Ricardo Abreu Blaya,¹ Juan Bory Reyes,² Richard Delanghe,³ and Frank Sommen³

Abstract

Let ℝ0,m+1(s) be the space of s-vectors (0≤s≤m+1) in the Clifford algebra ℝ0,m+1 constructed over the quadratic vector space ℝ0,m+1, let r,p,q∈ℕ with 0≤r≤m+1, 0≤p≤q, and r+2q≤m+1, and let ℝ0,m+1(r,p,q)=∑j=pq⨁ ℝ0,m+1(r+2j). Then, an ℝ0,m+1(r,p,q)-valued smooth function W defined in an open subset Ω⊂ℝm+1 is said to satisfy the generalized Moisil-Théodoresco system of type (r,p,q) if ∂xW=0 in Ω, where ∂x is the Dirac operator in ℝm+1. A structure theorem is proved for such functions, based on the construction of conjugate harmonic pairs. Furthermore, if Ω is bounded with boundary Γ, where Γ is an Ahlfors-David regular surface, and if W is a ℝ0,m+1(r,p,q)-valued Hölder continuous function on Γ, then necessary and sufficient conditions are given under which W admits on Γ a Cauchy integral decomposition W=W++W−.

1. Introduction

Clifford analysis, a function theory for the Dirac operator in Euclidean space (), generalizes in an elegant way the theory of holomorphic functions in the complex plane to a higher dimension and provides at the same time a refinement of the theory of harmonic functions. One of the basic properties relied upon in building up this function theory is the fact that the Dirac operator in factorizes the Laplacian through the relation . The Dirac operator is defined by , where and is an orthogonal basis for the quadratic space , the latter being the space equipped with a quadratic form of signature . By virtue of the basic multiplication rules(1.1)valid in the universal Clifford algebra constructed over , the factorization is thus obtained.

Notice that is a real linear associative algebra of dimension , having as standard basis the set where , , , and , the identity element in .

Now let be open and let be a -function in . Then, is said to be left monogenic in if in . The equation gives rise to a first-order linear elliptic system of partial differential equations in the components of By choosing as an orthogonal basis for the quadratic space , then inside , thus generates the Clifford algebra . It is then easily seen that(1.2)where . If the -valued -function in is decomposed following (1.2), that is where and are -valued -functions in , then in (1.3)where is the Cauchy-Riemann operator in and is the Dirac operator in .

Obviously the system (1.3) generalizes the classical Cauchy-Riemann system in the plane: it indeed suffices in the case to take -valued and -valued.

Left monogenic functions in are real analytic, whence by virtue of , they are in particular -valued and harmonic in .

As the algebra is noncommutative, one could as well consider right monogenic functions in , that is satisfies the equation in . If both and in , then is said to be two-sided monogenic in .

Notice also that through a natural linear isomorphism (see Section 2), the spaces and of smooth -valued functions and smooth differential forms in may be identified. The left and right actions of on then correspond to the actions of and on , where and denote, respectively, the exterior derivative and the coderivative operators. For the sake of completeness, let us recall the definition of and on the space of smooth -forms in , (see [1]).

For with where , , and are defined by(1.4) A smooth differential form satisfying in was called in [2] a self-conjugate differential form.

It thus becomes clear that through the identifications mentioned (see again Section 2) a subsystem of (1.3) corresponds to a subsystem of self-conjugate differential forms and vice versa. For instance, for fixed, the study of left monogenic -vector valued functions thus corresponds to the study of -forms satisfying the Hodge-de Rham system and .

Let us recall that the space of -vectors in () is defined by(1.5)For an account on recent investigations on subsystems of (1.3) or, equivalently, on the study of particular systems of self-conjugate differential forms, we refer to [2–10].

Now fix , take such that and , and put(1.6)The present paper is devoted to the study of -valued smooth functions in which are left monogenic in (i.e., which satisfy in ). The space of such functions is henceforth denoted by . The system defines a subsystem of (1.3), called the generalized Moisil-Théodoresco system of type in .

To be more precise, let us first recall the definition of the differential operators and acting on smooth -valued functions in . Call the space of smooth -valued functions in and put for ,(1.7) Note that is -valued while is -valued and that through the isomorphism , the action of and on corresponds to, respectively, the action of and on the space .

If is written as(1.8)we then have that the generalized Moisil-Théodoresco system of type reads as follows (see also Section 2):(1.9)Note that for and fixed, the system (1.9) reduces to the generalized Riesz system Its solutions are called harmonic multivector fields (see also [11]). We have(1.10)Furthermore, for , and fixed, the system (1.9) reduces to the Moisil-Théodoreco system in (see, e.g., [3]):(1.11)In the particular case, where , and , the original Moisil-Théodoresco system introduced in [12] is reobtained (see also [4]).

In this paper, two problems are dealt with; we list them as follows.

(i) To characterize the structure of solutions to the system (1.9).

It is proved in Section 4 (see Theorem 3.2) that, under certain geometric conditions upon , each corresponds to a harmonic potential belonging to a particular subspace of the space of harmonic -valued functions in .

The proof of Theorem 3.2 relies heavily on the construction of conjugate harmonic pairs elaborated in Section 3.

(ii) To characterize those which admit a Cauchy-type integral decomposition on of the form(1.12) where is the boundary of a bounded open domain in and denotes the space of -valued Hölder continuous functions of order on , . Putting , the elements and should also belong to and as such should be the boundary values of solutions and of (1.9) in and , respectively.

In Section 5, this problem is solved in terms of the Cauchy transform on , being an -dimensional Ahlfors-David regular surface (see Theorem 4.2).

In order to make the paper self-contained, we include in Section 2 some basic properties of Clifford algebras and Clifford analysis. For a general account of this function theory, we refer, for example, to the monographs [13–15].

2. Clifford Analysis: Notations and Some Basic Properties

Let again be an orthogonal basis for and let be the universal Clifford algebra over . As has already been mentioned in Section 1, is a real linear associative but noncommutative algebra of dimension ; its standard basis is given by the set and the basic multiplication rules are governed by (1.1). For fixed, the space of -vectors is defined by (1.5), leading to the decomposition(2.1)and the associated projection operators .

Note in particular that for , and that for , .

An element is therefore usually identified with .

For , the product splits in two parts, namely,(2.2)where is the scalar part of and is the 2-vector or bivector part of . They are given by(2.3) More generally, for and (), we have that the product decomposes into(2.4)where(2.5) Another useful decomposition of may be obtained by splitting it “along the -direction,” as indicated in (1.2). This in fact means that we split following and that within , the Clifford algebra is generated by the orthogonal basis of . denotes the space to which the original quadratic form of signature on has been restricted.

Following the decomposition (1.2), the element is then often identified with the so-called paravector .

Let us also recall that if is open and is an -valued -function in , then is said to be left monogenic in if in , being the Dirac operator in .

As already mentioned in (1.3), by putting , being the Dirac operator in , we have for ,(2.6)Let us recall that a pair of -valued harmonic functions in is said to be conjugate harmonic if is left monogenic in (see [16]).

Notice also that, when defining the conjugate of by , we have that .

If is a subspace of , then and denote, respectively, the spaces of left monogenic and harmonic -valued functions in . As we have that .

In particular, for such that , with , we have put in Section 1 (see (1.6)), and .

Furthermore, for fixed, a natural isomorphism(2.7)may be then defined as follows.

Put for ,(2.8)where for each with , and for all .

By means of the decomposition (2.1), may be extended by linearity to , thus leading to the isomorphism , where as usual .

It may be easily checked that the action of the exterior derivative and the co-derivative on then corresponds through to the left action of and on . For the definition of and (resp., and ) we refer to (1.4) and (1.7). In fact, taking into account the relations (2.5), the expressions (1.7) mean that for ,(2.9) Consequently, for , splits into(2.10)It thus follows that for , the system is given by (1.9).

Obviously, for , , while for , . Finally, notice that and that hence, as mentioned in Section 1, through , the left action of on corresponds to the action of on . We thus have on that .

The following notations will also be used:(2.11) Let us recall that if is contractible to a point, a refined version of the inverse Poincaré lemma then implies that(2.12) are surjective operators.

For the inverse Poincaré lemma and its refined version we refer to, respectively, [1, 17]. For more information concerning the interplay between differential forms and multivectors, the reader is referred to [17, 18].

Obviously, all notions, notations, and properties introduced above may be easily adapted to the case where is the orthogonal projection of on and and are the Dirac and Laplace operators in .

3. Conjugate Harmonic Pairs

Let be as in Section 1, let with , and decompose each following (1.2), that is(3.1)where and .

Then, with(3.2) Now suppose that , that is, is a conjugate harmonic pair in in the sense of [16]. Then, as already stated in (1.3),(3.3)By virtue of (2.10) and (3.2), the equations in (3.3) lead to the systems(3.4)(3.5)From (3.5) it thus follows that implies that in .

We now claim that, under certain geometric conditions upon , given , harmonic and -valued in , the condition in is sufficient to ensure the existence of a , harmonic and -valued in , which is conjugate harmonic to , that is .

In proving this statement, we will adapt where necessary the techniques worked out in [16] for constructing conjugate harmonic pairs.

Let again denote the orthogonal projection of on . Then, we suppose henceforth that satisfies the following conditions (C1) and (C2):

(C1) is normal with respect to the direction, that is, there exists such that for all , is connected and it contains the element ; (C2) is contractible to a point. The condition (C1) is sufficient for constructing harmonic conjugates to (see [16]), while the condition (C2) ensures the applicability of the inverse Poincaré lemma and its consequences in (see [17]).

As is well known, classical results of cohomology theory provide necessary and sufficient conditions for the validity of the inverse Poincaré lemma in . For convenience of the reader, we restrict ourselves to the condition (C2), thus making the inverse Poincaré lemma applicable for any closed or coclosed form in ().

Now assume that harmonic and that -valued in satisfies the condition in .

Put(3.6)where is a smooth -valued solution in of the equation(3.7)As is surjective (see [19]), such indeed exists and any other similar solution of (3.7) has the form where .

Fix a solution of (3.7). Then by construction, the corresponding determined by (3.6) belongs to (see [16]).

We now prove that there exists such that in ,(3.8)To this end, first notice that, as by assumption in , we have that in , whence .

As is surjective (see also (2.12)) there exists such that , that is, satisfies in the relations(3.9)Furthermore, put . Then, on the one hand,(3.10)while on the other hand(3.11)Consequently, in and is -valued in .

Now define by(3.12)Then by construction, and clearly in , .

Furthermore, as in , is -valued and obviously is -valued. As moreover , we get that is -valued.

Consequently, as , , that is, is a conjugate harmonic pair in .

We have thus proved the following theorem.

Theorem 3.1. Let be open and normal with respect to the -direction and let be contractible to a point. Furthermore, let be given. Then, admits a conjugate harmonic if and only if in .

Remarks

(1) If , then in , thus implying that the condition is automatically satisfied and that in constructing , no correction term should be added to (i.e., we may take in (3.12)).

(2) It is of course tacitly understood that if , then in the expression of (see (3.2)), the first term is taken to be identically zero in .

(3) The systems (3.4) and (3.5) show a lot of symmetry.

The following theorem (Theorem 3.2) holds, the proof of which is omitted.

Theorem 3.2. Let be open and normal with respect to the -direction and let be contractible to a point. Furthermore, let be given. Then, admits a conjugate harmonic if and only if in .

4. Structure Theorems

Assume that are such that and that with .

This section essentially deals with the construction of harmonic potentials corresponding to solutions of the generalized Moisil-Théodoresco system.

We start with the following lemma.

Lemma 4.1. Let be open and contractible to a point and let . The following properties are equivalent:

(i) , (ii) there exists such that .

Proof. It is clear that if , then is -valued. As moreover , ; whence (ii)⇒(i) is proved.

Conversely, assume that and put . From it follows that(4.1)By a refined version of the inverse Poincaré lemma (see [17]) we obtain from the first equation in (4.1) that there exists such that in (4.2)Analogously, the third equation in (4.1) implies the existence of(4.3)such that in (4.4)Put . Then,(4.5)and by virtue of (4.2) and (4.4),(4.6) But where is -valued and harmonic in . As is surjective (see [19]), there exists that such that .

Put . Then, clearly and(4.7)Finally, put . Then, is -valued and(4.8)As is obviously harmonic in , the proof is done.

Remarks

(1) In the case where and , we have that in (4.1) the equation is automatically satisfied. Putting , take such that and define by . Then, .

In the case where , the equation is automatically satisfied. An analogous reasoning to the one just made then leads to an appropriate such that .

(2) Obviously, in the case where as well as , the technique suggested in Remark (1) then produces such that .

(3) A particularly important example where as well as occurs when . Indeed, put for given real valued smooth functions in , ,(4.9)Then, for ,(4.10)Both equations in (4.10) give rise to the same system to be satisfied by , namely(4.11)The system (4.11) is the Fueter system in for so-called left regular functions of a quaternion variable; it lies at the basis of quaternionic analysis (see [20, 21]).

We have taken this example from [2], where it was proved in the framework of self-conjugate differential forms. We have inserted it here because it demonstrates how quaternionic analysis can be viewed upon as part of Clifford analysis in , namely as the theory of special solutions to a generalized Moisil-Théodoresco system in of type .

(4) In the case where and , Lemma 4.1 tells us that, given , there exists such that . This result was already obtained in [3, Lemma 3.1].

Theorem 4.2. Let be open and normal with respect to the -direction, let be contractible to a point, and let . The following properties are equivalent:

(i) , (ii) there exists with in such that .

Proof. (i)→(ii). Let and put, following (1.2), . Then, the pair is conjugate harmonic in with and .

Associate with the harmonic -valued potential given by (3.12), that is,(4.12)where in , , and .

As moreover in (see Theorem 3.1), it thus follows from (4.12) that in . Consequently, with and .

From , it is then easily obtained that is independent of and that in , , that is, . By virtue of Lemma 4.1, there exists such that ; whence .

Put . Then by construction,

(i),(ii),(iii); whence (i)→(ii) is proved.

Conversely, let with . Then clearly .

Remarks

(1) Theorem 4.2 tells us that each admits an - valued harmonic potential in satisfying .

(2) Let , that is, we take and . Then from Theorem 4.2, it follows that the following properties are equivalent:

(i),(ii)there exists with such that .

This characterization was already obtained in [3, Theorem 3.1].

5. Cauchy Integral Decompositions

Let be a bounded open subset with boundary where is a rectifiable closed Jordan curve such that for some constant and this for all and , where is the closed disc with center and radius and is the 1-dimensional Hausdorff measure on . Furthermore, let and let , .

In classical complex analysis, the following jump problem (5.1) is solved by means of the Cauchy transform:

“Find a pair of functions and , holomorphic in and with , such that are continuously extendable to and that on (5.1)where in (5.1), .”

Let be the Cauchy transform on , that is, for ,(5.2)where is the outward pointing unit normal at and is the elementary Lebesgue measure on .

Then, the following fundamental properties hold (see, e.g., [22]):

(i) is holomorphic and of the class on with ; (ii) Plemelj-Sokhotzki formulae:(5.3)where for ,(5.4)define the Hilbert transform on ; (iii) on .

It thus follows that the answer to the jump problem (5.1) is indeed given by .

The decomposition (iii) thus obtained is known as the Cauchy integral decomposition of on .

Now let be a bounded and open subset of with boundary . Then, in Clifford analysis, for suitable pairs of boundaries and -valued functions on , the Cauchy transform is defined by(5.5) where the following conditions hold.

(i) , , is the fundamental solution of the Dirac operator , where is the area of the unit sphere in . is -valued and monogenic in . (ii) is the outward pointing unit normal at . (iii) is the -dimensional Hausdorff measure on . For the definition of , see, for example, [23, 24]. In what follows we restrict ourselves to the following conditions on the pair (see also the remarks made at the end of this section).

(C1) is an -dimensional Ahlfors-David regular surface, that is, there exists a constant such that for all and ,(5.6)where is the closed ball in with center and radius and is the diameter of .For the definition of AD-regular surfaces, see, for example, [24, 25]. (C2) , , being the space of -valued Hölder continuous functions of order on .

Under the conditions (C1) and (C2), the following properties hold (see, e.g., [26–28]):

(i) is left monogenic in and ; (ii) (Plemelj-Sokhotzki formulae) the functions determined by(5.7)belong to , where(5.8)the integral being taken in the sense of principal values; (iii) , . It thus follows that, given a Hölder continuous -valued density on , the jump problem (5.9)

“Find and , belonging to and which are the boundary values of left monogenic functions and in, respectively, and with such that on (5.9) is solved by considering the Cauchy transform . Indeed, we can take in and in .

Now let again be a triplet satisfying and with , and let .

As is -valued, it is easily seen that is -valued. Consequently, if the jump problem (5.9) is formulated in terms of -valued Hölder continuous functions , , and on , then if we wish to solve it by means of the Cauchy transform , restrictions on have to be imposed, namely, in we should have(5.10) The very heart of the following theorem (Theorem 5.1) tells us that the conditions (5.10) are necessary and sufficient. Although the arguments used in proving Theorem 5.1 are similar to the ones given in the proof of [29, Theorem 4.1], for convenience of the reader we write them out in full detail.

Theorem 5.1. Let be open and bounded such that is an -dimensional Ahlfors-David regular surface and let with and . The following properties are equivalent:

(i) admits on a decomposition , where belong to and moreover are the boundary values of functions with , (ii) , (iii) , (iv) and in .

Proof. (i)→(ii). Assume that , where and satisfy the conditions given in (i). Then(5.11)In view of the assumptions made on , we have that in , in and that in .

Consequently,(5.12)As with (ii) is proved.

(ii)→(iii): Trivial.

(iii)→(iv). Let us first recall that is left monogenic in with .

According to the decomposition(5.13)and by the assumption made on , it follows from [1] that(5.14)Furthermore, as and split into an and an , respectively, into an and an multivector, we obtain from [22] that in (5.15) Moreover, as by assumption is -valued, by virtue of the Plemelj-Sokhotzki formulae, we obtain that on (5.16) Furthermore, and are Hölder continuous on .

It thus follows that and are left monogenic in and continuously extendable to . Painlevé's theorem (see [30]) then implies that and are left monogenic in .

Finally, as , we obtain by virtue of Liouville's theorem (see [31]) that and in .

(iv)→(i). First note that, as , by means of the Plemelj-Sokhotzki formulae, we have on that(5.17)In view of the assumption (iv) made, the functions defined in by obviously belong to and they satisfy all required properties.

Remarks

(1) In the last decades, intensive research has been done in studying the Cauchy integral transform and the associated singular integral operator on curves in the plane or on hypersurfaces in . Two types of boundary data are usually considered, namely a Hölder continuous density or an -density .

In this section, we have formulated the jump problems (5.1) and (5.9) in terms of Hölder continuous densities. The reason for this is that in proving some of the equivalences stated in Theorem 5.1, the continuous extendability of the Cauchy integral up to the boundary plays a crucial role. This becomes clear for instance when use is made of Painlevé's theorem in proving the implication “(iii)→(iv)”.

Note that for (), the continuous extendability of was already obtained in 1965 by V. Iftimie in the case where is a compact Liapunov surface (see [32]). For an overview of recent investigations on conditions which can be put on the pair , being a continuous density on , we refer the reader to [28, 30, 33–38]. In particular, we wish to point out that the introduction and the references in [35] contain a detailed account of the historical background of the jump problems (5.1) and (5.9).

(2) The case and was dealt with in [39]. For open, bounded and connected with -boundary such that is also connected, a set of equivalent properties was obtained ensuring the validity of the Cauchy integral decomposition for given.

(3) If where(5.18)then , that is the condition (ii) in Theorem 5.1 is satisfied.

Analogously, if where(5.19)then and so the condition (ii) in Theorem 5.1 is again satisfied.

Acknowledgments

The central idea for this paper arose while the second author was visiting the Department of Mathematical Analysis of Ghent University. He was supported by the Special Research Fund no. 01T13804 of Ghent University obtained for collaboration between the Clifford Research Group in Ghent and the Cuban Research Group in Clifford analysis, on the subject Boundary value theory in Clifford Analysis. Juan Bory Reyes wishes to thank the members of this Department for their kind hospitality. The authors also are much grateful to the referees: their questions and remarks contributed to improve substantially the final presentation of this paper.

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