transform ofT in the sense

that, for f e V, (2.6)

('T(1)- f) _

J eiT J

f) dx dd .

1-I

We first of all note that the convergence of the integral on the righthand side of (2.6) poses no problem. Indeed. we consider the integral I(x) = f c "'{ f (l;)a(x, t;) dl;, observing that this function is continuous

and O&I-N) at infinity, for every integer N. This last property is established using integration by parts. The distribution T is thus well-defined, and (2.6) makes sense- The proof of the identity (2.6) is found by writing

(1T(1), f) =JT(f) dx =

(2) f 1(x)dx.

09''

We give an example. Let 0a, A E A, be an orthonormal basis of infinitely differentiable wavelets (arising from the Littlewood-Paley multiresolution approxdmation). Let 0 E V(lll;") be a function of mean 1 and,

for A = 2-j (k + e/2), k E Z", c = let us put BA(x) = 2"j0(23x - k).

e") E {0,1}" \ {(0.....O)},

9.2 Pseudo-diferentw.l operators and Calderon-Zygmund operators 83 Consider the operator T defined by (2.7)

T f (x) _ > 0,\ (x) (a, VGA) (f ipA) . AEA

where aEB°m . The symbol a(x, (2.8)

a(x, ) = E

of T can be calculated immediately and we get

x(7)0(21 x - k)e

i2-it.(2- x-k)

moo'

d1,

where a(A) = 2n1/2 (a, ?P,\) E l°° (A), because a E Br0 are those for which The only values of j for which c12j < c22j, c2 > c1 > 0 and, for fixed j and x, the sum over k is restricted by the compact support of 0. These two remarks let us find a bound f o r I , Y. a (x, ) I and obtain the required estimates. We have T(1) = 0 and tT(1) = a E Bed . Note that in this example, the transpose operator tT is exactly the pseudo-product of a and f. The operator T is continuous on L2(R") if and only if a E BMO. We are about to come to the main definition. If T is an operator,

,ti

defined in the weak sense, and if the distribution-kernel of T satisfies (2.3), then we shall write T(x") = 0, modulo the polynomials of degree < jai = rn, if, for every function 9 E D(]R"), satisfying f x'39(x) dx = 0, for every multi-index 13 of height 1/31 < m, we have f tT(g).c' dx = 0. We note that, outside the support of g(x), we have This can be seen by ob'Tg(x) = f S(y, x)g(y) dy = serving that g = 1 r1=,"+1 olgy, where fy' = (e/(9xl )"y' ... (a/8xn)'Y° O(Ixl-n-",,.-1).

and 9,y E D(1R").

Definition 1. We say that a Calderon-Zygmund operator T belongs to the class A,,. if the following three conditions are satisfied. (2.9) The kernel K(x, y) corresponding to T is infinitely differentiable off the diagonal and satisfies JO K(x, y) I < C(a,13)I x for a, /3 E N". yI-"-I"I-10

(2.10) For all a E lY", T(x") = 0, modulo the polynomials of degree < lal. (2.11) For all a E N", tT(x") = 0, modulo the polynomials of degree < lal. The following theorem provides a characterization of A.

9 Examples of Calderon-Zygmund operators

84

Theorem 1. An operator T belongs to A.,,. if and only if one of the following conditions is satisfied.

(2.12) For every 'y > 0, T belongs to the algebra Op.M y.

(2.13) The symbols of T and tT satisfy condition (2.2). The significance of this characterization is that it relates a description in terms of kernels to one in terms of symbols and to a characterization of the corresponding matrices.

Corollary. The vector space A... is an algebra of operators. Indeed, OpMy is an algebra for each ry > 0. Let us turn to the proof of Theorem 1.

(2.14)

,,a

Essentially, this proof continues our examination of the clams A. which we started in Chapter 8. That is. estimates for the entries of the matrix corresponding to T E A,,. are found by repeating, word for word, the calculations of Chapter 8, but using (2.10) or (2.11) and applying Taylor's formula of order m to expand the fimction f , of Lemma 3 in Chapter 8, about x0. The details are left to the reader. Once we know that T belongs to OpMy, for all -y > 0, it is easy to work out the symbol of T and to check (2.2). the calculation does not involve any difficulties and is left to the reader. Lastly, we suppose that the symbol of T satisfies (2.2). Then Lemma 2 tells us that the kernel K of T satisfies (2.3). We get T(xa) = 0, modulo the polynomials of degree < lal, using the technique of approximating T by T9, which we have already described. Repeating this argument for the transpose of T concludes the proof of Theorem 1. A result similar to Theorem 1 was obtained by G. Bourdaud (1211). Instead of (2.2), we use symbols a(x, l;) E C°°(R" x R") satisfying Ids

«(a,)3)(1+I,I)1IH0i.

Using the terminology due to L. Hormander, c belongs to the class of symbols S° 1(R" x R"). Then we write T E B,,,; if the symbols of both T and its transpose satisfy (2.14). The operators T E B, are characterized by estimates of wavelet coefficients. But, to do this, we need to use the orthonormal basis consisting of via, A E A9, j E N and dk, defined by mk(x) = 0(x - k), k c Z. We note that A is the disjoint union of r0 = Z" and the Ai, j E N and that, by abuse of language, we can write & instead of 0k, for a E I'e. Associated with A E Aj, j E N, are the dyadic cubes Q(a) defined by

23 - k E [0, 1)', where a = 2-i (k + E/2), k E Z", E E E = {0,1 }" \

9.2 Pseudo-differential operators and Calder6n-Zygmund operators 85

{(0, .... 0)}. On the other hand, if A E A0, the corresponding cube is 'u3

simply A + [0,1)}z.

a8,

Lastly, we introduce a distance on the collection of all dyadic cubes of side < 1 which are involved. If Q and R are two of these cubes, we let A(Q, R) denote the greatest lower bound of the numbers .\ > 1 such that AQ contains R and AR contains Q (where AQ has the same centre as Q. but the diameter of AQ is A times the diameter of Q). Then d(Q, R) = 1092 .\(Q, R) is a distance on our set of dyadic cubes. From this, we obtain a metric on A by writing d(,\,.\') = d(Q(A), Q(A')) + d(e, s'), where d(e, e') = 0 or 1, depending on whether e = e' ore 0 E' (e, e E E). The characterization of operators T E B,,, is then given by the following theorem.

.p,

pp+

Theorem 2. Let T : S(R")

''J

S'(IR'S) be an operator which is weakly defined. A necessary and sufficient condition for T to belong to B. is that the matrix a(,\, ,V) = 1.

G`S 04'

Ic a3S(x, y) I < C(N, a, 8) Ix - yI-N as long asIx - yJ > 1.

Having given some motivation, with these examples, for studying the we return to the fundamental problem of the symbolic calalgebra culus in the algebra of operators tip. According to A. Calderon, a sym-

bolic calculus for a Banach algebra B is a continuous homomorphism coo

X : B - C, where C is a Banach algebra which is "simpler" than B and where X has the property that b E B is invertible in B if and only if c = X(b) is invertible in C. The Banach algebras B and C may be non-commutative. .00

BOG

In our case, the algebra B is 13.,, and C is C(L2(lRn), L2(IR' )), the algebra of all continuous operators on L2(lR). The homomorphism X is the continuous injection of B into C. We ask whether this injection

9 2 Pseudo-differential operators and Calderon-Zygmund operators 87

is a symbolic calculus for

This comes down to knowing whether

an operator T E Bo which is invertible as a continuous operator on L2(Rn), has as its inverse an operator T-' E 8,,,. We shall give a decidedly negative answer to this question. We remark that every operator L E B,o is a Calderdn-Zygmund operator and is thus bounded on LP(R'), for 1 < p < cc.

Theorem 3. For every p > 0, there exists an operator T e Bx which is an isomorphism of L2(Rn) with itself, while the inverse T-' is not bounded on LP(R'). coo

This result was first discovered by P. Tchamitchian. We give a different example, due to P.G. Lemarie. 2-a-'Zn \ 2-'Zn and that A can be considered as Recall that A2 =

Nam

N>"

the disjoint union of to = Z' and the Aj, j c N.

We define a mapping 0:A->Aby 0(k)=kif kEI'o=Znand E2-;-'

+ e2-3-') = k2-; + E2-i-2 if A = k2 E A;, E E E. + We note that 6(A) E Aj+1 and that 0 defines an injective mapping of A 0(k2-i

into A.

Let U : L2(Rn) - L2(1Rn) be the partial isometry corresponding to

0, which is defined by U.k = dk, cbk(x) = di(x - k), k E Z". and

.-:

car

U0.\ = U tJo(a). Since the distance from A to 8(A) is bounded as A runs through A, the operator U belongs to B... If z is a complex number with JzI < 1, then T = 1-zU is invertible on

L2(Rn). We shall show that, for p > po('zl), the operator T-' cannot be bounded on LP. Thus T-' cannot be a Calder6n Zygmund operator and, a fortiori, T-1 cannot belong to B,,,). We have T-1(f) _ >o zkTk(f) and, if f = vp,\, this gives T-' Eo zkV)9k(a). It is then a straightforward matter, using Theorem 1 of Wavelets and Operators, Chapter 6, to m1culate the LP norm of this function and to verify that, if JzJ26 > 1, where 6 = n/2 - n/p, then

-

AEI

T-' VLp G+"

When 1 < p < 2, there similarly exists an operator T E A= which is an isomorphism on L2(lRn) but whose inverse T-' is not bounded on Lp.

To see this, we essentially repeat the preceding argument. by defining tar

an operator U by U(Vr,,) = 0, if A f 0(A), and U(*.\) _ V)e-'(a) otherwise. Then, for JzJ < 1, T = 1 - zU is invertible on L2(Rl) and the same calculation as above gives a proof that T-' is not bounded on LP fin'

2n(1/2-1/p)_ when Jzi > The algebras ..4. and B,(, may seem pathological. In fact, A,, is oco

necessarily involved when we verify that the waveletsa, A E A, aris-

"!',

9 Examples of Calderdn-Zygmund operators

88

'S'

.-.

ing from the Littlewood-Paley multiresolution approximation, form an unconditional basis of a classical space B of functions or distributions (such as the Besov spaces). For such a verification, it is essential to consider all continuous operators T on L2(lR) that are diagonalizable, with respect to the orthonormal basis 7pa, and to show that these operators are bounded on the space B. But these operators belong to Ate, which is why we carry out a systematic examination of the continuity of the operators of A,- on various spaces of functions or distributions. (We shall do this in the next chapter.) Further, if V),\ and via, A E A, are unconditional wavelet bases belong-

ing to the Schwartz class S(Rn), the unitary operator U : L2 -~ L2,

defined by U(f).\) = a, A E A, belongs to A. The algebra A, is thus unavoidable, once we become interested in orthonormal bases of wavelets.

The following result is an application of the above remarks. chi

Proposition 2. There exist three fiuctions, '01, 02 and 03 in S(R2) such that the wavelets .-:

23 3(23x-k), x E R2, k E Z2, j E Z 23 V;2 (23 x - k), form an orthonornial basis of L2(R2), but such that there is no r-regular multiresolution approximation (r > 1) from which these wavelets can be obtained. '+'

23 Vii(23x-k),

p5"

A recent result of P.G. Lemarie-Rieusset, completed by P. Auscher, shows that there is no such counter-example in dimension 1. More precisely, let Vi(x) be a function in the Schwartz class S(R) such that 2'/2b(2jx - k), j, k E Z, is an orthonormal basis of L2(R). Then there

+.a

nib

must necessarily be a function fi(x) E S(R) such that the ¢(x-k), k E 7G, 2j/2,0(2Jx together with the - k), j > 0, k E 7G, form an orthonormal basis of L2(R). In other words, in the context of the Schwartz class, in dimension 1, every orthonormal wavelet basis arises from a multiresolution approximation. For the proof of Proposition 2, we start with the orthonormal basis of wavelets -ia, A E A, in dimension 2 constructed by the tensor product method from the Littlewood-Paley multiresolution approximation. We shall apply the unitary operator whose symbol is (/1(l, where C =1;+ivl, in other words, U = RI+iR2i where Rt and R2 are the Riesz transforms. 'fl

Then the example of Proposition 2 is given by the wavelets ba = U(V)a)-To "''

see this, let Vj, j E Z. denote the multiresolution approximation from which the ba are constructed. If there is a multiresolution approximation giving rise to the ia, then that can only be f /j = U (Vj ). We shall show that Vj, j e 7G, is not r-regular if r > 1. Fn'

.V3

9.3 Calderon's improved pseudo-differential calculus

89

We argue by contradiction. Suppose that there exists h E V0, such that h E L2 fl L' and that h(x - k), k E Z2, form an orthonormal basis of V0. Then we have h = dxC/SCI, where x(e, r7) is 27r-periodic in each coo

variable, because Vo = U(Vo) and because of the characterization of FVo. Since the sequence of functions h(x - k), k E Z2, is orthonormal, it follows that E jh(l; + 2kir)12 = 1. But the same condition is satisfied

by 4 so jx(l;,ri)j = 1, almost everywhere. Now, in the construction

v,'

.-:

of the Littlewood-Paley multiresolution approximation, > 0, if < , 'q < 7r. Since h E L' (1R2), h is continuous on the square Se = [--7r, 7r]2 and the same is true for the product x(1;, r))(t + ivy)/ t2 -+q2. We now form the one-parameter family r£, 0 < e < 1, of contours EBS0, where tSo is the oriented boundary of So. We calculate the winding numbers about 0 of the image curves x(E8So). When E > 0 is small + y12 + o(1), where c is a constant of enough, x(C, q) = c(t - ivj)/ modulus 1. Thus the winding number of x(EaSo) is -1 for small enough e. On the other hand, if E = 1, we use the periodicity of x to see that opposite sides of the square are traversed in opposite directions. It is thus clear that the winding number of x(USo) about 0 is 0. On So \ {0}, the function x is continuous and of modulus 1. We now need only note that, for all values of e > 0, 0 f eOSo, so that X(EaSo) is contained in Izi = 1. We have obtained a contradiction. 'C7

'^.

?^C

1-+

'"'

'fl

'C"

3 Commutators and Calderon's improved pseudo-differential calculus coo

f'!

The pseudo-differential calculus is like that mythological bird, the phoenix, which is reborn from its own ashes. Its first birth was at the end of the 1930s, the founding fathers being Giraud ([119]) and Marcinkiewicz ([183]). The second birth took place at the end of the 1950s and clearly benefited from the theory of distributions, developed by Schwartz during the 1940s.

The third birth, or renaissance, is the one to claim our interest. In

Ego

order to deal with linear partial differential equations having coefficients which are only slightly regular and, above all, in order to approach the ..,

problem of the regularity of solutions of non-linear partial differential equations, Calderon decided to make the pseudo-differential calculus include the operators A of pointwise multiplication by functions a(x) which are only slightly regular with respect to x (the precise meaning is given below). Of course, Calderon wanted to keep what had been gained during the previous decades: the classical pseudo-differential operators ...

s.'

'6o E-+

TEOpSi0.

9 Examples of Calderon-Zygmund operators

90

tea,

The difficulty of any pseudo-differential calculus lies in that two comX67

mutative algebras of operators confront each other: the algebra X of operators of pointwise multiplication by functions a(x) of a given regularity (infinite differentiability, in the usual case) and the algebra Y of differential operators with constant coefficients and of those which can be algebraically formed from them (that is, the convolution operators Iii

gyp, +,A

T E Op S174e).

E=+

1'3

Their confrontation is expressed by the fact that the operators S E X do not commute with the operators T E Y. That lack of commutativity means we must calculate the commutators [S, T], where S E X and T E Y. The simplest example comes from Leibniz's rile, which gives the commutator [A, D3], where A is pointwise multiplication by the C' (or Lipschitz) function a(x) and D; = 8/8x3. Then Ay = [A, D3] is the operator of pointwise multiplication by a, (x) = -8a/8x3, which belongs {.,

to L°°(1Rn). spy

55,

iii

Calderdn tried to extend Leibniz's rule to the case [A, T], where A is the preceding operator and T E OpS,,0. In 1965 ([37]) he showed that this commutator was always bounded on L2(lRn) when a(x) is a Lipschitz function and, in a way. this does extend Leibniz's rule. Calderdn's theorem can now be obtained very simply, using the T(l) theorem. We shall present this proof and, following Calderdn, we shall use it to show that there exist algebras of pseudo-differential operators which have minimal reguarity with respect to x. IS,

i~'

1-3

Theorem 4. Let A be the operator of pointwise multiplication by a (fl

Lipschitz- function a(x). Then, for every classical pseudo-differential operator T E OpSI,o of order 1, the commutator [A, T] is a CalderonZygmund operator. Conversely, if [A, T] is bounded on L2(IRn), for T = 8/8x3, 1 < j < n, then 8a/8x3 E L°°(IR') and a(x) is thus a Lipschitz function.

.J1

To prove Theorem 4, we find it convenient to make two simplifications. On the one hand, we may suppose that a(x, 0) = 0, by replacing a(x,1;) by a(.c, l:) - o-(x, 0), if necessary. In terms of the operators, this means that we are altering T = a(x, D) by an operator of pointwise multiplication by a function of x and this does not affect the commutator [A,71 -

.n.

The other simplification is to replace T by En T3D3, where D3 = + tnan(x, t;), 8/8x3. This amounts to writing a(x, l;) = llal (x, l;) + where a3 (x,1;) E S° 0. This can be done, because a(x, 0) = 0, which enables us to use classical results on ideals of differentiable functions

9.3 Calderdn's improved pseudo-differential calculus

91

(in fact, we need to return to the proofs in order to get the necessary

estimates on the aj (x, )). Let us first set out the formal structure of the proof. The following considerations will be rigorously justified later. Let Q C Rn x R be the open set y :It x. The restriction to fl of the distribution-kernel S(.r, y) of T is an infinitely differentiable function such that yl-n-l-lal-ICI 1O'& S(x, 9)1 < C(a, /3)Ix -

As a consequence, the restriction K(x, y) to f of the distribution kernel of [A,T] is (a(x) - a(y))S(x,y), and satisfies

IK(., y)I < CIIVall.Ix - yl-n , I ((9/Ox.i)K(x, y)I Ila,u1l2 < dR""2IlVail00 Similarly, n

IITA(u)112 II0(au)/a.rjII2 0, is the operator defined by the Cauchy kernel (2ir)-1(z(x) - z(y))-1, where z(x) = x - iba(x), x E R. The curve I', whose parametric representation is z(x), x E R, is thus the graph of the Lipschitz function -6a(x). The problem has become that of the continuity of the Cauchy kernel (2ir)-1 PV fr f (w)(z - w)-1 dw on L2(I' ds), where ds is the arclength

0-4

measure on the Lipschitz graph IF. This continuity problem will be resolved at the end of this chapter but a further, systematic, account will be given in Chapter 12. Calderon exploited this relationship, between higher order commutators and complex analysis, to the utmost ([39]). The relationship with complex analysis disappears entirely once we work in R" and replace each DkH by a convolution operator Tk whose symbol r(e), # 0, satisfies the following estimates of "homogeneous fir"

type" :

(5.2)

lO"rwi< C(a)I0;-I°I

where

i 0 and a E N.

Ill

We again let A,,-, Ak be the operators of pointwise multiplication by Lipschitz functions a1(x), ... , ak(x). With this notation, we have Theorem 8. The iterated commutators I'k = [A1i [A2, ... , [Ak, TkJ ...JJ are all CaIderdn-Zygmund operators and the norms jjI'kIj of IPA; : L2 -L2 satisfy (5.3)

IIFkII S C(k, n, r)flVa1IIm ... IIVakII, ,

°'i

where the constant C(k, n, r) depends, in fact, only on the constants C(a) of (5.2).

C3'

The structure of this statement allows us to use a standard technique of approximation to Tk in order to prove the theorem. To do this, we replace the symbol r(C) by the symbols X E D(R") equals 1 in a neighbourhood of the origin. The symbols rm satisfy (5.2) uniformly in m, which will let us pass to the limit, once the theorem has been proved under the supplementary hypothesis that r(C) E D(PJ') and that r(C) is zero in a neighbourhood of the origin. The kernel K(x, y) of Tk is a convolution kernel. Qualitatively, K(x, y) is an infinitely differentiable function which is O(Iy - xI-N) for every N 1, as ly - xj tends to infinity. Quantitatively, we have boy

(5.4)

jiTOK(x, y)l < C(a, $)l x So the kernel L(x, y) of rk is (a1(x)-a1(y)) (ak(z)-ak(y))K(x, y) and it is clear that L(x, y) satisfies the Calderon-Zygmund estimates. To prove that I'k is continuous on L2(W'), we use induction on k and apply the T(1) theorem.

98

9 Examples of Calderon-Zygmund operators

We first show that Fk is weakly continuous. To that end, we calculate Fk(f ), where f is a C1 function of compact support. The calculation is done by writing Tk = Sk1)D1 + + Sk" 1 D", where the symbols of the Sk') satisfy (5.2), with k-1 instead of k, and where Dj denotes (a/axj). Then, writing Sk't (r - y) for the kernels of the Skjl, we get

Fk(f) _ - f(ai(r)-an(y)) We can then integrate by parts, which gives two kinds of term. In the first kind, w e differentiate o n e o f the a j (x) - a j (y), j = 1, ... , k. This leads to terms of the form r,(2) C 19y; f l .... , J, whose L2 norms can be estimated by the induction hypothesis. The

r k-l

C 49yj f

1

second kind of term is that in which f is differentiated. The kernel we then use is (a,(x) - an(y))... (ak(x) - ak(y))Sk(A(x - y). It is bounded in modulus by CIIValll=... IIVakll.Ir - yl-n+', and the estimate is trivial, because f is a Cl function of compact support. s''

This way of organizing the argument gives a precise formulation of the general definition of Fk (f) when f is a C1 function of compact support. At the same time, it gives the weak continuity of Fk. Exactly the same kind of argument gives rk(1). If we go through the steps above, we get rk (1) in the form of a linear combination of terms

Fk-I(D_,ai), where D., = (a/axj). By the induction hypothesis, these terms are in BIM. Finally, IF,. has exactly the same structure as I'k, except that Tk is replaced by'Tk. So'Fk(1) E BMO.

6 Takafumi Murai's proof that the Cauchy kernel is L2 continuous We return to dimension 1 and to the "historical" Calderon commutators. We begin with a Lipschitz function A : R C, with which we associate the distributions PV((A(x) - A(y))'/(x - y)k+l) belonging

to S'(R2). So we arrive at the operators Ik : S(111) -. S'(111) whose distribution-kernels are the distributions we have just described. Thus

(A(x) - A(y))k 9(x)f (y) dy dx . (r - y)k+1 whenever f and g are C1 functions of compact support. The L2 continuity of Fk is a consequence of the results of the previous

III

(rk(f), 9) = lim ela

Jj1X-Yj>C

9.6 Murai's proof that the Cauchy kernel is L2 continuous

99

section. From this, we go on to deduce that, for every f E L2(R) and almost all x E IR, 7C'

(A(x) -A(y))k

lim

(x - Y) k+1

Eto J Ix-5I>e

f(y) dy

exists.

By using the general results of Chapter 7 (Cotlar's inequality) we

see that it is enough to prove that the limit exists when f is a C' O

O

o

function of compact support. In that case, we can integrate by parts, observing that (a/ay)(x-y)-k = k(x-y)-k-1, to get, as in the previous section, two kinds of term. One kind involves the operator rk_1 and is dealt with using the induction hypothesis and the fact that Al f E L2. The other kind of term comes from differentiating f and leads to an absolutely convergent integral. The integrals of these terms tend to 0 I

II

H

O

O

w

O

F

o

almost everywhere because leio

I

A(x+e)-A(x-

_

I

A'(x)

2E

w

H

for almost all x E R. So, for every f E L2(R), every k E N and almost all x E 1R', w

w

O

))kf(y)dy-

w

w

rkf(x) =limfix-yl>e (A(x) ejo

(+1

(x - y) k+1 O

We make one final observation: about the growth of the norms Ilrk II O

of the operators I'k : L2(R) -, L2(R). Using the obvious estimates on the size and regularity of the kernel (A(x) - A(y))k/(x - y)k+l, we get IIrkII < vCkIIA'IIk. This method does not give a value for the constant, which comes from two multiplicative factors. The first is due to the presence of a constant C0, which we do not know how to evaluate with any precision, in the statement I

o

(6.1)

IITII A almost everywhere on I and BI(x) > A(x) on I.

Let E C [a, b] he the compact set {x : BI(x) = A(x)} and let Sl = [a, b] \ E. It is worth noting that a E E. On the other hand, b need not belong to E. This means that Il (assuming it is non-empty) consists of open disjoint intervals (ak, bk) and, possibly, an interval (a', b]. Having established the above notation, here is the rising sun lemma. Lemma 7. BI (x) = Ax + ck on every connected component IL of fl and the measure IfZI of 0 satisfies 101 < (M - m)(M - A)-IIII.

To prove the rising sun lemma, we put f (x) = A(x) - Ax and F(x) _

102

9 Examples of Calderdn-Zygmund operators

BI(x) - Ax. Thus F(x) = sup. f (ak) = F(ak), choose ck E (ak, bk) such that f (ck) > f(ak) and dk E (ak- ck] such that f (dk) =

sup{ f (x) : x E [ak, CO. But this implies that f (dk) = F(dk), so that dk E E, which contradicts the definition of (ak, bk). To conclude this part of the proof, we need to consider the case where

r-'

(a', b] is a connected component of 9. Then F(b) > f (b) and F(a') = E a'). We see that F(b) = f (c), for some c E [a', b). If F(b) > f (a'), then a' < c and F(c) = f (c), contrary to the definition of 0. Hence, F(b) = F(a'). To find an upper bound for 191, we observe that A(a) = BI(a). Thus A(b) - A(a) < Br (b) - Br (b) =

J

b B'r (x) d.r

/'a

_ f B',(x)dx+ J B'I(x)dx E

S

< MIEI + All = Mill - (M - A)ISZI We shall use this lemma to examine the behaviour of the kernel EA(x, y) when x and y belong to the same interval I We replace the function A by the function BI of the rising sun lemma (the sequel will

H

show why this is a useful thing to do.) To simplify the notation. we drop the subscript I from Br and get (6.9) IEA(x. y) - EB(x, y)I < M(x -y)-2(dist(.c, E) + dist(y, E)) , because I Br(x) - A(x)I = BI (x) - A(x) < M dist(x, E) . To evaluate the upper hound of the norms of the Calderdn-Zygmund operators TA when 0 < A' < M, we introduce the following estimating function:

u(A, I) = jf EA(x, y) dydx = IITA(Xr)IIL1(r)

-

This was first considered by Journe in a result which anticipated the T(1) theorem. We then write a(A) = sup, IIIa(A, I) and, lastly, put z(M) = sup{a(A) : 0 < A'(x) < M}. (6.10) We aim to prove the following lemmas.

Lemma 8. There exists a constant Co such that (6.11)

IITA(1)IIBMO 0, T(M) < 4r(2M/3) + C, (1 + M) -

(6.12) E-+

These lemmas easily give (6.7). Indeed. we start from the continuity of the operators TA, when IIA'II,,, < 1. The continuity is a direct result of the T(1) theorem. applied to the. Calderon commutators, as we have indicated. Repeated application of the estimate (6.12) gives T(M) < C2(1+M)5, as can be seen by considering the intervals (3/2)k < M < (3/2)k+i cad

Then Lemma 8 provides IITA(1)IIBM0 (1/4)111. But we shall also require the function B(x) to satisfy M/3 < B'(x) < M on I. This will let us replace M by 2M/3. by substituting B'(x) - M/3 for B'(x), and thus =reduce the slope" -

Let us give the details of this algorithm. We write I = [a, b] and

b-a

Al

distinguish between the cases m = A(b) - A(a) > M

-2

and

m=

A(b) - A(a) < M

b-a

2

9 Examples of Colderdn-Zygmund operators

104

a(A, I) = a(A, 1),

0 < A'(x) < M and

A(bb

NIA

In the first case, we apply the rising sun lemma directly, with A = M/3. and get 101 < (3/4)111. In the second case, we replace A(x) by A(x) = Mx - A(x). Then

- a (a) > 2

Fr.

which brings us back to the first case. Applying the rising sun lemma to A, or A, with A = M/3, gives us a function which we call B (rather than BI) to simplify the notation. To

finish, we put b(x) = B(x) - M/3, getting B'(x) E [0,2M/3]. We get u(B, I) = a(B, I) _< r(2M/3)111, by the definitions of r and a. To compare a(A, I) with a(B, 1), we intend to prove the basic inequality

a(A, I) < a(B, I) + E u(A, Ik) + C(1 + M)III, Zlw

(6.14)

k>O

4'b

where the Ik are the connected components of the open set (1 C I 2f~

produced by applying the rising sim lemma to A or A. with A = M/3. Once (6.14) has been established. we can conclude the proof of the lemma. Indeed, a(A,Ik) < a(A)IIkI and E IIkI < (3/4) 111 imply that (6.15)

a(A, I) < r(2M/3)III + 3a(A)III +C(1 + M)III -

We divide both sides of (6.15) by III and take the supremum, over all intervals I, of the left-hand side. This gives MIA

(6.16)

a(A) < r(2M/3) + 3a(A) +C1 + M)

But we know that a(A) is finite, because all the operators TA are Calderon-Zygmund operators. It follows that (6.17)

a(A) < 4r(2M/3) + 4C(1 + M)

and it is enough to take the supremum, over all functions A with 0 < A'(x) < M, of the left-hand side of (6.17) to get (6.12). We still have to prove (6.14). We know that (6-18)

EA(x, y) = EB(x, y) + R(x, y)

where the error term R(x, y) satisfies (6.19)

IR(x, y)I < M(x - y)-2(dist(x, E) + dist(y, E)).

We also know that, on each of the intervals Ik which make up fl. we have B(x) = Ax + ck (with A = M/3) so that, for x and y belonging to the same Ik, we have EB(x, y) = eia(x - y)-n.

9.7 The method of rotations

105

These two properties are enough to prove (6.14). Indeed,

a(A, I) =

EA(x, y) dyl dx

I

J J r

r

If

EB(x, y)

dyl da; +

r

r

R(x, y)

dyl dx

=a(B.I)+n. To estimate the error term y, we let G denote the set of ordered pairs (x, y) E I X I which do not belong to the union of the sets Ik x Ik. In other words, x and y do not belong to the same interval Ik. Then

71 < f f !R(x, y)I dx dy +

E k>o

R(.c, y) dyl dx.

l

rJk

The upper bound f fG I R(x, y) I dx dy < 4M I I l follows immediately from (6.19).

We are left with the terns ak = frk frk R(x, y) dyI dx. To deal with 5°c

them, we go back to (6.18), which gives

,,,If EA(x, y) dyl dx + f If EB (x, y) dyl dx . k ak < k k The first of the integrals is v(Ik) and the second can he calculated, by the remark after (6.19), to give frk I frk (x - y)-1 dyl dx < 7rIIkj, because the Hilbert transform is unitary on L2(Rn).

7 The Calderon-Zygmund method of rotations '.7

By the method of rotations, we can construct Calderbn-Zygmund operators whose actions on L2(R'") are like taking the mean (or Bochner integral), over all directions, of Calderdn-Zygmund operators acting on functions of one real variable. 'r^

We are now in a position to think of the method as the prototype of the transference methods developed systematically by Coifman and {C'

Weiss ([77]). ".s

For example, Calderdn and Zygmund used the method of rotations to establish the continuity of the Riesz transforms R 1 < j < n, on every LP(R' ), 1 < p < oc, as a corollary of the corresponding result for the Hilbert transform. The method of rotations, as well as the other techniques described in this section, is based on the description of operators in terms of their kernels. That is, we start from a function K(x, y) which, to begin with, will belong to L°° (R' x 1R'), to avoid the problems associated with

9 Examples of Calder6n-Zygmund operators

106

singular integrals. We then define the operator T by

Tf(x)=JK(xy)f(y)dy,

(7.1)

where f belongs, for example, to the space E of continuous functions of compact support. We shall let Sn-1 denote the emit sphere in III" and write w,,,_1 for its surface area. Theorem 9. Let K(x, y) E L°° (I[?n x R") define an operator T by (7.1). For each xo E R" and every v E S' 1, let T(.(,,,) be the operator, acting on l inctions of one real variable, whose distribution-kernel is k(s, t; xo, v) = Is - tIi-1K(ao + sv, xo + tv)

for s, t E R.

at p satisfy 1 < p < oo (the extreme values are not excluded), and suppose that the operators T(.(,,,) are uniformly bounded on LP(R) with operator norms (on LP(IR)) not exceeding 1. Then T is bounded on Lp(I[l;n) and the norm of T does not exceed w,,,_ 1 /2.

Put g = T f . We integrate using polar co-ordinates, centre x, to get

9(x) = f K(x,y)f(y)dy =

2f

=

2

,

I TO K(x, x + tv) f (a +

tv)ItI"-1 dt

00

} da(v) JJJ

s,-19v(x)da(v),

where 9v (x) =

fxx+tv)+tt11dt. 00

We shall show that II9v(x)II Lp(R') 0, `."

(7.7)

II2 0,,,)f112 s CIIf1I2.

r-3

Combining (7.6) and (7.7) gives IITEII _< C'. Then the antisymmetry 0, for of the kernel guarantees that (T' f , g) tends to (T f , g), as a all f,g c S(]R'1). The uniform estimate IITEII < C' thus gives the weak convergence of the TE to T E £C(L2, L2).

Here is a remarkable application of this method, due to Calderon. Theorem 11. Let n, m > 1 be integers and suppose that A : 1[in -> 1Rnz is a Lipschitz function. Let F : Rm --+ R be an infinitely differentiable odd function. Then the antisymmetric kernel (7.8)

K(x,y) = F

\

A(x) - A(y)1 Ix - yl J

Ix

- yI-n

defines a bounded operator on L2 (R'). To see this, we apply the method of rotations, which brings us back to

the case n = 1. Next, we observe that Ix-yI-1 F((A(x)-A(y))/Ix-yI) = (x-y)-'F((A(x)-A(y))/(x-y)), because F is an odd function. We then take advantage of the inequality I A(x) - A(y) I < M I x - y I to replace F on the ball Jul _< M of 1R', by an infinitely differentiable periodic function,

of period 4M in each variable. We expand this periodic function as a Fourier series to get, for Jul < M,

F(u) =

a(k)eiak-u

kEZ'

where b = -7r/2M and the a(k) are rapidly decreasing. The kernel (x-y)-1F((A(x)-A(y))/(x-y)) can thus be written as > a(k)Gk(x, y), y)-lezSk-(A(x)-A(y))/(x-y), The norm of the opwhere Gk(x, y) = (x erator Gk defined by the kernel Gk(x, y) does not, therefore, exceed C(1 +bIkIIIA'II,)5 and the series >a(k)Gk is absolutely convergent in ,C(L2, L2).

We finish this section with two applications of Theorem 11, drawn respectively from potential theory and from complex analysis. We shall give more details of the first example in Chapter 15.

Let A : IR" --+ R be a Lipschitz function, let S C R'" be its graph and let d r be the surface measure on S. We start from an electric charge density g E L2(S, dv) and let V(x), x E 1R"1, be the potential created

9.7 The method of rotations

109

by the charge density. We suppose that n > 2. Then (7.9)

V(x) = en fix - yj-f.+19(y) ma(y) -

Now, consider the electromagnetic field arising from the potential V. Ignoring the normalization constants, we write E _ -VV Let 1 denote the vector (0, 0, ... , 0,1) E Rn+1. It is well-known in

electrostatics that E(x) has a discontinuity as it passes through the surface S. For x E S, we set (7.10)

E+(x) = lim E(x + E1) and E_(x) = lim E(x - E1) . Elo

CIO

These limits exist, for almost all x E S. as we shall show in Chapter 15. Finally, the discontinuity of the electromagnetic field is given, for almost

all xES,by (7.11) E+(x) -E_(x) ='Y"9(x)n(x), where n(x) is the upwards (1 - n(x) > 0) unit vector normal to the

surface S. We then have the following result.

Theorem 12. If IIVAII= < M < oo, then, for everyg E L2(S,da), we have (7.12)

(is IE+(x)I2 dc(x) J 1/2 < QM, n) s

/

I

IsI9(x)I2 da(x) //J 1/2

where C(M, n) depends only on the upper bound M and the dimension n.

To prove this result, we use an algorithm to write E+(x) in terms of g(x). The algorithm is that of the Calderon-Zygmund operators and gives

(7.13) E+(x) = ry"9(x)n(x) +PV c f(x - y)I x - yl-n-19(y) da(y) ear

Of course, it is the second term on the right-hand side of (7.13) that is problematic. We deal with it by using the parametric representation of S given by x = (u, A(u)), u E R', and the vectorial kernel (x - y)Ix yl-n-1, restricted to L2(S), becomes A(u) - A(v) u-v (Iu - v12+(A(u) _A(v))2)(n+l)/2 , (In - v12+(A(u)_A(v))2)(n+1)/2 and this then acts on L2(R'). The continuity of this operator is then a consequence of Theorem 11. Stein and Weiss discovered how the Hardy spaces of functions holomorphic in the upper half-plane could be generalized to R" x (0, oo).

110

9 Examples of Calderon-Zygmund operators

They replaced the pair (u(z),v(z)) of real and imaginary parts of a holomorphic function by the gradient of a harmonic function. Identity (7.11) then appears in the guise of a generalization of Plemej's formula, which occurs in the following context. Let r be a rectifiable curve, without double points, in the complex plane parametrized by the arclength s which runs through the whole

real line. We suppose that limR jz(s)I = lim8, lz(.s)1 = oo. Let 121 and 122 be the open sets whose boundaries are F. We define the Hardy spaces H2(111) and H2(122) as the closures in L2(I',ds) of the rational functions P(z)/Q(z) which are zero at infinity and whose poles belong to 122 and 121i respectively.

The problem is then to determine whether L2(r. ds) = H2(111) + H2(122) , where the sum is direct, but not necessarily orthogonal. We shall exandue this question systematically in the course of Chapter 12. Here, we just want to remark that (7.14) is equivalent to the Cauchy kernel's defining a bounded operator on L2(I', ds), by (7.14)

Tf(z)=PV z7r1

1

z- W

f(w)dw.

Using the parametrization given by the arclength. everything reduces to knowing whether (z(s) - z(t))-1 defines a bounded operator on L2(R). A rectifiable curve is a Lavrentiev curve if there exists a constant b > 0 such that, for all s and t, we have Iz(s) - z(t)I > bas - tj. With this definition, we can state

Theorem 13. For every Lavrentiev curve, the Cauchy kernel defines a bounded operator on L2(F, ds) and this operator is a Calderc n-Zygmund operator. To see this, we let F be an infinitely differentiable and odd function on R2, which coincides with 1/z when b < jzj < 1. We can then write 1

F r z(s) - z(t)

` sZ(S) - z(t) and it is enough to apply Theorem 11.

1

t J S- t

10 Operators corresponding to singular integrals: their continuity on Holder and Sobolev spaces

1 Introduction The kernels, IX - yl-"+l, 0 < A < n, whose Fourier transforms in the sense of distributions are c(n, A)1e1-a, define the fractional integration operators. Both the kernels and their Fourier transforms are locally integrable and it is easy to prove, without further ado, that convolution with IxI-"+a is a continuous operator from C9 to CB+a, for all s > 0. The C8 spaces are the homogeneous Holder spaces. For convenience, we once again give their definition.

If 0 < s < 1 then f E C8 when f is Holder of exponent s, that is, when f is a continuous function modulo the constant functions and If(x) - f(y)I 1,when Ix -yI>1(and a,QEN").

coo

Fir

To apply Theorem 1, we must make sure that T(xa) = 0, modulo the polynomials of degree not exceeding a. This condition can be written in the form P. (-r), where Po(x) is a polynomial whose degree is not greater than or. To reduce to this special case, it is enough to write where #i(x) = 0 in a as neighbourhood of 0 and 00 E D(R"). This induces a decomposition a(x, D) = T = To + T1. The operator To is smoothly regularizing and Theorem 1 applies to Ti. In view of (3.1), we can pass from the homogeneous to the inhomogeneous spaces and confirm that T E OpS°,1 is continuous on all inhomogeneous Besov spaces BB'q, where s > 0 and 1 < p, q < oo. The operators in question are, in general, unbounded on L2 (W ). The next example is due to Calderon. Let a : R -> C be a Lipschitz function with IIa'II= < M. We consider the distribution-kernel ..a

K(x,y) = PV((a(x) - a(y))/(x - y)2) and the operator T : D(R) --' .-.

D' (lit) defined by this kernel. Then T is continuous on L2 (R") (Calderon,

1965), but not on the Sobolev space H", when s > 0, or on its homogeneous version b'. By the same token, T is not continuous on C". On the other hand, the distribution-kernel a(x) - a(y) - (x - y)a'(y) = 9 a(x) - a(y) Kl (x, y) = PV

(x_y)2 ay x-y leads to a better operator Ti, in the sense that T1 is bounded on the

Singular integrals on Holder and Sobolev spaces

116

Sobolev space H", for 0 < s < 1 and on the homogeneous Holder space I-+

C", when 0<s21y-xI

Iu-xI- 2aif(u)- f(x)I2du

We conclude the estimate of f f I y1(.r, y) I2Ix - yI -n-2s dx dy by integrat-

ing with respect to y, u and, finally, x to give

'-'

C," JJf(u) - f(x)121u - xI_28 dudx = C"N(f). The case of 92 is similar. We suppose that the support of is contained in Jul < 10 and get

I92(x,y)I 5 C fu-x15101Y-x1 Ix - uI-nIf(u) - f(x)I du. I

We then introduce an exponent f3, such that 0 < Q < Q, and write Ix - ui-n = Ix - uI-n/2+0Ix - uI-n/2-13. After applying the CauchySchwarz inequality, we conclude as above. 193(x,Y)I

1, so if the operator Q(x, D) were continuous on H8(R), s > 0, it would be continuous on L2(R) and vice E-,

versa. But tT(1) = Er(e-Z2-'x - 1) and this function does not belong to BMO. On the other hand, T(1) = 0 modulo the constants and, for -ti

10.6 Continuity on ordinary Sobolev spaces

123

that matter, T(xk) = 0 modulo polynomials of degree not exceeding k, '`3

for allkEN. The selfsame operator T cannot be continuous on the inhomogeneous

Holder spaces C3. Indeed, let 0 E D(R) be a function which is 1 on [-1,1]. For any f E L°O(IR), define g by g(1=) = 0()f(). Then g is in C" for all s > 0. If T were continuous on C8 then T would be continuous

on L°°, because T(f) = T(g). But the continuity of T on LO" would 'L7

imply its continuity on L2 (Chapter 8, Proposition 9). In this example, we see that the behaviour of the kernel of T as l y -xl tends to infinity appears to be bound up with the continuity of T on inhomogeneous spaces. The following result gives the precise relationship.

°,y

Theorem 2. Let T be an operator in 4, -y > 0. Suppose, further, that the kernel K(x, y) of T satisfies the following estimates for l x - yl > 1. (6.1)

IoK(x,y)l 0, put N8(f) = (21r)-"V2(fl f(e)I2Ie128 d)1/2.

Lemma 2. There is a constant C(s, n), s > 0, n _> 1, such that, for every R > 0 and every function f in the Sobolev space H8 whose support is contained in lxl < R, 11f112 Ok f and T1(f) = E T1(8k f ). The support of TI (8k f) is contained in the ball Ix - kI < R and the observations above lead to

IITi(f)IIH= 5 C(> IITi(0kf)II .)1/2 < C'(> IITi(0kf)IIBa)1/2 5C"(EIlekfIIB$)1/2

..q

We still have to reduce the general case to that in which T1(xa) = 0 for IaI < m. We do it in the manner of the T(1) theorem, correcting T1 by an operator which we know to be bounded on C" and H". So we start with a function w(x) E D(R't) whose mean is 1, but which is such that f xaw(x) dx = 0, if 1 < at < M. Let L1 be the operator with kernel 'w(x - y). Clearly L1(xa) = ma(t), for IaI < m. On the other hand, L1 is bounded on the inhomogeneous spaces C" and H", because these spaces are invariant under pointwise multiplication by C'' functions. Lastly, T1 = Ti -L1 satisfies the hypotheses of Theorem 2 and T1(xa) = 0, for IaI < rn. This proof also extends to the case of the inhomogeneous Sobolev

-.r^

spaces WP-', where 1 < p < oo and 0 < s < y. In particular, the E-,

pseudo-differential operators T whose symbols belong to S°'1(IR') are bounded on for every s > 0 and all p c (1, oo).

7 Additional remarks We return to the homogeneous Sobolev space B" = B2'2 and suppose

that 0 < s < 1. Let ry > s and suppose that T is an operator in G.y. We could ask under what conditions T extends to a continuous linear operator on B". By Theorem 1, this happens if T(1) = 0, modulo the constant functions. But this condition is not necessary. Following Stegenga ([2161), we introduce the Banach space E. of functions,, modulo

10.7 Additional remarks

125

tea.

lip

Eon

the constant functions, such that the pseudo-product rr(,l, f) satisfies II ir(p, f) II E' < CIIf IIB. If s = 0, E, coincides with 13MO. We then show ([191]) that the necessary and sufficient condition for T E Lry to extend continuously to b' is that T has to be weakly continuous on L2(R") and that T(1) must belong to E,. Here is a condensed proof of this result. If T E L., and if T is weakly continuous on L2, it is easy to show that ,6 = T(1) belongs to #0",'. Then the operator R0(f) = 7r(/3, f) also is an element of Ly (for each y > 0) and is weakly continuous. The difference So = T - R0 satisfies Sp(1) = 0 and is weakly continuous. Because of this, SR is continuous on B8 (Theorem 1). Thus T is continuous if and only R0 is, which is what we wanted to establish. Let us consider the special case where T is the operator of pointwise multiplication by a function rn(x). The kernel K(x, y) corresponding to T vanishes and T is in Ly. T is continuous on B8 if and only if T is weakly continuous and T(1) E E,. But the weak continuity means that m(x) is in L. The pointwise multipliers of the homogeneous Sobolev space k are thus the functions m(x) E L°° fl E,. This is the statement coo

.p°

!`,

of Stegenga's theorem ([216]). We should remark that E, reduces to {0}

if s > n/2. Here we have only dealt with the case 0 < s < 1. The case s > 1 remains open.

11 The T(b) theorem

1 Introduction We have already mentioned that David and Jour n's remarkable T(1) theorem has a drawback. It cannot be used to show that. the Cauchy kernel on a Lipschitz curve is L2 continuous. The criterion does not work 8E-

for the following reason: if z(x) = x+ia(x) and -M < a'(x) < M, then the integral PV ff'.,,(z(x) - z(y))-1 dy is not a BMO(R) function that we already know. This is because, in complex analysis, it is natural to integrate with respect to the measure dz(y) and not the measure dy. Indeed f ° (z(x) - z(y))`' dz(y) = 0, modulo the constant. functions. This extremely simple remark leads to the following conjecture. Let b(x) E L°°(Rn) be a function with lReb(x) > 1 almost everywhere. Let K(x, y) be an antisymmetric kernel such that I K(x, y)1 < CO Ix -

Via.

VII

yI n and J(8/8xj)K(x,y)J < C1Ix - yj-n-1, for 1 < j < n. Suppose

't7

further that f K(x, y)b(y) dy is in BMO(Rn). Then PV K(x, y) defines an operator which is continuous on L2(lR' ). This would mean that the continuity of a singular integral operatorcould be established by just one function b, as long as We b > 1. To avoid the circular argument of assuming that the operator under consideration is defined on arbitrary functions b E L°°, when we do not even know whether the operator is a Calderon Zygmund operator, we restrict our attention to kernels K(x, y) which satisfy the extra, qualitative condition (1 + Ix - yl)n+1K(x, y) E L°°(Rn x ll$n). In 1984, G. David, J.L. Journe and S. Semmes proved this conjecture,

11.2 Statement of the fundamental geometric theorem

127

which was the first formulation of the T(b) theorem. P. Tchamitchian later found another, much more geometrical, version in terms of wavelets

adapted to the measure b(x) dx. The T(b) theorem then arises as a corollary of the existence of the wavelet basis and this approach runs parallel to the argument we used to prove the T(1) theorem.

2 Statement of the fundamental geometric theorem Let Vj, j E Z, be a multiresolution approximation of L2(lR") which is r-regular, for some r > 1. We say that V1 is real if the function 0 and the wavelets,k corresponding to the multiresolution approximation are real-valued.

Even though it is not strictly necessary, we shall suppose that there is an exponent -y > 0 and that there is a constant C > 0 such that. (2.1) if jal < r. I8a-0(x)I S Ce-'rl=I It follows that, for Ial < r, we also have (2.2)

C'2"h/22uIaIe-7'I2s=-kl

Ia" VG;k(x)I
0. These hypotheses are satisfied in many examples. (2.1) and (2.2) could be replaced by the usual hypotheses of fast decrease at infinity, but then the proofs below would become unnecessarily complicated. We fix a function b(x) E L-(WL), with We b(x) > 1. and define the symmetric bilinear form B : L2(1Rn) x L2(W') -* C, by

B(f,g) =

(2.3)

J]fin

f(x)g(x)b(x) dx.

We further define f3(f, g) = B(f, g). The bilinear form B has the following important property: WJ2eB(f,f)? 11f112

(2.4)

21

(OD

'L1

which has been studied systematically by T. Kato in [151]. This property replaces the usual (strict) positivity properties of the sesquilinear form defining a Hilbert structure. We let W J C V1+j denote the linear subspace defined by '3(f, g) = 0 for all g E V;. If b(x) = 1, this reduces to the subspace Wj which we used in Chapter 2 of Wavelets and Operators.

11 The T(b) theorem

128

mow;

Ins

w''

Theorem 1. L2 (III") is the direct sum of the linear subspaces W1, j E Z. This means, firstly, that there are constants C2 > Cl > 0, such that, for every sequence f y E W j satisfying F "Q00 f II2 < oo, 1/2

°vlJ

0

(2.s)

C,

1:11f 12

ao(t11f

_ 0. T is b-ac_cretive, for b > 0, if We (T(x), x) > blIx112 This is the same as writing T + T* > bI (where I denotes the identity operator) in the sense of self-adjoint operators. A b-accretive operator is an isomorphism of H. Indeed `'"

o1,

6lIxII2 < ate (T(x),x) < IIRe (T(x),x)I < IIT(x)1IIIx1I .

.,,

"'9

:°..

-a'

So bIIxll < IIT(x)II and, similarly, bUIxDI < IIT*(x)II. It follows that T is an isomorphism. Let Q : H x H -> C be a bicontinuous sesquilinear form. This means: that 1 /(x, y)I < CIIxII IIyII; that, for each yo, [3(x, yo) is a continuous linear form on H; and that, for each xo. 13(xo, y) is a continuous linear form on H. We say that 13 is b-accretive if Re ,8(x. x) > blIxII2 for every x E H. If

11.3 Operators and accretive forms

129

,Q is 6-accretive, there exists an operator T : H - H which is 6-accretive and such that ,O(x, y) = (T(x), y) for all x, y E H. This remark gives us the following lemma.

Lemma 1. For each continuous linear form l : H -> C, there exists a unique element a E H such that 1(x) = 3(x, a). Indeed, by the Riesz representation theorem 1(x) _ (x, b) = (T(x), a), where T*(a) = b. This equation has a solution, since T* is an isomorphism.

The following result helps with Theorem 1.

Proposition 1. Let H be a Hilbert space, let /3 : H x H -> C be a 6accretive sesquilinear form (b > 0) and let V be a closed linear subspace of H. Define W by (3.1)

W={xEH:/3(x,y)=0 for ally E V}.

Then H is the direct sum of V and W. Moreover, the operator norm of the oblique projection of H on V parallel to W depends only on 6 and the constant involved in the continuity of/3.

We shall check that every c E H can be written uniquely as c = a + b, where a E V and b E W. Then /3(c, v) = (i(a, v) for every v E V. To find a E V, we apply Lemma 1 to the linear form 0(c, v) = 1(v), defined on V. (We have to replace H by V and restrict /3 to V x V. The restricted sesquilinear form is still 6-accretive.) Lemma 1 gives us a unique a E V such that 1(v) = /3(a, v). This concludes the proof of Proposition 1. Returning to Theorem 1, we thus get V.7+1 = V? + Wj and the norms of the oblique projections Pj : Vj+1 -> Vj, parallel to Wj, are uniformly bounded.

We proceed to a group of remarks about the symbolic calculus of accretive operators. Following T. Kato ([151]), we have

Proposition 2. Let T : H -' H be a 6-accretive operator. There exists a unique accretive operator S such that S2 = T. We denote this operator by T1/2. It is 6'-accretive for a certain 6' > 0, which depends only on 6 and IITII. Finally, the inverse T-1/2 of T1/2 is given by (3.2)

T-1/2 =

x Jo

(A + T)-1.\-1/2 dA .

Indeed, the spectrum u(T) of T is contained in the compact subset of the complex plane defined by Wez > 6, IzI IITII. The function z-1/2 is holomorphic in a neighbourhood of the spectrum. It follows that the right-hand side of (3.2) defines an operator whose square is T-1.

11 The T(b) theorem

130

To show that T1/2 is b'-accretive, for a certain S' > 0, we use the following lemma, whose very simple proof is left to the reader.

Lemma 2. Let T : H

H be a b-accretive operator. Then T-1 is

6IITII-2 accretive.

From the right-hand side of (3.2), we get We (T-1/2x, x) >

#!f f A + b (A

A-1/2 dA =

IIT'II)2

clIxII2.

Thus T-1/2 is c-accretive and, applying the lemma, so is T'/2. A different method of calculating T-1/2, which avoids the functional calculus, uses the following observation

Lemma 3. If T is b-accretive and if A >

(26)-1 IITII2, then

IIT - All < A.

(3.3) Indeed,

IIT -All = II(T* - A)(T - A)II1/2 and

(T*-A)(T-A)=T*T-A(T*+T)+A2 C1 > 0 such that, for any choice of coefficients a3, 1/2

(4.1)

C1 (laI2) jEJ

1/2

G F_ ajej 2> m(3, k)SjSk

i

r 6,E IS1I2

i

k

Let H be a complex Hilbert space. Let B : H x H - C be a jointly continuous bilinear symmetric form: B(x, y) = B(y, x) for all x, y E H. Let e3, j E J, be a Riesz basis of H. We consider the matrix B whose coefficients are B(e3, ek), j, k E J. This matrix is bounded on 12(J). Indeed, if £, E l2(J) and yi E 12(J), we have

i E C171kB(ej, ek) = B(x, y), where x =

.i ei and y = F- Ilk ek. Thus

CIIxIIIIyfl s C'(> I(7 I2)112(1: 171k We shall make systematic use of the following result.

12)1/2.

I

Proposition 3. Let H be a Hilbert space over the complex field and let B : H x H -, C he a jointly continuous, symmetric, bilinear form.

Suppose that ei, j E J, is a Riesz basis of H such that the matrix (B(ej,ek))(j,k)EJxJ is b-accretive, for some S > 0. Then there exists C'"

a Riesz basis fj, j E J, such that B(fj, fk) = 0 or 1, depending on whether j 0 k or j = k. If, moreover, J = Z' and if, for a certain exponent a > 0 and some constant C > 0, we have (4.3)

IB(e2,ek)I
3, we can integrate by parts, because the kernel is regular in x and the integral of by is zero. The model is not accurate for two reasons. The scale is 2-3 and not 1 (the scale is defined by the functions Oat and z1',) and, more importantly, (51i and 1' decrease rapidly at infinity rather than having compact supports. To reduce to the case j = 0, we use a change of scale which does not alter the hypotheses about K(x, y) and b(x) in the slightest. To reduce to the case of functions of compact support, it is enough to make the following observation.

Lemma 6. For every integer n, there exists R(n) > 0 such that the following property holds. For each y > 0, there is a constant C'(y, n) such that, if f (x) satisfies If(x)I < e-71x1,

IVf(x)l
1 almost. everywhere.

We require the existence of a constant C1, such that, for each ball BC1Rn, (9.4)

dx 0)

and we split b into b1 + b2 + b3i where b1 is the product of b by the characteristic function of B, b2 is the product of b by the characteristic function of the annulus A defined by r < Ix - xol < 2r, and b3 is the product of b by the characteristic function of Ix - x0I > 2r. Now, as we have often observed, if Ix - xoI < r, then ITb3(x)-Tb3(xo)I 1, so we get I'YB I < C" After these variations on the statement of Theorem 4, we come to the proof. It follows that of the T(1) theorem word for word except for the standard wavelets being replaced by the wavelets 4GA whose cancellation is adapted to the function b.

'N.

per

We begin with the construction of the pseudo-products. They are adapted to the function b E L°C(Rn) non-linearly. Starting from 0 E BMO(Rn), we intend to construct a continuous linear operator T on L2(IIRn), whose kernel, restricted to x 0 y, satisfies the usual estimates and which, moreover, is such that T(b) _ 0 and tT(b) = 0. 'lb construct T, we mimic the usual pseudo-product construction. Firstly, we put 9A(x) = 2niO(2?x - k), where 9 E D(Rn), f O(x) dx = 1, where A = 2-3 (k + e/2), e E E. If we set w(A) = (f b(x)OA(x) dx)-1, .Vi

'19

then Re f b(x)9A(x) dx > 1, so that Iw(A)I < 1. The wavelets A adapted to b(x) are those that we constructed earlier. We now define

ir(Q, f) =

w(A) (.f, 9A) (fl. bA)b, , AEA

II'

where (u, v) = f u(x)v(x) dx. We first of all check that the kernel of the operator T (f) = 7r(,8, f) satisfies the standard estimates. The integral f dx vanishes and we have O(IxI-n-1) at infinity. This is the same as saying that b(x)4'A(x) is a molecule of the Stein and Weiss space Hl. Note that A(x) itself is not a molecule and does not even belong to H'. Lastly, I ((3, W , \ )1 < C2-n''2II0IIBMo The rest is easy and the kernel 30i

K(x. y) = E w(A)7(A)2"!9(23y AEA

satisfies the usual estimates when I-y(A)I < C2-" /2. We move to the L2(lR') continuity. Taking Theorem 2 into account. we must show that I(f,9A)IZI(,o,r ,A)IZ < (9.7) AEA

The proof of (9.7) reduces to verifying the inequality (9.8)

1((3. bA)12 < CIQI , Q(A)cQ

for each dyadic cube Q. Once (9.8) has been established, Carleson's X00

inequality gives (9.7). To get (9.8), we refer to the proof of Theorem 4 in Wavelets and Oper-

11.10 The space Hb

145

can

ators, Chapter 5. The estimate (9.8) is a consequence of the "Plancherel formula" \)12,

112

211

(9.9)

I(f,bw,

11f112 = AEA

ear

III

vii

given by Theorem 2, together with the localization of the wavelets and, lastly, the condition f b(x)z A(x) dx = 0, which enables us to eliminate the floating constants which arise in the definition of the space BMO. We have T(b) = 3, because ?r(/3, b) = IaEA(16, s A)z A =13 (from the Corollary of Theorem 2). Also, 'T (b) = 0, because f via (x)b(x) dx = 0. The proof of the T (b) theorem then repeats that of the T (l) theorem word for word. Using pseudo-products to correct T, we reduce to an operator R which satisfies conditions (9.1), (9.2), (9.3) and (9.6) and is obtained by calculating the matrix coefficients T(J1, a') _ and showing that `--i

'-'

(9.10) IT(a, a' )I
a(x)). The Hardy space H'(921) consists of the functions F(z) which are holomorphic in 11, and such that

supJ IF(z+ir)I ds < oo. z>o r The Hardy space H' (112) is defined similarly. If F E H'(01), then

11.10 The space Hb

147

limTpj F(z + i-r) = F(z) exists in L' (I') norm and almost everywhere. Further, for any zo E fZl, 1 J2-7rz

z-zo dz,

so the boundary values completely determine the function F E H' (521). All that is left to do is to describe the Banach space consisting of these boundary values. To do this, we use the parametrization of I' by the abscissa x (where

z = x + ia(x), -oo < x < oo) and, with b(x) = 1 + id(x), we get F E H'(f21) if and only if F(z(x)) = f (x) E Hb (IR)

(10.3)

and, for each z2 E %, III

f00

(10-4)

f (x) Z(X) - z2

z(x) dx = 0. :L1

In other words, we may identify H' (fl,) with a closed subspace of Hb . The same applies to H'(fl2). Note that Hb is the direct sum of the two closed subspaces H' (521) and H' (112). Indeed, the decomposition f = fl - f2 of an arbitrary function f E Hb, as the difference of f, E H' (521) and f2 E H' (SZ2), may be obtained by applying the Cauchy operator (27ri)-' fr (C - z)-' f (t;) dS to f and then taking boundary values (respectively from above and from below). The continuity of this operator on Hb is a result of its continuity on L2(IR)) and of Theorem 3 in Chapter 7. Indeed, the kernel K(.c, y) = z'(x)(z(x) - z(y))`' has the necessary regularity with respect to y and satisfies fK(x, y) dx = 0, modulo the constant functions. To pass from these formal considerations to a precise statement, it is enough, once more, to approximate the kernel K by regular kernels Ke,T, defined by (r;

Ke,7 (x, y) =

z'(x) z(x) - z(y) ± ie

-

z'(x) z(x) - z(y) ± iT

A noteworthy subspace of Hb is the space generated by the special atoms, which we denote by b"'. This space consists of those functions f E Hb of the form a(a)a, where EXEA la()r)I2-y/2 < oo and the sum of the series gives the norm of f in Bl°'. If we let S(zo) denote the distance from r of a point zo and we consider the case where b(x) =

z'(x) = 1 + ia'(x), with -M < a(x) < M, then the function 6(zo)(z za)-2, restricted to r, belongs to B°'1 and the norm of b(zo)(z - xo)-2 in B?'1 does not exceed C(M). These "special atoms" form a total set

11 The T(b) theorem

148

in B°". More precisely, each function f E Bo" is of the form

f (.x)=E 00

(10-5)

0

) zk

(z(x)2 - zk '

Oc

where zk v r and E IakI C belongs to the dual of B' (111), it is enough to be sure that 7-,

'-'

,t3

.:7

p''

g(z) dzl < C. (z - w)2 But . fc.4(z)(z - w)-2 dz = -2irig'(w). so sup,irEf22 b(w)19'(w)1 < (10.6)

!Z'

b(w) I

C/2-1r'

by (10.6). The fact that we can restrict our considerations to functions which are holomorphic in 122 is a consequence of the duality of H2 (1l )

""L

'C7

and H2(112). It is as if the functions (z-w)-2, w E 512. were wavelets in H2(111) and !e+

thus, by duality with H2(112) and other spaces of holomorphic functions

11.11 The general statement

149

on 12i led to an analysis of those spaces, classifying them by simple conditions. Of course, the roles of 52I and S22 are symmetric. At the same time, these same functions (z - w)-2, to E 12i are the "atoms", by the approriate linear combinations of which we can represent those holomorphic functions on 521.

673

To end this chain of ideas, B>°° is the natural dual of B°'1. A distribution S belongs to B°o°° if and only if I(S, Jia)I < C2-J/2, A E A,, j E Z. In fact, BO,- is defined modulo the function b(x). Then, when b(x) = 1 +ia'(x), every distribution S E B0°° may be written uniquely as S = bS1 + bS2, where S1 E B°°(521) and S2 E B°°(522).

11 The general statement of the T(b) theorem Let T : D(R")

D'(R") be a continuous linear operator. We sup-

pose, firstly, that T is weakly continuous, which is a necessary condition

for L2 continuity. We suppose, further, that the restriction to x 34 y of the distribution-kernel S(x, y) of T is a function L(x, y), satisfying IL(x,y)I -< CoIx - yI-n. Then T has a natural extension to the space of Holder functions of exponent 17 > 0 and compact support. If f and g are two such functions, then (T (f ), g) is well-defined.

Next, we make use of the function b to state the extra conditions which will be necessary and sufficient for T to be continuous on L2. We assume that b E L°°(1R') and 2e b(x) > 1. We suppose, moreover, that L(x, y) = b(x)K(x, y)b(y), where K(x. y) satisfies (9.1). (9.2), and (9.3). '-'

We can then state a necessary and sufficient condition for the L2 continuity of T. This condition involves, once again. T(1) and tT(1). We must define these as mathematical objects. Letting A, A E A, denote the collection of the wavelets of Theorem 2 (of regularity r > 1),

H,,

we want. to make sense of the expression To do this, we start with the most academic case of (T(1), u), where u is a Lipschitz function of compact support, satisfying fu(x)b(x) dx = 0.

Then 'T(u) = a is a distribution, but, outside the support of u, we have U(X0) = f b(x)K(x, xo)b(xo)u(x) dx = O(Ixol)

t''

since we can use the kernel's regularity with respect to x and the fact that f u(x)b(x) dx = 0. As a result, the integral (a, 1) converges. To pass from the special case (compact support) to the general (Lipschitz functions u of exponential decrease and such that f u(x)b(x) dx = 0) we use Lemma 6.

11 The T(b) theorem

150

The wavelet coefficients Q(a) = (T(1), a) are thus well-defined and, if these coefficients satisfy Carleson's condition EQ(A)CQ lQ(a)12 < C1jQj,

we write T(1) E BMOb_ The same remark holds for the coefficients ('T(1), Ia). We have arrived at the statement of the T(b) theorem. Theorem 5. With the above notation, a necessary and sufficient condition for T to extend as a continuous linear operator on L2(R') is that T is weakly continuous, T(1) E BMOb and tT(1) E BMOb. The proof of Theorem 5 is a simple variant of that of Theorem 4 and we shall not detain the reader with the details.

12 An application to complex analysis

Ink

As an application of Theorem 5, let us return to the example of the Cauchy kernel operating on the space L2(F, ds), where r is a Lipschitz curve. It is no longer necessary to approximate by integrable kernels. Instead, we proceed as follows. As above, we let a(x) denote a real-valued Lipschitz function of a real

variable x. We then have -M < a'(x) < M. We then form the kernel K(x, y) = z'(x)z'(y)/(z(x) - z(y)). There exists a constant Co such that

yr.)

IK(x,,y)I < CoI.c-yl-1 and, because the kernel K is also antisymmetric, we can define the distribution S = PV K. This distribution belongs to S'(1R2) and is the distribution-kernel of a continuous linear operator

°,'

T : D(R) -+ D'(IR). To establish that T is continuous on L2(R), it is enough to verify that T(1) = 0, modulo z'. This identity depends, essentially, on Cauchy's formula. We know that T(1) E In other words, T(1) is a continuous linear functional on B°'1 which we can evaluate by calculating (T(1), a), where a(x) _ b(zo)(z(x) - zo)-2, zo v F. Such evaluations are enough, because the set of such atoms is aAwe total subset of B°". Finally, (T(1), a) = lim

(z - zo)-2(z - ww)-1 dzdw = 0,

z-newl>e as can be seen by integrating first with respect to z and then with respect CIO

to w.

13 Algebras of operators associated with the T(b) theorem > 1. Let We suppose, once more, that b c L°°(WI) and that We T be an operator which satisfies the hypotheses of Theorem 5. Can

c+9

'''

11.13 Algebras of operators

151

.a:

T be extended as a continuous linear operator on Hb ? The expected response is that this will be the case if T is continuous on L2(R") and if tT(1) = 0. But these two conditions lead to a continuous linear operator with domain H6 and range in H' (the standard space of Stein and Weiss). To get the conclusion we want, we transform the problem. We set T = BL, where B is the operator of pointwise multiplication by E''

b(x).

Can

bye

The distribution-kernel of the operator L, restricted to x , y, is thus of the form K(x, y)b(y), where K(x, y), as usual, satisfies (9.1), (9.2), and (9.3). From the continuity of the operator T on the space L2(l8') and from the condition 'L(b) = 0 (modulo the function b), it follows that L is continuous on the space Hb . This can be shown by writing T = MB and applying Theorem 3 of Chapter 7 to M. This same operator L is continuous on the usual BMO space if L(1) _ 0, modulo the constant functions, as is shown by the corollary to Theorem 3 of Chapter 7. Let G be the collection of operators L of the above type which are continuous on Hb and BMO. We shall verify that G is a subalgebra of C(L2, L'). To verify this, we make use of the method of Chapter 7. We let (T(a, denote the matrix of L with respect to the wavelet basis 1/ia, a E A, of Theorem 2. If 0 < Q < -y (y specifies the regularity with respect to x and y of the kernel K(x, y) corresponding to L), then 7-(,\,,\') belongs to the algebra M0. Conversely, if this matrix belongs to MO, then the distribution-kernel S(x, y) of L is ..a

1: 1: X

(y)b(y)

a1

(fl

and it is easy (by repeating calculations for the case b = 1, word for word) to see that L E C (and that K(x, y) T(\, )r')z ,\, satisfies (9.1), (9.2), and (9.3) with fl = y). Just as in the case b = 1, we can introduce the Banach algebras A(b, y), 0 < -y < 1, consisting of the operators L whose matrices with respect to the basis z/ia, )r E A, belong to M.y. The definition does not depend on the choice of basis. However, in the general case, the algebras A(b, y) are not self-adjoint. This is what prevents us using Cotlar's lemma in the proof of Theorem 5. In fact, if T E A(b, y), then T is continuous

on Hb and BMO. As a result, the transpose of T is continuous on H1 and BMOb. This makes it necessary to conjugate the transpose by the operator B defined by pointwise multiplication by b(x), in order to return to an operator belonging to A(b, y). The algebra C can equally well be interpreted by the use of vaguelets.

11 The T (b) theorem

152

Indeed, L E G if and only if, for every family wA A E A, satisfying (6.3),

(6.4), and (6.5), the functions w,, = L(wa) satisfy the same estimates (with, possibly, different values of the exponents a and /3). This way of looking at things has the advantage of showing that the L2 continuity of the operators L E G follows from Proposition 5. Indeed, we take f E L2 and decompose it as EaEA o(a)f,,, with (EAEA la(,\)12)1"2 < CIIf112; then L(f) = EAEA a(A)w.\, where wa = L(z'),). To conclude, it is enough to apply Proposition 5.

14 Extensions to the case of vector-valued functions We shall need the Hilbert space analogue of Theorem 4 when we study

Hardy spaces corresponding to a Lipschitz graph (Theorem 5 of Chapter 12). For this, we have to introduce the space BMO(W, H), where H is

a Hilbert space. A function f : R' -+ H belongs to BMO(R', H) if IIf (x) II H is locally square-integrable and if there is a constant C, such that, for each ball B C R", there exists a vector -y(B) E H with 1/2

CIBI f IIf(x)-y(B)IIHdx)

Rn he a Lipschitz function. We consider the kernel (.rn - yn) ay. (y) a(x) - a(y) - (xi - yl)"(y) K(x, y) _ [ix - y12 + (a(x) - a(y))21(n+l)/2 The question that Calderdn asked was whether there existed a con-

i-3

^,^

tinuous linear operator T : L2(Rl,dx) -p L2(Rn,dx) which could be defined by T(f) = PV fK(x, y) f (y) dy, for f E L2(Rn). The case n = 2 is given directly by Theorem 2, since, in this case,

c"'

rn'

tea.

K(x, y) is the imaginary part of the Cauchy kernel (z(x) - z(y))-1 dz(y) associated with the Lipschitz curve z(x) = x + ia(x). By the results of Chapter 9, we know that T is. indeed, continuous on L2(l$n). We shall see in a moment that PV f K(x, y) f (y) dy exists almost everywhere, for f E L2(Rn). What we intend to do is to describe a generalization of the T(b) theorem which applies directly to the double-layer potential without involving the method of rotations. First of all, let us recall the definition of the Clifford Algebra A,,,,. This algebra is a vector space of dimension 2'm over R. A basis of A,,, is consists of vectors es, where S is an arbitrary subset of the set {1, 2,..., m}. For the time being, this is just a notation which lets us assign a subscript to 2'm vectors. We now show how multiplication is defined on this basis.

If S = 0, es is the unit element of A,,,. We want A,,, to be an where we have writen el associative algebra generated by instead of a{l}, etc. The relations in A,,, are generated by ee = -1, 1
1. If b(x) = bo(x) + bl (x)el + m = 1, then Al is just the field of complex numbers and the condition reduces to We b(x) > 1. The operators which we consider are defined by A-valued distributionkernels S(x, y). They act on A-valued functions in L,24(Rn). This action arises from the asymmetry of the product S(x, y) f (y) calculated in the sense of the multiplication on A.

The norm of x = Esases is IxI = (iS IasI2)i/2. We suppose that S(x, y), restricted to x # y, satisfies S(x, y) = b(x)K(x, y)b(y) where (15.1)

I K(x, y) I A < Co lx - Y1_'

(15.2)

JK(x', y) - K(x, y)I A < C1Ix' - xI I x - yl-9L-ry , for J x' - xI < Ix - y[/2, and, similarly, (15.3)

IK(x,y')-K(x,y)IA

6 holds ([97], [98]). Tchamitchian's construction has been the starting point for very ac-

'L7

``'

'L7

tive lines of research, for the most part as yet unpublished. R. Coifman, P. Jones, and S. Semmes modify the Haar system h1, I E .1, in the simplest and naivest possible way, to construct h# (x) satisfying fh#(x)b(x) dx = 0. They decide to put h#(x) = n'1 on the left half of I and h#(x) _ /31 on the right half. The support of h* is I. By a judicious choice of these constants, h*, I E J, is a Riesz basis of L2(R) and can replace Tchamitchian's basis in the proof of the T(b) theorem ([64])

12 Generalized Hardy spaces

1 Introduction Let I' be a rectifiable Jordan curve in the complex plane, passing through the point at infinity. Let z(s) denote the arc-length parametIz(s) = oo. We shall, in fact, suppose rization of r. Then that, whenever zo lies off F, then, for any 1 < p < oo, we have fr, Iz(s) zzl-P ds < oo. It will be convenient to suppose that F is oriented. The Jordan curve theorem tells us that the complement of IF in C splits into two connected components 1l and 522. What we shall study is the generalized Hardy space HP(521), which consists of the functions F(z) which are holomorphic in ill and have, in a very precise sense, a trace 6(F) on F which belongs to LP(r, ds). The space HP(521) can be identified with a closed subspace of LP(Sll), which we shall also, by abuse of language, call HP(5l1). We recover the

holomorphic function F : fl -4C from its trace f = 0(F) by Cauchy's formula (1.1)

F(z) = 1

f (C)

27ri rS-z

'

when ft1. is the domain "to the left of' the oriented curve r. In fact, matters are not quite as simple as we have suggested and there are three possible definitions of HP(521). The definitions coincide when SZl is a Smirnov domain; it is known that this condition does not depend on the exponent p E (1, oo). Calderon's problem is to know whether, for 1 < p < oo, the space

12 Generalized Hairly spaces

158

^"'

NN.

cps

LP(r) is the direct sum of the spaces HP(SZ1) and HP(12), when these spaces are realized as subspaces of L"(I') by means of their respective trace operators 01 and 02. Thanks to a theorem of David ([931), we now can characterize the curves 1' for which Calderon's problem has a positive response. The curves in question are those which are regular in the sense of Ahlfors: they are rectifiable curves such that, for a certain constant C > 1, for all z0 E C, and for all R > 0, the measure of the set F(zo, R) of those s E l1 for which Jz(s) - zol < R is not greater than CR. This condition requires I' not to have "too many zigzags" and is violated, for example, by the rectifiable graph IxI" sin(1/.r), when 1 < a < 2. The object of this chapter is to prove David's theorem. There are now two ways to achieve this. One uses the resources of real analysis and has already been alluded to in Chapter 9. The other is based on complex analysis and gives the simplest proof of the continuity of the Cauchy kernel on Lipschitz curves. David's proof is a geometric variant of the methods introduced by D. Burkholder and R. Gundy under the name of "good ) inequalities". This proof needs a starting point: the continuity of the Cauchy kernel on Lipschitz curves. We shall therefore start with the special case of Hardy spaces corresponding to a Lipschitz domain. Calderon's theorem, either in the form of the identity L2(r) = H2(1i) + H2(112), or as the continuity of the projection of L2(t) onto H2(111), will get three different proofs (among them the so-called "shortest proof' of P. Jones and S. Semmes). We shall then follow David and prove his theorem by real-variable methods. On the way. we shall show how the algebras of operators defined by the T(b) theorem are related to algebras associated with singular holomorphic kernels K(z,w) acting on L2(r). coo

C7'

cwt

"'?

con

G',

2 The Lipschitz case Once and for all, t C C is the graph of a Lipschitz function a : IR --+ R. We have ds = dx and, if -M < a'(x) < M, it follows that (1+a'2(x))112

dx < ds < (1 + M2)1/2 dx. The space L2(F, ds) may thus be identified with L2(lil, dx). E.`

We let S21 denote the open set given by y > a(.r) and let !12 be that given by y < a(x)The definition of the Hardy spaces relies on the idea of parallel curves. We observe that, for every T > 0, 1' + iT lies in !l1, which allows us to make the following definition. .°m

,+'

12.2 The Lipschitz case

159

Definition 1. Let F(z) be a function which is holomorphic on 121. We say that F(z) E HP(1l1), for some 0 < p < oo, if

r

f r>o \ r+ar Pl

l I /P

IF(z)Ipds I

/

=

oo.

We could equally well have written supr>O(f1. IF(z+ir)IPds)1/P < o0 V

For the time being, we shall restrict attention to the case 1 < p < 00 and shall prove the following theorem.

Theorem 1. Let 1 < p < oo. If F E HP(121), then the functions Fz E LP(I'), defined by FT(z) = F(z + ir), where z E F and z > 0, converge, almost everywhere and in LP norm, to a function, which we denote by 0(F) and call the trace of F on F. The trace operator 0: HP(1l1) -, LP(F) is an isomorphism of HP(S21) with a closed subspace of LP(I'), which, by abuse of notation, we again denote by HP(SZi).

For each function f E LP(1', ds) the following conditions are equivalent:

(2.1) there exists a sequence R3 (z), j > 1, of rational functions which vanish at infinity, are holomorphic in a neighbourhood of 521i and converge to f in LP(I', ds) norm; f() rC-z

(2.2)

d(=0

forallzES2z;

f E HP(11) Finally, if the equivalent properties (2.1), (2.2), or (2.3) are satisfied, we have, for all z E Ili, (2.3)

(2.4)

F(z) = 1 j f2--zdo

where f = 0(F).

The meaning of the equivalence of (2.1) and (2.2) is that 121 is a Smirnov domain. It is not true that (2.2) implies (2.1) when r is an ...

arbitrary rectifiable Jordan curve, as Lavrentiev has shown ([155]). The deep meaning of Theorem 1 is that it is a form of the "maximum principle" for functions F(z) E HP(121). What happens is as follows. If we know, a priori, that F(z) belongs to HP(11), without being in a position to calculate the HP(111) norm of F(z), then we may use the estimate given by (fr If (z) IP ds)1/P. If, on the other hand, we lack that

a priori information (namely, that F E HP(1t1)), this procedure may not work. The classical counter-example is 121 = {x + iy : y > 0} with F(z) = (z + i)-'e ", whose boundary values are in L2(R) but which does not belong to H2(12, ).

12 Generalized Hardy spares

160

Much of the line of argument we take in this section is due to the above difficulty.

The proof of Theorem 1 relies on an approximate version of (2.4), given by the following lemma. (We shall write 1 for stl and I'r, for r + iT, to simplify the notation.)

Lemma 1. Let F(z) be in Hp(f2). Then, for each r > 0 and every zo=xo+ia(xo)+ivE&1, ( 2. 5)

F( zo ) =

1

and

0 =27ri1

( 2 . 6)

F(() d(

.

27rz Jrr ( - zo

r

Jrr S - zo

(

d

if v > T if v < T .

To establish these two identities, we reduce to the standard Cauchy formula, replacing rT by the oriented boundary of the rectangle given

by -R < r < R and a(x) + r < y < a(x) + r'. Then we let R and r'

.-.

tend to infinity. The integral along the top boundary can he bounded directly by

Ir Iz +( ir'

i-r') Il ds < (/ I

where, for 1/p + 1/q = 1,

E(T')=

if Iz+ir' - zoI-Qdsl

1/Q

which tends to 0 as r' tends to infinity.

The integrals on the vertical boundaries are dealt with by a technique described in Titchmarsh's treatise ([229]), which we now describe. We make the vertical boundaries "vibrate", by letting R run through the interval [M, M + 1]. We then take the means of the Cauchy formulas we have used. This lets us replace the integrals by double integrals which are bounded above, when we take the definition of HP(cl) into account. The reader is referred to [229]. The proof of (2.6) is identical and is left to the reader. The second ingredient, which we shall use systematically, is the following observation.

Lemma 2. Let KT (z, w), with z, w E r, be a function satisfying T IKT(z,w)I 0. So it is enough to show that f r, F(z)G(z) dz = 0. To do this, we use the same change of contour as in Lemma l to show that the value of this integral is independent of r. Finally, we note that limr.«,(f rr IF(z)lv ds)'/T = 0, indeed, if z e r, IF(z + ir)l < C.F*(z) E LP(P), which allows us to apply Lebesgue's dominated convergence theorem.

Thus limT-a frr F(z)G(z) = 0, which concludes the proof of the lemma. We return to the proof of Theorem 1, letting R(S2) denote the algebra of the restrictions to 11 of rational functions which vanish at infinity and whose poles lie outside P. To characterize the closure of R(Q) in LP(P), we use the Hahn-Banach

E3.

theorem. Let g E LQ(F) be a function such that jr g(z) f (z) ds = 0, for every f E R(S2). Since dz = z'(s) ds and lz'(s)l = 1. we may replace g(z) by h(z) = g(z)/z'(s). Then fi. h(z) f (z) dz = 0, for f E R(Sl). In particular, frr h(z)(z - xe)-1 dz r= 0, when zp . We then put

H(z) =21ri1

h(w) dw =

i W- Z

1 r W- Z 27ri I

1

1

W - z*

h(w)dw,

where z = u + ia(u) + iv E S2 and z* = u + ia(u) - iv

. Lemma 2 applies again and sup{IH(z)l : Wez =u,z c 121 0, continuous on Qrrn z > 0, and has a trace (in the obvious sense) on the real axis. What is more, the trace is the limit of the functions F(x + iE), for e > 0. But

12.3 Hardy spaces and conformal representations

163

F(z) does not lie in H2(sl), because F(x+ir) = eTe-ix/(x+iT+i): the + 1), which tends to infinity with T. However, we have the following result.

n o r m of this function is % / i - r e

!''

cps

row

Lemma 4. Let F(z) be a function which is holomorphic and bounded on IL Then F has a trace on r = tIZ: limT,o F(z + iT) exists for almost all z E t. If the trace belongs to LP(r), then F(z) lies in HP(1Z). 4''

N..

To see this, we first replace F(z) by FE(z) = F(z)/(1 - iez), which does lie in HP(1l), since F is bounded on Q. Hence the trace of FE(z) exists, which gives the first part of the lemma. Theorem 1 gives (2.12)

\

/r ` IF(z)Ids " 0} onto an open set I whose boundary is an

..j

oriented Jordan curve r passing through the point at infinity. We shall also use 4) to denote this conformal representation, which extends to a homeomorphism of R onto r. This homeomorphism is increasing if Il lies "to the left" of the oriented curve F. Let a : IR -' IR be a Lipschitz function (so I1a'L1(.. < M < oo). Let r be the graph of a, oriented from left to right. Il = {(x, y) : y > a(x)} is the open set "above" I' (that is, to the left of the oriented curve r). Let E be the sector of the complex plane defined by x > 0 and Jyj < Mx. The next theorem, due to Calderon and improved by C. Kenig, tells

12 Generalized Hardy spaces

164

+(++

us that, for each y > 0, the curves 4)(x + iy), x E R, are graphs of Lipschitz functions of slope not greater than M. More precisely and keeping the notation above, we have the following result.

Theorem 2. We can choose a branch of log 4)' (z) such that tan-1 M I9'm log4''(z)I < On = for all z E P.

y-!

To prove Theorem 2, we start with the case where I' is a polygonal line ending with two half-lines whose slopes do not exceed M. Then the conformal representation 4) is provided by the Schwarz-Christoffel formulas ([211]). We can find N real numbers cl < c2 < ... < cN (N is the number of vertices of 1'), a real number -y, and N real exponents ryj, 1 < j < N, such that 4)'(z) = e47 fN(z - cj)' U. We choose z7 so that is holomorphic in P with P = 1. The "angles" y and 'yj are then related to the slopes of the polygonal line r: indeed, if cj < x < cj+1, we have arg 4)'(x) = y + 7r(yj+1 + - - - + ryv); if x < cl, we have arg 4)'(x) _ +'yrr); and if x > cN, we have arg 4)'(x) = -y. Thus y + 7r(y1 + 0,wehave CC

(3.1)

V(z)

V

oo (x - u)2 + v2

4'(x) dx .

The complex numbers 4)'(x) lie in the sector E (by construction). Since E is a convex sector and 4)'(z) is a convex linear combination of the 4)'(x), the number 4)'(z) also lies in E. We arrive at the conclusion of Theorem 2 by composing log (defined on a conical neighbourhood (excluding 0) of E) with 4)'(z). We can pass to the general case by approximating r by a sequence I'j of polygonal lines such that the open sets f above rj increase to fl. To achieve this, it is enough that the rj are graphs of piecewise affine functions aj(x) which decrease to a(.a). To construct the a j (x), we consider the subdivision formed by the .32

points x = k2-j, where k E Z and Iki _< j2j. We put y(k,3) _

-'o

a(k2-3) + 2M2-3 and we let rj denote the polygonal line whose nodes are (k2-9, y(k, j)) and whose ends are formed by half-lines of slope M and -M, respectively. It follows immediately that the functions aj (x) form a sequence which decreases, uniformly on each compact set, to a(x). We then establish that the conformal representations 4) j converge to fem."

12.3 Hardy spaces and conformal representations

165

4. by returning to the case where P is replaced by the open unit disk D and Il becomes a bounded open set. We use the following remark (which is a consequence of Schwarz's lemma).

Lemma 5. Let D be the open unit disk, let 1z be a bounded open simply-connected set and let l l be an increasing set of simply-connected open sets such that SZ = U; >o Sk Fix zo E SZo and let 4)j : D -+ Ii, be the conformal representation

normalized by 43 (0) = zo and 4 (0) > 0. Then the sequence -!D? (0) is increasing and the functions converge uniformly on each compact subset of D to the conformal representation 4 : D -+ Q, normalized by zo and -1)'(0) > 0. Coming back to the case of P and an unbounded 12, we have 4)? (z) E E, so V(z) E E and the theorem has been proved. Theorem 2 gives log $'(z) = u(z) + iv(z), where supzcp I v(z)I < 0o
./'

the means of I$'I and IVI-1 on I is uniformly bounded. So W(t) = I4P'(t)l belongs to Muckenhoupt's class A2 (Chapter 7, section 8): the constant which appears in the definition of the class only depends on M. In other words

sup (FI I w(t) dt)

(j_) < C(M) < oo . ti,

(3.7)

As a consequence (Chapter 7, section 8), there exist an exponent b = 6(M) > 0, 6 < 1, and a constant C'(M) such that w(E)

6

< C' \IEI

rye

w(I) III (where w(E) = fE w(t) dt etc.). Writing (3.8) with w replaced by 1/w and taking account of the condition

(L w(t)

dt)

UE w

dt 1 > IEI2 ,

coo

we get

w(E) > C,, (J1)

(3.9)

-

Applying (3.8) and (3.9), with E = [0, 1], I = [0, t] (where t > 1), or E = [-1, 0], I = It, o] (where t < -1), we get

s

G1(M)Itl8


C2(M)ItI2-6

if t(o) = 0.

C3(M)h2-6(1 + lti)-2-F26,

where C1(M) > 0, C2(M) > 0, and C3(M) > 0 depend only on M. Finally, if 0 < h < 1, we get

I'(t + h) - t(t)I < C4(M)h5(1 +

(3.12)

ItI)2-26.

To arrive at these estimates, we need b

Ja

V(t) dt >!

jb

e' '(t) dt >_

+ M2 j l'(t)I dt a

12 Generalized Hardy spaces

168

and, having made this remark, the inequalities follow immediately. {,a

We are now equipped to pass to the limit. We begin with a real, measurable function v(t) such that lIvjI,, = eo < it/2. With it, we fly

,Q,

associate a sequence v, of step functions with compact supports, such that 11v,11. < 00 and v, (t) v(t) almost everywhere. We then construct -t1 : P -+ C, by requiring 00

t

z

vi (t) dt + C3 ,

C; E R, z E P, 0.'

log ,D3 (z) = - J

We log-tj' (i) = 0, and 4,(0) = 0. The functions -t, are univalent in P and they satisfy the estimates '3'

(3.10), (3.11), and (3.12), uniformly in j. To conclude, we apply the following lemma (after two appropriate bilinear transformations).

'4l

Lemma 7. Let F1, j E N, be a sequence of univalent functions on 'zI < 1, which are continuous on Izl < 1 and converge uniformly on IzI _< 1 to F. We suppose that F,(ea°) and F(ea0), 0 < 0 < 23r, are 'vim

closed Jordan curves, which we denote by F1 and r. Then F is univalent and F(D) is the inside of F.

IA..

t%]

This lemma is a variant of Rouche's theorem. To use it, we first consider the auxiliary functions Since F3 = -'D, (l18) is a polygonal line through 0, with slope not exceeding M,

HIV

._.

we have Ii + 41(t)i > c(M)(1 + 14)j(t)l) > c'(M)(1 + jti)' and, thus, I(i + 43(t))-'j < (c'(M))-1(1 + Iti)-a. Moreover, the functions $1(t)

,L"

....

coo

are equicontinuous: by passing to a subsequence, if necessary, we may suppose that the functions converge uniformly on every compact set. As a consequence. the converge uniformly on the whole real line. The maximum modulus principle now shows that the functions (i + 4'1(z))-' converge uniformly on P. To get to the situation in the lemma, it is enough to put z = i(1 - ()/(1 + (), S E D. We have proved the following theorem. l".

Theorem 3. If v(t) E L°°(R) is a real-valued function with norm III

+.a

IIvii= = 80 < 3r/2, then the h/rolomorphic function $, defined by

iz

1+t2)v(t)dt, (.L

t gives a conformal representation of the half-plane P onto the open set 11 above a Lipschitz graph r whose slope nowhere exceeds tan 00. On the way, we have established the following properties. The boundary values log 4'(t), -oo < t < oo, exist almost everywhere and I4'(t)I is

12.3 Hardy spaces and conformal representations

169 +N'

a weight function belonging to Muckenhoupt's class A2. The Lipschitz

graph t is represented parametrically by z = $(t), t E R, and we thus have ds = IV(t)I dt. The mapping t s is an increasing homeomorphism which preserves sets of measure zero. o-7'

fir'

Let us return to the Hardy spaces associated with Lipschitz open sets. We suppose that a(x) is a real-valued Lipschitz function of one real variable. We have IIa'IID < M < oo and let H denote the open set above the graph r of a(x). Let iP : P - 1 be a conformal representation

v0.

which extends to an increasing homeomorphism of JR onto F.

Let F(z) = P(z)/Q(z) be a rational function, vanishing at infinity, whose poles do not lie in Ti. Then G = (Fo f )' "/' belongs to the Hardy space HP(P). To see this, we must compute su IG(x+zy)Ir dx. 00

f

We use the change of variable z(t) = 4(t + iy), which leads to the '.4

notation F, C Il for the curve defined by this parametric representation. We know that I'y is a graph whose slope is nowhere greater than M. As a consequence, fry I P(z) I pI Q(z) I _p ds is bounded above by a constant, independent of y. Once we have verified this, we calculate the Hr norm of G by

( f IG(t)Ipdt)1/P= (f IF(z)Ipds1/p r

0o

l

The mapping taking F to G is isometric and thus extends to an isometry between HP(C) and HP(P).

To show that this closed subspace is the whole of HP(P), we use .-.

duality. We let g(x) E LQ(R, dx) denote a function such that 00

(3.13)

f°' o

4 V1/rg(x) dx = 0,

F E H"(1) .

"C3

We now change this integral into an integral along r and (3.13) becomes fl, F(z)h(z) dz = 0, where (h o 4')i' = ciii"Irg(x), which gives (h o $)V1/a = g(x). Theorem 1 now tells us that h(z) is in HQ(1). By the direct part of our argument, g(x) belongs to HQ(P), and hence the functions (F o d))4V1/1' are dense in Hr(P). We have established the following result.

Theorem 4. Let 1 < p < oo. If 4 : P

Tom

1 is a conformal representation of the tipper half-plane onto the open set Il above a Lipschitz graph t, normalized as above, then F - (F o $)c'l/P is an isometric isomorphism of HP(1Z) (considered as a closed subspace of LP(F)) with

HP(P). We shall get a more complete statement by also considering the extreme cases p = 1 and p = oo.

12 Generalized Hardy spaces

170

When p = 1, the arguments we used to prove Theorem 1 no longer work.

Recall that F E H' (fl) if F is holomorphic in fl and there exists a VIA

constant C such that sups>U fr I F(z + iy) I ds < C.

`-'

If F E H1(C ), we form F,,,,(z) = F(z + i/m), m > 1. and these functions F,,, are in HI(Il). Furthermore, they have an obvious trace on

r and satisfy

Fm(z) =2irz1 J

F- () dc,

r-i/2m (- Z

0.,

..f

coo

s].

if z E i U r. This can be established by following the proof we gave for the corresponding assertion of Theorem 1. From this relationship, we can deduce that F,,, E H°'([), but the L°° norms of the functions Fm are clearly not bounded. We then set f,,, _ (Fm o f)V, which is holomorphic on P, and, for each e > 0, we put f.,. (Z) _ f. (Z) (1 + EV(z))-I(1 -

iez)-2

"r1

Since V(z) E E, we have IV(z)(1 + EV(z))-II < C/E and it follows that I fr,e(Z)I < C(r71,E)I1 - iEZI-2. Thus f,,,,, (z) E HI (P), which is a closed subspace of L'(]R). But (F.,, o 0)V belongs to LI(]R) and I(1 + Ec')-1(1 - iez)-21 < 1. Lebesgue's dominated convergence theorem shows that (F. o 0)0' E

HI(P). gyp

Having got this far. we use a classical result ([239], Theorem (7.22) of Chapter 7), namely, the factorization of a function in HI (P) into the product of two functions in H2(P). This gives (F,,,, o 4')O' = g,,,h,,,., "ti

CSI

where g,,,, hm E H2(P), and where IIg..=112 = 111 4"112 = II (Fm o') 'III =

"I'

fr IF'.(z)I ds. We then define G,,, and H,,, by (G,,, ,)$11/2 = g,0 and (H,,, o .t).t11/2 = h,,,. We thus have F,,, = G,,,H,,, and both Gr, and H, belong to H2(Q). Further,

r

r

r

f IG'm(z)I2ds= f IHrn(Z)I2ds= j IF=(z)I ds.

r

r

e+y din;

Inequality (2.10) shows us that, if G,,(z) = sup,>a IG,,,(z+iT)I, z E t, coo .:'

then IIGm.IIL2(r) (a2 - 1)-1/2 and if m(C) is holomorphic and bounded on IqI < /3I1;'I, then (4.7) is satisfied.

In our application to the theory of operators, we must suppose that

(a2 - 1)-1/2 > M = IIa'II0 (recall that the Lipschitz curve r is the ...

C".

graph of the Lipschitz function a(x)). This will enable us to choose 13 so that (a2 - 1)-1/2 > a > M. The algebra of symbols we consider is the union, over all a satisfying 1 < a < (1 + M2)1/2/M, of the classes defined by (4.7). This union can also be regarded as the algebra of the functions which are holomorphic and bounded on a sector G, defined by

too

gyp'

1']I < LICI, where 1#1 > M. We denote this algebra of symbols by SM. With each symbol m E SAf we associate a distribution S which is the

fit"

coy

inverse Fourier-Laplace transform of m. Computing S is simplified by the approximation technique which we now describe. For each a > 0 and each m c SM, we put mE(e) = e-E!£!m(l:). For the holomorphic extension m(C) of if we write S = + i77, then e-E<m(C) when C > 0, whereas mE(C) = eECm(C) when t; < 0. It follows immediately that the mE(Ij), a > 0, form a bounded subset of SM. (21r)-1 f If M E SM, we now define S.,(x) by S.,(x) = converge to m(e) in the sense of tempered distribuThe symbols tions and the SE tend to S, the inverse Fourier transform of m. We shall characterize the distributions S, obtained in this way, as the boundary values of certain holomorphic functions.

12 Generalized Hardy spaces

174

For y > 0, we define H,y and H7 as follows: F E H,Y if F(z) is holomorphic in y > ylxl and satisfies IF(z)I 0, corresponds to the operator which takes f = F+ + F_ (where F+ E H2(111) and F_ E H2(112)) to nom

(J7

.`!

F+ (z + ie) which we consider as a function in L2 (1, ds). A further application of the T(b) theorem gives the following theorem of C. Kenig ([157]).

Theorem 6. Let S21 be the open set above the graph r of a Lip:Z"

: R -' R. Then the norms (fr IF(z) 12 ds) 1/2 arid (f ffl IF'(z)I2(y - a(x)) dx dy) 1/2 are equivalent on H2(S11).

schitz function a

We first check that there exists a constant C = C(M), M =

12 Generalized Hardy spaces

176

such that (4.11)

(f in IF'(z)I2(y - a(x)) dxdy)112 < C(M)(f IF(z)12 ds) 112, sI r

for F E H2(!11). We shall then show that a strong version of this inequality automatically implies the converse inequality.

To establish (4.11), we consider the kernel taking values in H = L2(0, oo), which is defined by IK(z, w) = K(z, w, t) = t1/2(z + it - w)-2,

z, w E r. Then, if f belongs to H2(fll), Jr

K(z,w,t)f(w)dw = -2trit1/2f'(z+it).

The inequality (4.11) thus follows from a stronger property, namely, the continuity of the operator T : L2(1') --+ L2 (1'), whose kernel is IK(z, w).

t,.

To avoid using singular integrals, it is convenient to replace T by a sequence of truncated operators defined by the kernels KN (z, w, t) _ K(z, w, t), when N-1 < t < N, and Kn,(z, w, t) = 0. otherwise. The continuity of T is then an immediate consequence of the Hilbert space version of the T(b) theorem. The verification of conditions (14.2) and (14.3) of Chapter 11 is done by the sane deformation of contour as was used earlier in this section and illustrated by figure 1. The details are left to the reader. Once the continuity has been established, we prove the converse to inequality (4.11) by using the following remarkable identity. 2i °°f F'(w + it) (4.12) F(z) = dw dt, r (Z+ it- u1)2 u

f it

for F E H2(f21).

Let us first prove (4.12). We write z + it - w = z + 2it - (w + it), which gives

f

Jr (F+ it± wit))2 z

/'

dw - Jr+zt

r F(+()

(-

d( = 27riF"(z + Zit)

G".

Then we get 00

,,,

tF"(z + 2it) dt = -4 F(z) "C3

i

"y1

h'.

by integrating by parts. As for the proof of the converse inequality, the continuity of the operator T : L2(r) --+ L2 (r), gives us, via the adjoint operator, the following result: for every function f (w, t) E L2(f21), W E r, t > 0,

g(z) = jj t1 12(w + it - z)2 f (w, t) dw dt

12.4 The operators associated with complex analysis

177

is in L2(F) and (4.13)

1191IL2(r) 5 CIIf IIL2(1 ) ,

where C depends only on M = IIa'Iloo-

Naturally, the same result holds for 522i the open set underneath F. This amounts to changing i into -i in the definition of g and putting h( z)=C(f)(z)=

fj

tl/2(w-it-z)2f(w,t)dwdt.

We now return to (4.12) and write f (w, t) = t1/2F'(w+it). This gives

F = 2ilr ' C(f) and IIFIIL2(r) 5

(cff tIF '(z+it)12dxdy)

1/2

as required. Before leaving Kenig's theorem, we reformulate it in the language of wavelets. For each ( V r, we consider the function OS E L2(I'.ds), defined by i1 (z) = (dist((, r))1/2(S - z)-2. This function has the regularity and localization properties that we have come to expect of wavelets. But the cancellation of s (z) is expressed by jr, 0( (z) dz = 0. Moreover, the functions ibb (z) are not normalized in the usual way: it would be

necessary to replace (dist(C,r))1/2 by (dist((,r))3/2 to achieve a normalization such that cl < II1CIIL2(r) < c2Our wavelets are indexed by C E C and we consider (continuous) linear combinations of wavelets of the form (4.14)

9(z) = ff

(z)a(() df drl ,

e

where the coefficients a(() satisfy (4.15)

ffi

)J2 dth h < oo. cad

We then have our reformulation of Kenig's theorem.

Theorem 7. With the above notation, there is a constant C = C(M), M = IIa'II,,, such that II9IIL2(r) < C(f f Ia(S)I2d dr7)1/2. Further, the continuous linear operator C : L2(1R2) -+ L2(r), which takes a to g, is surjective.

Before proving this theorem, we remark that it provides a particularly simple way of decomposing an arbitrary function g E L2 (r) into a sum 91 +g2, where gl E H2(521) and g2 E H2(112).

Indeed, g = C(a) and it is enough to split a into al + a2, where a, has support in 521 and a2 has support in 522. We then put gl = C(al) and g2 = C(a2). Let us proceed to the proof of Theorem 7.

12 Generalized Hardy spaces

178

The continuity of C is an immediate consequence of (4.13). It thus follows from Theorem 6, together with the identity L2(r) = H2(Slt) + 112(112). The fact that L is surjective uses the same ingredients. We first decompose g into 9.1 + 92, where gt E H2(111) and g2 E H2(112). The identity (4.12) then lets us expand gt and 92 as wavelets. Kenig's theorem and Caldenin's theorem thus imply Theorem 7. The approach of this section has been to use the T(b) theorem to prove everything- But the T(b)-theorem was proved only in 1985, some time after both Calderon's and Kenig's theorems were known. The following diagram summarizes the relationships between the theorems. T(b)-theorem

Theorem 5

[1985]

4 Kenig's theorem

4 Calderbn's theorem [1981]

[1977]

Theorem 7

5 The "shortest" proof We now want to give another proof of Calderc n's theorem. In the Spring of 1987, P. Jones and S. Semmes ([64]) discovered that it was possible to prove Calderon's theorem very simply, using only Theorem 6. Kenig discovered Theorem 6 in 1977 (it was published a year later). So, in a sense, Calderdn's theorem lay unnoticed in Kenig's Ph.D. dissertation. Let us state when the theorems were discovered:-

I Calderon's theorem (1977), II Kenig's theorem (1977), III Coifman, McIntosh, and Meyer (1981): Calderon's theorem in its strongest form, IV the T(b)-theorem (1985). Diagramatically, Calderon (1977) .4. [G David]

Kenig (1977)

full Calderon theorem.

After either Calderon's 1977 theorem, or Kenig's theorem of the same year, the obstacles in the way of a proof of the full form of Calderon's

12.5 The shortest proof

179

theorem were mainly of a psychological nature. We were amazed by this

fact, which is why we are giving an account of all the approaches to Calderon's theorem. To appreciate the "shortest" proof of Jones and Semmes, we must make the proof of Kenig's theorem independent of the T(b) theorem, since Calderon's theorem is also an immediate corollary of the T(b) theorem.

But there have "always" been several direct proofs of Kenig's theorem.

The first is due to B. Dahlberg ([81] and [82]). Another can be found in Kenig's thesis. Using either of these methods, a simple proof of the L2 continuity of the Cauchy kernel on Lipschitz curves can be obtained. We shall describe that proof now. To start with, we establish the continuity of the operator C of Theorem 7, without using Calderon's theorem, namely, the decomposition L2(I') = H2(121) + H2(522). That is, we begin with a function f (z, t) E L2(121), where z E IF, t > 0, and where (z, t) E IF x (0, oo) is identified with z + it E 521. We next form ao

tl/2

9f (w)=IL

(5.1)

(z-wfit)2f(z,t)dzdt

and claim that (5.2)

II9±IIL2(r) Iz - wI/2 + dist(, F)/2.

12 Generalized Hardy spaces

180

Putting a A b = inf (a, b), we observe that, for z, ( E f and w E 512, dist(w, r) < Iw - zI A Iw - (I. Lastly, for every a > 0, there is a constant C(a) such that, for a, b E C,

r > 0, and R> 0, (53)

Iw-alAlw-bI

t

C(a) (lb

dzbdu

R)-n

(r A

- al + R +

r)3-«

We are left with estimating

IL

s11R(z)If(z)IIf)I

r))-a R(z () _ (dist(z, r))'/2(dist((, r))1i2(dist(z, r) A dirt((, r))3-«

(Iz - (I + dist(z, r) + disc((, To do this, it is enough to apply Schur's lemma, that is. to verify that there exists a constant C, such that f f R(z, () dx dy < C. The symmetry of the kernel R(z. () will then let us deduce the L2 continuity which we require. The verification is simple, because it is enough to use x and t = y - a(x) as variables to reduce the expression to J00 J00 dxdt (I

00

x - (I + t+r)3-'

and it is immediate that this is bounded above. We have just verified that, if

9t (w) = I 1o"0 (z

-tw+it)2f(z,t)dxdt,

then 1/2

If(z,t)I2dsdt) via

(5.4)

II9±IIL2(r) 0, and we let E be the set {s E IIt : z(s) E Al. Then we let z1 = z(s1) be a point of r such that Izo - z1I = 10r. Such an s1 can be found, since Iz(s) - zoo takes all the values between 0 and oc when s runs through the real line. Having done this, we consider an interval I, centre s1: such that, for s E I, we

have jz(s) - z(s1)l < r/2. Let a be the argument of zo - z1. Then, if z E r fl A and (= z(s), s E I, the modulus of z - { lies between 8r and 12r, while the argument of z - C is in [a - 7r/4, a + lr/41. This implies that, for every function f (s) > 0 with support in E and for every t E I, C (6.9) dsl IJ JE f (s) ds, (s) (t) where C > 0 is a constant which is easy to calculate. We apply this remark to the characteristic function f of E and apply the hypothesis that Z(sf

g(t) = J z(s)

z(t)

ds

12.7 Transference

185

restricted to r has an LP norm dominated by that off . These inequalities give r-1+1IPIEI < C'IEI1/P, or AEI < C"r, as required. The rest of this chapter is devoted to the converse, that is, the proof

that (6.4) = (6.8). Here is the outline of the proof. The geometric definition of Ahlfors regular curves implies that, at every scale, a regular curve I' has good Lipschitz copies. More precisely, when (6.4) is satisfied, there exists a constant M such that. for every interval r C IR, we can find an orthonormal co-ordinate system RI and a Lipschitz function al, satisfying 11a' 11., < M and such that the parametric representation of the graph r1 of al in the co-ordinate system RI, denoted by zr(s), satisfies z1(s) = z(s), for s E E C 1, with (6.10) f EI > yIII, -y > 0, like M, being a constant which depends only on the constant C > 2 which appears in (6.4).

The second notable point of the proof is the possibility of transferring the estimates which apply in the case of Lipschitz curves onto the general

curves described by Theorem 8. This "transference" method depends on techniques of real analysis, invented by Calderon and Zygmund and then developed by Burkholder and Gundy. These methods are similar to those described in Chapter 7. The next section is about this line of attack.

7 Transference CS.

Let C* denote C\ {0} and let K be a function defined on C* which is homogeneous of degree -1 and odd. We also suppose that K is infinitely differentiable on V. Let µ be a positive, regular, Borel measure on the complex plane. For every e > 0, we form the truncated operator TE defined by spy

TE f (z) =

(7.1)

K(z - w) f (w) dp(w),

when f is a continuous function of compact support. We define the maximal operator T* by (7.2)

T. If (z) = sup IT. If (z) e>O

..Q

The problem of characterizing the measures p such that, for every kernel K satisfying the hypotheses above, the operator T* is bounded on L2(dp), is still open.

12 Generalized Hardy spaces

186

We intend to show that the operator is bounded when p is the arclength measure on an Ahlfors regular curve r. We know that this is the case when r is a Lipschitz graph. Furthermore, the hypotheses on K and the inequality we intend to establish are invariant under rotation and translation. We can therefore allow arbitrary displacements on Lipschitz graphs. The corresponding arc-length measures dp will uniformly be "good measures" for which II T* f II2 < Gill I12. The constant C depends only on the Lipschitz norm we use.

coo

We shall restrict our attention to a collection E of positive, regular, Borel measures p. To say that p E E will mean there are constants Cl and C2 such that, for each disk D of radius r in the complex plane, we have p(D) < C2r with, further, p(D) > Cir, when the centre zo of D lies in the support of p.

`-'

pip

..+

I+1

Proposition 2. Let It and or be measures in E. Suppose that, for every 1 < p < oo, the operator T.,, is bounded on LP(dp). Then T* : L"(dp) L"(&) and T,, : LP(do) -> L"(dp) are both continuous. The proof of the proposition depends on a series of lemmas very similar coo

to those used in Chapter 7. If p E E and f E Ll (dp), we define the maximal function Mµ (f) by

M(f)(z) = sup 1 If (w)I dp r>0 r z-wl'a

Lemma 10. If IL E E, then there exists a constant C = C(p) such that, for every zo E C and each r > 0,

r

(7.5)

If (z)l_ dµ(z) < CM,, f (zo) Iz-zol?r 1z - zoll

.

The proof is immediate. It is enough to decompose Iz - zoo ? r into dyadic annuli 2'r < Iz - zoo < 2a+1r and to apply the definition of Mµ f to each of these. The significance of the next lemma is that it is enough to know the coo

maximal operator T,, on the support of µ to be able to evaluate it coo

everywhere. This confirms our intuition, because it is on the support of p that the situation is at its worst. coo

Lemma 11. Let p E E. There exist constants C, and C2 (depending on µ and K) such that, for every z E C and every continuous function f of compact support, (7.6)

T* f (z) < C1141µ(T* f) (z) + C2M, f (z) .

To establish (7.6), we first replace z by zo and the left-hand side of (7.6) by T,' (zo). We now want to find an estimate, uniform in e > 0, of II'E f(zo)I

12 Generalized Hardy spaces

188

Let D(e) be the open disk Iz - zoI < c and let D(e/2) be the disk of half the radius. We write f = f1 + f2, where fn (z) = f (z), on D(e), and fu (z) = 0, otherwise. Then / TE f (zo) = T°f2(zo) = JK(zo - w)f2(w) d##(w)

We then observe that, for every z E D(e/2), ITS f2(z) - T1'f2(zo)I < CMµf (zo)

(7.7)

(we could even replace the right-hand side by CM," f (zl ), where zi E D(e/2), a remark that we shall use later). The above inequality follows immediately from Lemma 10 and the fact that

IK(zo - w) - K(z - w)I 4d, we take the mean of the inequalities (7.8) with respect to the restriction of p to D(e/2). By the definition of E, p(D(e/2)) and e are of the same order of magnitude and we get Ilrf (zo) I < CMJ. f (zo) + C'Mo f (zo) . If d/2 < e < 4d, we observe that ITE f (zo) - T4d f (zo) I < CMS, f (zo), and we are back to the preceding case.

Finally, if 0 < e < d/2, we necessarily have TJ = T4d, which brings us back, once again, to the first case. We have proved Lemma 11. We now return to the proof of Proposition 2. We suppose that, for

1 < p < oo, T,, : LP(dp) -+ LP(dI) is continuous. Then (7.6) and Lemma 9 give the continuity of 7;` : LP(dp) -+ LP(dv). In some sense, the continuity of T. : LP(dv) -, LP(dp) is the dual of the preceding result. The truncated operators TE : LP(dp) --+ LP(dv) are uniformly bounded, because the maximal operator Ti" is continuous as an operator from LP(dIL) to LP(do)_ By duality, it follows that the truncated operators 7E : L4 (do,) -+ Ly(dji) are also uniformly bounded (1/p + 1/q = 1). We still have to pass to the maximal operator 7°, for which we again need some "real-variable" techniques.

12.8 Calderdn-Zygmund decomposition of Ahlfors regular curves 189 converges weakly to an operator

Let ej be a sequence such that 7

that we shall denote by T°. So, for f E LQ(da) and g E LP(dp), (7'°f, g) = 7lim (TE f, g)

We intend to verify that the pointwise inequality (7.9)

Ta f (zo)
0. Then there is a compact subset E C I such that. for x. z' E E. (8.5)

x

(-(x' - x) AT) 2! 211

and

If(E)I >

(8.6)

2

Lemma 12 means that, if the fluctuations of f have a total outcome 1 which is positive, then f mast be increasing, with a slope of at least 1/(2111), on an appreciable subset of 1, whose size is measured by (8.6).

In our application of the lemma, the function f (x) will also be Lipschitz and satisfy 11 f'II= < 1. Then (8.6) will give IEI > If (E)1 > 1/2. In what. follows, we shall extend the restriction of f to E as a function g, defined on the whole real line and constrained to be linear on the contiguous intervals of E (and to be of the form (1/2111)x + con each of the remaining half-lines). In this case, we clearly get (8.7)

x' > x = g(x') - g(x) > 2III(x - x),

for all x', x E R, and (8.8)

g(x) = f(x)

We return to Proposition 3.

for x E E.

12.9 The proof of David's theorem

191

'..

ego

Let r = [a, b] and let K C F be the arc of r between z(a) and z(b). We denote the diameter of the compact set K by dK. Since F is an Ahlfors regular curve, iIl < CdK. Let a1 and b1 be two points of r such I°'

that jz(bi)-z(al)I = dK. Since jz(bl)-z(al)I : b1 -al, it follows that bl - a1 > c(b - a) for c = C-1 > 0. We now forget about the interval pp'

I, replacing it with [al, b1], and look for the set E in [al, b1]. We can further simplify the notation and omit the subscripts, putting a = al and b = bl. We may suppose, from now on, that there is a constant con

c'>0such that (z(b)-z(a)I>c'(b-a). A translation followed by a rotation now lets us assume that z(a) = 0 °..

andz(b)=l>c'(b-a)>0.

fin'

We finally write z(s) = x(s) + iy(s), in order to apply Lemma 12 to the function x(s)So we have an increasing function a(s) and a constant c" > 0 such that (s') - k(s) > c"(s' - s) whenever s', s E R satisfy s' > s. Further, c(s) = x(s) for s E E, where JEJ > vIfl, with v > 0 being a constant. We put Cr(s) = c(s) + iy(s), where z(s) = x(s) + ill(s). It is no longer true that s is the arc-length of the curve represented parametrically by (,(s). But the curve r,, given by {I(s), is the graph of a Lipschitz function C5.

Leo

and (I(s) = z(s) when s E E. By the construction of a(s), we have c"(s' - s) (s') - c(s) < s' - s, so that the arc-length t of the curve rI and the parameter s are related by c1(s' - s) < t' - t c2(s' - s), where c2 > e1 > 0 are constants.

All that is left to do is to modify rI at the points s

E so that,

Fly

for each complementary interval (s9, s ), the arc-lengths of rI and of r between z3 = z(sj) and zj' = z(s,) are the same. These arc-lengths are both of the same order of magnitude as sil - s or x,' - x3 (where, again, xj = x(s3) and x,' = x(s,)). The modification is very simple. It consists of replacing I'r, for x9 x < x,, by a polygonal line z_,wwz9, where wl = u9+ivf, uj = (xJ+xj)/2, and where v9 is chosen appropriately. Doing this does not alter the Lipschitz character of FI, and we have completely proved the proposition. f3.

ewe

a°'

9 The proof of David's theorem Let r be an Ahlfors regular curve. We let K(z) denote a homogeneous kernel on C*, of degree -1, odd, and infinitely differentiable. We know

that, if a : R -- R is a Lipschitz function satisfying iia'",,,, < M, then the kernel

K(a, s, t) = K(s + ia(s) - t - ia(t)) ,

s, t E R,

1 2 Generalized Hardy spaces

192

defines a continuous linear operator on LP(R), for 1 < p < no, and that the norm of this operator depends on the function a only to the extent that M does. If we put z(s) = s + ia(s), for s E R, we see that the same conclusion holds for K(a(z(s) - z(t))), when IA = 1, because K(z) and K(Az) satisfy the same hypotheses. By the general theory of Calderon-Zygmund operators of Chapter 7, the maximal operator corresponding to K(a, s, t) is also hounded on L)(R). This maximal operator is not the operator we considered in the earlier sections of this chapter, but Lemma 11 enables us to pass from one to the other. Lastly, if u is the arc-length measure along a Lipschitz curve, the operator T,, : LP(dli) LP(du) is bounded and the operator norm depends only on M (and K). Applying Proposition 2, we can transfer this estimate. Let IF be an Ahlfors regular curve and o, the arc-lengthF measure on r. Then (9.1)

IIT* (f)IILP(dc) S CII J IILP(dl+)

and (9.2)

IIT* (f)IILP(d,.) R, belongs to LP; (9.5)

there exist e > 0, -y(c) > 0, and 0, with 0 < 0 < 1. such that ,l3 < (1 + e) -P, and, for every A > 0.

I{x E l : u(x) > A +eA and v(x) : y(e)a}l :5131l.c E R: u(x) > A}I.

Then u E LP(R) and (9.6)

IIuIIP < ((1 + e)-P - fl)- /P(y(e))-m llvllp

The proof is identical to that given in Chapter 7. even though the fir'

hypotheses are stated in a slightly different way. In our case, f is a function of compact support (in the complex plane), u(s) = (T, f)(z(s)) and v(s) = (Mol f IT(z(s)))m/'', where 1 < r < p. The

...

12.9 The proof of David's theorem

193

measure a is, as we have said, the arc-length measure along the Ahlfors regular curve with are length parametric representation z(s). If we manage to verify (9.5), then David's theorem will follow from (9.6). By construction, u(s) is lower semi-continuous and vanishes at infinity.

n''

The set Q. defined by 1 = {s : u(s) > A}, is thus a bounded open set and hence the union of disjoint open intervals (a., b,). As usual, we let E3 be the set of s E (a.,,bj) such that u(s) > A + eA, where E > 0 will be chosen below. We may restrict to the case where there exists e E (a3, bj) such that v(e) < rya and we then intend to show that I Ej I < 8(b3 - a. ). Summing the inequalities then gives (9.5). .-.

As ever, we write f = fl+f2, where f, (z) = f(z), if Iz-z(a3)I < 213, where 13 = bj - aj, and fl(z) = 0, otherwise. Then, on repeating the proof of (7.6), we get (9.7)

T'f2(z) ae/2 is not greater than vlj/2. The set A, = Kj \ R. satisfies IA.I > W., /2 and, if s E 03, we have

12 Generalized Hardy spaces

194

T; fi(z(s)) < Ae/2 and T* f2(z(s)) < A + AE/2, giving I."f(z(s)) < A+ea. The number f3 E (0,1) is thus 1 - v/2 and, taking 0 < e < f3 - 1. (9.5) is proved as, indeed, is David's theorem.

10 Further results David and Semmes are in the process of extending David's theorem on rectifiable curves to livpersiirfaces in R"+1 Here is a result due to Semmes ([212]).

We consider an orientable surface S C R'+' which separates R ^O'

into two open connected components 121 and Sl2 and such that there is a constant K > 1 so that, for all x E S and for all R > 0, we have (10.1)

K-'R" -,

a,, 1 < j < k. These hypotheses are that the aj belong to the Wiener algebra A(R') consisting of the Fourier transforms of the functions in L'(IR"). Passing from A(]R") to L°`(R") remains a major difficulty, because we still have only partial results: sufficient conditions on the multilinear symbol which enable us to establish the Milder inequalities. To show that these conditions are, indeed, sufficient, we apply David and Journe's T(1) theorem. The same approach will be used in Chapter 14, to deal with Kato's problem about the domain of the square root of an accretive, differential operator, written in a divergent form.

2 The general theory of multilinear operators Let us begin by recalling the definition and properties of the Wiener algebra A(IR" ). It consists of those functions f which are the Fourier transforms of functions f E L'(IR") and, by definition, the norm off in A(IR") is the norm off in Lu(l "). Thus A(lR) is a (dense) subalgebra of C0(IR"), the algebra of all continuous functions on R" which vanish at infinity. Furthermore, the norm of g E A(Rn) is invariant under translation and dilation. The Wiener algebra A(Rn) is the natural space to start from, for the theory of multilinear operators which commute with translations and

dilation. moo'

Indeed, let R : A(R") - A(R") be the operation of translation by x, that is, R. f (y) = f (y - x). Similarly, for S > 0, we define Db : A(Rn) -, A(Rn) by Dbf(y) = f(b-1y). In the next proposition, we write A for A(IR") and L2 for L2(IR"). Also, in what follows, for

198

1 8 Multilinear operators

a E A it will be convenient to write a E Ll (It") for the function whose (inverse) Fourier transform is a.

Proposition 1. Let 7r : Ak x L2 - L2 be a (k + 1)-linear operator which is continuous and satisfies (2.1)

ir(Rxa1,... , Rak, Rf) = Rxir(al, .. , ak. f) ,

for all x E R", a1,.., at E A, and f E L2. Then there exists a function T E L°° (R'"(k+l) ), called the symbol of rr, such that (2.2) ir(a1,... , ak, f) = (27r)n(k+l)

fJei(r(n,e)a(r)J(e) dr7 4

where 77 = (rri, ....77k), f1 = 771 + ... + rlk and 6(r7) = al (rll) ... dk(77k).

If, moreover, 7r commutes with dilations, in the sense that (2.3)

7r(Dba1i...,Dsak,Dsf) = Dsir(a1,...,ak, f),

for all 6 > 0, then T is homogeneous of degree 0: (2.4) T(b77, b f) = -r (?7, ) for all 6 > 0. Conversely, if T E L°° (IRn(k+1) ), then (2.2) defines a multilinear operator which commutes with translations (in the sense of (2.1)) and (2.5)

IIir(al,.--,ak,f)112 _
0. We can find a function f , with the above properties, which does not vanish on IxI < R. Further, the weak convergence of A. to 0 lets us replace Si+2([3m.) by So((3m) and then by the constant cm = So

(0). The error terms, introduced in this way, can be written

as a scalar product of 3,,,. with an Hl function. We thus arrive at

CAD

condition (4.3). Conversely, if (4.2) and (4.3) hold, then we easily replace (4.3) by the convergence, in L2(WI, (1 + IxI)-n-1 dx), of the functions 0,,,. - So([m) to the function J3-So(/3). We then conclude that the functions 7r([3,,,, f) converge to 7r(,6, f) in L2, when f has the special properties as above. We are now in a position to state the fundamental theorem.

Theorem 2. Suppose that, together with the notation and hypotheses C>3

of Theorem 1, we have T (i1 i ... , qk,1;) = 0, whenever one of the qi = 0

(1 < j < k). Then the multilinear operator in : Ak X L2 - L2, defined by (2.2), satisfies .i7

(4.4)

IIir(bl,...,bk,f)112

0, h E R". We define the unitary action U9 of G on L2(IR") by U9f (.r) =

13.5 The general theory of holomorphic functionals

211

c}.

..,

ova

ear

^1'

6-" l2 f (6-1(x - h)). For the action of G on L°°(R?), it will be appropriate to use the operators V9b(x) = b(6-1(x - h)), 6 > 0, h E R". We are interested in the holomorphic functionals (5.1) F: S2 A which obey the commutativity rule (5.2) for every b E Il and g E G, F(Veb) = U9F(b)Uy 1 and the continuity rule (5.3) if 0 < r < 1 and if the bj E S2 satisfy llb2 (lam < R and converge strictly to b, then the operators F(bj) converge strongly to F(b). We can express the latter condition more symmetrically by requiring that, if the operators B, : L2 -. L2, given by pointwise multiplication by the functions bj (x), converge strongly to the operator B (pointwise multiplication by b(x)), then F(bj) converges strongly to F(b). This condition means that, in a certain sense, F(b) resembles the operation of pointwise multiplication by b(x). As in the multilinear case, the condition is indispensable if we want to start our analysis of F(b) by restricting to the case where b is in the Wiener algebra A(R' ). The holomorphic functionals we have just defined are analysed in the following theorem. Synthesis, on the other hand, pasts difficult problems which we shall describe later.

Theorem 3. A holomorphic functional F : S2 -- A, which satisfies (5.2) and (5.3), has the form F(b) (5.4) where, for b1 i ... , bk in the Wiener algebra A(W'). (5.5) Tk(b1,....bk)(f)(x)

(27r)n(k+1)

f

e

Tk1911, .

. 77k,

X 61(771) ... ak (?lk )f (C) [1711 ... d71k
0, (5.7)

T k (A71i , ... , ask, AC) = Tk (7h .... , Ilk,

)

Further, there exists a constant C > 0 such that, for all k E N, every choice of b1, ... , bk in L°°(Rn), and each f E L2(Rn), (5.8) 1jTk(bii...,bk)(f)112 < Cklibill, ... IRbklloollfll2. Finally, the operators Tk inherit the continuity property (5.3) from the functional F.

1 3 Mvltilimear operators

212

Before proving Theorem 3, we must remind the reader that holomorphic functionals F on an infinite Banach space B may have the following pathological behaviour. It is possible to find functionals which are entire on the space LO° (R) but which are not bounded on a given ball centre 0. Here is an example. We consider the linear forms

Ak(f) _ jl f

(x)e{dx,

for k E N and f E L°° (Ill;), and we put

x

F(f) _ 1:(ak(f))k 0

This functional is entire, because Ak (f) - 0 as k -i oo. But it is not bounded on the ball IIf IIx :5 2. If it were, 2 e-`k°F(e'e f) dO would also be hounded (u{j2 niformly in k). Thus eOF(eief)

sup

dOIIf II< 21

would be a hounded sequence. But this sequence is 27r2k. We return to Theorem 3. A holomorphic functional F on the unit ball IIbIIx < 1 is continuous at 0 and thus bounded on a neighbourhood of 0, on some ball IIbIIx < b. But we have no control over the value of b > 0. We put j27r Tk (b)

dO

when IIbIIx < b, and thus have IITk(b)IIA e > 0), and KE,R = 0 otherwise, be the tnmcated

.-.

3a.-

kernels. Since T(b) is a Calderon-Zygmund operator, the truncated operators TE,R defined by KE,R are uniformly bounded on L2(I[l;) and con-

{''

verge strongly to T(b). These truncated operators TE_R(b) are clearly holomorphic on IIbII= < 1 and are uniformly bounded on IIbII0 S r < 1. Since the truncated operators converge strongly to T(b), the latter is also holomorphic on IIbII< < 1.

We expand the operator T(b) as the series EO (-1)krk(b), where the The Fk(b) are defined by the kernels PV((B(x) - B(y) )k/(x kernels of the corresponding miiltilinear operators are PV (Bi(x) - B1(y)) ... (Bk(x) - Bk(y)) y)k+i).

(x - Y) k+1

where Bi = bi E L°O(IR),... , Bk = bk E L°O(I18" ). The corresponding multilinear symbol is 1 / 77k) _ I'

III

TOD,

M 77, ... 77k

13 Multilinear operators

214

where O-M) = f(C + a) - f(0

This multilinear approach was the basis of all subsequent work on Calderon's programme, but did not give the estimate (5.8), which Calderbn had obtained in 1977 by complex variable methods ([38]). In fact, we can even improve the estimate (5.8) if we have a bound for the norm T(b) : L2 -* L2, when IIbii= < 1 and IIbii... is close to 1. Using the results of Chapter 9. we get

IIT(b)II < C(1- Ilbll=)-5 . Since

rk(b)

1

j27r

e-4kOT(eaeb) d9,

2-7r

we

t

Ilrk(b)II S C(1- Ilbll00)-5, if IIbII0 < 1. We then bring the

homogeneity of rk into play and suppose that I1bI100 < 1- (k + 1)-1. In

this case, we get llrk(b)ll. < C(1 + k)5, and this estimate extends to the case 11b1I00 < 1, since (1 - (k + 1)-1)-k < C. This estimate is not optimal ([53]). We do not know what the best estimate is. On the other hand, the optimal estimate of IIT(b)II is known and is given by C(1- I1b1I.)-112, as has been shown by David ([94]) and Murai ([194]).

We conclude the proof of Theorem 4 by verifying that. if the bj converge strictly to b, and if Ilbj ll,,. < r < 1, then the corresponding operators T(bj) converge strongly to T(b). We again write T(b) = Eo (-1)krk(b), where we have shown that Ilrk(b)II < C(1 + k)511b11.

The series of operators which appear in T(b?) and T(b) are absolutely convergent, uniforn ly in j. It is thus sufficient to verify that, for each fixed k, the operators rk(bj) converge strongly to rk(b). Since we already have uniform estimates for the norms of the operators involved, we only need verify that, for each C' function f, of compact support, rk (bj) [ f ] converges to rk(b)[f] in L2 norm. Let [-T, T] be an interval containing the support of f . If Ix I > T + 1, we can write

rk(bj)[.f](x) = f

(Bj(x) - Bj(y))k f(y) dy (x - Y) k+1

This gives

Irk(bj)[f](x)I rIIbII,,,, for every b E L°°(IR). This inequality may be proved by means of the following lemma.

Lemma 1. Let T : L2(11r) --> L2(R') be a bounded linear operator with distribution-kernel K(x, y). For xo E R", and b > 0, write g(x) _ b-1 (x - xo). Then b'K(b.c + xo, by + x0), is the distribution-kernel of T(x°.6) = UgTUU i, whose norm is the same as that of T.

In our application, K(x,y) = PV((B(x)-B(y))/(x-y)2), where B is ear

a primitive of b. We let b > 0 tend to 0: the weak limit of the operators T(2°,s) is s-b(xo)H, where H is the Hilbert transform. This weak limit exists at each xo for which b(xo) is the derivative of B(x). We thus get 7rIlbllc < pr,(b)II, as claimed.

6 Application to Calderon's programme

cat

The sharper form of Calder6n's symbolic calculus has the following goal: to achieve precise results about the commutator of a classical pseudo-differential operator (essentially a convolution operator) with an operator of pointwise multiplication by a function with prescribed regularity. We shall give examples where the optimal results can be obtained by the multilinear calculus developed in section 3. Let us start with a one-dimensional example, in which the pseudodifferential operator is the Hilbert transform H.

-i7

13 Multilincar operators

216

Theorem 5. Let b(x) be a locally integrable function of the real variable x and let B be the operator of pointwise multiplication by b(x). Then the commutator [H, B] is bounded on L2(R) if and only if b(x) is in BMO. In that case, if the functions bj e BMO converge to b in the a(BMO, H1) topology, then the corresponding commutators [H, B?] converge to [H, B] in the weak operator topology.

Let us start by verifying the necessity of the condition b E BMO. Let I = [a, b] be an interval of length I and let J be the interval [b+1, b+21].

We let K(x, y) be the kernel of T = [B, H], which we suppose to be bounded on L2(IR). One way of testing the L2 continuity is to consider the auxiliary function g(x) = f, K(x, y)(x - y) dy, x E I, and to verify

that II9II0(r) < C13'2. Indeed, if we write x - y = .c - xo + xo - y and g(x) = 91(x) + 92(x), where gl(x) = (x - xo)T(xi) and 92 (X) = T((xo --)kj), then the continuity of T on L2 gives the stated inequality. In our situation,

r

"dam

9(x) =

(b(x) - b(y)) dy = l(b(x) - cl)

J and

ll9IILa(r) < C1312

is equivalent to the condition b E BMO. We now show that the condition b c BIM is sufficient. We start. with the special case where b is in the Wiener algebra A(R). The general case then follows via the weak continuity property described in Theorem 5. If f E L2 and iff brE A(R), then c.0

[H, B]f = -7ri J J e"`(C+' (sgn(Q + r) - sgn

t.7

= -7ri ff et "(+,l) (sgn(+ rl) - sgn.)b(r )9lo

dy d

\ /

f(

) dl d ,

where co E D(IR) equals 1 on [-1, 1]. The reason why this identity is valid is that sgn(!; + rl) = sgnC if 19l < 111, and it is only in this case that ¢o((/rl) # 1. We let ir(b. f) = Tb(f) be the bilinear operator defined by the symbol which is clearly homogeneous of degree 0 and infinitely differentiable except at (0, 0). By Theorem 2, Tb is bounded on L2 (R), when b is in BMO(R). Further, [H, B] = [H, Tb], by the bilinear identity above.

Thus (6.1)

for b c A(IR).

II [H, B] II < CIIbIIBMO,

13.6 Application to Calder6n's programme

217

To pass to the general case, we consider the trilinear form

J[H,B](f)gdx = JH(bf)gdx - JbH(f)gdc

_ -J bf H(9)dx-J bH(f)gdx

_ -Jbadx where

h=9H(f)+fH(9) If f,g E L2(R), then h lies in L'(R) and I fbhdxl 5 CIIbIIBMO, for b E A(R). As a consequence, h belongs to the Hardy space H' (III) and we have obtained the estimate 119H(f)+fH(9)IIHI

(7-5)

... 0(7)k + C))c(tC)m(t)

and (3.1) follows immediately. It is worth remarking that 7r(77, C) is homogenous of degree 0 if m(t) is a constant function. cow

We obtain the second statement of Proposition 5 by a method of separation of variables which we have used before, in Section 3. We

13 Mudtidinear operators

224

shall use it now in a slightly sharper form, described in the following lemma, for which we need to fix some preliminary notation. We let D(RP X 1R) denote the Schwarz space of compactly supported,

infinitely differentiable functions. We give it the usual topology. A bounded subset B of D(IRP x ]R9) is a set of functions f E V whose compact supports are contained in a fixed compact set and whose successive derivatives satisfy uniform bounds.

Lemma 2. For each bounded set B C D(RP x Rq), there exist two sequences gj(x) and h,(y), 3 E N, belonging to bounded subset,,; BI and B2, respectively, of D(RP) and D(lR) such that every function f E B may be written in the form CO

f (x, y) = E w9 (f )g3 (x)h3 (y)

(7-6)

0

where w3 (f) is a rapidly decreasing sequence.

As we have already had occasion to remark, we get (7.6) by the following three operations. Firstly, we periodify f in each variable, using a sufficiently large period T. Secondly, we expand the periodified function as a Fourier series. Finally, we use a cut-off function to separate f (x, y) from the parasitic terms accompanying the periodification. The three steps are all linear, so, if f (0, y) = 0, for all y, then gj (0) _ 0, for all j E N.

Let us now analyse the multilinear operators of Theorem 1, using McIntosh's formal calculus. We let w(g,C) be a function in D(Rn(k+I)) taking positive or zero values, which we shall suppose to be radial. We

further require w(77, C) to vanish in a neighbourhood of 0, but not, of course, to be identically zero outside that neighbourhood. Multiplying w by a positive constant, if necessary, we can assume that fO° L,) (h7, tl;)t-1 dt = 1, for (17, t;) # (0, 0). We now write ""J

J0

ire(tr1, t) tt , y,7

7r(?J, C) =

where 7rt(r7, C) _ 7r(t-177, t-IC)w(r7, ) . 1r'

Since (3.1) holds for all N > 1, the functions 7rt(1;, 77) form a bounded subset of D(R7 (k+l)). We shall analyse these functions, using Lemma 2 k/'

and the new variables 771 + - . - + 71k + C, 712 +

+ 77k + 1;, ... ,17k + t;, t;.

More accurately, we shall use the obvious generalization of Lemma 2 in which the two variables x and y are replaced by the k+ 1 variables which we have just defined. Looking at the case 7r(77, l;) = 0 when 27, +

+ 77k + C = 0, we get the

13.7 McIntosh'a theory of mnltilinear operators

225

expansion (7.7) t(27,

)_

co

mi (t)ai>s (rJl + ... { k + ) ... ak,J (1lk + C)b; (C) ,

0

where (7.8)

II m.3 (t) II r,' (o,o >) decreases rapidly,

(7.9) a1,j(0) = 0, and all the derivatives of a1,1 vanish at 0, and

mar.

(7.10) the functions a3,1, ... , akj and bj run through a bounded subset of D(R"). If we now put b3(C),

j(t') = a13(t'),

we obtain 7x(77, C) as a series of operators arising out of the McIntosh formal calculus, except for needing k different functions ¢. The cases where 7r(77, ) = 0 for 172 + - - - + = 0,. - -, = 0 follow by the same method applied to the terms 7x(77. C)X' (77, 6), ..-,7r(27, C)xk (71, )-

This concludes the proof of Proposition 5. It thus seems that Theorem 1 and McIntosh's approach are equivalent routes to the same results. This is not quite the case. Going back and forth between functions and Fourier transforms always requires too many hypotheses: for example, one cannot get the correct estimates for the Caldercin commutators in the formalism of Theorem 1, because the corresponding multilinear symbols are not sufficiently regular. On the other hand, the success of McIntosh's programme first became apparent in the analysis of these commutators. During his visit to Paris in 1980-91, McIntosh rsuggested studying them by using the identity (7.11)

PV

oo (73(x)

J

00

-,8(y))k

f(y) dy

(x - yr)

= PV J 7. [(I + itD)-1B]k(I + itD)-1 f

dt

t where D = -i(d/dx), b(x) E L°°(IR), B is multiplication by b(x), and /3(x) is a primitive of b(x). This identity was the starting point for the first proof of the L2 con-

tinuity of the Cauchy kernel on an arbitrary Lipschitz curve ([65]).

In fact (I + itD)-1 = Pt - iQt, where Pt and Qt are defined, as above, using O(x) = e-HxI/2 and 7/;(x) _ -sgnxe-1"I/2. The function,O does not, however, have the regularity that we imposed above. Now, note that Pt is an even function of t, Qt is an odd function, and PV f iQt)B]k(Pt - iQt)t-1 dt can thus be split into a sum of terms each of which contains the operator Qt at least once.

13 Multzlinear operators

226

We recognize our old friends, the operators of Theorem 7. When the original proof was written, we did not have the T(1) theorem and the analysis of the operator on the right-hand side of (7.11) was carried out by relating the operator, using certain tricks, to quadratic functionals of the form (f °C JQt(BPt)k fl2t-1 (lt)1/2. These latter were estimated using appropriate "Carleson measures". This means that all the recipes to get the L2 continuity of the Cauchy kernel on Lipschitz curves use essentially the same basic ingredients: the Carleson measures. If we go back to the

proof of the T(1) theorem, they appear in the proof of the continuity of the pseudo-product of a BMO function with an L2 function.

8 Conclusion

r--

Despite the results described in this chapter, we have only a rudimentary understanding of the multilinear operators 7r : (LOO )k x L2 - L2 which commute with translations and dilations. To illustrate this, let us consider, for k = n = 1, the bilinear symbol

1i!

7r(77, C) = sgn(77-1;). We do not know whether the corresponding bilinear

tea,

operator is bounded, as an operator from LOO x L2 to L2. If we were to restrict ourselves to functions 7r (7,1, l;), defined on R2 \ {0}, and homogeneous of degree zero, we might dream of finding a Banach space E of 27r-periodic functions such that the norm of lr(cos 0, sin 0) in the Banach space E was equivalent to the norm of the operator it : L°° x L2 - L2 corresponding to the symbol. Even using Theorem 1, we know only that E C LOO (R/2irZ) and that, for sufficiently large regularity r > 0, Cr is a subspace of E. Going back to the special case 7r(77,1;) = sgn(77-t;) and using a duality argument (as we did in Theorem 5) we may reformulate this problem in the following way. If f and g are in S(R), we put dt h(x) = PV f f (x - t)g(x + t) t 00 aid

and try to see whether there is a constant C such that IIhJ11 < Cf l f II2119002-

We end by remarking that the multilinear operators whose symbols satisfy the hypotheses of Theorem 1 have other noteworthy continuity properties, namely IIir(al,...

, ak,1)1 k 0 depends only on the dimension. To do this, we shall write the operator in a form using the resolvent of L, together with a series of multilinear operators which, after some work, can be analysed using David and Journ6's T(1) theorem ([96]). Towards the end, we shall return to dimension 1, where a version of Kato's conjecture gives precisely the operator defined by the Cauchy kernel on a Lipschitz curve.

2 Square roots of operators

Fr'

We start with the self-adjoint case. Let H be a Hilbert space, with inner product ( , ). Let V C H be a dense linear subspace and T : V -+ H a linear operator. We say that T is symmetric if (T f, q) = (f, Tg), for all f, g E V. The operator T is self-adjoint. with domain V, if one of the two following equivalent conditions is also satisfied: 11

too

(2.1) V is a complete normed space for the norm (IIT(f )II2 + (2.2) T + iI : V -+ H is an isomorphism.

Suppose, further, that (T f, f) > 0 for all f E V. Then there exists a unique, positive, self-adjoint operator S such that S2 = T. The domain W of S is the completion of V for the norm defined by ((T f, f)+II f Lastly, in a sense that we shall clarify,

II2)1/2.

ra

S= 1TJ (T + A1)-1 A-1/2 dA.

(2.3)

7

o

To make the me ring of (2.3) clear, we consider a much more general situation, in which T is no longer self-adjoint and positive, but has the following properties:

(2.4) T is defined on a dense linear subspace V C H and takes values in H;

(2.5) for all A > 0, T +.I : V - His an isomorphism; (2.6) there is a constant C > 1 such that, for all A > 0, we have

II(T+AI)-1II 0, in the sense of the holomorphic symbolic calculus.

Figure 2

We observe that the spectrum of T + eI is contained in a + E. We define the holomorphic function z1/2 in C \ (-oo, 0] (the cut is indicated in Figure 2) and we let IF be a contour contained in (St + e) \ (-cc, 01, as suggested by Figure 2. Then, putting T.. = T + cI, we get (2.7)

(T +

eI)-1/2

=

1 J((I 27ri

T)-1c-1/2 d(

The integral is absolutely convergent, with values in £(H, H). It does not depend on the choice of contour r, because ((I - Ti)' -is holomorphic in 11 + E.

Let us verify that (T + aI)-1/2(T + aI)-1/2 = (T + eI)-1. To do this, we use two distinct contours F1 and I'2i which are of the same type as F and satisfy the condition I(1> 6 > 0, as S1 describes t, and (2 describes I'2. We write down (2.7) for each of r1 and I'2 and multiply the right-hand

14 Square roots of accretive differential operators

230

sides together. This gives 1

41r

JS jl

(Ci7 - TE)-1((2I - Te)-'C1 1/2(2 1/2 d(1 d(2 -

To compute this integral, we use the identity -Ti)-'

((II -T£)-1((2I

= ((2 -(1)-' [(C'I -Te)-' -

((21-Tf)-'] .

This leads to two integrals. If as we may suppose, r1 is contained in an open convex region bounded by 12, we have 1

2Iri 1.

((2 - (1)-' 5212 d(2 = 0.

while

- C1)-1(1 2rri fl (Cz So we are left with 1

2rri

12 d(1 = -C2

1/2

CzEd(.

_ j'(c21_T)-'2_T-' z

Since T,-' : H ---+ V is an isomorphism, TE1/2 : H --> H is an injective operator. Indeed, if T "2(x) = 0, then T£ 1(x) = TE 112 (T 112(x)) = 0.

sox=0.

To simplify the notation, we set A = T,-1/2 and B = TeTE-1/2 We know that BA = I and from this we shall deduce that the domain of the (unbounded) operator B coincides with the image of A. We already know that. Im A C Dom B. Let y c DomB. We put By = z = (BA)z =

Bx, where x = Az. Thus B(y - .e) = 0. But TE and TE 1/2 are both injective, so B is as well. Therefore y = x and Dom B C Im A.

Let us verify that the domain is independent of e > 0. We write Be = TETE 1/2 and show that BE - B,, E £(H, H), when 0 < e < r, < oo.

To simplify the proof, we observe that, by an obvious deformation of the contour, we can reduce (2.7) to 00,

(2.8)

TE 1/2 = 1 fo (TE + AI)-1a''/2 dA

Next, we note that the £(H, H) norms of the operators T(T + AI) are uniformly bounded in A > 0. To see this, it is enough to write T(T + AI)-1 = I - A(T + Al)-' and apply (2.6). From this, it follows that T(TE1/2 _TT 1/2) E L(H,H), which implies Bf - B,, E C(H.H). We have thus established the first part of the following lemma.

Lemma 1. The image W of (T + aI)-1/2 : H -- H is independent of a > 0 and this image is the domain of (T + eI)(T + eI)-1/2 which. from now on, we shall denote by (T + al)1/2. On V it is also true that (T + 6i)1/2 = (T + eI)-1/2(T + eI).

14.2 Square roots of operators

231

mil

To prove the second assertion of the lemma, we show that T1/2Te = TE on V. Indeed, every p E V may be written as p = T-' (x), for some x c H. As a result, 1/2)(7

TE'12T i12(y) = TE,2T1,2T (c) = 7e'2(Tf7'f = T112(TTE ' )TE ''2 = TI/2T-1/2(X)

1/2T- 1/2)

=TTE 112TE 1/2(x)=x thus concluding the proof of the lemma. We have already indicated that TE/2 - T,'2 is continuous on H, for 0 < e < q < oo. More exactly, IITE,2

(2-9)

- T1/211 < C27,

where C is a constant. The proof, which we leave to the reader, depends on (2.8).

From (2.9), we deduce that, if x liras in the common domain W of the operators TE 12, then lime je TE12 (x) exists. The convergence is to be

understood as convergence in H and the limit is denoted by T'/2(x). In what follows, we shall need an integral representation formula for calculating T1/2 directly. This is given by the following proposition, which also summarizes the preceding discussion.

Proposition 1. With the hypotheses (2.4), (2.5) and (2.6), we define the operators T 112, fore > 0, by (2.8). The image W = TE 1/2(H) is independent of e > 0 and contains V. For all x E V, the integral fo T(T+AI)-' (x)A-'12 dA is an H-valued, Bochner integral, equal to f °O(T + AI)-'T(x)A-1/2 dA. For .c E V, we may define (2.10)

T'V2(x) _ ? 7r

Jo

T(T+AI)-1(x)A-1/2dA.

E-+

The operator T1/2 : V -+ H then extends to W as an operator, still denoted by T1/2, whose square on V is T.

To show that the integral is a Bochner integral, we split f0O into fo + ff O. To deal with the first integral, we observe that T(T +AI)-' _ 1 - A(T + AI)-' which, by (2.6), gives IIT(T + AI)-'II < C. Thus the integral fo T(T + AI)-1(x)A-'/2 dA converges for all x E H. As far as the second integral is concerned, we observe that, for all x E V, we have

T(T+AI)-'(x) = (T+AI)-T(x), and thus IIT(T+AI)-'(x)II = O(c-'). This ensures the convergence at infinity. To prove (2.10), we return to TE = T + cI, e > 0, .c E V, and to the

232

14 Square roots of accretive differential operators

integral (2.11)

T112(x) = 1 7r

Jo

TC(TE + 1\I)-1(x)A-1/2 d),.

We shall apply Lebesgue's dominated convergence theorem. We again

observe that, for 0 < \ < 1, IITE(TE + AI)-111 < C and, for A > 1, AI)-111 < Ca-1, uniformly in e > 0. Also, in the second case, if x E V, then 11TE(x)11 _< C, for 0 < e < 1. Thus the right-hand side of (2.11) converges to that of (2.10). To conclude these remarks, we should note that the set of x for which the right-hand side of (2.10) is a Bochner integral is, in general, strictly contained in the domain W of T'/2. That is, for an arbitrary element x c W, the integral (2.10) is not necessarily a Bochner integral. This difficulty will be apparent when T is an accretive differential operator. 11TE +

3 Accretive square roots In what follows, we shall restrict to the special case where T : V H is what T. Kato ([151]) calls a maximal accretive operator. This means that, for all x E V, (3.1)

RRe(T(x), x) > 0

and (3.2)

I +T : V ---> H is an isomorphism.

Let us begin by verifying that properties (2.5) and (2.6) of the preceding section hold. We shall establish a slightly more precise result. Lemma 2. An operator T : V - H is maximal accretive if and only if. (T+AI)-111 < A-1. for all A > 0, T+,\]': V - H is an isomorphism and II J"'

So what distinguishes the maximal accretive operators from those studied in the previous section is that the constant C of (2.6) is 1. Suppose that (3.1) and (3.2) are satisfied: let us establish (2.5). For

that, it is enough to check that S = I - (1 - \)(I + T)-1 : H , H is ELI

an isomorphism, because T + AI = S(I + T) and then T +,\I will be an isomorphism.

Clearly, S : H -- H is continuous. To show that S is an isomorphism, it is enough to show that there exists a constant 6 > 0 such that, for all

xEH, (3.3)

Re (S(x), x) ? 611x112 .

To establish (3.3), we use (3.2) to write x = (T + I)(y), for some

14.3 Accretzve square roots

233

y E V. This gives Re (S(x), x) = Re ((T + AI)(y), (T + I)(y))

=

'\11Y112 + (A +

1) Re (T(y), y) + IIT(y)112

> b(IIy112 + 2 Re (T(y), y) + IIT(y)112)

= blIxli2 ,

where b = min(A,1). We now show that II(T

Since T +,\I: V --' H is an isomorphism, we can put x = (T + AI) (y). We must show that +,\I)-1x1111x11

II(T + AI)(y)II > \11y11, for all A > 0. But II(T + AI)(y)112 = IIT(y)112 + 2A to (T(y), y) + A211YI12 2A211yI12-

Conversely, suppose that II(T+AI)-111 < A-1, for allA > 0. Then, for all x E H, IIT(x)112 + 2A Re (T(x), x) + A2IIxI12 >- A211x1I2 If Re (T(x), x)

were strictly negative, for some x c H, then the preceding inequality would not hold, for some small enough value of A > 0. Similarly, we can prove the following result.

Lemma 3. An operator T : V -> H is maximal accretive if and only '~l

if I + T : V -, H is an isomorphism and S = (I - T)(I + T)-' is a contraction.

An operator S is a contraction if IISII < 1. The salient point is to show that (3.1) is equivalent to 11(1 - T)xII < 11(1 +T)xII, when (3.2) holds. To do this, it is enough to square both sides of the inequality and expand. Part of the symbolic calculus on maximal, accretive operators is given by a classical theorem of von Neumann, which we now recall.

Theorem 1. Let H be a Hilbert space, S : H and P(z) = c0 + c1z +

H be a contraction. + c,,,,zm be a polynomial. Then

11P(S)11 :s sup IPWI Izj IIxI12/2, for all x E H.

Indeed, if we put y = Tx, then the lemma is a matter of verifying that

Re(y,(I- S)y) ? 2(y-Sy,y-Sy), which is equivalent to IIS(y)II (zk(Skx, y) +

zk((S.)kx,

y)) .

1

The series converges because IISII S 1 and IzI < 1. Putting y = x gives 00

zk(Skx,x) = Me((I-zS)-1x,x)-IIxII2 >_ 0,

BZ(x,x) = IIxII2+2 to by Lemma 4.

I'he proof of the Cauchy-Schwarz inequality now gives IB.(x,y)I 0, continuous on ate z > 0, and tend to a limit as IzI - oo. An obvious consequence of von Neumann's theorem is the following proposition.

Proposition 2. Let T : V - H be a maximal aecretive operator. Then there exists a unique algebra homomorphism X : A - C(H, H) such that X((A+z)-1) _ ()\I+T)-1, for all A > 0, and such that IIX(f)II S 111II=,

forfEA. To see this, we use the transformation S = (I- T) (I +T)-1 to reduce to the corresponding statement, in which A is the disk algebra and S is a contraction. The proposition then follows from von Neumann's theorem and the fact that the polynomials are dense in the disk algebra. 1..

t^'

For certain applications, we have to enlarge the algebra A a little, replacing it by the algebra B of bounded holomorphic functions on $2e z > 0 which are continuous on $2e z > 0. We no longer require a limit at infinity. G].

.."

If f E B, then f (z) belongs to A, for all e > 0. As a consequence, the operators ff.(T) = X(ff) form a bounded family.

14.3 Accretive square roots

235 4.'

Let us show that, for each x E H, fE (T) (x) converges to a limit which,

by definition, will be f (T)(x). We may restrict attention to x E V, because V is dense in H. So we write x = (I + T)-' y, for some y E H. Everything works out as if f (z) were replaced by g(z) = f (z)/(1 + z).

gyp'

G.0

'.J

off

Since the functions g(z)/(1 + Ez) converge to g(z) in the uniform norm on A, the operators fE (T) (I + T) -' converge to g(T) in operator norm. The strong convergence of the f6(T) follows. We leave the reader the task of verifying that the extension of X : A -i £(H, H) to B is still an algebra homomorphism. As an application of these ideas, consider the function f (z) = e-'Z, for t > 0, and form St = f (T). We have constructed the semigroup of contractions with infinitesimal generator -T. This sernigroup has the property that St converges strongly to the identity as t --- 0. Conversely, every strongly continuous semigroup of contractions is generated by a maximal accretive operator. The interested reader is referred to [151]. We return to the problem of square roots. ..,

If T is maximal accretive, then T(T + XI)-' is itself accretive, for any A > 0. Indeed, we can write this operator as I - A(T + )I)-' and A(T + AI)-' is a contraction. Every linear combination with positive coefficients of the operators C3.

...

Fn'

C++

6V.

T(T + AI)-' is still accretive and we thus have 3.e (T'/2x, x) > 0, for all .e E V. By continuity, this property extends to W, the domain of T'12. But we still must show that T1/2 : W --- H is maximal accretive. In other words, we must show that T1/2 + I : R'--+ H is an isomorphism. Recall that W is defined as the image of (T + I)-1/2. So it all comes down to showing that (T'/2+I)(T+I)-'/2 : H --+ H is an isomorphism. To do this, we compare (T'12+I)(T+I)-'12 with (TI/2+I)(T+I)-1h/2. Now TE"2 : W --- H is an isomorphism, as we established in the ..N

previous section. Further, T1/2 is accretive, as is TE 1/2. The operator I + Te 1/2 : H - H is thus an isomorphism. This operator may also

U~'

Fn'

`gym

be written in the form (1 + TE /2) TE "2, so I + TE "2 : W -i H is an isomorphism. Thus (TE/2 + I)(T + I)-'/2 is an isomorphism. We now show that there is a constant y > 0 such that, for 0 < E < 71, where g > 0 is sufficiently small, and for all x E H, (3.4)

Imo

Few

II(T''2 + I)(T + I)-'I/2xll ? yIIkII To establish (3.4), we first remark that II(TE/2 + I)yII2 > IIT 2yII2 + IIyII2, for all y E W, since T1/2 is accretive. But we already know that c.;

.-.

IIT1"2 - T1/211 < C f. Also, T'/2 and (T + I)1/2 differ by a bounded operator on H. Lastly, IIxii + IIT'12xII and 11 (T + 1)112X11 are equivalent '6b

norms on W. Putting these facts together gives (3.4).

236

Obviously,

14 Square roots of accretive differential operators II(Tf,2

+ I)(T + 1)-'/Ill < Co, if 0 < e < 1.

Finally,

(T1/2 + I)(T + I)-1/2 is an isomorphism of H with itself, because the norm of the difference between this operator and (T''2 + I)(T + I)-1/2 does not exceed Cl f, whereas Co and -y > 0 do not depend on e > 0. We summarize the above:

Proposition 3. Let T : V --' H be a maximal accretive operator. Then there is a maximal accretive operator, which we write as T1/2, whose domain W contains V and whose square is T. In particular,

W = (I +T)-1/2H and

I jT(T+AI)_lA_h/2dA. !Z.

(3.5)

T1/2 = As we have already remarked, the integral on the right-hand side is a Bochner integral when applied to x E V. Proposition 3 may be sharpened. In fact, there exists a unique maximal accretive operator L whose square is T. The reader is referred to [1511, where this assertion is proved.

4 Accretive sesquilinear forms We should not give applications without first constructing a maximal accretive operator. Throughout this section, we shall use Ho to denote the Hilbert space that we have hitherto denoted by H. We shall write HI C Ho for a dense linear subspace which is itself a Hilbert space for an inner-product ( , ) and norm . The inner product and norm on Ho will be denoted by

Ink

( , ) and We suppose that the injection H1 y Ho is continuous: there is a

III

constant C such that lxI _< ClIxII, for allx E H1. We have in mind an application in which Ho = L2(lR" ), H1 is the Sobolev space H' (R), ), and (f, g) = f' f (x)g(x) dx + fR V f - Vg dx. The Riesz representation theorem shows that, for every y E Ho, there

`..

is an element z E H1 such that (x, y) _ (x, z), for all x E Hl. V6e put z = J(y) and it is a triviality to show that J : Ho --+ Hl is linear. continuous and injective. We let H2 C H1 denote the image J(H0) and we have

(x, y) = (x, J-' (y)) , x E H1, y E H2 . In the concrete example, J = (I - A)-' and H2 is the Sobolev space (4.1)

H2(R,,). Finally, we let T : H2 --- Ho be the operator J-1.

14.4 Accretive sesquilinear forms

237

All this has been done so that T is a self-adjoint positive operator. This preliminary construction is a special case of what we shall now do.

We replace (x, y) by a sesquilinear form B : Hl x H1 - C, that is, a form satisfying the following properties: (4.2)

B(ax + f3y, z) = aB(x, z) + $B(y, z)

and

B(x, ay + /3z) = aB(x, y) +, B(x, z) for x, y, z E HI and a,,8 E C. We shall further suppose that there exist two constants C > 7 > 0 such that (4.3)

(4.4)

IB(x,y)I < CIIxIIIJyII

and (4.5)

2eB(x,x) >'vIIx112.

Ignoring Ho for a moment, we see that there exists an isomorphism S : H1 - H1 such that B(x, y) = (S(x), y), for all x, y E H1. Using the isomorphism T : H2 -- Ho that we constructed above, we put TB = TS. The domain of TB is the subspace V C H1 defined by the condition S(x) E H2. By our construction, S : V --+ H2 is an isomorphism (between these two subspaces of H1) and TB : V -+ Ho is an isomorphism. By (4. 1), for x E V and yE Hi, (4.6)

B(x, y) = (TBx, y)

and, for all x E V, (4.7)

2e (TBx, x) > 'YIxI2.

It is worth remarking that TB and V depend non-linearly on B. The operator TB is maximal accretive, with domain V. We have established the following result.

Proposition 4. Let B : H1 x H1 - C be a sesquilinear form satisfying (4.2) to (4.5). Then there exist a subspace V C Hl and a maximal accretive operator TB : V -' Ho such that, for all x E V and y E Hi, (4.8)

B(x, y) = (TB W, Y)

Word of mouth attributes the following question to T. Kato ([1511). Is it true that the domain W of the accretive square root TB'2 of the operator TB, which we have just constructed, coincides with the space Hl? In other words, is the domain of the square root the domain of the form? McIntosh constructed the first counter-examples and thus showed that

238

14 Square roots of accretive differential operators

we could not expect a positive answer in complete generality. We therefore need to restrict ourselves to the special case of square roots of operators T = - div A(x)V, where A(x) satisfies the conditions given in the introduction- It is this restricted problem which we call Kato's conjecture and we shall describe it in detail.

5 Kato's conjecture We start with an nx n matrix A(x), whose entries ajk(x), 1 < j, k < n, are in L'(Rn). We shall suppose that there exists a constant b > 0 such that, for each vector ((I, ... , (n) E C7,

(5.1)e

n

n

1

1

ja-k(x)C7(k ? 6(1(112 + ... + I(nI2)

for almost all x e Rn. We consider the sesquilinear form B, defined on H1(Rn) x H1(R"z) by n

(5.2)

B(f, 9) _

"a

aik(x) GX, Jan

k de .

We then have (5-3)

IB(f,9)I c CIIVfII21IV9I12

and

(5.4)

Re B(f, f) > III Vf II2 We are not in the situation of the preceding section, because II Vf II2 is not the norm of f in H'(W'). We right ourselves by considering the -

form B(f, g) = B(f, g) + f'R f g dx. Using B(f, g), we construct the maximal accretive operator TB, with domain V. This operator can be written as TB = TB + I and we have B(f,g) = (TB(f ), 9) , f E V, g e H1. The operator is maximal accretive and its domain is V. Returning to the construction of a maximal accretive operator via an accretive form, we immediately verify that the domain of TB is the vector space V C H'(R") of those f for which div(A(x)Vf) is in L2(R"). We note that A(x)Vf is a vector whose n co-ordinates belong to L2(Rn). Its (5.5)

divergence must therefore be calculated in the sense of distributions.

This characterization of the domain of TB is compatible with the general rules defining the domain of an operator obtained by the composition of unbounded operators: at each stage of the composition, all the calculations must be well-defined. So we may write TB (f) = - div(A(x)V f) .

14.6 The multilinear operators of Kato's conjecture

239 ['h

Kato's conjecture is that the domain of the accretive square root of TB is H'(JY"). We shall prove the following theorem.

Theorem 2. For each integer n > 1, there exists a constant e(n) > 0 such that, if III - A(x)II < e(n), the secretive square root T11' of TB may be written in the form

T'2 = ERj(A)D?,

D2 = -ixi , where i and where the operators R; (A) are functions of A(x), which are holomor(5.6)

phic on the open set III-A(x)II,,. < e(n) in (L°O(Rn))'2 and take values in the algebra C(L2(lRn), L2(Rn)) of bounded operators an L2(IRn).

Further, the operators RI (I), ... , Rn(I) are the usual Riesz transforms.

Finally, the operators R3 (A), 1 < j < n, extend as continuous operators from U'(lRn) to LP(IRn), for 2 < p < oo, and from L°°(IRn) to BMO(IRn).

do`

The theorem evidently implies that H' (Rn) is contained in the domain W of T a2. Let us show that the theorem, once we have established it for the matrix A(x) and its adjoint A*(x), implies that W = HI(]Rn). Let B* denote the adjoint of the form B. B* (f, g) = B(g, f) - Then

the adjoint of TB is TB. and the adjoint of TB112 is T'.2. To verify

this statement, all we need do is go back to the integral representation formula (2.10).

E-+

So B(f, 9) = (T'B f, g) = (T'B,2T /2f, g) = (T ,2f, T .2g), if j is in the domain V of TB and g E HI (lRn). We can extend the identity B(f,g) = (T(2f,T .2g) to H'(lR") x H'(lR"). Putting g = f, we get (5.7) bllvf112 < cB(f, j) _ file (Tt2 f,T 2 f) 0, depending only on the dimension, such that the norms IIR?,,,,(A)II of the operators Rjm(A) : L2(R") -- L2(Rn) satisfy

for all m > 1. IIRjm(A)II < CtII A(x) - III' The operators Rjo(A), 1 _< j < n, do not depend on A and are the usual Riesz transforms. (6.2)

Once we have established (6.2), it will follow immediately that T1/2 : H' (lR') -+ L2 (]R") is continuous, for CIIA(x) - III ,, < 1.

The operators Rj,...(A) are multilinear in B = I - A-1 and will be analysed by McIntosh's formal calculus (Chapter 13, section 7). The analysis of their structure will enable us to deal with a more general case than that needed for Kato's conjecture. We return to identity (3.5). We substitute for the variable of integration A by putting \ = t-2, where t > 0. The reason for this change of variable is that t has an important geometrical significance.

",,

We shall obtain the multilinear expansion of T1/2, from the expansion of the resolvent (I + t2 T)-I, by integrating with respect to t. To compute this resolvent, we shall write t2T as the following composition of unbounded operators: t2T = UAV. where

A is the operator of pointwise multiplication by the matrix A(x), and

all

U = -tdiv : E -, L2(Rn), where E C L2(Rn) is the domain of ."y

the divergence operator. 'J'

A few obvious remarks will be useful in what follows. With the nota tion we have introduced, I + UV = I - t2O, where L = a2/ax1 +___+ '°O

a2/ax22, and (I+ UV)-lU = Qt = -(I -t2O)-ltdiv. The operator Qt is a convolution operator with symbol -itt"n -itS1 can

(1 +t2I`I2'...' 1+t2ICI2 Similarly, the operator Zt = I - tVQt is a convolution operator. It is

14.6 The multilinear operators of Kato's conjecture

241

a classical pseudo-differential operator of order 0 whose matrix symbol is

C (k_

1+t2fE ))k1

where bjk=O,if j34 k,and1,if j=k. .'t1 ...

rye

The symbol of the Riesz transform R, is tj/lfl. If we put Pt = (I t21)-1, we may write Zt = ((bjkl - (I - Pt)RjRk))j k=1 . Let us clarify the meaning of t. The operator Pt is the operation of con-

`-'

volution by 4t, where Ot(x) = t-"4(t-lx) and Qi E L1(R"). More precisely, if Ix' > 1, then l f (x)I+iV0(x)I < C,,,.Ixl -"", for all integers m > 1,

but if lxI < 1, then 10(x)l < CJxl-"+1 and IV4(x)l < Clxl-n. This Q.,

means that O(x) is essentially supported by the unit ball and that, in fir'

essence, the "radius of influence" of Pt is t. The same remark is valid for Qt, apart from two things. Qt(f) = f *vpt, where 41t = t-",0(t-1x). This time, is vector-valued, belongs to L' (]R"), and satisfies f tb(x) dx = 0,

-6-9

whereas before f 41(x) dx = 1. On the other hand, 0(x) has the same

II.

properties of regularity and localization as 41(x). The case of the operator Zt is similar, except that the singularity at the origin is more pronounced. Indeed, there is a tempered distribution Z E S'(]R") such that the distribution-kernel of Zt is given by t-"Z(t-1(x y)). The restriction of Z to IxI > 1 coincides with that of a function in the Schwartz class S(R"), Z is infinitely differentiable on ]R" \ {0}, and the radius of influence of Zt is t. We shall use the fact that Zt = pt + irt, where the distribution-kernel

'°C

may'

of the singular part pt is supported by Ix - VI < t and where in is the operator of convolution with wt = t-"w(t-lx), where w(x) E S(]R"). We return to (I + t2 T)-1. Recall that A is the operation of multiplication by the matrix A(x). We put A-1 (x) = I - B(x) and let B denote the operation of multiplication by the matrix B(x). We shall expand (I +t2T)-1 as a series of powers of the operator B, using the identity (6.3)

(I + UAV)-'UA = Q(I - BZ)-1,

where U, A, and V are now arbitrary elements of an associative algebra,

where Q = (I + UV)-'U, Z = I - VQ, and we suppose that all the inverses we have written down actually exist.

The proof of (6.3) is an exercise in algebra which we leave to the reader. The particular choices of the operators U, A, and V that we need play no role in that proof. In our situation, Z = Zt : (L2(IIt"))" -. (L2(P"))" is bounded and the operator norm of Zt is independent of t. As a consequence, the norm of

242

14 Square roots of accretive differential operators

BZ = BZt is strictly less than 1, if III-A(x)II, < e(n), where e(n) > 0 is sufficiently small. We can, therefore, expand the right-hand side of (6.3) as a Neumann series >o Q(BZ)n`. Going back to (3.5), we get

T1"2 = > Rj (A)Dj

(6.4)

where

=UJ=Qt(I-BZt)-1;=-t Lm, 0 Nom'

(6.5)

0

Rn(A)

with (6.6)

Lm = f Qt (BZt )m

dt

t We shall show that there exists a constant C = C(n) such that o

R''

IILmII 0 and 0. In the special case of Kato's problem, m(l:) is one of the functions We let R : L2(Rn) -, L2(lR) denote the convolution operator whose symbol (or multiplier) is m(l;). Then R is a Calderon-Zygmund operator and thus bounded on LP(R'), for 1 < p < oo.

The final hypothesis is that we can decompose the operator Rt = (I - Pt)R into Rt = pt + 7rt, where the distribution-kernel of pt has support in Ix- yj < t and lrt is the operator of convolution with wt(x) _ t-"w(t-ix), where w(x) is in the Schwartz class S(RT).

14.6 The multilinear operators of Kato's conjecture

243

We note that the operator pt is bounded on LP(Rn), for 1 < p < oo, and that its norm is independent of T. be operators of pointwise multiplication by functions Let Bt,..., bl (x), ... , b,.(x), for some integer rn > 1. We suppose that Ilb11100 1, ... , Il bm l l . < 1, so that Bl, ... , Bm axe contractions on L2 (Rn ). With this notation and these hypotheses, the following result provides a generalization of Theorem 2.

Theorem 3. There exists a constant C, depending only on the functions 0, bi, and the operator R, such that, for all integers m > 1 and every function µ(t) E L°° (0, oo), the integral

J °° QtB1R, ... B,,,,RtFt(t)dtt 0

converges strongly to an operator Lm: L2(Rn) _, L2(Rn) whose norm does not exceed C'nllpll,,,,. Fbrther, there exists a constant C' such that the restriction L,,,, (x, y) of the distribution-kernel of Lm to the complement of the diagonal satisfies (6.7)

f

ILm.(x,y)-Lm(x',y)Idy T.) Proceeding as before, we have to calculate the limit of fR fT I(q)I2I f (C)I2t-1 < dt By (6.10), we can use Lebesgue's dominated convergence theorem to conclude.

The study of the operators L,,, thus reduces to that of the operators L,(2). J.L. Journe showed that the kernel Ll,nl (x, y) of Llnl satisfied the conditions of a Calder6n Zygmund kernel. This led him to use the T(1) Since Wt(1) = 1, we have theorem to establish the L2 continuity of (6.11)

L,nl(1) = Lm-1(bm)

This means that the induction argument we use must zigzag between the operators Lm and LM( and will require (6.7). The proof of (6.7) will be direct, making no appeal to induction. Now (6.7) gives us the will let us establish constant C'. Then the T(1) theorem, applied to

14.7 Estimates of the kernels of the operators L2)

245

that (6.12)

IIL12)II

CoIILm-iII +CIm-1

which will, in turn, lead to (6.13)

0 such that We a(x) > 6 > 0 and We b(x) > 6 > 0. With these hypothe es, the operator T = BDAD satisfies (2.6) and we can define T'/2 by applying Proposition 1. In collaboration with C. Kenig ([160]), we showed that '+-

T1/2 = J(A, B)D, where J(A, B) is an isomorphism of L2(R) with itself.

Now, if A = B, then J(A, B) is the Cauchy operator on the Lip-

i.1

schitz graph whose parametric representation is given by the primitive of 1/a(t), t E R. On the other hand, if B = I, T'/2 is the Kato operator. The square roots of accretive differential operators and the Cauchy operators on Lipschitz curves thus belong to the same family. In a sense, this has been confirmed by the approach we have followed in this chapter: J.L. Journe's method, which consists of relating Kato's operator to a singular integral operator. Finally, we draw the reader's attention to two other solutions of Kato's problem ([62] and [105]), obtained under the same restrictions as Theorem 2.

...

15 Potential theory in Lipschitz domains

1 Introduction Calderdn's research programme was motivated by the study of elliptic partial differential equations in domains with irregular boundary. Calderon's method consists of replacing the partial differential equation on the interior by a pseudo-differential equation on the boundary. But if the boundary is only Lipschitz, the nature of this equation changes: the operators which appear are no longer pseudo-differential, but are of the new type discussed in Chapter 9. When Calderdn inaugurated his programme, there were two difficulties. The very existence of the operators, needed for the method, was problematic. And, supposing that, in the fullness of time, such operators could be constructed, it would be necessary to solve the equations on the boundary that this process led to. For the solution, one would have to invert some operator and so would need a symbolic calculus. The regrettable absence of such a symbolic calculus was signalled in Chapter 9, and is the second problem of Calderdn's programme. We shall illustrate these remarks by examining a classical problem, which goes back to Poincare, Neumann, and Hilbert. This is the solution, by the double-layer potential method, of the Dirichiet and Neumann problems for a domain ft, in R'+1. When Il is a bounded, regular, open set, the operators of Calderon's method are classical pseudo-differential operators; they are also singular integral operators whose kernels are given by a "double-layer potential".

256

15 Potential theory in Lipschitz domains

c..

After the reduction given by Calderdn's method, resolving a Dirichlet orNeumann problem just amounts to inverting an operator of the form 1/2 + K acting on the boundary. The ambient Banach space will be L2(812, do), where dR is the surface measure on the boundary 812 of 0. When 11 is a bounded, regular, open set (of class C1+a, for some

a > 0), the operator K : L2(81l, do,) - L2(812, do) is compact, so

(fl

(fl

a.,

Fredholm theory (which was invented for this purpose) allows us to invert f/2 + K. When 12 is just of class C', the principal difficulty is to prove that K is continuous on the space L2(812, da). Once this continuity had been established (Calderon, 1977), Fabes, Jodeit, and Riviere ([104]) observed that the operator K was still compact, so that the rest could

chi

be done just as in the regular case. However, K no longer was a classical pseudo-differential operator. Finally, for 1i a bounded, Lipschitz, open set, the continuity of K was established in 1981 in [65]. It follows from the results of Chapter 9. However, the operator K is no longer compact. This essential difficulty was overcome by G. Verchota. He showed that 1/2 + K was invertible, using the Jerison and Kenig energy inequalities ([136]). After carefully stating Verchota's results, we shall examine only the special case where 12 is the open set above a Lipschitz graph. We shall give a complete proof for that case. The modifications needed for the general case are ingenious, but do

not need any new ideas. The reader who is interested may consult Verchota's thesis ([232]).

2 Statement of the results

-;m

In all that follows, Il C Rn+1 is a bounded, connected, open set. We say that 12 is a Lipschitz open set, if the following conditions are satisfied.

Firstly, the boundary 812 of 12 has only a finite number of connected components. Secondly, for each xo E 812, we can find an orthonormal co-ordinate frame 7Z(xo), with origin xo, two numbers E, 97 > 0, and a Lipschitz function 0 : R' -, R such that, if C = C(7Z(xo), E, rj) denotes the solid cylinder given by xi + - - - + xn < e and -27 < xn+1 < 71, then (2-1)

C l12 = {x = (x', xtz+1) : 1x'9 < E and !b(x') < xn+1 < 7j}

and (2.2)

C f, 812 = {x = (x',

Ix'1< E and

O(x')}.

2 Statement of the results

257

".3

04,

ego

Since 812 is a compact set, we need only a finite number N of cylinders Cl, ... , CN of the above type, to get a cover of 81z by the corresponding open cylinders. To deal with local problems on the boundary 81l of 11, it therefore is enough to work in the interior of one of these cylinders. We may then fix the orthonormal frame R, ignore the limitation corresponding to a and suppose that Il is defined globally by xn,+1 > O(x'), x' E Rn, xn+1 E R

Cs'

0

.ti

s'3

and that 0 is globally Lipschitz, that is, that IIVOII. < M, where M is a (finite) constant. With this simplification, we fix another constant M' > M and we attach a cone I'(x) of non-tangential approach to each point x E M. The cone 1'(x), with the exception of the point x itself, is contained entirely in 12 and is defined by yn+1 - xn+l ? M'Iy' - x9. 'ti

a...

fn'

The geometric significance of this cone is that there is a constant 6 > 0 Fn'

such that the distance from y E r(x) to 812 is not less than SIy - xj. When y tends to x, while remaining in I'(x), y is "relatively far" from

QED

chi

'L7

10'

RCS

the other points of the boundary. Our simplification has created an unbounded Lipschitz open set. In the case of a bounded Lipschitz open set, the approach cones have to be cut off. We still use r(x), x E 812, to denote them. They have a uniform angle a > 0, a uniform height T > 0 and their geometric significance is the same as in the unbounded case. The construction of these cut-off cones is very carefully described in ([232]), to which we refer the reader. The details do not have any part to play in what follows. We now look at the Dirichlet problem, which is to find a function u which is harmonic in ft, with given boundary values. Recall that u is harmonic if Du = 0, where A = 82/8x2 +... + 82/8xri 1. We intend to resolve Dirichiet's problem with boundary values g(x) in L2 (811, dv), where du is surface measure on the boundary. One might think that this problem is not well-defined because, on the one hand, g(x) is only defined up to a set of measure zero on 812 while, on the other, it is not sufficient to know the non-tangential limits, up to a set of measure zero, of a function u which is harmonic in 12, in order to determine u. A counter-example is given in 11 = { (x', xn+i) E Rn+i : xn+> > 0} by u(x) = u(x', xn,+i) = xn+i(Ix'I2 + x7+1) (n+1)/2 Then a is harmonic in 12 and u(x', 0) = 0 everywhere except when x' = 0. To exclude such pathological solutions of Dirichlet's problem, it is enough to introduce a condition like that of Lebesgue's dominated convergence theorem. This consists of restricting attention to those functions u which are harmonic in 12 and such that the maximal function C7'

Fn'

'z'

f'!

`'C

r++

`-'

Fr'

".3

...

.c7

1 5 Potential theory in Lipschitz domains

258

u*(x), corresponding to non-tangential approach to the boundary, is in L2(8f1, da). To do this, we make the following definition

Definition 1. Let u be a function which is harmonic in Q. We Write

...

u E 7-12p) if a*(x) E L2(812, do.), where

u*(x) = sup Iu(y)I -

(2.3)

^F1

yer(x) elf

The space 7-12(11) is independent of the particular choice of truncated cones I'(x). Now to describe the trace operator 0 : 712(11) --> L2(BSl,dv). It is clearly defined as the usual restriction to 812 of u E 7-12(11) when the

In the general ease, we say that u has a non-tangential limit at x E an if limu(y) exists, as y -+ x with y c r(x). If u E 72(11), then, for all harmonic function u extends to the boundary by continuity.

x E aft, except for a set of da measure zero, u(y) has a non-tangential iii

limit, which we denote by u(x). The function defined in this way belongs to L2 (811, du) and a different choice of non-tangential approach cones would give the same trace for almost all .c E an. The trace operator 0 : 7-L2 (52) -' L2 (852, da) is, in fact, an isomorphism between these two spaces. This theorem, due to B. Dahlherg, depends on the equivalence of the harmonic measure to the surface measure on as1 ([81] and [82]). Dahlberg's theorem gives a theoretical solution of the Dirichlet prob,.O

lem, which consists of finding u E 7{2(11) which is a solution of the

E-+

equation 0(u) = g, for a given g E L2(812, dv). We shall deal with this same Dirichlet problem by a different route, which will give an algorithm for the solution. To state the Neumann problem, we introduce the space 7L (11) of functions u which are harmonic in 11 and such that v(x) = supyer(.) Nu(y)I

coo

An

(x)

y&mr(x) Vu(y) n(x) .N.

(2.4)

tip,

~'"

belongs to L2(852, dc). Then the non-tangential limit of the gradient Vu(y) exists for almost all x E 852 (with respect to the surface measure dv), as y E r(x) tends to x. Solving Neumann's problem is to find, for a given g E L2(811, dv), a function u E 7-l2 (St) such that au/8n = g, almost everywhere on 811 (with respect to da). Here, n(x), x E 811, denotes the vector at x which is normal to 811. This vector exists for almost all x E 811 and we have put

To state the main results of this chapter, we must now introduce the double-layer potential. It is constructed in the following way. We start

2 Statement of the results

259

with the fundamental solution -w,; 1(n - 1)-1IxI-'°+1 of the Laplace equation in R"+1 (w is the surface area of the unit sphere Sn C R"+1) 70,

The fundamental solution enables us to calculate the potential generated by a charge distribution. A double-layer distribution is a distribution S, with support in 881, defined on the test functions g E D(III"+1) by X67

°~+

(S,9) = -J f(y)Lg(y)da(y), where f (y) E Ll (OIZ, da) is a given density.

The potential at x E IZ created by the double-layer distribution of density f on 8f1 is thus (2.5)

)C f (x) =

(Y - x) 1 wnf ast I11- xI n()i f (y) da(y)

If u and v are functions which are harmonic on a bounded regular open set Ii, and if a and v are C' in a neighbourhood of the closure Il of Il, then Green's theorem (whose use we shall justify in section 5) gives

f

1a

`

au "an -van da = 0.

f31

Applying this remark to v(y) _ -wn 1(n -1)-1 Ix - yI -"+1, excising a ball centre x, radius e, from SZ, and passing to the limit as a -+ 0 gives (2.6)

u(x) = Ku(x) + (n Ix - yI-n+1 a (y) da(y) . -11)w. Jan `J'

This identity means that /Cu(x) is a reasonable approximation to u. For example, if u is identically 1, we get K (u) = 1. By our construction, 1C f (x) is a harmonic function on IZ, when f lies in Ll (Oil, da). From now on, we shall restrict attention to the case where f e L2(BIZ, da), a choice which we shall justify after the event. It is not at all obvious that K f (x) belongs to 7{2(11), when f lies in L2(BIZ, da). It was to prove precisely this result that Calderon engaged on the closer investigation of operators defined by generalized singular integrals. The following theorem describes the properties of the operator 1C. bop

Theorem 1. Let IZ be a Lipschitz domain. Then the operator 1C, defined by the double-layer potential, is continuous, as an operator from L2(ant, da) to 7.12(0). Moreover, for each function f E L2(Ofl, da), we have, for almost all x E Oil (with respect to surface measure da), (2.7)

y- ire

K f (y) _ (2I + K)f (x) ,

where K is defined on L2(Il,L_X>E da) by (2.8)

K f (x) = n Elim

(y x) I y - xI

11) f () da(y) -

15 Potential theory in Lipschitz domains

260

The operator K is thus continuous on L2(811, dc) and its definition is obtained from a singular integral calculated entirely on M. The singular integral exists almost everywhere on 81 (equipped, as always, with the surface measure do). The solution to the Dirichlet problem is then given by the following statement.

Theorem 2. Let fl again be a Lipschitz domain. Then the "boundary" operator

2I + K : L2(8f1, da) --, L2(8fl, dv) is an isomorphism. If g lies in L2(8l, da), then the harmonic function u = K(I/2 + K)-'g is the unique solution of the following Dirichlet problem:

Au = 0 in fZ

(2.9)

when u* E L2(on.der)

and (2.10)

lim u(y) = g(x) v-x,yEr(z)

for almost all x E 01.

To deal with the Neumann problem, let K* : L2 (81l, da) -> L2 (851, dv) denote the adjoint of the operator K. We write f(j for the unique function in L2 (ell, da) satisfying

r fodQ=1, L and let L(011, dv) denote the subspace of functions of zero mean in (2I-K*)fo=0

and

L2 (8f1, dc).

The single-layer operator S : L2(BSt, da) (2.11)

S f (x) = - (n -11)wJ

7j2 (0) is defined by

Ix - yf -n+If (y) da(y),

for x E fl. With this notation, we can state Theorem 3.

Theorem 3. Let 11 be a Lipschitz domain. Then, for each function f c L2(8fl, dc), u = S(f) lies, in ?{i(1). Secondly, the operator 1/2 - K* : Lo (8f1, da) -. L2(8fl, dg) is an isomorphism.

Lastly, for each g E L02(8fl, do), the function u = -S(I/2 - K*)-lg .r,

is the unique solution of the following Neumann problem: (2.12) Au = 0 in fl and IDul* E L2(8flj , da) ; (2.13)

an = g almost everywhere da and

u fo da = 0. sp

Some explanantions are called for. We already know that K(1) = 1. As a consequence, for every f E L2(8il, dv), the function h = (1/2 -

15.3 Almost everywhere existence of the double-layer potential 261

K*)f necessarily has zero mean. Thus 1/2 - K* could not possibly be an automorphism of L2(ffl, da), but could, at best, be an isomorphism of Lo (?St, da) with itself.

In (2.13), au/8n = g, almost everywhere da, clearly means that g(x) is, for almost all x E 8I, the non-tangential limit of Vu(y) - n(x), as y E r(x) tends to x. Next, the condition fan u fo da = 0 arises because u = S(h) with fosl h da = 0. Indeed, S(fo) is a constant and, since S : L2(BSl, da) L2(852, da) is self-adjoint, f u fo da =fast S(h)fo du =fan hS(fo) da= cfas1hda=O.

Finally, if u, v E 7-1 (S1), by Green's theorem, we have fan (u(Ov/8n) coo

v(au/an)) da = 0. The details of passing to the limit, needed in the Lipschitz case, will be given later. Taking v = 1, we do get fail (au/8n) da = 0, for u E 7{i(Sl). The condition g E L2()ft, da) is, of course. necessary for the Neumann problem to have a solution.

3 Almost everywhere existence of the double-layer potential l`7

We now turn our backs on the general case and study the situation where Sl is defined globally by t > 0(x), t E R, x E 1R", and where 4(x) is a real-valued Lipschitz function on the whole of R", satisfying

a37

°+d

11 VOID < M < oo. To ease the burden of notation, we denote the points of W'+1 by X = (x, t) and Y = (y, s), where x, y E 1R" and s, t E R. We shall also write V for the graph of 0 : lR" --. R. Theorems 1, 2, and 3 will be proved in this setting. L'(V, da) has to be replaced by L2(V, da) in Theorem 3, because the constant function I is no longer square-integrable. Nor does the function fo exist any more. The changes necessary for the bounded case are ingenious, but of a more technical nature. So it seems sensible to refer the interested reader to [232].

The first remark is about the form of the double-layer potential. If X, Y E V, then (Y - X) - N(Y) (3.1) WnIY-Xjn+1 do (Y) = K(O; x, y) dy, where (3 . 2)

K(A; x is ) _

0(x) - 4(y) - (x - y) - VOW Wf1(I x - y12 + (0(x)

-.O(y))2)1n+11/2

We have written N(Y) for the unit vector normal to V at Y, oriented downwards, that is, away from S 1.

With this notation, the analogue of Theorem 1 is

15 Potential theory in Lipschitz domains

262

Theorem 4. For each function f E L2(lR, dx), (3.3)

To(f) (x) = E.LO lim f

Ix-yI>

K(O; x, y)f (y) dy

exists almost everywhere and the operator To defined by (3.3) is bounded on L2(R").

The operator Tm will be called the singular part of the double-layer potential. From Chapter 8, we know that, to prove such a result, it is sufficient to establish the existence of the limit when f lies in the Schwartz class S(Rn) and to show that there is a constant C such that, for -g.

7 (f)(x) = supl f

(3.4)

K(O; x, y) f (y) dyl , I7-vI>c

we have (3.5)

II(f)112 E

The right-hand side of (3.6) is an absolutely convergent integral. To prove (3.6), we fix x and apply the divergence theorem to

(x-y Ix - y1,,

A

.rte

Jx-yI?E div

- O(Y) ) IX-VI

f (y)) dy .

The divergence is taken with respect to the variable y. Now Wn

X-Y divlIx n

-

,

,

y)

15.3 Almost everywhere existence of the double-layer potential 263

The proof of (3.6) thus depends on the resulting surface integral's tending to 0 with E. Indeed, this surface integral is, up to sign,

I{E)= f n

f(x+EV)A(

0(x+ev)-41(x))der (v),

where da(V) is the surface measure on the unit sphere Sn-1. Since 0 is differentiable at x, the dominated convergence theorem applies and 1£r I (E) = f (X) J

n_ 1

\(v - V 4)(x)) da(v).

To conclude, we observe that A is odd. We pair up the antipodal points and get limE1o I(e) = 0. The estimate (3.5) of the maximal operator T,* basically comes from the general theory of Calder6n-Zygmund operators. We must, however. make some adjustments, because the kernel K(¢1; x, y) does not have the regularity with respect to y that is part of the hypotheses of Chapter 7. We start by splitting up the kernel K(41; x, y) as n

Wn (Ko(x, y) -

Kj (x, y) 11 a' t(y))

where 40) - 41(y)

Ko(x,y) =

(y))2)(n+l)/2

/

x - yl2 + (41(x) -

and

xs - yj

Kj(x, y) =

. (,))2)(n+1)/2

(lx - yl2 + (41(x) Since the operators act on functions f E L2(Rn), we may absorb L°° functions into such f, which lets us reduce to the operators To and T?, defined by the kernels PV Ko and PV K; .

p'°

We can thus use the results of previous chapters. On the one hand, Theorem 11 of Chapter 9 gives the basic L2 estimate. On the other, the theory of Chapter 7 gives the continuity of the corresponding maximal operators. In order to deduce the results of Theorem 1, we must compare the principal value with the limit defined by non-tangential convergence. To do this, we use the following remark. Lemma 3. With the notation above and for t # 41(x), t - 41(y) - (x - y) - V O(y) (3.7) JR_

t)2)( +1)/2

f (y) dy

Qy - xl2 + (41(y) -

= 2 sgn(t - O(x))f (x) - j

(y - x) - V f (y) A(#(y) - t) dy. ly - xl

^ly - xln

15 Potential theory in Lipschitz domains

264

`''

can

To prove this result, we repeat the method of Lemma 2. That is, we start with the identity x - y A( #(y) - t)) _ 4'(y) - t - (y - x) - VOW d iv t)2)(n+1>/2 (Ix - yin Ix - yl (ly - x12 + (4(y) Regrouping the integrals of Lemma 3 gives

x-y

I(e) fin,

Ix-yI?e

¢(x)-t

div(Ix - yin A( lx - yl ))f (y) dy

which becomes, by the divergence theorem,

I(e) = s-n+l J

(3.8)

y-xI_C

f (y)A(o(Y) xlt) do(y)

Once again, we use Lebesgue's dominated convergence theorem to calculate limE--,o I(e). It is convenient to change the surface integral so

that it is taken over the unit sphere Sn-1. We observe that A(±oo) = f fo (1 + t2)-(n+l)/2 dt, that Wn_1 fo (1 + t2)-(n+l)/2 dt = wn/2 and that this gives hm£_.o I(e) = (wn/2) sgn(o(x) - t) f (x). Let us go back to Theorem 1. We intend, for the first stage, to use

R,9049.

".7

the definition of the operator K given by (3.3) and to prove (2.7). After that, we shall indicate why the operator K may just as well be given

by (2.8). We should observe that, in (2.8), ly - x1 > e has become '+.

off

ly - xl2 + (O(y) - ¢(x))2 > e, because of our change of notation. To establish (2.7), we follow our accustomed route. We verify that this identity holds when f belongs to the Schwartz class S(R) and we then prove an estimate for the corresponding maximal operator. The non-tangential approach cone is given, via a parameter a > M >

IIV'II,, by r(xo) = {(x,t) : t - '(xo) >- alx - xoI}.

fin'

"CS

LS,

...

fin'

Let F(x, t) denote the left-hand side of (3.7). This satisfies F(x, t) _ -wnjC f (x, t). Let us calculate the limit of F(x, t) as (x, t) E r(xo) tends to (xo,4'(xo)). We make the change of variable y - x = u - xo in the integral on the right-hand side of (3.7), which then becomes (u-xo)I Vf(uu x-xo)cb(u+ (3.9) o)-t)du. ln 1U - oI IRn Since f E S(Rn) and we can restrict to the ball I x-xoI < 1, Lebesgue's dominated convergence theorem applies when calculating the limit of (3.9), as x -4 x0 and t - to = ¢(xo). The non tangential convergence plays no part in this, and the limit is

(u - xo) Vf (u) A (u) - to du

Iu - xoln (Iu - xol ) Je" However, the geometry of non-tangential convergence is crucial in proving the maximal estimate.

15.3 Almost everywhere existence of the double-layer potentzal

265

We Put F*(xo) = sup(x,t)Er(zo) IF(x,t)I and we intend to show that there is a constant C such that, for all functions f E L2(V, de), (3.10)dz`

Uot.IF(X)12

1/2

e and by 0 otherwise. Then, if a -1(t - 4(xo)) = e and (x, t) E I'(.eo), we can write (3.11)

t - O(y) - (x - y) - VOW t)2)(n+l)/2

(ly - x12 + (4(y) -

= KE(O; xo, y) + RE(xo, x, y),

where, for a certain constant C = C(M. a), we have if I xo - yJ < e,

JRE(xo, x, y) I < CC-', and JRE(xo, x, y)l CeJxo -

yJ-n-l

if Jxo - yI >- e.

Getting the estimates for RE (xo, x, y) is a simple geometrical exercise,

using the fact that t - ci(xo) > aix - xol and that fr(y) - (6(xo)J

May - xol,with a >M>0. Furthermore, E

f

Ixo - yI-''If(y)ldy:5 Cnf*(xo),

where f* (xO) is the Hardy-Littlewood maximal function of f . Thus (3.12)

F, (x) O

x, y)f (y) dy I

Inequality (3.12) is consistent with the results of Chapter 7. To conclude, the operator defined by the kernel K(O; x, y) can be split into n + 1 Calderdn-Zygmund operators preceded by n operators of multiplication by L°°(Rn) functions. Thus K*f E L2 whenever f E L2 (Chapter 7, Theorem 4). Finally F, the non-tangential maximal function corresponding to the double-layer potential, also belongs to L2 when f does. To finish the proof of Theorem 1, let us see how the principal-value operator defined by (2.8) differs from that given by (3.3). We put X =

15 Potential theory in Lipschitz domains

266

(x, O (x)), Y = (ry, 4S(y)), write N(Y) for n(y) in (2.8) and consider

(Y - X) N(Y)

AEf (x) - J

Ix-yI>E

IY - XI

f (y) du(Y)

Jx-vI?rwnK(4i x, y)f (y) dy - JR(e)

(Y - X) IY - N(Y) f (y) da(Y), XI-+1

(Ix-yI2+(4(x)_¢l(y))2)1/2 > E.

where R(E) is defined by Ix-yI < E and In particular,

Ix - yI > (1 +

M2)-1'2E,

if y E R(E), and J (Y - X) N(Y)IIY - XI-n-1 < CE'. The volume of R(E) is exactly CE', and we deduce from this that IoE f (x)I < Cf"(x), where f* (x) is the Hardy-Littlewood maximal function of f . To show that lune10 DE f (x) = 0 almost everywhere, for f E L2(Rn), it is enough to consider the case where f E S(l[i;'"). In the end. this amounts to showing that 1 to JR00 (YIl,

X). N(Y) da(Y) = 0. I"+1

We start by observing that the function N(Y) = N(y, 0(y)) is in L°°(lR' ). Thus, almost every point x E R" is a Lebesgue point for this function. That is, E-n fx_,1<E IN(Y) - N(X) I dy tends to 0 with E. Since E < IY - X I _< (1 + M2)1/2E on the shell R(E), we can replace N(Y) by N(X), whenever y is a Lebesgue point of N(Y). Now observe that XI"+1 X) &(Y) = 0 , lio JR(C) If(Yfor each point x where 0 is differentiable. This is because the function YIYI-1-1 is odd and the range of integration R(E) is essentially symmetric in X, for such x (the volume of the symmetric difference of R(E) and its symmetric hull is O(E")). The details are left to the reader. We have finished the proof of Theorem 1.

4 The single-layer potential and its gradient Before proving Theorem 2, we establish the analogue of Theorem 1 for the gradient of the single-layer potential. The methods used are the same as those of the previous section and we use the same notation. The points of lll;"+1 are denoted by X = (x, t), where x E lR' and

15.4 The single-layer potential and its gradient

267

t E R. The set V C Rn+1 is the graph of the global Lipschitz function 0 : R' - R, with IIo4'II. < M < oo. Finally, dQ denotes surface measure on V. If n > 3, the single-layer potential arising from f E L2 (V, dQ) is defined

in R-+1 \ V by (4.1)

Sf (X) _ (n -11)w" JV IX - YI-n+1 f (Y) da(Y) .

E-4

This is just the convolution of the function -(n-1)-lwn 1 fv IXI-n+1, a fundamental solution of the Laplace operator, with the surface measure f da, which is the single-layer distribution in question. In dimension 3 (n = 2), this integral may diverge at infinity, for an

arbitrary f E L2(V,da). To make it converge, it is enough to fix a

X30

point X0 off V and to replace the kernel IX - YI-"+1, throughout, by IX - YI-"+1 - IXo - YI "+1. Indeed, the only things that matter are the partial derivatives of S f : the choice of X0 is of no importance.

If X = (x, t) lies off V, we calculate the gradient of Sf at X by (4.2)

V S f (.r) =

n

`,,

differentiating under the inteJ ral sign to get

I X Yjn+r

f (Y) da(Y) .

What happens as X approaches V? In the following theorem, we no longer choose the open set ] above V. For each (xo, 4'(xo)) E V, we let r±(xo) denote the two cones defined, respectively, by t - ¢(xo) > aix - xoI and t - c5(.co) < -aix - xoI (where a > M is a fixed constant). We shall attempt to determine the limit of V S f (X) as X tends to Xo = (xo, ci(xo)) while staying within T+(xo) or r- (xo)

Theorem 5. If f E L2(V,de), then VSf(x) has a limit as X E r±(xo) tends to Xo = (xo, 4'(xo)). The limit is given by (4.3)

f1f(Xo)N(Xo)+

n PV

der(Y),

...

WO O where N(Xo) is the unit normal vector at Xo, pointing downwards. Further, the maximal operators sup,,Ert(,o) IVSf(X)I are bounded on L2(V, do).

As in the previous section, we split the proof of Theorem 5 into two parts. Firstly, we establish that the non-tangential limit and the principal value exist when f belongs to a suitable dense subspace E of L2(Rn). To pass from the special functions f E E to general functions, we need, secondly, to prove an L2 estimate for the maximal operator correspond-

ing to this problem. But the second stage is identical to that of the

15 Potential theory in Lipschitz domains

268

previous section. We shall concentrate on the first part. Let us define E. As in the previous section, it is a matter of integrating by parts to make the singularity of the kernel (X - Y) I X - YI -'x-1 disappear. But, first of all, we must undertake certain transformations of the kernel. (x)I2)-1/2 and consider the tangent vectors We put w(x) = (1 + IV at X = (x, 4(x)) E V, defined by T1(X) = (w(x), 0, 0, - . - , 0, w(x)OclOxi )

T2(X) = (0,w(x),0,---,0,w(x)O0/0x2),

(0,0,0..... w(x)' w(x)C9OlC9xn) , together with the unit normal vector N(X), given by .... , w(x)

1

90

w(x))

Q.1

-)0

N(X) = (w(x) j_

The lengths of these vectors he between (1 +

M2)-1/2 and 1. The de-

terminant det(Ti,... ,T,,, N) has value -(w(x))'"-1. The absolute value M2)-(9"-1)/2, so the inverse of the determinant is thus greater than (1 + F..

matrix of (T,,.. ., T, N) belongs to L°° (1[l:'°). Let (71 , ... , Tom, N) denote the dual basis of (T1, ... , T,,, N). Then every vector Z E lR'+1 may be written (4.4)

+(Z-N)N.

Z = (Z-T)7i

The vectors 71.... , 7n belong to L°° (R') . We apply (4.4) to Z = (X - Y) IX - Y I -"-1 and then put K(X, Y)

IX

YY+1 N(Y) and Kj (X, Y) = IX YY+1 .Tj(Y) ,

r-1

for 1 < j < n. This gives us the noteworthy decomposition (4.5)

IX ''C

The action of the operators of pointwise multiplication by T*(Y) and N(Y) on L2(R'") may be ignored, because the vectors involved belong to L°°(R'"). So the problems of convergence that we need to resolve reduce to the corresponding questions about the kernels Kj (X, Y) and K(X, Y). But these kernels have a simpler structure than the kernel we started with. Q'(

Indeed, du (Y) = and

1

a

n-lc7ya

(IX - YI-"+1) dy

15.4 The single-layer potential and its gradient

Y) du(Y) _

269

t - 0(y) - (x - y) - V (y) dy. - YI2 + (t - (,))2),n+1)/2

((Ix

The question whether fv K(X, Y) f (Y) du(Y) converges non-tangentially is thus resolved by Theorem 1. It remains to deal with each fv Kj (X, Y) f (Y) da(Y). Since X V V, we may integrate by parts when f is in the Schwartz class S(lRn). This gives (4.6)

JvKi(X,Y)f(Y)da(Y) = -n 1 1

w,

L IX-YJ"-18f

dy,

(y, ¢(y)) E V and X V V. As X = (x, t) tends to Xo = (x°, O(x°)), the limit of the right-hand side of (4.6) may be calculated by applying Lebesgue's dominated convergence theorem, after substituting y - x = u - xo. The non-tangential convergence is irrelevant and the calculation is similar to that of the where Y

previous section. We get (4.7)

-n-1 1

j

n

(I xo

- yI2 +

0(y))2)-(n-1),219f dy. ayi

We then suppose that ¢ is differentiable at x° and proceed to an integration by parts in the opposite direction, but replacing f R,, in (4.7) by fl__ O-YI>E" This means that we can apply Green's theorem and, finally,

after a calculation similar to that in the previous section, arrive at (4.8)

PV

r

x°,i - yi +

Ut^ (Ixo - y12 + (4(xo) -

(,))2),n+1)/2

(y) dy.

This term is one of the principal-value distributions which appear in Theorem 5.

The jump term which appears in (4.3) comes from K(X, Y), that is, from the double-layer potential which we have already analysed in Theorem 1. We have proved Theorem 5. Here is a corollary. Corollary. With the hypotheses and notation of Theorem 5, for every function f E L2(V, da) and almost all X E V, we have (4.9)

..lhn VS f(Y) - N(X) = f f(X) + K* f(X) ,

Y-X 2 where K* : L2(V, du) -+ L2(V, do,) is the adjoint of the operator K, where the choice of sign + is the opposite of the sign oft - 0(y), for Y = (y, t), and where n.t.lim means non-tangential limit. To prove this corollary, we let T : (L2(V, da))'"+1 -. L2 (V, da) be the vector-valued Calderon-Zygmund operator whose distribution-kernel is PV((Y - X) / I Y - X I n+ 1) By Theorem 5, this operator exists. Let N :

15 Potential theory in Lipsehitz domains 'd'

270

,,may.

L2(V,da) --, (L2(V,da))n+L be the operator of pointwise multiplication by the vector N(Y). Then

PV Iv IYY XI +1 N(Y)f (Y) da(Y) = T N f (X) . Thus K = TN and, passing to the adjoints, K* = N*T*. But T* = -T and it follows that pH..

Kf (X) _

(4.10)

K*f (X) = wI N(X) . PV f

v I YX X

+1 f (Y) da(Y),

'"'

which lets us identify the second term on the right-hand side of (4.9). We have proved the corollary. By abuse of notation, we shall write (e/8N)VSf(X) for the left-hand side of (4.9). Again, it will he necessary to state which side of V we are working from. We can now draw some conclusions about the Dirichlet and Neumann problems.

For the Dirichlet problem, let us take an arbitrary function f E L2(V, da) and suppose that there exists a function g E L2(V, da) such y7'

that (1/2 + K)g = f. Then Theorem I tells us that u = K(g) is the solution of the Dirichlet problem. Similiarly, if h E L2 (V, da) is a function such that (-1/2 + K *)h = f ,

then v = S(h) gives the solution of the Neumann problem, as (4.9) shows, when we work from the open set Il above V. It remains to show that the operators I/2+K : L2(V, da) -. L2(V, da) and -1/2 + K* : L2(V, du) --+ L2(V, da) are isomorphisms. At present, we do not have a symbolic calculus for generalized Calderon-Zygmund operators, moreover, K is not compact. The only resource we have is, following Verchota, to use the remarkable energy estimates of D. Jerison and C. Kenig.

5 The Jerison and Kenig identities

c5.

We continue with the same notation. We suppose that f E L2(V, do), we write u = S(f) for the single-layer potential generated by f , and we study the gradient Vu of u. This gradient is defined in Rn+1 \ V and, as it crosses V, suffers a discontinuity normal to V. Thus, for almost all X E V, we can define the tangential gradient Vtu(X) by projecting the non-tangential limit of Vu(Y), as Y -' X, onto the tangent plane at X. The tangential gradient defined in this way does not depend on the side from which the limit of V is taken As before, we let N(X) denote the downwards-oriented unit vector normal to V at X. With this notation, we get

15.5 The Jerison and Kenig identities

271

Theorem 6. IfV is a Lipschitz graph, if f E L2(V, da), and if u = S(f ), then 2

(5.1)

J

IVtuJ2Nda = J (8N) Ndv+2Jv (

) Vtudv

and

(5.2)

Ilu

2

f f) Nda+ 21 (a j VtuI2Nda =Jv(a + I

+ f)Btudv.

Let us begin by observing that (5.2) follows from the proof of (5.1) by applying that proof to the open set SZ_ below V, whereas, to prove (5.1) itself, we have to consider the open set St lying above V. When we replace I by St_, N is replaced by -N. Then the normal derivative of the single-layer potential changes sign, but also involves a discontinuity. In keeping the same normal vector in (5.2) as in (5.1), the jump term in (4.9) is responsible for the change from eu/8N to 8u/ON + f. So it is enough to prove (5.1). We first cheat a little, by pretending that Il is a bounded, regular, open set and that V = 852. We now write Vtu = Vu - (49u/ON)N and (5.1) reduces to the vector equation (5.3)

fanIVu12Nda=2 fan Vu-

-do.

To establish (5.3), we first change it to a scalar identity by taking scalar products with an arbitrary constant vector e E R"+1 It is enough to check that (5.4)

Jao(t2N e- 2(Vu-e)(Vu-N))de=0.

We then apply the divergence theorem and see that it is enough to verify

that div(IVuI2e - 2(Vu e)Vu) = 0 on Q. But this divergence equals -(Vu - e)Au, which is zero, since u is harmonic. To apply these considerations, we approximate to our original si by an increasing sequence IL of bounded, regular, open sets. The boundary OIl,,, will be the union of two subsets V,,, and W,,,,. In addition, we suppose that f is continuous, with compact support. We consider (5.3) for the open sets 52,,,,, with u = S f being fixed and the operator S being taken with respect to V (not V,,,.) Then we shall show that (5.5) (5.6)

IV_ IVtd2N,nda= - J IVU12Nde, Iv,.,

VU

On ON,,,

du,,, - rVu ON cu d., Jv

and that the corresponding integrals over W,,, tend to 0. From this point, we get (5.3) by a simple passage to the limit from the corresponding identity for the harmonic function u, restricted to 11,,,.

15 Potential theory in Lipschitz domains

272

We now describe the method of approximation and justify the limiting process. '.y

Let On be a sequence of real- or complex valued Lipschitz functions defined on lit". We say that this sequence converges strictly to a Lipschitz function ¢, if there exists a constant C > 0 such that, for every integer in E N, we have IIV4),,,,IIo < C and 04),,,(x) - V45(x), almost everywhere. Under these conditions, there exist normalization constants c,,,, such that the functions (x) + c,,, converge to 4)(x), uniformly on compact sets. So let us start with a Lipschitz function 0 and construct the 4,,, by G".

convolving with m"g(mx), where g is an infinitely differentiable function of compact support and mean 1. Then the functions 4,,, converge strictly to 0. Changing the meaning of the subscript m, if necessary, and possibly adding a constant en to each ¢,,,, we may suppose that ¢(x) < dim(x)
n/p. Theorem 1 tells us

16.2 A first example of lineanzation of a non-linear problem

279

that the infinitely differentiable functions operate on this algebra, in the sense of symbolic calculus. Since L1"5 is not an algebra any more when

s = n/p, the choice F(t) = t2 must be excluded in this case, which is why we must suppose that F is sub-linear. To prove the theorem for s > n/p, we put f j = Sj (f) and write

(2.4) F(f) =

F(fo)+(F(f1)-F(fo))+...+(F(fj+i)-F(fj))+---

The series in (2.4) is a telescopic series which converges uniformly to

F(f)The term F(fo) presents no difficulties, because fo and all its derivatives belong to Lr. It follows that F(f0) and all its derivatives belong to LP, as long as F(0) = 0. We then write (2.5)

F(fi+1) - F(fi) = m,°i(f)

where (2.6)

mj(x) = 10 F'(fj(x) + toj f(x)) dt. 0

The information we need about the functions f j is contained in the following lemma.

Lemma 1. If f E L1''(R") and ifs > n/p, or ifs = n/p and F' E LO° (IR"), then there exist constants Ce,, a E N", such that, for all j E N, we have II& mjIj°o < Ca271a1.

We assume this result for the moment and continue with the proof of the theorem. Consider the linear operator L defined by M

(2.7)

,Cu(x) _

m.7 (x)Oju(x) . 0

is the The operator L belongs to Cep S10,1. Indeed, Eo mj symbol of ,c and the estimates defining an element of S° 1 are precisely those given by Lemma 1. The operator L is thus bounded on all L1"7,

ii:

where r > 0, 1 < p < oo (Chapter 10, section 6). In particular, £(f) E LP,' when f E LA8. We still have to prove Lemma 1. Let us start by looking at the case s > n/p. Then IIfj II= < C, since f E L°O(R"). Bernstein's theorem then gives II8' f j II, < C2(j+1) I"I . To show that F(f j + tE 1(f)) satisfies inequalities of the same type, we apply the following lemma.

Lemma 2. If the functions gj satisfy Ilogj II. < C,01'4, where Ca does not depend on j, then the functions F(gj) = hj satisfy Il8hj IIo < Ca' 2j1-1.

16 Paradiferential operators

280

v''

An amusing way to see this is to consider the auxiliary functions g'1(x) = g, (2-3x), which satisfy II&93Iloo < Ca. Then the functions h3 = F(9?) satisfy Ilc h?II,,o < Ci,, as can be seen by applying the Faa di Bruno formula for the derivative of the composition of functions. We now recall that formula.

To compute a&F(g), where a = (al, ... , an), we let q denote an 'C7

integer with 1 < q< IaI and split the vector a E Nn into all the possible vector sums 81 + . + /3q, where t1, ... , Qq E Nn. We then form all the corresponding products F(9) (g)ad' g .. aNg. We then take the sum over all the possible decompositions of a, keeping q fixed, and then take the sum over all values of q E [1, IaI]. This gives a'F(g).

If s = n/p and F' E L'(W ), then, clearly, m., E L°°(l' ). We C7'

obtain the bounds on the derivatives of m1 by observing that IIa'f3II.,, < C,231 «1, when IaI > 1, even though it is not the case that II!3 II. CO-

3 A second linearization of the non-linear problem 'ti

The paraproduct 7r(a, f) of the tempered distributions a and f is defined by (3.1)

7r(a, f) _

Sa_2(a)03(f). 2

It is not at all obvious that the series converges in the topology of S'(lRt), but we can verify that it converges, by applying the following simple lemma.

Lemma 3. Let u3, j E N, be C°°(Rn) functions of polynomial growth satisfying

(3.2) there exist constants /3 > a > 0 such that the support of the Fourier transform of the distribution u3 lies in a23 < ICI < /323;

(3.3) there exist a constant C and an exponent m such that I%(x)1 < C2-2 (1 + Ixl)t, for all j E N. Then J:o u_,(x) converges to a tempered distribution in S'(R'). Conversely, if S is a tempered distribution, u3 = 01 (S) satisfies (3.3). Under the given conditions, IS,_2(a)(x)I < C2p1(1 + Ixl)p, for some integer p, and the support of the Fourier transform of S3_2(o) is contained in the ball B1, centre 0 and radius 23-2. Similarly, I A3 (f) (x) I < C292(1 + Ixi)q and the support of .T(.j(f)) is contained in the dyadic Thus, the support of the Fourier shell F, defined by 23-1 < ICI < transform of the product S?_2(a)01(f) is contained in B, +F1, that 23+1.

16.3 A second linearization of the non-linear problem

281

f32j, where a = 1/4 and 6 = 9/4. The

tai

is, in the region a2j
o i'>o

= E E A.i(a)O:i'(f) + ,j5j'-3

j'<j-3

=ir(a,f)+7r(f,a)+

Aj(a)Aj, (.f) +

Aj(a)Ai'(f) Ij-j'1:52

EO.u-E(a)oj(f) AEI 0. Suppose that f : lRn Then

R belongs to LP-(R") and that F E C' (R) satisfies F(0) = 0.

(3.4)

F(f) _ ir(F'(f ), f) + g

where g E LP,'+'(lR").

The purpose of the paradifferential calculus is to localize, as precisely as possible, the singularities of solutions of non-linear partial differential equations. But these singularities will be relative to a threshold of regularity: everything which is more regular than the threshold is not analysed and is considered as an error term. In (3.4), g is such an error term. It is worth describing a holomorphic variant of (3.4), because it involves the bilinear operation which foreshadowed the paraproduct in Calderon's work. If f and g are holomorphic functions in the (open) upper half-plane P and if f and g vanish at infinity, in a certain sense, Calderon defines

16 Paradiferiential operators

282

a function h, which is holomorphic in P and satisfies h'(z) = f (z)g'(z) and h(ioo) = 0. We put II(f, g) = h. Now, let f be a function which is holomorphic in P and which, together with its derivative f', belongs to the holomorphic Hardy space H2 (p). We let K denote the closure of f(P) in C. Let F be a function which vanishes at 0 and is holomorphic in a neighbourhood of K. Then (3.5)

F(f) = H(F'(f ), f) .

Indeed,

F(f)(ioo) = F(0) = 0

and

d F(f) = F'(f (z)) f'(z) .

Theorem 2 generalizes, and slightly improves, Bony's theorem ([16]), where p = 2 and s + r was replaced by s + r - e, for arbitrary e > 0. The proof of (3.4) consists of comparing the operator G of (2.7) (which arose in the first linearization) with the operator Ta, defined by TQ(f) _ rr(a, f). We need the followimg lemma.

Lemma 4. Let f belong to LP's(R' ), where r = s - n/p > 0 and let c be defined by (2.7). If a = F'(f ), then L -T,, belongs to the Hormander class OpSj,i.

Assuming this, the proof of (3.4) is immediate. The first lineariza-

tion gave F(f) = F(S0(f )) + £(f), where F(S0(f )) may already be considered as an error term, since it belongs to all the LP,' spaces, for m > 0. So, if we put R(f) = £(f) - T. (f ), by the lenuna and section 6 of Chapter 10, R(f) E LP,'+r To prove, as asserted by Lemma 4, that R belongs to Op Si i , we compute its symbol p(x, ). It is given by 00

p(x' ) _

4j (4

(2-3

) + mo(x)1'( ) + mi

2

where (3.6)

41(x) =f F'(Sj(f)+toj(f))dt-Sj-2(F'(f)).

We want to show that there exist constants C,,, a E N", such that, for all j E N, (3.7) . II&'4j(x)Ij= < The terms and of lead to functions which belong to all the Lp'm spaces, for m > 0, so we need take no further notice of them. We concentrate on (3.7). It will be enough to establish these inCa2-j''23Ial

equalities for jal > r and for a = 0. The classical logarithmic con-

16.3 A second linearization of the non-lznear problem

283

vexity inequalities will take care of the cases in between, that is, when

1 r. We no longer use the fact that q1 (x) is the difference of two terms, but show separately that

IIaSj-2(F'(f))II= < C.2-jT2j1(rI

(3.8)

and (3.9)

II Y'F'(Sj(f)+tA3(f))Iloo r, we have 1 +... + 2j(kr1-T) < C2j(I"l-T), which concludes the proof of Lemma 5. We move to the proof of (3.9). We shall change the notation a little by putting f j = S j (f) +to j (f ). We shall keep only the following properties

of the fj: (3.11)

IIfj+1 - fj II. < C2-'T

for j E N

and (3.12)

fjII.randjEN.

We then apply Lemma 6. Let f j, j E N, be a sequence of infinitely differentiable functions satisfying the estimates (3.11) and (3.12). Then, for any function F E C- (R), F(fj) still satisfies (3.11) and (3.12), the constants C and C,, being replaced by new constants C' and C. The statement is obvious for (3.11) and we shall concentrate on (3.12).

16 Paradiferential operators

284

The only difficulty is that the derivatives (?F(f,), IaI > r, involve derivatives of the form aO fi, with IQI < r, to which (3.12) does not apply. The remarks which follow are designed to get round this difficulty. First of all, we observe that (3.11) and (3.12) give (3.13) for all a E IYn_ llO'(fi+1- fill. < CC271a12-r'

In our particular situation this gives no new information. However, with the hypotheses of Lemma 6, (3.13) follows by convexity from the cases

a=0andlal>r. Thus (and from now on we may ignore (3.11)) if IQI < r llO f, U C be an infinitely differentiable function of N real variables ui, ... , UN. We suppose that F(O) = 0 and let Fi denote the N partial derivatives t5F/Sui, 1 < j < N.

Let r = s - n/p > 0, for some p with 1 < p < on, and let f = (fl,..., fN) be an RN-valued function, belonging to LP,9(Rn). Then F(f) = F(f1i... , fN) has the following expansion as a sum of paraproducts.

Theorem 3. Under the above hypotheses, N

F(f)rr(F'i(f),fi)+g,

r4.

(3.17)

where g E LP,s+r

The proof of Theorem 3 is the same as that of Theorem 2 and thus left to the reader.

16.4 Paradiffereniial operators

285

Before finishing this section, we give a lemma which will be useful for studying paradifferential operators.

Lemma 7. Let fj, j E N, be a sequence of functions in C°°(R") whose behaviour as j oo is described by the following conditions:

(3.18) there exist r > 0 and a constant CO such that, for all j E N, we have Ilfj Ilcr < Co; (3.19) there exists a family of constants Cp such that, if 1,61 > r, then, for all j E N, we have 118pfj11. < Cp20IPI-T).

Then we can decompose f, as

fj = gj + hj ,

(3.20)

where

(3.21) the Fourier transform of gj has support contained in the ball ICI < 2j-'o, (3.22)

for all 3 E N,

IIg; Ilcr _< CO,

and (3.23)

for all,Q E N'2 .

IIOhj II0 j-10

The estimates (3.21) and (3.22) are now obvious. For (3.23), we use (3.19); further, Ilol(fj)lL0 < Cm2(j-l)m2-jr for every integer m E N, which, combined with Bernstein's ineqality, gives (3.23). We may remark that, conversely, (3.22) and (3.23) imply (3.19).

4 Paradifferential operators Let r be a strictly positive real number which we shall keep fixed. Following Bony, we shall construct an algebra of operators, contained in G. Bourdaud's algebra, and a symbolic calculus which will enable us to invert the operators in our algebra, modulo operators which are regularizing of order r, that is, operators whose symbols lie in the class Si,i (IIY" x IIS"), which we used in the proof of Theorem 2. An operator which is regularizing of order r is a continuous mapping from LP'e(R")

286

16 Paradiferential operators

.7'

pay

to LA9+''(R"), as long as s + r > 0, for 1 < p < oo (Chapter 10, Section 5), and there is a similar statement for Besov spaces. Bourdaud's algebra consists of operators T which, together with their adjoints T*, belong to (gyp S1 1. This is described in Chapter 9, but has the disadvantage of not having a reasonable symbolic calculus. In a way, Bourdaud's algebra is too big and we are going to improve it by incorporating additional regularity with respect to x into the symbols we use. That regularity is measured by the fixed real number r > 0. It is no extra effort to define symbols of order m, although the calculations we shall need to carry out involve only operators of order 0. Lastly, in the definition which follows, C' will denote the inhomogeneous Holder space: the norm of f in C' includes the L°° norm. We have written enough about these spaces in the preceding chapters for the reader to need no further details.

Definition 1. Let r be a positive real number. A function a(x,

)

belongs to the class A,"` of symbols if u(x, ) E C°° (R" x R") and the two following conditions hold:

(4.1) there exist constants C(cr), cr E N", such that the hinctions of the variable x defined by i3{ a(x, ) all belong to C'(R") and satisfy ),n-dal

10'0-(x, )IIC-(R-) < C(a) (1 + I (4.2) there exist constants C(a, E3), a, 13 E N", such that, if IQI > r,

I0'a (x,O I < Qn,")(1 + In other words, we get regularity with respect to x "for free", as long as that regularity is not greater than r. After that, we have to pay. If r = oo, we are back to the usual class S. Using a symbol of this type, we define the operator a(x, D) by the usual formalism (4.3)

a(x, D)[e`x £] = a(x,

The next definition is closer to Bony's approach ([16]).

Definition 2. The set B consists of the symbols o-(x, ) E A;" which satisfy the following condition: for each fixed ., the support of the (partial) Fourier transform ofa(x, ), regarded as a function of x, is contained in the ball IM < Ie /10. We could just as well define B,'." without specifying that a(x, C) E A', but requiring (4.1) to hold, as well as the last condition of Definition 2. Condition (4.2) then follows automatically, as can be seen by rereading the proof of Lemma 7.

16.4 Paradifferential operators

287

The reader is invited to check that, for all real m,

gin c mCS'n CAm+r. r 1,0

1,1

Clearly A, C A', if s > r, and Br is contained in Am, as we have already remarked.

If m = 0, we write Ar and Br, instead of A° and B. As in the usual case, a(x, t;) belongs to A;." or B"`, if and only if belongs to A,. or Br, which enables us to reduce (1 + many problems to the case m = 0. The following lemma clarifies the relationship of Ar to B.

Lemma 8. For a function a(x, t;) E C' (R' x R"), the two following properties are equivalent: (4.4) a(x, k) E Ar ; (4.5) a(x, t; ) = -r(x, 0 + p(x,

wherer(x,t;) E Br and p(x,t;) E Sj,i. As we have already remarked, ST is contained in Ar. It is then clear that (4.5) implies (4.4). Conversely, we use the notation of the Littlewood-Paley decomposition to write 1 = 46(x') + Eo ,0(2-1t) and then (4.6)

al (x,)

a(x, ) = a(x, 0

where o,(x,t;) = a(x,t),0(2-it).

We then use Lemma 7 to decompose each of the functions a,(x, t; ) with respect to x. The importance of Ar lies in its being an algebra under ordinary multiplication. More precisely, we have Lemma 9. Let al (x, t), ... , UN (X, t;) be N real-valued symbols belong-

ing to Ar. For every function F E C°°(RN), F(ai (x, t), ... , aN(x, t)) lies in Ar.

There is no difficulty about the proof of (4.1), while the verification of (4.2) is identical to the proof of Lemma 6, with 2i replaced by 1 + Jkl. Corollary 1. Let al (x, t;) and a2 (x, t;) be two real- or complex valued

symbols in Ar. Suppose that there is a a constant 6 > 0 such that ja2 (x, t;) I > 6 whenever a1(x, k) # 0. Let us define a3 (x, k) by putting

a3(x,t;) = 0, when ul(x,t;) = 0, and a3(x,k) = al(x,l;)/a2(x,t;) otherwise. Then a3(x,t;) also belongs to Ar. Indeed, let F(u, v) be an infinitely differentiable function of two real variables it and v, which coincides with (u+iv)-1 when (u2+v2)1/2 > 6.

16 Paradifferential operators

288

We let a2 (x, t;) denote the real part of a2 (x, e) and let 12 (x, ) denote its imaginary part. We then form F(a2(x, t), /32(x, t)). The new symbol belongs to A,., by Lemma 9, and the same is true for the product al (x, t) F(a2 (x, t), l32 (x, )) = as (x, e)

Corollary 2. Let u(x, l;) be a symbol in

Suppose that, for some

lim inf Ia(xo, Ao)I > 0.

(4.7)

a-o0

Then there exist an infinitely differentiable function u(x), which is 1 at xo, an infinitely differentiable function v(k), satisfying v(M) = for

-

It;I > 1, A > 1, and v(ato) = 1, if A > A0 > 0, and, lastly, a symbol -r(x,t) E A, such that (4.8)

Ink

a(x,0T(x, ) = u(x)v()-

Indeed, let 26 be the liminf of (4.7). The regularity of

given

by (4.1), implies that Ia(x',k') - a(x,l;)I < e, if Ix' - xI < r7(e) and 6 > 0 if Ix - xoI < ri, Ie - I/ICI S 17(e)- In particular, We choose u E D(R) equal to 1 at xo and zero outside Ix - xoI -Similarly, the function v(E) will be zero in a neighbourhood of 0 and outside a cone of revolution with vertex 0, generated by the ball Y7.

tea,

- xoI 27 with, further, v(A o) = 1, if A > A0 > 0. It is then enough to apply Corollary 1 to the quotient u(x)v(l;)/a(x, ).

5 The symbolic calculus for paradifferential operators The symbolic calculus for paradifferential operators follows from the next result.

Theorem 4. Let r(x,1:) be a symbol in the Hormander class Sot 1 and let a(x, k) be a symbol in Br, for some r > 0. Then 7(x, D) o a(x, D) = y(x, D) + p(x, D), (5.1) where (5 . 2)

` Y (x ,

r) InI'

Let p denote the difference v - 5. The Fourier transform of p has compact support and vanishes in a neighbourhood of 0. Denoting the operator of convolution with 2"3p(23x) by R?, we see that >° 0?_3(a)Ri(f)

is an error term. Indeed, its symbol is E3 ...

i.4

and this < C2-3,. Now put sri(a, f) = E2 Sj_2(a)(V3+i(f) - V3 (f)) and let us verify that, up to an operator whose symbol lies in Si i , sri and it coincide. Their difference is exactly E2 S3 _2 (a) (R3+s (f) - R3 (f) ), a series to which we apply the Abel transform. Modulo a trivial operator we get the error term - E3 i3_3(a)R3(f ). After this, the comparison of 7rl (a, f) with *r(a, f) is easier. Their because

.r.

`''

belongs to Sj.

.t"

difference belongs to (gyp Si, i .

We return to the wavelets and to Proposition 9 of Wavelets and Opcoo

erators, Chapter 2. To simplify the notation a little, we restrict our attention to dimension 1. With the notation of that chapter, we have `'J

(7.5)

E3 = S3 + RR + RR

16 Paradifferential operators

296

where Si is the operator of convolution with 218(21x), and (q 1, on [-27r/3,27r/3] and = 0, if 4ir/3. The operators R,+ and Ry are of the form M,N,, where N, is an operator of convolution with 2'(2'x) or 21 C(21 x). Here i and r; belong to S(R) and their Fourier transforms vanish in a neighbourhood of 0 and have compact supports. On the other hand, M, is the operator of pointwise multiplication by et2ii2-x. With this notation, we have

Lemma 12. Ifa(x) is in C'(R), the paraproduct ir(a, f) may be defined by Eo E1(a)D, (f ), modulo an operator belonging to Cep SI,i . Here, again, we replace Ej (a) by the principal term of (7.5), namely

Sj(a). As for the error terms, we have IIRI (a)1k0 < C2-j', because a(x) E C'.We get bounds for the derivatives of R; using Bernstein's inequalities.

This leaves E°° S, (a)D, (f). We have D1=E1+1-E1=S1+i-S3+R+1-Rt +R.7--+i-R; The principal term is >o Sj(a)(Sj+1 - S,)(f), that is, one of the forms of *r (a, f). The two error terms are dealt with by Abel's transform. We let 11 (a, f) denote the form of the paraproduct given by Lemma 12. For this, we have (7.6) 11(a,V),\) = E, (a)%ia , if ,\ E A,, j E N, and

II(a,4k)=0,

(7.7)

forkEZ.

We write aj (x) = Ej (a) (x) and suppose, as before, that 0 < r < 1. We then let k denote the diagonal operator with respect to the wavelet basis, defined by and II.(4k)=0, ifkEZ. if AEA1, Then H(a, Va)

(a,(x) - a,(A+)),Ga(x) = g1,k(2jx)oa(x) .

Since a(x) E C', we have 191,k(u)I < C2-1'Ju - (k + 1/2)1' and, since 0 < r < 1, we get, for every integer l > 1, (7.8)

`

d) q_,.k(u)

C12-'j.

The symbol of the difference rl(a, -) - I , is E°O ry1(2?x, 2-1"t) b(2-1 e), where 00

gj,k(u)V(u -

7j(u, v) _ k=-oo

k)e-'(u-k)v

16.7 Paraproducts and wavelets

It is easy to see that

297

('0 yn

and it follows that ll(a, ) - I7a E OpSj,i. The final modification is to replace a, (A) by a(A), which is immediate, < C2-''-7. We have proved Theorem 6.

because Ila - all

If m < r < in + 1, we would have to replace the operator Ta of Theorem 6 by

nn)

(a(A)'h ... + `x

mf)ma(m)(A) f V x(X)

and a similar proof would give T. - Tare) E Op Si,i

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