Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Ztirich E Takens, Groningen
1575
Marius Mitrea
Clifford Wavelets Singular Integrals, and Hardy Spaces
SpringerVerlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona
Budapest
Author Marius Mitrea Institute of Mathematics of the Romanian Academy P. O. Box 1764 RO70700 Bucharest, Romania and Department of Mathematics University of South Carolina Columbia, SC 29208, USA
Mathematics Subject Classification (1991): 30G35, 42B20, 42B30, 31B25
ISBN 3540578846 SpringerVerlag Berlin Heidelberg New York ISBN 0387578846 SpringerVerlag New York Berlin Heidelberg CIPData applied for 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from SpringerVerlag. Violations are liable for prosecution under the German Copyright Law. 9 SpringerVerlag Berlin Heidelberg 1994 Printed in Germany SPIN: 10130077
46/3140543210  Printed on acidfree paper
to D o r i n a
Table of Contents Page Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 1: Clifford Algebras
. . . . . . . . . . . . . . . . . . . . .
IX 1
w
Real and complex Clifford algebras
. . . . . . . . . . . . . . .
1
w
Elements of Clifford Analysis . . . . . . . . . . . . . . . . . .
5
w
Clifford modules
. . . . . . . . . . . . . . . . . . . . . . .
11
Chapter 2: Constructions of Clifford Wavelets . . . . . . . . . . . . . .
16
w
Accretive forms and accretive operators
. . . . . . . . . . . . .
17
w
Clifford Multiresolution Analysis. The abstract setting
. . . . . . .
18
w
Bases in the wavelet spaces . . . . . . . . . . . . . . . . . . .
23
w
Clifford Multiresolution Analyses of L2(IRm)  C(n )
. . . . . . . .
26
w
Haar Clifford wavelets
. . . . . . . . . . . . . . . . . . . . .
30
Chapter 3: The L 2 Boundedness of Clifford Algebra Valued Singular Integral Operators
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
w
The higher dimensional Cauchy integral
w
The Clifford algebra version of the
. . . . . . . . . . . . .
T(b) theorem
42 43
. . . . . . . . . .
53
. . . . . . . . . . . .
60
. . . . . . . . . . . . . . .
61
. . . . . . . . . . . . . . . . . . . . . .
70
Chapter 4: Hardy Spaces of Monogenic Functions w
Maximal function characterizations
w
Boundary behavior
w
Square function characterizations
w
The regularity of the Cauchy operator
. . . . . . . . . . . . . . . .
VII
. . . . . . . . . . . . . .
73 82
Chapter 5: Applications to the Theory of Harmonic Functions . . . . . . .
87
w
Potentials of single and double layers . . . . . . . . . . . . . . .
87
w
L 2  e s t i m a t e s at the boundary
90
w
Boundary value problems for the Laplace operator mains
w
. . . . . . . . . . . . . . . . . in Lipschitz do
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
A BurkholderGundySilverstein type theorem for monogenic functions and applications
References
. . . . . . . . . . . . . . . . . . . . . . . .
98
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
106
Notational Index
. . . . . . . . . . . . . . . . . . . . . . . . . .
113
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . .
114
VIII
Introduction As the seminal work of Zygmund [Zy] describes the state of the art in the mid 30's, much of classical Fourier Analysis, dealing with the boundary behavior of harmonic functions in the unit disc or the upperhalf plane, has initially been developed with the aid of complexvariable methods. The success of extending these results to higher dimensions, the crowning achievement of Zygmund, CalderSn and their collaborators, was largely conditioned upon devising new techniques, this time of purely realvariable nature (see e.g. [St], [To]). Then why Clifford algebras? The basic motivation is that within this algebraic framework we can still do some sort of "complex analysis" in ]R~, for any n, which turns out to be much better suited for studying harmonic functions, say, than Several Complex Variables. For instance, any harmonic function is the real part of a Clifford analytic one and, on the operator side, the double layer Newtonian potential operator is the real part of the CliffordCauchy integral. At the heart of the matter lies the fact that while in general the square root of the Laplacian A  012 + .   + 0,, 2 is only a pseudodifferential operator in 1~ with its usual structure, by first embedding R n into a Clifford algebra we can do better than this, and realize A 1/2 as a first order, elliptic differential operator, of CauchyRiemann type (though, a Clifford algebra coefficient one). The hypercomplex function theory has a long history, and its modern fundamentals have been laied down by Moisil, Teodorescu and Fueter (among others). However, much of the current research going on along these lines originates in the work of Coifman, McIntosh and their collaborators. This book is conceived as a brief, fairly elementary and reasonably selfcontained account of some recent developments in the direction of using Clifford algebra machinery in connection with relevant problems arising at the interface between Harmonic Analysis and Partial Differential Equations.
Our goal is to provide the
reader with a body of techniques and results which are of a general interest for these areas. Strictly speaking, there are no essentially new results, although perhaps some proofs appear for the first time in the literature. Yet, we believe that this presentation
IX
is justified by the point of view we adopt here. The text is by no means intended to be exhaustive and the topics covered rather reflect the interests and the limitations of the author. No specific knowledge of the subject is expected of the reader, although some familiarity with basic elements of classical Harmonic Analysis will help. The plan of the book is as follows.
Chapter 1 contains some preparatory
material about Clifford algebras and Clifford analysis. The presentation is as concise as possible, yet aimed to give a sufficiently rich background for understanding the algebraic formalism used throughout. More detailed accounts on these matters can be found in [BDS] and [GM2]. The scope of the next two chapters is to treat Clifford algebra valued singular integral operators. The underlying idea for proving L2boundedness results ([Tc], [CJS]; cf. also [Day2]) is fundamentally very simple. It consists of representing the given operator as an infinite matrix with respect to a certain Clifford algebra valued Riesz bases in L 2, whose specific properties ensure that this matrix has an almost diagonal form, i.e. the entries die fast enough off the main diagonal. Then a familiar argument based on Schur's lemma yields the result. In Chapter 2 we construct such Riesz bases with wavelet structure, called Clifford wavelets, adapted to some Clifford algebra valued measures in R n, and having a priori prescribed properties of smoothness, cancellation and decay.
In
particular, to deal with the higher dimensional Cauchy singular integral operator on a Lipschitz hypersurface ~ in R n, we produce a (Clifford)weighted Haar system which incorporates the information concerning the geometry of ~ (cf. [CJS], [AJM]). Once this is accomplished, one can work directly on ~ just as easily as if it were flat. Consequently, the L2boundedness of the higher dimensional Cauchy integral operator on ~ follows exactly as in the more classical case of the Hilbert transform in ~ (see e.g. [Ch]). This is worked out in detail in Chapter 3. Here we also outline the proof of the L2boundedness for a more general class of Clifford algebra valued singular integral operators satisfying the hypotheses of a Clifford T(b) theorem. This is done in the same spirit as before, i.e. essentially as a corollary of the existence of some suitable bases of Clifford wavelets. A natural setting for studying the boundedness, regularity and boundary behavior of the CliffordCauchy integral on Lipschitz domains is a type of Hardylike spaces
of Clifford analytic functions which we discuss in Chapter 4. There is an interesting connection between these and the classical H p spaces as introduced by Stein and Weiss [SWl] in that any system of conjugate harmonic functions can be identified with (the components of) a Clifford analytic function. Altogether, the results presented here can be regarded as a partial answer to the problem posed by Dahlberg in [Dah3] inquiring about the possibility of extending to higher dimensions the theory developed in C via conformal mapping techniques from [Ke]. As for Chapter 5, we submit that the classical boundary value problems for the Laplace operator in Lipschitz domains can be very naturally treated with the aid of Clifford algebra techniques. An abstraction of the main idea is that to any reasonable harmonic function one can append a "tail" so that the resulting function is Clifford analytic and has roughly the same "size" as the initial one. We also discuss several other applications, including a BurkholderGundySilverstein type result which is very close in spirit to the original theorem (cf. [BGS]). Several exercises outline further developments and complement the body of results in each chapter. I would like to express my sincere appreciation and gratitude to the people with whom I have discussed various aspects of this book during its elaboration. In particular, thanks are due to Bjhrn Jawerth who actually suggested the writing of this book, for reading preliminary drafts and for his many constructive suggestions. Several enriching discussions with Alan McIntosh, Paul Koosis, Richard Delanghe and Margaret Murray are also acknowledged with gratitude. Last but not least, I wish to thank Professor Martin Jurchescu for the trust, inspiration, guidance and moral support he generously (and constantly) gave me over the years.
XI
Chapter 1 Clifford Algebras This chapter is an overview of some basic facts concerning Clifford algebras (cf. also [BDS] and [GM2]; see also [MS] for some related historical clues). Here we set up the general formalism commonly used in the sequel. w
REAL
AND COMPLEX
CLIFFORD
ALGEBRAS
D e f i n i t i o n 1.1. The Clifford algebra associated with •n, endowed with the usual Euclidean metric, is the extension of R n to a unitary, associative algebra R(~) over the reals, for which (1) x 2 = [x[ 2, for any x e ~ ; (2) ~(n) is generated (as an algebra) by ~n; (3) R(.) is not generated (as an algebra) by any proper subspace ofI~ ~. By polarization (1) becomes xy+yx
= 2(x,y),
(1.1)
for any x, y E ] ~ , where (., .} stands for the usual inner product. In particular, if {ej}jn=l denotes the standard basis of ]~n, (1) is equivalent to ejek + ekej = 2 5jk.
(1.2)
In other words ej2 =  1 for any 1 <. j < n, . and eje k. = eke.j for any 1 < j ~ k < n. Consequently, by (2), any element a C ]~(n) has a representation of the form I
a=Eaiei,
aIER.
(1.3)
I
Here ~
indicates that the sum is performed only over strictly increasing multiindices
I, i.e. ordered ltuples of the form I = (i1,i2,... ,il), with 1 _< il < i2 < " " < il ~ n,
where 0 < l < n. ~ r t h e r m o r e , ei stands for the product ell "el2 ". 9.eft. By convention, eo : e0 : 1. We also set e I : (1)tefi  eil_l " . . . 9eft, so that e i eI = e l e i ~ 1. We shall see momentarily that the condition (3) is in fact equivalent to e(1,2...... ) ~ =[=1, hence, in evendimensional cases it becomes superfluous. However, note that in general one must impose (3) to ensure the uniqueness of the Clifford algebra 1~(~). This can be seen from the following simple example. One can embed II~3 either in I~ the skew field of quaternions, by identifying el, e2, e3 with i , j , k, respectively, or in H @ IE this time by identifying el, e2, e3 with (i, j ) , (j, i), (k, k), respectively, and both embeddings satisfy (1) and (2) in Definition 1.1. Here i , j , k are the usual imaginary units in ]E, i.e. i 2 = j2 =  1 , i j =  j i
= k.
T h e o r e m 1.2. T h e Clifford algebra •(n) exists and is u n i q u e l y d e t e r m i n e d up to an algebra i s o m o r p h i s m .
Before proceeding with the proof of this result, we point out that the Clifford algebras ~(0), lI~(1) and R(2 ) are the real numbers, complex numbers, and quaternions, respectively. P r o o f . Let us first indicate the proof of the uniqueness part. Obviously, it suffices to show that if a in (1.3) is zero, then necessarily aI = 0 for all I. To this effect, we use the following identity (whose proof we postpone for the moment) 2_n~..~,el a e I = I
~ ao, if n is even,
(1.4)
[ a~ k a(1,2,...,n)e(1,2 ...... ), if n is odd.
The minimality of ]~(~) ensures that 1 and e(1,2,...,n) are linearly independent over the reals so that, at any rate, (1.4) implies that a~ = 0. Using the above reasoning with e l a in place of a, for arbitrary I, the conclusion follows.
As far as the existence of such an algebra is concerned, we shall produce an example in the matrix algebra
.]~2nx2n(]~).
Consider the matrices ej := E~,
k are inductively defined by j = 1, 2 , . . . , n, where, for each 1 < k < n, {E~k }j=l
1)0
and, in general, for l < k < n  1 , Ek+l:= (El 0
l_<j_
~:k+l '
and
~k+l
:=
(00/2 /2 k
k)
It is easy to see that {ej}j satisfy (1.2). Also, since the trace of eI is zero for all I # O, it is easy to see that {eI}x are linearly independent over IR. Thus, we may take IR(~) to be the subalgebra of M2,, x2~ (IR) consisting of all matrices a of the form (1.3).
9
P r o o f o f (1.4). Let I' I denote the cardinality function. Since for I, J _ {1, 2, ..., n}, elej = (1)[IIJlIINJlejel, we have, for an arbitrary x = ~ j l x j e j in ]R(n),
~:'eixe'= ~ ' X j e l e j e ' = I I,J
~;'(1)lIllJllINJIxjej. I,J
Going further, for a fixed J C {1, 2, ..., n}, iJi niJi ~'(1)I/IIJIIINJI = ~ ~ ~~' (1)(i+j)lJI i I i=0 j=0 Izl=i+j IINJ]=i IJI "[JI = ~ ~ (l~JlJl(l~i(lJl1)C i C j t J t ] [JI nIJI i=0 j=0 = (1 + (1)lJI)nlJl(1 + (1)lJl1) IJI = 0,
unless either n is odd and J = {1, 2, ..., n}, or J = 0. In these cases we end up with 2 n and, hence, (1.4) follows.
9
Note that this proof actually gives more. First, dim N(~) = 2 n and we have a natural conjugation on ]R(~), denoted by :, given by the usual transposition of matrices in M2•
(IR).
In the sequel, it will be also useful to embed IR~+1 into IR(n) by identifying (x0, x) C ]Rn+l  IR @ ]Rn with x0 9e0 + x e IR(n) (note that, by (1) in Definition 1.1, 1 ~ ]R~), and call these elements Clifford vectors. An important observation is that any Clifford vector X has a multiplicative inverse, given by X 1 = X/]Xi 2. The multiplicative group generated by all Clifford vectors in ll~(n) is called the Clifford
group.
We define the real part of a E II~(~) as Re a : = a o , if a is as in (1.3), a n d endow ~ ( . ) with the n a t u r a l Euclidean metric [a[ 2 : = Re (a~) = Re (~a). It is n o t difficult to check t h a t [xyl <_ 2~lixiIY[, for a n y x , y C ]R(~), a n d t h a t ]xy] = ]xiiY[ if at least one of x, y belongs to the Clifford group of R(~). We also note t h a t a similar c o n s t r u c t i o n works for the complex case, too. T h e resulting 2 ~ d i m e n s i o n a l complex algebra will be d e n o t e d by C(~). In the last p a r t of this section we shall discuss the e x p o n e n t i a l m a p
~ expx :=
xk k~'
x E R(n).
k=0 In general, for x E ]~(n), ]expxl < exp (2nl[x]), b u t if x is a Clifford n u m b e r t h e n this e s t i m a t e becomes more precise: ]exp x[  exp (Re x). O u r m a i n result concerning this f u n c t i o n is the next theorem. Theorem
1.3. exp : ~ n + l
~ I~n + l \ {0} is welldefined and onto.
A n i m p o r t a n t consequence is the existence of the N  t h root for Clifford vectors. Corollary
1.4. For each integer N >_ 1 and each u E R ~+1, the equation x N = u
has a solution in R n+l. P r o o f . I f u = 0, we pick x = 0. Otherwise u  e x p y for some y E ]~,+1, a n d t h e n we m a y take x :exp ( y / N ) . Proof of Theorem
9
1.3. It is easy to see t h a t the p r o d u c t of two Clifford vectors
is again a Clifford vector if a n d only if they commute.
Using this observation a n d
proceeding inductively the welldefiniteness of exp follows. Let us i n t r o d u c e one more piece of n o t a t i o n . For a r b i t r a r y u E R n + l , we set
n(u) :=
; e x [ x , z]},
i . e . . A ( u ) is the smallest closed, c o m m u t a t i v e C *  s u b a l g e b r a of ]~(~) which contains u. We now fix u E ~ + 1 ,
u ~ 0, a n d prove the existence of Clifford vector x so t h a t
exp x = u. First, we claim t h a t if u has Re u > 0, t h e n there exists x in .A(u) such t h a t exp x  u. To see this, we write u = M ( 1 + (u  M ) / M ) for a n a r b i t r a r y positive
constant M , so t h a t if [u  M[ < M 1 we can define
y := Z
e
k=l Since exp y = u/M, x := y + log M will do. Hence, proving the claim comes down to finding such a M.
To this effect, it suffices to note that, as a function of M,
[ ( u  M)/M[ 2 = [u[2/M 2  2Re u/M + 1 attains its minimum 1  (Re u)2/[ul 2 < 1 at M := [u[2/(Reu) > 0. Next, we treat the case u0 := Re u ~ 0. For any real number a we have that
( a + ~ ) u = a u + [ u [ 2 9 n + l , R e ( a u + [ u ] 2) = a u 0 + [ u l 2 and R e ( a + ~ ) We axe looking for an a such t h a t auo+[ul 2 > 0 and a + u 0
=a+u0.
> 0. If u0 = 0 any
positive number will do, whereas for u0 < 0 any a from the interval (  u 0 , [u[2/uo) does the job. The only case when this interval degenerates is for u = u0, but then e.g. x := log (  u 0 ) + ~rel solves the problem. Taking a with these properties, the above claim then shows t h a t we can find two Clifford vectors x y E .A(a+~)
9 .A(au + ]u[ 2) = .A(u) with e x p x = (a + ~)u, and
= .4(u) with e x p y = a + ~ .
Since z , y
9 .A(u), it follows that x
and y commute so t h a t u  (exp x)(exp y )  I = exp (x  y), and the proof is complete.
E x e r c i s e . Use the identity (1.4) to describe the center of the Clifford algebra R(,.). Before closing this section, let us introduce a notational convention which will be constantly used in the sequel. The estimate F ~ G, for two quantities depending on some p a r a m e t e r s C S, signifies t h a t there exists a positive constant C such that
F(s) <_C G(s), for all s 9 S. ~ r t h e r m o r e , F ~ G stands for F ~< G and G ~< F . w
ELEMENTS OF CLIFFORD ANALYSIS
In this section we shall work in the Euclidean space ~ n + l , assumed to be embedded in the Clifford algebra R(,,). Let / , g be two locally bounded, Clifford valued functions defined in an open domain ~t q I~n+l. Inspired by Pompeiu's "d~riv~e ar~olaire" ([Po2]), we introduce the Clifford derivative of the ordered pair (f, g) at a point X 9 ~ by
D(f[g)(X) := lim foQ f ngda Q.Lx ffQ d Vol
More specifically, (f, g) is Clifford differentiable at X if there exists an element c C I~(~) such that, for any e > 0, there exists an open neighborhood U C ~ of X so that
oQfngdacVol(Q)
for all rectangles Q of •n+l with X E Q c U and such that, for some a priori fixed positive number C, Vol (Q) _~ C diam (Q)~+I. Here n is the outward unit normal to the boundary of the rectangle Q, da is the surface measure of OQ, and Vol stands for the usual Lebesgue measure in ~ + 1 . Also, the top integrand must be interpreted in the sense of pointwise multiplication of Clifford algebra valued functions. Let us call the ordered pair (f, g) absolutely continuous on [2 ([Ju]) if for any rectangle Q c [2 and any e > 0 there exists 5 > 0 such that
Z
f fngd~r < e, ieJ JOQi
for any finite rectangular subdivision (Qi)iel of Q, and any subset J C I for which
E ~ j Vol (Q~) _< 5 It is easy to check that if, for instance, both f and g are locally Lipschitz continuous then (f,g) is absolutely continuous.
The importance of the notion of
absolute continuity resides in the following. Theorem
1.5. If (f, g) is absolutely continuous on [2, then D(f[g) exists at almost
any point of[2. Moreover, D(fIg) is locally integrable on [2. S k e t c h o f p r o o f . ([Ju]) For any rectangle Q of ]RT M which is contained in [2 set p(Q) := sup {/e~/ ~ 0o~ fngda;(Qi)i~xfiniterectangularsubdivisionofQ}, (1.5) so that
fOQ fngda
~ p(Q) < boo for any Q. Also, since Q ~+ fOQ fngda is
rectangleadditive, i.e. fOQ fng d~r = ~iEl foQ, fng dcr for any rectangle Q and any rectangular subdivision (Qi)ie! of Q, so is p. Next, we extend the action of p to the collection of all compact subsets of [2 by setting
k iEl
where the infimum is taken over all finite collection of rectangles (Qi)ir included in f~ and having mutually disjoint interiors. As p is rectangleadditive, this extension is consistent with the initial definition of p. Also, due to the absolute continuity of (f, g), p becomes continuous in the sense that p(K,)
) p(K), whenever {Kv}~ is a nested sequence of compacts in f~ such
that M,K~ = K. For any multiindex a E Nn+l and for any ~ E N, we introduce Q~,~ := [0,2v] u+l + 2  v a , and Iv := {a r Nn+l; Q~,~ c_ f~}. Also, for any realvalued, compactly supported function ~ E C0(f~), we set I~,(~) := {a e I~, ; supp qo N Q~.,. r o } and P~(~o):=
U
Q~,""
aGL, (~)
It follows that Pv+i(~) _C P.(~o) for any v and N~P~(qD) supp ~. If we now introduce sv(qa) :=
E
~(2"a) /
~elv@)
f ngdcr,
JOQ~,~
then s~ is it(linear and satisfies
Finally, we define # : Co(fl)
) R(n ) by setting #(~) := lira s~(~),
where the existence of the limit easily follows from the uniform continuity of ~. Since # is R  l i n e a r and satisfies [#(~)[ <_ p(supp ~)l1991Ic~, we conclude that # is a Clifford algebra valued Radon measure on 12. Next, we fix an arbitrary rectangle Q c ~2 and take qo~ E C0(f~) a sequence of realvalued functions such that 0 _< qo~ _< 1 on g/, ~v = 1 in a neighborhood of Q, supp ~u+l c_C_supp ~ and fq~ supp qa~ = Q. From the defnition of/z,
~(~v)
 JO/Qf n g do
~ p(supp ~u)  p(Q),
hence, by the continuity of p, fo d# = foQ fng do, for any Q. Using this and once again the absolute continuity of (f, g), we infer that # is absolutely continuous with respect to the Lebesgue measure. Therefore, if h E Ll(Q, loc) denotes the RadonNikodymLebesgue density of # with respect to the Lebesgue measure, we have that
/oQfngdo'=j?d#=//Q
h"
Using this and Lebesgue's differentiation theorem we finally obtain that D(flg ) = h, and this completes the proof of the theorem.
9
The next lemma, which in the complex case goes back to Pompeiu [Po3], (cf. also ['re], [JM]), can be though of as the higher dimensional analogue of the classical LeibnitzNewton formula on the real line. Recall that a bounded Lipschitz domain in R ~+1 is a bounded domain whose boundary is locally given by the graph of a Lipschitz function (see e.g. [Gr], [Ne]). Lemma
1.6. Let ~ be a bounded Lipschitz domain in ~n+l and let f,g be two
Clifford algebra valued, continuous functions on ~ such that D(flg) is also continuous on ~. Then
~ofngda
= j ~ D(flg) dVol.
(1.6)
S k e t c h of P r o o f . We shall use a Goursat type argument. Reasoning by contradiction and assuming ]fo~fngdo' fff~ D(f]g)[ _> c > 0, an usual partitioning argument yields a sequence of nested domains (wj)j, with ~ j •j = {X0}, for some X0 E fl, and Vol (wj) ~ 2J('~+UVol (~), such that
fo fngdo'/f wj
j
D(f[g) ~ 2J(n+l)e.
Dividing by Vol (wj) and using the fact that ~ 1
fO~ f ngdo" + D(flg)(Xo) by f f ~ D(flg ) + D(f]g)(Xo) by the continuity of D(flg),
definition, whereas ~
we finally contradict the original assumption.
9
If f, g are Lipschitz continuous, say, it is easy to check the Leibnitz rule D(f[g) 
D(f[1)g + fD(l[g), and we shall simply set Dg := D(l[g) and fD := D(f[1). Note that Lemma 1.6 gives
~
fngdo'= / ~ { ( f D ) g + f(Dg)}dVol.
(1.7)
We also set D f :=
D(TI1) and
f D :=
D(llT).
It should be pointed out that at any
!
point of differentiability X 9 ~ of f = ~_,iflel, we have
i
/:o
J
and
(/
_
nl5,,15,O/, (X)e e x j=o ~xj
The verification is straightforward. Note that, by linearity considerations, it actually suffices to treat the case of a scalar valued function f. We can also assume that the point of differentiability is the origin of the system. In this later case, expanding f into its first order Taylor series around the origin
f(X)
=
f(O) + E xj(Ojf)(O) + o ( I Z l ) , J
and using the easily checked fact that foQ xjn da
=
Z = (xj)j 9 ]]~n+l
ejVol (Q), for any j, the conclusion
follows. Going further, simple calculations give that the Laplace operator h in IR~+1 has the factorizations
A
DD =
=
(1.8)
DD.
Following Moisil and Teodorescu [MT], we shall call f left monogenic (right mono
genic, or twosided monogenic, respectively) if D f = 0 (fD = 0, or D f = f D = O, respectively). Note that, by (1.8), any monogenic function is harmonic. Our basic example of a twosided monogenic function, the so called Cauchy kernel, is the fundamental solution of the operator D
E(X)
1 :=  
~
X iXl,~+ x ,
X
9
R ~+1
\ {0),
(1.9)
where aN stands for the area of the unit sphere in IR~+1. This can be readily seen from (1.8) and E = DFn+x = F..+ID, where
1
1
(1n)~nlXI nl'
X#0,
F~+I(X)
~loglXl, x # o ,
~=1,
n>2,
is the canonical fundamental solution for the Laplacean in
I~ n + l .
In fact, our next
result shows that any left (or right) monogenic function which is /Rn+Lvalued is necessarily twosided monogenic. n Proposition 1.7. Let F = uo  ~ j = l ujej be a •n+lvalued function defined on a
open set f~ o f ~ n+l. The following are equivalent: n (1) The (n + 1)tuple U := (U J)j=o is a system of conjugate harmonic functions in
fl in the sense of MoisilTeodorescu [Mo3], [MT] and SteinWeiss [SWI], i.e. it satisfies the so called generalized CanchyRiemann equations div U = 0 and curl U = 0 in ~; (2) F is left monogenic in f~;
(3) F is right monogenic in ~; (4) The 1form w := uodxo  uldx~  ...  undxn has dw = 0 and d*w = 0 in f~, where d and d* are the exterior differentiation operator and its formal transpose, respectively. In addition, if the domain ~ is simply connected, then the above conditions are further equivalent to
(5) There exists a unique (modulo an additive constant) real valued harmonic n function U in ~2 such that (~ZJ)j=o = gradU in f] (i.e. F = DU).
The easy proof is omitted. Lemma 1.6 applied to f and g := E ( X  .) in ~2 \ B e ( X ) yields, after letting e go to zero, the Clifford version of Pompeiu's integral representation formula ([Poll, [Mol,2], [Te]). Thereto 1.8. Let f~ be a bounded Lipschitz domain in ~ + 1 .
If f and D f are
continuous on ~, then f(X) = Cf(X) +T(Df)(X),
X e ~2,
wh ere
Of(X) :  
l f o a [y Y_ XXln+ , n ( Y ) f ( Y ) d~(Y), o,
X E a,
and 1//~ T f ( X ) : : a,~
X
Y
[)~~,]~+af(Y ) dVoI(g),
10
X C ~2.
A similar formula for the left action of D holds as well. As a corollary, let us note the Cauchy type reproducing formulas ([Di], [MT])
Y  X~~+1 n(Y)f(Y)da(Y), f(X) = ~l o [ 0a [17
X E a,
(1.10)
X e f,
(1.11)
if f is left monogenic in f , and
X 1 da(g), I(X) = a~l fo a f(Y) n(g)iN Y_ Xln+ if f is right monogenic in f~.
For f right monogenic and g left monogenic in f , we also obtain from (1.7) the Canchy type vanishing formula
~0 f ( X ) n ( x ) g ( X ) da(X) = 0. fl
(1.12)
E x e r c i s e . Let fl be a bounded domain with C ~ boundary. * Prove that C maps Coo(Off) into Coo(fl) and that T maps Coo(~) into itself. 9 Show that D(Tf) = f on Coo(K). 9 Use this and the identity (1.8) to solve the Poisson equation Au = v in ~, for arbitrary realvalued data v E Coo ( f ) . w
CLIFFORD MODULES
The "Clifordized" version V(,) of an arbitrary complex vector space V is defined by ~):=V@C(~)
=
x=
x1
.
I
Thus ~n) becomes a twosided Clifford module (that is, a twosided module over the ring q n ) ) , by setting I
otx :~ Z ~ I,J
I
Q ejel'
xo! := ~ oljxI Q eiej, l,J 11
!
for x = YT~Iz I  el E V(n ) and a = ~~.) aa e.l E C(,.). Moreover, if (V, I1" II) is a normed vector space, then we endow V(,) with the Euclidean norm
'
:
x
'[IziII 2
(.)
If W c_ V(n) is a left(or right)submodule of V(,.), then any morphism of Clifford modules L : W + C(,) is called a Clifford functional of W. The collection of all Clifford functionals of W will be denoted by W*. Consider now 7 / a complex Hilbert space (fixed for the rest of this section) and let (., .) be the corresponding inner product on 7/. Then ?/(n) becomes a complex Hilbert space when endowed with t
[=,x] := IIxll(% = ~
2
>,11 := Z ' ( x , , ~ , ) ,
I
I
if x = y}.} x,  ez C 7/(.). We also introduce the following C(.)  v a l u e d form on 7/(.)
<x, ~> := ~ ' ( x , , y.l) e~7, l,J
if x = y~.} xi  e1 E 7/(,0, Y = Y]) YJ  <1 E 7/(.). Finally, suppose that 7l has an involutive structure, i.e. there exists a conjugate linear isometry of 7/, denoted by 7, such that x = x for any x in 7/. We extend this involution to 7/(.) by introducing
:= Z ' ~ 7  7 ,
if x = ~7'~,
I

e, c 7/(.).
I
Call an element x C 7/(n) selfadjointprovided g = x. Similarly, a twosided submodule V of 7/(n) is said to be selfadjoint if V := {~; x C V} = V. The main, elementary properties of these objects are collected in the next proposition. Proposition
1.9. For x, y C 7/(,~) and a E C(,) the following hold.
(1) (x, y} = ~ [ ~  x , y]el. In particular, Fle (x, y} = Ix, y].
(2) Re<~,.> = I1<1(\) In particular, I]x[]~.)~ I(x,x)l.
12
(3) I<x,y>L <_ IIxll~.)llyllt~). (4) I f 5 e ]~nA1 C C(n), then H5XH(n) : [5Ll[xH(n).
In particular 1lSxll(,~) < 2~lsIIIxll
IIotxll(n) <~ 2n1/2lo~lllxlltn ). (5) <~, y} = ~<x, y>,
<x, ~y) = <~, y>g.
Also, ( ~ , y> = <~, y~) and <~=,y) =
(6)
=~,
~=x~,
x~=~,
(~,~)=(~,y).
P r o o f . Let us consider (4), for example. We have IIxSH~,~) = Re (xS, xS} = Re <xSX, x} = 1512Re (x, x} = 1512Ilxll~,~). Thus, llSxll(n)= Remark.
115~11(~)= II~XIl(n> = IXlll~ll
15IIIxll(n) .
The notion of monogenicity has a natural extension in the context of
3/(n)valued functions. More specifically, F : t2 + 7t(n), where ft is an open set in R ~+1, is called left monogenic provided (F(.), h) is left monogenic in the usual sense in ft, for any h in 7/(n ). Similarly, F is called right monogenic provided (h, F(.)) is right monogenic in ft for any h C 7/(,,). Adopting this convention, one can readily see that the results of the previous section continue to hold for 7/(n)valued functions as well. The next proposition is the analog of the usual Riesz representation theorem. P r o p o s i t i o n 1.10. Let 7l be as before and let V be a dosed left(or right)submodule of 7/(.,).
Then for any L C V* there exists a unique element a ~ V such that
L(x) = (x,a) (or L(x) = (a,5}, respectively) for all x C V. Moreover, Ilall(~) ~ NLIE.
P r o o f . The usual form of the Riesz representation theorem yields the existence of an a in V such that Re L() = [.,a] = Re (.,a). Then, for any x e V, L(x) = E ' R e I
(KlL(x))ei = E ' R e I
L(~x)ei : E'Re
(K[x, a)ai
I
: E'Re I
( V ( x , a})el = (x, a).
As for the uniqueness part, we simply remark that (x, a) = 0 for any x C V implies I[al[~n) = Re (a,a} = 0, i.e. a = 0. Finally, from the properties of (.,) given in Proposition 1.9, the norm of L is easily seen to be equivalent with ISa[](~).
13
9
C o r o l l a r y 1.11. If V is a dosed leftsubmodule, say, of ~(n ) and B : V x V 9 C(,,)
is a continuous form, additive in each variable and such that 13(ax, y) = aB(x, y) and B(x, ay) = 13(x, y)ff, for a E C(n), x, y E V, then there exists a unique continuous endomorphism T of V so that B(x,y) = (Tx, y),
x , y e V.
(1.13)
In what follows, B will be referred to as a continuous C(,~)sesquilinearform on V. B is called nondegenerate if for any x E V
~(~, u) = 0 v v ~ v ~ = , ~ = 0, and B(U, ~) = 0 VU ~ V r
x = 0.
Note t h a t for a continuous C(n)sesquilinear form B on V, the operator T given by (1.13) is an automorphism of V if and only if B is nondegenerate. Also, we call B a
gnondegenerate form on V, provided sup
yEV
I~(x,y)l > gllxll(n>,
and
Ilyll(,,)_
sup
yEV
I~(y,~)l ~> gll~ll(n>,
I1~,11(.)<1
for any x E V. Actually, any nondegenerate form is gnondegenerate for some 5 > 0. Finally, call B symmetric if i~(x, y) = 13(y, x), for any x, y E V. Corollary
1.12. If V is a dosed leftsubmodule of ~l(,~) and 13i, i = 1,2, two
continuous, nondegenerate C(n)sesquilinear forms on V, then there exists a unique continuous automorphism T of V such that z l ( T x , y) = z 2 ( x , y ) ,
f o r a l l x , y c y.
P r o o f . The results discussed above ensure the existence of two continuous automorphisms Si of V for which Bi(x, y) = (Six, y), i = 1, 2. Take T := $11S2.
9
E x e r c i s e . Let V be a normed complex vector space, and let X be a leftsubmodule of V(=). Then any continuous Clifford functional ~ of X extends to a continuous one
~o : V(n )   4 C(n), having comparable norm with the initial functional.
14
Hint: Use the classical version of the HahnBanach theorem to extend first the real part of T as a continuous morphism of complex vector spaces Re ~ : V(n) ~
C(n),
then check that ~ = ~ ) Re T ( ~  )el is in fact a morphism of Clifford modules. E x e r c i s e . Prove that (X(n))* ~ (X*)(~). Consider ~r(,~) : X(~) ~
TJ(n)X :~
X(~) defined for any x = ~~]I xI  ei E X(,,) by
S x o  eo, if n is even,
I xo 
e0 + x{1,2,...,n}  e{1,2 ...... }, if n is odd.
Also, for S C X(n), let (S) be the smallest twosided submodule of X(~) containing S. E x e r c i s e . If Y is a twosided submodule of X(n), then Y = (r(n)Y). In particular, if n is even, then any twosided Clifford submodule of X(n ) is of the form ]~n) for some linear subspace Y C_ X.
15
Chapter Constructions
2
of Clifford Wavelets
The aim of this chapter is to present constructions of systems of Clifford algebravalued waveletlike bases adapted to a Clifford algebra valued measure b(x) dx in R m, where b : IRm ~
IR'~+1 C C(~) is an essentially bounded function having intergal
means bounded away from zero (e.g. Re b(x) _> 5 > 0 will do). Because the complex Clifford algebra C(n) is noncommutative, a distinguished feature of such a system is that it should be in fact a system of pairs of Clifford algebra valued functions, say
L k {l~)j,k}j,
and
{O~k}j,k , called Clifford wavelets. These Clifford
wavelets must have some adequate smoothness, the cancellation properties R ( oLj,k, Oj,,~,)b = ,Sj,j,,Sk,k,, and, also, form a Riesz frame for L 2, i.e. R L f = ~~(f, ei,k)bej,k V'ORj,k\/eLj,k,f)b, =/__,
ilfll 2 ~ ~
L I(f, eRj,k)bl 2 ~ ~l(ej,k,f)bl
2,
for any L2integrable, Clifford algebra valued function f. Here the pairing {, ")b is defined by (fl, f2)b := ]~m fl(x)b(x)f2(x) dx. In the first part of this chapter, w167
we shall closely follow Meyer and
Tchamitchian ([Me], [Tc]) and prove the existence of such systems of Clifford wavelets satisfying additional smoothness and decay properties: L On Oj,k, j,k E d"(R'~)(n),
for an arbitrary, a priori fixed, nonnegative integer r, and
10'~e~k(~)l + 10'~e~k(~)l ,%<2k(m/2+k'l)exp(~l 2kx Jl), 16
Vlo, I _< ~.
for some a > 0. Of course, one cannot hope to obtain these functions by the usual dilationtranslation operations performed on some initial O, but the above estimates suggest that everything happens as if this is possible. Finally, in the last part (w
a generalization of the classical Haar wavelet
system to the Cliffordalgebra framework is presented (cf. [CJS]). Applications will be discussed in the following chapters. w
ACCRETIVE FORMS AND ACCRETIVE OPERATORS
Let 7t be an arbitrary complex separable Hilbert space, fixed throughout this chapter, and let V be a closed left(or right)Clifford submodule of 7/(n ). Following Kato [Ka], we call a continuous endomorphism T of V aaccretive on V if
Re[Tx, x] >_allxll~n),
x 9 g.
Obviously, if T is (faccretive, then T is an isomorphism of V. Also, we call a form
B : 7/(~) x "/L(~)   + C(,~)
5accretive on 7/(~) if the following conditions are fulfilled. (1) B(.,.) is Clifford bilinear, i.e.
B(ax + fly, z) = aB(x,z) + 13B(y,z), and
B(x, ya + zfl) = B(x, y)a + B(x, z)fl for all a, fl 9 C(~) and x, y, z 9 7/(n ). (2) B(,) is continuous on V, i.e.
[B(x,y)l ~ ]]xil(,~)iiYii(n), uniformly for
x, y 9 7/(~). (3) R e B ( x , 5 ) ~_ 5]lxi[~n), for any x 9 7/(,.). The next proposition provides us with a basic example in this respect. P r o p o s i t i o n 2.1. Ifb 9 ~n+l r
C(~) has Re b > 0, then the form
B(x, y) : : (xb, y),
is (Re b)accretive.
17
y 9
Proof. The only nonobvious property is (3). But for an arbitrary x 9 V, Re B(x, 5) = Re (xb, x) = ~Re {(xb, x) + (xb, x)} 1
= ~ R e { ( x b , x) + (x, xb)} = ~Re 1 {(xb, x} + (xb, x)} 1
= ~Re (~(b + ~), x) = Re b ae is, 5) = Re blfxll~), and the conclusion follows.
[]
Finally, note that for any faccretive form B(, .) on 7/(~), B(., 7) is a 3nondegenerate form on H(~). w
CLIFFORD MULTIRESOLUTION ANALYSIS. THE ABSTRACT SETTING
Our first result is the core of the algorithm we shall set up in the next section. P r o p o s i t i o n 2.2. Let V C_ H C 74(~) be closed, twosided submodules of Tl(~), let B be a 3nondegenerate form on H, and consider X L:={x 9149
X R:{x 9
B(y,x)Oforally 9
Then :
(1) X L is a closedleft submodule of Tt(,~), X R is a dosed rightsubmodule in 7"l(n) and X L (~ V = V @ X R = H (nonorthogonal sums).
(2) The oblique projection operators from H parallel to V onto X L and X R, respectively, denoted by ~rL and zrR, respectively, are continuous morphisms with operator norms bounded by a constant depending solely on n, [[B][ and 3.
(3) Letting
w := {5 9 H ; (x, y) = 0 for all y 9 V}, then W = H @ V (orthogonM difference) and, consequently, ~rL, 7rR project W isomorphically onto X L and X R, respectively. In addition, the operator norms of lrL and 7rR are bounded from below by a constant depending only on n.
18
P r o o f . From the nondegeneracy of B we immediately get X L • V C_ H. By "Riesz lemma" (Proposition 1.10), there exist two leftCliffordlinear continuous operators S :H ~
V and T : V   + V such t h a t B ( h , v ) = ( S h , ~ ) , for any h E H , v E V,
and B ( u , v) = (Tu, ~), for any u, v E V. Also, the nondegeneracy of B implies t h a t T is actually an automorphism of V. Let h be an arbitrary element in H and set v := T  1 S h E V. For any u E V we have
B(v,u) = (Tv,~) = (Sh,~) = B(h,u),
proving t h a t in fact X L @ V = H. Likewise, V @ X R = H. Moreover, "ItL = I  T  1 S and ]ISI[ ~ IIBI], [IT1][ < 5 1, so t h a t (2) follows. As for (3), simply note t h a t on one hand Ker 71"L  Y and V M W  {0}, while on the other hand any x E X L has a decomposition x = v @ w E V @ W and therefore
X = 7rLx  7fLy q 7rLw .= 7rLw.
Thus 7rL[w is an isomorphism. In addition, if x E W , then x  7rLx belongs to V, hence (x, x  ~rLx) = 0. Consequently,
[Ixll(2n) < I(X,X>I~ [<X, 7rLx>] <,~
i.e.
Ilxll(~)~
]]4(~)ll~rLxII(n),
HTrLx[[(n).
9
We also note t h a t if V and H in the above proposition are selfadjoint, then by (6) in Proposition 1.9, W is selfadjoint too. We now make the following definition. D e f i n i t i o n 2.3. A Clifford multiresolution analysis of 71(n ) ( C M R A for short) is any
increasing sequence {Vk}+~ of closed, twosided submodules of ~(,~) such that
A+cc,z _oovk = {0},
and
+oc V} is dense in 71(n), U_oo
together with a continuous Clifford bilinear form B on 71(n ) such that, for some ~ > O, B(, 7) v~•
is a 5nondegenerate form on Vk, for any k E Z.
19
Also, i f for each k, X kL, X R, W k and 7rk, L 7rk R are as in Proposition 2.2 when one takes V := Vk1 and H := Vk, then { x kL }k, { x f } k
will be called the wavelet spaces
o f this C M R A . Obviously, UVk and •Wk have the same closure in 7/(~), therefore @Wk is dense k k in 7/(~), and the next proposition tells us that something similar happens for the wavelet spaces of a C M R A of 7/(n). 2.4. With the above notations, the nonorthogonal sums 9 X kL and k G X f f are dense in 7@0. k
Proposition
P r o o f . Since UVk is dense in ~(~) it suffices to show that each Vk is included in YkL := the closure of @ X L in 7/l~.,, Let v be an arbitrary element of V. For an arbitrary, l
XkL_.l (~ ... @ X~_ N 9 WbN,
and write v = x l + x2 + ... + XN + VN with xj C X kL j for 1 < j < N and v N 9 Y~_ N. If we can prove t h a t VN ~
0 weakly in 7/(n) as N tends to +oo, then we get
oo
v = E x j 9 the weak closure of y L in 7/(n), j=l i.e. v 9 y L as desired, by the HahnBanach theorem. But
,Sll~Nlltn) <:
sup IB(~v,w)l weVk_~ Ilwll(.)_<1
=
sup weVk_~ llwll(n)
IB(v,w)l < Ilvll(,o,
since B ( X kL_ j , V k _ N ) = 0 for j < N , so that [IvNl[(n) < const < + o e for all N. Now the limit ~ of any weakly convergent subsequence of {VN} N has the property that e V k  N for all N , therefore ~
9
nNVk_N
=
{0}, and we are done.
9
D e f i n i t i o n 2.5. Let V a closed left(or right)submodule o f 7l(n). Call a system o f vectors { v j } j in V a left(or right)Riesz basis for V i f the mapping
{~j}~, > ~Jv j
(or{~Aj~*~W~j,~espectively) J 20
(2.1)
is a continuous isomorphism between g~  C(n) and V.
Note that the quality of being a left(or right)Riesz basis is preserved under the action of a continuous isomorphism of Clifford modules. Given a CMRA of 7/(n), our goal is to construct (if possible) a pair of systems of vectors L R {@j,k}j,k and {Oj,k}j,k, called dual pair of wavelet bases, having the next properties: (1) For each k, oLk belongs to xLk for any j; (2) For each k, O~,k belongs to X ~ for any j;
(a)
{
L
is a leftRiesz basis for 7/(~);
(4) {O~,k}j,k is a rightRiesz basis for U(~); (5) They are dual to each other with respect to B, i.e. B ( O jLk , O j , ,Rk , ) = 5j,j,Sk,k, ,
for all j , k , j ' , k ' .
In particular, these conditions will imply that any x in 7/(~) has the representations R L Oj,kB(Oj,k, x),
B(z, O.i,k)Oj, k = j,k
j,k
and, moreover, that
%,k)l j,k
Z W(o~,k,x)l 2. j,k
With the same notation as in the definition of CMRA, we define the following (possible unbounded) operators T L, T R : G W k k
> 7/(~),
setting T L : @~rL,
and
T R : @lr~.
k
k
Their importance is emphasized by the next result.
21
Proposition
L j {oj.k}
2.6. Suppose that for a given C M R A there exist a leftRiesz basis
in x kL a n d a
rightRiesz basis
R {%,k}J in
X ~ , both uniformly in k (in the
sense that the norms of the isomorphisms in (2.1) and the norms of their inverses are
uniformly bounded in k), for which  R
B(
Then {
= ~j,j'~k.k',
for all j, k, j', k'.
and {O~.k}j. k are a dual p a i r of wavelet bases for this C M R A if and
only if T L and T R are bounded operators on ~(n). P r o o f . Taking x E 7/(,),
with xk := ~ j aj,koLk E X ~L,
we have Ilxkll],,) ~ Z;j I~j,~l 2, uniformly in k. Next,
there exists y = GYk E q~W~ such t h a t zrLyk = xk for all k, hence I]xdl(n) ~ IlYkl[(,,) k k uniformly in k, and
Ityllo, : Z tly ll , Z LIxllo) Z Z Io , l2k
k
k
j
Moreover, T L y = x so t h a t T L is continuous if and only if
~j,kO~k
5 ~ I'~j,~l2. (n)
(2.2)
j,k
Analogously, T R is continuous if and only if
j,k
(n)
j,k
The fact t h a t (2.2) and (2.3) also imply the reverse inequalities, and therefore {oLk}i,k and {162
are shown to be left and right, respectively, Riesz bases for ?/(n), is
22
obtained from a standard duality argument. Writing
j,k
\ j,k
j,k
/
Zo5 5 j,k ~<
(.)
j,k
i~j,ki2
(n) L
. (,.)
and taking the supremum of both sides when ~ I~j,kl 2 = 1, the conclusion follows. 9 w
B A S E S IN THE WAVELET SPACES
Consider {V~}k a CMRA of 7/(n) such that Vk is selfadjoint for each k E Z (and, hence, so are the corresponding (Wk)k). Also, assume that we are given a family of vectors {r (1) {r
(2) Cs,~ = r
for which: live in Wk for each k;
for all j, k;
(3) For any k, {r162
= •j,j' for all j,j';
(4) For any x e Wk, x = E j ( x , Oj,k)Oj.k = E j Oj,k(~bj,k,x);
(5) For any x e Wk,
X 2(.)
~ E j I(x,r
2.
Note that actually there always exists a family of vectors with the above properties.
L R R for all j,k. E X~L and OR j,k := 7rkCJ,k E X k, Set oLk := IrkCj,k
Thus, for
each k, {oLk}j is a leftRiesz basis in X L, while {03R.k}j is a rightRiesz basis in X ~ . Consequently, it makes sense to define a family of bounded operators
ck:x~
~x~,
kez,
by setting
Finally, we introduce a C(n)sesquilinear form on X~, denoted {, .}, by putting
(2.4)
23
Again from the fact that {Oj,k} L j is a leftRiesz basis for X L, we infer that this form is welldefined, continuous, symmetric and nondegenerate on X L. In addition, Re {.,} becomes an inner product on X L which yields the same topology as the one inherited from 7/(n). Next, for each k 9 Z, consider the C(~)sesquilinear form B(., Ck()) on X L, and note that for any y 9 X L
IB(x, C k y ) l ~
sup
sup
IB(x, Cky)l ~ ~IICkYlI(,~/ ~ ~IlYlI(.)
zeX~
zeV~+l
Ilzll(~)<1
II~ll(n)_
Here we have used B ( X L, Vk) = 0 plus the &nondegeneracy of B on Vk+l. With a similar trick
sup IB(y, Ckx)[~ sup [B(Y,X)I~IlYlI(n), 9ex2 ~x~" I1~11(~)_<1
I1~11(,)<1
L Thus, by Corollary 1.12 there exists a i.e. this is a &nondegenerate form on X k" continuous automorphism U~ of X L such that {Ukx, y} = B(x, Cky),
for all x, y 9 X L.
(2.5)
L e m m a 2.7. If the form B is 5accretive on ~t(n), then Uk is 5accretive on XLk
endowed with the inner product Re {., .}. P r o o f . The key observation to be made here is that Ckx  ~ 9 Vk for any x 9 X L. This is because if x = ~ j ~j0~,k, then
J
J
J
J
by the selfadjointness of Cj,k's and Vk. Thus, for any x 9 X L, using B ( X L, Vk) = 0 we get
Re{Ukx, x} = Re B(x, Ckx) = Re B(z,~) >_ 5][xl]~n), where the last inequality follows from the 5accretivity of B.
9
P r o p o s i t i o n 2.8. If for each k we set oLk := UkIOL k and O h := 0~,, then {O/,k} jL is a leftRiesz basis for xLk , {Oj,k}JR
is a rightRiesz basis for X~, both uniformly in
k, so that L Oj',k,) 1t = (~j,j'(~k,k', for all j, k, j', k'. B(Oj,k,
24
P r o o f . The first part of the statement follows from the fact that for any k E Z
II k If, IIuf~ll ~ c < +~, for some constants c depending solely on n, [IBII and 5. Further, by (2.4);
L ' Oj,.k) R B(Oj,k .= e ( w ' e ~ , c ~ e } . ~ )
, ,.k, oL j,,~,l = ai,,,. : roL
Finally, the Borthogonality between various levels in k is easily seen from X # C Vk+ 1 and B(Vk+I,XkR,,) = 0 if e.g. k t > k + 1, etc.
9
We conclude this section with the following technical result which is needed later. P r o p o s i t i o n 2.9. Let V be a d o s e d Ieftsubmodule of ~(n), 13 a C(n)sesquilinear form on V, and { v j } j a leftRiesz basis of V so that
/3(vj,vj,) = 6j,j,,
for all j, j'.
(2.6)
f f V is equipped with the inner product Re/3(., .) and T is a 5accretive automorphism o f ( V , R e 13) such that
I/3(Tvy,vk)l
< exp(alj
k[) for some a > 0 and all
j, k, then, for some a t > O, we also have
]/3(Tlvj,vk)l <~ e x p (  a ' [ j  kl) ,
for all j, k.
P r o o f . Let us first remark that (2.6) implies that/3 is nondegenerate and symmetric, therefore Re/3(, ) is, indeed, an inner product on V. Moreover, for x = y~j a i r j E V,
one has Re/3(x, x) = E j Re ~ j ~  E , I~jl 2 ~ Ilxll~). Now, for a fixed, large M > 0, we write
k=0 The series is absolutely convergent in the strong operator norm on (V, Re B(,)) since, for suitably large M, w := I[(T  M)/M][
< 1, as one can readily check
using the 5accretivity of T. Let now { t k ( j , j ' ) } j , j , be the matrix representation of
25
( T  M ) k + I M (k+l), k > O, with respect to the orthonormal basis {vj}j in (V, Re B). Inductively, we see that
tk(j,j') = ~~ " " ~ Jl
to(j, jl) to(jl,j2) . . . to(jkl,j').
(2.7)
Jk1
Using this together with Ito(j,j')l < exp (a[j  J'l), we conclude that there exist some positive constants C and ~ such that
Itk(j,J')l < C~exp (  ~ l J  J'l),
for all k,j,j'.
(2.8)
On the other hand,
Itk(j,j')l = Re 13 ( ( ~ M  )
k vj, vj, )
<_ •k,
(2.9)
so that a logarithmically convex combination of (2.8) and (2.9) shows that there exist 7>0and0
1 for which
[tk(j,j')[ < Akexp (7[J J'[),
for all k,j,j'.
(2.10)
Consequently, returning to (2.7), (2.10) gives us
vj,vj,
IB(Tlvj,vj,)l ~ ~_~ k
~ ~_,ltk(j,J')l k
)~k exp (TlJ  J'l) ~ exp (TlJ  J'l),
and the proof is complete. w
9
CLIFFORD MULTIRESOLUTION ANALYSES OF L2(]I~m) @ C(n )
The particular context we shall discuss in this section is 7 / : = L2(~m), with the involution given by the usual conjugation of complexvalued functions, and
B(.f,g) := L m f(x)b(x)g(x)dx,
26
f,g e L2rN t m~/(n),
(2.11)
where b : 1Rm ~
ll~n + l C
with Re b(x) _> 6 > 0. Note that,
C(n ) is a L~
according to Proposition 2.1, B is a gaccretive form on L2(]Rm)(n). Consider now {V~.}k a multiresolution analysis of L2(R m) ([Me]), that is, a family {V~} k of closed subspaces of L2(N "') for which: (1) V/+~V~ = {0} and U+_~cV~ is dense in L2(Rm); (2) For any k 9 Z, f(x) 9 V~ .r
f(2x) 9 V/.+I;
(3) For any j 9 Z, f(x) 9 V~ ~
f ( x  j) C V~;
(4) There exists r
 j)}j is an orthonormal basis for Vd.
9 Vd such that {r
We make the supplementary assumptions that r E C"(IRm) for some nonnegative integer r, and that all its partial derivatives have exponential decay at infinity, i.e. there exists a certain constant x > 0 so that
10ar
<~ exp (  x l ~ l ) ,
x E R TM,
for any multiindex a having [a[ _< r. From the standard theory (see [Me]), let us recall that the functions {r
given
by r
:= 2~m/2r
 j),
k E Z, j E Z " ,
(2.12)
form an orthonormal basis for V~, and that there exist 2 TM  1 functions {~r162 in V1~, having the same type of regularity and decay as r which form an orthonormal basis for the wavelet space W~ := V~ G Vd. In particular, {r162162 with
r
:= 2km/2r162
 j),
k E Z, j E Z m,
(2.13)
is an orthonormal basis for W~ := V/k+l O V~.. i In the sequel, we shall identify r with r  e0, r
with %b~ e0, etc. We shall also assume that V,'k = VZ.t for any k E Z.
Setting Vk := V~  C(,~), Wk := Ws  C(,), and then taking {Vk}k together with the form (2.11), we then obtain a selfadjoint CMRA for L2(IRm)(,). The main result of this section concerns the existence and regularity of a dual pair of wavelet bases for this CMRA. T h e o r e m 2.10. For the above CMRA of L 2(IRm)(n) there exists a dual pair of wavelet
bases {O L e,j,k}e,j,k which are rregular in the sense that e,j,k}e,j,k and {O R L On~,j,k E C~(L~'")(,~), O~,j,k,
27
/or aD j, k, e,
and there exists some ~ > 0 so that
Io~eLj,k(~)l + IO~e~j,k(~)l ~
2 k('n/2+t~t)exp(~12% 
j]),
for all j, k, e and all multiindices a with [a[ _< r. P r o o f . Starting with the
{Ipe,j,k}e,j,k from (2.13), the algorithm presented
functions
in the previous section allows us to construct two families of (left and right, respectively) Riesz bases, {O~,j,k}~, j L
in X L and {O~j,,k}~.j in Xff, both uniformly
in k, for which L
R
B ( O e,j,k, O d,j',k' ) = 5e,etSj,j,Sk,k,.
Now we use a version of the aforementioned algorithm, this time starting
with
{r
(from (2.12)), to produce for each fixed k 9 Z a leftRiesz basis {eLk} j and a rightRiesz basis {r
for Vk such that R B(r L Cj,,k) = ~j,j'.
More specifically, we can take
r
:= Cj,k and r k := s k l C j , k where Sk is the unique
continuous leftCliffordlinear operator Sk : Vk 4 Vk such that ( S k f , g) = B ( f , y), for all f, g in Vk (Sk is the analog of Uk from (2.5)). Since
r
= Z(s~
1
r162162
l
a simple application of Proposition 2.9 shows that {r L
and {r
have the
same smoothness and decay as the initial r Moreover, TckL f = f _ ~f~ B ( f , Cj,k)C~,k, R L J
and
R
Cj,kB(Cj,k, f),
I c YCk.
J
Returning now to our old O's, recall that in fact we can take 
:= O~j,k = 7rkR l~e,j, k _ 1Pc,j,k  ~
R r162
L
Ce,j,k),
l
so that the regularity and decay properties of O~j,k immediately follow from the L's corresponding ones for 1~e,j,k ,S, e j,k
and q5j,k R's "
28
As for ee,j,k L
:
rrloLe,j,k = tJk
U~ 1 (TrkL r
(recall that Uk has been introduced in (2.5)), a similar argument holds,
although we have to invoke Proposition 2.9 one more time (all technicalities have been taken care of in the previous section). According to Proposition 2.6, all that remains to be proved is the boundedness of the operators T L, T R. Note that the distribution kernel of e.g. T L is K(x,y)
L
:j,k
L
x
y.
e.
This is easily checked to be a standard kernel, therefore the L2boundedness of T L can be obtained using a Clifford algebra version of the celebrated T(1) theorem of David and Journ6 [D J] (see also the next chapter). However, the computations are completely analogous to those for the scalar case (see [Me] and [Tc]), hence we omit them.
9
C o r o l l a r y 2.11. With the above notations, for all j, k, e we have
P r o o f . The constant function 1 belongs to the L2(R '~ e~l~ldx)(n)closure of Vk and, consequently, everything follows from B ( X L, Vk) = B(Vk, X~) = O.
9
Exercise. Let Qj,k stand for the dyadic cube {x C ]~n ; 2kx _ j C [0, 1]n}, and let Hl(ll~n) stand for the usual Hardy space (see e.g. [St]). Prove that for a
s e q u e n c e {ce,j,k}c,j, k
of elements from C(n) the following are
equivalent:
2
(1) .4 := ( ~ j e z n E k e Z ~ 2 n k l c ~ j , k l XOj,k)
(2) B := E
112
E nl(]~n);
zEk z E, c,,j,kO,,j,kL 9 bill(if{n)(.);
(3) C := Ej~Z" EkeZ Ee O~j,kC,,j,k e bHl(~n)(n); Moreover, if the above conditions are fulfilled, then I[AIIL~ .~ ]IBI]bH~ ~ IlCllbH~. In particular, {O~,j,k}~,j, L k and {O~j,k}~,j,a are unconditional basis for bHl(R~)(n). Remarks. L (~R t 1  C(,~) for all e,j,k. (1) Since X L, Xff C Vk+l, we have that OE,j,k, e,j,k E Vg+
(2) Using the exponential decay of the O's as before, we can get higher order vanishing moments for O's provided the initial multiresolution analysis is
29
suitably chosen. If we take Vd to be e.g. the mfold tensor product of the compactly supported real spline functions of order r+2 in L2(]~) having integer breakpoints, then we have
[ JR
xaOLj,k(x)b(x)dx=Oand [ rn
~
xab(x)O~j,k(x)dx=O,
Via I < r + 2 .
rn
(3) The same results continue to hold if the exponential decay is replaced with a rapid decay. Finally, let us mention that the main theorem of this section can be adapted to contain the case of a dyadic pseudoaccretive function b, i.e. a L ~ , Rn+lvalued function whose integral means over dyadic cubes are greater than a certain fixed, positive 5. More specifically, we note the following result from [AT]. T h e o r e m 2.12. For any dyadic pseudoaccretive function b in R m, there exists a
CMRA of L2(~m)(n) with B(., .) given by (2.11) for which one can construct a dual L {oRj,k} , with small regularity, i.e. for some 0 < r < 1, pair of wavelet bases { 0 e,j,k}' one has that for any N C N, there exists CN > 0 such that, for all j, k, e,
fo
fo
j,k(x)

,k(x)l <_ CN2 m/2(1 + r2%  il) mN,
x e R
o Lj,k(v) l < CN2k(m/2+r)lx _ y]r(( 1 + [2kx _ j [ )  m  N + (1 + ]2ky _ j ] )  m  N ) ,
's 9 x, y E R m, and similar estimates for O ne,j,k The idea is to start with a very simple CMRA, e.g. the one associated to the Haar system. One cannot conclude here because of the lack of regularity of the Haar system, but a suitable perturbation of this special case will do. w
HAAR CLIFFORD WAVELETS
First let us introduce some notation. For any k E Z let 5rk denote the collection of all dyadic cubes Q, Q=Qk,v={xE~m;2kvi<xi~2k(vi+l),
30
i = 1 , 2, ..., m},
v E Z m,
with sidelength
l(Q) := 2 k, and set 9c := k~Z~k. Each dyadic cube Q e ~" has 2m
"children" {QJ}2~ 1 : {Q' e ~k+l ; Q' c Q). For Q cube and A positive constant, AQ will stand for the cube having the same center as Q and sidelength The CMRA of
)~l(Q). We also set XQ for the characteristic function of Q.
L 2[~m~ ~ J(n) we shall work with throughout this section consists of
Vk := { f C L2i]i~m~ ~ J(n), 9f piecewise constant on the dyadic cubes of ~'~}, for k E Z, together with the 5accretive form B(., .) given by (2.11). This time, following [CJS], we shall produce explicit expressions for a system of Clifford wavelets having 0regularity, i.e. a system of
Haar Clifford wavelets. In fact, for this particular
context, we can perform our construction in a slightly more general form. Suppose that, for some 5 > 0, the L~Cfunction b : I~m is actually
) ~n+l from the definition of B(, .)
5dyadic pseudoaccretive, i.e. it satisfies 1
Ir~ ~ b(x)dx >5,
(2.14)
for any dyadic cube Q in ]l~m; here IQI denotes the Euclidean volume of Q. Note that in this case, by Lebesgue's differentiation theorem, one has
IIblllL~ ~_ 51.
Next, we introduce
m(Q):fQb(X)dx, Our hypotheses on b imply that
QE~.
m(q) e IRn+l and Im(Q)l ~ IQI. For each Q e ~
Cn 12mi we first construct a family of 2m  i functions in Vk+I, denoted by "WQ,i~r , such that
(1) IR~OQ,~(x)b(x)d~
= fR~b(x)0q,~(x)d~
(2) f ~ 0 q , ~ ( x ) b ( ~ ) 0 q , ~ ( ~ ) d ~
= ~
= 0,
for all
i = 1,2,...,2 m 
1;
i,j.
Actually we shall take
OQ,i : : ai
)
XQJ  bi+l XQ,+I, j:l
31
(2.15)
for some ai, bi E C(,~), i = 1,2,...,2 m  1, suitably chosen. It is visible from (2.15) that unless OQ,i and OQ,j have the same pair of subscripts, one of them is constant on the support of the other one. Thus, (1) automatically implies (2), at least for i r j. However, (1) is fulfilled if we choose
ai
m(Q j
:=
bi+l
and
if we have ~ j =i l m ( Q J )
:=
m(Qi+l) 1,
for
i =
1,...,
2m 
1,
r O, for i = 1, 2,...,2 m  1. This is taken care of in the
following elementary lemma. L e m m a 2.13. Consider N vectors in a normed vector space (V, H"I[) and let S denote
the norm of their sum. Then there exists an enumeration of them, say Vl, v2,..., vN, SO that IlVl + v 2 ~ ... + Vii I ~> S / N for i = 1, 2, ..., g . P r o o f . We proceed inductively. Let wl,w2, ..., wN be an arbitrary enumeration of the given family of vectors. Since
N
N
~
Ei:l j~r wj > i=~/(j~r wj) = ( N  l )
~lk= N
wk = ( N  1 ) S ,
we infer the existence of an index i0 for which
S~:=
j~ciowJ > N N 1 S "
For {wj}j#io we use the induction hypothesis and get an enumeration {wj}jr {v *.~1v1 such that Ilvl + v2 + Ji=l
"'"
+ viii > S ' / ( N 
1) > S / N for i = 1, 2, ""~ N 
we have to do now is to rename Wio to be vN. Since in our situation
• m(Qj)
=
Im(Q)J ~ IQ[,
j=l
it follows that one can enumerate the children of Q such that i
j_~,~(QJ)
~ FQI,
for i = 1, 2, ..., 2 m  1.
32
=
1. All
9
As for the case i = j in (2), introducing
M(Q,i) := f OQ,i(x)b(x)OQ,i(x) dx, JR m
a direct calculation shows that
In particular ]M(Q, i)[ ~
IQ11, and
M(Q, i)is a Clifford vector.
Finally, we the define Haar Clifford wavelets by normalizing the 0's
 
: = OQd M(Q, i) U2,
Note that unless they vanish, 0 ~ , i and
i=1,2,...,2 m1
(2.16)
i = 1, 2, ..., 2m  1.
(2.17)
@~,i take
on values in the Clifford group of
N(~).
§
§
0
t
0 FIGURE 2.1.
0
§
The three Haar Clifford wavelets
living in the same dyadic cube for m = 2. The main result of this section is the following.
Theorem
2.14.
With the above hypotheses, {@~,i}q,i and {
give,, by (2.16)
and (2.17) satisfy: (1) e L Q,i,  Q,i 6 Vk for MI k E Z, Q E 7k and i = 1, 2, ..., 2 " 
1;
(2) suppe~,i, s,ppe~,, c_ r and te Lq,,I, legit, s I#11/2, (3) f~m e~,~(x)b(~) dx = fRm b(~)e~,~(=) d= = 0 rot all Q 9 7, i = 1, ..., 2 m  1; (4) fRm e~,dz)b(=)e~,,~,(=)
dz = ~Q,Q,~,~,, for ali Q, Q', i, i',
(5) { eLQ,i}Q,i is a leftRiesz basis for L2(Nm)(n) and {e~,i}Q,i is a rightRiesz basis
for L2(~m)(n).
33
P r o o f . The only thing that we still have to check is (5). For each k E Z we consider
X L := { f E Vk+l;
f f(z)b(x) dx =
0, for any Q E ~k},
Xff := { f E V~+I ;
/Q b(x)f(x) dx =
O, for any Q E ~k}
JQ
and
We claim that { o LQ,i}, with Q E ~k and i  1, 2, ..., 2 m  1, is a leftRiesz basis for X L uniformly in k E Z. Restricting our attention to one dyadic cube Q E ~k and using the explicit expressions of the 
we readily see that
XQ2 is
spanned by
XQ~
and 04,1 in the set of C(n)valued functions on R'* with its natural structure as a left Clifford module. Continuing this inductively, we see that any characteristic function
XQi is
spanned by
XQ1 and
O~2,1, 9,  Q,2,~1" Now, if f is the restriction to Q of a
function from X L, we have 2rn_x
i=l
The fact that
B(f, 1)
< ,~
since
= 0 implies that /~1  0. Moreover,
Z7 o8,,
11o~,~112,lion,ill2 ~< 1.
" lTlllOQ,ill2 ~< ~
i
17/I, i
Finally, since there are only finitely many 71's, the g2_
sum is comparable with the elsum, so Ilfll~ ~ Z1171t 2 and this proves the claim. A similar result is valid for 
also.
At this point, by Proposition 2.6, everything is reduced to proving the estimates: 2m1
E Z
QE~" i1
2
L2q~ t "~:(~),
(2.18)
for f E L2(~m)(,0.
(2.19)
uniformly for f E
2rn1
O0,i)l < lrfll'~, uniformly QE.T" i=1
34
To this end, we introduce the projection operators A L, /k kR
/~L : L2(]l~m)(n)
) X L,
/~ff : i2(]~m)(n) ~
XkR
by setting 2m1 L
::
Z
2m1
B(:, OQ,i)(~Q R L .i'
QE,Yk i=l
. := Z &kf
QE.Tk
~
R L f). Oo,iB(Oo,i,
(2.20)
i=1
Clearly, 2m1 L
2
[B(L eQ,i)l
(2.21)
QE~k i=1
and
R 2~ IIA~Ylh
2m1
~
~ IB(O~,~,/)[2,
(2.22)
QESr~ i=1
uniformly in f E L2(IRm)(n) and k C Z. Next, we consider the so called conditional expectation operators (left and right, respectively) E k, L EkR with respect to the aalgebra generated by ~k and the Cliffordalgebra valued measure b(x)dx, i.e.
if x c Q E .Tk, and
Efff(x):=m(Q)l(/Qb(y)f(y)dy),
i f x E Q c ~'k,
respectively. The relation between these operators and A L, A R is the following. L e m m a 2.15. We have that /~L _= EkL1 _ ELk and /X kR = Ek+IR  ERk" P r o o f . By restricting our attention to one dyadic cube Q E .Tk at a time, we easily see that B(EL+i f  ELf, 1) = 0 and that EL+If  E L f is constant on each dyadic subcube of Q, i.e. Ek+lfL _ ELkf E X k.L Since {O~,i} is a leftRiesz basis for X L, it suffices to show that both A kLf and Ek+lf  E kL f have the same coefficients with respect to this basis, or even that B(E~+ i f , O~,j) = B ( f , O~),j), since E L f is constant o n Q.
35
However, if 
= ~ i ; k iJX O ' and
E k L+ l f
= ~~4fqim(Qi)lxO',
where we set
fQ, := ]Q/i1 fQi fb, then
B(Ek+:f L , OQd . ) : ~~ lQil fQi .~i :
,~io
, f b.k{ dx : B(f, 
).
i
B~fore we come to the proof of (2.18) and (2.19), we consider the case b(x) = 1 on 1Rm. For Q E Y and i  1, ..., 2 m
9
2m/2 (
hQ:= IQI1/2 ~

1, we set
i )1/2
I~~XQ,_XQ,+ 1
7 v=l
i The family { h Q}Q,i is easily checked to be an orthonormal basis for L2(IRm) with the
standard inner product (., .). In this special case, we denote A L ( o r / k f ) and ELk (or Ekn) b y / k k and Ek, respectively. As in the general case, /l k : Ek+l  Ek. Moreover, since 2m1
aks =
( f , hQ)hQ, ' '
Z QEDCk i = 1
we have that
+oc
~,~ IAkfl 2dz = Ilfll~,
SC
L2(Rm)(r~).
(2.23)
P r o o f o f (2.18) a n d (2.19). Note that E L f = Ek(fb)Ek(b) 1. Hence,
IAkLfl = I ELk + l i 
E~fl =
IEk+l (fb)E~+l@)I _ Ek(fb)Ek@)ll
N IEk+l(ib)Ek+l(b) 1  Ek(fb)Ek+l(b)ll + IEk(fb)Ek+l(b) 1  Ek(fb)Ek(b)ll
(2.24)
< IAk(fb)l + IEk(ib)l]Ek+l(b) 1  Ek(b)ll
< IA#<(/b)l +
IEk(fb)llAk(b)l,
since IE~(b)l ~ 1.
In order to complete the proof of (2.18) and (2.19), the estimates (2.21), (2.22), (2.23), and (2.24) show that it is enough to prove that +c~
L IEk('fb)121Ak(b)12dx< The proof of this is more or less standard.
(2.25)
We shall follow Christ [Ch] with very
minor alterations. We make the following definition.
36
Ilfll~.
Definition
2.16. We call a sequence of positive, measurable functions on •m,
w = {wk)keZ, Carleson if
QeJ:r~
wk(x)dx {keZ;2 _
< qoo.
We need the following two (essentially) well known lemmas. L e m m a 2.17. For every b in BMO(Rm)(n), {]Ak(b)]2}lr is Car]eson, with norm not
exceeding (a multiple of)]lbil2Mo . L e m m a 2.18. Suppose w = {wk}k is Carleson. Then, for any 1 < p < o%
uniformly for f E Lv(~m)(n). Accepting these results, the inequalities (2.18) and (2.19) immediately follow and the proof of Theorem 2.14 is therefore complete.
9
For the reader's convenience, we shall include the proofs of these lemmas. P r o o f o f L e m m a 2.17. We need to estimate
for an arbitrary, fixed, dyadic cube Q. Since Ak annihilates constants, we may suppose that bQ, the integral mean of b over Q, is in fact zero. Write b E BMO(~m)(n) as b = b0 + boo with bo = XQb, and boo = XR~\Qb. Then, by (2.23) and the J o h n Nirenberg inequality,
IQI' SQ ~
fR, ~ IA,(b0)()l'dIbl'
IA,(b0)()l'd _< IQ11
2k_
=lQl'llbolt~ = IQll/q
kEZ
Next, we want to estimate IA~(b~)(x)l for a fixed x E Q with l(Q) < 2 4. We note that there are unique dyadic cubes Qr E ~k and Q" E ~k+l which contain x, and
37
these cubes must be contained in Q. Since b~ = 0 on Q, we have
dx 
b~ dx = O, r
and the proof is complete. P r o o f o f L e m m a 2.18. For an arbitrary 1 < p < co, we have
m
IEkf(x)lV~k(x) dx = p
~ wk({x; IEkf(x)l > A})ApldA
(2.26)
k
For each fixed A we choose a maximal collection {Q j} of pairwise disjoint dyadic cubes among the dyadic cubes Q such that
~Q~ ~ f(x) dx > A . If we let Mr(x)
:= sup~ezlEkf(x)[ , then clearly [{z; [Mf(x)[ > A}[ = ~~j [Qj["
Furthermore, if [Q[1 f Q f d Z
> A for some Q E Fk, then, by the maximality of the
Q/'s, Q is contained in exactly one of the Qj's. As a consequence,
k
j
{k;2k
j
< II~llc I{~; IMf(x)l > ),}1. Inserting this in (2.26), the LPboundedness of maximal fufiction gives
%[]
The last result we prove in this section is that the just constructed Haar Clifford wavelets are an unconditional basis of U~(R'~)(,~), for 1 < p < oo. T h e o r e m 2.19. For 1 < p < oo and f : R m
) C(n) locally integrable, the following
are equivalent: (1) f e Ifl(Nm)(n); 2m1
(2) f = E Q ~ T E ~ = I (3) A(x)
:=
B(f, OQ,i)OQ, R L i with convergence in LP(Nm)(~);
(~Qe~= V'2m1 /'i=1 [B(f, @~ Q iJ~
(4) A'(x):= [~vL.,Q~7 '
x  ' 2 ' ~  1
~,=I
Q iktx~/ 2~ J I/2 E Lp(Rm);
IB(f, O~,gl2lQI  1 XQ(Z) ) 1/2 c L'(R m)
38
Moreover, if the above conditions are fulfilled, then
II:IIL" ~ IIAIIL" ~ IIA'IIL'Also, similar results are valid for 0 R's. P r o o f . By (2.20) and Lemma 2.15, (2) is equivalent to f = E ~ e z ( E L + I  EL)f in LP(IRm)(,~). It is not difficult to see that the sequence of bounded operators in /2, {EL}keZ, satisfies
E L ____+{ I , ask 0, as k
>+oo > 0%
in the strong operator norm. Therefore (1) *=:> (2). We consider next the equivalence (1) *::* (3) and introduce the operators
T~(:)(x) := Z Z ~q,,B(:, O~,,)O~,,(x),
:e : n
L,,
Qe~r i where w = {wO,i} , with Q E .T" and i = 1,2, ...,2 m  1, is a sequence of =t:1. Clearly, for any such w, T~ is a bounded operator in L2(Nm)(~). We claim that in fact T~ is also of weaktype (1, 1). To prove this claim, for a given f E L 1 fq L 2 and ), > 0, we perform the Calder6nZygmund decomposition for f at the level A (cf. [St]). Hence, we can write f  g + b, where the "bad" b part is decomposed further as b = ~ i bj, where suppbj C QJ e Jr, fQj bj = 0 and ~ j ]QJ[ ~
A111fllL1. Consequently,
Using the vanishing moment property and the precise localization of O's and bj's, we see that this sum has only zero terms for Q ~ QJ. In particular, suppT~(bj) C QJ which, in turn, implies that
I{x; IT~(b)l > A}l ~ ~~ IQil ~ A111f]lL 1, J and the claim is proved.
The usual interpolation argument then yields the /2
boundedness of T~ for 1 < p _< 2.
39
The dual range is dealt with by a fairly standard duality argument, which we include for the sake of completeness. Let T~* be the adjoint of T~ with respect to the form B(.,), i.e. T~o* is the unique continuous rightCliffordlinear operator in L 2 for which
B(T,,f,g) = B(f, T j g ) ,
f,g E L 2.
As before, we can display the kernel of T~* in terms of O's, and using the same argument we get that To~* is LPbounded for 1 < p < 2. Now, if 2 < p < c~ and q is its conjugate exponent, we have
I(T~f, 9)1 = [B(T~f, bl g)l = IB(f, T~*(bl g))l = I(f, bTo,*(bl g))l < IIfIILp IIbT,., 9(b 1 g)llLq < IIflIL,'IIglILq, since 1 < q < 2 and b1 C L ~. Hence the equivalence
[[T~f]IL~ ~ ][fHLp, and since T J = I, we get
IIT~fEILP ,~ IIIIrLP, uniformly in w C {1, +1} 7x{1'2,''''2m1}. Finally,
we integrate this equivalence against the measure given by d# :=

d#(w) on {  1 , 41} ~x{1,2`''''2"*U
where dz~ is the probability measure on {  1 , 41} taking the
value 1/2 on {41}, and then use Khintchine's lemma asserting that on the measure space ( {  1 ,
41}'7:x{1,2'"2"~l},d#) any L p norm is equivalent to the L 2 norm (see
also [Me]). The conclusion follows, and the proof of (1) ~ : ~ (3) is complete. Obviously, (4) implies (3). To see the converse implication, we first note that
IB(f,O~,i)O~,i(x)l ~ IB(f, 0~,i) I IO~,i(x)l, since the nonzero values of 0~, i are in the Clifford group of ]~(~). Thus, any dyadic cube Q has a children Q~ on which
A(x) ,~ A~(x). Finally, one can use Lebesgue's differentiation theorem to conclude that in fact
A(x) .~ A'(x) for a.e. x E ]~m.
9
Remarks. (A) It should be noted that a LittlewoodPaley type estimate of the form
2\ 1/2
is valid under more general circumstances, e.g. as part of a Cliffordmartingale theory as developed in [GLQ].
40
(B) In the construction of the Haar Clifford wavelets the family of all dyadic cubes in ll~'* can be replaced by a more general system )r = UkjVk subject to the following set of conditions (all constants involved being independent of k): (1) For each k C Z, ~k =
{Qk,j}j
is a countable partition of ~'~ consisting
of measurable sets of finite Lebesgue measure which satisfy (diam Q)m < const IQ ] for all Q E ~k(2) If Q E 5rk and QI E ~k+l are not disjoint, then necessarily QI c Q and 1 <: const ~ [Q]/IQ'[ ~ const < +oo. (3)
b(x) is a L~function which takes on values in ~ + 1
and for which (2.14) holds
for any Q E ~. This allows us to handle the case when b is a
pseudoaccretivefunction,
i.e. b is a
L~function for which there exist ~ > 0 and C > 0 such that for any x E ll~n and any r > 0, one can find a cube Q c
BT(x) with r < Cl(Q) ~Q]/Q b(y)dy > 8.
for which
(2.27)
Roughly speaking, this means that for any cube Q there exists another cube Q~ at about the same place and of about the same size for which (2.27) holds. The idea (due to David) is to set up a class of "cubes" .T by suitably correcting each dyadic cube such that, in the end, for any "cube" Q E 9r, (2.27) is fulfilled (eventually for some smaller 8; see [CJS]). (C) The construction also works on spaces of homogeneous type (see e.g. [CW]).
41
Chapter The
L 2 Boundedness Singular
3
of Clifford Integral
Algebra
Valued
Operators
The L 2  b o u n d e d n e s s of the higher dimensional Canchy integral operator on a Lipschitz hypersurface can be reduced, by means of the rotation method of Calder6n and Zygmund, to the special case of a Lipschitz curve and in that case one is in a position to use the celebrated result of Coifman, McIntosh and Meyer [CMM]. However, it is natural to try to prove this without relying on the rotation method, working directly on the surface and the first results in this direction are due to Coifman, Murray and McIntosh ([Mu], [Mc]). To explain our approach, we first need to introduce some notation.
We shall
work in the Euclidean space IRn+l canonically embedded in the Clifford algebra ]R(n). Let g : IR~ ~
IR be a Lipschitz continuous function, i.e. for some M > 0 one has
[g(x)  g(Y)l < Mix  Yl, for all x, y 9 IRn, and denote by E its graph:
r, := { x = (g(~), 5) ; ~ 9 IRn} g ~t~+l ~ R(~). The exterior normal N(x) := (  1 , Vg(x)) is then a welldefined Clifford vector for almost all x 9 IRa , and R e g ( x ) =  1 , [N(x)I ~ 1. Also, n := N/IN[ is the unit normal on E. Formally, in local coordinates, the Cauchy singular integral operators on E are defined by
z(~)~[n§ l dy, cL I (x) := ~~01imlan f~ ,~f (y) g (y) IZ("z(y) ~  Z('~) and
CRf(x) := lim 1 JfR z(x)5 ....... ~ 0 ~Z . I zz( (~y:) ~(T): ~F§ 1iv tyJJ~yJ ~y, where an is the area of the unit sphere in IRn+l z(x) := (g(x),x) E ]Rn+l C R(n), and f is a C(n)valued function on ]Rn. Note that the above integrands are to be understood in the sense of multiplication in C(n).
42
We shall follow the second part of [CJS] and prove the L2boundedness of C L and C R. The idea is to adapt the standard inner product for L2(IR~) to the geometry of E by considering the Clifford bilinear form (, ">x:
~ := /R ~ fl(x)N(x)f2(x) dx
defined for Cin)valued functions fl and f2. Considering the Haar Clifford wavelets constructed in the previous chapter which correspond to the 1accretive, L~function
N(x) and expressing the Cauchy integral as a matrix operator with respect to this basis, the boundedness will be a consequence of (a version of) the ordinary Schur's lemma. It is interesting to note that, in the commutative case i.e. for n = 1, essentially the same reasoning yields even more, namely David's theorem ([Davi D concerning the boundedness of the Cauchy integral on the so called chordarc curves (cf. also [CJS]). In w
we discuss in detail the boundedness of the Cauchy singular integral
operator while in section w
we briefly indicate how these techniques may be used
to prove the Clifford algebra version of the T(b) theorem. w
THE HIGHER DIMENSIONAL CAUCHY INTEGRAL
As much as possible, we shall keep the notations introduced so far. For a fixed 5 ~ 0 and for f C n2(Rn)(n) , we define the operators C L and C~ by
1 s
.......
J~y)lv [y) [z~y) Z('~ ~1n+l dy,
X C ~n,
1 O"n
z ( y )  z(x) + iz('~ z(~)q..~[n+lN(Y)f(y)dy,
x C I~n,
:=
0"~
:=

z(y)  z(x)  5
and c
I(x)
respectively. Let {O~,i, O~,i}O, i be the Haar Clifford wavelets associated to the Clifford bilinear form {', ">:~. The main estimates in the proof of the L2boundedness of these operators are contained in the next theorem.
43
T h e o r e m 3.1. Let 0 < e < sn and, for any dyadic cube o f N '~, let w(Q) :=
1 E IQI:
Then we have 2n1
sup w(Q)' ~
L O~),,j)s < C < +oo, ~ , w(Q') /\ c L~o Q,i,
Q' j = l
Q,i
2n1
L O~,,j)S < C < +oo, supw(Q')  1 ~ ~ w(Q) \/cLo ~ Q,i, Q',J Q i=1
(3.2)
with C independent off. Similar estimates are also valid for C~. We shall postpone the proof of this for a moment and first derive some of its consequences. First recall the following version of Schur's test. L e m m a 3.2. Assume that the rows and the columns of an infinite matrix A = (aij)ij
satisfy
Wi1Z
[aij[wj <~ C
< +oo
for each
i,
J and
wT1 Z
[alj[wi < C < +oo
for each j,
i
for some constant C > 0 and some positive numbers (wj)j. Then A defines a bounded
operator on g2(N) with norm _< C. A combination of this and Theorem 3.1 immediately yields the following. C o r o l l a r y 3.3. The operators C L and C~ are bounded on L2(~;~n)(n), with bounds
independent of & Using the Poissonlike decay of the kernel of Cfi  cL~ one can readily see that
(C L

cL_~)f ~ f in L p as 6 ~
cL(cLcL_~)f = c L + J
0, for any f E L2(]Rn)(~).
) c L f in L 2 as 6' ~
Moreover, since
0, we see that the limit lim~+o C ~ f
exists for any f E L2(llC~)(n). In fact, the next theorem (whose proof will be completed in the next chapter) shows that we can be very precise about the limit operator. For a fixed, 5 > 0, we introduce the truncated Hilbert transforms H fi and H R by
H L f ( X ) : 2 fl x an
_y[>$
X 1 dS(Y), f ( Y ) n ( Y ) iy Y_ Xln+
YEE 44
X 9 P.,
and
H~f(X) ::

2 fix

r
_yl>_~ Y EY3
v_=x XI + l n ( Y ) I ( Y ) es(yl,
Y
X
~,
where dS is the canonical surface measure on E. Simple, direct computations involving the Cauchy kernel E ( X ) show that, for 6 > 0,
cLf(x)H~(foz1)(z(x))
~ f*(x),
x E A n,
uniformly in 6. Here 9 is the standard HardyLittlewood maximal operator. A similar estimate holds for the difference between C~ and H~.
In particular, {HL}~ and
{H~}~ are bounded in L2(II~n)(~) with bounds independent of & As a consequence, the Cauchy kernel is a Calder6nZygmund kernel; see the Appendix of [CMM]. With this and some standard arguments from the theory of the Calder6nZygmund operators, as presented in e.g. [Me], we get the following. T h e o r e m 3.4. The Cauchy singular integral operators, or Hilbert transforms on ~,
H L and H R defined for any f E LP(E, dS)(~) and Mmost all X E ~ by
H L f ( X ) :  lim H L f ( x ) ~~+0
and H R f ( X ) : =
lim H ~ f ( X ) ~++0
are welldefined and bounded on L2(E, dS)(n). Moreover, lim c L f ( x ) = ~(=t=I+ H L ) F ( X )
&+:t:o and
lim C ~ f ( x ) = ~(=t=l + H R ) F ( X ) ,
8+t=o
at almost any x E I~n, where F := f o z 1, X := z(m) E E. The rest of this section is devoted to presenting the proof of Theorem 3.1. First we shall establish some estimates on the "size" of the image of a Hear Clifford wavelet under the action of the Cauchy integral. We emphasize that all the constants involved are independent of ~.
45
L e m m a 3.5. For any dyadic cube Q, and any i = 1, 2, ..., 2"  1, we have :
c~'e~,~(a:) <
l(Q)lQl~/21x 
XQI('~+~),
(3.3)
r 2Q,
ifx
and cn l(Q) if x E 2Q, <~ [Ql1/21~ dist (x, OoQ)'
cLo~'i(x)
(3.4)
where OoQ : = Uj2" = l O~j w , with {QJ}j the "children" of Q, and XQ is the "leftmost corner" of Q, i.e. for Q  Qk,,, XQ := 2kv.
Here c~ denotes a constant which
R as well. depends only on n. Analogous estimates hold for CRO a Q,j P r o o f . Inequality (3.3) is a consequence of the fact that O's have vanishing moment
IcLo~,,(x)l
=
iz
7zg;
.
.
.
.
.
.
Hence, (3.3) follows by using the meanvalue theorem to estimate the the expression inside the parentheses and by noting that for x • 2Q, dist (x, Q) ~ [x  XQ]. Next we consider (3.4). If x E 2Q,'we have
Jc~o~.,~(.)l ~
f~"L
. 'g" "
2'~
~< 'Q[1/2 E
j=l
.
z(y)  z(x)  a
dy
z(y)  z(x)  a
N(Y) lz(~z(~ ~l~+l J
Let d := dist (x, OoQ) <_ gist (x, OQJ), for all j .
dy
.
If x E 2Q \ Q, we majorize each
integral in the above sum by
~d
which gives a b o u n d of the right order. Assume now that x E QJ for some j . This time we split each integral as
/Q.i=/X_Yl>a+~X_Yl
46
=:I+II'
where X := z(x), Y := z(y). Since
I~  yl
~
Ix
 YI, we get that I has the right
size in the same way as before. As for II, we note that if ~2_ is the domain in R "+1 below the surface E then, by the monogenicity of the Cauchy kernel,
II = 
fix Y[d YEf~
Y  X YX6 IY x [ IYX~I ~+~ dad(Y),
where dad is the standard surface measure on the sphere of radius d centered at the origin of ]~+1. Hence,
IIII <
fix  Y l = d
1 dad(Y)=
IX  YP
o,,,
yER,,.+ 1
which completes the proof of the lemma. L e m m a 3.6. For f E Lc~omp(R~)(~) satisfying the cancellation condition (f, 1)s = 0
we have (cL f, 1>~ = O. Also, if
These are both simple consequences of Cauchy's vanishing theorem (see
Chapter 1) applied to the domains fl_, ~+ located below and above E, respectively, and to the functions c L f , C~Rf which are right monogenic in a neighborhood of ~_ and left monogenic in a neighborhood of fl+, respectively. That this works is guaranteed by the good decay of the functions at infinity (as in the proof of the previous lemma, the vanishing moment of ] actually improves the decay of c L f and
C~/).
9
Our next lemma shows that C L and C~ are essentially adjoint to each other with respect to the form (, .)~. oo )(n)" In L e m m a 3.7. We have ( e L f , f')~ = (f, C ~R f I)r~, for any f, ff E Lcomp(]~
particular,
r (C~L eo,~,e~,,j)~
= _(
, c~R eo,n j)~.
P r o o f . This is immediate by Fubini's theorem since the double integral
/R n/~tn ~ . . . . . .
z(y)  z(x)  6
is absolutely convergent.
47
N(x)H(x) dxdy
Let us now consider a dyadic cube
Qk,,,. We let (I) be the linear rescaling mapping
of Qk,. onto the s t a n d a r d unit cube in N ~, i.e. ~ ( y ) := w : = dyadic cube Q we set Q* := ~(Q).
2kyv. For an arbitrary
We have t h a t Q* is still dyadic as long as
l(Q) < 2 k. Furthermore, if z*(w) := 2kz(2k(w + v)), for w C ~ , one can readily see that z* is a biLipschitz application with constants comparable to those of the initial z. Let ~* be the graph of z*. As a general rule, we agree that the superscript 9 is used to label objects related to 2" which are analogous to those we have constructed in connection with ~. Direct calculation shows that
=2
OQ,i(y),
=z
t.,2_k~i~Q,ity),
and (t~L* t:~L*
t:~R* \
L L
R
~2k~':'Q*,i, '='Q,.,fl~* = (C~ OO,i, OO,,j)r.
In addition,
w*(Q*) = 2nk(W2~)w(Q), so t h a t 2n1
w.(Q'.)I
~
~
{QE~;I(Q )<_I(Q*)} i=l ,
w.(Q.)[/cL*oL. R. \ 6 Q',i, Q)Q'*,j)2"
!
2n_l =w(O')I
Z w(O) ](CI(Q')~OQ,i, L L O~, j)~
Z
{Q~;t(Q)
\ ~
Q,i, O~0,1]j)s
< C < +~,
{QeJC;l<_l(Q)}i=1 and that, for any i, 2nI
\"~[0,1] ,i, (i
Qj/~ < C < +o%
{QESC;I(Q)<_I}j = l where C may depend on II ~7 gilL ~ but is independent of ~ and
i,j. If we combine
these estimates with the similar ones needed to complete the proof of (3.2), we see
48
that we actually must prove that for all
i,j,
E w(O) (C50Q,i, L L ORo,i]nj)E ~_ C < +oo OeJ=
(3.5)
and L R R E w(O)I(O[o,a],~,i, C,~ OOj)~ < C < +0%
QE~
with C possibly depending on H V
(3.6)
gilL~ but not on 5.
The proofs of (3.5) and (3.6) are virtually the same, and we confine ourselves to showing e.g. (3.5). We need to discuss several cases. C a s e I.
l(Q) "large" and Q "clearly" disjoint from the standard unit cube: l(Q) >_1
and 2Q fq [0, 1]" = o . Using (3.3) and that [x 
XQI >~[XQ[,we get
L L R w(Q) (C~OQ,i,6)[o,1]n,j)~ < IQli/2~l(Q)lQI1/21XQl(n+l).j
(3.7)
Now, if Q = Qk,,, our hypotheses imply that v ~ 0 so that, as e > 0, the righthand side of (3.7) is majorized by OO
E
E zkn(1]2~)2k2kn/2zk(n+l)N(n+i)
k=O vr vEZ n
which proves that the corresponding piece in (3.5) satisfies the right estimate. Case II.
l(Q) is "large", i.e. l(Q) >_ 1, but [0, 1]n n 2Q r o .
Note that [0, 1]'~ Cl 2Q r 0 implies that there exists a fixed nonnegative integer M0 (3 ~ will do), such that for any
k, .Tk contributes with at most M0 dyadic cubes
to this case. Now, by L e m m a 3.7, (3.3) and (3.4), L L R R w(Q) (C50Q,i, O[0,i]n,j)E = w(Q) I(OL Q,i, C~R O[0,1]n,j)E
49
~'[Q[1/2e[Q'I/2(JQN2[O,1]n[C~O~~ [Oal~
'cRo~,I]n,J (x)'dx)
,,\2[o,1]~ [xl(n+l)dx
dist (x, 0o[0, 1]'~)
IQI~.
Hence,
w(Q) (cLo~,I, O[Ro,1],~j>s < Ir ~. Using this, the part of (3.5) corresponding to this case can be estimated by 0~
00
k=O
vEfinite set
k=O
as desired. C a s e I I I . l(Q) "small" and Q "clearly separated" from the standard unit cube:
l(Q) < 1 and 2Q fq [0, 1]n = g . This time we have, using (3.3) again,
\ ~ q,i, [0,1] ,j/~[ < [Q11/2c [0,1]~
< l(Q)lQIlClxQl(n+l). Assume that Q = Qk,v.
Since 2Qfq[0,1] ~ = o , we must have [vl > 2 k. The
appropriate part of the sum in (3.5) is therefore majorized by +oo
+oo [vi(n+l) ~ E 2k(ne1)< +00,
E 2k2kn(1e)2k(n+l) E k=0
Ivi>2k
k=0
provided 0 < e < 1/n. There remains C a s e I V . l(Q) < 1 and Q c [3, 3] n. Let us first analyze the situation when Q C [3, 0] x [3, 3] n1.
R note that since (~0,1]n,/ is a linear combination To estimate /\ c L$ o LQ,i,O[o,1]",j}21, of the characteristic functions of the "children" of the standard unit cube, it is enough to control
50
for an arbitrary fixed "child" Q' of [0, 1]n. The first possibility is that the boundary of Q has no common points with the hyperplane {x 1 = 0}. Then we may use (3.3) in (3.8) combined with the fact that
Ix  XQI ~ [zlQI for x E [0, 1]~ to dominate the integral by a multiple of
l(Q)[QIm
r(n+l)r~ldr =
t(Q)IQImlziQVI.
Hence, the contribution to the sum in (3.5) from this part has the upper bound +c~
~_r 2kn(1/2e)2k2kn/22k k=0
 2 k+l
~
too
~
Iv, 11 ~ ~
32k
k2 k('"') < +oo,
k=0
i~2,...~n
forl/n>e>0. The second possibility is that Q is adjacent to the hyperplane {x 1 = 0}. Let us write Q'  Q~ (3 Q'2 with Q~  2Q N Q' and Q~ = Q' \ Q1 On the Q~ part we still use (3.3) to majorize the integrand in (3.8). This and the fact that Ix  XQI > l(Q) show that
/Q, cLo~.,i(x) dx < I(Q)IQ[ '/2 f, Ix  XQV(n+l)dx JQ~
< l(O)lQI1/2
Q)
r'~lrnldr ~ IQI~/2.
For each fixed k > 0 there are 0(2 k(n1)) dyadic cubes which fit into this case, and ~or
{oo
2kn(1/2d2k'~/22 k(n1) ~ k=0
2 k(1~r
< +oo,
k=0
for 1/n > e > 0. Hence, we conclude that this part of the sum in (3.5) satisfies the right estimate as well. As for the Q~ part in (3.8), using (3.4) and dist (x, OoQ) >_ x 1 > O, we find
dist (x, OoQ) dx ~ IQI U2
< ]Q]I/21(Q)'~I
log ~~ dx
f Z(Q) log ~~c,~ dx 1 = IQI1/20(I(Q)'~I+'~), dO
51
since, for any a E (0, 1), [t(log(1/t) + 1)[ ~ t ~ uniformly for t C (0, 1). Choose a such that ne < a < 1. Then +c~
E
+~
2kn(U2~)2kn/22k(nl+~)2k(n1) = E
k=0
2k(~n~) < +oo.
k=0
We have now finished the proof when Q c [3, 0] • [3, 3] n1. Next, we use the invariance of the boundedness of
Q
( C 5 0 Q , i , XQ' )E
under translations, permutations of coordinates, and symmetries with respect to the coordinate axes. This allows us to reduce the problem to the one we just finished, whenever Q and Q' have disjoint interiors. In fact, the only remaining possibihty we need to check in Case IV is when Q _c Q' = [0,1/2] n. On account of Lemma 3.6 we have xE0,,/21)
=
L L XR"\2[O,1/2]n)E ~,~
L X2[o,1/2],~\[o,1/2],~)E =: I + II. ] /\ c Lti( ~ Q,i,
(3.9)
Now II can be estimated by decomposing 2[0, 1/2] ~ \ [0, 1/2] ~ as an union of finitely many cubes which are disjoint from Q and for which we can apply the argument described in the preceding paragraph. To estimate I in (3.9), let BR be the ball of radius R and centered at the origin of R ~. We have that (C 5L0 QL, i ,
XRn\2[0,1/2]n )E
=
lira R~oo
= lim
R~oo
L XBR\2[O,1/2]n) E (C~L OQ,~, L (OQ,i, C~R XBR\2[0,U2]n )~
lim I"~I{OL'i' C~R XBR\2[O,1/2I  CRxBR\2[O,1/2] n(xQ))~. . = R.~ oo Since I~C~XBR\2[O,1/2],(X)I ~< 1 for x e [0, 1/2] n, j = 1, 2, ..., n, uniformly in R, we may bound the last limit from above by ]Q]U21(Q). Each ~'k contains O(2 kn) dyadic cubes inside [0, 1/2] ~ and +c~
+cr
E 2kn(1/2e)2~'~/22k2kn = E 2  k ( 1  ~ ) < +OC, k=0
k=0
52
for 1/n > e > 0. This finishes Case IV. The proof of (3.5) is therefore complete, hence so is the proof of Theorem 3.1.
9
A more careful account of the way the Lipschitz constant II V gilLoo
Remark.
intervenes in the upper bound for the operator norm of H L and H a on L 2 reveals that this proof actually gives
IIHLlIop, IIHnllop ~ const (1 + II V gllL~) 8. w
THE CLIFFORD ALGEBRA VERSION OF THE T(b) THEOREM
In this section we shall briefly discuss the Clifford algebra form of the T(b) theorem ([MM], [DJS]) together with some of its immediate extensions and corollaries. Other versions and applications (apparently the first ones) can be found in [Sel,2]. See also [Dav2] and [GLQ]. More specifically, consider the Clifford bilinear forms
b, := fRn fl(x)b~(x)f2(x) dx,
: 1,2,
defined for C(m)valued functions fl, f2, where we assume that the functions bl, b2 are pseudoaccretive on I~" and Clifford vectorvalued. We introduce the right Clifford modules biD(m ) := {bif; f E D(m)}, i = 1,2, where, as usual, 7) stands for the class of smooth, compactly supported functions on R n. Similarly, we define 7)(m)bl, i = 1, 2. Consider T a continuous morphism of rightCliffordmodules from bl:D(m) into (:D(m)b2)* (the duality sign refers to the leftCliffordmodule structure of 7)(re)b2), and say that T is associated with a standard kernel if for some Clifford algebra valued continuous function K(x, y) defined for x # y, x, y E R n, and for some number 5, 0<5_
onehas:
(i) IK(x,y)l <~Ix  YIL for x # y;
(2) IK(~, y)  K(~, Yo)L+ IN(y, ~)  K(yo, ~)l ~< lY yol~l 9 yln~, for all ~ # y with lY Yol < 89 (3) T(bl~O)(r
~l;
: IIR~•162
y)bl(y)~o(y)dxdy for any ~o, r e T)(m)
with disjoint supports.
53
By analogy with the previous usage, we write
T(bl~p)(r
= (r T(blqo))b.~.
T t, the transpose of T, is the unique continuous morphism between the left Clifford m o d u l e s / ) ( , 9 6 2 and (bl:D(m))* such that, for any ~, r E :D(m), one has
(r T(bl~)}b2 = (Tt(r
~O)b~
Note that T t is associated with the s t a n d a r d kernel K(y, x). Introducing the change of variables operators A~,tf := tn/2f(( 9 x)/t), for x E II~'~ and t ). 0, we say t h a t the T has the weak boundedness property with respect to
bl and b2 if the operators .A~,tb2Tbl.A~,t 1 axe bounded from :D(m) to D~(m) uniformly in x and t. If T is associated with a s t a n d a r d kernel, we can define T(bl) as a Clifford leftlinear flmctional defined on the space
by setting
T(bl)(r
:= (T(blT), r
+ JJR~•
r
y)  K(xo, y))bl(y)(1  ~(y)) dxdy,
for some ~ E T~, ~  1 in a neighborhood of s u p p r
Likewise, we define Tt(b2).
We are now almost ready to state the Clifford algebra version of the T(b) theorem. To this effect, recall that a locally integrable function f belongs to BMO(]~ n) if
sup
If(x)  fol d~ < + ~ ,
O where the supremum is taken over all cubes Q in ~n, and fQ is the integral mean of
f onQ. 54
T h e o r e m 3.8. With the above notations, T associated to a standard kernel has an
extension as a bounded operator on L2(R'~)(m) if and only if (1) T(bl) E BMO(I~n)(m); (2) Tt(b2) e BMO(Rn)(m); (3) T has the weak boundedness property with respect to bl and b2. S k e t c h o f P r o o f . One can readily adapt the proof of the necessity of (1)  (3) from the scalar case (PeetreSpanneStein; see e.g. [Jo]) to this somewhat more general context, therefore we consider the sufficiency. In fact, it is enough to treat only the special situation when T(bl) = Tt(b2) = 0 (in which case, the theorem is actually due to McIntosh and Meyer [MM]). The general case, T(bl), Tt(b2) e BMO(Rn)(m) reduces to the previous one by subtracting off some paraproductlike operators. More precisely, setting 171(f)
:=
E k
AL,21TIb ~EL, 1 iblt~ k ~, k 1)) klk 1 J)
and //2(f) :' E
t (b2))Ek_l(b L,2 21 f), /~kL , l(T
k where AL'i ~ k and E kL'i are defined as in section w
with respect to bl, i = 1,2, we
would like to conclude that the operator S := T 
171  172t falls within the scope
of the special case we have mentioned first. It is not difficult to see that 171 and [I2 are bounded operators on L 2 and that S(bl) = St(b2) = O. The only problem is that this operator will not be associated with a standard kernel, but the logarithmicrate blow up near the boundaries of dyadic cubes is good enough to control it in the same fashion as before (cf. also [Day2]). Returning to the case T(bl) = Tt(b2) = 0, let us consider the systems of Haar Clifford wavelets {~Lk, r
ftx/L 1 associated to the pseudoaccretive and "t j,k, II/R j,k~J,~ In order to estimate (f,T(blg))b~, we
functions bl and b2, respectively (see w write R
L
(~j,k)blCj,k,
f = E(f, j,k R
j,k
55
L
and then formally expand
blCj',k R j,k '
(3.10)
j,k j',k' where Cj~.;~k, := <~Lk, T(blkO~,k,))b2. Using the properties of these dual pairs of wavelet bases one can produce estimates for ICJ.;~k,I analogous to the ones established in the previous section, so that Schur's lemma allows us again to conclude that the infinite matrix
2 is bounded on gNxN  C(m)" The boundedness of T on L2(Rn)(m) follows then from
(3~i0).
[]
Remarks.
(A) Another equivalent way of expressing the conditions ( 1 )  ( 3 ) in the above theorem, which more explicitly emphasizes the translationdilation invariance of the theory, is
fB s k(m,y)bl(y)dydy
(3.11)
s f. b2(x) (x,y)ex ex
(3.12)
IBI,
uniformly for all balls B C IRn (cf. also [Me]). (B) Theorem 3.8 admits various extensions some of which we shall use in the sequel. For further reference, we note that any operator T associated with a standard kernel which is (extendible to) a bounded operator on L 2 is in fact bounded on any L p (and even on LP~, with the weight w in the Muckenhoupt class Ap), for all 1 < p < oc. Also, when suitably interpreted these results continue to hold for 7/(m)valued standard kernels, where 7 / i s some HAlbert space. Let us illustrate these ideas by considering a generalization of the usual Cauchy kernel on a Lipschitz hypersurface analyzed in the previous section. Although valid in more generality, we restrict ourselves to the version needed later. Let E : {(g(x), x) ; x E Rn}, for some realvalued, Lipschitz continuous function 9 on R n. For 0 < a < ~  arctan (H V gllL~), take Fa to be the upright circular cone in the upperhalf space N~_+1 having overture a and whose vertex is at the origin of
56
the system. Also, let 7/ be a Hilbert space and K(X, Y) be a "//(,)valued function defined for Y  X ~ F~, satisfying
IIK(X,V)ll{n) < tx  V l ~,
(3.13)
for V  X r
and having the property that for any h E 7/there exists e = e(h) > 0 such that, for any X C R n+l,
D(K(., X), h) = 0 on IR"+1 \ (  F a  e + X),
(3.14)
(K(.,X),h)D = 0 on 1~,~+1 \ (ra + e + X).
(3.15)
Let us make the notation L~(~, wdS) for the Banach space of (classes of) measurable functions on ~ which are 7/(n)valued and LPintegrable with respect to the weighted surface measure wdS. T h e o r e m 3.9. If the kerne/K(X, Y) of the integral operator
Tf(X):=p.v. f
/(Y)K(X,Y)dS(Y),
satis~es (3.13), (3.I4) and (3.15), then 7 is bounded on n ~ ( r ,
X e ~,
~dS)
for any 1 < V < o~
and any w CAp. In fact, as it will become more apparent in the next chapter, the continuity of the operators of the type described above can be nicely expressed in terms of some weighted Hardy spaces of monogenic functions, 7/~(f~), naturally associated to ~/, the domain in ]R'~+1 lying above ~ (see w
for precise definitions). More specifically, the
following holds. T h e o r e m 3.10. With the above hypotheses, for any h ~ 7/{,.), the operator
Tf(X) :=/z(h,f(Y)K(X,Y))dS(Y),
X E f~,
maps L~ (E, wdS) boundedly into H~(fl). For kernels of the form K(X, Y) = ~ ( X  Y), with ~(X) right monogenic and satisfying [~(X)I < IX[ n in IRn+l \ Fa, direct proofs of these results can be found
57
in [LMS]. We shall present here an alternative argument based on an idea of Meyer [Me] which utilizes Theorem 3.8. Proof of Theorem
3.9. To see that
K((g(x), x), (g(y), y))
is a standard kernel, it
suffices to show that for all h E 7/(n), Ilhll(~) = 1,
IVx(K(X, Y), h)I <~IX  yI .1, VyK X,Y in
uniformly for X, Y c E (the estimate for
is completely similar). Fix h E 7/(n )
with Ilhll(.) = 1, two disjoint points
E, and set d := l d i s t ( Y 
X, Fa).
Cauchy's integral representation formula (1.10) gives
IVxl ~< __f,xzl=~ NK(Y  Z, Y)]i(~)i(vE)tY  X  Z)i dS(Z) <~ IX  YI  n  l ,
by (3.13), since [ V E[ ~< d  •  1 and IZI ~ IX  YI ~ d on the contour of integration. Let us now prove that for any surface bail B, which without any loss of generality is assumed to be of the form Br := E rG B((g(O),O);r), one has
fB~ IB. n(Y)(K(X'Y)'h}dS(Y)
dS(X) ~ dS(Br), (,~)
(3.16)
uniformly for h E 7/(n) with Hhi[(,~) = 1. To this end, using Cauchy vanishing therorem (1.12), we can deform the contour of integration in the innermost integral to tgCr \ E, where C, is the cylinder Cr := {X + t ; X E Br, 0 < t < r}. Therefore the lefthand side of (3.16) can be majorized by
iB~ i~+B~ I(K(X' Y)' h),dS(X)dS(Y)+ IB~ iv ,(K(X, Y), h), dS(X)dar(Y), where V is the "vertical" part of
OC~ and dar its
canonical surface measure. Let
I, II
respectively denote the two terms from above.
Y), h)l <~ilK(X, Y)lt(,O <~iX Y[~, we have that the ~ r n ~ dS(B~) 1, hence I <~ dS(B~). Also, by an appropriate
Using I(K(X, in I is
integrand change of
variables,
ss <
i.S. 
1
,.\B. Ix YI"
dxdy <~r n
(!yleil
<1~t<2 I~  i YI" :~ER n
58
dxdy ) ,
where the last estimate was obtained by projecting ~ onto R n. Since the rightmost integral is finite, the conclusion follows.
9
59
Chapter Hardy
4
Spaces of Monogenic
Functions
In this chapter we study the Clifford algebra version of the weighted Hardy spaces on Lipschitz domains 7tP(ft), 1 < p < oo, w CAp, introduced by Kenig in [Ke] (cf. also [CMM], [GM12]).
Although presented in a somewhat more implicit form,
such a monogenic H P  t h e o r y also appears in the work of Li, Mclntosh and Semmes [LMS]. See also [Sel], [Se3]. To motivate this extension, let us recall that for
n/(n

1) < p < oo, say, the
classical H p space for the upperhalf space (SteinWeiss [SW] and FeffermanStein [FS]) is the collection of all systems of conjugate harmonic functions u = (u 3)j=0 on ~_+1 so that
I1~11/~,:= sup t>0
t}xR n
I~(x)lPd~
<
+~.
By Proposition 1.7, this is the same as saying that the Ilia+lvalued, monogenic is uniformly LPintegrable on hyperplanes parallel to
function
F
0 I ~ +1.
Thus, it is not artificial to consider spaces of arbitrary Clifford algebra
= uo 
~~i u j e j
valued, monogenic functions on Lipschitz domains, subject to the same type of growth restriction as above.
Actually, these monogenic Hardy spaces naturally arise in
connection with several other important problems like, for instance, the boundedness, regularity and boundary behavior of Clifford algebra valued integral operators, to mention only those of concern for us in this chapter. The layout of the chapter is as follows. Several alternative descriptions of 74~P(f2), including maximal function and square function characterizations, are presented in w
and w
context in w
The classical Riesz theory on boundary behavior is extended to this Finally, in w
we indicate how one can measure the regularity of
the higher dimensional Cauchy integral and related operators in terms of these Hardy spaces.
60
w Let g : ~n
MAXIMAL FUNCTION CHARACTERIZATIONS
> R be a Lipschitz function and denote by E C ~n+l its graph.
As usual, we assume that ]Rn+l is embedded in the 2ndimensional Clifford algebra R(n) Set 12~=for the domains in ~n+l which lie above and respectively below E. For brevity, we shall sometimes write ~ instead of 12+. Let n be the outward unit normal of 12, defined d S  a l m o s t everywhere on E, where We introduce
dS is the surface measure.
the higher dimensional (left and right, respectively) Cauchy integral
operators by 1/z YX cL f ( x ) : an f(Y)n(Y) iy _ Xin+ 1 dS(Y),
X E R n+l \ E,
CRf(X) := c~~ 1 ~ IY Y X ~ +ln(Y)f(Y) dS(Y),
X E 11~.n+l \ E,
where f is a Clifford algebra valued function on E. For an arbitrary application F : fl•
+ C(,~) we define its
nontangential
maximal function tifF : E   + [0, +oc] by
N'F(X) :=
IF(Y)[,
sup
X e E,
YEX=I=P.
where P~ stands for the cone {(x0, x) E IRn+l;x0 > Ixltana}, for a fixed a e (0,~) having HVgll~ < [0,
tana. We also introduce the radial maximal functions Frad : E +
as
Srad(X) := sup I F ( X + 5)1. ~>0
For 0 < p < oc and w a nonnegative, locally integrable function on E, we set
IIFII B := sup
5>0
IF(x =t=6)[Pw(X) dS(X)
}1,,
and define the weighted Hardy spaces of monogenic functions
7/P~(gl+) : {F left monogenic in ~ + ; IIFIIT/~< +c~},
(4.1)
]CP(12+) : {F right monogenic in l~+; ]IFII?~ < +r
(4.2)
61
Recall that w EAv, the Muckenhoupt class, if
sup ( 1 [ ~ d x )
( 1 /Q
x
,~p1 <
where the supremum is taken over all cubes Q in ~n and g(x) := w((g(x), x)), x E I~". For the basic properties of Ap weights the reader may consult the excellent exposition in [GF]. Our first result collects several equivalent characterizations of these Hardy spaces. T h e o r e m 4.1. Let 1 < p < oo and w E Ap. For a left monogenic function F in ~,
the following are equivalent: (1) F e 7/~(~);
(2) There exists f E LP(E, wdS)(,~) such that F is the (right) Cauchy integral extension o f f , i.e. F ( X ) = CRy(X) for X in ~; (3) .MF E LV(E, wdS); (4) Frad E LP(E, wdS); (5) F has a nontangential boundary limit F+(X) at almost any point X E E, i.e.
there exists
F+(X) :=
lim
YEX+F~, Y~X
F ( Y ) for a.e. X E E,
and F is the (right) Cauchy integral extension of its boundary trace. In addition, if any of the above conditions is fulfilled, we also have
][FII~E ~ IINFIILE ~ I[F+IIL~ ~ IIF~adIILE" Remark.
(4.3)
Obviously, analogous results are valid for right monogenic functions, for
functions defined in ft_, and even for 7/(n)valued monogenic functions in 12• where 7/is some Hilbert space. P r o o f . The nontrivial implications are (1) =v (2) ~ (3) ~ (5) =v (1). To deal with the first one, we need the following lemma.
62
P L e m m a 4.2. Let F ~ 7~(f2) and set F~ := F(. q 6) in f2  6, for ~ > O. Then
F~ = CRF~. Accepting this for the moment, we may use Alaoglu's theorem for the bounded sequence {F~}~ in LP(E, wdS)(n) to extract a subsequence, which we denote by the same symbol {F~)& which is weakly convergent in LP(E, wdS)(~) to a certain function f C LP(~, wdS)(n). Now, if we fix X E f2+ and let ~ tend to zero in the equality
1 f~ [y Y  X F~(X) = ~ X•+l n(Y)F~(Y)dS(Y), we obtain F(X) = CRf(X), as desired. Next, we consider the implication (2) ~ transforms from w
(3).
Recall the truncated Hilbert
and the usual HardyLittlewood maximal o p e r a t o r . ,
1 N E) /B r(x)ns [f(Y)l dS(Y), f*(X) := supr>0dS(Br(X)
X e ~.
It is a wellknown fact that 9 is a bounded mapping of LP(E, wdS) for any 1 < p < oc and any w CAp (cf. e.g. [Jo], [GF]). L e m m a 4.3. With the above notations, one has
Af(CRf)(X) < suplH~f(X)l + f * ( X ) ,
(4.4)
e>O
uniformly in X E Y~, and f E LP(~,wdS)(n). Accepting this result and recalling Cotlar's inequality
sup IH~f(X)I < (HRf)*(X) + f*(X) e>0
(cf. e.g. [Jo]), we see that if F = CRf in f~ for some f E LP(~,wdS)(n), then
IWFIIL~ = IW(CRf)IIL= < IIIIIL~ <
+~
(4.5)
by the LP~boundedness of the HardyLittlewood maximal operator and the Hilbert transform. Note that the above reasoning also takes care of (5) ~ (1).
63
Finally, we prove that (1)(4) imply (5). Using (2) and following a classical pattern (see e.g. [Jo]), we must prove the boundedness of the maximal operator associated with this type of convergence (i.e. the nontangential convergence to the boundary) together with the almost everywhere pointwise convergence for a dense suhspace of LP(E, todS)(n). The first part is implicitly contained in Lemma 4.3. As for the second part, take f to be Lipschitz continuous, compactly supported on E, so that for any e > 0 we have lim YEX+F~ Y> X
C R f ( Y ) : Yex+r. lim 1 ~ cr~ zl>_e[Z Z y ~ +an(Z)f(Z) dS(Z) YrX
+
lim 1 [.. Y~x+r~ a,~.'lazl
Z  Y
IZ YF+I n( Z) f (Z) dS( Z).
Y~X
Since f is Lipschitz and since
[ Y  XI < I Y  ZI and I z  x I < d i s t ( Z , X + F ~ ) ,
(4.6)
uniformly for X, Y E E, and Z E X + Fa, we get
IzZ 4
 s(r)) ds(z) <~[
alX Zl
Iz  Yl"lf(Z)  f ( r ) l dS(Z) < ~11V/IIL~,
hence this part goes to zero as e ~ O. On the other hand, by the monogenicity of the Cauchy kernel,
~n
ZI<~ [Z _
+In(Z)dS(Z) :
an
Z[=~ IZ~K]~+lIg   X l ZE~2_
and this last integral tends to 1 for a.e. X E E if we first make Y + X and then e~
O. Summarizing, we have shown that lim
Cnf(Y) = ~ { f ( X ) + HRf(X)},
YEX+Pa Y~X
64
for a.e. X E E.
(4.7)
This proves the first part of the assertion in (5), i.e. the a.e. existence of
F+(X).
Since A f F 9 LP(E, wdS), using Lebesgue's dominated convergence theorem one can readily see that the sequence {F,} 5 introduced in Lemma 4.2 converges in L~ to F +. Finally, letting 5 go to zero in F, =
CnF,, we obtain that F can be recovered as the
(right) Cauchy integral extension of its boundary trace. Finally, to get (4.3), just use (4.5)
IIF+IILE
IIFII E <IIF: dlILE < IIHFIILE = liH(CRF+)IILE
IIF+IILL 9
Hence, modulo the proofs of lemmas 4.2 and 4.3, the proof of Theorem 4.1 is complete.
Proof of Lemma
4.2.
Clearly, it suffices to prove the statement corresponding
to 5 = 0, assuming that F is left monogenic in a neighborhood of ft. To this end, let us first introduce some notation. {(t,x);
For r, s > 0, let C~,~ denote the "cylinder"
g(x) < t < g(x) + s, Ixl < r} and set E~ for E f/C.... Now fix~X 9 f~ and
choose r, s large enough such that X 9 C~,,~. As F is left monogenic in a neighborhood of the closure of
C~,s, Cauchy's reproducing formula gives
+ ~1
Jr~~ iy2 Y  X +~s ~1;~+1n(Y)F(Y + s) dS(Y)
+l
~noC.+~o ~ Z  X + t ,
Iz
= : I + II + III,
Z ' F ( Z + t ) dtd#~(Z) (4.8)
where Z' is the projection of Z E ]~n+l onto {0} • ]~n and #~ is the canonical measure on C~,or Since
I ~
CnF(X) as s, r
~ oc, all we have to check is that both II, III
converge to zero when s, r tend to oo. A simple application of HSlder's inequality gives that, if
lip + 1/q = 1, then
IlI[ <~~ [ X  Y  s [  n l F ( Y
+ s)IdS(Y)
65
The rightmost factor from above is comparable with ~~)q" dy, fR. (s ~~q/P(Y) where x, y denote the projections of
(4.9)
X,Y, living in ]RTM, onto IRn identified with
{0} x IRn. Estimating (4.9) is fairly standard. Since ~ E even
A v, we get ~q/P E Aq and
"~q/P E Aq_~, for some e > 0, and decomposing the domain of integration as
a disjoint union of annuli of the form {y;
kt < Ix  y[ < (k + 1)t}, where k E N,
an elementary calculation reveals that this integral is
O(s e(w'p'n)) as s
~ oo (see
also [GF], [To]). Using this and the fact that F E Nv(f~), the conclusion is that
[II[ = O(s ~) for a certain small positive e, uniformly in r. The idea to estimate somehow.
III is to make the "vertical" part of 0C~,8 "vibrate"
More concretely, taking the integral average of (4.8) over the interval
[r, 2r], we obtain
F(X)=rj~
Idr+r
The first integral still converges to
Ildr+r
IIIdr.
CRF(X), while the second one is O(s~). To
estimate the third one we need a result of a geometric nature. L e m m a 4.4.
dS ..~ dr  d#r in the sense that they are absolutely continuous with
respect to each other and the RadonNikodym derivative is (essentiMly) bounded by some finite, strictly positive constants from above and from bellow, respectively. Taking this for granted for a moment, we can write
<1 ~ dS(Z)) 1/p " r fo (f~ 'F(Z +t)lpw(z) /
.(:,.
z[_>l (t [I~  zl)q"
Arguing as before, the product of the inner integrals is L ~ in the variable t E ll~+ so that, we finally get
~f2~lIIdr
66
<~
Letting r first, and then s, tend to oo, we conclude the proof of the Lemma 4.2. To prove L e m m a 4.4, for any regular Borelian measure ~ on E, let z*(~) be the measure defined by
z(x)
z*(L,)(E) = ,(z(E)),
for measurable sets E C ]R" (recall t h a t
: = (g(x), x), x 9 tRn). Clearly, it suffices to check t h a t
First,
z*(dS)
= (1 + ] V
g]2) 1/2dx ~ dx = dr  dw~
z*(dr  dp~) .~ z*(dS).
where
dw~
measure of the sphere of radius r centered at the origin of ]Rn. local paraaneterization for this sphere, so that
h(t)
:=
is the surface Let
(g(f(t)),](t))
f(t)
be a
becomes a
local parameterization for E n C~,ao. It is not difficult to see t h a t z*(dr  d#~) {det
( {Oj h, Okh})j& } l /2 dr  dt
dr  dwr ~
and
( {Ojf , Okf })j,k } l /2 dr  dr,
{det
where
(, .} stands for the usual inner product in IR". To conclude, we need to prove that these two determinants are comparable. Set A for the n x (n  1) m a t r i x having the vectors Since
Off,
j = 1,2, ...,n, as columns, and B for V g viewed as a 1 x n matrix.
{Ojh, Okh} = (Ojf, Okf} + {Vg, Ojf}(vg, Okf},
in question become
AA t
and
AA ~ + (AB)(AB) t,
the matrices of the determinants
where the superscript t denotes the
transpose. Some elementary linear algebra gives us t h a t
det
and since ldet
A(I + BBt)A ~ =
(I + BBt')[
det
AA t det (I + BB~),
~ 1, the proof is complete.
9
Finally, we return to Proof of Lemma
4.a.
In the next calculation we fix an arbitrary point X E E,
an arbitrary point Y E X + Fa, and set e := IX  Y[. Using (4.6), we have t h a t
l o b ( v )  89
I is comparable to
1J~lx _zi<elf(Z)ldS(Z) + /Ix <~ 7g
_zL>l](z)lI
The first integral above is clearly majorized by
i z ,Zyl Y ,+l
f*(X)
ZX IZ_Ty~+ 1 [ dS(Z).
which is of the right order.
Finally, use the meanvalue theorem to dominate the second integral by
II(z)l Ixzlo+' es(zl, 67
which in turn is further estimated as
cx)
s
/ [f(Z)l IX _ ~in~ 1 dS(Z) k=OJ2ke<[XZl<2k+le oG oo If(Z)IdS(Z) ~ ~ 2kf*(X) ,~ f*(X) Z 2k(n+l)e'n f k=0 JIXZI<e2k+I k=0 and the proof of the lemma is complete.
9
Exercises. 9 Using the mean value property of F, prove directly the pointwise estimate A l p ( x ) < (Frad)*(X), uniformly for X E E. 9 Prove that the inclusion operator of 7/P(~) into the space of all left monogenic functions on 12 (endowed with the topology of uniform convergence on compact subsets) is compact. 9 Prove the bounded domain version of Theorem 4.1. 9 Part of the Theorem 4.1 also continues to hold in the case p = 1, w E A1. More precisely, for F left monogenic in f) consider the following assertions: (1) A f F E LI(E,wdS);
(2) Frad e LI(E, wdS); (3) F has a nontangential limit F + at almost any point of ~, F + belongs to
Ll(Z, wdS)(n) and CRF + = F; (4) There exists f e Ll(E, wdS)(n) such that H R f E LI(E, wdS)(,~) and F = CRf in ~. Prove that (1) ~
(2) ~ (3) =~ (4). The implicatlon (4) ~
(1) appears to be still
open for general Lipschitz graphs. 9 Prove that for any function F which is left monogenic in 12 and has A f F E L I ( ~ , wdS), where w E A1, there holds the cancellation property f~ n F + dS = O. As the next theorem shows, these weighted Hardy spaces also turn out to be the right ranges for the higher dimensional Cauchy operators acting on Lp(F~, wdS)(,~). T h e o r e m 4.5. For 1 < p < oo and w a nonnegative, locally integrable function on ~,
the higher dimensional Cauchy integral operator Ctr maps LP(~, wdS)(n) boundedly
68
onto TiP(f2) if and only if w belongs to the Muckenhoupt statement is valid for CL also.
c/ass
P r o o f . We have already seen in the proof of Theorem 4.1 that bounded operator mapping
LP(~],wdS)(n)onto
Ap. An analogous
Cn
is a welldefined,
7~P(Q).
As for the converse, let us assume that, for some nonnegative locally integrable function w on E, Cn :
LP(E, wdS){n)
> 7/P(f~) is welldefined and bounded. Recall
that, for any function f which is e.g. Lipschitz continuous and compactly supported on E, one has
lim CRf(X + 6) = 21{f(X) + ~~+0 If we now set
F,i(X)
:=
CRf(X
HRf(X)},
for a.e. X E ~.
+ (f), 5 > O, we infer that
sup IIF~IILG = [[CRfllng ~ IIflI/G, 8>0
so that, an application of Fatou's lemma gives
It(Z+ Therefore
Hn(~. ).
HR)f[IG~ li~ni~fIIF~IIL~~
H R : LP(~,wdS)(n) ~
LP(Y.,wdS)(n)
[IfllG.
is a bounded operator, hence so is
For the remaining part of the proof we follow Coifman and Fefferman [CF].
Fix two cubes Q, Q' in IRn having the same sidelength I and such that dist (Q, Q') = l, otherwise arbitrary. Also, set Q, := Now, for any f E LPomp(E,
wdS)
z(Q)
and Qt, :=
z(Q')
where
z(x)
:= (g(x), x).
with supp f C Q, and any X E Q~,, we have
[HR(~f)(X)[ ~ fQ. [f(Y)IIX  Yt '~ dS(Y) >~iQIl fQ If(x)l dx, where f is the composition of f with z. Consequently,
fQl
f[ p~dz ~ fQ ]fl po.'dS ~
II/II~G~ IIHR(~f)tI~ (4.10)
69
Making / = XQ in the above inequality we obtain ~(Q) > U(Q'), hence u(Q) ~ ~(Q'), by symmetry. Plugging now f = ~ v~l in (4.10), a direct calculation gives that
i.e. ~ 6 Ap.
9
Before we conclude this section, it is important to point out that, for the upperhalf space case, a substantial part of Theorem 4.1 also carries over to the range (n  1)/n < p < 1. More precisely, we have the following. T h e o r e m 4.6. Let (n  1)/n < p <_ 1 and w 6 A ~,v . Then, for a left monogenic n1
function F in R'~+1, the following are equivalent: (1) F E qlP , ~ + {~nq 1~ ,, .
(2) A f F 6 LP(R~,oJdx); (3) Era d 6 Lp(Rn,wdx). In addition, if any of the above conditions is fulfiled, then F has a nontangential limit F + ( X ) at almost every point X 6 R n, and
IIFII~ ~ IIXFrlL~ ~ IIF+IIL~ ~ IlF~dllL~Moreover, 7lv~( I ~ • ) is a complete metric linear space when endowed with the distance (F, G) ~
[]FGIlPin, and the embedding TlP(R~ +1) r
C~(R~+I)(,), where we have
endowed the later space with the usual topology, is continuous. As expected, a similar result is valid for right monogenic functions. Exercise. Prove Theorem 4.6. Hint: Recall a basic lemma due to Stein and Weiss ([SW2]), namely that if F is left (or right) monogenic, then IF[ ~ is subharmonic for any c > (n  1)In. w
BOUNDARY BEHAVIOR
Let B L be the (nontangential) trace operators mapping functions F from 7/P(~+) into their nontangential boundary limits F • 6 LP(E, a;dS)(n). Similarly, we define B~R acting on/Cv~(~+). Note that, according to Theorem 4.1, these operators are welldefined. The next result, whose proof is implicitly contained in the previous section, describes the way these trace operators and the Canchy operators link up.
70
T h e o r e m 4.7. For 1 < p < cx~ and w 9 Av, the following hold:
(1) B~: K:P(a+) ~ LP(E, wdS)(~) are bounded operators; P (2) c L B R, = I on lQo(~2+); (3) ~ c L = ~(+I + n L) on LP(~, ~dX)(~) (elemeIj formulae; cr. [It1); (4) Y~C L = I on Im 13R; (5) KerC L = Im B~. Similar results are valid for I3L, , too.
In the sequel, it is useful to have intrinsic characterizations of the spaces Im BL and Im B~. Using (2) and (3) above, simple considerations show that in fact one has the following descriptions. Corollary 4.8. We have Iml3 L = { f 9 LP(E, wdS)(n) ; H L f = i f } , I m B R = { f 9 LP(E, wdS)(~) ; g R f = + f } .
It is then justified to introduce the following notations 7/L(E,p,w) := {f 9 LV(E,wdS)(~) ; HL f = + f } ,
7/~(E,p,~) := {f 9 LP(E,wdS)(~) ; g R f = i f } . Theorem 4.7 has also some immediate consequences which deserve to be stated separately. Corollary 4.9. With the same hypotheses as in Theorem 4.7 one has: (1) H L H L : H n H n = I on LY(E,wdS)(~); (2) ( H L f , f')~Z = (f, H R f ' ) E , for f 9 LP(E, wdS)(n) , f ' 9 Lq(E, wdS)(n) , where l i p + 1/q = 1, w := w q/p 9 Aq, and the pairing (., .)~ defined by
(f, f')~ : = / ~ f ( X ) n ( X ) f ( X )
dS(X).
(4.11)
(3) L~(r~,~dS)(n) = n~(r~,p,~) ~n+R(z,v,~) and /.Y(E, wdS)(,0 = 7tL (E, p, w) @ "//_R(E,p, w), (Calder6n's decompositions);
(4) nL. (r~,;,~) _~ ns177
u.R(r~,p,~) ___~:s
Next, we present a duality result.
7t
Proposition
4.10. If 1 < p, q < co are conjugate exponents and a; 9 Av,
w := w q/p 9 Aq, then we have the isomorphisms R
*
Here * refers to the corresponding left or right C(,,)module structure. P r o o f i Starting with an arbitrary functional r in 7l~(E,p, w)* and using the results described in the last part of Chapter 1, we see that there exists g 9 Lq(E, wdS)(n) such that r
= (f,g)2 for all f 9 7/L(E,p,w). Next, we use Calder6n's decomposition
(in Corollary 4.9) to write
g = g  r 9+ 9
q,
r n+'(r,, q,
so that, by (2) in Corollary 4.9,
(f,g+)E = (HLf, g+)E = (f, HRg+)E =   ( f , g + ) E ' = (f,g_)E.
Therefore (/,g+)r. = 0, hence r
Since the pairing (',')r~ is non
degenerate we have that the mapping r ~,~ g_ is welldefined and in fact this is an L isomorphism between 7/+(E,p,w)
*
and ~_R(E, q, w), etc.
9
Let T~L(f]+) and ~ R ( ~ + ) be the left and right, respectively, Clifford modules spanned by { E ( X )}XEf2:F (recall that E ( X ) := ~ X / ] X p +1 is the usual Cauchy kernel). E x e r c i s e . Prove the following Smirnovtype characterizations
7/L(E,p,w) = the L~  closure of {r12 ; r 9 ~L(y~• H~(F,,p, 0;) = the L v  closure of {r[r. ; r 9 7~n(f2+)}. Note that, in particular, T~L(fl+) ~+ ~ P ( f l + ) densely, etc. E x e r c i s e . Recall the usual Hardy space HI(]~"). Prove that { f 9 Ll(It~n)(n) ; H L f C LI(IR~)/,)} = { f E LI(R~)(n) ; H R f 9 Ll(~n)(,0 }
= Hl(I~n)(n).
72
w
SQUARE FUNCTION CHARACTERIZATIONS
The main results of this section give descriptions of the Hardy spaces of monogenic functions obtained in terms of the natural extensions of the classical Lusin areafunction and LittlewoodPaley gfunction to the Clifford algebra framework. T h e o r e m 4.11. Assume that 1 < p < oo and w E Ap. Then
\ P12
,~ 1Iv
and
la0F(X • t)12tdt)
w(X) d S ( X ) )
uniformly for F 9 7lP~(f~+). Similar results are valid for right monogenic functions. For an arbitrary Clifford valued function F defined in ~t+ we introduce the Lusin
areafunction by
A+(F)(X) :=
//X
\
[OoF(Y)[2IX  Y I I  ' d Y ) 1/2 ,
X 9 f~+,
and its radial analogue, the LittlewoodPaley gfunction
g+(F)(X) :=
(~0ooIOoF(X
• t)12tdt
)1/2,
X 9 12+.
Whenever clear from the context, we shall drop the subscripts 4. The key estimates nedded in the proof of the above theorem are formulated in the next lemma. L e m m a 4.12. For any 1 ( p ( oo and w EAv, one has
Ilf j: HLflIL~(E) ~ [Ig•
~ [IflIL~(~),
(4.12)
[If q HLfIIL~(E) ~ IIA•
~ IIflIL~(E),
(4.13)
and
73
uniformly for f 9 LP(E, wdS)(~). The proof of the above lemma is accomplished in several steps. The idea is to use the Hilbert space valued version of the Clifford T(b) from the previous chapter in a suitable context. Let us first deal with the areafunction. To this end, we introduce the space KS of (classes of) measurable functions h : Fa ~ C such that
Ih(Z)12lZ[1~dZ
Ilhll~ :
< +c~,
and note that if we consider the operator
Sf(X)(Z) := j ~ f(Y)(OoE)(X  Y 4 Z) dS(Y), then A+(f)(X)
=
I!S(fn)(X)lll~), x
9 ~.
X e E, Z C V~,
Thus, the above estimate for the
area function will be a consequence of the boundedness of the operator S from
LP(F,,wdS)(n) into L~(F~,wdS), the space of Ks(n)Valued, LPintegrable functions on E.
I n turn, by Theorem 3.9, this comes down to checking that the kernel
K ( X , Y ) ( Z ) := (OoE)(X  Y 4 Z) satisfies the conditions (3.13)(3.15). However, (3.14) and (3.15) are simple consequences of the monogenicity of the Cauchy kernel, and we are left with (3.13). By definition,
IIK(X'Y)II(") = ffr I(O~
Y 4 z)IZIZ]a~dZ'
X, Y E ~,,
ot
and since for Z G Fa w e have m a x {IZ[, IX  Y[} < IX  Y + ZI, w e get
I(00E)(X  Y 4 Z)I 2 ~< IX  Y 4 Z12(n+1) ~< (IZI 4 IX  YI)  2 ( n + l ) .
Now if Z = (t, z), projecting everything into (0, oo) x I~n, we are led to considering the integral
fo~176
tln
n (t2 + Izl 2 + IX  YI2) n+l dt dz
which, integrated first with respect to z and then with respect to t, is easily seen to do not exceed (a multiple of) IX  Y12". This concludes the proof of the boundedness of S.
74
The treatment of the corresponding estimate for the g  f u n c t i o n is essentially the same. More specifically, this time we take K: := L2((0, oc), t clt) and consider
7Lf(x)(t) := Oo(CLf)(X :t:t),
X E E, t > 0.
(4.14)
As g(f)(X)  [ITLf(x)]1(,) and since the kernel associated with the integral operator TL is seen to satisfy (3.13), (3.14), (3.15), the conclusion is once again provided by Theorem 3.9. To obtain the bound from below, we rely on a remarkable identity, however not surprising for the reader familiar with some elements of LittlewoodPaley theory. To state it, recall the Clifford bilinear form (., ")z introduced in (4.11). Also, let T ~ denote the operator introduced in (4.14) in which C/r is used in place of CL. Lemma w
:=
4.13. Let 1 < p,q < oc be conjugate exponents and let w E Ap,
w q/v E Aq. Then, for any f E LP(E, wdS)(,~) and any ff E Lq(E, wdS)(n),
we have
(TLf, T_Rf'}z t dt =  ((I + HL)f, f')E.
(4.15)
To see how this can be used to conclude the proof of Lemma 4.12, we write
I<(I + HL)f'f'>zI <~fO~ JZ ITZf(x)(t)llTfif'(X)(t)ltdtdS(X)
< I[7+LSIIL~(~,.~dS)IIf'ILL~. Taking the suppremum over all ff E Lq(E,wdS)(n) with IIf'IIL~,  1, we get (4.12). Finally, we note that the areafunction can be handled similarly or, alternatively, one can use the pointwise estimate g+(F) < .A:~(F) on E (see e.g. [St]). Next we return to the proof of Lemma 4.13.
Here we shall adapt an one
dimensional calculation from [DJS]. We proceed in a sequence of steps. S t e p 1. For any F E 7tP(f2) and any fixed t > 0, a0F(. + t) belongs to 7{P(t2). Moreover,
t(OoF)(X + t) 3 O, as t ~ cx~ or t ~ O, for i.e. X E }2.
75
(4.16)
To justify this, observe that, by differentiating the Cauchy reproducing formula
F ( X + t) = (CLF+)(X + t) with respect to t and then multiplying both sides by t, we get
t(OoF)(X + t) = ~ F+(Y)n (Y)t(OoE)(Y  X  t)dS(Y) (recall that E(.) is the Canchy kernel). Now since t(OoE)(Y  X  t) decays in fl like the usual Poisson kernel in IR~.+1, a wellknown argument gives It(OoF)(X + t)l < (F+)*(X), uniformly in X G E and t > O. Consequently,
sup Jt (OoF)(. + t)l L~ < HF+IIL~'
(4.17)
p
hence, in particular, (OoF)(. + t) E 7/o~(ft), for any t > 0. The first convergence in (4.16) is easily seen by using once again the Poissonlike decay of t(OoE)(X + t) in gt. More specifically, a routine estimate gives
]t(OoF)(X + t)l < [[F+I[L~
(fR . (]x tqw(Y)q/P ,~l/q, ~+1 d y ]l Yl2 + t2) rq
where x 6 ~n is such that X = (g(x), x). The last integral from above receives the same treatment as (4.9) so that, we finally get
]t(OoF)(X + t)l < t `"/q [~q/P( BI (x) )]X/qlIF+IILs, for some small, positive e. This estimate yields the first part of (4.16). The limit for t + 0 is a bit more subtle. First remark that, as a limiting case of the Canchy vanishing formula
(1.12),
f n(Y)(OoE)(YXt)dS(Y)=O,
XeE,
t>O.
Using this, one can easily check that, for all X E E,
f f ( Y ) n ( Y ) t(OoE)(Y  X  t) dS(Y) = 0 J~ 76
if e.g. f is Lipschitz continuous, compactly supported on E (see Stein [St] p.6263). Moreover, once again due to the Poissonlike behavior of t(OoE)(X + t) on f~, sup f f(Y)n(Y) t(OoE)(Y  X  t) dS(Y) <~if(X). t>0 JE Since w E Ap, we see that the maximal operator canonically associated to the type of convergence in question is bounded on LP(E, wdS).
Thus, the usual argument
completes the proof of Step 1. S t e p 2. If F E 7/P(~), then for all X E E and t > 0,
(O~F)(X + 2t)  jf (OoF)(Y + t) n(Y) (OoE)(Y  X  t) dS(Y).
(4.18)
This is simply obtained by differentiating
(OoF)((X + s) + t) = f (OoF)(Y + t) n(Y) E(Y  X  s) dS(Y) with respect to s, and then making s = t. S t e p 3. For any f E LP(E, wdS)(n) we have
sup f N tO2(cLf)( x + 2t) dt e,N>O de
Lp 5
II/IIL~,
(4.19)
and, for almost every X E E,
lim
t O g ( c L / ) ( X + 2t)dt = 
(I + H L ) I ( X ) .
(4.20)
e~+O
N~+oo To prove this, we integrate by parts twice
Thus (4.19) is a consequence of (4.17) and (2) ~ (4) in Theorem 4.11, while (4.20) follows from (4.16) and the Plemelj formulae. S t e p 4. Here are the last details of the proof of L e m m a 4.13. For two arbitrary functions, f
E LP(E,wdS)(n) and f ' E Lq(~,wdS)(n), where liP + 1/q = 1, 77
W : m Coq/P, let US write the identity (4.18) for F := cLf, multiply both sides on the right by n(X) if(X), and then integrate the resulting formula on E against dS(X). The resulting equality reads
~ O~(cL f ) ( X + 2t) n(X) f' (X) dS(X)
Oo(CRI')(Y
= j f Oo(CLf)(Y + t) n(Y)
t) dS(Y) = (TLf, T_Rf')E.
All we need to do now is to integrate this identity against f o t dt. Then, permuting the integrals in the lefthand side and using (4.20), we immediately get (4.15) (all the technical problems have been taken care of in Step 3).
9
Next, we we shall prove the converse of Theorem 4.11. T h e o r e m 4.14. Let 1 < p < cc and CoE Ap. For any left monogenic function F on
~, the following conditions are equivalent: (1) .A(F) E LV(E, wdS) and limt~o~ F ( X + t)  0 for some X E E;
(2) g(F) E LP(E, CodS) and l i m t  ~ F ( X + t) = 0 for some X E E; (3) F belongs to 7lP~(f~).
Analogous results are valid for right monogenic functions as well. P r o o f . We only need to show that (1),(2) ~ (3). Let F be as in (2) (the reasoning for F as in (1) is completely similar). Consider the Hilbert space/C := n2((O, oz), t dt) and the left monogenic/C(n)valued function U on fl defined by
u(x)(t)
:=
OoF(X+
t),
x c fl, t > o.
Note that Urad(X) = g(F)(X), hence Urad E LP(E, wdS). According to Theorem 4.1, U has a nontangential boundary trace on E, U + E L~(E, wdS), and it is easy to see that
U+(Y)(t) = OoF(Y + t),
for a.e. Y E E and t > 0.
We now claim that
t[OoF(X + t)[ ~< (U+)*(X),
(4.21)
uniformly for t > 0 and X E E. To see this, note that there exists a constant 0 < A < 1 depending only on ~ such that B~t(X + t) C f2 for any X E E and any t > 0. Using
78
the meanvalue theorem for monogenic functions, we have
IOoF(X + t)l
1
< IBat(X + t)l
fs
~,(x+,) 10oF(W)[dW
(writing W := Y + s, with Y E E and s > 0)
JI~NBAt(X) \ a ( 1  A ) t
10oF(Y + s)l es
aS(Y)
(using HSlder's inequality in the innermost integral) 1/2
< t n1 [
([(l+A)t
100F( Y b s)12s ds
dS(Y)
J2nBAt(X) \a(1A)t
5 tn1[
J~NBAt(X) 5 tl(u+)*(X),
thus the claim. In particular,
c3oF(. +
llU+(g)li(n) dS(r)
p t) C "H~(f~) for any fixed t > 0. Now take
0 < 5 < N < oo, arbitrary otherwise. If we can prove that
IIF(. + 5)  F(. + N)IIL~ < const
< +oo
(4.22)
uniformly in 5, N, and that lim F( + t) = 0, then Fatou's lemma will give t~
IIF(. + 5)IILg ~< liNm~nfIIF ( + 5 )  F (  + N)IILg < 1, P i.e. F E 7/~(ft) and we are done. To this end, for a fixed X E E, we write /* N
F(X + N )  F ( X +5)= L
OoF(X +t) dt
=tOoF(X +t)l ~ =: By (4.21), I above belongs to Lv(E,
tOgF(X +t)dt
I + II.
wdS)(n) uniformly
in 5 and N, so we only need to
control the second term in a similar fashion. The idea is to use the fact that
79
OoF(.+t)
belongs to 7/P(12) and, therefore, one can still use the identity (4.18). Integrating both sides of this identity against
fN t dt
yields
~NtO~F(X +t) dt = 4 [ [N/2 OoF(Y + t)n(Y)(OoE)(Y  X  t) tdt dS(Y) J~ J5/2 G(Y)(t) := OoF(Y + t) n(Y) X(512,N/2)(t), for Y K(X, Y)(t) := (OoE)(Y  X  t), we can continue with
and, by introducing the kernel
f = ] (G(Y),
C E, t > O, and
K(X, Y)) dS(Y)
=: s a ( x ) , where the pairing (.,.) refers to the Hilbert space E(n ) (see w
It easy to check
that the integral operator S (or rather its formal transpose) satisfies the hypotheses of Theorem 3.10, so that
~5N t O~F(X + t)dt
<~118GIILp ~ IIGIILp(F~,~dS)~ IIg(F)HL~, L5
and (4.22) follows. Furthermore, standard arguments show that l i m t  ~
F(X + t)
exists and is independent of X C E hence, by hypothesis, this limit is zero. The proof of Theorem 4.14 is therefore complete.
9
We point out that quadratic estimates of the type presented in this section can in turn be used to prove LPboundedness results for convolution singular integral operators with even more general kernels than the Cauchy kernel and without using the W(b) theorem. See [LMS] and [LMQ]. Exercises. 9 Prove an analogous "identity" to (4.15) for operators built in connection with the areafunction. More precisely, for fl E
LP(E, wdS)(~)
and f2 E
consider
uL, II(X)(Z) := Oo(cLfl)(X •
u f2(x)(z)
:= n(x)l/2n(W 80
)mOo(CL f2)(x • z),
Lq(E, wdS)(~),
where X E ~, Z E F~ and W• is the "projection" of X 4 Z on ~, i.e. W• is the unique point on E for which X 4 Z  W• is parallel to e0. With these notations, prove that
9fr /ld+J1, L'e U J2]~ Rt \ IZllndZ
~
](fl,f2)~].
(4.23)
Use this to give a direct proof to the areafunction estimates in Lemma 4.12. 9Give a direct proof to the implication (i) =* (3) in Theorem 4.14. 9 Let F be a left monogenic function in ~, such that
g(F) E LP(Z,wdS), where
i < p < c~ and w E Ap. Show that this implies that limt+~ F(X + t) exists and is independent of X E ~l. 9 Prove that
OoF can
be replaced by the whole gradient • F
in Theorem 4.11 and
Theorem 4.14. 9 Show that one can also use higher order area and 9  functions in Theorem 4.11 and Theorem 4.14. Formally, for a positive integer k and a function F defined on ~+, the areafunction of order k is given by
A~:(F)(X)
:
(//x
I(OkoF)(y)12lx Y[2klndY
, X E ~,
and the g  f u n c t i o n of order k is defined by
g~, ( F ) ( X ) :=
(/0
I(OkoF)(X + t)12t2kldt)
, X E E.
9 Let P~0 denote the usual Poisson kernel in IR~+1, Bj the j t h Riesz transform in
IRn = OIR~_+1, 2 fR n ] xx j y [y"~Ifty) j "" "d y' Rjf(x) := o'n Also, set R0 := I and assume that 1 < p
z E
I~n•
j = 1 , 2 , . . . ,n.
< c~ and w E Ap. Prove that a s y s t e m
of conjugate harmonic functions u = (uj)j belongs to H~, the weighted version of the Hardy spaces for the upperhalf space (as defined in the introduction of this chapter, but with dx replaced by wdx) if and only if there exists f E LP(]R~, wdx) such that uj(xo, x) = ((Rjf) * Pzo)(X) for all j. Moreover, HUHHg~, HfilL p, hence H~ is isomorphic with LP(]Rn, wdx).
81
Hint: Let F := uo  ~~j ujej and let f = fo  Y]j f j e j be its boundary trace. Now, by Corollary 4.8, H L f = f and since H L =  ~']~jn=_1 R j e j , it follows that
j=l
l<j,k
j~l
Working componentwise, everything reduces to f j = R j ( f o ) , for all j, etc. 9 Let 1 < p < oo. For a ~n+lvalued function f defined on I~n the following are equivalent: (1) f is the nontangential limit to the boundary of a function from the SteinWeiss
H v space (regarded as a subspace of ~L(IR~_+I)); A (2) f C LP(I~n)(n) mad f(x)(1 + ix/Ix]) = O, x E ~'~, where A denotes the Fourier transform, and i is the usual complex imaginary unit. In particular, when n = 1 and p  2, note that this contains the classical result of Paley and Wiener (see e.g. [Ht~) asserting that a function in L2(R) is the nontangential boundary limit of an analytic function in the Hardy space H2(I~]_), if and only if its Fourier transform vanishes a.e. on the interval (  0 % 0). 9 Prove Theorem 3.10. 9 Prove Theorem 3.10 without the assumption (3.14). Hint: Show that any K which satisfies (3.13) and (3.15) can be written as
K = ~IKlel
where, for each I, K1 satisfies (3.13), (3.14) and (3.15).
For this
and related matters see [LMQ], [Ta]. w
THE REGULARITY OF THE CAUCHY OPERATOR
For a Clifford algebra valued, locally integrable function f on E, we define
(v~f)(g(x),x)
:= ~7~[f(g(x),x)] in the distributional sense on R ~. The action of
V n naturally extends to functions F defined in f~ by letting
(v~zF)(g(x) +t,x):=
v ~ [ F ( g ( x ) + t,x)],
x e ~'~, t > O.
Next, for 1 < p < cr and w E Ap, we introduce the homogeneous Sobolev space /)~
wdS)(,~) as the vector space of all locally integrable functions f on E such that
82
(each component of) v z f , taken in the distributional sense, belongs to LP(E, wdS)(n). We endow this space with the "norm"
"f"L~'* := ( / ~ 'Vz[f(g(x),x)][Pw(x)dx) 1/v.
Also, we set L p,I(E, wdS)(n) := L p'* (E, wdS)(n) fh LP(E, wdS)(n) and endow this space with the obvious norm []f][LV,~ := [[f[[L~'* + NfHLv~ 9 Clearly, Lv,I(E,wdS)(n) is a Banach space, whereas LP,*(E,wdS)(~) is a Banach space modulo constants. Note that V~ is a welldefined operator on these spaces. Also of interest for us are the following versions of (4.1):
7/v~'*(f~) := {F Clifford valued, left monogenic in ~ ; OjF G 7/P(f~) for all j},
and
. ] . ~ p , l ( ~ ) : "~P'*(a] w \ J
f') ~]'~P(~'~)= { F e nP(~~),' OjF e ~'~w(a), P
which we endow with the natural "norms"
for all j},
IIFIl~5,* : E~_0 IlOjFfl~
and
]IFLIn~,I :
p,1
][F[]7/~,* + HF]I~/v, respectively. Likewise, we define/CP'*(~) and/C~ (~). The next lemma essentially asserts that, for a function monogenic in a domain of R n+l, the derivatives in only n linearly independent directions actually control the entire gradient. Lemma
4.15. For any left or right monogenic function F in ~ we have that
[ ~7~ F[ ~ [~FI, where ~7 stands for the usual gradient in IR~+1. Proof.
Assume that F is e.g.
left monogenic.
Note that Oj[F(g(x) + t,x)] =
OoFOjg + OjF, j = 1, ..., n, and, thus,
j~=e~Oj[F(g(x) +t,x)] n
I v:~ FI ~
=
00F
ej0 g + Z~ ej0jF j=l
j=l
= IOoFI 1 
ejOjg , j=l
83
where the monogenicity of F has been used to derive the last equality. Now, since
1 Z
ejOJg ~ 1 ,
j=l
we obtain that I V~ El > 100FI 9 With this at hand, [OjF l < 1~7~ FI for all j immediately follows. Lemma
9 p~*
4.16. Any F E 7l~ (f~) has a nontangential boundary limit F+(X) at
almost any X E ~, the limit function belongs to LP'*(~,wdS)(n) and ~7~.(F +) = (vy.F) + in the distributional sense. P r o o f . If F E 7/w (~2), then Af(VF) E LP(E, wdS) so that A/'(VF)(X) < +oo at almost every point X E ~. For such a point X,
F(Y)=F(Z)+
F((VZ)s+Z)ds,
]I, Z E F ~ + X .
Keeping Z fixed and letting Y approach X nontangentially, Lebesgue's dominated convergence theorem ensures the convergence of F(Y).
Moreover, it is easy to see
now that the fimit function is actually locally integrable on ~. As for the last part in the lemma, let ~b be an arbitrary test function in II~~. By means of Theorem 4.1 and repeated applications of the Lebesgue dominated convergence theorem, we have (v~F+,r
lim F(g(x) + t,x)V~r
=  [
JR n t~O
= lim [
t~o JR"
= lim [
F(g(x)+t,z)V~r
V~[F(g(x) + t, x)]r
dx
lim ~7~[F(g(x) + t,x)]r
dx
t+o J R "
= [
JR~ t~o
= ( ( w F ) +,
r
where (., .) is the usual distributional pairing. Corollary
9
4.17. The operator 13L mapping functions into their nontangential
boundary traces is welldefined and bounded from 7lo~P'*(f~)into LP'*(E, wdS)(n). A similar statement holds true for the action of 13L from 7i~p,1 (~) to L p,1 (~, wdS)(n). The main result of this section is the following.
84
T h e o r e m 4.18. Let 1 < p < oe andw E Ap. The Canchy operatorsC L, CR extend as
P *([2), and between LP,I(E, wdS)(n) bounded operators between L p'* (E, wdS)(,~) and "lid and n~':(a). Proof. Let f be a realvalued, Lipschitz continuous, compactly supported function on E. Also, let Jr(x) := f(g(x), x), x e R n. In local coordinates, the right, say, Cauchy integral extension of f is given by 1 /R !g(y_) g (__x_)_t ,x  y) cR/(g(x) + t, x) = ~  {(g(y) _ g(~) _ t)2 + 1~  y12} ~
(1, ~7g(y) ) f (y)dy,
for t > 0 and x C R n. Therefore, straightforward integrations by parts show that
CR f(g(x) + t, x) = ~ / ( g ( x ) , ~) j:l +
~. /......d
j=l
dy
IX :y~
ej
oj/(v)
crn(n 1) 9fRn {(g(y)
g(x)~;
[xy]2} v@dy
l_<j
V~(Cnf)(X) = L'R.i(Of/OTj)(X),
X e gt,
(4.24)
j=l where {Tj(X)}j is an orthonormal frame for the tangent hyperplane to E defined at almost any X E E. An inspection of the kernels of the operators {7~j}j shows that they can be treated via the T(b) technology set up in Chapter 3 to obtain
IIN'(~jf)IIL~ <~ IlfilL~, for all j. Utilizing this and Lemma 4.15, we have IIAr(VCR/)IILg ~ IW(Vrs
< ~ IWnj(OI/OTj)IIL~ J
<~
II(OZ/OTj)IILg <~ II V~. fllLg. J
At this point, the usual density argument completes the proof of the theorem.
85
9
C o r o l l a r y 4.19. For 1 < p < oo andw CAp, the Hilbert transforms H L, H R extend
as bounded mappings of LP,*(E,wdS)(~) and of LP'l(E,wdS)(~). Proof.
If f is a Lipschitz continuous, compactly supported function on E, then p.1
F := CRf belongs to ?/~ (fl) by the previous result. Also, a combination of Theorem 4.1 and Corollary 4.17 yields 1
( v ~ F ) + = ~ V ~ (f + H R I ) so that, once again by Theorem 4.18,
]IHLf]]L~ '* <~ I] V ~ (f4H f)HL~ 4II~
f]lL~ <~ II~(V~F)]ILE 4IIV~ fllL~ <~ Hf]]Lp,'.
The second part follows similarly. Exercise.
9
For 1 < p < co and w E Ap, consider the family {S~}~>0 of bounded
operators on LP(E,wdS)(n) ,
s ~ y ( x ) := ( c L / ) ( X + ~) + (cL y ) ( X  ~),
X C ~, ~ > O.
Show that the family {Ss},>0 is a oneparameter strongly continuous semigroup of operators on LP(E, wdS)(,,), and that the domain of the infinitesimal generator of the semigroup is L v'l (E, wdS)(n).
86
Chapter 5 Applications to the Theory of Harmonic Functions In this chapter we shall see that the techniques developed so far almost exclusively within the Clifford algebra framework have also important applications to several seemingly not directly related problems. The departure point is w
dealing with layer potential operators on Lipschitz
domains, treated as close relatives of the higher dimensional Cauchy integral operator. Some quantitative expressions of the Cauchy vanishing theorem for monogenic functions are obtained in section w
before discussing the classical boundary value
problems for the Laplace operator in Lipschitz domains (section w
These results
are due to Dahlberg, Jerison, Kenig and Verchota ([Dahl], [JK], [Ve], [DK]), and our Clifford algebra approach allows us to treat in a rather simple and unified manner the Dirichlet, Neumann and the regularity problem for the Laplacian. Finally, in w
we discuss a Clifford algebra version of the celebrated theorem of
Burkholder, Gundy and Silverstein ([B G S]), essentially asserting that only "half" of a twosided monogenic function determines the size of the entire function. As a corollary of this result, we give a simple proof of a theorem of Dahlberg [Dah2] concerning the LPnorm equivalence of the Lusin areafunction and the nontangential maximal function of harmonic functions in Lipschitz domains, 0 < p < c~. Several times in the sequel we shall specialize some of our previous results by taking w  1. Whenever the case, we shall simply omit it as an index (e.g. write 7/P(ft), etc).
Also, when no ambiguity is likely to produce, we shall denote the
functions in 7/P(12) or/CP(ft) and their boundary traces by the same symbols. w
POTENTIALS OF SINGLE AND DOUBLE LAYERS
Consider the harmonic Hardy spaces
H~(ft) := {u realvalued, harmonic in f/; Afu e
87
LP(E,wdS)}
which we endow with the norm IblIHs := IIXUlIL~, and H pp'*(a) := {u realvalued, harmonic in ft; Oju 6 HP(a), for all j},
Hs
::
Hs
p~
n H~,
(~),
endowed with IlullH~," :: E j II%UllH~ and IlullH~,l := IlullH~ + IbllH~,', respectively. The so called double layer potential operator 7) is formally defined for scalarvalued functions f on E by
1 f~ (YX,n(Y))
V f ( X ) : : ~
f(Y) dS(Y), X e
IY  X] n+l
]1~n+l \ ~,
where (., .) stands for the usual inner product in ]1{n+l. Also, the singular double layer
potential operator 1r is defined for scalarvalued functions f as the principal value singular integral rf(X)
:= lim 2 f e~0O"n
Y6E IXYl>e
X,n(Y)>
.
(]7~p+1 f(Y) dS(Y),
XEE.
T h e o r e m 5.1. For 1 < p < o~ and w E Ap, ~ is a bounded mapping of LP(E,wdS), LP'*(E, wdS) and LPa(E, wdS), whereas 7) maps these spaces boundedly into HV~(f~), H~v'*(9) and HP~a(f2), respectivel3:
Furthermore, we have the classical jumprelations lim ~Pf(Y) = ~ { + f ( X ) + / U f ( X ) } , YGXIFo, Y,X
for almost any X 6 E and M1 f G LP(E, wdS). P r o o f . For a function f G LP(E, wdS), writing out the Cauchy integral as
cLf(x) =~1 ~.~_o~E(yjxj)nj(Y) _ IX _ yl,,+l f(Y) dS(Y)
+ ~
j=l
{ 1 ~(yjxj)no(Y)+(yoxo)nj(Y)f(y) ~ IX  YI "+1
dS(V)
}
ej
{ < 1 ~ (yj  xj)nk(Y)  (yk  xk)nj(Y)f(Y)dS(Y)}ejek, + ~<j<~<~ E _ _ I x  Y I n+~
88
where X = ~ j xjej 9 ~;{n+l \ p,,, y = ~ j yjej 9 P, and n = ~ j njej 9 ]~n+l, we infer that 7) is simply the real part of CL (or Cn) when acting on scalarvalued functions. Similarly, the singular principal value double layer potential operator on P~ is the real part of either Hilbert transform on ~. Thus, the results in Chapter 4 yield the theorem.
9
For n _> 2, the single layer potential operator is defined by
Sf(X) .
'L
1
crn(n  1)
IX  r l "1 f ( Y ) dS(Y),
X 9 ~n+l.
(5.1)
Actually, as it stands, the integrand in (5.1) might not be absolutely convergent for arbitrary functions f in L2(p,, dS), say. To remedy this, we shall replace the kernel
IX  YI n+l by 1
IX  r l n'
IXo

1 v l u1
where X0 is an arbitrary fixed point in I~n+l \ P,. Since we shall be mainly concerned only in the gradient of S f ( X ) , the particular choice of X0 will not play any role in the sequel. T h e o r e m 5.2. Let 1 < p < oo and w E Ap. (1) The operator ,.g is a welldefined, bounded mapping of LP ( P~,wdS ) into HP~'*( f~)
and of LP(S, codS) into Lp'* (S, wdS);
(2) For any f in LP(S, dS), lim Yex• Y~X
c%Sf ,. ,  ~  n ( r ) :=
lim ( v S f ( Y ) , n ( X ) > = Yex+ra YaX
{~:f(X) + IC*f(X)},
for almost all X E P,, where lC*f(X) := lim 2 / ecO (7n
(X_~)} YEE IxYI>~
I~~l
f ( Y ) dS(Y),
XE~,
is the formal transpose of lC; (3) For any f E LP(P,, wdS) and almost any X E S, lim K T T S f ( Y ) = lim ~TT3f(Y), YExr~ Y~x+r~ Y~X Y~X where V T is the tangential gradient operator, ~TT := V  n(O/On).
89
(5.2)
P r o o f . Differentiating under the integral sign gives
D,.qf(X) = 1 fE o~
i.e. D S f
XY
IX _ y [ . + l f ( Y )
dS(Y),
X EIR n + l \ E ,
= eL(fg) = CR(gf), as nn = nn = Inl 2 = i a.e. on E. From this, (1)
immediately follows. As for (2), using R e { n H R ( f i f ) } =  E ' f ,
lim
YEX+Fa Y+X
we have
lim ( ( D S f ) ( Y ) , ~ ( X ) ) YEXiFc~ Y~X = lim R e { n ( X ) (  D S f ) ( Y ) } YEX• Y~X
(Y)=
 1Re {n [T nY  H R ( n f ) l ) ( X ) = l { : ~ f ( X ) + K~*f(X)}.
To see (3), for X E E and t E ]~ \ {0}, set A ( X + t) :  n ( X ) C R ( ~ f ) ( X + t). A simple inspection shows that the components of ~T~qf coincide with those of  ( A  Re A)fi, when restricted to the boundary. If we now recall from the Plemelj formulae (section w
that the jumpdiscontinuities of A across the boundary occur
precisely within its real part, we are done. w
9
L2ESTIMATES AT THE BOUNDARY
First we note a boundary cancellation property for monogenic functions. Recall that ~2(~) stands for the Hardy space of right monogenic functions in ~t with square integrable nontangential maximal functions. L e m m a 5.3. For any F E 712(~t) and any G E K:2(~), one has
~ Proof.
FnGdS
= 0.
(5.3)
There are several ways to see this. One would be to use Proposition 4.10
from which our lemma directly follows. Instead, we could also use Canchy's vanishing theorem and a limiting argument similar to the one presented in the proof of Lemma 4.2.
9
90
For an arbitrary Clifford algebra valued function F we now introduce F+ := 2 ( F + F), resembling of the real part and the imaginary part, respectively, of a complex valued function. An easy corollary of Lemma 5.3 is the following. L e m m a 5.4. For functions F in 7/2(12) n/C2(V~), one has
Re/rF(nF)•
F+nFdS = Re
F ~ F + e S = 4~
P r o o f . Everything is readily seen from 2(Fn)+ = F n 4 ~
Re~lFI2eS. = F n 4 g F , Lemma 5.3
and the identity Re ( ~ Y F ) = Re ( F ~ F ) = i R e ( F ~ Y + F ~ Y ) = 1Re{F(~ + n)F} = i r e (FF)(2 Re n) = IYI2Re n.
The main result of this section is the following. T h e o r e m 5.5. We have
IIFLIL:(Z> ~ IIF~IIL:(~) ~ IlFniiL:(~) ~ [[(Fn)+lli:(Z) ~ I[(nF)+HL=(~) IIg(F)[IL:(~) ~ IIg(F+)llL2(~) ~ [[A(F)IiL2(~)~ IIA(F+)i[L:(~),
unfformJy for F ~ ~t:(g~) n K:2(~). P r o o f . The first four equivalences are immediate from the identities derived in the previous lemma, the fact that Re n < C < 0 almost everywhere on cgf~ and Schwarz inequality. The last three equivalences are obtained in a similar fashion, this time starting with e.g. the /C(,)valued twosided monogenic function U(X)(t) := c3oF(X + t), where/C := L2((0, co), tdt) and X E ~. Finally, Theorem 4.11 gives the missing link, and the proof is complete.
9
91
It is interesting to point out that for the special case in which F = ,9f, for some scalarvalued function f C L2(E, dS), Theorem 5.5 reduces to the estimates used by Jerison, Kenig and Verchota which, in turn, go back to the work of Rellich [Re] (cf. also [Ne] and [PW]). Our next result shows that Theorem 5.5 automatically extends for the larger range 2  00 < p < 2, for some 0 > 0 depending only on f2. This is done by a purely real variable argument due to Dahlberg, Kenig and Verchota ([DKV], [DK]). T h e o r e m 5.6. There exists Oo depending only on n and the Lipschitz character of
a (i.e. I[ V gllL~) such that, for any twosided monogenic function F in 7~v(a) with 2  Oo _< p _< 2, the following estimates hold
[[FI[LV(Z) ~ [[F.pILv(Z) ~ [[(Fn)+[[LV(~) ~ H(nF)•
).
P r o o f . We shall prove only the first equivalence, the rest being completely analogous. Fix e > 0, A > 0, and set G := F(.+e), Oh := {X E E ; A/'G(X) > ),}. Let us consider the "tent region"
U XEE\O~ As G vanishes at infinity, Oh is a bounded, open set so that OT)~ is a Lipschitz hypersurface which coincides with E outside of a compact subset of E. Also, the Lipschitz character of 0T.~ depends only on the Lipschitz character of E. It is not difficult to see that G E 7/2(T;~) (see also w
so that, by Theorem 5.5,
Note that [G• __ A/G __ )~on OT.~\E by construction, thus, as dS(OT.~\E) ,.~ dS(O)~), these estimates imply
~\0~lOl2dS~ ~\0~JG• 92
+ A2dS(Ox)"
Therefore, if p := 2  0 , by Theorem 4.1,
<'~"O[~176
\0,~'G4'2d~)d/~tOfo rx~/~O+ldS(OA) d'~ 0
Hence, for 0 sufficiently small, fr~ WGI2~
<~f~ la~l 2~
uniformly in e > 0.
Recalling that G = F( + ~) and letting e go to zero, we are done.
9
The last result of this section can be regarded as the harmonic analogue of the corresponding estimates for monogenic functions from Chapter 4. L e m m a 5.7. Let u := D f in ~2 for some f in L2(N, dS). Then
[[UllL:(~) ~ [IAfuliL~(~) ~ [[UradtlL2(r~)~ IIA(u)liL~(r ) ~ [Ig(u)llL~(~). Proof.
Set F
:=
cLf
E
(5.4)
,].~2(~) and G := c R f G /C2(Y~). It is easy to see that
F + G = 2u so that Lemma 5.3, Lemma 5.4 and Schwarz inequality give
IIFll2=(~)~ A FnP dS = A Fn(P § a)dS ~ IIFllL~(~)I[ulIL=(~). Therefore, Theorem 4.1 yields
tlXulIL2(~)~ IWFII/2(~) 5 IIFllL2(~)~IlullL2(~)Similar reasonings, this time starting with the Hilbert space valued harmonic function
U(X)(t) := OoDf(X + t), complete the proof of the lemma. w
9
BOUNDARY VALUE PROBLEMS FOR THE LAPLACE OPERATOR IN LIPSCHITZ DOMAINS
In this section we shall discuss the results of Dahlberg, Jerison, Kenig and Verchota ([Dahl], [JK], [DK], [Ve]) concerning the solvability of the classical boundary value problems for the Laplace operator on Lipschitz domains.
93
More specifically, we study the Dirichlet problem for A, the Laplacian in (n + 1) coordinates, in f~
/
(D)
Au = 0 in ll, Afu 9 L2(E, dS),
[. u[2 = f 9 L2(E, dS), the Neumann problem A u = 0 in t2, A/'(Uu) 9 L2(E, dS),
(N)
0hu7 = f 9 L2(E, dS), on E
and the regularity problem
/ (R) [
Au=0inf~, .h/(Vu) e L2(~, dS), ~Ou j : = ~ jOf j forallj,
where, in (R), f E L2'*(E, dS) and {Tj)j is an orthonormal frame for the tangent plane at almost every point of E.
Also, the above boundary traces should be
understood in the sense of the nontangential limit to the boundary. The basic idea is to look for the solutions expressed as layer potential extensions of a certain boundary density. Then everything reduces to inverting the corresponding singular integral operators. Let us first deal with (D). For r E L2(E, dS) we set u := 7)r in f). Since ul~z = 89 + K:)r we must solve 89 + ~ ) r = f, i.e. we have to invert the operator I k ]C on L 2 ( 2 , dS).
T h e o r e m 5.8. The operators I i / C : L2(E, dS) ~
L2(E, dS) are invertible.
P r o o f . Actually we shall show the invertibility of I + K:*, the formal transpose of I + K:. We first claim that II(I + ]C*)flIL~(E ) ~ I[(I


]C*)fIIL2(E),
(5.5)
uniformly for f E L2(E, dS). To see this, for a fixed f E LP(E, dS), we consider the function F := D S f so that F[~+ E 7~2(9t+). In addition, F is twosided monogenic
94
(in fact, even I~'~+avalued).
It is easy to see that (F+n)+ = (::FI + )~*)f and
that (F+n)_ = ~w(Sf), where F + are the boundary traces of F[n~. Consequently, Theorem 5.5 and the fact that ~ T ( S f ) is continuous across Z (cf. (3) in Theorem 5.2) yield the claim. Since
[[fIIL=<E) ~< II(I /C*)flJ/2(~) + I1(I + ]~*)fllL2<~.) 5
IlYllL:(~),
it follows that IlfJLL2(~) ~ II(I • ]C*)fHL2(~),
(5.6)
uniformly for f in L2(2, dS). To conclude, we use the dilation invariance of (5.6) together with a simple form of the continuity argument. Setting T~ for the operator I +/C* in which the Lipschitz function g(x) has been replaced by x ~+ sg(x), 0 _< s <_ 1, we see that T~ varies continuously with respect to s and is bounded from below uniformly in s (cf. (5.6)). Consequently, the index of T8 is independent of s. As T1  I + K;* and To = I, the conclusion follows. Theorem
9
5.9. The Dirichlet problem (D) has the unique solution
u(X) :=
Ix
P r o o f . We are left with proving the uniqueness part. However, this will be a direct consequence of the a priori estimate
II:IIL:< >
IIX:IIL2< ),
(5.7)
uniformly for u in H2(fl). There are several ways to see (5.7). One can for instance employ L e m m a 5.7 (see also the last exercise in this section). Another possibility is to use the fact that any u E H2(fl) is of the form R e F for some R ~ + l  v a l u e d function from 7/2(fl) (actually F is unique as IOoFI = I ~ (ReF)I; see also w
Taking this
for granted for the moment, on account of Theorem 5.5 and Theorem 4.1 we have
95
The proof is complete.
9
Now we consider the Neumann problem (N). Due to the properties of the single layer potential operator described in the Theorem 5.2, it is natural to seek a solution for (N) in the form u := Sr for some convenient r in L2(E, dS). T h e o r e m 5.10. The Neumann problem (N) has the unique (modulo additive
constants) solution
( n  12) ~
it(x) 
~ I x  Y 1I ~l[( I + K*)lf](Y)dS(Y),
X 9 fL
Proof. As before, looking for solution in the form of a single layer potential, the existence part amounts to inverting the corresponding singular integral operator in
L2(E,dS). In fact, this has already been done in Theorem 5.8. Furthermore, the uniqueness part will be a simple consequence of the following a priori estimate
~
L~(~) ~ IIAf(vu)llL~(~),
(5.8)
uniformly for u 9 H2,*(~). To see this, we set F := Du, so that DF = DDu =/Nu = t
0 in ~. Thus, F is a ~ + l _ v a l u e d (hence, twosided) monogenic function in ~. Since AfF = N'(Vu) 9 L2( p,, dS), Theorem 4.1 gives that F C 7/2(t2) M K72(f~). Finally, as
Ou/On = (Du, ~} = (F, ~} = (Fn)+, Theorem 5.5 concludes the proof of (5.8).
9
The first step for treating the regularity problem is the following. T h e o r e m 5.11. The operator ,9 : L2(P., dS)
>L2'*(E, dS) is invertible.
P r o o f . The key element is the boundedness of 8 from below, which can be seen from
IIfllL=(~) s uniformly for f
OSS On
9 L2(E, dS).
L2(~)
~ II VT (Sf)llL~(~) ~ IlsflIL~,'(E),
With this at hand, the invertibility follows from a
continuity argument similar to the one used in proof of Theorem 5.8.
9
Finally, we are in a position to prove the following. Theorem
5.12. The regularity problem (It) has the unique (modulo additive
constants) solution it(X) .
(n X)gr n
IX _ g [ n  l ( $  l f ) ( Y )
96
dS(Y),
X 9 fL
P r o o f . The existence is clear from Theorem 5.11, while the uniqueness follows from the a priori estimate
II VT ullLz(r~)~ IIH(Vu)IIL2(~/,
(5.9)
uniformly in u E H2'*(f2). As for (5.9), if we set F := Du, we see that
I(VTu)I =
I(Fn)_l, so that the conclusion is provided by Theorem 5.5 and Theorem 4.1.
9
Exercise. Show that the oblique derivative problem
{
A u = 0 in f2,
X(Vu)
9 L2(E,
dS),
(00u)l~ = f e L2(E,
dS),
has a unique solution. Hint: Existence follows by shwoing that the operator
f +(OoSf)l~. is invertible.
Uniqueness is provided by the a priori estimate
IIX(Vu)IIL~(~) ~ Ila0ullLZ(~) which, in turn, follows from Theorem 5.5. Exercise. Prove that the operators 5=1 +/C are invertible on L2'*(E, dS). Hint: Prove the identity K:S  SK:*. Exercise. Show that any u E H 2 (~) is of the form :Dr for some scalar valued function f in n2(E,
dS).
Hint: Let {f2~}~ be a nested sequence of smooth domains exhausting f~ in a suitable way. Use the maximum principle for harmonic functions to show that, with selfexplanatory notation,
"Dv[2(I q/~v)l(~l]Eu)] ~ ula.,
for all ~,,
so that, by a weak* convergence argument, one can find f E L2(E,
dS)
with u = 79f
in fL Remark.
In this section we have sketched the L p theory for the boundary value
problems for the Laplace operator on f2 only for p = 2. However, similar results are valid in Lp for certain larger ranges of p's (cf. [ D a h l ] , [DK], [Ve]). In particular, the
97
Dirichlet problem (D) is uniquely solvable for any f E LP(Z, dS) with 2  E < p < oc, while the same holds true for the Neumann problem in the range 1 < p < 2 + e. Here e is a small, positive constant, depending only on the domain f2. Note that, at least the 2  e < p _< 2 part, also follows from Theorem 5.6 and the arguments above. Actually we can do better than this as Theorem 5.8 automatically extends to L p for p in a small interval around 2.
More specifically, we have the
following result due to Calder6n ([Ca]). T h e o r e m 5.13. Let T be an operator which maps measurable functions on ~ into mesurable functions on ~ and is bounded on any Lv(E, dS) for p near 2. I f T :
L2(~, dS)
> L2(~, dS) is bounded from below, then T : LP(~, dS)   ~ LP(~, dS) is
also bounded from below for p near 2.
Note that, in particular, if T is an isomorphism of L2(E, dS), i.e. both T and T* are bounded from below, then actually T is an isomorphism of LP(~, dS) for p in a small, open interval (2  e, 2 § e). Proof.
Let s
be the Banach space of all bounded linear operators on Lp(~, dS).
Set A := T * T  e and B := I[[A][z:~ + A. For some small e > 0, the operator A is selfadjoint and positive, hence IIB[1s _ 89 IIAI1s
limsup p42
IIBIILp_ IIBIIc2 <
it follows that T * T = (e + 89163
Since by the RieszThorin theorem
e +
lllAtlc2,
+ B is actually invertible (via a Neumann series)
in L p for [p  2[ small. From this, the conclusion easily follows. w
9
A B U R K H O L D E R  G U N D Y  S I L V E R S T E I N TYPE THEOREM FOR MONOGENIC FUNCTIONS AND APPLICATIONS
In the classical setting of one complex variable, the theorem of Burkholder, Gundy and Silverstein ([BGS]) asserts that a holomorphic function belongs to the Hardy space 7/P(IR 2) if and only if the nontangential maximal function of its real part belongs to LP(IR), 0 < p < o0. Recall .A(F) and g(F), the are and g  f u n c t i o n of F, respectively (see w this section we shall prove the following.
98
In
T h e o r e m 5.14. Let 0 < p < oo. Then, for a twosided monogenic function F in f~ such that limt~oo F ( X + t) = 0 for some X E E, the following are equivalent:
(1) N ' F e LP(E,dS); (2) Fra d e LP(~,dS);
(3) .a(F) e L~(r~, dS); (4) 9(F) e / i f ( E , dS); (5) N'(F•
9 LP(E, dS);
(6) (F:t:)rad 9 Lv(E, dS); (7) ,A(F+) 9 LP(E, dS); (8) g(F+) 9 LV(E, dS). In addition, if any of these conditions is fulfilled, then a/so
[[NF]]L,(~) ~ ][Frad][Lv(~) ~ ][.A(F)I[LV(~) ~
]]g(F)NL,(~)
HAf(F+)ilLp(~:) '~ [[(F•
Ilg(fi)liL,'(~) ~ IIA(F~:)IIL,(rO.
In particular, if 1 < p < 0% then F belongs to nP(f~) and IIFII~p is ~3so equivalent with any of the above twelve LPnorms.
We first recall some essentially wellknown estimates. Recall that the superscript * stands for the usual HardyLittlewood maximal operator. L e m m a 5.15. For any function u harmonic in f~ one has
V u(X + t)l
~,~tl[(d~[~t[P/2)*(X)]2/P, t1.]kf(u)* (X),
V u(X + t)l
V . ( X + t)l s tI~/P]W~I]Lp(~), V " ( X + t)l 5 tl[(g(u)P/2)*(x)] 2/;, V u ( X + t)l
0 < p < 2,
0 < p < o0, o < ; < 2,
5 t~g(u)*(x),
v ~,(x + t)l
tln/P[]g(u)][Lp(E),
0 < p < oo,
uniformly for X E E and t > O.
The proof of the lemma is straightforward and goes along the same lines as in the upperhalf space case presented in [FS].
99
Exercise. Prove it! P r o o f o f T h e o r e m 5.14. We first treat the case 0 < p < 2. The idea of proof (cf. also [ K o l , Ko2]) is to use the fact that the
L 2 theory is valid in arbitrary Lipschitz
domains to extend the result in the range 0 < p < 2 via some "good A" inequalities. Finally, for the dual range, we present an argument based on the invertibility of the double layer potential operator. Note that for any twosided monogenic function F, we have
~Fe
= ~ D ( F 4  F ) = 12 D F =
Let us first assume that N'(F•
9 LP(E,
1
i
~(D + D)F = OoF.
(5.10)
dS). A convex combination of the estimates
presented in Lemma 5.15 yields that, for any 0 < a < 1, IV
F+(X + t)l ~< tI~/P[(AfIF•
uniformly for X 9 E and t > 0. Fix e > 0 and set G := F(. + e). Since
C(X) =
fo~C
• X + t dr,
(
)
by the boundedness of the maximal operator Gr~o(X) _< const(e, a, p, n)[(Af[F+ [P/2)*(X)](Z2a)/P
9 Lp/(1~)(E, dS).
As a 9 (0, 1) is arbitrary, we can use Theorem 4.1 to infer that G 9 7/2(~2). From now on, we shall keep the notations from the proof of the Theorem 5.6 with only one exception, namely O.x, which is taken to be this time
o~ := {x 9 r~; N(c~)(x) > ~}. By the above reasoning, G C 7~2(T~) so that, by Theorem 4.1 and Theorem 5.5, in which we take ~r to be the corresponding nontangential maximal operator for the Lipschitz domain T;~ (i.e. A/I corresponds to a "sharper" cone Fa,), we have
100
Note that, once again by construction, IG• _
\oh
IG•
<
\oh
[Af(G•
<_2
t dS(Ot) dt.
Thus, as before, the above estimates amount to
Finally, we use Chebyshev's inequality to transform this into the weak type estimate P
dS({X 9 ~ ;
IAf'G(X)I > ),}) <~ dS(O;~)
+ ,k2Jo tdS(Ot)dt
which, in turn, after multiplication with )~p1 and integration against f o d ) , , yields
IWF(" + ')IIL,'(~) < IIN"F( + QIILp(:~) < IIArF.4(+ ~)llz,,(~.). Letting e 4 0 and using Lebesgue's monotone convergence theorem, we obtain the equivalence
IIAfFIILp(E) ~ IIAf(F•
Next, we turn our attention to the area and g  function. First, with Jr(G) instead of G and A/'G in place of Af(G•
the same arguments as before give that
ilg(F)IILp(=) < IIA(F)IIL,,(s) < ]I..'V'FIIL,~(~,)For the converse inequalities we follow the same line, taking this time for G•
so that, we arrive at
IIAfGIILp(~) < IIAfg(G)llLp(~ ).
g(G)
to stand
Therefore, by Lebesgue's
monotone convergence theorem it suffices to show that
II.'V'g(a)llL,(=) < IIg(C)llLp(=). tdt), OoG(X + t).
To this effect, we once again consider'the Hilbert space 7/ := L2((0, +oo), the 7/(n)valued harmonic function U, defined in ft by
U(X)(t)
not difficult to see that
[.,v'u]~/2(x) ~< [(u~,d)p/2]*(x), 101
x e
:=
and It is
(cf. e.g. [FS] p.170). Since
IIg(x)II(.)
g(G)(x),
=
we infer Ur~4(X) <~ g(G)(X).
Consequently,
IIArg(G)IILp(~) < IIUr~alIL,,(~) < IIg(G)IIL~(~), by the L2boundedness of the maximal operator. At this point, we have shown that (5) implies (1)(8). If we now assume that (8) holds true, then (5.10) and standard arguments show that
g(F) E LP(E, dS).
From
this point on we proceed as before, with no essential alteration, and obtain that (1)(8) are true. The other implications, as well as the various LVnorm equivalences, are either simple or are immediately implied by what we have proved so far, and this completes the proof of the theorem in this case. Next, we treat the case 2 < p < oo. In this situation, everything is readily seen from Theorem 4.1 and Theorem 4.14 except that (5) implies any of (1)(4) or (6)(8). Suppose that H F + C LP(E,
dS).
From the uniqueness in the Dirichlet problem with
L P  d a t u m and a standard weak* compactness argument we infer that F+ = 7)f • in ~, for some scalar valued f + E LP(E,
dS).
Using this, we get
tlfillL~(~.> s H(I + ~)fillL~(~> ~ IlF+IILp(~> s IIArF+IILp(~). Therefore, as 7) =
ReC L, we
may write
IIA(F+)II/~<E) s IIA(cLf=~)IIL~(~) s IIH(cLf+)IILa~> < IIf•
IIX(F+)IIL~(~)
Consequently, (5.10) and standard arguments imply that
IIA(F)IIL~<~) ~ IIA(F~=)IIL~<~,>s IlJV(Fi)II/~(~> < + ~ . Thus, by Theorem 4.14, F C 7/P(Ft) and tlA(F)IILV < JIA/'(F~)IILp <
IIHFIILV
11.4FHLP, which completes the proof of the equivalences in the 2 < p < oo case. Note that because I + K is actually invertible on LP(G, dS) for 2  r < p < oo, the constants appearing in the above equivalences do not blow up as p approaches 2. Finally, for the last part of the theorem, we once again invoke Theorem 4.1 and we are done.
9
102
An useful observation is t h a t a R " + l  v a l u e d monogenic function is automatically twosided monogenic (see Proposition 1.7) so that the above theorem is valid for such functions. As an application, we shall give a simple proof of the graph version of a theorem of Dahlberg [Dah2] concerning the norm equivalence between the area and the nontangential maximal function of a harmonic function in a Lipschitz domain. First we need the following. Lemma
5.16. Let 0 < p < oo and let u be a harmonic function in ~ such that
A/'u 9 L~(E, dS).
Then there exists a unique I~n+lvalued monogenic function F
in ~ which dies at infinity and such that Re F = u. The same conclusion is valid if Urad E L P ( ~ , d S ) , and even i f l i m t  ~ u ( X + t) = 0 and A ( u )
9 LP(E, dS), or
l i m t  ~ u ( X + t) = 0 and g(u) 9 LP(E, dS), respectively. P r o o f . By Proposition 1.7, we can construct F in fl by setting
s ( x + t) := whereX 9
f
(~)(x
+ ~) e~,
andt >0.
9
An easy consequence of this and Theorem 4.1, which is worth mentioning, is the wellknown fact t h a t any harmonic function in HP(f~), 1 < p < oo, has a nontangential b o u n d a r y trace on E. The i m p o r t a n t thing, see from Theorem 5.14, is t h a t F has roughly the same "size" as u, i.e. IIHFIILP ~ IW~IILp (or IIe~allLp ~ II~r~dllLP, IIA(F)IIL~ ~ IIA(~)IIL~,
IIg(F) IIL~ ~ IIg(~)IIL~, respectively). An immediate corollary of Theorem 5.14 and Lemma 5.16 is the following version of the result of Dahlberg [Dah2] alluded before. C o r o l l a r y 5.17. Let u be harmonic in f~ and normalized such that l i m t  ~ u ( X +t) = 0 for some X 9 E. Then, for 0 < p < co, the following are equivalent:
(1) X ~ e L~(X, dS); (2) ~r~a 9 LP(S, dS); (3) A(u) 9 LP(E, dS); (4) g(u) 9 LP(E, dS). 103
In addition, if any of these conditions is satisfied, then
II./~'UlILp(E) ~ IlUradllLp(E)~ IIA(u)IILp(2) ~ I[g(u)l[L~(2). Finally, we discuss one more application, also due to Dahlberg [Dah2] (actually it was Kenig who first realized that Dahlberg's square function estimates for harmonic functibns can be obtained from the much simpler square function estimates for monogenic functions [Mcl]). C o r o l l a r y 5.18. For any harmonic function u in 12 which dies at inlinity, one has
f
'u'2 dS"~ / f
' V u'2dist(X,E) dVol.
P r o o f . Obviously, this is an equivalent formulation of IlUllL2(E) ~, IIg(u)llL2(~,), which is proved e.g. in Theorem 5.5. Exercise.
9
For 1 < p < oc and w E Ap, prove a weighted version of Lemma 5.16
and Corollary 5.17 (it should be pointed out that these results are actually valid for 0 < p < cx~ and w E A ~ ; cf. [ b a h 2 ] , IBM]). Remark.
For 1 < p < c~ and w E Ap, a proof of the fact that Afu E LP(I~n,wdx)
implies .A(u) E LP(]R'~, wdx) for the upperhalf space case can be seen more directly from the fact that any u E HP(R~_+1) is of the form u = Pt * f, f E LP(Rn, wdx) (with Pt standing for the usual Poisson kernel; see e.g. [GW] p.llT). Indeed, setting F := Pt * (f + HLf) we have that F is left monogenic in R~ +1, vanishes at infinity and ]00F[ = I V u]. Thus, by Theorem 4.14,
tI'A(u)IILp = II'A(F)IILp "~ II'/kfFIILP~ ~ Ill § HLflIL~ ~ IIIIIL~ ~ II'h/'UlILP~9 Exercise.
Use the results of the last two chapters to give a simple proof of the
FeffermanStein ([FS]) characterization of the Hardy space H l(]Rn). More specifically, prove that if Rj is the j  t h
Riesz transform in I~n and Pt denotes the usual Poisson
kernel for R~_+1, then a complex valued function f E L I ( ~ n) has A/'(Pt * f) e LI(L~n) if and only if R j f E LI(I~n) for j = 1,2, ...,n. Hint: Consider F := c L f in ~_+1 and note that F is the harmonic extension of its boundary trace FIOR~_+I= l ( f _ ~ j
ejt~jf), i.e. F = 1 p t * ( f  ~ j
104
ejRjf). Now, the
direct implication is given by Theorem 5.14, whereas the converse one is seen from Theorem 4.6. Exercise.
Use Lemma 5.16, Theorem 5.14 and Theorem 4.1 to obtain an integral
representation formula for arbitrary harmonic functions in HP(f~).
105
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112
Notational
Index
ej, ei: 1,2
Re: 4
]~(~), C(~): 1,4
V(.), 11" 11(,0:11,12
D, D: 5,8,9
E(X): 9
~,<:
Jrk: 30
5
EL, EkR: 35
Af, Af: 35
cL c~: 43
OL Q,i, OnQ,i.. 33
H L, Hy: 44,45
H L, HR: 45
E: 42
dS: 61
N(y), n(y): 42
(', "}z: 43,71
(',')b:
F., AfF: 61
53
~(m),
~ ~:~(u•
CL, cR: 61
9 61
Frad: 61
Bf, Bf: 70
F+: 62
f*: 63
.A(F), g(F): 73
Rj, P~0:81
Vz: 82
p,* p 1 74~ (~), 7/j (fl): 83
H~P(~), HP'*(n), H~P'I(~): 87,88
~f,/Cf: 88
VT: 89
F+: 91
113
Subject Index accretive 17,24,25,27 pseudo
30,31,41,53,55
areafunction (Lusin)
73,75,78,81,87,103
BMO
37,54,55
CMder6n's decomposition
71
Calder6nZygmund decomposition
39
operators
45
method of rotation
42
Carleson sequence
37
Cauchy kernel
9,54,72,74,80
reproducing formula
11,65,76
vanishing theorem
11,47,76,90
integral operator
43,44,61,68,70,82,85
Clifford algebra
1,5,16,42,61
bilinear form
17,19
differentiation
5
functional
12,14,54
group
3,33,40
module
11,15,53
Multiresolution Analysis
18,19,27
vectors
3,4,42
wavelets
16
Coifman, McIntosh and Meyer's theorem
42
Conditional expectation operators
35
114
Cotlar's inequality
63
Dahlberg's area theorem
104
Dirichlet problem
87,94,95,98
double layer potential operator
88,89,100
dual pair (of wavelet bases)
21,27,30
dyadic cubes
30
Haar Clifford wavelets
30,43,55
Hardy spaces
57,6062,6870,70,78,8385,98
HardyLittlewood maximal operator
45,63,99
Hilbert transform
44,45,63,71,86,89
Homogeneous type (spaces of)
41
jumprelations (Plemelj)
71,77,88
Lipschitz domain
61,87,93,103
function
56,61
LittlewoodPaley gfunction
73,75,78,81,98,99,103
theory
75
monogenic (left, right, twosided)
9,61,70,73,78,83,90,91,99
Muckenhoupt class
56,62,69
nontangential boundary trace
45,62,70,84,94
maximal function
61,62,70,87,94,99,103
Neumann problem
94,96,98
Pompeiu's formula
10
principal value
45,88
radial maximal function
61,70,99,103
real part
4,91,103
regularity of the Cauchy operator
82
of a CMRA
27
problem
94,96
115
Riesz basis (left, right)
20,24,33
transforms
81,104
Schur's lemma
44
sesquilinear form
14
single layer potential operator
89
square functions
73,104
standard kernel
53
system of conjugate harmonic functions
10,60
T(1) theorem
29
T(b) theorem
53,55,74,80,85
weak boundedness property
54
116
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Vol. 1557: J. Az6ma, P. A. Meyer, M. Yor (Eds.), S~.minaire de Probabilit6s XXVII. VI, 327 pages. 1993. Vol. 1558: T. J. Bridges, J. E. Furter, Singularity Theory and Equivariant Symplectic Maps. VI, 226 pages. 1993. Vol. 1559: V. G. Sprind~uk, Classical Diophantine Equations. XII, 228 pages. 1993. Vol. 1560: T. Bartsch, Topological Methods for Variational Problems with Symmetries. X, 152 pages. 1993. Vol. 1561 : I. S. Molchanov, Limit Theorems for Unions of Random Closed Sets. X, 157 pages. 1993. Vol. 1562: G. Harder, Eisensteinkohomologie und die Konstruktion gemischter Motive. XX, 184 pages. 1993. Vol. 1563: E. Fabes, M. Fukushima, L. Gross, C. Kenig, M. R6ckner, D. W. Stroock, Dirichlet Forms. Varenna, 1992. Editors: G. Dell'Antonio, U. Mosco. VII, 245 pages. 1993. Vol. 1564: J. Jorgenson, S. Lang, Basic Analysis of Regularized Series and Products. IX, 122 pages. 1993. Vol. 1565: L. Boutet de Monvel, C. De Concini, C. Procesi, P. Schapira, M. Vergne. Dmodules, Representation Theory, and Quantum Groups. Venezia, 1992. Editors: G. Zampieri, A. D'Agnolo. VII, 217 pages. 1993.
Vol. 1543: A. L. Dontchev, T. Zolezzi, WellPosed Optimization Problems. XII, 421 pages. 1993.
Vol. 1566: B. Edixhoven, J.H. Evertse (Eds.), Diophantine Approximation and Abelian Varieties. XIII, 127 pages. 1993.
Vol. 1544: M.Schiirmann, White Noise on Bialgebras. VII, 146 pages. 1993.
Vol. 1567: R. L. Dobrushin, S. Kusuoka, Statistical Mechanics and Fractals. VII, 98 pages. 1993.
Vol. 1545: J. Morgan, K. O'Grady, Differential Topology of Complex Surfaces. VIII, 224 pages. 1993.
Vol. 1568: F. Weisz, Martingale Hardy Spaces and their Application in Fourier Analysis. VIII, 217 pages. 1994.
Vol. 1546: V. V. Kalashnikov, V. M. Zolotarev (Eds.), Stability Problems for Stochastic Models. Proceedings, 1991. VIII, 229 pages. 1993.
Vol. 1569: V. Totik, Weighted Approximation with Varying Weight. VI, 117 pages. 1994.
Wol. 1547: P. Harmand, D. Werner, W. Werner, Mideals in Banach Spaces and Banach Algebras. VIII, 387 pages. 1993. Vol. 1548: T. Urabe, Dynkin Graphs and Quadrilateral Singularities. VI, 233 pages. 1993. Vol. 1549: G. Vainikko, Multidimensional Weakly Singular Integral Equations. XI, 159 pages. 1993. Vol. 1550: A. A. Gonchar, E. B. Saff (Eds.), Methods of Approximation Theory in Complex Analysis and Mathematical Physics IV, 222 pages, 1993. Vol. 1551: L. Arkeryd, P. L. Lions, P.A. Markowich, S.R. S. Varadhan. Nonequilibrium Problems in ManyParticle Systems. Montecatini, 1992. Editors: C. Cercignani, M. Pulvirenti. VII, 158 pages 1993. Vol. 1552: J. Hilgert, K.H. Neeb, Lie Semigroups and their Applications. XII, 315 pages. 1993. Vol. 1553: J.L ColliotTh61~ne, J. Kato, P. Vojta. Arithmetic Algebraic Geometry. Trento, 1991. Editor: E. Ballico, VII, 223 pages. 1993.
Vol. 1570: R. deLaubenfels, Existence Families, Functional Calculi and Evolution Equations. XV, 234 pages. 1994. Vol. 1571: S. Yu. Pilyugin, The Space of Dynamical Systems with the C~ X, 188 pages. 1994. Vol. 1572: L. G6ttsche, Hilbert Schemes of ZeroDimensional Subschemes of Smooth Varieties. IX, 196 pages. 1994. Vol. 1573: V. P. Havin, N. K. Nikolski (Eds.), Linear and Complex Analysis  Problem Book 3  Part I. XXII, 489 pages. 1994. Vol. 1574: V. P. Havin, N. K. Nikolski (Eds.), Linear and Complex Analysis  Problem Book 3  Part II. XXII, 507 pages. 1994. Vol. 1575: M. Mitrea, Clifford Wavelets, Singular Integrals, and Hardy Spaces. XI, 116 pages. 1994.