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 0, there exists an open neighborhood U C ~ of X so that
oQfngdacVol(Q)
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) /
[email protected])
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 [#(~)[ 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 ~):
[email protected](~)
=
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  gllxll(n>,
and
Ilyll(,,)_ gll~ll(n>,
I1~,11(.) 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 ,% 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 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, )
0and0 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
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 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)
[email protected])I _ Ek(fb)
[email protected])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# 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 ),}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 ~[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 ~ ~
32k0. 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 ~,~ 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 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
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~
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 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 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) 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
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 = 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,
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