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 rulesvalid 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 thatwhere . If the -valued -function in is decomposed following (1.2), that is where and are -valued -functions in , then in 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 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 byFor 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 putThe 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 , 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 aswe then have that the generalized Moisil-Théodoresco system of type reads as follows (see also Section 2):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 haveFurthermore, for , and fixed, the system (1.9) reduces to the Moisil-Théodoreco system in (see, e.g., [3]):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 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 decompositionand 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,where is the scalar part of and is the 2-vector or bivector part of . They are given by More generally, for and (), we have that the product decomposes intowhere 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 ,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 isomorphismmay be then defined as follows.

Put for ,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 , Consequently, for , splits intoIt 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: Let us recall that if is contractible to a point, a refined version of the inverse Poincaré lemma then implies that 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 iswhere and .

Then, with Now suppose that , that is, is a conjugate harmonic pair in in the sense of [16]. Then, as already stated in (1.3),By virtue of (2.10) and (3.2), the equations in (3.3) lead to the systemsFrom (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