Wavelets and Singular Integrals on Curves and Surfaces

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Ztirich F. Takens, Groningen

Subseries: Nankai Institute of Mathematics Tianjin, P. R. China (vol. 7) Advisers: S. S. Chern, B.-j. Jiang

1465

Guy David

Wavelets and Singular Integrals on Curves and Surfaces

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest

Author Guy David Universit6 Paris-Sud Math6matiques (B~t. 425) 91405 Orsay C6dex, France

Mathematics Subject Classification (1980): 42B20, 42B25

ISBN 3-540-53902-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-53902-6 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper

INTRODUCTION

These notes are the transcript of a series of lectures that were held in the Nankai Institute of Mathematics, in June 1988, as a part of the program on harmonic analysis. This book consists of three parts devoted to the following topics : wavelets, Calder6nZygmund operators, and singular integral operators on some curves and rectifiable subsets of IRn. Our aims for the three parts axe slightly different. The first part is intended to be an introduction to the theory of wavelets, and will insist mostly on the construction of various orthonormal bases of L 2 . The second part is centered on a necessary and sufficient condition for the L2-boundedness of certain singular integral operators on

~ n (the so-called T(b)-theorem), and some of its

applications. Although the result is not very recent, the t~chniques to prove it have not been described, up to now, in too many sources. Thanks to a recent work of R. Coifman and S. Semmes, we shall be able to give a full, conceptually very simple proof of that theorem. The final part is a survey of the author's main area of interest : questions related to the description of those kdimensional subsets of ~ n on which analogues of the Cauchy kernel define bounded operators on L 2 . The area is quite recent, but there are already sufficiently many results, often of a fairly technical nature, to justify the description attempted in Part 3. Unfortunately, there will not be enough room for a complete description of all recent results, but we shall try to allude to most of them, and to give a precise account of the fundamental techniques involved. This book is not intended to be exhaustive. We shall try instead to explain in details a few of the "real-variable methods" that have appeared after the books [St], [CM1] and [J6]. The lectures in Nankai and, to a lesser extent, these notes, were prepared for an audience whose main center of interest would be close to classical analysis. This means that, in many cases, we shall avoid spending time on some of the classical results of harmonic analysis. A fair knowledge of these results, although not really required, will often help the reader understand our motivations, for instance. The first part is about wavelets. Actually, we shall only consider orthonormal wavelets, i.e. wavelets that form orthonormal bases of L2(IR n) . These bases are obtained from one, or a finite number of functions ~ , by dyadic translations and dilations. For instance, the Haar system is a basis of this type, but the corresponding functions x~ are not, in this case, continuous functions. The interesting wavelets will be asked to have both a good decay at infinity and some reasonable amount of smoothness. The main advantage, from our point of view, is that the

VI decomposition of a function

f

in such a basis then looks a lot like a Littlewood-Paley

decomposition of f , but of course is simpler and also uniquely determined. In practical terms, this will imply that the size of the coefficients of f, as well as the type and speed of convergence of the series that gives f back, will depend in a very simple way on the smoothness and decay properties of the function f. Also, the fact that the wavelets have enougtl decay makes the "wavelet transform" (i.e., the mapping that sends f to its coefficients in the wavelet basis) a local transform : one does not need to know a function on the whole real line to have an idea of its coefficients near 0. This is of course an advantage over the Fourier transform (which does not have these two nice properties). Most of Part 1 will be devoted to the construction of various orthonormal bases. A few of the properties of the wavelet transforms will also be mentioned, but we shall restrict ourselves to those which have a direct connection to harmonic analysis. This means that most of the applications of wavelets, in particular the practical ones, will be omitted. The author is aware that this is a very incomplete choice of topics, but does not think he has enough time, or competence, to treat much more material. Moreover, the interested reader will easily find all the missing information in the very good books by Y. Meyer entitled "Ondelettes et op6rateurs" [My4]. Incidentally, most of Part 1 was prepared out of a preliminary version of [My4], which French-speaking readers might as well consult directly. Part 1 begins with a small introduction to the notion of wavelets. All our constructions of wavelets will use the concept of "multiscale analysis" introduced by S. Mallat and Y. Meyer. This concept is presented in Sections 2 and 3. The orthonormal basis of L2(IR) associated to a given multiscale analysis is constructed in Section 4. This gives, for instance, Y. Meyer's "LittlewoodPaley wavelet" ~

,~ (IR), or wavelets arising from spline functions. In Section 5, we shall see

how build a basis of L2(/R n) using one-dimensional wavelets and a tensor product construction. It is also possible to associate an orthonormal basis of L2(IR n) to a multiscale analysis even when it does not come from a tensor product ; we shall see how in Section 6. Section 7 is devoted to the construction of I. Daubechies' compactly supported wavelets. In Section 8, we shall briefly describe some of the properties of wavelets that make them so attractive (this section can be read before the previous ones). A few words will also be said abo~t the existence of various extensions. The second part of these notes is about singular t,ntegral operators. What we now call Calder6n-Zygmund operators (see Definitions 1-1, 1-2 and 1-5) is a much wider class than the first examples that were first studied by Calder6n and Zygmund in the fifties. Of course, many of the properties of the early examples remain true in the more general case : for instance, the LPboundedness, l 0. The formula is

(2)

f:

fo ° ¢ ~ * ¢ ~ * f

dt T

'f°r

fEL2(R~).

Now, we can see (2) as a way to associate coefficients to f , and then reconstruct f from the coefficients : for a > 0 and b E R '~, call ~ , ~ = a-"/2~(z~b) (the "wavelet" with "scale" a and "centered" at b). The "wavelet transform" can be defined by

(3)

F(a, b) = / f(x) ~a,b(x)dx,

and (2) can be seen as the "inversion formula"

(4)

f(z) =

/

F(a, b)~a,b(~)

dadb

a>O bER

'~

(we leave the details as an exercise). This formula is far from being recent. The reader should be warned that, for most of the applications of wavelets , the formula (2) would be quite sufficient. However, introducing orthogonal wavelets will, in most cases, make computations much simpler and, in some cases, the fact that one has a basis will be really necessary. Note that, with a more clever choice of ~b, one can simplify (2) a little, and in particular replace the integral by a discrete sum (see [FJ1,2,3]). Also, reproducing formulae such as (2) were introduced later (and independently) by A. Grossman (for theoretical physics reasons) and J. Morlet (for oil prospection reasons). We shall look more specifically for functions ~b, defined in R , such that the functions ~bj,k(x) = 2H2~b(2Jz- k), j,k C Z are an orthonormal basis of L2(R). Furthermore, we shall ask ~b to be rapidly decreasing, and reasonably smooth. Let us rapidly explain why. An example of function ~b would be 1~o,1/2] - 111/2,1], which would give the Haar basis. When we take a function f , and write it as a series E cj,k~bj,k, the partial sums are never continuous (even if f is very regular), and will, in general, converge to f only in L~-norm. If the function ~b is smooth, then the partial sums are smooth too, and it is possible to hope that if f is regular, then the partial sums will converge to f in a much smaller space than L 2. Without getting ahead of ourselves, let us mention that it is indeed the case : if ~b is smooth enough (and rapidly decreasing) and f is in the Sobolev space H s, then partial sums will indeed converge to f for the topology of H m ! A connected fact is that many function spaces will be characterized (again if ~b is smooth enough, and rapidly decreasing) by the fact that the coefficients ej,k = f ~j,k vanish at a certain rate when j tends to +co. We shall be a little more precise later. Before we start constructing wavelets , let us mention a little bit of history. The first occurence of "smooth, orthogonal wavelets ", is due to O. Stromberg, in 1981 : he found, for each integer r, a function ~b of class C r that decays exponentially at co, and such that the ~bj,k's are an orthonormal basis of L2(R) [Sg]. A function ~b E S(R) with the same property was discovered (again independently) by Y. Meyer (in 1985), and rapidly generalized to r~ dimensions by P.G. Lemari6 [LM]. The approach that will be described here is quite different from the initial ones. It relies on the notion, introduced by S. Mallat, of "multiscale analysis".

2. M u l t i s c a l e A n a l y s i s .

The following notion will help us understand the construction of the wavelets. It will also make the proofs a little longer : in some cases, we could just pull out a formula for ¢, and ask the reader to check that it works (actually, the first proofs looked a little like this to the unexperienced reader). Let us mention, before we start, that the construction below is due to S. Mallat and Y. Meyer. D e f i n i t i o n 2 . 1 . - - A multiscale analysis of L 2 ( R n) is an increasing sequence Pry, j E Z, of closed subspaces of L 2 (R'~), with the following properties : (5) j ~ . V,. = {0} and Je~-UV,. is dense

(6)

e vi if and only if l(2 ) e 5 + , ;

(7) f(x) e Vo if and only if f ( x - k) E Vo for each k • Z "~ ; (8) There is a function g • Vo such that the functions g(x - k), k • Z n, form a Riesz basis of Vo. Recall that (8) means that the g(x - k) generate a dense subspace of Vo, and that there is a C > 0 such that

for all £2-sequences ak (or all finite sequences ak). Equivalently, it means that (ak) ~-* ~ Ctkg(x -- k) defines an isomorphism from £2(Z'~) kEZ" onto V0. Comments : - If we k n o w Vo, (6) tells us h o w to define the Vj's, and so we only need to check (5),

(7), (8) and the monotonicity of the Vi's. - The function g is not unique. analysis.

Many different g's could give the same multiscale

- A trivial consequence of (6) and (7) is that f ( x ) E Vj if and only if f ( x - k2 - i ) E Vi for all k E Z'*. - Here is a vague interpretation of the role of the Vi's. For each j , call P1 the orthogonal projection on Vi ; (5) says that P i f --* 0 when j ~ - o o , and P i f --~ f when j --* +oo (in both case, with strong convergence [Exercise[). Each j corresponds roughly to the size 2 - i , and the projection P i f we'll give details about f , up to the size - 2 - i . So, PYf should give more and more precise details (we'll see this later, in examples). Example

1 :

splines of order r .

For this example, the dimension is n = I. W e define V0 = {f E L2(R) : / is of class C r-1 and the restriction of f to each interval ]k, k + 11 is a polynomial of degree _< r). For instance, if r -- 0 we get step functions, whereas r = 1 gives piecewise affine functions that are continuous. The spaces Vj are defined using (6), and the properties (5) and (7) are clear. We'll see soon that (8) is true, and even that we can pick g = X * X ... * X (r + 1 times), where X is the characteristic function of [0,1].

D e f i n i t i o n 2 . 2 . - - Let r ~ N . W e T say that the multiscale analysis (Vj) is r-regular if one can choose the function g in (8) so that

(9)

la"g(~)l 0 and all multiindices a such that [a[ < r.

Note that we only ask Oag to be in L c~ but not necessarily continuous. If we believe that g = 27 * ... * Z gives a Riesz basis for V0, then splines of order r give a M.S.A. (multiscale analysis) which is r-regular. Example 2 : Let us try Vo = { f E Lz(R) : f i s supported on [-Tr, lr]} in dimension 1. It is easy to check that Vo is stable by translations by integers and that (5) is true (with the obvious definition of Vh). Also, the Vj's are an increasing sequence. The problem with this example will be to find a suitable function g. If we look for an orthonormal basis of 110, we might as well use the Fourier transform, because the l[_r,rl e ik~, k E Z, are an orthogonal basis of V0. Applying ~r-1 gives a function g ( x ) -- -sin,vx - 7 ~ " But g does not decay rapidly at oo. Furthermore, no other choice of g E Vo would both give a Riesz basis, and enough decay at oo (we'll leave this as an exercise). So we'll have to exclude this example from the authorized MSA's. Note, by the way, that since g(0) = 1 and g(k) = 0 f o r Akg(x - k) is quite easy: ~1¢ = f(k).

k¢

0, writing f E Vo as

k

Example 3 : This one is a smoothed-up version of Example 2 ; this is the example that will give Y. Meyer's basis. Let 0(~) be a function in D(R) = C ~ ( R ) , with the properties (10) 0 _< 0(~) _< 1 everywhere ; (11)

0 is even ;

(12)

O(~) = 1 for ~ E r-2~ ~1. t 3 ' 31'

(13) 0 ( ~ ) = 0 out of t[ - 48 r ' 34_~x 1 and 1

(14) o2(~) + o;(2~ - ~) -- 1 for o 0, f lg(x)2l(1 + ( x ) ) 2 ~ d x < + c o , and so ~ is in the Soholev space H M ( R " ) . Let 0 E C ~ ( R n) be such t h a t O(x) - 1 on B ( 0 , 2 0 n ) . For each M > 0, the sequence k ~ II~(00(~ - k ~ ) l l ~ ~ in e2(z ") [exercise ; use Leihnitz'rule and the fact that if functions have disjoint s u p p o r t s and their s u m is in L 2, then their L2-norms form a l 2sequence]. By Soholev's embedding's theorem, we see t h a t for all M ~ > 0, the sequence k -+ II~(00(~ - k~)llo~, is also in e2(Z'~). This is enough to conclude t h a t the series

~ Ig(( + 27rj)[2 converges uniformly on jEZ ~ each compact. T h e same thing would be true if ~ were replaced by any of its derivatives. Therefore, G is in COO(Rn), and we proved L e m m a 3.1. Since we know t h a t G > c > 0 a.e., it follows t h a t G(~) -1/2 is a well-defined, positive and Coo function ]the expert reader will note the similarity with G r a m ' s process]. As (15) would suggest, let us try to define 99 by (16)

~(~) = m ( ~ ) ~ ( ~ ) ,

where

m(~) : G(~) - 1 / 2 .

Lemma 3.2.-The function 99 defined by (16) satisfies the regularity conditions (9) with the same r as g, and 99(x - k), k E Z '~, is an orthonormal basis of Vo. ~ akc ik~, kE ~.,* where the ak's are decaying as fast as we wish at infinity. It is then an easy exercise to check t h a t 99{x) : ~ a k g ( x + k), and its derivatives of order _< r, have the same decay as the derivatives of order n (because n > 2) in the real valued-case, and 2 "*+1 - 1 > n in the complex case. Consequently, the image by (2"*/2rno,n)neE, of [0, 2r]'* has zero surface measure and, since it is compact, there is a small ball, centered on the sphere, which does not meet the image [2"*/2(mo,n),eE,}([O, 27r]'*). We shall use the following lemma. 6.3.-Let S be the unit ball of R 2" (respectively C2"), and B a small ball centered on S. There is a mapping F, defined and C °o on S \ B, with values in the set of unitary, 2"* × 2"* matrices with real (respectively complex) coefficients, and such that for all x C S \ B, x is the first row of f ( x ) .

Lemma

Let us first see how to deduce T h e o r e m 6.1 from this lemma. Let us take, for our matrix ((2'~/2me,,)), the function F((2~'/2rno,,)) given by the l e m m a [where B is a ball that does meet (2"*/2rno,,([O, 2r]'*)) (2"*/2mo,tl(R'*))]. T h e n all the me,n are Coo, 2~rperiodic in all directions, and the matrix 2"*/2U(~) of L e m m a 6.2 is unitary for all ~. It follows t h a t the functions ¢c, ¢ E E ' defined by (21) and (22) give an o r t h o n o r m a l basis =

15 of V1, and so the ¢ ~ ( x - k), e E E and k E Z n, are an o r t h o n o r m a l basis of W0. Also, each rn~(~) is C °°, and so the functions Cr defined by (21) satisfy (9) because ~o does (we have seen the a r g u m e n t a few times already). Thus T h e o r e m 6.1 follows from L e m m a 6.3. Let us prove the lemma in the complex-valued case (in the real case, one would just remove the bars). Call q = 2 n - 1 and let ( Z l , . . . ,zq,zq+l) denote the coordinates of a vector z E S c C q+l. We can safely assume that B is centered at ( 0 , . . . ,0, 1). Our first step will be to find q vectors w l , . . . ,wq, so t h a t Z, W l , . . . ,Wq form a basis of C q+l. Let us take for w I the (3" + 1) at column vector of the matrix

z2

a 0

0 a

... ...

zq Zq+l

0 --~l

... ...

a --Y~q

zl

(23)

A(z) =

where a > 0 is chosen as a function of the size of B. We just have to show t h a t det (A(z)) 0 for all z ~ B , if a is small enough. Exercise:

Prove t h a t det (A(z)) = (--l)qq-l{•q--l([zl] 2 J r ' ' ' ~-[Zq] 2) -- olqzq+l}.

Let us check t h a t if z ~ B and a is small enough, det A(z) ~ O. If this was not true, one would have aZq+l = [Zl[ 2 + . - . + IZq[2, and so Zq+l :> 0 and also, since [Zq+l[ _< 1, [1 - Zq+l[ 2 = [zll 2 + - . . + [zq[2 >0 such that, for each integer r > 0, there is a multiscale analysis of L 2 ( R ) , of regularity r, such that the functions ~o and ¢ of Section 3 and Section 4 are compactly supported, with supports in [-Cr, Cr].

16 Note t h a t finding a ¢ E C ~ is impossible (exercise : it would follow f r o m the fact t h a t the ¢),k form an orthonormal basis t h a t all moments of ¢ are zero, which is impossible if ¢ is supported in [ - M , M ] by density of the polynomials). The fact t h a t ¢ has c o m p a c t support is a nice property, because it makes the "wavelet transform" completely local (instead of almost local). One can hope t h a t this p r o p e r t y will be useful for practical problems (for instance involving time !). The proof we are going to see is not the original one. As usual, it was c o m m u n i c a t e d to me by Y. Meyer, and it involves ideas of S. Mallat, Y. Meyer, J-P. K a h a n e and Y. Katznelson. The construction is also similar to Tchamitchian's work mentioned above. The idea is the following. If (Vi) is a MSA of L 2 ( R ) , we have seen in Section 4 t h a t there is a 2~r-periodic, C ~ function mo(~), such t h a t ~(2~) -- mo(~)~(~) (this is (17)). We shall start from a function m0, and try to build a MSA with it. T h e point is t h a t the size of the supports of ¢ and to is easier to control from m0. So, let us give ourselves a function m0 with the following properties :

(25)

{

m0(~) is C ~ and 2 ~ r - periodic, Im0(e)l 2 + Imo(e + )12 = 1 for all too(0) = 1.

We have seen in L e m m a 4.2 that the first conditions are necessary if we want m0 to arise from (17) for a MSA. It would not be too hard to check t h a t the third condition is necessary too (because of (17), one only needs to check t h a t ~(0) -- 1). Since we are only interested in the converse, we will leave the verification to the reader. If we want to build a MSA from too, we will need a function to, and here is a good candidate : (26)

=

m0C2-J

).

j=l Note t h a t the infinite p r o d u c t converges because m0(0) -- 1 ; indeed, m 0 ( 2 - i { ) = 1 + O ( 2 - J ) , so that ~ L o g m 0 ( 2 - i { ) converges uniformly on c o m p a c t a , and 3' so ~({) is well-defined. We still want to check t h a t ~ ( x - k), k E Z, is an o r t h o n o r m a l system, and t h a t its closed span V0 is part of a MSA. Finally, we shall have to choose m0 so t h a t the resulting to is regular enough, and compactly supported. If we want ~p to be compactly supported, we'Ll have to choose for m0 a finite trigonometric sum, because

mo(~) = ~ O~keik~, where On the other hand, is compactly supported. which are s u p p o r t e d in the supports, i.e. [ - 2 c ,

ak = ~1 X to(])~o(x x - - + k)dx

is also c o m p a c t l y supported.

suppose m0 is a (finite) trigonometric sum, and let us show t h a t to By (26), to is the (infinite) convolution p r o d u c t of the [2Jrh0(2Jx)], [ - c 2 - J , c 2 - J ] , so that the support of ~a is contained in the s u m of 2c]. [Exercise : check the argument, and in particular make sure

17 it has a sense distributionwise, l Finally, if ~o is c o m p a c t l y s u p p o r t e d and m0 is a finite trigonometric sum, ¢ is also c o m p a c t l y s u p p o r t e d because of (20). So we'll choose for m0(~) a trigonometric polynomial. Let us come back to the p r o b l e m of defining a M S A from a function m0(~) defining

(25). Lemma

7.2.--

If ~ is defined by (26) for a function m o satisfying (25), then [[~o][2 ~, then one of the ~/21 falls in +~ - ~ - - , - ~ [ and the product is zero. One can see the definition of g below as an a t t e m p t to imitate this example, and choose g as large as possible near 1 and as small as possible near 7r, so t h a t the infinite p r o d u c t will have more chances to drop sharply. To define g, select a large k E N , and call Ck =

{/0"(sin t) 2k+l dt }-1

We shall define g by (32)

g(~) = 1 - ck r]n~(sin t) 2k+1 dt.

Note t h a t g is a trigonometric polynomial of degree < 2k + 1, t h a t 0 < g(~) < 1 everywhere and g(~) > 0 on ] - ~r, ~r[ by our choice of ck, and t h a t g(~) + g(~ + It) = 1. So we'll just have to check t h a t oo

G(~)= Lemma

7.6.--

1-I 9 ( 2 - j ~ ) j=l

We have ck < Cx/k.

satisfies

(31)

for some

s>k/C.

21 Choose a small a > O, and write

f,_~÷a/V~

1 _> (sin t)2k+'dt > -2a ~ [sin ck ~-a/v'~

(2

~k)] 2k+1

+

V~ COS Since (2k+l)

Log

(

cos

~(2k+l)~-

~ 2 a 2,

we obtain e~-1 > ( C v ~ ) -1, and so the lemma is true. Lemma

7.7.--

If k is large enough,

(33)

[4~ k g(t) _ 1.

22 Before proving this lemma, let us show how it implies that G(~) satisfies (31). A change of variables shows that sup

f(~)...f(~--f)

sup

Fi(t) =

i1 < t < ~ 1

2~+-2 a - l r + a -1 we get the function m0(~) required for Lemma 7.4, therefore proving the theorem. Thus we only need to prove Lemma 7.8. To do so, it is more convenient to consider the function h(t) = f(t) f(2t). If we can prove that h(t) h ( 2 t ) . . , h(21-1t) 0 such t h a t , for each d y a d i c interval I,

~

Icj,k[2 0 and x C ~ n , call ~It,z the operator of translation by x and dilation by t defined by ~qt,zf(u) = t'~12f(tu - x). We shall say that the operator T

: P ~

P~ is weakly bounded if the operators

~ t-,1z T ~ t , z are uniformly bounded from P to P ' for t > 0 and x C IRn. This means that, for each compact set K, there is a suitable seminorm NI" Ill such that

I
l---

c

II1 .r III III g III

for all f and g supported in K. Let us give an equivalent definition. For each B C ]Rn and each integer q > 0, define, for f supported in B,

(8)

N~(f) = R n/2 ~

R I'~IIIO'~f Iloo, l,~l_ 0 and 6 > 0 such that, for all x E ]R n and r > 0, there exists a c u b e Q such t h a t dist(x, Q) < Cr, t r < d i a m Q _< Cr, and such t h a t (10) We'll even say t h a t b is "special paraaccretive" if (10) is true for all dyadic cubes. E x e r c i s e . Check t h a t - if b is paraaccretive, then I b I_> 5 a. e. ; - in dimension 1, e iz is not paraaccretive ; - cubes can be replaced by balls in the above definition, without changing the notion. T h e next t h e o r e m says that, in T h e o r e m 1.9, the function 1 can be replaced by any paraaccretive function b. T h e definitions are slightly more complicated, because it is more natural to define T from bD to (bP) I in this case.

31

Notation. If b E L ~ , we shall denote by Mb the operator of pointwise multiplication byb.

Theorem 1.11 ([DJS]). Let bl and b2 be two paraaccretive functions. Let T be a (bounded) operator from bid to (b2P)~. Suppose that T is associated to a standard kernel K(x,y) in the sense that < Tf, g > = f f K(x,y)f(y)g(x)dydx whenever f ~ btP and g E b2P have disjoint supports. Also suppose that Mb2TMba is weakly bounded, that Tbl E B M O and Ttb2 E BMO. Then T extends to a bounded operator on L2(IRn). This t h e o r e m is actually a generalization of a result by A. McIntosh and Y. Meyer [McM] : the i m p o r t a n t special case when Tbl = Ttb2 = 0 for accretive functions bl and b 2. Their proof uses a formulation in terms of interpolation of the L2-boundedness of the Cauchy integral on Lipschitz graphs [CMM]. Remarks 1. We didn't define Tbl (or Ttb2) in this case. This time, it will be a linear form on b2 P, defined modulo an additive constant. This means t h a t < Tbl, b2~o > will be defined if is a test function such t h a t f ~o(x)b2(x)dx = O. The definition of < Tbl,b2ta > is similar to w h a t was done in Definition 1.4. One picks a function X E P such t h a t X =-- 1 on a neighborhood of supp ~a, and one writes

< Tbl,b2~o > = < T(blX),b2~o > + f ] KCx, y)[b,(1- x)]Cy)b2(x)~o(x)dy dx = < T(blX),b2~ > + f / { K ( x , y ) - K(xo,y)i{bl(1 - X ) } (y)b2(z)~(x)dy dx. The last integral converges because of (3), as before. 2. For b o t h theorems, one actually gets a control on T, like I[T I1L2,L2~_k I] Tbl IIBMO +k II Ttb2 IIBMO +kCo + kC, where Co is the constant of (2) and (3), and C comes from the weak boundedness (C is any constant t h a t shows up in an inequality (9)). This easily follows from the proof, or from Baire's Category theorem. 3. T h e converse of T h e o r e m 1.11 is, as for the T l - t h e o r e m , a direct consequence of T h e o r e m 1.6. 4. Paraaccretivity is, in some sense, an optimal notion : one can show t h a t if b is such t h a t the t h e o r e m is true for bl = b2 = b (and all operators !), then b is paraaccretive (see [DJS]). The next sections are devoted to giving a short, new proof of the T b - t h e o r e m which is due to R. Coifman and S. Semmes [CJS]. It is also a nice coincidence t h a t it uses ideas related to wavelets.

32 2. F i r s t s t e p o f C o i f m a n - S e m m e s ' a Riesz basis

proof :

This section is needed only for the Tb-theorem. If bl -- b2 -- 1, then we shall construct the Haar system in this section ! Given a paraaccretive function b, we intend to build a Riesz basis of L2(~'~). We shall first consider the case of a "special paraaccretive" function b, i.e. a b o u n d e d function such that, for some b > 0, (10) is true for all dyadic cubes. Let us first define projection operators : for k E ~ , let E k be such t h a t E k f ( x ) ---- 1

(11)

/q f(t)dt,

where Q is the dyadic cube of side length 2 - k t h a t contains x. (This is the projection operator associated to the Haar s y s t e m ; note t h a t it is a typical martingale projection.) Also define the difference D k = E k + l - E k . Now define corresponding projection operators "relative to b" : 1

(12)

-

-

F k f ( x ) - EkbCx)

where again Q is the dyadic cube of size 2 - k containing x ; note t h a t the first denominator is b o u n d e d below because of (10). Finally, call Ak ----Fk+l -- Fk. Let us check a few easy facts. First, note t h a t f o f ( t ) b ( t ) d t = fQ F k f ( t ) b ( t ) d t for all dyadic cubes Q of size 2 - k (the "martingale property;'~. Next, (13)

F i F k = Fi,.,k.

If k < j, then F k f is constant on each cube of size 2 - k , and so it is constant on cubes of size 2 - i , and F i does not change it. If k > j, we use the martingale property above for Fk : the integral of Fk on cubes of size 2 - k is the same as the integral of f ; this remains true for cubes of size 2 - i and so F j F k f = F i f . (14)

A j A k = bj,kA i .

Just expanding gives A i A k = (Fi+ x - F j ) ( F k + 1 - F k ) = F j + I F k + 1 - 1 ? j F k + l -- F i + I F k ÷FjFk. If j > k, w h a t remains is Fk+l - Fk+l -- Fk + Fk = 0 ; if j < k, one also gets 0 (by s y m m e t r y ) and if j = k, one is left with F j + I - Fj - Fj ÷ F i = A j . If u and v are c o m p a c t l y supported and k ~ £,

(15)

/

-- O.

33 We can assume k > l. T h e n A t v is constant on each cube of size 2 - k . On such a cube Q, preserve the integral against b, and so fo(Aku)b -----0. We just have to sum

Fk+l and Fk orI Q.

Let us now prove a square function estimate. Lemma

2.1. There is a

(16)

0 -1

constant C > 0 such that,

for

all f E L 2,

II/II~ Z = ~ n a'~b'Tfln" We are given a first vector v0, and have to "complete the basis". First consider v~ -- {w = (wn) : ~ Wnao,,~ n -- 0}. This is a space of dimension 2 '~ - 1 ; the restriction to v~ of the bilinear form < , > 8 cannot be zero, because it w o u l d imply t h a t each element of v0a- is "orthogonal" to the whole space (but the linear form < w,. >~ does not identically vanish unless w = 0). So one can find two elements wl and w2 of v~, such t h a t < w l , w 2 > ~ 0. If b o t h < wl~wl >~ and < w2,w2 >~ vanish, then < wl + w 2 , w l + w2 > Z ¢ 0. In any case, we can find a vector vl C v~ such that < v l , v l > ~ ¢ 0 and, after normalization, one can even assume t h a t < v l , v l > ~ - - 1 (we are implicitly assuming here t h a t I -- {0, 1 , . . . } ; otherwise, we would give Vl is a more complicated name). Let us do the same thing a g a i n : v~ Mv~ is a (2 '~ - 2 ) - d i m e n s i o n a l space, and (unless n = 1) the bilinear f o r m < , > p cannot vanish on this whole space. So we can find Wl and w2 E v~ M v~ such that < w l , w 2 > ~ 0. T h e n there is a v2 C v~ n v~, with < v2, v2 >Z-- 1. Applying this argument enough times, one gets vectors re, e E I, such t h a t < re,re, > ~ - - 6c,~,. Translating things back in terms in terms of functions, we just proved the existence of the h~'s. Note t h a t the ve's are independent : if ~ )~v~ = 0, t h e n taking the twisted scalar p r o d u c t with re, gives ~e, --- 0. Hence, the h ~ ' s are independent from each other, and are also independent from the constant function (coming from v0). L e r n m a 2.2. Call VQ the space of a11 functions f that are supported on Q, constant on each Q , , and such that f fb = O. Then every f E VQ can be written

(20)

S =


b

eel. where < f , g > b = correctly chosen, (21)

S fgb

is a notation for the twisted scalar product. Also, ff the h~'s are

c - ' II S I1_
bl < C II f

for all f E VQ, and a constant C that does not depend on Q or e.

36 To prove (20), just note that, since the h~'s are independent, they form a basis of VQ. So every f E VQ can be written f = ~ Ceh~, and then

/,

hQhQb = C~,

f hqb = E

(by (18) and (19)), which gives (20). A short glance at the construction of the h~'s is enough to convince oneself that one can choose the ve's so that their coordinates (the ae,,'s) are less than C

/

min I ~ , l t r/

}"

, h~, Q

•

and

c-' II: II~ i~'.

Also, all the cubes of a given generation have roughly the same size, and they form a partition of ]R n. Finally, if R is a cube of generation k, all the cubes of generation k + 1 that meet R are contained in R, and the number of these cubes is < C. We can now apply the same argument as in the beginning of this section, and get a Riesz basis h~, where this time Q runs in the new set of cubes, and for each Q, e is in a set [(Q) (which this time does not always have the same number of elements). The formulas (22) and (23), in particular, are still true in this context.

38 3. P r o o f o f T h e o r e m

1.11 when

Tbl = Ttb2 : 0

To simplify notations a little, we shall do the proof when b 1 : b2 = b (we'll say a few words later on how to modify the argument in the general case). We'll also suppose that b is "special paraaccretive', the modifications to obtain the general case would be trivial. The idea of the proof is, simply, to estimate the coefficients of the matrix of TMb in the basis (h~) of the previous section, and to show that they decrease rapidly enough away from the diagonal. Let us isolate in a first lemma the only part of the proof where we'll need the regularity estimate (3) on the kernel of T. L e m m a 3.1. Let Q be a dyadic cube, and let xQ be its center. Let h be supported on Q,

and such that (26)

fQ h b = O. Then, 17"h(x)

for x ~[ 2Q,

I_< C lO la+ l

-xQ I-"- li h It ,

where T = MbTMb. In particular, if h is one of the h~ 's, we get (27)

I :Fh~(z) l< C I Q 1½+~l x - xq I- " - 8

for x • 2Q.

To prove the lemma, write

Th(x) = b(x) f o K ( x , y)b(y)h(y)dy = b(x) / { K ( x , Jq

y) - K ( x , xQ)]b(y)h(y)dy,

and (26) follows from (3) ; (27) follows from (26) and the fact that [I h~ lloo< C [ Q ]-1/2. R e m a r k 3.2. This is the only place where (3) is used (we'll use this remark in Section 4). Call CQ, R = < T h Q , h n > . Because h~ and h n a r e not smooth, we did not say yet how to define this number. When the closures of Q and R are disjoint, we can of course use the kernel of T ; when T is the principal value operator associated with an antisymmetric kernel, it is easy to check that the formula (4) can still be used. For the general case, we shall have to use a small limiting argument and the weak boundedness of T. We postpone this definition to the proof of Lemma 3.4. Since the indices e and e~ do not play any role in the estimates, we shall simplify and write CQ,R = < 7"ho,hn > instead (the e's will reappear only when we need them). Let us summarize the estimates on the CQ,R that we want. L e m m a 3.3. Let Q and R be dyadic cubes, and let the R , ' s be the cubes of the next generation that are contained in R. Then

(28)

if I Q II R I ~, then we simply can replace the last line by C I Q 1½+~l R I-1/~l R I dist(Q,R) '~+~, which gives (29). Next, suppose that I Q I- l < C I Q ll/21R 1-1/2 . For < Thq, hi >, one can use a small limiting argument, which is very similar to the argument in Lemma 3.4 below (but simpler), and thus left as an exercise, to show that

~_ C / / [ x - y [-nl Q l-1/21R [-i/2 1Q(y)13Q\Q(x)dxdy = < ThQ, hR-c >, where c is the value of h R on R0. Note t h a t this is not quite conform to our definition of Ttb (the first remark after the s t a t e m e n t of T h e o r e m 1.11) : we said t h a t < T(bhQ),b > is defined when hQ is a test function such t h a t f hQb = O. Here, we still have f hQb = 0, but hQ is not smooth. A small covering a r g u m e n t allows one to extend the definition to the case when hQ is a finite s u m of characteristic functions of cubes (the idea is t h a t in this case, the singularity of hq is not too bad). Once again, we leave the details to the reader, because we shall give a slightly m o r e complete a r g u m e n t in L e m m a 3.4 below. Note t h a t h R -- c vanishes on R0 D Q, and so we'll be able to use the kernel again :

IC~,R I< C [

IThQ(x) IIR 1-1/~dx+ [

J2 QAR~

1, f m = 1R * where Cm has integral 0 and is supported a partition of unity composed of functions f m = ~keI(m)gin,k, where each 9m,k is this with a set of indices I(m) such t h a t

FI(m) I--- c2-m("-x),

(34)

because fm is supported on {x : dist(x, OR) < 2 - m + l r } . Also, we can m a n a g e so that each gm,k satisfy

(3s)

II 0 gm,k

C (2-mr) -I l for all

a E

~l".

Now, let us estimate each < 7"gm,k,¢ > • We choose a point x0 in the s u p p o r t of gin,k, and write ¢ = ¢1 + ¢2, where ¢1 is supported in B(xo, lOr2-m), and ¢2 is zero on B(xo,br2-m). We can easily manage to have [ ¢2 [-< C, and [[ Oa¢l [[oo< C(2-mr) -[al for all a. Applying the weak boundedness of T, we get [< 7"gm,k,¢l > [ < C2 -mnrn. On the other hand, l< Tgm,k,¢2 >[_< C f f D I x - y I-'~ d y d x , w h e r e D = { ( x , y ) : I Y - Xo I < 2 - m + l r , I x - Xo I~ 5r2-m and x E 2R}. This is less t h a n c2-mnLog(m+ 1)r ~, and so I< Tgm,k, ¢ >[_< C 2-mnLog(m + 1)r ". Summing on k (using (34)) and then on m gives l< T 1 R , ¢ >1< Cr" = C I R I . This proves L e m m a 3.4 and, by the same token, L e m m a 3.3. R e m a r k . L e m m a 3.4 seems a little long to prove (although the proof does not require too much thought), but it seems t h a t one very often has to prove something like this. For instance, Y. Meyer's " c o m m u t a t i o n lemma" (see [Myl]) is proved very much like this, and its use seems unavoidable in some instances. The usual way to solve the problem is of course to leave the easy proof as an exercise. The estimates of L e m m a 3.3 will be enough to imply t h a t T is bounded. We want to apply Shur's lemma, but we cannot s u m our estimates on I CQ,R I directly, because there ~S are m a n y more small cubes t h a n large ones. So we'll sum the [< :Fh~,,ho, >[, for each Q and e, against an appropriate weight that depends on the size of Q~. Let us first take sums

42 on sets of cubes of the same size. Let Q be a given cube, with sidelength r. Call ~j(Q), j ~ 2~, the set of all dyadic cubes with sidelength equal to 2it. L e m m a 3.5.

(36)

C2 -sÈn if j > 0 j < 0 and ~ # 1 C2-i~/2 I J { 2J if j < 0 but ~ = 1.

X: (ICQ,RI+ICR,~I)-- 0 (so that [ R ]->l Q ])There are only a finite number of R's such that dist(Q,R) 2Jr from Q), note that for a given k _> 0, the number of cubes R such that dist(Q,R) ~ 2k2/r is < C2 '~k. For each of these cubes,

I CQ,R [b=
O. Suppose that, for some C >_ O, the matrix ((Ci,j)) satisfies

Lemma

(39)

~ I Cid

[ wi b 2 : < ~~..q ,..Q >, where T : Mb~TMb~. The computations are then done exactly as above.

4. E n d o f t h e p r o o f : p a r a p r o d u c t s To complete the proof of the Tb-theorem, we shall build some sort of a paraproduct. Here, too, ideas related to wavelets will help us : we shall use a simple modification of the paraproduct using wavelets defined by Y. Meyer. To simplify notations, we shall still consider the case when bl = b2 = b ; the general case would only require minor modifications. Let us first define the equivalent of the function ~ of Part I. For each Q, call

OQ(x) = ( / Q b ( t ) d t l - l lQ(x).

(41)

Note that ]10Q ][oo< C [ Q [-1, and that f #Qb -~ 1 ; if b is not "special-paraaccretive', Q runs through the modified cubes of the end of Section 2. Next, let us recall how one proves that the "wavelet coefficients" of a function of BMO satisfy a Carleson measure condition. L e m m a 4.1. If/3 E B M O , and if the C~ = in the basis (h~Q), then, for each cube R,

(4~)

f

~(x)h~Q(x)b(x)dx are the coefficients of fl

~ ~ I c~ Isb . J

45 Consequently, it foIlows from (23) that

~ QCR

I C~ 12< C II ( ~ - mRI3)IR II~ tends to f l~(x)b(x)g(x) for each g C P such t h a t f bg = 0. A similar meaning is given to Ptb. Note t h a t if T is a SIO, and Tb = fl, then < T(blR), bg > tends to f l?(x)b(x)g(x), too, for each g E D such t h a t f bg = O. This can easily be checked from the definition of Tb (Remark 1 following T h e o r e m 1.11). To prove L e m m a 4.4, we shall do the c o m p u t a t i o n formally, and leave the details of the limiting a r g u m e n t (involving the large cube R) to the reader. First,

e'b(y) = f because each

f

: o

Q

h~Q(x)b(x)dx is zero. Also,

Q

~

Q

(by definition of OQ). This is B(x) by definition of the C~'s, and (22). We also need to show t h a t P is bounded. Lemma

4.5. P is bounded on L 2, with a norm tends to 0 when g is a test function such t h a t f g(x)b(x) = O. A similar estimate is true for T *t, and so we can apply the proof of Section 3, and obtain that T t is bounded. Since P1 and/)2 are bounded, we get the boundedness of T by substraction. If the functions bl and b2 were different, one would have to define the paraproducts a little differently. For instance, the kernel of PI would be

Pl(x,y) = E

E OQhQ (z)OQ(y),

(Q,~) e l where the set of cubes Q (replacing the dyadic cubes when bx or b2 is not "special paraaccretive') is adapted both to bl and b2 [the construction of such a set is just an iteration of the end of Section 2]. The rest of the proof is the same. We finally completed the proof of Theorem 1.11.

5. C o m m e n t s o n

Tb, s p a c e s o f h o m o g e n e o u s t y p e .

Let us start with a comment on the proof. A first proof of (part of) the Tb-theorem using wavelets was given by Tchamitchian [Tc3]. However, this proof was not quite as simple. The proof given above is quite stricking, because it gives directly the "full" Tbtheorem. A first glance at the number of pages could make the reader mistakenly think that it is not so simple, after all. Let us suggest to the reader that would be tempted by such a thought to go and have a look at [DJS] ! Also, we did our best to give all the relevant details about the proof : it would be much shorter ff we had restricted ourselves to the case when bl -- b2 -- b is accretive and defined on the real line. For a compact proof, see [CJS]. Another nice feature about this proof is that it extends rather easily to spaces of homogeneous type. We do not wish to give a detailed explanation, and so we'll not even say what a space of homogeneous type is. We refer to [CW], [McS1] and [McS2] for this. The point is that the notion of a singular integral operator extends nicely to such a space (see [McSI& 2]) and there is even an extension of the Tb-theorem to the case of spaces of homogeneous type. If we look at the proof given in Sections 2, 3 and 4, we see that we only used the existence on ]R n of something like dyadic cubes. If the space of homogeneous

48 type E has a family Rk, k E 2Z, of partitions of E with the properties below, the proof given in the previous sections will extend to give a T b - t h e o r e m on L2(E) : we used the fact t h a t - if Q 6 )~k and Q ' E £k, for a k < k', then Q contains Q ' as soon as Q (1 Q ' ¢ 0 ; - all the cubes in R~ have comparable mass : Q, Q ' c ~k implies t h a t I Q I< C I Q ' I ; - each cube of Rk contains < C cubes of Rk+l ; - each cube Q E Rk has a "sufficiently small b o u n d a r y " . We used the last condition in L e m m a 3.4, to make sure t h a t 1Q would be regular enough to allow the use of the weak boundedness, and in L e m m a 3.6 to control the n u m b e r of small cubes R t h a t are close to Un OQn. A condition that would be enough, for instance, would be the existence of an a > 0 such that, if Q is one of the cube and r is its diameter, then [ { x : dist(x, OQ) < 2-kr} [< C2 -k~ [ Q t .

The existence of such cubes is not hard to establish in the special case of an Ahlforsregular subset of some ]R n, i.e. a closed subset E of ]R '~ with a measure # (generally the restriction to E of the k - d i m e n s i o n a l Hausdorff measure) such that, for some positive real n u m b e r k > 0 and some C > O, C - l r k < # ( E M B ( x , r ) ) < Cr ~ for all x E E and r > 0. A construction is given in [Dv6], but the a u t h o r is not very proud of it (it is more complicated t h a n necessary), and since it is used in a few different places in this book, a slightly simplified version will be added as an appendix. W h a t about all the other spaces of homogeneous type, then ? It turns out t h a t another construction can be given, and t h a t this construction extends to all spaces of homogeneous type (see [Ch.2]) ! Moreover, M. Christ's construction is not much more complicated. He uses it to prove a variant of the Tb-theorem, b u t with variable b (see [Ch.2], but also [Ch. 1] for a statement of the result). A final c o m m e n t on this proof : the fact t h a t we do not have to deal with complicated approximations of the identity makes it easier to extend the t h e o r e m to the case of matrixvalued kernels. You will see an example of this in P a r t III, T h e o r e m 6.7. Let us not insist more.

6. A p p l i c a t i o n s We'll make this chapter a little shorter t h a n it should he. T h e kind reader is referred to [My4] and [Ch.1] for most applications.

49

A. More wavelets We start with two results at the interface between singular integrals and wavelets. T h e o r e m 6.1. Let ¢ be a C 1 - f u n c t i O n on IR, with rapid decay at oo, and such that the 21/2¢(2ix - k), (j, k) E ~2, form an orthonormal basis of L2(IR) (for instance, take Y. Meyer's wavelet). Then the 2 i ¢ ( 2 1 x - k) form an unconditional basis of the atomic space HI(1R).

Proof. One needs to show that if f E H 1 is a finite s u m of the form cj,k[2J¢(2J

j,k

- ,)1 =

c,¢,

I

(with a self-explanatory change of notations), and if (el) is of modulus < 1, then £ = E I £ICI•I satisfies I[ f~ IIHI ~ not depend on f or (er) !]. Call T~ the operator which sends ¢ I to e~¢l. Te is n o r m < 1 because the 2J/2¢(2Jx - k) are an orthonormal

a sequence of complex numbers C II f IIHI [with a C t h a t does well defined on L 2, and has a basis of L 2. The kernel of T~ is

~-~Y,k 2-JeJ,k¢(21x -- k) ¢(21y - k), and so one checks easily t h a t Te is a singular integral operator. Finally, T~I = 0, and it follows from the R e m a r k 3 after T h e o r e m 1.6 that the Te's are uniformly b o u n d e d on H I. (Hence we proved the theorem.) We leave the estimates on the kernel as an easy exercise. Remark. We stated this theorem because it is rather easy ; the same sort of proof would have shown t h a t the ¢ i ' s are an unconditional basis for m a n y other spaces (for H I, however, the existence of such a basis is not so recent). T h e o r e m 6.1 also generalizes to H 1 of the bi-disk (the idea is to use the theory of Calder6n-Zygmund operators on p r o d u c t spaces). See [Le]. Our second result is a counterexample due to Lemari~ (a first example was found, shortly before, by P. Tchamitchian [Tcl]). The question was the following : if T is a b o u n d e d singular integral operator, and is invertible on L 2, is it true t h a t it is also invertible on L p, 1 < p < + c ~ ? The answer is no : for each p, there is an operator T like t h a t which is not invertible on L p. Note t h a t one cannot hope a better counterexample : an easy use of complex interpolation shows t h a t if T is b o u n d e d on L p for P0 _< P -< 4Po 1 for some P0 < 2, and is invertible on L 2, then it is also invertible on L p for p in a small neighborhood of 2 (which of course depends on the norms of T on the LP's). This was observed by Calder6n [Ca2]. Here is the counterexample. Consider Y. Meyer's basis of L2(]R) (for instance) 2i/2¢(2Jx - k), and index it by dyadic intervals I = [k2 - j , (k + 1)2-J], so as to get a basis ( ¢ I ) i dyadic. If I is a dyadic interval, call ¢ ( I ) = [0,2 i+1] if I = [0,2J] and ~ ( I ) = 0 otherwise. Now let To be the only operator such t h a t To¢l = 0 if ~ ( I ) = 0, and To~bl = ¢¢(x) otherwise. Clearly, To is a b o u n d e d operator on L 2, with n o r m 1. Also, the kernel of To is

50 ~]j. ~b[0,2j+,](x)d]10,2j](y) ; i t satisfies (2) and (3) by the usual computation, and so To is a SIO. Thus, if 0 < r < 1, the operator T --- 1 - rT0 is a SIO, and is invertible on L~(lR). The inverse is T -1 = ~ k > o rkTok, and in particular T-1(¢I°,1} ) = Z rkOek([o,']) = Z rk2-~/2¢(2-~x)" k>o k>o We leave the fact that, given a p < 2, one can find r ,< 1 such that this function is not in L p as an exercise (one can do it directly, or use the characterization of L p by wavelet coefficients given in Part I, § 8, Example 2). If one wishes a counterexample for p > 2, one can use the transposed operator (which corresponds to taking ¢([0, 2k]) :: [0, 2~-1]). In spite of this example, it is possible to prove positive results concerning the inversion of singular integral operators. See [Tc4], for instance. B. T h e C a u c h y i n t e g r a l a n d r e l a t e d o p e r a t o r s . In most of the following examples, the operators will be principal value operators defined by an antisymmetric kernel (Example 1.3). For these operators, we said in Section 1 that the weak boundedness property is always satisfied. Since T t l = - T 1 , one will only have to check that T1 C B M O to apply the T l - t h e o r e m (a similar remark can be formulated concerning the Tb-theorem). E x a m p l e 6.2. (Calder6n's commutators). Let A be a Lipschitz function on the real line (i.e., a function such that I A(x) - A(y) I~ C I x - y I). Set g n ( x , y ) = (A(~)-A(Y))" (~_~).+l , and let Tn be the principal valeur operator defined by Kn. E x e r c i s e : prove that Tn is bounded on L 2 ( ~ n ) , with a norm < C n+l II A' I]~o for some C_>0. Hint : do this by induction. After checking that K satisfies (2) and (3) with Co _~

C(n + 1) I] A' lifo, show that T,,(1) --- T~-I(A') and conclude. C o m m e n t s . The first proof of this estimate dates from Calder6n [Call, in 1977. In their famous paper [CMM], Coifman, McIntosh and Meyer proved a polynomial estimate, namely II Tn ]1~_ C(1 ÷ n 4) I] A' li~, ; this estimate was then improved by Mural [Mul], by a completely different technique. The best estimate, up to now, is due to Christ and Journ6 : for each e > 0, (44)

II Tn II_ ~ C¢(I+ II A' Hoo)T M

(see [ChJ]).

E x a m p l e 6.3. (The Cauchy integral on a Lipschitz graph).

51 Let A

:

IR -* IR be Lipschitz.

T h e antisymmetric s t a n d a r d kernel K(x,y) =

[x + iA(x) - y - iA(y)] - l defines a principal value o p e r a t o r CA. Exercise • Write

I [I + i A ( ~ - A(Y) ] - I K(x,Y) -- x _ y -- y

n

~

--

'

n

'

deduce Calderdn's result from the estimate of E x a m p l e 6.2 : CA is b o u n d e d as soon as II A' IIoo is small enough. • Prove Coifman, Mclntosh and Meyer's result directly : CA is b o u n d e d for all Lipschitz A's ( H i n t : c o m p u t e CA(1 + iA')). Comments. T h e first proof of the boundedness of CA when II A' Iloo is small did not use c o m m u t a t o r s . Calderdn [Call showed t h a t the II CtA II, where A is given and 0 _< t < 1, satisfy a differential relation which allowed h i m to estimate CA ; the estimates on c o m m u t a t o r s are a consequence of the boundedness of CA. Since the first proof of the boundedness of CA in the general case [CMM], it has become fashionable to give new proofs of the boundedness of Ca. T h e reader is invited to produce one or two of his own. For a nice collection, see T. Mural's book [Mu2] ; for the "shortest" proof, see [CJS] (it is also in [Mu2]). T h e best estimate on the n o r m of CA is (45)

II CA II a0 -4 II A' II~ 2 . The idea is to approximate G a r n e t t ' s example (see Part III) by a lipschitz graph [Dv3]. Note t h a t it is not a bad idea to have a very good estimate for CA, because m a n y other operators can be constructed from CA. See E x a m p l e 6.5 below, for instance. E x a m p l e 6.4. (Chord-arc curves). A rectifiable, connected curve r C C is called chordarc if, for some constant C > 0 and any two points A , B E r , the length of the piece of r connecting A to B is less than C times I B - A I • (This is for an open curve r ; a small modification, left to the reader, has to be done for closed curves.) Let I" be a chord-arc curve, and denote by ds the arc-length measure on r . One can try to define a Cauchy integral on r by Crf(x) = p . v . f r z@~f(w)ds(w) for f E L~(F,ds}. E x e r c i s e . Show t h a t Cr is a bounded o p e r a t o r on L 2 ( r , d s ) . (Do not pay too much attention to the definition of the principal value.) Hint : call t ~ z(t) a p a r a m e t r i z a t i o n

52 of I" by arclength. The problem is equivalent to showing t h a t K(x,V) = [z(x) - z ( v ) ] - 1 defines a b o u n d e d operator T on L 2 (JR), and this is easily done because Tz' = 0 and z' is "special p a r a a c c r e t i v e ' . Example

6.5. Let A : IR -~ IR N be a Lipschitz function, and let F be a C °o function

from ]R N to C. T h e k e r n e l

1--L--F x--y \(A(z):A(y)] ~ u /

defines a b o u n d e d singular i n t e g r a l o p e r a t o r

on L 2 (JR). One can find a first proof of this fact in [CDM]. The idea is to write F as a Fourier series, and then to s t u d y the case of F(z) = ei(~1¢~+'''+xN~N). One is trivialy reduced to the case of one dimension, and then one writes the exponential e i~ in terms of an integral of objects like (1 + ix)-I (using the C auchy formula). One gets an estimate on the operator with kernel 1-2-exn z-y -

(iB(~)-B(u)] \ z-y ]

which grows polynomially in ]] B ' ][oo, and this estimate

is enough to sum the operators coming from the Fourier series of F, provided t h a t the Fourier coefficients have enough decay. Of course, the condition " F E Coo " is not needed ; the better one can estimate the n o r m of CA, the less derivatives will be needed on F. So it is advisable to use Mural's result (45). For a reasonably sharp estimate, see [DS1]. Using this example, and the "rotation method" to get operators on L 2 (IRn), one can get quite a collection of Calder6n-Zygmund operators. E x a m p l e 6.6 ("Stable kernels"). Let K(x,y) be a s t a n d a r d kernel on the line ; we'll say t h a t K is a Calder6n-Zygmund kernel if there is a b o u n d e d s i n such t h a t K is its kernel. We'll say that K is "stable" if, for all Lipschitz functions A : ]R --* IR, the kernel K(x,y) A(z)-A(u) is a Calder6n-Zygmund kernel (so, taking A(x) x, K is Calder6nX--y Zygmund). It was first observed by Journ6 that if K is an antisymetric Calder6n-Zygmund kernel, then it is stable. E x e r c i s e : use the T l - t h e o r e m to prove it. One can easily characterize the Calder6n-Zygmund kernels t h a t are stable. First, the notion only depends on the kernel, and can be studied on truncated kernels K(x, y)l{e

=

r,:s(,,)do,

where

To:f (u) = fit I>, K ( u ' u + tO)f(u + tO) l t Id-1 dt can be controlled by the 1-dimensional result. The reader who is not too familiar with the m e t h o d of rotations will find descriptions in m a n y sources, and a proof of the exercise can be found in [Dv5], p. 245. R e m a r k . W h e n k(x) is odd and homogeneous of degree - d , most of the exercise can be avoided : one can prove the result by a direct application of Example 6.5 and the m e t h o d of rotations.

PART III SINGULAR

INTEGRALS

ON CURVES AND SURFACES

--oo0oo--

1. I n t r o d u c t i o n a n d n o t a t i o n s We would like to extend Coifman, McIntosh and Meyer's nice result to as many higher-dimensional objects as possible. In the following, we'll be given 0 < k < n, and a k-dimensional object $ c ]R n. We'll always call $ a "surface", but we do not assume any smoothness on $ : $ is a set. Also, S will come with a non negative Radon measure on lR'~, such that supp ~u -- $. For instance, we could take for S a smooth k-dimensional surface, with for ~ the k-dimensional surface measure on $. The singular integrals we wish to consider are generalizations of the Cauchy integral (defined on curves of the complex planes). D e f i n i t i o n 1.1. The function K(z), defined on IR"\{O}, will be called a "good kernel" if g ( - z ) = -K(z) for all z # 0, g is Coo and (1)

I V J K ( x ) I< C ( j )

[xl -k-j

for all j_>O.

R e m a r k 1.2. The choice of the class of "good kernels" is not extremely important: we only have to take it small enough, so that defining singular integrals on lipschitz graphs will not be a problem. We could also have taken the slightly larger class where the antisymmetry condition is replaced by (2)

sup

0<e<M

[

g(to) [tl k - l d t O.

56 We'll see soon t h a t /z E A is necessary if we want all g o o d kernels to define b o u n d e d operators on L2(lRn,dtz). Let us first define a truncated and a maximal operator. For e > 0 and f E Cc(IR'~), let

(4)

= fix

K(x - y)f(y)dl (y)

and

(5)

sup

I

e>O

If the maximal operator T~ is b o u n d e d on L2(IR n, d#), the T~'s are uniformly bounded, and we'll be able, if we want, to define an operator T by extracting a weakly convergent sequence. On the other hand, studying T~ will not be harder t h a n just showing the boundedness of some limit of T~'s ; in fact, we'll prove later that, with a small additional hypothesis on #, the L 2 - b o u n d e d n e s s of such a limit implies the boundedness of T*/ z " Our main question, in its general form, is the following : for which measures # is T~ b o u n d e d on L2(]Rn,d#) ? W h e n k = 1 and n = 2, we could consider the special case of K ( z ) = ~, ' and ask for which measures # the Cauchy kernel defines a b o u n d e d operator on L2(dtz). By the way, we are just talking about L 2 for convenience, but L p, 1 < p < + c o , would do as well (and give the same measures #), assuming/z is in the class ~ of Definition 2.3 below. We'll see on the way t h a t the question is not so simple, even when k -- 1 and n = 2. Let us at least justify our decision to restrict to measures # E A. P r o p o s i t i o n 1.4. Let Ki, i E I, be a finite family of good kernels such that, for some 6 > 0 and all x ~ O, [ Ki(x) [> i5 [ x [-k for some i E I. I f # is a non-atomic measure such that the T~'s corresponding to the various K i ' s are uniformy bounded (in i and e), then /zEA. A minor variant of this result is proved in [Se3] allegedly using a well-known method; let us give here a slightly different proof. Let us suppose t h a t the 7~'s are uniformly bounded, and t h a t / , ~/A, and let us find an atom. L e m m a 1.5. There is a constant Co such that if Qo c N n is a cube, r0 /s its sidelength, mo ~- #(Qo) and Ao moro k, then one can find a cube Q1 c Qo, of radius rl = ro/lO0, =

and such that m I = b t ( Q 1 ) > (1 - ~ - ) \

rn 0 .

-'o/

Of course, this will be useful only when A0 is large enough ! Let us prove the lemma by contradiction. Split Q0 into C~ cubes of size Ci-~r0 ; one of the cubes (call it R) is such t h a t # ( R ) > C T n m o . Also, if the lemma is not true for Q0, one can find another cube S C Q0, of size C ~ ' r o , and at distance >_ ~

from R, with a measure/z(S) _> [ ~ m 0 ] t"o

J

C 7 ~.

57 Take the test-function f = l s , and look at the T~f(x), for e < < r, on the cube R. If C1 is large enough, then there is an i E I such t h a t the corresponding T~f satisfy o-k on R [this is because, the Ki(x, y) being good kernels, each of t h e m [ T~f(x) 1>_ U/z(S)r ~ is almost constant when y varies in S]. So

II TLt

uCR)I/ uCS)ro

>_

ClZCR)l/2 (S)l/2ro k II .," llz, ~(R)'/zU(S)X/Zro k _ _(1 - c._¢_~? .) mi > 1-m'2, if A0 is large enough. T h e n ~i+1 = (100)km~+~ ~ i r n , > 10AI, as promised. We get a b e t t e r e s t i m a t e for rni t h a n what was just written :

ml > (1 - A I C ° I ) ( 1 - A ~ )

too> -

1-~(100)

.-.

1-

too:>--

2

m

if we choose A02 > 10Co, for instance. T h e n Ni Qi, which is reduced to one point, has a m e a s u r e >_ ~0_ > 0, and the proposition is true. R e m a r k 1.6. If # E A and K is a good kernel, one can define T~f(x) everywhere for any f e L2(d/z) (use Schwarz' inequality, and the estimate

-ut>,

Ix - y I

du(y)

=

oo

I x - y 1-2k d#(y) < ~ C(2Q)-2k(2Q) k < +oo). £=0

E x a m p l e 1.7. Let z : ]Rk --* ]R n be a continuous function. We think of z as being the p a r a m e t r i z a t i o n of some surface. We can define a measure # on ]R n by # ( f ) -- f ~ . f d# -f~tk f(z(x))dx (for f e ¢c(Nn)). We'll often call # the " m e a s u r e associated with z". Saying t h a t / z E A amounts to saying t h a t

58 for all w E JR" and all r > 0.

2. C a l d e r 6 n - Z y g m u n d

techniques

We shall need extensions to our context, of some of the classical results of "Calder6nZ y g m u n d theory". By lack of time, we'll only recall the m a i n steps of the proof ; if the reader has problems filling the gaps, he will find all the details in [Dvl], [Dv2], or [Dv5]. One of the useful tricks t h a t will be used here is distinguishing between the measure # t h a t comes into the definition of T~, and the measure (call it a) on which we want to integrate T~ (x). This is a little like studying a bilinear form instead of a quadratic form. The advantage is t h a t we'll be able to change # (or o) without affecting the other measure. We need an easy extension of the usual maximal theorem concerning the HardyLittlewood maximal function. D e f i n i t i o n 2.1. If # is a nonnegative R a d o n measure and f is measurable, define a maximal function by f

M~.f(z) =

sup r -k ] r > 0 Slz-wl O, a constant ff > 0 such that, for all ~ > O, (11)

kt({x e X : u ( x ) > )~ + eA and v(x) A}).

Then u C LP(X,d,)

and [] u

llp 0, and call fl = {x • IR n : u(x) > ).} ; fl is open because T~ is lower semicontinuous. Cover I2 n s u p p # by the balls B(x) = B (x, ½dist(x, fie)) , where x • s u p p # . A covering l e m m a of Vitali type (see [St], p. 9-10) gives a sequence of points xi such t h a t the B(xi) are disjoint, but fl n supptz C U i ~ 1 1 0 B ( x i ) . If we prove t h a t , for each i, (18)

#({x • B(xi)

0

: u(x) < )t + eA or v(x) > "~A}) _> ~#(B(xl)),

then (11) will follow because s u m m i n g on i gives 0 # ( ( x • fl : u(x) < )~ + e,k or vCx) > qA}) > ~ E # ( B C x , ) ) i (because B(xi) is centered on s u p p # and tt • E ) Thus, we only need to find, for each e > 0, a i. If v,(x) > qA for all x • B(xi), there is nothing is a ~ • Y(xl) such t h a t v(~) < -/A. Write f = f l + f2, with fx = 1B(~,Rlf, and constant C1. For f2, one uses the regularity of T~f2(:c) < T~f(a) + CM~f(~) for all x • S(xi) as an exercise.) Let us choose a point a • 3B(xi)

(19)

0

> -~ E t t C 1 0 B C x / ) ) d

> -~tt(fl). This is (11), with r / = -~. ff > 0 such t h a t (18) is satisfied for each to prove. So we can suppose that there where R = Cldist(~,12 c) for some large K to show t h a t , if C1 is large enough, and a • 3 S ( x i ) . (The verification is left n fl c, so t h a t we get

eA T ; A ( x ) _< A + ~ - for all x ~ B ( x , ) ,

62 provided we choose ? small enough. Therefore, we only need to prove t h a t

(20)

•

:




Let us use the c o m p a c t set E and the measure a of the hypothesis. Since/~(E) > 8#(B(xi)), . e)~ we only have to show t h a t #({x E E : T~fl(x) > ~-}) < ~#(B(xl)). By (14), the left hand side is less t h a n a({x • E : T~fl(x) > ~ } ) < 2re-rA -~ [[ T~f, [[~'(d~) • By (16), 7'* is b o u n d e d on LP(da) for 1 < p < +c~. By L e m m a 2.5, T~ sends r < Lr(dtz) to n r ( d a ) , and so /~({x • E : T~fl(x) > ~-}) _< Ce-r)~ -r ]l fl ][Lr(dp)-Ce-r)~-'Rk{M~,([ f ]r)}(~) < C,-r)~-~#(B(x,))q~)V < ~#(B(x~)) if we choose 7 small enough. This proves (20), and so (11) and the proposition follow. Let us say how we intend to use Proposition 3.2. Let $ C IRa be given with a measure # • ~ such t h a t $ = supp #. D e f i n i t i o n 3.4. We shall say t h a t S "contains big pieces of Lipschitz graphs" (and often write $ C B P L G ) if there are constants 0 < 8 < 1 and M > 0 such t h a t , for each x E S and r > 0, there is a compact set E C S N B ( x , r ) , with # ( E ) > 8r k, and which is contained in the image by a linear isometry of IRa of the g r a p h of some M-lipschitz function A : IR k ~ IRn-k. R e m a r k 3.5. T h e notion is really a notion concerning S, and not #, because if S C IRa is the s u p p o r t of some measure # E ~--~.,then/~ is equivalent to the k-dimensional Hausdorff measure on S (the verification is a s t a n d a r d covering argument, which is left as an exercise). Of course, the p r o p e r t y of Definition 3.4 does not change when/~ is replaced by any measure a such t h a t C - 1 # < a < C/z ! C o r o l l a r y 3.6. If S CBPLG, and I~ 6 ~ is such that supp lz = $, then T~ is bounded on LP(S,dlz) for 1 < p < +c~. To prove the corollary, one applies Proposition 3.2, with for a the k-dimensional surface measure on the image by an isometry of the Lipschitz graph of the definition. T h e estimate (15) then comes from L e m m a 2.6, and (14) follows because b o t h 1E/z and l e a are equivalent to the k-dimensional Hausdorff measure on E (see the r e m a r k above, or the a r g u m e n t in [Dv5], p. 252). E x e r c i s e . Show that i f z : IR k --* IRa is a bilipschitz m a p p i n g (i.e. satisfies C -1 I u - v l< I z(u) - z ( v ) t< C l u - v I for some C > 0), then S = z(iR k) C B P L G (and so Corollary 3.6 applies to S). [Warning : this is more challenging t h a n most of the previous exercises. If you don't find, just go on reading !]. This is not the most striking application of Corollary 3.6, because the boundedness of the singular integral operators on S can be proved directly, using the p a r a m e t r i z a t i o n of S and the rotation m e t h o d , as for L e m m a 2.6 or E x a m p l e 8.5 of Part II.

63 R e m a r k 3.7. One can imagine a more complicated corollary. Let us say t h a t " $ (CBP)2LG" if $ "contains big pieces of surfaces that C B P L G uniformly" (with a definition similar to Definition 3.4). A second use of Proposition 3.2 gives, exactly like in the proof of Corollary 3.6, that T~ is bounded on all LP($). One can of course go further, and prove the boundedness of T~ whenever $ (CBP)'~LG for some m (with obvious definitions). We shall come back to this in later sections. We conclude this section by a remark on the existence of principal values. Up to now, we tried to avoid talking about "the operator defined by the kernel K ' , because the integral f g ( x - y ) f ( y ) d ~ ( y ) does not converge in general. It turns out that, once we know that T~ is bounded, very little is needed to conclude that T~ converges to an operator T~ as e tends to 0. P r o p o s i t i o n 8.8. Suppose k is an integer, K is a good kernel and t~ E ~ is such that T~ is bounded on L2(dl~). Also suppose that the support S of# is a rectifiable set. Then, for all f C L 2 ( d , ) , T~,f(x) = lim T~f(x) exists for t~-almost every x. ¢--+0

t,.

There are various (non-trivially) equivalent definitions of a rectifiable set, and we'll choose the one that makes the proposition easy. We'll say that S is rectifiable if it is contained in the union of a set of measure 0 and a countable number of (rotated) Lipschitz graphs. For more information about rectifiable sets, see [Fe], [Mt] or [Fa]. E x e r c i s e . Show that if $ CBPLG, or even if S ( C B p ) m L G (see Remark 3.7), then $ is rectifiable. We stated an L2-result here, but the corresponding LP-result also holds, even up to p = 1 included. The proof of Proposition 3.8 (or of its LP-variant) only requires fairly standard techniques, so we'll leave it as an exercise, with the following instructions. 1. First reduce to the ease when # is the restriction to S of the k-dimensional Hausdorff measure (remember Remark 3.5). 2. Next, note that it is enough to prove that lim T,~f(x) a. e. for a dense class of ~---*0

r-

functions f. Indeed, given f E L2(d#), it is enough to show that

=1 lim ~:~Ct ---,*0

-

T~" f(2:) I

has an L~-norm which is equal to 0. This is shown by writing O f _ Og + 2T~.(f - g), and by choosing g in the dense class and close enough to f. This argument is a fairly standard way to use maximal estimates (see [St], p. 8 or 45, for instance). 3. Now suppose the result was established when S is a Lipschitz graph. Take for your dense class the linear combinations of functions supported on the intersection of $ with Lipschitz graphs (here we use the rectifiability of S). Since elira T~f(x) always exists when ---*0 rx ~ supp f , the general case will follow.

64 4. Prove the proposition when k -- 1 and S is a straight line. For this case, compactly supported s m o o t h functions are a nice enough dense class. 5. Show that, when k is odd, the operator of Example 6.7 of P a r t II is a principal value operator in the classical sense : for f E n 2, lim fl~ vl>~ g ( u , v ) f ( v ) d v exists for ~:---* 0

a . e . u . To do this, first note t h a t each time the boundedness of an o p e r a t o r is established, the corresponding maximal operator is also bounded, because of Cotlar's inequality (or (III.10)). Therefore, one only has to follow the proof and apply the d o m i n a t e d convergence theorem as often as needed. 6. Conclude. You m a y have to use the fact t h a t the g r a p h of a Lipschitz function has tangent planes almost everywhere to go from the operator of Example II.6.7 to the operator of L e m m a 2.6.

4. R e g u l a r c u r v e s a n d L i p s c h i t z g r a p h s Let us pause in a m o m e n t to see how the " g o o d A m e t h o d " works when k = 1. Let us first consider the case of a connected, rectifiable curve F ; for simplicity, we shall restrict to the case when F has infinite length. A - Regular curves. D e f i n i t i o n 4.1. A (connected, rectifiable) curve F is regular if there is a constant C > 0 such that, for all x E ]R '~ and r > 0, the total length of F M B(x, r) is less t h a n Cr. The notion is due to Ahlfors [Ah]. If # is the arclength measure on F, we see t h a t F is regular if and only if # E A. Because F is connected,/~ also satisfies (8) automatically, and so # E ~ . Example.

Chord-arc curves are regular ; a parabola in ]R 2 is regular (but not chord-arc).

T h e o r e m 4.2. Let F C ~ n be a regular curve, and ~ the arclength measure on F. Then T~ (as defined by (4) and (5)) is bounded on LP(F,d/~) for 1 < p < +oo, when K is any "good kernel" (see Definition 1.1, with k -- 1). Note t h a t Proposition 1.4 says t h a t the condition is also necessary. To prove the theorem, we shall appeal to Corollary 3.6, and show t h a t F C B P L G . Let x E F and r > 0. Call z(s), 0 < s < so a parametrization by arclength of the piece of r between x and the first time r gets out of B(x, r). Thus, z(0) = x and [ Z(So) - x I= r. Since F is regular, we get so _~ C0r for some Co > 0. We shall find our big piece of Lipsehitz graph in z([0, so]). After a possible change of coordinates, we can assume t h a t x = 0 and z(so) = (r, 0 , . . . 0 ) . Let 6 = 2 CIo "

65 Lemma

4.3. There are a s e r E C [0, so] and a Lipschitz function h : [0, so] -~ ~t, such

that h(s) = zl(s) on E (zl(s) is the/irst coordinate of z(s)),

(21)

IE

I>

r

and (22)

g < h' if[E I> -~. To prove L e m m a 4.3, let us try the function h(s) =

sup [zl(t) - ~ft] + ~s. O f(bk) (with equality, except p e r h a p s on the last

66 component). Then f ( s o ) - f(O) = r - 6so >_ r/2, and so ~ < f o ° f ' ( t ) d t = f E f ' ( t ) d t + ~k

f~: f ' ( t ) d t

A(0). Pick A(x) = sup [A(t) + 4 t ] - gx4 (see Figure 2) ; then O_cn 1/2, therefore showing that Murai's estimate (inequality (45) of Part II) is sharp (again see [Dv3] for more details). R e m a r k 5.2. The fact that the Cauchy integral is not bounded on LZ(d/z) shows that the connectedness assumption of Definition 4.1 and Theorem 4.2 was not superfluous. If r n is the boundary of kn x kn, and #n is the arclength measure on r,~, we see that #n E uniformly in n ; however, since #n ~ / ~ , the T~.'s are not uniformly bounded ! One can go a little further, and build a counterexamaple in higher dimensions. For instance, let S~ be the 2-dimensional surface of ]R3 defined by Sn = kn x k,t x {1} u

69 Fn x [0, 1] U

[lR2\(kn x

k,)] x {0} (see the picture).

A sketch of the surface S2 Note that the surface measure on S,~ is a measure/zn which is in ~ , uniformly in n, but it is not hard to find a kernel K such that the corresponding T~. 's are not uniformly bounded on L2(/zn). Thus, in higher dimensions, something stronger than conneetedness is required (see Sections 9 and 10 for further comments).

6. Three classes of surfaces We wish to describe, rather rapidly, three classes of K-dimensional objects in ~ n where something positive can be said about the operators T~'s. A - Chord-arc

surfaces

with

small constant

This first example was introduced by S. Semmes ([Se3] and [Se4]). For this example, k = n - 1, and also one is interested in more refined properties than the boundedness of the T~'s. We shall try to say, as fast as possible, what the point is (and leave all the definitions, precise statements, and proofs). Let $ be a hypersurface in IRk+l. We shall make the a priori assumption that S is smooth (including at c¢), oriented, and separates IR k+l into two connected components (call them 12+ and f l - ) . Let d/z denote the surface measure on S. One can define on S the analogue of the Cauchy operator. This time, the kernel takes values in the Clifford algebra (instead of C when k = 1). Let us call ¢s the resulting operator (see Semmes' papers for definitions).

70 One of the nice things about the Clifford algebra setting is that there is a notion (we'll call it "Clifford-analyticity') that generalizes analyticity. One can then define Hardy spaces H2(12 +) and (H2(fl - ) of traces on S of Clifford-analytic functions in fl + and f l that have boundary values in L2(S,d#), and one can show that L2($,d~) is the direct sum of H2(fl +) and H2(fl - ) if and only if Cs is bounded on L2(S,d#). The kernel of Cs is K ( x - y ) , where K is Clifford valued, C °°, odd, and homogeneous of degree - k , and so Cs is one of the operators of the previous sections. S. Semmes asks a more precise question : when is L2($, d~) the "almost-orthogonal" sum of H2(n+) and H2(f] +) ? By "almost orthogonal', we mean that

(27)

[< f , g >[___e IIf IIIIg II

for all f G H2(fl +) and g E H 2 ( n - ) , and where e is small enough. In terms of operators, this means that a slightly different version Cs of the CliffordCauchy operator is "almost antiselfadjoint" (i.e. I[ Cs + C; [[< e' for an e' ~ e). T h e o r e m 6.1. ([Se3], [Se4] and [Se7]). Let 5 be as above. If (27) is true with a small enough e, then there is an Q ~ Ce such that (28) the unit normal n(y) to $, pointing towards fl +, is in B M O ( S, d#), with norm ~_ el. Conversely, if (28) is true for a small enough el, then there is an E ~_ Cel such that (27) is true. The definition of B M O ( $ ) is not a surprise : we ask for (29)

f

.(BCx, r))-' !

JB (~,r)

where nx,r :

IN(y) -

fBc ,r nCy)d Cy), • e S and

I d.Cy) _< El, > 0

R e m a r k 6.2. When k -- 1, the result says that the Cauchy operator on a rectifiable Jordan curve F is almost anti-self adjoint (or the Hardy spaces are almost orthogonal) if and only if F is a "chord-arc curve with small constant" (with the definition of II, 8.4, we should say "constant close to 1"). This was essentially proved by Coifman and Meyer [CM2] (also see [Dv01 for the "only if" part). For this reason, S. Semmes decided to call the surfaces that satisfy (27) (or (28)) "chord-arc surfaces with small constant" (we'll write CASSC's).

R e m a r k 6.3. There are many other equivalent geometric definitions of CASSC's. For instance, (27) implies (30) for all x C S and r > 0, there is a hyperplane H such that each y ~ $ N B(x, r) is at distance < e2r from H ;

71 (31) for all x E $ and r > 0, I P(S M B ( x , r ) ) - akr k I_< ~3r k, where akr k is the volume of a ball of radius r in IRk . (32) if D(x, y) denotes the geodesic distance between x and y E S, then I x - y (1 q - E 4 ) I ~ - y l ;

I< - D(x, y)
0, there is a surface S, which is the image of an es-Lipschitz graph by an isometry of IRk+l, and such t h a t # ( S N B ( x , r ) N go) < esrk. All these properties are various ways of saying that S looks very much like a hyperplane at all scales. N6te t h a t (33) makes it easy to show t h a t Cs is b o u n d e d (once (33) is deduced from (28)!). Of course, they are all true for chord-arc curves. One thing we are missing here, however is a very good p a r a m e t r i z a t i o n (we'll talk a b o u t this in Section 10). Finally, (31) and (32) (with e3 -b e4 small enough) imply (27), too. B - w-regular surfaces Let us mention now a generalization of T h e o r e m 4.2 above. T h e main difficulty of studying surfaces is avoided by assuming we have a very nice p a r a m e t r i z a t i o n of S. D e f i n i t i o n 6.4. Let w C Aoo(IR k) be a weight of the M u c k e n h o u p t class. T h e function z : IRk __~ IRn will be called "w-regular" if there is a constant C _> 0 such t h a t

I z'Cx)I~ cw(x) ~/~

(34) and

(35)

I {~ e IRk : z(~) ~ B(w, r)} I~d~< c r k

for all w E IRn and r > 0. C o m m e n t s . T h e condition (34) is m e a n t "distributionwise'. Since w is a weight in A~o, it is equivalent to

(36)

I zCx) - z(y) I< c

wC~)d~

,

where B = B(x, [ x - y [), for instance. The reader will find all needed knowledge on Aoo-weights in [Jg] or [GR]. He can also content himself with the case when w = 1 (in this case, z is Lipsehitz). If we define a measure # on IRn by p ( f ) = f ~ f ( z ( x ) ) w ( x ) d x , condition (35) means t h a t tz e A. Furthermore, (34) (or (36)) implies t h a t p E ~ . T h e o r e m 6.5 [Dv5]. Let z : IR k ~ IR" be w-regular, and define # by fR~ f d # = fRk f o Z W dx. Then the operators T~ defined in Section 1 are bounded on nP(d#) for l to denote the effect on the effect on g (by an action on the left) of the image of f by T (acting on the right). In other words, if K were integrable, we would have < fT, g > = f f f(y)b(y)g(x - y)b(x)g(x)dx dy.

76 We now use the Ak's to write f = ~

aQfq,

and the A~¢'s to write g = ~

13Qgq, so

that < fT, 9

>= E Z c~Q/~Q,< fQT, gQ, Q

>.

Q'

The coefficients CqQ, = < fQT, gQ, > can be estimated like in P a r t II, Section 3, using the s t a n d a r d estimates on K, the properties of dyadic cubes, and the facts t h a t T1 = T t l = 0 and f fQb = f bgQ = 0. The result is t h a t the matrix with coefficients H CQQ, II defines a b o u n d e d o p e r a t o r on £2, and t h a t

I1< f T , g >][< (7

I~Q I~

I / ~ 12

_< c II f I111g II.

This proves the boundedness of Cs. As we said earlier, the proof of S. Semmes also works for other kernels t h a n the Cauchy-Clifford kernel K. The idea is to use the T b - t h e o r e m again, and to c o m p u t e the image of the unit normal by an integration by parts. If the result is the image of a b o u n d e d function by an operator we already know how to treat, we are in business. The details can be found in [Se5]. E x e r c i s e . Complete the proof of the T b - t h e o r e m in the case of a function b with values in IR k+l C Ck(lR). Use a p a r a p r o d u c t with a kernel of the form

P(x, y) = E c~QfQ(x)Oq(y) Q (where the f Q ' s are obtained by decomposing a function of B M O with the A~'s), or of the form

P(x, y) = E aQOQ(x)fq(y) q (where the

fQ's

are obtained with the Ak'S).

7. F i n d i n g b i g p i e c e s : c l o u d s a n d s h a d o w s Let us come back to the good-~ method. We are given a "surface" $ C ]R n, and a measure # C Y]~ s u p p o r t e d on $, and we wish to apply Corollary 3.6, so we wish to show that $ C B P L G . A first step in t h a t direction is to show t h a t $ "has big projections" in the following sense : there is a 0 > 0 such t h a t for each x E $ and r > 0, there is a k-dimensional vector space V such that, if ~r is the orthogonal projection on V, then (39)

I r(S f1B(x,r))]>

Ork

77 (where I ] denotes the Lebesgue measure on V). It is clear that, if S CBPLG, then S "has big projections" : if E C $ n B ( x , r ) is contained in the graph of an M-Lipschitz function from V to V ±, then ] ~r(E) ]> C-I(M + 1)-lg(E). In most known cases, showing that S has big projections is the easy part. For instance, if S is in S(k) (see Definition 6.6), we can take for V the hyperplane orthogonal to the line L that contains the points xl and x2 of Definition 6.6. Indeed, if L' is any line that meets B ( x l , C - l r ) and B ( x 2 , C - l r ) , then n' N $ N B(x,2r) is not empty. From this, we deduce that r ( S N B(x,2r)) contains a ball of radius C - l r centered at ~r(xl), which gives the "big projection" for 2r (see the picture).

i

Let us assume now that we proved that S "has big projections". We still have to find a piece of Lipschitz graph in S A B(x, r) or, equivalently, a subset E c S A B(x, r) such that ~rl~ is bilipschitz (the "Clouds and shadows problem"). A condition is given in [Dv6], that allows in certain cases to solve the "Clouds and shadows problem". The condition is not very pleasant, and the proof by a stopping time argument is not too simple, so we'll just mention a corollary : T h e o r e m 7.1. If $ = z(iR k) for some w-regular function z : IRk -+ IR'~, or if $ E S(k), then $ CBPLG (see Definitions 6.4, 6.6 and 3.4 for the jargon). This theorem gives a new proof of Theorems 6.5 and 6.7. One can also define higher codimension analogues of S(k) for which Theorem 7.1 also holds (see [Dv6]). We shall see in Section 8 a faster proof when $ = z(iR k) for an w-regular z. In the case of S(k), there is another proof, too ([DJe]), which also gives a little more : T h e o r e m 7.2. Let $ ~ S(k). Assume that 0 E S, and that the bail B with center ( 0 , . . - , 0 , 1 ) and radius C -1 is entirely contained in one of the connected components of IRk+I\s (call this component 0). Then there is a constant C1 > O, that depends only on C and the constants of Definition 6.6, and a C1-Lipschitz function A, defined on B(0, (2C)-1) C IRk, and such that the graph F of A satisfies :

(40)

r c 0

78

and (41)

u(r

n

s) > c i -1

(see the picture).

oC The proof of T h e o r e m 7.2 is reasonably simple, but we shall not have enough time to give it. T h e new piece of information is t h a t F is "above" $, and this allows one to obtain the following harmonic measure estimates. C o r o l l a r y 7.3. Let $ E S(k), and 0 be one of the components of IRk+lk$. If 0 is a Non-Tangential Access domain, then the harmonic measure on O0 (relative to O) is in the Muckenhoupt class Aoo with respect to surface measure on $. We refer to [DJe] or [JK] for the definition of NTA domains. Let us mention t h a t S. Semmes has another proof of Corollary 7.3, obtained independently by a m e t h o d related to the "corona construction" [Seg]. This m e t h o d had been used by P. Jones and S. Semmes to treat the case of the image of a bilipschitz m a p p i n g from IR k into IRk+l. Semmes obtained the special case of Corollary 7.3 when b o t h sides of S are N T A at the same time as [DJe], and the general case a little later. The condition $ E S(k) is not necessary for the proofs of T h e o r e m 7.2 and Corollary 7.3 ; in particular, the conclusion of the corollary is still true if the k-dimensional Hausdorff measure on S is in ~ , if for each x E S and r > 0, there are two k-dimensional disks D1 and D2, with radius ~,r at distance _< r from x, and t h a t are contained in different components of I R k + l \ $ (of course, the NTA condition on Harnack chains is still needed, too). The idea of the proof is to compare harmonic measure on 0 with harmonic measure on a Lipschitz domain fl C 0 such t h a t I 0fl N O0 I is relatively large. One uses the t h e o r e m to find fl, and techniques of [JK] to conclude. More details can be found in [DJe].

79 8. B i l i p s c h i t z m a p p i n g s inside L i p s c h i t z f u n c t i o n s The following theorem of P. Jones will allow us to solve the "clouds and shadovs problem" for regular mappings. T h e o r e m 8.1 [Jn2]. Let I be the unit cube ofIR k and f : I ~ IRk be a Lipschitz function. For each e > O, there is an integer M = M(I I V f IIoo,e), a constant C(I I V f IIoo,e) and sets E l , " "EM C I such that

(42)

I f(x)

- f ( y ) I> C(ll vf

Iloo, ~)-' I • - v I

whenever z and y are in the same Ei, and (43)

I f(I

n E~ n . . . n E ~ ) I ~, then there is a compact set E C I such that [ E [> 0(11 v f Iloo,~) > 0 and I f(~) - fCy) I > C(ll v/11~,6)

-~ I~ - v I for x,y E E.

The corollary clearly follows from the Theorem (choose e = ~). A first proof of the Corollary was given in [Dv6], but it is a little less engaging. Let us now use the corollary to prove that if z : IR k --* IR'~ is 1-regular (i.e., w-regular with the weight w = 1), then $ = z(IR k) CBPLG. The general result (when w is any Aooweight) would require slightly different versions of Theorem 8.1 and Corollary 8.2, which are both true (and can be deduced easily from these results), but we decided to simplify the statements and take w = 1. Let w E $ and r > 0 be given. Choose a point x0 E IRk such that z(zo) = w. If C is large enough, the cube Q centered at z0 and with sidelength ~ is such that z(Q) c B ( w , r). L e m m a 8.3. There is a k-dimensional subspaee V of iR n such that, if ~r is the orthogonal projection onto V, [ ~(z(Q)) 1> ~ I Q I. Of course, we do not want ~ to depend on Q. If ~ were allowed to depend on Q, the lemma would be completely trivial : one can find a point x in the interior of Q, such that z is differentiable at x ; then the derivative of z at z is of rank k (because z is 1-regular), and if V is chosen well, ~r(z(Q)) will contain a small ball around ~r(z(x)) (by elementary degree theory). The fact that we can choose ~ independent of Q and z (provided the regularity constants of z stay bounded) now follows from a simple compactness argument, which is left as an exercise (take a sequence (z,t, Qn) for which the best ~n tends to O, extract a

80 convergent subsequence after normalization, and use the argument above and some more elementary degree theory). More details can be found in [Dv6], lemme 10, p. 96. Apply Corollary 8.2 to the function f ( x ) = 7r(z(x)) (possibly after an affine change of variables if Q is not the unit cube). We get a s e t E c Q, with [ E I> 0 ] Q [, and such that r o z is bilipschitz on E. The set /~ = z(E) satisfies # ( 2 ) _> Ork/C, and rl~ is also bilipschitz, so/~ is the piece of Lipschitz graph we were looking for.

Let us now prove Theorem 8.1. We shall need a way to control the situations where ] f(x)-f(y) I< 6 ] x - y [ (6 > 0 i s a small constant, to be chosen later). The general idea is that, in such a case, either ] f ( B ) ] is very small for some ball B containing x and y, or else f is not close to being affine on B, and so f " must be large somewhere in B. We intend to prove that the second possibility cannot occur very often by associating to B a large "wavelet coefficient" for f~. To simplify notations, let us assume that f is 1-Lipschitz, and also is defined on the whole IR k. We need a few notations before we start the stopping time argument. Let us fix a constant C > 1 ; we'll say that two dyadic cubes Q1 and Q2 c I are "semi-adjacent" if[ Q1 [=[ Q2 ], and if ] Q1 ]l/k[ Q1 [1/2/C. Therefore, [ Q1 ]_ C ~ QI EC1

[ a S ]2( C (since ][ V f [[oo< 1). If we choose N large enough, we'll

QCCI

get [ B [< ~ , w h e r e B -- {x e I 1-Lipschitz, I f ( B ) I < ~.

: x b e l o n g s to more than N cubes of ~1}. Since f is

Call G the complement in I of B U (~-~q~82 Q ) " We just have to split G into M sets

El,'" "EM on which f will be bilipschitz. We want to associate, to each cube Q ~ ~q'e~'~ Q~ a certain sequence a(Q) of O's and l's ; the length of the sequence will be called £(Q). We want to define a(Q) by induction, starting from I, and then defining a(Q), for each Q, in terms of a(Q*) (Q* is the father of Q). First, let us introduce more notations. For each n _> 0, call J[n the set of all dyadic cubes Q E ~¢1 of sidelength 2 - n that are not contained in any cube of ~2. Call Pn the set of all (unordered) paris (Q1, Q2), where QI and Q2 are semi-adjacent cubes of .~, ; choose any order on Pn, and call P,~,I, P,,,2, "" "Pn,e""

82 the elements of P,~. We are now ready to define a(Q) for Q of sidelength 2-'*, assuming this was done for cubes of sidelength 2 - ( n - l ) (we take for a(I) the e m p t y sequence). C a s e 1. If Q ~ ~'1, and is not contained in any cube of ~2, we simply set a(Q) = Otherwise, we will define (or change) a(Q) as follows. We take the set P , in order, and define (or modify) a(Q1) and a(Q2) (where pn,t -- (Q1, Q2)) successively for £ -- 1,.-. Suppose we already went through Pn,1,'"P,,,t-1, and let Pn,~ -- (Qx,Q2), where Q1 and Q2 are semi-adjacent cubes of ~1. The cubes Q~ and Q~ are not contained in any cube of ~2, and so a(Q~) and a(Q~) are well defined. C a s e 2. First suppose that £(Q~) = £(Q~). A - If ~(Q~) ¢ a(Q~), we leave a(Q,) = c~(Q*~) and a(Q2) = a(Q[). B - If a(Q~) = a(Q~) = ( e l , " "e~Cq;)), we set a(Qx) = (cx,-"ElCq;),0) and a(Q2) =

(E~,-- .E~(Q;), 1). C a s e 3. Now suppose that £(Q~) ¢ l(Q~). If l(Q~) > l(Q~), a(Q~) = and (Q2) ' * •• * = --Q(Q~)), we keep a(Q1) = a(Q1), and set a(Q2) = (el,. • . e ~' ( Q ~ ) , e'),where e' ~ et(Q~)+l. If l(Q1) * < t(Q2) * , we exchange Q1 and Q2, and do as above. If the same cube Q appears in more than one pair Pn,t, we do not exactly do as above, but replace a(Q*) by the last defined value of a(Q) (so, most of the time, a(Q) will change a few times during the definition process). Note that, if Q c Q', then the sequence a(Q) starts with the sequence a(Q'). Also, if x is in the good set G, and I D Q(D . .. D Q(n) D • • • is the sequence of dyadic cubes containing x, then the sequence or(Q('*)) is well defined because x ~ [-Jc2 Q, and changes less than CN times during the whole definition process (it only changes for those n's such that Q(n) E ~1, and even for those n's, it changes less than C times because there is _< C pairs of semi-adjacent cubes containing Q(n)). Therefore, for each x E G, the limit of the a(Q('~))'s exists, and has length _ CN. Call a(x) this limit. Let us call E the set of all sequences of length < CN, composed of O's and l's. For each i C E, let Fi = {x C G : a(x) = i}, and let us check that, for each i, (44) Let yE was not

If(x)-f(Y)

I>Co 1EIx-yl 2

for x, y e F i .

x,y E Fi for some i, and call Q1, Q2 two semi-adjacent cubes such that x E Qx and

Q2 [such cubes exist, provided the constant C in the definition of semi-adjacent cubes large enough]. Since a(x) = or(y), the sequence a(Q1) is the beginning of a(Q2) if it is longer, and otherwise a(Q2) is the beginning of a(Q1). We made sure, when defining a(Q1) and a(Q2), that this would never happen if Q1 and Q2 axe in £1. So Q1 (or Q2) is

83 not in ~'1, and this implies (44) by definition of ~'1. This concludes the proof of Theorem 8,1; because the number of sets Fi is less than CN2 CN. E x e r c i s e . State and prove an analogue of Theorem 8.1, where Lipschitz functions are replaced by functions satisfying (34) (or (36)) for some weight weAoo. Deduce a proof of the fact that z(]R k) C B P L G when z is w-regular. R e m a r k . The proof above has the advantage of using very little of the structure of ]Rk. This is used in [DS4] to extend Theorem 8.1 to some functions defined on a regular set, and to give one more proof of the fact that every $ C S(d) contains big pieces of Lipschitz graphs.

9. S q u a r e f u n c t i o n s , g e o m e t r i c l e m m a a n d t h e c o r o n a c o n s t r u c t i o n In this section, we wish to take advantage of a small delay in the preparation of the manuscript to present a few more results (some of them are actually posterior to the lectures). We shall not have time to see any proof, but we'll try to introduce a few notions that are relevant to our study of singular integrals on subsets of ]R n. A - Square function estimates Let # E ~ , and $ C ]R n be the support of #. An easy argument using Rademacher functions shows that, if all the good kernels K of Definition 1.1 give operators T~ that are bounded on L2(d#), then the following square function estimates hold. Let (~ be the class of all odd functions ¢ E ¢~(]R n) ; then for each ¢ E (~, there is a constant C = C¢ such that

P

0, define/3(x, r) by (47)

~(x, r) = inf

sup

r - l d i s t ( y , P),

P yGSABCx,r )

where the infimum is taken over all k-dimensional affine subspaces P of ]R n. The number $(x, r) measures how well S can be approximated, near x and at the scale r, by k-planes. If S is a Lipschitz graph, for instance, a very brutal estimate only gives fl(x,r) < C < 1, but the point is that, for most couples (x,r), ~ ( x , r ) is much smaller than that. The $ ( x , r ) ' s were introduced in [Jnl], to obtain a quite interesting new proof of the boundedness of the Cauchy integral operator on Lipschitz graphs and Ahlfors-regular curves. In particular, he used the following estimate (now known as the "geometric lemma") to compare S to straight lines. T h e o r e m 9.2 [Do], [Jnl]. If S is a Lipschitz curve in IR 2, then ~(~" , . ~ 2 ~ measure on $ x IR+, i.e.

ESnB(X,R)

O. This theorem is actually an easy consequence of a result of Dorronsoro on affine approximations of functions. P. Jones' main contribution is not the fact that he re-discovered it, but that he had the idea to use it in the context of singular integrals on curves. His idea of measuring the regularity of S with the function fl(x, r) (again, some sort of "geometric Littlewood-Paley theory") has been the source of a lot of recent work (see Theorem 9.4 and 9.5 below, for instance). In higher dimensions, it was noted by X. Fang [Fn] that the analogue of (48) does not always hold for Lipschitz graphs. Fortunately, only minor modifications are needed. For l 2, or even if w C Aoo(]R k) is such that there exists an w-regular mapping, then w has to satisfy the following "strong-A~o condition" : there is a constant C > 0 such that, if B C IR k is a ball and q is a p a t h t h a t joins the center of B to its boundary, then

f..,lwl/k _:>_c-l { fBW} 1/k .

(64)

We do not know, however, if every strong-Aoo weight(or even every strong-Aoo weight with II log w IIBMO small enough) is such t h a t there exists an w-regular mapping. For more details on strong-Aoo weights, see [DS2], and for a partial result in this direction, see [Sell]. It seems reasonable to conjecture that every CASSC of dimension k > 2 admits an w-regular parametrization (or even a quasisymmetric one). Reasonable parametrizations were found by Semmes [Se4], but they do not quite have the right scale invarianee ; also, note that part h) of T h e o r e m 9.5 only gives an w-regular parametrization of a set which is larger t h a n $. One can try to relate Sobolev estimates on $ to the boundedness of singular integrals on L2($). Let us give an example of Sobolev estimates. Suppose t h a t $ E S(k), and make all the necessary a priori assumptions. Define

gI(x)

= sup ~-~ r)>O

~-k

NB(z,r)

I I(Y) - f(x) 12 a.(y)

and

o~I(x,~)

= i~f

~-~

nB(x,r)

I I(y) - A(yl I~ e . ( y )

,

where f is, say, of class C 1 on $ and the infimum is taken over all Clifford-analytic, affine functions A. Semmes proves t h a t

91 Furthermore, if each of the two connected components of ]Rk+l\$ is a NTA domain, then the two quantities above are equivalent to fs ] df ]2, where df denotes the differential of f on S. In the general case, $ might have points where it chokes, and then I f(x) - f(y) I could be large even if x and y are close and ] df I is small (think about a curve which is not chord-arc), and we cannot expect as much. It is still possible to give an estimate, by controlling the number of choking points of S. We refer to [SeS] for more details. It is not clear whether we should expect the right Sobolev estimates to imply that singular integrals are bounded. This looks like a reasonable way to replace the assumption of connectedness (or local connectedness) which implies the boundedness of singular integrals when S is Ahlfors-regular and one-dimensional. We know that, when k >_ 2, local connectedness is not enough : see [Dv6, p. 113] for the picture of an example. It would probably be interesting to study a little more the theory of differentiability of functions on one of our preferred surfaces S (for instance, is there anything like Littlewood-Paley theory on S ?). Note that the corona decomposition already gives some information. Theorem 9.5 does not tell us everything we wanted to know about singular integrals on surfaces. The most important problem is that conditions a) and c) use much more kernels and functions ¢ than we would like. For instance, if k -- 1 and n -- 2, we would really like to characterize the Ahlfors-regular sets S such that the Cauchy kernel (alone !) defines a bounded operator on L2(S). We do not even know if it is enough to consider, in a), all the odd kernels K that are homogeneous of degree - k . When k = 1 and n -- 2, knowing whether the boundedness of T~ when K is the Cauchy kernel implies the boundedness of all other T~'s would have nice applications to the theory of analytic capacity. Even if we decide to consider all good kernels, the equivalent conditions of Theorem 9.5 are not necessarily easy to check. In fact, it seems that in all known examples, it is just as simple to prove directly that S C B P L G ! Here is an example of a class of surfaces, for which we do not know whether Theorem 9.5 applies. Suppose $ is a k-dimensional surface, homeomorphic to ]R k, contained in lR k+l, and such that $ -- supp ~ for some # E ~-~. Also suppose that for each x C S and r > 0, there is a topological ball of dimension k, contained in $ NB(x, r), but which contains S MB(x, r). What can be said about S ? When k -- 2, Semmes used a conformal mapping and modulus estimates to show that $ is the image of lR 2 by a quasisymmetric mapping (and so, S CBPLG), but in higher dimensions nothing is very clear. Some consequences of Theorem 9.5 have, at this moment, surprisingly indirect proofs. Let us give a few examples to amuse the reader. To prove that a), or b), is invariant when S is replaced by its image under a bilipsehitz mapping, it seems that we need the equivalence with one of the last five conditions of the theorem. Similarly, if we want to know that, if S "contains big pieces of sets that contain big pieces .... of Lipschitz graphs", then S contains big pieces of sets that contain big pieces of Lipschitz graphs, or even d), e), f), g) or h), we seem to have to go through singular integral operators ! Finally, Proposition 3.8 tells us that, if the equivalent conditions of Theorem 9.5 are satisfied, then e----*0 lim T,~f(x) cexists almost everywhere for every good kernel K. It is not clear how to prove this directly (and in particular under the weaker assumption that T~ is bounded for a single K). We still do not know whether the equivalent properties of Theorem 9.5 also imply that S CBPLG. Since $ satisfies a weak geometric lemma, this is equivalent to asking whether

92 S has big projections. Using condition g), we see that it would be enough to prove the following. If S is the image of IRk by a bilipschitz mapping, and if 0 > 0 is given, there is a constant t / > 0 (that depends only on k, n, the bilipschitz constant and 0) such that, whenever x E S, r :> 0, and E C S M B(x, r) is a compact set satisfying ~(E) > Ork, there is a k-plane Y such that I Try(E) [> vr k (~r~ denotes the orthogonal projection on Y). Amusingly enough, this does not even seem to be known (or known to be wrong) when

k=landn=2! Let us finally quote a variant of Vituschkin's conjecture on sets of vanishing analytic capacity. Let us restrict to k = 1 and n = 2 (the question is probably hard enough in that case !). Suppose/~ E ~ , S = supp #, and 0 e S, and consider So -- S M B(0,1). The Favard length of S0 is the number FL(So) = f o I ~ro(So) [ dO, where, for each 0 E [0,~r], 7r0 is the orthogonal projection on a line that makes the angle 0 with the real axis. Is it true that if FL($o) > a > 0, then there is an image by a rotation of a Lipschitz graph, F, such that #(F N So) _> l / > 0 ? We would like v, and the Lipschitz constant of F, to depend only on a and the Ahlfors-regularity constant of #. Would it help if we supposed that the Favard length is large at all scales ? What about supposing that I ~ro(So) I> _ a >0 for all 0 in a small interval ? Of course, we are interested in this question because the existence of F would imply that the analytic capacity of $ is > 0. If these questions are not hard enough, one can always try to say something sensible in the case when # E A, but # is not necessarily in ~ . For instance, does a variant of the implication e) =~ h) (the traveling salesman theorem in higher dimensions) hold when one does not assume that S = supp ~z for a # E ~ ? I hope, too, that at this stage, the reader will be tempted to add a few arrows of his own (or, better, a few boxes) to the diagrams of Appendix II.

APPENDIX I C o n s t r u c t i o n o f d y a d i c c u b e s o n a r e g u l a r set

Let/~ E ~ (see Definitions III.2.3 and III.1.3), and let ,~ be the support of )--~. We want to construct a family ~ of"dyadic dubes" with the properties (54)-(57) described in Section III.9. The proof that follows is a minor modification of the argument in [Dv6] ; for an extension to spaces of homogeneous type (or a different proof), see [Ch 2]. Let us start by constructing balls with reasonably small boundaries. Let r} > 0 be a small constant (to be chosen later). L e m m a A . 1 . For each x E $ and j E 2~, there is a radius r ~ ( 2 - J , (1 + ~ ) 2 - i ) , and such that (A1)

ball By(x) centered at x and with a

#({V E Bj(x) : dist(y, S\Bi(x)) < T122-i}) +/z({y E $\Bi(z) : dist(y, Bi(x)) < 17~2-i})
C~/-1, so that h(x) E A(j). Note that I h(x) - x [< 2 - i + 2 . For each y e A(j), we set (A3)

Di(Y) = S N { BJ'(y) U [ Ua(~)=~B~'(x)]} "

94 The

Di(y), y E A(j), are a new partition of $, and they satisfy D 1(y) < 2 - i + 3 ,

(A4)

diam

(A5)

I~(Dj(y)) >

Co'2 -ik,

and (A6)

#({u E Di(y ) : dist(u, $\Di(y)) < r/22-1}) +/~({u E S\Di(y ) : dist(u, Dj(y)) < r/22-i})
0, and let

F = {u • Qi(x) : dist(u, $\Qi(x)) _ 0, let B(d) be the set of all y • A(i+ Nd) such t h a t Qi+Nd(Y) meets F, and let Fd = (JB(d) Qi+Nd(Y). Obviously, Fo ~ FI " " ~ Fd ~ F. Now suppose that r < 2 -N(d+1) and let y • B(d). If u • Fd+l n Qi+Nd(Y), then d i s t ( u , f ) < 24-]2 -N(d+l) and so gist(u, S\Q]+Nd(Y)) iS less t h a n gist(u, S\Qi(x)) if u • Qi(x) and than dist(u, Qi(x)) otherwise. Consequently, dist(u,S\Qi+Nd(y)) < 24-12 -N(d+l) + r2 -1 < ~22-Nd2-1 by our choice of N. By (A9), #(Fd+I N Qi+Nd(Y))