Generalized Moisil-Théodoresco Systems and Cauchy Integral Decompositions

Abstract

Let ℝ^(𝑠)_0,𝑚+1 be the space of 𝑠-vectors (0≤𝑠≤𝑚+1) in the Clifford algebra ℝ_0,𝑚+1 constructed over the quadratic vector space ℝ^0,𝑚+1, let 𝑟,𝑝,𝑞∈ℕ with 0≤𝑟≤𝑚+1,0≤𝑝≤𝑞, and 𝑟+2𝑞≤𝑚+1, and let ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1=∑^𝑞_𝑗=𝑝⨁ℝ^(𝑟+2𝑗)_0,𝑚+1. Then, an ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1-valued smooth function 𝑊 defined in an open subset Ω⊂ℝ^𝑚+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 ℝ^𝑚+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 ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1-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 ℝ^𝑚+1 (𝑚≥2), 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 𝜕_𝑥 in ℝ^𝑚+1 factorizes the Laplacian Δ_𝑥 through the relation 𝜕²_𝑥=−Δ_𝑥. The Dirac operator 𝜕_𝑥 is defined by 𝜕_𝑥=∑^𝑚_𝑖=0𝑒_𝑖𝜕_𝑥𝑖, where 𝑥=(𝑥₀,𝑥₁,…,𝑥_𝑚)∈ℝ^𝑚+1 and 𝑒=(𝑒_𝑖∶𝑖=0,…,𝑚) is an orthogonal basis for the quadratic space ℝ^0,𝑚+1, the latter being the space ℝ^𝑚+1 equipped with a quadratic form of signature (0,𝑚+1). By virtue of the basic multiplication rules𝑒²_𝑖𝑒=−1,𝑖=0,1,…,𝑚,_𝑖𝑒_𝑗+𝑒_𝑗𝑒_𝑖=0,𝑖≠𝑗;𝑖,𝑗=0,1,…,𝑚, (1.1)valid in the universal Clifford algebra ℝ_0,𝑚+1 constructed over ℝ^0,𝑚+1, the factorization 𝜕²_𝑥=−Δ_𝑥 is thus obtained.

Notice that ℝ_0,𝑚+1 is a real linear associative algebra of dimension 2^𝑚+1, having as standard basis the set (𝑒_𝐴∶|𝐴|=𝑠,𝑠=0,1,…,𝑚+1), where 𝐴={𝑖₁,…,𝑖_𝑠}, 0≤𝑖₁<𝑖₂<⋯<𝑖_𝑠≤𝑚, 𝑒_𝐴=𝑒_𝑖1𝑒_𝑖2⋯𝑒_𝑖𝑠, and 𝑒_𝜙=1, the identity element in ℝ_0,𝑚+1.

Now let Ω⊂ℝ^𝑚+1 be open and let 𝐹∶Ω↦ℝ_0,𝑚+1 be a 𝐶₁-function in Ω. Then, 𝐹 is said to be left monogenic in Ω if 𝜕_𝑥𝐹=0 in Ω. The equation 𝜕_𝑥𝐹=0 gives rise to a first-order linear elliptic system of partial differential equations in the components 𝑓_𝐴 of 𝐹=∑_𝐴𝑓_𝐴𝑒_𝐴. By choosing 𝑒=(𝑒₁,…,𝑒_𝑚) as an orthogonal basis for the quadratic space ℝ^0,𝑚, then inside ℝ_0,𝑚+1, ℝ^0,𝑚 thus generates the Clifford algebra ℝ_0,𝑚. It is then easily seen thatℝ_0,𝑚+1=ℝ_0,𝑚⊕𝑒₀ℝ_0,𝑚, (1.2)where 𝑒₀=−𝑒₀. If the ℝ_0,𝑚+1-valued 𝐶₁-function 𝐹 in Ω is decomposed following (1.2), that is 𝐹=𝑈+𝑒₀𝑉 where 𝑈 and 𝑉 are ℝ_0,𝑚-valued 𝐶₁-functions in Ω, then in Ω𝜕_𝑥𝐹=0⇔𝐷_𝑥𝜕𝐹=0⇔^𝑥0𝑈+𝜕_𝑥𝜕𝑉=0,_𝑥𝑈+𝜕_𝑥0𝑉=0, (1.3)where 𝐷_𝑥=𝑒₀𝜕_𝑥=𝜕_𝑥0+𝑒₀𝜕_𝑥 is the Cauchy-Riemann operator in ℝ^𝑚+1 and 𝜕_𝑥=∑^𝑚_𝑗=1𝑒_𝑗𝜕_𝑥𝑗 is the Dirac operator in ℝ^𝑚.

Obviously the system (1.3) generalizes the classical Cauchy-Riemann system in the plane: it indeed suffices in the case 𝑚=1 to take 𝑈ℝ-valued and 𝑉ℝ𝑒₁-valued.

Left monogenic functions in Ω are real analytic, whence by virtue of 𝜕²_𝑥=−Δ_𝑥, they are in particular ℝ_0,𝑚+1-valued and harmonic in Ω.

As the algebra ℝ_0,𝑚+1 is noncommutative, one could as well consider right monogenic functions 𝐹 in Ω, that is 𝐹 satisfies the equation 𝐹𝜕_𝑥=0 in Ω. If both 𝜕_𝑥𝐹=0 and 𝐹𝜕_𝑥=0 in Ω, then 𝐹 is said to be two-sided monogenic in Ω.

Notice also that through a natural linear isomorphism Θ∶ℝ_0,𝑚+1↦Λℝ^𝑚+1 (see Section 2), the spaces ℰ(Ω;ℝ_0,𝑚+1) and ℰ(Ω;Λℝ^𝑚+1) of smooth ℝ_0,𝑚+1-valued functions and smooth differential forms in Ω may be identified. The left and right actions of 𝜕_𝑥 on ℰ(Ω;ℝ_0,𝑚+1) then correspond to the actions of 𝑑+𝑑^∗ and 𝑑−𝑑^∗ on ℰ(Ω;Λℝ^𝑚+1), where 𝑑 and 𝑑^∗ denote, respectively, the exterior derivative and the coderivative operators. For the sake of completeness, let us recall the definition of 𝑑 and 𝑑^∗ on the space ℰ(Ω;Λ^𝑠ℝ^𝑚+1) of smooth 𝑠-forms in Ω, 0≤𝑠≤𝑚+1 (see [1]).

For 𝜔^𝑠∈ℰ(Ω;Λ^𝑠ℝ^𝑚+1) with 𝜔^𝑠=∑_|𝐴|=𝑠𝜔^𝑠_𝐴𝑑𝑥^𝐴, where 𝑑𝑥^𝐴=𝑑𝑥^𝑖1⋀𝑑𝑥^𝑖2⋀⋯⋀𝑑𝑥^𝑖𝑠, 0≤𝑖₁<𝑖₂<⋯<𝑖_𝑠≤𝑚, 𝑑𝜔^𝑠 and 𝑑^∗𝜔^𝑠 are defined by𝑑𝜔^𝑠=_𝐴^𝑚_𝑖=0𝜕_𝑥𝑖𝜔^𝑠_𝐴𝑑𝑥^𝑖𝑑𝑥^𝐴,𝑑^∗𝜔^𝑠=_𝐴^𝑠_𝑗=1(−1)^𝑗𝜕_𝑥𝑖𝑗𝜔^𝑠_𝐴𝑑𝑥^{𝐴⧵{𝑖𝑗}}. (1.4) A smooth differential form 𝜔 satisfying (𝑑−𝑑^∗)𝜔=0 in Ω 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 0<𝑠<𝑚+1 fixed, the study of left monogenic 𝑠-vector valued functions 𝑊^𝑠 thus corresponds to the study of 𝑠-forms 𝜔^𝑠 satisfying the Hodge-de Rham system 𝑑𝜔^𝑠=0 and 𝑑^∗𝜔^𝑠=0.

Let us recall that the space ℝ^(𝑠)_0,𝑚+1 of 𝑠-vectors in ℝ_0,𝑚+1 (0≤𝑠≤𝑚+1) is defined byℝ^(𝑠)_0,𝑚+1=span_ℝ(𝑒_𝐴∶|𝐴|=𝑠). (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 0≤𝑟≤𝑚+1, take 𝑝,𝑞∈ℕ such that 0≤𝑝≤𝑞 and 𝑟+2𝑞≤𝑚+1, and putℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1=^𝑞_𝑗=𝑝ℝ^(𝑟+2𝑗)_0,𝑚+1. (1.6)The present paper is devoted to the study of ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1-valued smooth functions 𝑊 in Ω which are left monogenic in Ω (i.e., which satisfy 𝜕_𝑥𝑊=0 in Ω). The space of such functions is henceforth denoted by MT(Ω,ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1). The system 𝜕_𝑥𝑊=0 defines a subsystem of (1.3), called the generalized Moisil-Théodoresco system of type (𝑟,𝑝,𝑞) in ℝ^𝑚+1.

To be more precise, let us first recall the definition of the differential operators 𝜕⁺_𝑥 and 𝜕⁻_𝑥 acting on smooth ℝ^(𝑠)_0,𝑚+1-valued functions 𝑊^𝑠 in Ω. Call ℰ(Ω;ℝ^(𝑠)_0,𝑚+1) the space of smooth ℝ^(𝑠)_0,𝑚+1-valued functions in Ω and put for 𝑊^𝑠∈ℰ(Ω;ℝ^(𝑠)_0,𝑚+1),𝜕⁺_𝑥𝑊^𝑠=12(𝜕_𝑥𝑊^𝑠+(−1)^𝑠𝑊^𝑠𝜕_𝑥𝜕),_−𝑥𝑊^𝑠=12(𝜕_𝑥𝑊^𝑠−(−1)^𝑠𝑊^𝑠𝜕_𝑥). (1.7) Note that 𝜕⁺_𝑥𝑊^𝑠 is ℝ^(𝑠+1)_0,𝑚+1-valued while 𝜕⁻_𝑥𝑊^𝑠 is ℝ^(𝑠−1)_0,𝑚+1-valued and that through the isomorphism Θ, the action of 𝜕⁺_𝑥 and 𝜕⁻_𝑥 on ℰ(Ω;ℝ^(𝑠)_0,𝑚+1) corresponds to, respectively, the action of 𝑑 and 𝑑^∗ on the space ℰ(Ω;Λ^𝑠ℝ^𝑚+1).

If 𝑊∈ℰ(Ω;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1) is written as𝑊=^𝑞_𝑗=𝑝𝑊^𝑟+2𝑗,with𝑊^𝑟+2𝑗∈ℰ(Ω;ℝ^(𝑟+2𝑗)_0,𝑚+1),𝑗=𝑝,…,𝑞, (1.8)we then have that the generalized Moisil-Théodoresco system of type (𝑟,𝑝,𝑞) reads as follows (see also Section 2):𝜕_𝑥𝜕𝑊=0⇔^−𝑥𝑊^𝑟+2𝑝𝜕=0,_+𝑥𝑊^𝑟+2𝑗+𝜕⁻_𝑥𝑊^{𝑟+2(𝑗+1)}𝜕=0,𝑗=𝑝,…,𝑞−1,_+𝑥𝑊^𝑟+2𝑞=0. (1.9)Note that for 𝑝=𝑞=0 and 0<𝑟<𝑚+1 fixed, the system (1.9) reduces to the generalized Riesz system 𝜕_𝑥𝑊^𝑟=0. Its solutions are called harmonic multivector fields (see also [11]). We have𝜕_𝑥𝑊^𝑟𝜕=0⇔^−𝑥𝑊^𝑟𝜕=0,_+𝑥𝑊^𝑟=0. (1.10)Furthermore, for 𝑝=0, 𝑞=1, and 0≤𝑟≤𝑚+1 fixed, the system (1.9) reduces to the Moisil-Théodoreco system in ℝ^𝑚+1 (see, e.g., [3]):𝜕⁻_𝑥𝑊^𝑟=0,𝜕⁺_𝑥𝑊^𝑟+𝜕⁻_𝑥𝑊^𝑟+2=0,𝜕⁺_𝑥𝑊^𝑟+2=0. (1.11)In the particular case, where 𝑚+1=3, 𝑝=0,𝑞=1, and 𝑟=0, 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 Ω, each 𝑊∈MT(Ω,ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1) corresponds to a harmonic potential 𝐿 belonging to a particular subspace of the space ℋ(Ω,ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚) of harmonic ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚-valued functions in Ω.

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

(ii)To characterize those 𝑊∈𝐶^0,𝛼(Γ;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1) which admit a Cauchy-type integral decomposition on Γ of the form𝑊=𝑊₊+𝑊₋, (1.12) where Γ is the boundary of a bounded open domain Ω=Ω₊ in ℝ^𝑚+1 and 𝐶^0,𝛼(Γ;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1) denotes the space of ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1-valued Hölder continuous functions of order 𝛼 on Γ, 0<𝛼<1. Putting Ω₋=ℝ^𝑚+1⧵(Ω∪Γ), the elements 𝑊₊ and 𝑊₋ should also belong to 𝐶^0,𝛼(Γ;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1) and as such should be the boundary values of solutions 𝑊₊ and 𝑊₋ of (1.9) in Ω₊ and Ω₋, respectively.

In Section 5, this problem is solved in terms of the Cauchy transform 𝒞_Γ on Γ, Γ being an 𝑚-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 𝑒=(𝑒₀,𝑒₁,…,𝑒_𝑚) be an orthogonal basis for ℝ^0,𝑚+1 and let ℝ_0,𝑚+1 be the universal Clifford algebra over ℝ^0,𝑚+1. As has already been mentioned in Section 1, ℝ_0,𝑚+1 is a real linear associative but noncommutative algebra of dimension 2^𝑚+1; its standard basis is given by the set (𝑒_𝐴∶|𝐴|=𝑠,0≤𝑠≤𝑚+1) and the basic multiplication rules are governed by (1.1). For 0≤𝑠≤𝑚+1 fixed, the space ℝ^(𝑠)_0,𝑚+1 of 𝑠-vectors is defined by (1.5), leading to the decompositionℝ_0,𝑚+1=^𝑚+1_𝑠=0ℝ^(𝑠)_0,𝑚+1 (2.1)and the associated projection operators []_𝑠∶ℝ_0,𝑚+1↦ℝ^(𝑠)_0,𝑚+1.

Note in particular that for 𝑠=0, ℝ⁽⁰⁾_0,𝑚+1≅ℝ and that for 𝑠=1, ℝ⁽¹⁾_0,𝑚+1≅ℝ^0,𝑚+1.

An element 𝑥=(𝑥₀,𝑥₁,…,𝑥_𝑚)=(𝑥₀,𝑥)∈ℝ^𝑚+1 is therefore usually identified with 𝑥=∑^𝑚_𝑖=0𝑒_𝑖𝑥_𝑖∈ℝ^0,𝑚+1.

For 𝑥,𝑦∈ℝ⁽¹⁾_0,𝑚+1, the product 𝑥𝑦 splits in two parts, namely,𝑥𝑦=𝑥•𝑦+𝑥𝑦, (2.2)where 𝑥•𝑦=[𝑥𝑦]₀ is the scalar part of 𝑥𝑦 and 𝑥⋀𝑦=[𝑥𝑦]₂ is the 2-vector or bivector part of 𝑥𝑦. They are given by𝑥•𝑦=−^𝑚_𝑖=0𝑥_𝑖𝑦_𝑖,𝑥𝑦=_𝑖<𝑗𝑒_𝑖𝑒_𝑗(𝑥_𝑖𝑦_𝑗−𝑥_𝑗𝑦_𝑖). (2.3) More generally, for 𝑥∈ℝ⁽¹⁾_0,𝑚+1 and 𝜐∈ℝ^(𝑠)_0,𝑚+1 (0<𝑠<𝑚+1), we have that the product 𝑥𝜐 decomposes into𝑥𝜐=𝑥•𝜐+𝑥𝜐, (2.4)where𝑥•𝜐=[𝑥𝜐]_𝑠−1=12(𝑥𝜐−(−1)^𝑠𝑥𝜐𝑥),𝜐=[𝑥𝜐]_𝑠+1=12(𝑥𝜐+(−1)^𝑠𝜐𝑥). (2.5) Another useful decomposition of ℝ_0,𝑚+1 may be obtained by splitting it “along the 𝑒₀-direction,” as indicated in (1.2). This in fact means that we split ℝ^𝑚+1 following ℝ^𝑚+1=ℝ×ℝ^𝑚 and that within ℝ_0,𝑚+1, the Clifford algebra ℝ_0,𝑚 is generated by the orthogonal basis 𝑒=(𝑒₁,…,𝑒_𝑚) of ℝ^0,𝑚. ℝ^0,𝑚 denotes the space ℝ^𝑚 to which the original quadratic form of signature (0,𝑚+1) on ℝ^0,𝑚+1 has been restricted.

Following the decomposition (1.2), the element 𝑥=(𝑥₀,𝑥)∈ℝ^𝑚+1 is then often identified with the so-called paravector 𝑥=𝑥₀+𝑒₀𝑥=𝑥₀+𝑒₀∑^𝑚_𝑗=1𝑥_𝑗𝑒_𝑗∈ℝ⨁𝑒₀ℝ^0,𝑚.

Let us also recall that if Ω⊂ℝ^𝑚+1 is open and 𝐹 is an ℝ_0,𝑚+1-valued 𝐶₁-function in Ω, then 𝐹 is said to be left monogenic in Ω if 𝜕_𝑥𝐹=0 in Ω, 𝜕_𝑥=∑^𝑚_𝑖=0𝑒_𝑖𝜕_𝑥𝑖 being the Dirac operator in ℝ^𝑚+1.

As already mentioned in (1.3), by putting 𝐷_𝑥=𝑒₀𝜕_𝑥=𝜕_𝑥0+𝑒₀𝜕_𝑥, 𝜕_𝑥 being the Dirac operator in ℝ^𝑚, we have for 𝐹=𝑈+𝑒₀𝑉,𝜕_𝑥𝐹=0⇔𝐷_𝑥𝜕𝐹=0⇔^𝑥0𝑈+𝜕_𝑥𝜕𝑉=0,_𝑥𝑈+𝜕_𝑥0𝑉=0. (2.6)Let us recall that a pair (𝑈,𝑉) of ℝ_0,𝑚-valued harmonic functions in Ω is said to be conjugate harmonic if 𝐹=𝑈+𝑒₀𝑉 is left monogenic in Ω (see [16]).

Notice also that, when defining the conjugate 𝐷_𝑥 of 𝐷_𝑥 by 𝐷_𝑥=𝜕_𝑥0−𝑒₀𝜕_𝑥, we have that 𝐷_𝑥𝐷_𝑥=𝐷_𝑥𝐷_𝑥=Δ_𝑥.

If 𝑆 is a subspace of ℝ_0,𝑚+1, then ℳ(Ω,𝑆) and ℋ(Ω,𝑆) denote, respectively, the spaces of left monogenic and harmonic 𝑆-valued functions in Ω. As 𝜕²_𝑥=−Δ_𝑥, we have that ℳ(Ω,𝑆)⊂ℋ(Ω,𝑆).

In particular, for 𝑟,𝑝,𝑞∈ℕ such that 0≤𝑟≤𝑚+1, 0≤𝑝≤𝑞 with 𝑟+2𝑞≤𝑚+1, we have put in Section 1 (see (1.6)), ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1=∑^𝑞_𝑗=𝑝⨁ℝ^(𝑟+2𝑗)_0,𝑚+1 and ℳ(Ω,ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1)=MT(Ω,ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1).

Furthermore, for 0≤𝑠≤𝑚+1 fixed, a natural isomorphismΘ∶ℰ(Ω;ℝ^(𝑠)_0,𝑚+1)↦ℰ(Ω;Λ^𝑠ℝ^𝑚+1) (2.7)may be then defined as follows.

Put for 𝑊^𝑠=∑_|𝐴|=𝑠𝑊^𝑠_𝐴𝑒_𝐴∈ℰ(Ω;ℝ^(𝑠)_0,𝑚+1),Θ𝑊^𝑠=𝜔^𝑠⇔𝜔^𝑠=_|𝐴|=𝑠𝜔^𝑠_𝐴𝑑𝑥^𝐴, (2.8)where for each 𝐴={𝑖₁,…,𝑖_𝑠}⊂{0,…,𝑚} with 0≤𝑖₁<⋯<𝑖_𝑠≤𝑚, 𝑑𝑥^𝐴=𝑑𝑥^𝑖1⋀⋯⋀𝑑𝑥^𝑖𝑠 and 𝜔^𝑠_𝐴=𝑊^𝑠_𝐴 for all 𝐴.

By means of the decomposition (2.1), Θ may be extended by linearity to ℝ_0,𝑚+1, thus leading to the isomorphism Θ∶ℰ(Ω;ℝ_0,𝑚+1)↦ℰ(Ω;Λℝ^𝑚+1), where as usual Λℝ^𝑚+1=∑^𝑚+1_𝑠=0⨁Λ^𝑠ℝ^𝑚+1.

It may be easily checked that the action of the exterior derivative 𝑑 and the co-derivative 𝑑^∗ on ℰ(Ω;Λ^𝑠ℝ^𝑚+1) then corresponds through Θ to the left action of 𝜕⁺_𝑥 and 𝜕⁻_𝑥 on ℰ(Ω;ℝ^(𝑠)_0,𝑚+1). For the definition of 𝑑 and 𝑑^∗ (resp., 𝜕⁺_𝑥 and 𝜕⁻_𝑥) we refer to (1.4) and (1.7). In fact, taking into account the relations (2.5), the expressions (1.7) mean that for 𝑊^𝑠∈ℰ(Ω;ℝ^(𝑠)_0,𝑚+1),𝜕⁻_𝑥𝑊^𝑠=[𝜕_𝑥𝑊^𝑠]_𝑠−1,𝜕⁺_𝑥𝑊^𝑠=[𝜕_𝑥𝑊^𝑠]_𝑠+1. (2.9) Consequently, for 𝑊^𝑠∈ℰ(Ω;ℝ^(𝑠)_0,𝑚+1), 𝜕_𝑥𝑊^𝑠 splits into𝜕_𝑥𝑊^𝑠=[𝜕_𝑥𝑊^𝑠]_𝑠−1+[𝜕_𝑥𝑊^𝑠]_𝑠+1=𝜕⁻_𝑥𝑊^𝑠+𝜕⁺_𝑥𝑊^𝑠. (2.10)It thus follows that for 𝑊∈ℰ(Ω;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1), the system 𝜕_𝑥𝑊=0 is given by (1.9).

Obviously, for 𝑠=0, 𝜕⁻_𝑥𝑊⁰=0, while for 𝑠=𝑚+1, 𝜕⁺_𝑥𝑊^𝑚+1=0. Finally, notice that 𝜕_𝑥=𝜕⁺_𝑥+𝜕⁻_𝑥 and that hence, as mentioned in Section 1, through Θ, the left action of 𝜕_𝑥 on ℰ(Ω;ℝ_0,𝑚+1) corresponds to the action of 𝑑+𝑑^∗ on ℰ(Ω;Λℝ^𝑚+1). We thus have on ℰ(Ω;ℝ_0,𝑚+1) that Δ_𝑥=−(𝜕⁺_𝑥𝜕⁻_𝑥+𝜕⁻_𝑥𝜕⁺_𝑥).

The following notations will also be used:ker^𝑠𝜕⁺_𝑥={𝑊^𝑠∈ℰ(Ω;ℝ^(𝑠)_0,𝑚+1)∶𝜕⁺_𝑥𝑊^𝑠=0},ker^𝑠𝜕⁻_𝑥={𝑊^𝑠∈ℰ(Ω;ℝ^(𝑠)_0,𝑚+1)∶𝜕⁻_𝑥𝑊^𝑠=0}. (2.11) Let us recall that if Ω is contractible to a point, a refined version of the inverse Poincaré lemma then implies that𝜕⁺_𝑥𝜕⁻_𝑥∶ker^𝑠𝜕⁺_𝑥↦ker^𝑠𝜕⁺_𝑥,𝜕⁻_𝑥𝜕⁺_𝑥∶ker^𝑠𝜕⁻_𝑥↦ker^𝑠𝜕⁻_𝑥 (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 ^0𝑥0Ω^02𝑑𝑐⊂ℝ^𝑚 is the orthogonal projection of Ω on ℝ^𝑚 and 𝜕_𝑥 and Δ_𝑥 are the Dirac and Laplace operators in ℝ^𝑚.

3. Conjugate Harmonic Pairs

Let 𝑟,𝑝,𝑞∈ℕ be as in Section 1, let 𝑊∈ℰ(Ω;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1) with 𝑊=∑^𝑞_𝑗=𝑝𝑊^𝑟+2𝑗, and decompose each 𝑊^𝑟+2𝑗∈ℰ(Ω;ℝ^(𝑟+2𝑗)_0,𝑚+1) following (1.2), that is𝑊^𝑟+2𝑗=𝑈^𝑟+2𝑗+𝑒₀𝑉^{𝑟−1+2𝑗}, (3.1)where 𝑈^𝑟+2𝑗∈ℰ(Ω,ℝ^(𝑟+2𝑗)_0,𝑚) and 𝑉^{𝑟−1+2𝑗}∈ℰ(Ω,ℝ^{(𝑟−1+2𝑗)}_0,𝑚).

Then, 𝑊=𝑈+𝑒₀𝑉 with𝑈=^𝑞_𝑗=𝑝𝑈^𝑟+2𝑗ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚-valued,𝑉=^𝑞_𝑗=𝑝𝑉^{𝑟−1+2𝑗}ℝ^{(𝑟−1,𝑝,𝑞)}_0,𝑚-valued. (3.2) Now suppose that 𝑊∈MT(Ω,ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1), that is, (𝑈,𝑉) is a conjugate harmonic pair in Ω in the sense of [16]. Then, as already stated in (1.3),𝜕_𝑥𝜕𝑊=0⇔^𝑥0𝑈+𝜕_𝑥𝜕𝑉=0,_𝑥𝑈+𝜕_𝑥0𝑉=0. (3.3)By virtue of (2.10) and (3.2), the equations in (3.3) lead to the systems𝜕⁻_𝑥𝑉^{𝑟−1+2𝑝}𝜕=0,_𝑥0𝑈^𝑟+2𝑗+𝜕⁺_𝑥𝑉^{𝑟−1+2𝑗}+𝜕⁻_𝑥𝑉^{𝑟−1+2𝑗+2}𝜕=0,𝑗=𝑝,…,𝑞−1,_𝑥0𝑈^𝑟+2𝑞+𝜕⁺_𝑥𝑉^{𝑟−1+2𝑞}𝜕=0,_−𝑥𝑈^𝑟+2𝑝+𝜕_𝑥0𝑉^{𝑟−1+2𝑝}𝜕=0,_+𝑥𝑈^𝑟+2𝑗+𝜕⁻_𝑥𝑈^𝑟+2𝑗+2+𝜕_𝑥0𝑉^{𝑟−1+2𝑗+2}𝜕=0,𝑗=𝑝,…,𝑞−1,_+𝑥𝑈^𝑟+2𝑞=0. (3.4)(3.5)From (3.5) it thus follows that 𝑊∈MT(Ω,ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1) implies that 𝜕⁺_𝑥𝑈^𝑟+2𝑞=0 in Ω.

We now claim that, under certain geometric conditions upon Ω, given 𝑈=∑^𝑞_𝑗=𝑝𝑈^𝑟+2𝑗, harmonic and ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚-valued in Ω, the condition 𝜕⁺_𝑥𝑈^𝑟+2𝑞=0 in Ω is sufficient to ensure the existence of a ^0𝑥0𝑉^02𝑑𝑐, harmonic and ℝ^{(𝑟−1,𝑝,𝑞)}_0,𝑚-valued in Ω, which is conjugate harmonic to 𝑈, that is ^0𝑥𝑊^002𝑑𝑐=𝑈+𝑒𝑉₀^{0𝑥002𝑑𝑐}∈MT(Ω,ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1).

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

Let again ^0𝑥0Ω^02𝑑𝑐 denote the orthogonal projection of Ω on ℝ^𝑚. Then, we suppose henceforth that Ω satisfies the following conditions (C1) and (C2):

(C1)Ω is normal with respect to the 𝑒₀ direction, that is, there exists 𝑥^∗₀∈ℝ such that for all 𝑥∈Ω^{0𝑥002𝑑𝑐}, Ω∩{𝑥+𝑡𝑒₀∶𝑡∈ℝ} is connected and it contains the element (𝑥^∗₀,𝑥);(C2)^0𝑥0Ω^02𝑑𝑐 is contractible to a point. The condition (C1) is sufficient for constructing harmonic conjugates to 𝑈 (see [16]), while the condition (C2) ensures the applicability of the inverse Poincaré lemma and its consequences in ^0𝑥0Ω^02𝑑𝑐 (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 ^0𝑥0Ω^02𝑑𝑐. 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 𝜔^𝑠 in ^0𝑥0Ω^02𝑑𝑐 (0<𝑠<𝑚).

Now assume that 𝑈=∑^𝑞_𝑗=𝑝𝑈^𝑟+2𝑗 harmonic and that ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚-valued in Ω satisfies the condition 𝜕⁺_𝑥𝑈^𝑟+2𝑞=0 in Ω.

Put^0𝑥0𝐻^02𝑑𝑐(𝑥₀,𝑥)=^𝑥0_𝑥∗0𝑈(𝑡,𝑥)𝑑𝑡−ℎ^{0𝑥002𝑑𝑐}(𝑥), (3.6)where ^0𝑥0ℎ^02𝑑𝑐=∑ℎ^𝑞_𝑗=𝑝^{0𝑥002𝑑𝑐𝑟+2𝑗} is a smooth ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚-valued solution in ^0𝑥0Ω^02𝑑𝑐 of the equationΔℎ_𝑥^{0𝑥002𝑑𝑐}(𝑥)=𝜕_𝑥0𝑈(𝑥^∗₀,𝑥). (3.7)As Δ_𝑥∶ℰ(Ω^{0𝑥002𝑑𝑐};ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚)↦ℰ(Ω^{0𝑥002𝑑𝑐};ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚) is surjective (see [19]), such ^0𝑥0ℎ^02𝑑𝑐 indeed exists and any other similar solution of (3.7) has the form ^0𝑥0ℎ^02𝑑𝑐+ℎ, where ℎ∈ℋ(Ω^{0𝑥002𝑑𝑐};ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚).

Fix a solution ^0𝑥0ℎ^02𝑑𝑐 of (3.7). Then by construction, the corresponding ^0𝑥0𝐻^02𝑑𝑐 determined by (3.6) belongs to ℋ(Ω;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚) (see [16]).

We now prove that there exists ℎ^𝑟+2𝑞∈ℋ(Ω^{0𝑥002𝑑𝑐};ℝ^(𝑟+2𝑞)_0,𝑚) such that in ^0𝑥0Ω^02𝑑𝑐,𝜕⁺_𝑥(ℎ^{0𝑥002𝑑𝑐𝑟+2𝑞}+ℎ^𝑟+2𝑞)=0. (3.8)To this end, first notice that, as by assumption 𝜕⁺_𝑥𝑈^𝑟+2𝑞=0 in Ω, we have that 𝜕⁺_𝑥(𝜕_𝑥0𝑈^𝑟+2𝑞)(𝑥^∗₀,𝑥)=0 in ^0𝑥0Ω^02𝑑𝑐, whence 𝜕_𝑥0𝑈^𝑟+2𝑞(𝑥^∗₀,𝑥)∈ker^𝑟+2𝑞𝜕⁺_𝑥.

As 𝜕⁺_𝑥𝜕⁻_𝑥∶ker^𝑟+2𝑞𝜕⁺_𝑥↦ker^𝑟+2𝑞𝜕⁺_𝑥 is surjective (see also (2.12)) there exists ^0𝑥𝑊^{002𝑑𝑐𝑟+2𝑞}∈ker^𝑟+2𝑞𝜕⁺_𝑥 such that 𝜕⁺_𝑥𝜕𝑊⁻_𝑥^{0𝑥002𝑑𝑐𝑟+2𝑞}(𝑥)=−𝜕_𝑥0𝑈^𝑟+2𝑞(𝑥^∗₀,𝑥), that is, ^0𝑥𝑊^{002𝑑𝑐𝑟+2𝑞} satisfies in ^0𝑥0Ω^02𝑑𝑐 the relations𝜕𝑊⁺_𝑥^{0𝑥002𝑑𝑐𝑟+2𝑞}=0,𝜕⁺_𝑥𝜕𝑊⁻_𝑥^{0𝑥002𝑑𝑐𝑟+2𝑞}(𝑥)=−𝜕_𝑥0𝑈^𝑟+2𝑞(𝑥^∗₀,𝑥). (3.9)Furthermore, put ℎ^𝑟+2𝑞=𝑊^{0𝑥002𝑑𝑐𝑟+2𝑞}−ℎ^{0𝑥002𝑑𝑐𝑟+2𝑞}. Then, on the one hand,Δ_𝑥(ℎ^{0𝑥002𝑑𝑐𝑟+2𝑞}+ℎ^𝑟+2𝑞)(𝑥)=−(𝜕⁻_𝑥𝜕⁺_𝑥+𝜕⁺_𝑥𝜕⁻_𝑥)𝑊^{0𝑥002𝑑𝑐𝑟+2𝑞}(𝑥)=−𝜕⁺_𝑥𝜕𝑊⁻_𝑥^{0𝑥002𝑑𝑐𝑟+2𝑞}(𝑥)=𝜕_𝑥0𝑈^𝑟+2𝑞(𝑥^∗₀,𝑥), (3.10)while on the other handΔ_𝑥(ℎ^{0𝑥002𝑑𝑐𝑟+2𝑞}+ℎ^𝑟+2𝑞)(𝑥)=Δℎ_𝑥^{0𝑥002𝑑𝑐𝑟+2𝑞}(𝑥)+Δ_𝑥ℎ^𝑟+2𝑞(𝑥)=𝜕_𝑥0𝑈^𝑟+2𝑞(𝑥^∗₀,𝑥)+Δ_𝑥ℎ^𝑟+2𝑞(𝑥). (3.11)Consequently, Δ_𝑥ℎ^𝑟+2𝑞=0 in ^0𝑥0Ω^02𝑑𝑐 and 𝜕_𝑥(ℎ^{0𝑥002𝑑𝑐𝑟+2𝑞}+ℎ^𝑟+2𝑞) is ℝ^{(𝑟−1+2𝑞)}_0,𝑚-valued in ^0𝑥0Ω^02𝑑𝑐.

Now define 𝐻 by𝐻(𝑥₀,𝑥)=𝐻^{0𝑥002𝑑𝑐}(𝑥₀,𝑥)−ℎ^𝑟+2𝑞(𝑥). (3.12)Then by construction, 𝐻∈ℋ(Ω;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚) and clearly in Ω, 𝜕_𝑥0𝐻=𝑈.

Furthermore, as 𝜕⁺_𝑥𝑈^𝑟+2𝑞(𝑥₀,𝑥)=0 in Ω, ∫^𝑥0_𝑥∗0𝜕_𝑥𝑈(𝑡,𝑥)𝑑𝑡 is ℝ^{(𝑟−1,𝑝,𝑞)}_0,𝑚-valued and obviously 𝜕_𝑥(∑ℎ^𝑞−1_𝑗=𝑝^{0𝑥002𝑑𝑐𝑟+2𝑗}) is ℝ^{(𝑟−1,𝑝,𝑞)}_0,𝑚-valued. As moreover 𝜕_𝑥(ℎ^{0𝑥002𝑑𝑐𝑟+2𝑞}+ℎ^𝑟+2𝑞)=0, we get that ^0𝑥0𝑉^02𝑑𝑐(𝑥₀,𝑥)=−𝜕_𝑥𝐻(𝑥₀,𝑥) is ℝ^{(𝑟−1,𝑝,𝑞)}_0,𝑚-valued.

Consequently, as 𝐷_𝑥𝐷_𝑥=Δ_𝑥, ^0𝑥𝑊^002𝑑𝑐=𝐷_𝑥𝐻=𝑈+𝑒𝑉₀^{0𝑥002𝑑𝑐}∈MT(Ω,ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1), that is, (𝑈,𝑉^{0𝑥002𝑑𝑐}) is a conjugate harmonic pair in Ω.

We have thus proved the following theorem.

Theorem 3.1. Let Ω⊂ℝ^𝑚+1 be open and normal with respect to the 𝑒₀-direction and let Ω be contractible to a point. Furthermore, let 𝑈∈ℋ(Ω;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚) be given. Then, 𝑈 admits a conjugate harmonic ^0𝑥0𝑉^02𝑑𝑐∈ℋ(Ω;ℝ^{(𝑟−1,𝑝,𝑞)}_0,𝑚) if and only if 𝜕⁺_𝑥𝑈^𝑟+2𝑞=0 in Ω.

Remarks
(1) If 𝑟+2𝑞=𝑚+1, then 𝑈^𝑟+2𝑞_0,𝑚=𝑈^𝑚+1_0,𝑚≡0 in Ω, thus implying that the condition 𝜕⁺_𝑥𝑈^𝑟+2𝑞=0 is automatically satisfied and that in constructing 𝐻, no correction term ℎ^𝑟+2𝑞 should be added to ^0𝑥0𝐻^02𝑑𝑐 (i.e., we may take 𝐻=𝐻^{0𝑥002𝑑𝑐} in (3.12)).
(2) It is of course tacitly understood that if 𝑟=𝑝=0, then in the expression of 𝑉=∑^𝑞_𝑗=0𝑉^−1+2𝑗 (see (3.2)), the first term 𝑉⁻¹ is taken to be identically zero in Ω.
(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 Ω⊂ℝ^𝑚+1 be open and normal with respect to the 𝑒₀-direction and let Ω be contractible to a point. Furthermore, let 𝑉∈ℋ(Ω;ℝ^{(𝑟−1),𝑝,𝑞}_0,𝑚+1) be given. Then, 𝑉 admits a conjugate harmonic ^0𝑥0𝑈^02𝑑𝑐∈ℋ(Ω;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1) if and only if 𝜕⁻_𝑥𝑉^{𝑟−1+2𝑝}=0 in Ω.

4. Structure Theorems

Assume that 𝑟,𝑝,𝑞∈ℕ are such that 0≤𝑟<𝑚+1 and that 0≤𝑝<𝑞 with 𝑟+2𝑞≤𝑚+1.

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 Ω⊂ℝ^𝑚+1 be open and contractible to a point and let 𝑊∈ℰ(Ω,ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1). The following properties are equivalent:
(i)𝑊∈𝑀𝑇(Ω,ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1),(ii)there exists 𝐻∈ℋ(Ω,ℝ^{(𝑟+1,𝑝,𝑞−1)}_0,𝑚+1) such that 𝑊=𝜕_𝑥𝐻.

Proof. It is clear that if 𝐻∈ℋ(Ω,ℝ^{(𝑟+1,𝑝,𝑞−1)}_0,𝑚+1), then 𝑊=𝜕_𝑥𝐻 is ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1-valued. As moreover 𝜕²_𝑥=−Δ_𝑥, 𝑊∈MT(Ω,ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1); whence (ii)⇒(i) is proved.
Conversely, assume that 𝑊∈MT(Ω,ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1) and put 𝑊=∑^𝑞_𝑗=𝑝𝑊^𝑟+2𝑗. From 𝜕_𝑥𝑊=0 it follows that𝜕⁻_𝑥𝑊^𝑟+2𝑝𝜕=0,_+𝑥𝑊^𝑟+2𝑗+𝜕⁻_𝑥𝑊^{𝑟+2(𝑗+1)}𝜕=0,𝑗=𝑝,…,𝑞−1,_+𝑥𝑊^𝑟+2𝑞=0. (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 𝑊₋^𝑟+2𝑝+1∈ℰ(Ω;ℝ^{(𝑟+2𝑝+1)}_0,𝑚+1) such that in Ω𝑊^𝑟+2𝑝=𝜕⁻_𝑥𝑊₋^𝑟+2𝑝+1,𝜕⁺_𝑥𝑊₋^𝑟+2𝑝+1=0. (4.2)Analogously, the third equation in (4.1) implies the existence of𝑊₊^{𝑟+2𝑞−1}∈ℰ(Ω;ℝ^{(𝑟+2𝑞−1)}_0,𝑚+1) (4.3)such that in Ω𝑊^𝑟+2𝑞=𝜕⁺_𝑥𝑊₊^{𝑟+2𝑞−1},𝜕⁻_𝑥𝑊₊^{𝑟+2𝑞−1}=0. (4.4)Put 𝐻^′=𝑊₋^𝑟+2𝑝+1+𝑊₊^{𝑟+2𝑞−1}. Then,𝐻^′∈ℰ(Ω;ℝ^{(𝑟+1+2𝑝)}_0,𝑚+1⊕ℝ^{(𝑟+1+2(𝑞−1))}_0,𝑚+1) (4.5)and by virtue of (4.2) and (4.4),𝜕_𝑥𝐻^′=𝜕⁺_𝑥(𝑊₋^𝑟+2𝑝+1+𝑊₊^{𝑟+2𝑞−1})+𝜕⁻_𝑥(𝑊₋^𝑟+2𝑝+1+𝑊₊^{𝑟+2𝑞−1})=𝑊^𝑟+2𝑞+𝑊^𝑟+2𝑝. (4.6) But 𝑊=𝑊^2𝑟+𝑝+𝑊^∗+𝑊^𝑟+2𝑞, where 𝑊^∗=∑^𝑞−1_𝑗=𝑝+1𝑊^𝑟+2𝑗 is ℝ^{(𝑟,𝑝+1,𝑞−1)}_0,𝑚+1-valued and harmonic in Ω. As Δ_𝑥∶ℰ(Ω;ℝ^{(𝑟,𝑝+1,𝑞−1)}_0,𝑚+1)↦ℰ(Ω,ℝ^{(𝑟,𝑝+1,𝑞−1)}_0,𝑚+1) is surjective (see [19]), there exists that 𝐻^∗∈ℰ(Ω,ℝ^{(𝑟,𝑝+1,𝑞−1)}_0,𝑚+1) such that Δ_𝑥𝐻^∗=𝑊^∗.
Put 𝐻^′′=−𝜕_𝑥𝐻^∗. Then, clearly 𝐻^′′∈ℰ(Ω;ℝ^{(𝑟+1,𝑝,𝑞−1)}_0,𝑚+1) and𝜕_𝑥𝐻^′′=−𝜕²_𝑥𝐻^∗=Δ_𝑥𝐻^∗=𝑊^∗. (4.7)Finally, put 𝐻=𝐻^′+𝐻^′′. Then, 𝐻 is ℝ^{(𝑟+1,𝑝,𝑞−1)}_0,𝑚+1-valued and𝜕_𝑥𝐻=𝜕_𝑥𝐻^′+𝜕_𝑥𝐻^′′=^𝑞_𝑗=𝑝𝑊^𝑟+2𝑗=𝑊. (4.8)As 𝐻 is obviously harmonic in Ω, the proof is done.

Remarks
(1) In the case where 𝑟=𝑝=0 and 2𝑞<𝑚+1, we have that in (4.1) the equation 𝜕⁻_𝑥𝑊⁰=0 is automatically satisfied. Putting ^0𝑥𝑊^002𝑑𝑐=∑^𝑞−1_𝑗=0𝑊^2𝑗, take ^0𝑥0𝐻^02𝑑𝑐∈ℰ(Ω;ℝ^{(0,0,𝑞−1)}_0,𝑚+1) such that Δ𝐻_𝑥^{0𝑥002𝑑𝑐}=𝑊^{0𝑥002𝑑𝑐} and define 𝐻 by 𝐻=𝜕_𝑥(−𝐻^{0𝑥002𝑑𝑐}+𝑊₊^2𝑞−1). Then, 𝑊=𝜕_𝑥𝐻.
In the case where 𝑟+2𝑞=𝑚+1, the equation 𝜕⁺_𝑥𝑊^𝑚+1=0 is automatically satisfied. An analogous reasoning to the one just made then leads to an appropriate 𝐻∈ℋ(Ω;ℝ^{(𝑟+1,𝑝,𝑞−1)}_0,𝑚+1) such that 𝑊=𝜕_𝑥𝐻.
(2) Obviously, in the case where as well 𝑟=𝑝=0 as 2𝑞=𝑚+1, the technique suggested in Remark (1) then produces 𝐻∈ℋ(Ω;ℝ^{(1,0,(𝑚+1)/2−1)}_0,𝑚+1) such that 𝑊=𝜕_𝑥𝐻.
(3) A particularly important example where as well 𝑟=𝑝=0 as 2𝑞=𝑚+1 occurs when 𝑚+1=4. Indeed, put for given real valued smooth functions 𝑓_𝑖 in Ω⊂ℝ⁴, 𝑖=0,1,2,3,𝑊⁰=𝑓₀,𝑊²=𝑓₁(𝑒₂𝑒₃−𝑒₀𝑒₁)+𝑓₂(𝑒₃𝑒₁−𝑒₀𝑒₂)+𝑓₃(𝑒₁𝑒₂−𝑒₀𝑒₃𝑊),₄=𝑓₀𝑒₀𝑒₁𝑒₂𝑒₃. (4.9)Then, for 𝑊=𝑊⁰+𝑊²+𝑊⁴∈ℰ(Ω;ℝ^(0,0,2)_0,4),𝜕_𝑥𝜕𝑊=0⇔^+𝑥𝑊⁰+𝜕⁻_𝑥𝑊²𝜕=0,_+𝑥𝑊²+𝜕⁻_𝑥𝑊⁴=0. (4.10)Both equations in (4.10) give rise to the same system to be satisfied by 𝑓=(𝑓₀,𝑓₁,𝑓₂,𝑓₃), namely𝜕𝑓₀𝜕𝑥₀−𝜕𝑓₁𝜕𝑥₁−𝜕𝑓₂𝜕𝑥₂−𝜕𝑓₃𝜕𝑥₃=0,𝜕𝑓₀𝜕𝑥₁+𝜕𝑓₁𝜕𝑥₀−𝜕𝑓₂𝜕𝑥₃+𝜕𝑓₃𝜕𝑥₂=0,𝜕𝑓₀𝜕𝑥₂+𝜕𝑓₁𝜕𝑥₃+𝜕𝑓₂𝜕𝑥₀−𝜕𝑓₃𝜕𝑥₁=0,𝜕𝑓₀𝜕𝑥₃−𝜕𝑓₁𝜕𝑥₂+𝜕𝑓₂𝜕𝑥₁+𝜕𝑓₃𝜕𝑥₀=0. (4.11)The system (4.11) is the Fueter system in ℝ⁴ 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 ℝ⁴, namely as the theory of special solutions to a generalized Moisil-Théodoresco system in ℝ⁴ of type (0,0,2).
(4) In the case where 𝑝=0 and 𝑞=1, Lemma 4.1 tells us that, given 𝑊=𝑊^𝑟+𝑊^𝑟+2∈MT(Ω;ℝ^(𝑟)_0,𝑚+1⊕ℝ^(𝑟+2)_0,𝑚+1), there exists 𝐻∈ℋ(Ω;ℝ^(𝑟+1)_0,𝑚+1) such that 𝑊=𝜕_𝑥𝐻. This result was already obtained in [3, Lemma 3.1].

Theorem 4.2. Let Ω⊂ℝ^𝑚+1 be open and normal with respect to the 𝑒₀-direction, let ^0𝑥0Ω^02𝑑𝑐 be contractible to a point, and let 𝑊∈ℰ(Ω;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1). The following properties are equivalent:
(i)𝑊∈𝑀𝑇(Ω;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1),(ii)there exists 𝐿∈ℋ(Ω;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚) with 𝜕⁺_𝑥𝐿^𝑟+2𝑞=0 in Ω such that 𝑊=𝐷_𝑥𝐿.

Proof. (i)→(ii). Let 𝑊∈MT(Ω;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1) and put, following (1.2), 𝑊=𝑈+𝑒₀𝑉. Then, the pair (𝑈,𝑉) is conjugate harmonic in Ω with 𝑈∈ℋ(Ω;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚) and 𝑉∈ℋ(Ω;ℝ^{(𝑟−1,𝑝,𝑞)}_0,𝑚).
Associate with 𝑈 the harmonic ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚-valued potential 𝐻 given by (3.12), that is,𝐻(𝑥₀,𝑥)=^𝑥0_𝑥∗0𝑈(𝑡,𝑥)𝑑𝑡−(ℎ^{0𝑥002𝑑𝑐}+ℎ^𝑟+2𝑞)(𝑥), (4.12)where in ^0𝑥0Ω^02𝑑𝑐, Δℎ_𝑥^{0𝑥002𝑑𝑐}(𝑥)=𝜕_𝑥0𝑈(𝑥^∗₀,𝑥), ℎ^𝑟+2𝑞∈ℋ(Ω^{0𝑥002𝑑𝑐};ℝ^(𝑟+2𝑞)_0,𝑚), and 𝜕⁺_𝑥(ℎ^{0𝑥002𝑑𝑐𝑟+2𝑞}+ℎ^𝑟+2𝑞)=0.
As moreover 𝜕⁺_𝑥𝑈^𝑟+2𝑞=0 in Ω (see Theorem 3.1), it thus follows from (4.12) that 𝜕⁺_𝑥𝐻^𝑟+2𝑞=0 in Ω. Consequently, ^0𝑥𝑊^002𝑑𝑐=𝐷_𝑥𝐻=𝑈^{0𝑥002𝑑𝑐}+𝑒𝑉₀^{0𝑥002𝑑𝑐}∈MT(Ω;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1) with ^0𝑥0𝑈^02𝑑𝑐=𝜕_𝑥0𝐻=𝑈 and ^0𝑥0𝑉^02𝑑𝑐=−𝜕_𝑥𝐻.
From 𝜕_𝑥(𝑊^{0𝑥002𝑑𝑐}−𝑊)=0, it is then easily obtained that ^0𝑥0𝑉^02𝑑𝑐−𝑉 is independent of 𝑥₀ and that in ^0𝑥0Ω^02𝑑𝑐, 𝜕_𝑥(𝑉^{0𝑥002𝑑𝑐}−𝑉)=0, that is, ^0𝑥0𝑉^02𝑑𝑐−𝑉∈MT(Ω^{0𝑥002𝑑𝑐};ℝ^{(𝑟−1,𝑝,𝑞)}_0,𝑚). By virtue of Lemma 4.1, there exists 𝐻^∗∈ℋ(Ω^{0𝑥002𝑑𝑐};ℝ^{(𝑟,𝑝,𝑞−1)}_0,𝑚) such that ^0𝑥0𝑉^02𝑑𝑐−𝑉=𝜕_𝑥𝐻^∗; whence 𝑉=−𝜕_𝑥(𝐻+𝐻^∗).
Put 𝐿=𝐻+𝐻^∗. Then by construction,
(i)𝐿∈ℋ(Ω;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚),(ii)𝜕⁺_𝑥𝐿^𝑟+2𝑞=0,(iii)𝑊=𝐷_𝑥𝐿; whence (i)→(ii) is proved.
Conversely, let 𝐿∈ℋ(Ω;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚) with 𝜕⁺_𝑥𝐿^𝑟+2𝑞=0. Then clearly 𝑊=𝐷_𝑥𝐿∈MT(Ω;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1).

Remarks
(1) Theorem 4.2 tells us that each 𝑊∈MT(Ω;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1) admits an ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚- valued harmonic potential 𝐿 in Ω satisfying 𝜕⁺_𝑥𝐿^𝑟+2𝑞=0.
(2) Let 𝑊=𝑊^𝑟+𝑊^𝑟+2∈ℰ(Ω;ℝ^(𝑟)_0,𝑚+1⊕ℝ^(𝑟+2)_0,𝑚+1), that is, we take 𝑝=0 and 𝑞=1. Then from Theorem 4.2, it follows that the following properties are equivalent:
(i)𝑊∈MT(Ω;ℝ^(𝑟)_0,𝑚+1⊕ℝ^(𝑟+2)_0,𝑚+1),(ii)there exists 𝐿∈ℋ(Ω;ℝ^(𝑟)_0,𝑚⊕ℝ^(𝑟+2)_0,𝑚) with 𝜕⁺_𝑥𝐿^𝑟+2=0 such that 𝑊=𝐷_𝑥𝐿.
This characterization was already obtained in [3, Theorem 3.1].

5. Cauchy Integral Decompositions

Let Ω=Ω₊⊂ℂ be a bounded open subset with boundary 𝛾, where 𝛾 is a rectifiable closed Jordan curve such that for some constant 𝑐>0,𝐻¹(𝛾∩𝐵(𝑧,𝑝))≤𝑐𝑝 and this for all 𝑧∈𝛾 and 𝑝>0, where 𝐵(𝑧,𝑝) is the closed disc with center 𝑧 and radius 𝑝 and 𝐻¹ is the 1-dimensional Hausdorff measure on 𝛾. Furthermore, let Ω₋=ℂ⧵(Ω∪𝛾) and let 𝑓∈𝐶^0,𝛼(𝛾), 0<𝛼<1.

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

“Find a pair of functions 𝑓₊ and 𝑓₋, holomorphic in Ω₊ and Ω₋ with 𝑓₋(∞)=0, such that 𝑓_± are continuously extendable to 𝛾 and that on 𝛾𝑓=𝑓₊+𝑓₋, (5.1)where in (5.1), 𝑓_±(𝑢)=lim_{Ω±∋𝑧→𝑢}𝑓_±(𝑧),𝑢∈𝛾.”

Let 𝒞_𝛾 be the Cauchy transform on 𝐶^0,𝛼(𝛾), that is, for 𝑓∈𝐶^0,𝛼(𝛾),𝒞_𝛾1𝑓(𝑧)=2𝜋_𝛾1𝑡−𝑧𝜈(𝑡)𝑓(𝑡)𝑑𝐻¹(𝑡),𝑧∈ℂ⧵𝛾, (5.2)where 𝜈(𝑡) is the outward pointing unit normal at 𝑡∈𝛾 and 𝑑𝑠 is the elementary Lebesgue measure on 𝛾.

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

(i)𝒞_𝛾𝑓 is holomorphic and of the class 𝐶^0,𝛼 on Ω₊∪Ω₋ with 𝒞_𝛾𝑓(∞)=0;(ii)Plemelj-Sokhotzki formulae:𝒞^±_𝛾𝑓(𝑢)=lim_{Ω±∋𝑧→𝑢}𝒞_𝛾1𝑓(𝑧)=2(±𝑓(𝑢)+𝒮_𝛾𝑓(𝑢)),𝑢∈𝛾, (5.3)where for 𝑢∈𝛾,𝒮_𝛾1𝑓(𝑢)=𝜋PV_𝛾1𝑡−𝑢𝜈(𝑡)(𝑓(𝑡)−𝑓(𝑢))𝑑𝐻¹(𝑡)+𝑓(𝑢) (5.4)define the Hilbert transform 𝒮_𝛾 on 𝐶^0,𝛼(𝛾);(iii)𝑓=𝒞⁺_𝛾𝑓−𝒞⁻_𝛾𝑓 on 𝛾.

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

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

Now let Ω=Ω₊ be a bounded and open subset of ℝ^𝑚+1 with boundary Γ=𝜕Ω. Then, in Clifford analysis, for suitable pairs (Γ,𝑓) of boundaries Γ and ℝ_0,𝑚+1-valued functions 𝑓 on Γ, the Cauchy transform 𝒞_Γ𝑓 is defined by𝒞_Γ𝑓(𝑥)=_Γ𝐸(𝑦−𝑥)𝜈(𝑦)𝑓(𝑦)𝑑ℋ^𝑚(𝑦),𝑥∈ℝ^𝑚+1⧵Γ, (5.5) where the following conditions hold.

(i)𝐸(𝑥)=(−1/𝐴_𝑚+1)(𝑥/|𝑥|^𝑚+1), 𝑥∈ℝ^𝑚+1⧵{0}, is the fundamental solution of the Dirac operator 𝜕_𝑥, where 𝐴_𝑚+1 is the area of the unit sphere in ℝ^𝑚+1. 𝐸(𝑥) is ℝ⁽¹⁾_0,𝑚+1-valued and monogenic in ℝ^𝑚+1⧵{0}.(ii)𝜈(𝑦)=∑^𝑚_𝑖=0𝑒_𝑖𝜈_𝑖(𝑦) is the outward pointing unit normal at 𝑦∈Γ.(iii)ℋ^𝑚 is the 𝑚-dimensional Hausdorff measure on Γ. For the definition of ℋ^𝑚, see, for example, [23, 24]. In what follows we restrict ourselves to the following conditions on the pair (Γ,𝑓) (see also the remarks made at the end of this section).

(C1)Γ is an 𝑚-dimensional Ahlfors-David regular surface, that is, there exists a constant 𝑐>0 such that for all 𝑦∈Γ and 0<𝜌≤diamΓ,𝑐⁻¹𝜌^𝑚≤ℋ^𝑚(Γ∩𝐵(𝑦,𝜌))≤𝑐𝜌^𝑚, (5.6)where 𝐵(𝑦,𝜌) is the closed ball in ℝ^𝑚+1 with center 𝑦 and radius 𝜌 and diamΓ is the diameter of Γ.For the definition of AD-regular surfaces, see, for example, [24, 25].(C2)𝑓∈𝐶^0,𝛼(Γ;ℝ_0,𝑚+1), 0<𝛼<1, 𝐶^0,𝛼(Γ;ℝ_0,𝑚+1) being the space of ℝ_0,𝑚+1-valued Hölder continuous functions of order 𝛼 on Γ.

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

(i)𝒞_Γ𝑓 is left monogenic in ℝ^𝑚+1⧵Γ and 𝒞_Γ𝑓(∞)=0;(ii)(Plemelj-Sokhotzki formulae) the functions 𝒞^±_Γ𝑓 determined by𝒞^±_Γ𝑓(𝑢)=lim_{Ω±∋𝑥→𝑢}𝒞_Γ1𝑓(𝑥)=2(𝒮_Γ𝑓(𝑢)±𝑓(𝑢)),𝑢∈Γ, (5.7)belong to 𝐶^0,𝛼(Γ;ℝ_0,𝑚+1), where𝒮_Γ𝑓(𝑢)=2_Γ𝐸(𝑦−𝑢)𝜈(𝑦)[𝑓(𝑦)−𝑓(𝑢)]𝑑ℋ^𝑚(𝑦)+𝑓(𝑢),𝑢∈Γ, (5.8)the integral being taken in the sense of principal values;(iii)𝑓(𝑢)=𝒞⁺_Γ𝑓(𝑢)−𝒞⁻_Γ𝑓(𝑢), 𝑢∈Γ. It thus follows that, given a Hölder continuous ℝ_0,𝑚+1-valued density 𝑓 on Γ, the jump problem (5.9)

“Find 𝑓₊ and 𝑓₋, belonging to 𝐶^0,𝛼(Γ;ℝ_0,𝑚+1) and which are the boundary values of left monogenic functions 𝑓₊ and 𝑓₋ in, respectively, Ω₊ and Ω₋ with 𝑓₋(∞)=0 such that on Γ𝑓=𝑓₊+𝑓₋^” (5.9) is solved by considering the Cauchy transform 𝒞_Γ𝑓. Indeed, we can take 𝑓₊=𝒞_Γ𝑓 in Ω₊ and 𝑓₋=−𝒞_Γ𝑓 in Ω₋.

Now let again 𝑟,𝑝,𝑞∈ℕ be a triplet satisfying 0≤𝑟≤𝑚+1 and 0≤𝑝<𝑞 with 𝑟+2𝑞≤𝑚+1, and let 𝑊∈𝐶^0,𝛼(Γ;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1).

As 𝐸(𝑦−𝑥)𝜈(𝑦) is ℝ⁽⁰⁾_0,𝑚+1⊕ℝ⁽²⁾_0,𝑚+1-valued, it is easily seen that 𝒞_Γ𝑊 is ℝ^{(𝑟−2+2𝑝)}_0,𝑚+1⊕ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1⊕ℝ^{(𝑟+2+2𝑞)}_0,𝑚+1-valued. Consequently, if the jump problem (5.9) is formulated in terms of ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1-valued Hölder continuous functions 𝑊, 𝑊₊, and 𝑊₋ on Γ, then if we wish to solve it by means of the Cauchy transform 𝒞_Γ, restrictions on 𝒞_Γ𝑊 have to be imposed, namely, in ℝ^𝑚+1⧵Γ we should have[𝒞_Γ𝑊]_{𝑟−2+2𝑝}≡0,[𝒞_Γ𝑊]_𝑟+2+2𝑞≡0. (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 Ω⊂ℝ^𝑚+1 be open and bounded such that Γ=𝜕Ω is an 𝑚-dimensional Ahlfors-David regular surface and let 𝑊∈𝐶^0,𝛼(Γ;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1) with 0<𝛼<1 and 𝑝<𝑞. The following properties are equivalent:
(i)𝑊 admits on Γ a decomposition 𝑊=𝑊₊+𝑊₋, where 𝑊_± belong to 𝐶^0,𝛼(Γ;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1) and moreover are the boundary values of functions 𝑊_±∈𝑀𝑇(Ω_±;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1) with 𝑊₋(∞)=0,(ii)𝒞_Γ𝑊∈𝑀𝑇(ℝ^𝑚+1⧵Γ;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1),(iii)∑^𝑞_𝑗=𝑝[𝒞_Γ𝑊]_𝑟+2𝑗∈𝑀𝑇(ℝ^𝑚+1⧵Γ;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1),(iv)[𝒞_Γ𝑊]_{𝑟−2+2𝑝}≡0 and [𝒞_Γ𝑊]_𝑟+2+2𝑞≡0 in ℝ^𝑚+1.

Proof. (i)→(ii). Assume that 𝑊=𝑊₊+𝑊₋, where 𝑊₊ and 𝑊₋ satisfy the conditions given in (i). Then𝒞_Γ𝑊=𝒞_Γ𝑊₊+𝒞_Γ𝑊₋. (5.11)In view of the assumptions made on 𝑊_±, we have that 𝒞_Γ𝑊₊=0 in Ω₋, 𝒞_Γ𝑊₋=0 in Ω₊ and that 𝒞_Γ𝑊_±=𝑊_± in Ω_±.
Consequently,𝒞_Γ𝑊𝑊=⁺inΩ₊,𝑊₋inΩ₋. (5.12)As 𝑊_±∈MT(Ω_±;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1) with 𝑊₋(∞)=0, (ii) is proved.
(ii)→(iii): Trivial.
(iii)→(iv). Let us first recall that 𝒞_Γ𝑊 is left monogenic in ℝ^𝑚+1⧵Γ with 𝒞_Γ𝑊(∞)=0.
According to the decomposition𝒞_Γ𝑊=[𝒞_Γ𝑊]_{𝑟−2+2𝑝}+^𝑞_𝑗=𝑝[𝒞_Γ𝑊]_𝑟+2𝑗+[𝒞_Γ𝑊]_𝑟+2+2𝑞 (5.13)and by the assumption made on ∑^𝑞_𝑗=𝑝[𝒞_Γ𝑊]_𝑟+2𝑗, it follows from [1] that𝜕_𝑥([𝒞_Γ𝑊]_{𝑟−2+2𝑝})+𝜕_𝑥([𝒞_Γ𝑊]_𝑟+2+2𝑞)=0inℝ^𝑚+1⧵Γ. (5.14)Furthermore, as 𝜕_𝑥([𝒞_Γ𝑊]_{𝑟−2+2𝑝}) and 𝜕_𝑥([𝒞_Γ𝑊]_𝑟+2+2𝑞) split into an (𝑟−3+2𝑝) and an (𝑟−1+2𝑝), respectively, into an (𝑟+1+2𝑞) and an (𝑟+3+2𝑞) multivector, we obtain from [22] that in ℝ^𝑚+1⧵Γ𝜕_𝑥([𝒞_Γ𝑊]_{𝑟−2+2𝑝}𝜕)=0,_𝑥([𝒞_Γ𝑊]_𝑟+2+2𝑞)=0. (5.15) Moreover, as by assumption 𝑊 is ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1-valued, by virtue of the Plemelj-Sokhotzki formulae, we obtain that on Γ[𝒞⁺_Γ𝑊]_{𝑟−2+2𝑝}=[𝒞⁻_Γ𝑊]_{𝑟−2+2𝑝},[𝒞⁺_Γ𝑊]_𝑟+2+2𝑞=[𝒞⁻_Γ𝑊]_𝑟+2+2𝑞. (5.16) Furthermore, [𝒞^±_Γ𝑊]_{𝑟−2+2𝑝} and [𝒞^±_Γ𝑊]_𝑟+2+2𝑞 are Hölder continuous on Γ.
It thus follows that [𝒞_Γ𝑊]_{𝑟−2+2𝑝} and [𝒞_Γ𝑊]_𝑟+2+2𝑞 are left monogenic in ℝ^𝑚+1⧵Γ and continuously extendable to Γ. Painlevé's theorem (see [30]) then implies that [𝒞_Γ𝑊]_{𝑟−2+2𝑝} and [𝒞_Γ𝑊]_𝑟+2+2𝑞 are left monogenic in ℝ^𝑚+1.
Finally, as [𝒞_Γ𝑊]_{𝑟−2+2𝑝}(∞)=[𝒞_Γ𝑊]_𝑟+2+2𝑞(∞)=0, we obtain by virtue of Liouville's theorem (see [31]) that [𝒞_Γ𝑊]_{𝑟−2+2𝑝}≡0 and [𝒞_Γ𝑊]_𝑟+2+2𝑞≡0 in ℝ^𝑚+1.
(iv)→(i). First note that, as 𝑊∈𝐶^0,𝛼(Γ;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1), by means of the Plemelj-Sokhotzki formulae, we have on Γ that𝑊=𝒞⁺_Γ𝑊−𝒞⁻_Γ𝑊. (5.17)In view of the assumption (iv) made, the functions 𝑊_± defined in Ω_± by 𝑊_±=±𝒞_Γ𝑊 obviously belong to MT(Ω_±;ℝ^{(𝑟,𝑝,𝑞)}_0,𝑚+1) 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 𝛾 in the plane or on hypersurfaces Γ in ℝ^𝑚+1(𝑚≥2). Two types of boundary data are usually considered, namely a Hölder continuous density or an 𝐿_𝑝-density (1<𝑝<+∞).
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 𝑓∈𝐶^0,𝛼(Γ;ℝ_0,𝑚+1) (0<𝛼<1), the continuous extendability of 𝒞_Γ𝑓 was already obtained in 1965 by V. Iftimie in the case where Γ is a compact Liapunov surface (see [32]). For an overview of recent investigations on conditions which can be put on the pair (Γ,𝑓), 𝑓 being a continuous density on Γ, 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 𝑝=𝑞=0 and 0<𝑟<𝑚+1 was dealt with in [39]. For Ω⊂ℝ^𝑚+1 open, bounded and connected with 𝐶_∞-boundary Γ such that ℝ^𝑚+1⧵(Ω∪Γ) is also connected, a set of equivalent properties was obtained ensuring the validity of the Cauchy integral decomposition for 𝑊^𝑟∈ℰ(Γ,ℝ^(𝑟)_0,𝑚+1) given.
(3) If 𝑊∈𝐶^0,𝛼(Γ;ℝ⁺_0,𝑚+1), whereℝ⁺_0,𝑚+1=_𝑠even⊕ℝ^(𝑠)_0,𝑚+1, (5.18)then 𝒞_Γ𝑊∈MT(ℝ^𝑚+1⧵Γ;ℝ⁺_0,𝑚+1), that is the condition (ii) in Theorem 5.1 is satisfied.
Analogously, if 𝑊∈𝐶^0,𝛼(Γ;ℝ⁻_0,𝑚+1) whereℝ⁻_0,𝑚+1=_𝑠odd⊕ℝ^(𝑠)_0,𝑚+1, (5.19)then 𝒞_Γ𝑊∈MT(ℝ^𝑚+1⧵Γ;ℝ⁻_0,𝑚+1) 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.