Commutators and singular integral operators in Clifford analysis

Complex Variables, Vol. 50, No. 4, 15 March 2005, 265–281

Commutators and singular integral

operators in Clifford analysis

RICARDO ABREU BLAYAy* and JUAN BORY REYESzx

yFaculty of Mathematics and Informatic, University of Holguın 80100, Holguın, CubazFaculty of Mathematics and Computer Science, University of Oriente,

Santiago of Cuba 90500, Cuba

Communicated by H. Begehr

(Received 19 May 2004)

In this paper we develop a method for setting the compactness of the commutator relative to thesingular integral operator acting on Holder continuous functions over Ahlfors David regularsurfaces in Rnþ1. This method is based on the essential use of the monogenic decompositionof Holder continuous functions. We also set forth explicit representations of the adjoints ofthe singular Cauchy type integral operators, relative to a total subset of real functionals.

Keywords and phrases: Clifford analysis; Cauchy transform; Singular integral operator; AhlforsDavid surfaces

AMS Subject Classification: 30E20; 30E25; 30G35; 42B20

1. Introduction

At the present time Clifford analysis is an independent mathematical discipline withits own goals and tools. One of the most important indicators for such a developmentis the capacity of generalizing, unifying, and simplifying several other analytictheories that are developed for solving problems in higher dimensional spaces; see[12,19,21,24,30,32,33] for more information and further references.

It is shown in [29] how Clifford analysis can be used to prove the L2 boundednessof the double layer potential operator on surfaces with small Lipschitz constant,this method has been extended to all Lipschitz constant by McIntosh [27]. A keyidea here is that the double layer potential operator is the real, or scalar, part of thesingular integral operator over the surface in Rn.

*Corresponding author. Tel.: (+5324) 481302, ext. 44. Fax: (�5324) 468050. E-mail: [email protected].: (�5322) 633894. Fax: (+5322) 632689. E-mail: [email protected]

Complex Variables

ISSN 0278-1077 print: ISSN 1563-5066 online � 2005 Taylor & Francis Ltd

http://www.tandf.co.uk/journals

DOI: 10.1080/02781070410001732197

In [23], Viorel Iftimie sets up basic results on Cauchy transforms over domains in Rn,and establishes Sokhotski-Plemelj formulae for Holder continuous functions definedover compact Liapunov surfaces. Using these formulae, he is able to show that thesquare of the singular integral operator over such a suraface is, when acting onHolder continuous functions, the identity map. More recently, these results havebeen applied in [36–39] to deal with boundary value problems in R3 and Rn. In [9]and [1] the authors showed how these statements can be generalized to Ahlfors-David regular (AD-regular) surfaces lying in R3 and Rn respectively.

There is a long tradition in applying singular integrals for studying elliptic andparabolic boundary value problems. For proving the L2 (or Holder) boundedness ofsingular integral operators, one needs to show when they are invertible or at leastFredholm. A criterion for the Fredholm property of the singular integral equations,which are equivalent to the Riemann problems in Quaternionic analysis, is obtainedby Shapiro and Vasilievski in [36] (see also [7]). Analogous result has been reportedby S. Bernstein [5,6] in the Clifford analysis context. We will mention here that themethod used by these authors depends significantly on the possibility of building thesymbol of the corresponding operator.

It is well known that the Fredholmness of a linear operator is equivalent to theexistence of a two-side regularization of it to the ideal of compact operators. Moredetailed accounts of these topics are in [28,31].

As it was earlier remarked by Shapiro and Vasilievski, the Fredholmness of the socalled right handed Riemann operator follows easily from the Holder compactness ofthe commutator related to the singular integral operator. This result was obtained bythese authors over Liapunov surfaces. Hence, the surface is C1 with a Holder continuousderivative. This stronger smoothness condition gives sufficient cancellation for one todeduce that the operator is weakly singular. However, when one replaces Liapunov sur-faces by AD-regular surfaces (they can even be non rectifiable), we no longer have thecancellation property mentioned above and, consequently, one needs to find differenttechniques.

How to get the compactness of the commutator operator over surfaces complicatedgeometrically is discussed in this paper.

2. Preliminaries

In this section we shall briefly review the construction and the basic properties of theClifford algebra associated with Rn, as well as the basic elements of Clifford analysis.

2.1. Clifford algebra, Weyl operator and monogenic functions

The real Clifford algebra associated with Rn endowed with the Euclidean metric is theminimal enlargement of Rn to a real linear associative algebra R0, n with identity so thatx2 ¼ �jxj2, for any x 2 Rn.

It thus follows that if fejgnj¼1, is the standard basis of Rn, we must have

e2i ¼ �1, i ¼ 1, . . . , n

and

eiej þ ejei ¼ 0, i 6¼ j:

266 R.A. Blaya and J.B. Reyes

A basis for R0, n then consits of the elements

eA ¼ ei1 � � � eik , where A ¼ ði1, . . . , ikÞ � f1, 2, . . . , ng

is such that 1 � i1 < i2 < . . . , ik � n. By convention, e 6 0 :¼ e0 :¼ 1, is the unit element.Consequently, dimR0, n ¼ 2n and any element a 2 R0, n may be written as

a ¼XA

aAeA, aA 2 R:

We define the real, or scalar, part of a to be <a ¼ a 6 0. For a 2 R0, n the conjugation isdefined by a :¼

PA aAeA, where

eA ¼ ð�1Þkeik � � � ei2ei1 , if eA ¼ ei1ei2 � � � eik :

By means of the conjugation we endow R0, n with the natural Euclidean normjaj2 ¼ <ðaaÞ. An algebra norm may be defined by taking jaj20 ¼ 2njaj2.

For a 2 R0, n, we shall denote by Ma (respectively aM) the right (respectively left)multiplication operator by a.

We shall work in the Euclidean space Rnþ1 assumed to be embedded in the Cliffordalgebra R0, n, by identifying ðx0, xÞ 2 Rnþ1 ¼ R�Rn with x0 þ x 2 R0, n.

A R0, n-valued function u on an open set � � Rnþ1 is defined via:

uðxÞ ¼XA

uAðxÞeA,

where the uA’s are R-valued.Any continuity, differentiability or integrability property which is ascribed to u has to

be possessed by all its components uA. In this manner, the real Banach spaces of allcontinuous, k-time continuously differentiable and p-th power Lebesgue integrableR0, n-valued functions are denoted by C(G), CkðGÞ and LpðGÞ respectively, where Gcan be any suitable subset of Rnþ1.

In a formal way, we now introduce the following first order linear differentialoperator D called the Weyl (or Cauchy-Riemann) operator

D ¼ @x0 þXnj¼1

ej@xj :

It acts on C1ð�Þ, where � is an open subset of Rnþ1.If uðxÞ ¼

PA uAðxÞeA, then the action may be from the left, i.e.

Du ¼Xj,A

ejeA@uA@xj

,

Commutators and singular integral operators 267

or from the right,

uD ¼XA, j

eAej@uA@xj

:

Let � � Rnþ1 be an open set and let u 2 C1ð�Þ. Then we say that u is left monogenic in� if Du¼ 0 in �. Analogously, u is called right monogenic in � if uD¼ 0 in �.

The set of left monogenic (resp. right monogenic) functions in � is denoted by Mlð�Þ

(resp. Mrð�Þ).In particular, as DD ¼ �, u 2 Mlð�Þ or u 2 Mrð�Þ implies that u is harmonic in �.The basic example of a both left and right monogenic function, the so called

monogenic Cauchy kernel, is the fundamental solution of the Cauchy-Riemannoperator D, given by

eðxÞ ¼1

�n

x

jxjnþ1, x 2 Rnþ1 n f0g,

where �n stands for the area of the unit sphere in Rnþ1.

2.2. Basic integral formulae

Through the whole paper we shall employ the surface integration with respect to the n-dimensional Hausdorff measure Hn. This measure is defined in terms of the diametersof various efficient coverings and agrees with ordinary ‘‘n-dimensional surface area’’ onnice sets, see [20,22].

It is well known that one of the basic and principal analytical facts that forms thebasis of the Clifford analysis is the n-dimensional Stokes formula, which is usuallystated on domains with a sufficiently smooth boundary. It is not so obvious that thisformula remains valid even if the boundary is very complicated geometrically.Research on the problem of finding the most general form of the Stokes formula hascontributed greatly to the development of Geometric Measure Theory. The conceptof the exterior normal n(x) defined in [20] was used to establish the validity of thefollowing version of the Stokes formula:

Z�

’ðxÞ � nðxÞdHnðxÞ ¼

Z�

div ’ðxÞdLnþ1ðxÞ,

for every open subset � of Rnþ1 with boundary � such that Hnð�Þ <1. Here and inwhat follows the symbol Lnþ1 denotes the Lebesgue measure in Rnþ1. The principalgoal of the use in this section of the Hausdorff measure is therefore to state the basicintegral formulae as well as to investigate in particular the behaivor of singular integraloperators in this general setting.

The next lemma can be thought of as the Clifford analysis version of theStokes formula.

LEMMA 2.1 (Stokes) Let � 2 Rnþ1 be a bounded domain with the boundary � such thatHnð�Þ <1. Suppose that v, u 2 C1ð�Þ \ Cð�Þ. Then

Z�

vðyÞnðyÞuðyÞdHnðyÞ ¼

Z�

ððvDÞuþ vðDuÞÞdLnþ1:

268 R.A. Blaya and J.B. Reyes

COROLLARY 2.1 (Cauchy) If v is right monogenic in � and u is left monogenic in �, then

Z�

vðyÞnðyÞuðyÞdHnðyÞ ¼ 0:

Stokes formula applied to vðyÞ ¼ eðy� xÞ in � n B�ðxÞ, yields, after letting � to zero,the Borel-Pompeiu integral representation formula

uðxÞ ¼

Z�

eðy� xÞnðyÞuðyÞdHnðyÞ �

Z�

eðy� xÞDuðyÞdLnþ1ðyÞ,

for x 2 �. In particular, if u is left monogenic, we have the Cauchy- type reproducingformula

uðxÞ ¼

Z�

eðy� xÞnðyÞuðyÞdHnðyÞ:

In a quite analogous way, we can state the right-handed versions of these formulae.

3. Cauchy transform and singular integral operator on AD-regular surfaces

In this section we want to state some important properties of the Cauchy transform

ðCuÞðxÞ ¼

Z�

eðy� xÞnðyÞuðyÞdHnðyÞ, x =2�, ð1Þ

and its singular version, the principal value integral operator

ðSuÞðxÞ ¼ 2

Z�

eðy� xÞnðyÞðuðyÞ � uðxÞÞdHnðyÞ þ uðxÞ, x 2 �: ð2Þ

Of course, there are also right-handed versions for these operators (i.e. eðy� xÞ appearsin the rightmost part of the integrand). When it is necessary, we shall indicate whichoperator one is using by employing the superscripts l, r (e.g., the operators (1) and(2) will be denoted by C

l and Sl , respectively).

3.1. Ahlfors David regular surfaces

We will say that the set E in Rnþ1 is an n-set ifHnðEÞ < þ1, where as aboveHn denotesthe n-dimensional Hausdorff measure.

The geometric condition HnðEÞ < þ1 represents a natural condition without anyquantitative estimates on the size of the set E. Among n-sets, rectifiable sets, see [20],form essentially the largest class where many basic properties of smooth surfaceshave reasonable analogues. Properties such as, for example, existence of tangent

Commutators and singular integral operators 269

planes (defined in a measure-theoretic, approximate way), parametrization by Lipschitzmaps, and the establishment of the analogue of Lebesgue’s density point theorem stillhold in this context. All these properties are qualitative, without any estimates.

If � is a curve in the complex plane such that H1ð�Þ < þ1, it can be parametrizednicely by a Lipschitz function. For n-dimensional surfaces (n>1) one cannot, ingeneral, find such a nice parametrization.

Definition 3.1 A set E in Rnþ1 is said to be Ahlfors David regular (AD-regular) withdimension n if it is closed and there is a constant c>0 such that

c�1rn � HnðE \ Bðx, rÞÞ � c rn, ð3Þ

for all x 2 E and r>0, where Bðx, rÞ stands for the closed ball with center x andradius r.

In the sequel, for convenience, we shall denote by c certain generic constant notnecessarily the same in different occurrences.

The requirement that the set E is AD-regular can be viewed as a quantitative versionof the property of having upper and lower densities with respect to Hn, which arepositive and finite. For further information regarding the AD-regular sets, the readermay consult the exposition in [15–18,25,26,34,35].

The AD-regular curves in the plane are closely related with the singular integraloperator

f ðxÞ ! 2

Z�

eðy� xÞnðyÞf ðyÞdH1ðyÞ,

acting in Lpð�Þ. Based on the works of Calderon [13], Coifman, McIntosh and Meyer[14], David [15] proved that the Lpð�Þ – boundedness of the singular integral operatortakes place if and only if � is an AD-regular curve (in fact, only the inequalityH1ð� \ Bðx, rÞÞ � c r is essential, since in this case the lower bound is clear). Forn-dimensional surfaces, n>1, such a simple characterization seems impossible.

In [18] the authors studied singular integral operators on general finite dimensionalAD-regular sets. They showed that a large class of Calderon-Zygmund singular integraloperators is bounded on LpðEÞ, 1 < p < þ1, if and only if E is uniformly rectifiable.

Let us describe the remainder of this section. Although we suppose that the reader isfamiliar with the notion of Holder spaces C0, �ð�Þ, 0 < � < 1, in the subsection 3.2 wefirstly recall the definition. Then we shall give some theorems on the Holder bounded-ness of the Cauchy transform and its singular version on AD-regular surfaces, whichhave been obtained previously in [1,2] (see also [9–11]). For this reason they will begiven without proof.

3.2. Boundedness of the singular integral operator in Holder spaces

Let F be a compact set in Rnþ1, and let u(x) be a continuous R0, n-valued functiondefined on F. The modulus of continuity of the function u is the non-negative function!ðu, tÞ, t > 0, defined by the formula

!ðu, tÞ ¼ supjx�yj�t

fjuðxÞ � uðyÞj : x, y 2 F g:

270 R.A. Blaya and J.B. Reyes

In what follows, we will rather use the Stechkin’s rectification of !ðu, tÞ (see [4])

!uðtÞ ¼ t sup��t

��1!ðu, �Þ:

The function !uðtÞ, t>0, is increasing and !uðtÞ=t is not increasing.Equivalently, we may define the Holder spaces C0, �ðFÞ, 0 < � < 1 as the collection of

all functions u 2 CðFÞ for which the quantity

kukCðFÞ þ sup0<t�d

!uðtÞ

t�, d ¼ max

x, y2Fjx� yj,

denoted in the sequel by kuk�, is finite.Provided with the norm kuk�, the space C0,�ðFÞ becomes a real Banach space.In what follows, we suppose � to be an AD-regular surface with diameter d, which

bounds a bounded domain �þ 2 Rnþ1.

THEOREM 3.1 The singular integral operator S is bounded on the space C0,�ð�Þ, and it isan involution, i.e., S2

¼ I , where I denotes the identity on C0, �ð�Þ.

THEOREM 3.2 Let �� : ¼ Rnþ1 n ð�þ [ �Þ and let u 2 C0,�ð�Þ. Then Cu 2 Cð�� [ �Þand we have the following formulae

lim��3x!z2�

ðCuÞðxÞ ¼1

2ðSuðzÞ � uðzÞÞ ¼: �P�uðzÞ,

for any z 2 �.

Consequently, the following important corollary holds.

COROLLARY 3.1 The operators P� are mutually complementary projections in C0,�ð�Þ,i.e., bounded operators satisfying

ðP�Þ2¼ P�, PþP� ¼ 0 ¼ P�Pþ, and Pþ þ P� ¼ I :

Thus, we have the direct decomposition

C0, �ð�Þ ¼ PþC0,�ð�Þ � P�C

0,�ð�Þ,

and then, each function u 2 C0, �ð�Þ admits a unique decomposition of the formu ¼ uþ þ u�, where u� 2 P�C

0,�ð�Þ.

Remark 3.1 As it was remarked earlier by S. Bernstein (see [5–8]), even in the smoothcontext, it is difficult to compute the norm of the singular integral operator, in the spaceC0, �ð�Þ: However, taking into account the decomposition above, one introduces inC0, �ð�Þ a new norm

kuk? :¼ kuþk� þ ku�k�,

Commutators and singular integral operators 271

and then one can assure that the space C0,�ð�Þ also becomes a real Banach space andthe singular integral operator remains a bounded operator and has the norm 1.

We can adapt immediately the arguments given in [9] to get the following statements,which are generalizations of those obtained in the 3-dimensional case.

THEOREM 3.3 Suppose u 2 Cð�Þ. If the following integral converges uniformly as �! 0Z�nð�\Bðx, �ÞÞ

eðy� xÞnðyÞðuðyÞ � uðxÞÞdHnðyÞ, ð4Þ

then Clu 2 Cð�� [ �Þ and the ‘‘left-handed’’ Sokhotski-Plemelj formula

lim��3x!z2�

ðCluÞðxÞ ¼

1

2ðS

luðzÞ � uðzÞÞ ¼: �Pl�uðzÞ

holds.The analogous result holds for the right handed operators if the integral

Z�nð�\Bðx, �ÞÞ

ðuðyÞ � uðxÞÞnðyÞeðy� xÞdHnðyÞ ð5Þ

converges uniformly as �! 0.

Let us now introduce the real linear space M lð�Þ (rep. M rð�Þ) consisting of allcontinuous functions in � for which the integral (4) (resp. (5)) converges uniformlyas �! 0 (see [9]).

Thus we have the topological splitting M lð�Þ ¼ PlþM lð�Þ � Pl

�M lð�Þ and each func-tion u 2 M l admits a unique decomposition u ¼ ulþ þ ul�, where now ul� 2 Pl

�M lð�Þ.Moreover, the space M lð�Þ with the norm

kukl ¼ kulþkCð�Þ þ kul�kCð�Þ

becomes a real Banach space. The real Banach space M rð�Þ can be defined in ananalogous manner.

Remark 3.2 If u 2 C0,�ð�Þ, then u belongs to both spaces M lð�Þ and M rð�Þ. In fact, letu 2 C0,�ð�Þ, then

Z�\Bðx, �Þ

eðy� xÞnðyÞðuðyÞ � uðxÞÞdHnðyÞ

����������

�

Z�\Bðx, �Þ

!uðjy� xjÞ

�njy� xjndHnðyÞ � c

Z�\Bðx, �Þ

jy� xj��ndHnðyÞ:

Using Lemma 3.2 in [1] and taking into account the AD-condition for the surface �, thepreceding inequality yields

j

Z�\Bðx, �Þ

eðy� xÞnðyÞðuðyÞ � uðxÞÞdHnðyÞj � c

Z �

0

dt

t1��:

272 R.A. Blaya and J.B. Reyes

Finally, it is clear that the integral in the right hand side tends to zero when �! 0 andthen, u 2 M lð�Þ. Hence we have C0, �ð�Þ � M lð�Þ. The inclusion C0,�ð�Þ � M rð�Þ isobtained analogously.

From the above remark, it follows that the Holder space C0,�ð�Þ is embedded intothe spaces M lð�Þ and M rð�Þ. In the next section we are going to prove that this embed-ding is not only continuous but also compact.

4. Compactness in Ml ðCÞ and MrðCÞ

The standard functional analytic method leads immediately to the following state-ments. We establish only the left versions of the results, the right versions are totallyanalogous.

LEMMA 4.1 A set E � M lð�Þ is relatively compact if and only if the sets Pl�E are

relatively compact as subsets of Cð�Þ.

LEMMA 4.2 A set E � M lð�Þ is relatively compact if and only if the following twoconditions are satisfied

(i) E is bounded in M lð�Þ:(ii) Pl

�E are sets of equicontinuous functions.

LEMMA 4.3 The space C0, �ð�Þ is continuously embedded in the space M lð�Þ.

Proof Obviously, it is sufficient to prove the inequality kukl � c kuk�, with a constantc independent of u 2 C0, �ð�Þ.

We have

ulþðxÞ ¼ ðPlþuÞðxÞ ¼ uðxÞ þ

Z�

eðx� yÞnðyÞðuðyÞ � uðxÞÞdHnðyÞ:

Hence

julþðxÞj � juðxÞj þ1

�n

Z d

0

wuð�Þ

�d�

� kukCð�Þ þ sup0<t�d

wuðtÞ

t�1

�n

Z d

0

d�

�1��� ckuk�:

Similarly, we have jul�ðxÞj � ckuk�.Summarizing one has the desired estimate. g

THEOREM 4.1 The embedding operator of the space C0,�ð�Þ into the space M lð�Þ is acompact operator.

Proof Let E � C0, �ð�Þ be a bounded set. Then for any function u 2 E we havekuk� � c, where c is a universal constant.

Now, let us show that the set E satisfies the conditions i) and ii) in the Lemma 4.2.

Commutators and singular integral operators 273

The condition i) is obvious in virtue of the preceding lemma. For the proof of ii) itshould be observed that for any u 2 E, we have wuðtÞ � c t�. Moreover, the followingZygmund type inequality

!SluðtÞ � c

Z t

0

!uð�Þ

�d� þ t

Z d

t

!uð�Þ

�2d�

� �,

can be used to get the estimate !SluðtÞ � c t�: We omit the detailed discussion here

because it is completely analogous to the case of three dimensions (see [2]).Then, the equicontinuity of the sets Pl

�E follows directly in accordance with theobvious inequality

wul�ðtÞ �

1

2ð!uðtÞ þ !SluðtÞÞ: �

As it has been remarked above, for the right handed version one has the followingresult:

THEOREM 4.2 The embedding operator of the space C0,�ð�Þ into the space M rð�Þ is acompact operator.

5. Compactness of the commutator operators in C0,�ð�Þ

We shall prove in this section the compactness of the commutators

½Ma,Sl� :¼ MaS

l� S

lMa

½aM,Sr

� :¼ aMSr� S

r aM,

from the space C0, �ð�Þ into the space C0,�ð�Þ, where 0 < � � 1, 0 < � < 1, anda 2 C0,�ð�Þ.

This property was proved by Shapiro and Vasilievski in the case of Liapunov surfacesin the quaternionic analysis setting (see [35,36]).

In what follows, we shall need the Zygmund type estimates for the commutatoroperators.

THEOREM 5.1 Let aðxÞ 2 Cð�Þ be such that

Z d

0

!að�Þ

�d� <1:

Then for u 2 M lð�Þ we have the Zygmund type estimate

!½Ma,Sl �uð�Þ � ckukl

Z �

0

!að�Þ

�d� þ �

Z d

�

!að�Þ

�2d�

� �:

Similarly for u 2 M rð�Þ

!½aM,Sr�uð�Þ � ckukr

Z �

0

!að�Þ

�d� þ �

Z d

�

!að�Þ

�2d�

� �:

274 R.A. Blaya and J.B. Reyes

Proof Let x1, x2 2 �, jx1 � x2j ¼ 2t. In virtue of the decomposition u ¼ ulþ þ ul� it issufficient to show that

j½Ma,Sl�ul�ðx1Þ � ½Ma,Sl

�ul�ðx2Þj � ckul�kCð�Þ

Z t

0

!að�Þ

�d� þ t

Z d

t

!að�Þ

�2d�

� �:

We shall use the notation �rðxÞ ¼ � \ Bðx, rÞ.Then we have

1

2f½Ma,Sl

�ulþðx1Þ � ½Ma,Sl�ulþðx2Þg

¼

Z�tðx1Þ

eðy� x1ÞnðyÞulþðyÞðaðyÞ � aðx1ÞÞdH

nðyÞ

�

Z�tðx2Þ

eðy� x2ÞnðyÞulþðyÞðaðyÞ � aðx2ÞÞdH

nðyÞ

þ

Z�nð�tðx1Þ [�tðx2ÞÞ

ðeðy� x1Þ � eðy� x2ÞnðyÞulþðyÞðaðyÞ � aðx1ÞÞdH

nðyÞ

þ

Z�nð�tðx1Þ [�tðx2ÞÞ

eðy� x2ÞnðyÞulþðyÞðaðx2Þ � aðx1ÞÞdH

nðyÞ

þ

Z�tðx2Þ

eðy� x1ÞnðyÞulþðyÞðaðyÞ � aðx1ÞÞdH

nðyÞ

�

Z�tðx1Þ

eðy� x2ÞnðyÞulþðyÞðaðyÞ � aðx2ÞÞdH

nðyÞ:

Let us denote by I1, . . . , I6 the integrals in the right side of the above equality.Following the same technique used in the proof of the Theorem 8.1 in [2], we have

jIkj �

Z�tðxkÞ

!aðjy� xkjÞ

�njy� xkjn julþðyÞjdH

nðyÞ � c kulþkCð�Þ

Z t

0

!að�Þ

�d�,

for k¼ 1, 2,and analogously

jIkj � c kulþkCð�Þ t

Z d

t

!að�Þ

�2d�,

for k ¼ 3, 5, 6.Finally

jI4j � !að2tÞj

Z�nð�tðx1Þ [�tðx2ÞÞ

eðy� x2ÞnðyÞulþðyÞdH

nðyÞj:

In accordance with the Clifford analysis analogue of the Proposition 2 in [9], and theobvious inequality !uð2tÞ � 2!uðtÞ, we have

jI4j � c !aðtÞ maxx2�[�

julþðxÞj ¼ c!aðtÞkulþkCð�Þ:

Commutators and singular integral operators 275

Since !aðtÞ �R t

0 !að�Þ=�d�, summarizing one has the desired estimate

j½Ma,Sl�ulþðx1Þ � ½Ma,Sl

�ulþðx2Þj � c kulþkCð�Þ

Z t

0

!að�Þ

�d� þ t

Z d

t

!að�Þ

�2

� �:

For ul� the estimate may be proved by the same method. g

In accordance with the preceding theorem we thus have the following assertion.

COROLLARY 5.1 Let aðxÞ 2 C0, �ð�Þ. Then the commutator ½Ma,Sl� (resp. ½aM,Sr

�) is acontinuous operator from M lð�Þð resp. M rð�Þ into the space C0,�ð�Þ.

The following conclusion is an immediate consequence of the precedent corollary,Theorem 4.2 and its right handed version. We omit the details.

THEOREM 5.2 Let a 2 C0, �ð�Þ (0 < � < 1). Then the commutators ½Ma,Sl� and ½

aM,Sr�

are both compact operators from the space C0,�ð�Þ (0 < � � 1) into the space C0, �ð�Þ.

Remark 5.1 (About the Riemann boundary value problem in Clifford Analysis.) TheRiemann Hilbert problem and related ones were intensively studied in the last decade(see e.g. [1,5–8,10,36–39] and references quoted there). Following Shapiro andVasilievski’s paper [36], let us consider the algebra R generated by the operators S

l ,Ma and T , where a 2 C0,�ð�Þ and T is a compact operator on C0, �ð�Þ.

It follows from Theorem 5.2 that an arbitrary operator K from R is representable inthe form

K ¼ MaPlþ þMbPl

� þ T :

As it has been remarked by Shapiro and Vasilievski, here the right (not the left one)multiplication is essential, and for the operator aM the result will not be true. This isdue to the fact that left monogenic functions form a right but not a left Clifford module.

If the singular operator K is of normal type, i.e., the functions aðxÞ, bðxÞ are bothinvertible for all x 2 �, then in virtue of the Theorem 5.2, the operatorMa�1

Plþ þMb�1

Pl� is a two-sided regularizer of K to the ideal of compact operators

in C0, �ð�Þ. Consequently, under the normality condition the operator K is aFredholm operator (�-operator). See [28,31].

Equivalently, the Riemann boundary value problem for left monogenic functionsmay be reformulated as the characteristic integral equation

K0u ¼ MaPlþuþMbPl

�u ¼ f on �:

Of course, the operator K is a Fredholm (semi Fredholm) operator if and only if thecharacteristic part K0 has the corresponding property.

In [10], the authors discussed a reduction method to solve via successive approxima-tion the generalized singular integral equation

KðmÞ

0 u :¼ ðMa1Plþ � � �MamPl

þ þMb1Pl� � � �MbmPl

�Þu ¼ f ,

where the functions a1, . . . , am, b1, . . . , bm, and f are assumed to belong to C0,�ð�Þ.

276 R.A. Blaya and J.B. Reyes

Clearly, the operator KðmÞ

0 belongs to the algebra R. Let for example m¼ 2, sincePl

þ ¼ I � Pl�, we have

Kð2Þ0 ¼ Ma2a1Pl

þ þMb2b1Pl� þ T ,

where T :¼ �Ma1Pl�M

a2Plþ �Mb1Pl

þMb2Pl

� is a compact operator.Similarly, it is easy to see that

KðmÞ

0 ¼ Mam���a1Plþ þMbm���b1Pl

� þ T ,

where T is a compact operator.Consequently, if a ¼ am � � � a1 and b ¼ bm � � � b1 are both invertible, then the general-

ized singular operator KðmÞ

0 is Fredholmian.

6. Adjoint operators

There has been considerable effort in the literature (e.g., [36], [6,7]) to obtain theformula

ðSlÞ¼ nM S

l nM, ð6Þ

where the star denotes the adjoint operator in the Clifford valued Hilbert space L2ð�Þ.However, the situations in which the formula holds require fairly strong hypotheses onthe boundary �.

The scalar product in L2ð�Þ is given by

hu, vi ¼

Z�

u v dHn�1:

In particular, if � ¼ Rn�1, then nðxÞ ¼ en and

ðSlÞ¼ enS

len ¼ �e2nSl¼ S

l :

Consequently, the operators Pl� are orthogonal projections on L2ðRn�1Þ, i.e.

ðPl�Þ

¼ Pl

�.However, in the general case when � is a surface, the formula (6), does not allow to

express the dual nature of the singular operators Sl, Pl� and its right handed versions

Sr, Pr

�. Our main concern, in this section, is to show that the relevant adjointness here israther the relative one to the total subset of the real functionals defined by

h�, ui ¼ <

Z�

�ðyÞnðyÞuðyÞ dHnðyÞ:

Let us first briefly review some basic facts from Functional Analysis.

Commutators and singular integral operators 277

Let X be a real linear space. We denote by X 0 the space of all linear functionalsdefined on the space X. A space � � X 0 is total if the condition �ðxÞ ¼ 0, for all� 2 �, implies x¼ 0 (x 2 X). Evidently, the space X 0 is total.

Each total subspace of the space X 0 is called an adjoint space to X.Let � � X 0 be an adjoint space. To any linear operator A on X corresponds an

operator A defined on the space � with values in the space X 0:

ðAÞðxÞ ¼ ðAxÞ ðfor all x 2 X and for all 2 �Þ:

The operator A is called the adjoint operator to the operator A. We shall denote itby A

0, i.e. A0 ¼ A. Let us remark that ðA þ BÞ

0¼ A

0þ B

0 and I 0 ¼ I . A moreexhaustive information about these basic facts can be found in [31].

Next, we start by dealing with the determination of the adjoint operators to theoperators S

l and Pl�, defined on the subspace �0 � ðC0,�ð�ÞÞ0 of all real valued

functionals � of the form

h�, ui ¼ <

Z�

�ðyÞnðyÞuðyÞ dHnðyÞ,

where �ðxÞ 2 C0, �ð�Þ. This space may be identified with the space C0, �ð�Þ.

LEMMA 6.1 The subspace �0 is total.

Proof Fix a function u 2 C0,�ð�Þ, and suppose that h�, ui ¼ 0, for all � also in C0, �ð�Þ.Then, the equality

<

Z�

eA�ðyÞnðyÞuðyÞdHnðyÞ ¼ 0

remains valid for all � 2 C0, �ð�Þ and for any A � f1, . . . , ng.Taking into account the obvious identities eAeA ¼ e0 and <ðeA aÞ ¼ e0aA, the above

equality becomes Z�

�ðyÞnðyÞuðyÞdHnðyÞ

� �A

¼ 0, A � f1, . . . , ng:

Therefore we have

Z�

�ðyÞnðyÞuðyÞdHnðyÞ ¼ 0,

for all � 2 C0,�ð�Þ. This is enough to obtain the desired result u 0. g

Next we calculate the integral

h�, Slui ¼ <

Z�

�ðyÞnðyÞðSluÞðyÞdHn:

In accordance with the left and right handed Sokhotski-Plemelj formulae, we have

ðSluÞðyÞ ¼ ðPl

þuÞðyÞ � ðPl�uÞðyÞ

278 R.A. Blaya and J.B. Reyes

and

�ðyÞ ¼ ðPrþ�ÞðyÞ þ ðPr

��ÞðyÞ,

respectively.In virtue of the Cauchy theorem one hasZ

�

ðPr��ÞðyÞnðyÞðP

l�uÞðyÞdH

n ¼ 0:

Hence,

h�, Slui ¼ �<

Z�

ðPrþ� þ Pr

��ÞðyÞnðyÞðPlþu� Pl

�uÞðyÞ dHn

¼ �<

Z�

ðPrþ� � Pr

��ÞðyÞnðyÞðPlþuþ Pl

�uÞðyÞ dHn

¼ �<

Z�

ðSr�ÞnðyÞuðyÞdHn ¼ h�S

r�, ui:

Thus, we obtain the following useful relations, which express the dual nature of the leftand right monogenic function spaces. Compare with Lemma 5.1 in [3].

THEOREM 6.1 The adjoint operators to Sl and Pl

�, relative to the total subspace �0, are�S

r and Pr� respectively.

Acknowledgements

This paper has had a rather long period of preparation. The study of the topic coveredhere has been initiated while the second named author spent one of his study visitsat Gent University in 2000, and was eventually completed during the first author’svisit to the University of Oriente, Cuba, in March 2002. The authors are sincerelygrateful to Richard Delanghe and Frank Sommen for their various support of theClifford Analysis interest of the authors. They have been continuously providing uswith relevant information including references. The basic material reported here waspresented at the conference on Computational Methods and Function Theory heldin Aveiro, Portugal, in June 2001. The second author would like to thank ProfessorHelmuth R. Malonek for his invitation and financial support as well as the warmhospitality during this conference.

The authors also wish to express their sincere gratitude to Michael Shapiro for thereading of the first draft of this paper and various improvements and suggestions.During his visit to the University of Oriente in March 2002, we had useful discus-sions, especially about the Banach structure of the function space in the CliffordAnalysis.

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