# Generalized Moisil-Théodoresco Systems and Cauchy Integral Decompositions

Authors: Ricardo Abreu Blaya; Juan Bory Reyes; Richard Delanghe; Frank Sommen
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

---

## Abstract

Letℝ0,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−.


---

## Body

## 1. Introduction

Clifford analysis, a function theory for the Dirac
operator in Euclidean spaceℝm+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 ℝm+1 factorizes the
Laplacian Δx through the
relation ∂x2=−Δx. The Dirac operator ∂x is defined by ∂x=∑i=0mei∂xi, where x=(x0,x1,…,xm)∈ℝm+1 and e=(ei:i=0,…,m) is an
orthogonal basis for the quadratic space ℝ0,m+1, the latter being the space ℝm+1 equipped with a
quadratic form of signature (0,m+1). By virtue of the basic multiplication rules(1.1)ei2=−1,i=0,1,…,m,eiej+ejei=0,i≠j;i,j=0,1,…,m,valid in the universal Clifford
algebra ℝ0,m+1 constructed
over ℝ0,m+1, the factorization ∂x2=−Δx is thus
obtained.Notice thatℝ0,m+1 is a real
linear associative algebra of dimension 2m+1, having as standard basis the set (eA:|A|=s,s=0,1,…,m+1),
where
A={i1,…,is}, 0≤i1<i2<⋯<is≤m, eA=ei1ei2⋯eis, and eϕ=1, the identity element in ℝ0,m+1.Now letΩ⊂ℝm+1 be open and let F:Ω↦ℝ0,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 fA of F=∑AfAeA. By choosing e¯=(e1,…,em) as an
orthogonal basis for the quadratic space ℝ0,m, then inside ℝ0,m+1, ℝ0,m thus generates
the Clifford algebra ℝ0,m. It is then easily seen that(1.2)ℝ0,m+1=ℝ0,m⊕e¯0ℝ0,m,where e¯0=−e0. If the ℝ0,m+1-valued C1-function F in Ω is decomposed
following (1.2), that is F=U+e¯0V where U and V are ℝ0,m-valued C1-functions in Ω, then in Ω(1.3)∂xF=0⇔DxF=0⇔{∂x0U+∂x¯V=0,∂x¯U+∂x0V=0,where Dx=e¯0∂x=∂x0+e¯0∂x¯ is the
Cauchy-Riemann operator in ℝm+1 and ∂x¯=∑j=1mej∂xj is the Dirac
operator in ℝm.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 inΩ are real
analytic, whence by virtue of ∂x2=−Δx, they are in particular ℝ0,m+1-valued and
harmonic in Ω.As the algebraℝ0,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 isomorphismΘ:ℝ0,m+1↦Λℝm+1 (see Section 2), the spaces ℰ(Ω;ℝ0,m+1) and ℰ(Ω;Λℝm+1) of smooth ℝ0,m+1-valued
functions and smooth differential forms in Ω may be
identified. The left and right actions of ∂x on ℰ(Ω;ℝ0,m+1) then correspond
to the actions of d+d* and d−d* on ℰ(Ω;Λℝm+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;Λsℝm+1) of smooth s-forms in Ω, 0≤s≤m+1 (see [1]).Forωs∈ℰ(Ω;Λsℝm+1) with ωs=∑|A|=sωAsdxA, where dxA=dxi1⋀dxi2⋀⋯⋀dxis, 0≤i1<i2<⋯<is≤m, dωs and d*ωs are defined
by(1.4)dωs=∑A∑i=0m∂xiωAsdxi⋀dxA,d*ωs=∑A∑j=1s(−1)j∂xijωAsdxA∖{ij}. 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 Section2) 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 Ws thus
corresponds to the study of s-forms ωs satisfying the
Hodge-de Rham system dωs=0 and d*ωs=0.Let us recall that the spaceℝ0,m+1(s) of s-vectors in ℝ0,m+1 (0≤s≤m+1) is defined by(1.5)ℝ0,m+1(s)=spanℝ(eA:|A|=s).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 fix0≤r≤m+1, take p,q∈ℕ such that 0≤p≤q and r+2q≤m+1, and put(1.6)ℝ0,m+1(r,p,q)=∑j=pq⨁ℝ0,m+1(r+2j).The present paper is devoted to
the study of ℝ0,m+1(r,p,q)-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,ℝ0,m+1(r,p,q)). The system ∂xW=0 defines a
subsystem of (1.3), called the generalized Moisil-Théodoresco system of type (r,p,q) in ℝm+1.To be more precise, let us first recall the definition
of the differential operators∂x+ and ∂x− acting on
smooth ℝ0,m+1(s)-valued
functions Ws in Ω. Call ℰ(Ω;ℝ0,m+1(s)) the space of
smooth ℝ0,m+1(s)-valued
functions in Ω and put for Ws∈ℰ(Ω;ℝ0,m+1(s)),(1.7)∂x+Ws=12(∂xWs+(−1)sWs∂x),∂x−Ws=12(∂xWs−(−1)sWs∂x).
Note that ∂x+Ws is ℝ0,m+1(s+1)-valued while ∂x−Ws is ℝ0,m+1(s−1)-valued and
that through the isomorphism Θ, the action of ∂x+ and ∂x− on ℰ(Ω;ℝ0,m+1(s)) corresponds to,
respectively, the action of d and d* on the space ℰ(Ω;Λsℝm+1).IfW∈ℰ(Ω;ℝ0,m+1(r,p,q)) is written
as(1.8)W=∑j=pqWr+2j,withWr+2j∈ℰ(Ω;ℝ0,m+1(r+2j)),j=p,…,q,we then have that the
generalized Moisil-Théodoresco system of type (r,p,q) reads as
follows (see also Section 2):(1.9)∂xW=0⇔{∂x−Wr+2p=0,∂x+Wr+2j+∂x−Wr+2(j+1)=0,j=p,…,q−1,∂x+Wr+2q=0.Note that for p=q=0 and 0<r<m+1 fixed, the
system (1.9) reduces to the generalized Riesz system ∂xWr=0. Its solutions are called harmonic multivector fields (see also
[11]). We
have(1.10)∂xWr=0⇔{∂x−Wr=0,∂x+Wr=0.Furthermore, for p=0, q=1, and 0≤r≤m+1 fixed, the
system (1.9) reduces to the Moisil-Théodoreco system in ℝm+1 (see, e.g.,
[3]):(1.11)∂x−Wr=0,∂x+Wr+∂x−Wr+2=0,∂x+Wr+2=0.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
Section4 (see Theorem 3.2) that, under certain geometric conditions upon Ω, each W∈MT(Ω,ℝ0,m+1(r,p,q)) corresponds to
a harmonic potential L belonging to a
particular subspace of the space ℋ(Ω,ℝ0,m(r,p,q)) of harmonic ℝ0,m(r,p,q)-valued
functions in Ω.The proof of Theorem3.2 relies heavily on the
construction of conjugate harmonic pairs elaborated in Section 3.(ii)
To characterize thoseW∈C0,α(Γ;ℝ0,m+1(r,p,q)) which admit a
Cauchy-type integral decomposition on Γ of the
form(1.12)W=W++W−,
where Γ is the boundary
of a bounded open domain Ω=Ω+ in ℝm+1 and C0,α(Γ;ℝ0,m+1(r,p,q)) denotes the
space of ℝ0,m+1(r,p,q)-valued Hölder
continuous functions of order α on Γ, 0<α<1. Putting Ω−=ℝm+1∖(Ω∪Γ), the elements W+ and W− should also
belong to C0,α(Γ;ℝ0,m+1(r,p,q)) and as such
should be the boundary values of solutions W+ and W− of (1.9) in Ω+ and Ω−, respectively.In Section5, this problem is solved in terms of the
Cauchy transform 𝒞Γ 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 Section2 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 againe=(e0,e1,…,em) be an
orthogonal basis for ℝ0,m+1 and let ℝ0,m+1 be the universal
Clifford algebra over ℝ0,m+1. As has already been mentioned in Section 1, ℝ0,m+1 is a real
linear associative but noncommutative algebra of dimension 2m+1; its standard basis is given by the set (eA:|A|=s,0≤s≤m+1) and the basic
multiplication rules are governed by (1.1). For 0≤s≤m+1 fixed, the
space ℝ0,m+1(s) of s-vectors is
defined by (1.5), leading to the decomposition(2.1)ℝ0,m+1=∑s=0m+1⨁ℝ0,m+1(s)and the associated projection
operators []s:ℝ0,m+1↦ℝ0,m+1(s).Note in particular that fors=0, ℝ0,m+1(0)≅ℝ and that for s=1, ℝ0,m+1(1)≅ℝ0,m+1.An elementx=(x0,x1,…,xm)=(x0,x¯)∈ℝm+1 is therefore
usually identified with x=∑i=0meixi∈ℝ0,m+1.Forx,y∈ℝ0,m+1(1), the product xy splits in two
parts, namely,(2.2)xy=x•y+x⋀y,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 by(2.3)x•y=−∑i=0mxiyi,x⋀y=∑i<jeiej(xiyj−xjyi).
More generally, for x∈ℝ0,m+1(1) and υ∈ℝ0,m+1(s) (0<s<m+1), we have that the product xυ decomposes
into(2.4)xυ=x•υ+x⋀υ,where(2.5)x•υ=[xυ]s−1=12(xυ−(−1)sυx),x⋀υ=[xυ]s+1=12(xυ+(−1)sυx).
Another useful decomposition of ℝ0,m+1 may be obtained
by splitting it “along the e0-direction,”
as indicated in (1.2). This in fact means that we split ℝm+1 following ℝm+1=ℝ×ℝm and that within ℝ0,m+1, the Clifford algebra ℝ0,m is generated by
the orthogonal basis e¯=(e1,…,em) of ℝ0,m. ℝ0,m denotes the
space ℝm to which the
original quadratic form of signature (0,m+1) on ℝ0,m+1 has been
restricted.Following the decomposition (1.2), the element x=(x0,x¯)∈ℝm+1 is then often
identified with the so-called paravector x=x0+e¯0x¯=x0+e¯0∑j=1mxjej∈ℝ⨁e¯0ℝ0,m.Let us also recall that ifΩ⊂ℝm+1 is open and F is an ℝ0,m+1-valued C1-function in Ω, then F is said to be
left monogenic in Ω if ∂xF=0 in Ω, ∂x=∑i=0mei∂xi being the Dirac
operator in ℝm+1.As already mentioned in (1.3), by putting Dx=e¯0∂x=∂x0+e¯0∂x¯, ∂x¯ being the Dirac
operator in ℝm, we have for F=U+e¯0V,(2.6)∂xF=0⇔DxF=0⇔{∂x0U+∂x¯V=0,∂x¯U+∂x0V=0.Let us recall that a pair (U,V) of ℝ0,m-valued
harmonic functions in Ω is said to be
conjugate harmonic if F=U+e¯0V is left
monogenic in Ω (see [16]).Notice also that, when defining the conjugateD¯x of Dx by D¯x=∂x0−e¯0∂x¯, we have that DxD¯x=D¯xDx=Δx.IfS is a subspace
of ℝ0,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 ∂x2=−Δx, we have that ℳ(Ω,S)⊂ℋ(Ω,S).In particular, forr,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)), ℝ0,m+1(r,p,q)=∑j=pq⨁ℝ0,m+1(r+2j) and ℳ(Ω,ℝ0,m+1(r,p,q))=MT(Ω,ℝ0,m+1(r,p,q)).Furthermore, for0≤s≤m+1 fixed, a
natural isomorphism(2.7)Θ:ℰ(Ω;ℝ0,m+1(s))↦ℰ(Ω;Λsℝm+1)may be then defined as follows.Put forWs=∑|A|=sWAseA∈ℰ(Ω;ℝ0,m+1(s)),(2.8)ΘWs=ωs⇔ωs=∑|A|=sωAsdxA,where for each A={i1,…,is}⊂{0,…,m} with 0≤i1<⋯<is≤m, dxA=dxi1⋀⋯⋀dxis and ωAs=WAs for all A.By means of the decomposition (2.1), Θ may be extended
by linearity to ℝ0,m+1, thus leading to the isomorphism Θ:ℰ(Ω;ℝ0,m+1)↦ℰ(Ω;Λℝm+1), where as usual Λℝm+1=∑s=0m+1⨁Λsℝm+1.It may be easily checked that the action of the
exterior derivatived and the
co-derivative d* on ℰ(Ω;Λsℝm+1) then
corresponds through Θ to the left
action of ∂x+ and ∂x− on ℰ(Ω;ℝ0,m+1(s)). 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 Ws∈ℰ(Ω;ℝ0,m+1(s)),(2.9)∂x−Ws=[∂xWs]s−1,∂x+Ws=[∂xWs]s+1.
Consequently, for Ws∈ℰ(Ω;ℝ0,m+1(s)), ∂xWs splits
into(2.10)∂xWs=[∂xWs]s−1+[∂xWs]s+1=∂x−Ws+∂x+Ws.It thus follows that for W∈ℰ(Ω;ℝ0,m+1(r,p,q)), the system ∂xW=0 is given by
(1.9).Obviously, fors=0, ∂x−W0=0, while for s=m+1, ∂x+Wm+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;ℝ0,m+1) corresponds to
the action of d+d* on ℰ(Ω;Λℝm+1). We thus have on ℰ(Ω;ℝ0,m+1) that Δx=−(∂x+∂x−+∂x−∂x+).The following notations will also be
used:(2.11)kers∂x+={Ws∈ℰ(Ω;ℝ0,m+1(s)):∂x+Ws=0},kers∂x−={Ws∈ℰ(Ω;ℝ0,m+1(s)):∂x−Ws=0}.
Let us recall that if Ω is contractible
to a point, a refined version of the inverse Poincaré lemma then implies
that(2.12)∂x+∂x−:kers∂x+↦kers∂x+,∂x−∂x+:kers∂x−↦kers∂x−
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 whereΩ˜⊂ℝm is the
orthogonal projection of Ω on ℝm and ∂x¯ and Δx¯ are the Dirac
and Laplace operators in ℝm.


## 3. Conjugate Harmonic Pairs

Letr,p,q∈ℕ be as in
Section 1, let W∈ℰ(Ω;ℝ0,m+1(r,p,q)) with W=∑j=pqWr+2j, and decompose each Wr+2j∈ℰ(Ω;ℝ0,m+1(r+2j)) following (1.2),
that is(3.1)Wr+2j=Ur+2j+e¯0Vr−1+2j,where Ur+2j∈ℰ(Ω,ℝ0,m(r+2j)) and Vr−1+2j∈ℰ(Ω,ℝ0,m(r−1+2j)).Then,W=U+e¯0V with(3.2)U=∑j=pqUr+2jℝ0,m(r,p,q)-valued,V=∑j=pqVr−1+2jℝ0,m(r−1,p,q)-valued.
Now suppose that W∈MT(Ω,ℝ0,m+1(r,p,q)), that is, (U,V) is a conjugate
harmonic pair in Ω in the sense of
[16]. Then, as already
stated in (1.3),(3.3)∂xW=0⇔{∂x0U+∂x¯V=0,∂x¯U+∂x0V=0.By virtue of (2.10) and (3.2), the
equations in (3.3) lead to the systems(3.4)∂x¯−Vr−1+2p=0,∂x0Ur+2j+∂x¯+Vr−1+2j+∂x¯−Vr−1+2j+2=0,j=p,…,q−1,∂x0Ur+2q+∂x¯+Vr−1+2q=0,(3.5)∂x¯−Ur+2p+∂x0Vr−1+2p=0,∂x¯+Ur+2j+∂x¯−Ur+2j+2+∂x0Vr−1+2j+2=0,j=p,…,q−1,∂x¯+Ur+2q=0.From (3.5) it thus follows that W∈MT(Ω,ℝ0,m+1(r,p,q)) implies that ∂x¯+Ur+2q=0 in Ω.We now claim that, under certain geometric conditions
uponΩ, given U=∑j=pqUr+2j, harmonic and ℝ0,m(r,p,q)-valued in Ω, the condition ∂x¯+Ur+2q=0 in Ω is sufficient
to ensure the existence of a V˜, harmonic and ℝ0,m(r−1,p,q)-valued in Ω, which is conjugate harmonic to U, that is W˜=U+e¯0V˜∈MT(Ω,ℝ0,m+1(r,p,q)).In proving this statement, we will adapt where
necessary the techniques worked out in [16] for constructing conjugate harmonic pairs.Let againΩ˜ denote the orthogonal
projection of Ω on ℝm. Then, we suppose henceforth that Ω satisfies the
following conditions (C1) and (C2):(C1)
Ω is normal with
respect to the e¯0 direction, that
is, there exists x0*∈ℝ such that for
all x¯∈Ω˜, Ω∩{x¯+te0:t∈ℝ} is connected
and it contains the element (x0*,x¯);(C2)
Ω˜ 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 Ω˜ (see [17]).As is well known, classical results of cohomology
theory provide necessary and sufficient conditions for the validity of the
inverse Poincaré lemma inΩ˜. 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 ωs in Ω˜ (0<s<m).Now assume thatU=∑j=pqUr+2j harmonic and
that ℝ0,m(r,p,q)-valued in Ω satisfies the
condition ∂x¯+Ur+2q=0 in Ω.Put(3.6)H˜(x0,x¯)=∫x0*x0U(t,x¯)dt−h˜(x¯),where h˜=∑j=pqh˜r+2j is a smooth ℝ0,m(r,p,q)-valued
solution in Ω˜ of the
equation(3.7)Δx¯h˜(x¯)=∂x0U(x0*,x¯).As Δx¯:ℰ(Ω˜;ℝ0,m(r,p,q))↦ℰ(Ω˜;ℝ0,m(r,p,q)) is surjective
(see [19]), such h˜ indeed exists
and any other similar solution of (3.7) has the form h˜+h, where h∈ℋ(Ω˜;ℝ0,m(r,p,q)).Fix a solutionh˜ of (3.7). Then
by construction, the corresponding H˜ determined by
(3.6) belongs to ℋ(Ω;ℝ0,m(r,p,q)) (see [16]).We now prove that there existshr+2q∈ℋ(Ω˜;ℝ0,m(r+2q)) such that in Ω˜,(3.8)∂x¯+(h˜r+2q+hr+2q)=0.To this end, first notice that,
as by assumption ∂x¯+Ur+2q=0 in Ω, we have that ∂x¯+(∂x0Ur+2q)(x0*,x¯)=0 in Ω˜, whence ∂x0Ur+2q(x0*,x¯)∈kerr+2q∂x¯+.As∂x¯+∂x¯−:kerr+2q∂x¯+↦kerr+2q∂x¯+ is surjective (see also (2.12)) there exists W˜r+2q∈kerr+2q∂x¯+ such that ∂x¯+∂x¯−W˜r+2q(x¯)=−∂x0Ur+2q(x0*,x¯), that is, W˜r+2q satisfies in Ω˜ the
relations(3.9)∂x¯+W˜r+2q=0,∂x¯+∂x¯−W˜r+2q(x¯)=−∂x0Ur+2q(x0*,x¯).Furthermore, put hr+2q=W˜r+2q−h˜r+2q. Then, on the one hand,(3.10)Δx¯(h˜r+2q+hr+2q)(x¯)=−(∂x¯−∂x¯++∂x¯+∂x¯−)W˜r+2q(x¯)=−∂x¯+∂x¯−W˜r+2q(x¯)=∂x0Ur+2q(x0*,x¯),while on the other
hand(3.11)Δx¯(h˜r+2q+hr+2q)(x¯)=Δx¯h˜r+2q(x¯)+Δx¯hr+2q(x¯)=∂x0Ur+2q(x0*,x¯)+Δx¯hr+2q(x¯).Consequently, Δx¯hr+2q=0 in Ω˜ and ∂x¯(h˜r+2q+hr+2q) is ℝ0,m(r−1+2q)-valued in Ω˜.Now defineH by(3.12)H(x0,x¯)=H˜(x0,x¯)−hr+2q(x¯).Then by construction, H∈ℋ(Ω;ℝ0,m(r,p,q)) and clearly in Ω, ∂x0H=U.Furthermore, as∂x¯+Ur+2q(x0,x¯)=0 in Ω, ∫x0*x0∂x¯U(t,x¯)dt is ℝ0,m(r−1,p,q)-valued and
obviously ∂x¯(∑j=pq−1h˜r+2j) is ℝ0,m(r−1,p,q)-valued. As
moreover ∂x¯(h˜r+2q+hr+2q)=0, we get that V˜(x0,x¯)=−∂x¯H(x0,x¯) is ℝ0,m(r−1,p,q)-valued.Consequently, asDxD¯x=Δx, W˜=Dx¯H=U+e¯0V˜∈MT(Ω,ℝ0,m+1(r,p,q)), that is, (U,V˜) is a conjugate
harmonic pair in Ω.We have thus proved the following
theorem.Theorem 3.1.
LetΩ⊂ℝm+1 be open and
normal with respect to the e¯0-direction and
let Ω be contractible
to a point. Furthermore, let U∈ℋ(Ω;ℝ0,m(r,p,q)) be given. Then, U admits a
conjugate harmonic V˜∈ℋ(Ω;ℝ0,m(r−1,p,q)) if and only if ∂x¯+Ur+2q=0 in Ω.Remarks
(1) Ifr+2q=m+1, then U0,mr+2q=U0,mm+1≡0 in Ω, thus implying that the condition ∂x¯+Ur+2q=0 is
automatically satisfied and that in constructing H, no correction term hr+2q should be added
to H˜ (i.e., we may
take H=H˜ in
(3.12)).
(2) It is of course
tacitly understood that ifr=p=0, then in the expression of V=∑j=0qV−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 (Theorem3.2) holds, the proof of which is omitted.Theorem 3.2.
LetΩ⊂ℝm+1 be open and
normal with respect to the e¯0-direction and
let Ω be contractible
to a point. Furthermore, let V∈ℋ(Ω;ℝ0,m+1(r−1),p,q) be given. Then, V admits a
conjugate harmonic U˜∈ℋ(Ω;ℝ0,m+1(r,p,q)) if and only if ∂x¯−Vr−1+2p=0 in Ω.


## 4. Structure Theorems

Assume thatr,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.
LetΩ⊂ℝm+1 be open and
contractible to a point and let W∈ℰ(Ω,ℝ0,m+1(r,p,q)). The following properties are equivalent:
(i)
W∈MT(Ω,ℝ0,m+1(r,p,q)),(ii)
there existsH∈ℋ(Ω,ℝ0,m+1(r+1,p,q−1)) such that W=∂xH.Proof.
It is clear that ifH∈ℋ(Ω,ℝ0,m+1(r+1,p,q−1)), then W=∂xH is ℝ0,m+1(r,p,q)-valued. As
moreover ∂x2=−Δx, W∈MT(Ω,ℝ0,m+1(r,p,q)); whence (ii)⇒(i) is proved.
Conversely, assume thatW∈MT(Ω,ℝ0,m+1(r,p,q)) and put W=∑j=pqWr+2j. From ∂xW=0 it follows
that(4.1)∂x−Wr+2p=0,∂x+Wr+2j+∂x−Wr+2(j+1)=0,j=p,…,q−1,∂x+Wr+2q=0.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;ℝ0,m+1(r+2p+1)) such that in Ω(4.2)Wr+2p=∂x−W−r+2p+1,∂x+W−r+2p+1=0.Analogously, the third equation
in (4.1) implies the existence of(4.3)W+r+2q−1∈ℰ(Ω;ℝ0,m+1(r+2q−1))such that in Ω(4.4)Wr+2q=∂x+W+r+2q−1,∂x−W+r+2q−1=0.Put H′=W−r+2p+1+W+r+2q−1. Then,(4.5)H′∈ℰ(Ω;ℝ0,m+1(r+1+2p)⊕ℝ0,m+1(r+1+2(q−1)))and by virtue of (4.2) and (4.4),(4.6)∂xH′=∂x+(W−r+2p+1+W+r+2q−1)+∂x−(W−r+2p+1+W+r+2q−1)=Wr+2q+Wr+2p. But W=W2r+p+W*+Wr+2q, where W*=∑j=p+1q−1Wr+2j is ℝ0,m+1(r,p+1,q−1)-valued and
harmonic in Ω. As Δx:ℰ(Ω;ℝ0,m+1(r,p+1,q−1))↦ℰ(Ω,ℝ0,m+1(r,p+1,q−1)) is surjective
(see [19]), there
exists that H*∈ℰ(Ω,ℝ0,m+1(r,p+1,q−1)) such that ΔxH*=W*.
PutH″=−∂xH*. Then, clearly H″∈ℰ(Ω;ℝ0,m+1(r+1,p,q−1)) and(4.7)∂xH″=−∂x2H*=ΔxH*=W*.Finally, put H=H′+H″. Then, H is ℝ0,m+1(r+1,p,q−1)-valued
and(4.8)∂xH=∂xH′+∂xH″=∑j=pqWr+2j=W.As H is obviously
harmonic in Ω, the proof is done.Remarks
(1) In the case
wherer=p=0 and 2q<m+1, we have that in (4.1) the equation ∂x−W0=0 is
automatically satisfied. Putting W˜=∑j=0q−1W2j, take H˜∈ℰ(Ω;ℝ0,m+1(0,0,q−1)) such that ΔxH˜=W˜ and define H by H=∂x(−H˜+W+2q−1). Then, W=∂xH.
In the case wherer+2q=m+1, the equation ∂x+Wm+1=0 is
automatically satisfied. An analogous reasoning to the one just made then leads
to an appropriate H∈ℋ(Ω;ℝ0,m+1(r+1,p,q−1)) such that W=∂xH.
(2) Obviously, in
the case where as wellr=p=0 as 2q=m+1, the technique
suggested in Remark (1) then produces H∈ℋ(Ω;ℝ0,m+1(1,0,(m+1)/2−1)) such that W=∂xH.
(3) A particularly important
example where as wellr=p=0 as 2q=m+1 occurs when m+1=4. Indeed, put for given real valued smooth functions fi in Ω⊂ℝ4, i=0,1,2,3,(4.9)W0=f0,W2=f1(e2e3−e0e1)+f2(e3e1−e0e2)+f3(e1e2−e0e3),W4=f0e0e1e2e3.Then, for W=W0+W2+W4∈ℰ(Ω;ℝ0,4(0,0,2)),(4.10)∂xW=0⇔{∂x+W0+∂x−W2=0,∂x+W2+∂x−W4=0.Both equations in (4.10) give rise
to the same system to be satisfied by f=(f0,f1,f2,f3), namely(4.11)∂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.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
wherep=0 and q=1, Lemma 4.1 tells us that, given W=Wr+Wr+2∈MT(Ω;ℝ0,m+1(r)⊕ℝ0,m+1(r+2)), there exists H∈ℋ(Ω;ℝ0,m+1(r+1)) such that W=∂xH. This result was already obtained in [3, Lemma 3.1].Theorem 4.2.
LetΩ⊂ℝm+1 be open and
normal with respect to the e¯0-direction, let Ω˜ be contractible
to a point, and let W∈ℰ(Ω;ℝ0,m+1(r,p,q)). The following properties are equivalent:
(i)
W∈MT(Ω;ℝ0,m+1(r,p,q)),(ii)
there existsL∈ℋ(Ω;ℝ0,m(r,p,q)) with ∂x¯+Lr+2q=0 in Ω such that W=D¯xL.Proof.
(i)→(ii). Let W∈MT(Ω;ℝ0,m+1(r,p,q)) and put,
following (1.2), W=U+e¯0V. Then, the pair (U,V) is conjugate
harmonic in Ω with U∈ℋ(Ω;ℝ0,m(r,p,q)) and V∈ℋ(Ω;ℝ0,m(r−1,p,q)).
Associate withU the harmonic ℝ0,m(r,p,q)-valued
potential H given by (3.12),
that is,(4.12)H(x0,x¯)=∫x0*x0U(t,x¯)dt−(h˜+hr+2q)(x¯),where in Ω˜, Δx¯h˜(x¯)=∂x0U(x0*,x¯), hr+2q∈ℋ(Ω˜;ℝ0,m(r+2q)), and ∂x¯+(h˜r+2q+hr+2q)=0.
As moreover∂x¯+Ur+2q=0 in Ω (see Theorem
3.1), it thus follows from (4.12) that ∂x¯+Hr+2q=0 in Ω. Consequently, W˜=D¯xH=U˜+e¯0V˜∈MT(Ω;ℝ0,m+1(r,p,q)) with U˜=∂x0H=U and V˜=−∂x¯H.
From∂x(W˜−W)=0, it is then easily obtained that V˜−V is independent
of x0 and that in Ω˜, ∂x¯(V˜−V)=0, that is, V˜−V∈MT(Ω˜;ℝ0,m(r−1,p,q)). By virtue of Lemma 4.1, there exists H*∈ℋ(Ω˜;ℝ0,m(r,p,q−1)) such that V˜−V=∂x¯H*; whence V=−∂x¯(H+H*).
PutL=H+H*. Then by construction,
(i)
L∈ℋ(Ω;ℝ0,m(r,p,q)),(ii)
∂x¯+Lr+2q=0,(iii)
W=D¯xL;
whence (i)→(ii) is proved.
Conversely, letL∈ℋ(Ω;ℝ0,m(r,p,q)) with ∂x¯+Lr+2q=0. Then clearly W=D¯xL∈MT(Ω;ℝ0,m+1(r,p,q)).Remarks
(1) Theorem4.2 tells
us that each W∈MT(Ω;ℝ0,m+1(r,p,q)) admits an ℝ0,m(r,p,q)- valued
harmonic potential L in Ω satisfying ∂x¯+Lr+2q=0.
(2) LetW=Wr+Wr+2∈ℰ(Ω;ℝ0,m+1(r)⊕ℝ0,m+1(r+2)), 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;ℝ0,m+1(r)⊕ℝ0,m+1(r+2)),(ii)
there
existsL∈ℋ(Ω;ℝ0,m(r)⊕ℝ0,m(r+2)) with ∂x¯+Lr+2=0 such that W=D¯xL.
This
characterization was already obtained in [3, Theorem 3.1].


## 5. Cauchy Integral Decompositions

LetΩ=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∈C0,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 functionsf+ and f−, holomorphic in Ω+ and Ω− with f−(∞)=0, such that f± are
continuously extendable to γ and that on γ(5.1)f=f++f−,where in (5.1), f±(u)=limΩ±∋z→uf±(z),u∈γ.”Let𝒞γ be the Cauchy
transform on C0,α(γ), that is, for f∈C0,α(γ),(5.2)𝒞γf(z)=12π∫γ1t−zν(t)f(t)dH1(t),z∈ℂ∖γ,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)
𝒞γf is holomorphic
and of the class C0,α on Ω+∪Ω− with 𝒞γf(∞)=0;(ii)
Plemelj-Sokhotzki formulae:(5.3)𝒞γ±f(u)=limΩ±∋z→u𝒞γf(z)=12(±f(u)+𝒮γf(u)),u∈γ,where for u∈γ,(5.4)𝒮γf(u)=1πPV∫γ1t−uν(t)(f(t)−f(u))dH1(t)+f(u)define the Hilbert transform 𝒮γ on C0,α(γ);(iii)
f=𝒞γ+f−𝒞γ−f on γ.It thus follows
that the answer to the jump problem (5.1) is indeed given by 𝒞γf.The decomposition (iii) thus obtained is known as the
Cauchy integral decomposition off on γ.Now letΩ=Ω+ be a bounded
and open subset of ℝm+1 with boundary Γ=∂Ω. Then, in Clifford analysis, for suitable pairs (Γ,f) of boundaries Γ and ℝ0,m+1-valued
functions f on Γ, the Cauchy transform 𝒞Γf is defined
by(5.5)𝒞Γf(x)=∫ΓE(y−x)ν(y)f(y)dℋm(y),x∈ℝm+1∖Γ,
where the following conditions hold.(i)
E(x)=(−1/Am+1)(x/|x|m+1), x∈ℝm+1∖{0}, is the fundamental solution of the Dirac operator ∂x, where Am+1 is the area of
the unit sphere in ℝm+1. E(x) is ℝ0,m+1(1)-valued and
monogenic in ℝm+1∖{0}.(ii)
ν(y)=∑i=0meiνi(y) is the outward
pointing unit normal at y∈Γ.(iii)
ℋm is the m-dimensional
Hausdorff measure on Γ. For the definition of ℋm, 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,(5.6)c−1ρm≤ℋm(Γ∩B(y,ρ))≤cρm,where B(y,ρ) is the closed
ball in ℝm+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∈C0,α(Γ;ℝ0,m+1), 0<α<1, C0,α(Γ;ℝ0,m+1) being the space
of ℝ0,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)
𝒞Γf is left
monogenic in ℝm+1∖Γ and 𝒞Γf(∞)=0;(ii)
(Plemelj-Sokhotzki formulae) the functions𝒞Γ±f determined
by(5.7)𝒞Γ±f(u)=limΩ±∋x→u𝒞Γf(x)=12(𝒮Γf(u)±f(u)),u∈Γ,belong to C0,α(Γ;ℝ0,m+1), where
(5.8)𝒮Γf(u)=2∫ΓE(y−u)ν(y)[f(y)−f(u)]dℋm(y)+f(u),u∈Γ,the integral being taken in the
sense of principal values;(iii)
f(u)=𝒞Γ+f(u)−𝒞Γ−f(u), u∈Γ.
It thus follows
that, given a Hölder continuous ℝ0,m+1-valued density f on Γ, the jump problem (5.9)“Findf+ and f−, belonging to C0,α(Γ;ℝ0,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 Γ(5.9)f=f++f−”
is solved by considering the
Cauchy transform 𝒞Γf. Indeed, we can take f+=𝒞Γf in Ω+ and f−=−𝒞Γf in Ω−.Now let againr,p,q∈ℕ be a triplet
satisfying 0≤r≤m+1 and 0≤p<q with r+2q≤m+1, and let W∈C0,α(Γ;ℝ0,m+1(r,p,q)).AsE(y−x)ν(y) is ℝ0,m+1(0)⊕ℝ0,m+1(2)-valued, it is
easily seen that 𝒞ΓW is ℝ0,m+1(r−2+2p)⊕ℝ0,m+1(r,p,q)⊕ℝ0,m+1(r+2+2q)-valued.
Consequently, if the jump problem (5.9) is formulated in terms of ℝ0,m+1(r,p,q)-valued Hölder
continuous functions W, W+, and W− on Γ, then if we wish to solve it by means of the Cauchy
transform 𝒞Γ, restrictions on 𝒞ΓW have to be
imposed, namely, in ℝm+1∖Γ we should
have(5.10)[𝒞ΓW]r−2+2p≡0,[𝒞ΓW]r+2+2q≡0.
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.
LetΩ⊂ℝm+1 be open and
bounded such that Γ=∂Ω is an m-dimensional
Ahlfors-David regular surface and let W∈C0,α(Γ;ℝ0,m+1(r,p,q)) 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 C0,α(Γ;ℝ0,m+1(r,p,q)) and moreover
are the boundary values of functions W±∈MT(Ω±;ℝ0,m+1(r,p,q)) with W−(∞)=0,(ii)
𝒞ΓW∈MT(ℝm+1∖Γ;ℝ0,m+1(r,p,q)),(iii)
∑j=pq[𝒞ΓW]r+2j∈MT(ℝm+1∖Γ;ℝ0,m+1(r,p,q)),(iv)
[𝒞ΓW]r−2+2p≡0 and [𝒞ΓW]r+2+2q≡0 in ℝm+1.Proof.
(i)→(ii). Assume that W=W++W−, where W+ and W− satisfy the
conditions given in (i). Then(5.11)𝒞ΓW=𝒞ΓW++𝒞ΓW−.In view of the assumptions made
on W±, we have that 𝒞ΓW+=0 in Ω−, 𝒞ΓW−=0 in Ω+ and that 𝒞ΓW±=W± in Ω±.
Consequently,(5.12)𝒞ΓW={W+inΩ+,W−inΩ−.As W±∈MT(Ω±;ℝ0,m+1(r,p,q)) with W−(∞)=0,
(ii) is
proved.
(ii)→(iii): Trivial.
(iii)→(iv). Let us first recall that 𝒞ΓW is left
monogenic in ℝm+1∖Γ with 𝒞ΓW(∞)=0.
According to the decomposition(5.13)𝒞ΓW=[𝒞ΓW]r−2+2p+∑j=pq[𝒞ΓW]r+2j+[𝒞ΓW]r+2+2qand by the assumption made on ∑j=pq[𝒞ΓW]r+2j, it follows from [1] that(5.14)∂x([𝒞ΓW]r−2+2p)+∂x([𝒞ΓW]r+2+2q)=0inℝm+1∖Γ.Furthermore, as ∂x([𝒞ΓW]r−2+2p) and ∂x([𝒞ΓW]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 ℝm+1∖Γ(5.15)∂x([𝒞ΓW]r−2+2p)=0,∂x([𝒞ΓW]r+2+2q)=0.
Moreover, as by assumption W is ℝ0,m+1(r,p,q)-valued, by
virtue of the Plemelj-Sokhotzki formulae, we obtain that on Γ(5.16)[𝒞Γ+W]r−2+2p=[𝒞Γ−W]r−2+2p,[𝒞Γ+W]r+2+2q=[𝒞Γ−W]r+2+2q.
Furthermore, [𝒞Γ±W]r−2+2p and [𝒞Γ±W]r+2+2q are Hölder
continuous on Γ.
It thus follows that[𝒞ΓW]r−2+2p and [𝒞ΓW]r+2+2q are left
monogenic in ℝm+1∖Γ and
continuously extendable to Γ. Painlevé's theorem (see [30]) then implies that [𝒞ΓW]r−2+2p and [𝒞ΓW]r+2+2q are left
monogenic in ℝm+1.
Finally, as[𝒞ΓW]r−2+2p(∞)=[𝒞ΓW]r+2+2q(∞)=0, we obtain by virtue of Liouville's theorem (see
[31]) that [𝒞ΓW]r−2+2p≡0 and [𝒞ΓW]r+2+2q≡0 in ℝm+1.
(iv)→(i). First note that, as W∈C0,α(Γ;ℝ0,m+1(r,p,q)), by means of the Plemelj-Sokhotzki formulae, we have
on Γ that(5.17)W=𝒞Γ+W−𝒞Γ−W.In view of the assumption (iv) made, the functions W± defined in Ω± by W±=±𝒞ΓW obviously belong to MT(Ω±;ℝ0,m+1(r,p,q)) 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 curvesγ in the plane or
on hypersurfaces Γ in ℝm+1(m≥2). Two types of boundary data are usually considered,
namely a Hölder continuous density or an Lp-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 forf∈C0,α(Γ;ℝ0,m+1) (0<α<1), the continuous extendability of 𝒞Γf 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 casep=q=0 and 0<r<m+1 was dealt with
in [39]. For Ω⊂ℝm+1 open, bounded
and connected with C∞-boundary Γ such that ℝm+1∖(Ω∪Γ) is also
connected, a set of equivalent properties was obtained ensuring the validity of
the Cauchy integral decomposition for Wr∈ℰ(Γ,ℝ0,m+1(r)) given.
(3) IfW∈C0,α(Γ;ℝ0,m+1+), where(5.18)ℝ0,m+1+=∑seven⊕ℝ0,m+1(s),then 𝒞ΓW∈MT(ℝm+1∖Γ;ℝ0,m+1+), that is the condition (ii) in Theorem 5.1 is
satisfied.
Analogously, ifW∈C0,α(Γ;ℝ0,m+1−) where(5.19)ℝ0,m+1−=∑sodd⊕ℝ0,m+1(s),then 𝒞ΓW∈MT(ℝm+1∖Γ;ℝ0,m+1−) and so the
condition (ii) in Theorem 5.1 is again satisfied.



---

*Source: 746946-2008-05-12.xml*