Open Access

Hermitean Téodorescu Transform Decomposition of Continuous Matrix Functions on Fractal Hypersurfaces

  • Ricardo Abreu-Blaya1,
  • Juan Bory-Reyes2,
  • Fred Brackx3 and
  • Hennie De Schepper3Email author
Boundary Value Problems20102010:791358

DOI: 10.1155/2010/791358

Received: 18 December 2009

Accepted: 12 April 2010

Published: 20 May 2010

Abstract

We consider Hölder continuous circulant ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq1_HTML.gif ) matrix functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq2_HTML.gif defined on the fractal boundary https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq3_HTML.gif of a domain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq4_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq5_HTML.gif . The main goal is to study under which conditions such a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq6_HTML.gif can be decomposed as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq7_HTML.gif , where the components https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq8_HTML.gif are extendable to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq9_HTML.gif -monogenic functions in the interior and the exterior of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq10_HTML.gif , respectively. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq11_HTML.gif -monogenicity are a concept from the framework of Hermitean Clifford analysis, a higher-dimensional function theory centered around the simultaneous null solutions of two first-order vector-valued differential operators, called Hermitean Dirac operators. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq12_HTML.gif -monogenic functions then are the null solutions of a ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq13_HTML.gif ) matrix Dirac operator, having these Hermitean Dirac operators as its entries; such matrix functions play an important role in the function theoretic development of Hermitean Clifford analysis. In the present paper a matricial Hermitean Téodorescu transform is the key to solve the problem under consideration. The obtained results are then shown to include the ones where domains with an Ahlfors-David regular boundary were considered.

1. Introduction

Clifford analysis is a higher-dimensional function theory offering a generalization of the theory of holomorphic functions in the complex plane and, at the same time, a refinement of classical harmonic analysis. The standard case, also referred to as Euclidean Clifford analysis, focuses on the null solutions, called monogenic functions, of the vector-valued Dirac operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq14_HTML.gif , which factorizes the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq15_HTML.gif -dimensional Laplacian: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq16_HTML.gif . Here https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq17_HTML.gif is an orthonormal basis for the quadratic space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq18_HTML.gif underlying the construction of the real Clifford algebra https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq19_HTML.gif , where the considered functions take their values. Since the Dirac operator is invariant with respect to the action of the orthogonal group https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq20_HTML.gif , doubly covered by the Pin( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq21_HTML.gif ) group of the Clifford algebra https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq22_HTML.gif , the resulting function theory is said to be rotation invariant. Standard references for Euclidean Clifford analysis are [15].

More recently, Hermitean Clifford analysis has emerged as yet a refinement of the Euclidean case. One of the ways for introducing it is by considering the complex Clifford algebra https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq23_HTML.gif , equipped with a complex structure, that is, an https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq24_HTML.gif element https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq25_HTML.gif for which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq26_HTML.gif . In fact, it is precisely in order to ensure that such a complex structure exists that the dimension of the underlying vector space is taken to be even. The resulting function theory focuses on the simultaneous null solutions of two complex Hermitean Dirac operators https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq27_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq28_HTML.gif which no longer factorize but still decompose the Laplace operator in the sense that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq29_HTML.gif . Since the system indeed constitutes a refinement of the original Euclidean Dirac equation, the fundamental group invariance of this system breaks down to a smaller group; it was shown in [6] that it concerns the unitary group https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq30_HTML.gif . The study of complex Dirac operators was initiated in [710]; a systematic development of the associated function theory still is in full progress; see, for example, [6, 1115].

In [16] a Cauchy integral formula for Hermitean monogenic functions was established, obviously an essential result in the development of the function theory. However, as in some very particular cases Hermitean monogenicity is equivalent with (anti)holomorphy in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq31_HTML.gif complex variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq32_HTML.gif (see [12]), such a representation formula could not take the traditional form as in the complex plane or in Euclidean Clifford analysis. The matrix approach needed to obtain the desired result leads to the concept of (left or right) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq33_HTML.gif -monogenic functions, introduced as circulant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq34_HTML.gif matrix functions, which are (left or right) null solutions of a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq35_HTML.gif circulant matrix Dirac operator, having the Hermitean Dirac operators https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq36_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq37_HTML.gif as its entries. Although the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq38_HTML.gif -monogenic system thus arose as an auxiliary concept in Hermitean Clifford analysis, it was meanwhile also further studied itself; see also [15, 17, 18].

In the present paper, we consider Hölder continuous circulant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq39_HTML.gif matrix functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq40_HTML.gif defined on the fractal boundary https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq41_HTML.gif of a domain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq42_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq43_HTML.gif , and we investigate under which conditions such a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq44_HTML.gif can be decomposed as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq45_HTML.gif , where the components https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq46_HTML.gif are extendable to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq47_HTML.gif -monogenic functions in the interior and the exterior of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq48_HTML.gif , respectively. This type of decomposition (or "jump") problem has already been considered in Euclidean Clifford analysis in, for example, [1922] for domains with boundaries showing minimal smoothness, including some results for fractal boundaries as well. In [23] a similar decomposition problem for domains with fractal boundaries was considered in the Hermitean Clifford context, the approach, however, not being suited for a treatment of the circulant matrix situation. It turns out that the introduction of a matricial Hermitean Téodorescu transform is crucial to solve this problem.

2. Preliminaries

2.1. Some Elements of Hermitean Clifford Analysis

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq49_HTML.gif be an orthonormal basis of Euclidean space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq50_HTML.gif and consider the complex Clifford algebra https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq51_HTML.gif constructed over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq52_HTML.gif . The noncommutative or geometric multiplication in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq53_HTML.gif is governed by the following rules:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ1_HTML.gif
(2.1)
The Clifford algebra https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq54_HTML.gif thus is generated additively by elements of the form https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq55_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq56_HTML.gif is such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq57_HTML.gif , while for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq58_HTML.gif , one puts https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq59_HTML.gif , the identity element. The dimension of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq60_HTML.gif thus is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq61_HTML.gif . Any Clifford number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq62_HTML.gif may thus be written as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq63_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq64_HTML.gif , and its Hermitean conjugate https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq65_HTML.gif is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ2_HTML.gif
(2.2)

where the bar denotes the real Clifford algebra conjugation, that is, the main anti-involution for which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq66_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq67_HTML.gif stands for the complex conjugate of the complex number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq68_HTML.gif .

Euclidean space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq69_HTML.gif is embedded in the Clifford algebra https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq70_HTML.gif by identifying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq71_HTML.gif with the real Clifford vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq72_HTML.gif given by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq73_HTML.gif , for which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq74_HTML.gif . The Fischer dual of the vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq75_HTML.gif is the vector-valued first-order Dirac operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq76_HTML.gif , factorizing the Laplacian: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq77_HTML.gif ; it is precisely this Dirac operator which underlies the notion of monogenicity of a function, the higher-dimensional counterpart of holomorphy in the complex plane. The functions under consideration are defined on (open subsets of) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq78_HTML.gif and take values in the Clifford algebra https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq79_HTML.gif . They are of the form https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq80_HTML.gif , where the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq81_HTML.gif are complex valued. Whenever a property such as continuity and differentiability is ascribed to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq82_HTML.gif , it is meant that all the components https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq83_HTML.gif possess the cited property. A Clifford algebra-valued function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq84_HTML.gif , defined and differentiable in an open region https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq85_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq86_HTML.gif , is then called (left) monogenic in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq87_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq88_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq89_HTML.gif .

The transition from Euclidean Clifford analysis as described above to the Hermitean Clifford setting is essentially based on the introduction of a complex structure https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq90_HTML.gif . This is a particular https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq91_HTML.gif element, satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq92_HTML.gif . Since such an element cannot exist when the dimension https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq93_HTML.gif of the vector space is odd, we will put https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq94_HTML.gif from now on. In terms of the chosen orthonormal basis, a particular realization of the complex structure may be https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq95_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq96_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq97_HTML.gif . Two projection operators https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq98_HTML.gif associated to this complex structure https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq99_HTML.gif then produce the main objects of Hermitean Clifford analysis by acting upon the corresponding objects in the Euclidean setting; see [11, 12]. First of all, the vector space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq100_HTML.gif thus decomposes as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq101_HTML.gif into two isotropic subspaces. The real Clifford vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq102_HTML.gif is now denoted by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ3_HTML.gif
(2.3)
and its corresponding Dirac operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq103_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ4_HTML.gif
(2.4)
while we will also consider their so-called "twisted" counterparts, obtained through the action of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq104_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ5_HTML.gif
(2.5)
As was the case with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq105_HTML.gif , a notion of monogenicity may be associated in a natural way to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq106_HTML.gif as well. The projections of the vector variable https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq107_HTML.gif and the Dirac operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq108_HTML.gif on the spaces https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq109_HTML.gif then give rise to the Hermitean Clifford variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq110_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq111_HTML.gif , given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ6_HTML.gif
(2.6)
and (up to a factor) to the Hermitean Dirac operators https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq112_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq113_HTML.gif given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ7_HTML.gif
(2.7)

(see [6, 11]). Observe for further use that the Hermitean vector variables and Dirac operators are isotropic, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq114_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq115_HTML.gif , whence the Laplacian allows for the decomposition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq116_HTML.gif , while also https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq117_HTML.gif . These objects lie at the core of the Hermitean function theory by means of the following definition (see, e.g., [6, 11]).

Definition 2.1.

A continuously differentiable function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq118_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq119_HTML.gif with values in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq120_HTML.gif is called left Hermitean monogenic (or left https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq121_HTML.gif -monogenic) in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq122_HTML.gif , if and only if it satisfies in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq123_HTML.gif the system
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ8_HTML.gif
(2.8)
or, equivalently, the system
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ9_HTML.gif
(2.9)

In a similar way right https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq124_HTML.gif -monogenicity is defined. Functions which are both left and right https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq125_HTML.gif -monogenic are called two-sided https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq126_HTML.gif -monogenic. This definition inspires the statement that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq127_HTML.gif -monogenicity constitutes a refinement of monogenicity, since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq128_HTML.gif -monogenic functions (either left or right) are monogenic w.r.t. both Dirac operators https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq129_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq130_HTML.gif .

In what follows, we will systematically take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq131_HTML.gif to be a so-called Jordan domain, that is, a bounded oriented connected open subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq132_HTML.gif , the boundary https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq133_HTML.gif of which is a compact topological surface. Note that, in the case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq134_HTML.gif , this notion coincides with the usual one of a Jordan domain in the complex plane. For further use, we also introduce the notation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq135_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq136_HTML.gif .

2.2. Some Elements of the Matricial Hermitean Clifford Setting

The fundamental solutions of the Dirac operators https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq137_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq138_HTML.gif are, respectively, given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ10_HTML.gif
(2.10)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq139_HTML.gif denotes the surface area of the unit sphere in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq140_HTML.gif . Introducing the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq141_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq142_HTML.gif , explicitly given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ11_HTML.gif
(2.11)
it is directly seen that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq143_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq144_HTML.gif are not the fundamental solutions to the respective Hermitean Dirac operators https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq145_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq146_HTML.gif . Surprisingly, however, introducing the particular circulant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq147_HTML.gif matrices
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ12_HTML.gif
(2.12)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq148_HTML.gif is the Dirac delta distribution, one obtains that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq149_HTML.gif , so that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq150_HTML.gif may be considered as a fundamental solution of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq151_HTML.gif in a matricial context, see, for example, [8, 16, 18]. Moreover, the Dirac matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq152_HTML.gif in some sense factorizes the Laplacian, since
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ13_HTML.gif
(2.13)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq153_HTML.gif is the usual Laplace operator in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq154_HTML.gif . It was exactly this simple observation which leads to the idea of following a matrix approach in order to establish integral representation formulae in the Hermitean setting; see [15, 16]. Moreover, it inspired the following definition.

Definition 2.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq155_HTML.gif be continuously differentiable functions defined in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq156_HTML.gif and taking values in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq157_HTML.gif , and consider the matrix function:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ14_HTML.gif
(2.14)
Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq158_HTML.gif is called left (resp., right) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq159_HTML.gif -monogenic in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq160_HTML.gif if and only if it satisfies in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq161_HTML.gif the system
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ15_HTML.gif
(2.15)

Here https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq162_HTML.gif denotes the matrix with zero entries.

Explicitly, the system for left https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq163_HTML.gif -monogenicity reads:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ16_HTML.gif
(2.16)

Again, a notion of two-sided https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq164_HTML.gif -monogenicity may be defined similarly. However, unless mentioned explicitly, we will only work with left https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq165_HTML.gif -monogenic matrix functions. This matrix approach has also been successfully applied in [17, 24] for the construction of a boundary values theory of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq166_HTML.gif -monogenic functions.

Observe however that the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq167_HTML.gif -monogenicity of the matrix function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq168_HTML.gif does not imply the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq169_HTML.gif -monogenicity of its entry functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq170_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq171_HTML.gif . Nevertheless, choosing in particular https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq172_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq173_HTML.gif , the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq174_HTML.gif -monogenicity of the corresponding diagonal matrix, denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq175_HTML.gif , is seen to be equivalent to the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq176_HTML.gif -monogenicity of the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq177_HTML.gif . Moreover, considering the matricial Laplacian introduced in (2.13), one may call a matrix function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq178_HTML.gif harmonic if and only if it satisfies the equation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq179_HTML.gif . Each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq180_HTML.gif -monogenic matrix function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq181_HTML.gif then is harmonic, ensuring that its entries are harmonic functions in the usual sense.

In general, notions of continuity, differentiability, and integrability of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq182_HTML.gif are introduced by means of the corresponding notions for its entries. In what follows, we will in particular use the notations https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq183_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq184_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq185_HTML.gif for the class of Hölder continuous and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq186_HTML.gif -integrable circulant matrix functions, respectively.

2.3. Some Elements of Fractal Geometry

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq187_HTML.gif be an arbitrary subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq188_HTML.gif . Then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq189_HTML.gif its Hausdorff measure https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq190_HTML.gif may be defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ17_HTML.gif
(2.17)

where the infimum is taken over all countable https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq191_HTML.gif -coverings https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq192_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq193_HTML.gif with open or closed balls. Note that, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq194_HTML.gif , the Hausdorff measure https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq195_HTML.gif coincides, up to a positive multiplicative constant, with the Lebesgue measure https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq196_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq197_HTML.gif .

Now, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq198_HTML.gif be a compact subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq199_HTML.gif . The Hausdorff dimension of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq200_HTML.gif , denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq201_HTML.gif , is then defined as the infimum of all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq202_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq203_HTML.gif . For more details concerning the Hausdorff measure and dimension we refer the reader to [25, 26].

Frequently, however, see, for example, [27], the so-called box dimension is more appropriated than the Hausdorff dimension to measure the roughness of a given set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq204_HTML.gif . By definition, the box dimension of a compact set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq205_HTML.gif is equal to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ18_HTML.gif
(2.18)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq206_HTML.gif stands for the minimal number of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq207_HTML.gif -balls needed to cover https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq208_HTML.gif . Note that the limit in (2.18) remains unchanged if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq209_HTML.gif is replaced by the number of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq210_HTML.gif -cubes, with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq211_HTML.gif , intersecting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq212_HTML.gif . For completeness we recall that a cube https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq213_HTML.gif is called a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq214_HTML.gif -cube if it is of the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ19_HTML.gif
(2.19)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq215_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq216_HTML.gif are integers. The box dimension and the Hausdorff dimension of a given compact set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq217_HTML.gif can be equal, which is, for instance, the case for the so-called https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq218_HTML.gif -rectifiable sets (see [28]), but this is not the case in general, where we have that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq219_HTML.gif .

In what follows we will assume the boundary https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq220_HTML.gif of our Jordan domain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq221_HTML.gif to have Hausdorff dimension https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq222_HTML.gif . This includes the case when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq223_HTML.gif is fractal in the sense of Mandelbrot, that is, when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq224_HTML.gif .

3. A Pair of Euclidean Téodorescu Transforms

From now on we reserve the notations https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq225_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq226_HTML.gif for Clifford vectors associated to points in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq227_HTML.gif . We may then consider the Euclidean Téodorescu transforms https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq228_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq229_HTML.gif of a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq230_HTML.gif , assumed to be integrable in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq231_HTML.gif , given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ20_HTML.gif
(3.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq232_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq233_HTML.gif are oriented volume elements on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq234_HTML.gif , for which it is easily checked that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq235_HTML.gif . For the sake of completeness, we recall some basic properties of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq236_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq237_HTML.gif , which are generalizations to the case of Clifford analysis of the well-known properties established in the complex plane.

To this end, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq238_HTML.gif be a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq239_HTML.gif -valued function defined on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq240_HTML.gif , which satisfies a Hölder condition of order https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq241_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq242_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq243_HTML.gif , and denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq244_HTML.gif the so-called Whitney extension of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq245_HTML.gif from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq246_HTML.gif to the whole of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq247_HTML.gif (see [29]). We recall that the Whitney extension of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq248_HTML.gif is a compactly supported function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq249_HTML.gif for which it holds that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq250_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ21_HTML.gif
(3.2)

We then first formulate an auxiliary result.

Lemma 3.1.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq251_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq252_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq253_HTML.gif -integrable in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq254_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq255_HTML.gif .

Proof.

We only give the main lines of the proof; for details we refer the reader to [22, Lemma https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq256_HTML.gif ]. In the notation of [30], let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq257_HTML.gif be the Whitney partition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq258_HTML.gif by means of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq259_HTML.gif -cubes. We then have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ22_HTML.gif
(3.3)
On the other hand, (3.2) implies that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq260_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ23_HTML.gif
(3.4)
since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq261_HTML.gif . Now, invoking the fact that the number of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq262_HTML.gif -cubes appearing in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq263_HTML.gif is less than https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq264_HTML.gif , while by definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq265_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ24_HTML.gif
(3.5)
for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq266_HTML.gif , we arrive at
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ25_HTML.gif
(3.6)

the last series being convergent for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq267_HTML.gif . In view of the arbitrary choice of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq268_HTML.gif , this concludes the proof.

Now, take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq269_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ26_HTML.gif
(3.7)
and then it holds that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq270_HTML.gif ; whence, on account of the previous lemma, there exist exponents https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq271_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq272_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq273_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq274_HTML.gif -integrable in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq275_HTML.gif . From this observation it then follows that, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq276_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq277_HTML.gif as in (3.7), both https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq278_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq279_HTML.gif belong to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq280_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq281_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ27_HTML.gif
(3.8)

due to the fact that the Téodorescu transform maps the space of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq282_HTML.gif -integrable functions with compact support to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq283_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq284_HTML.gif (see, e.g., [5]). The following result then holds.

Proposition 3.2.

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq285_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq286_HTML.gif as in (3.7), consider
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ28_HTML.gif
(3.9)

With https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq287_HTML.gif denoting the characteristic function of the set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq288_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq289_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq290_HTML.gif are monogenic in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq291_HTML.gif and in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq292_HTML.gif , with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq293_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq294_HTML.gif , respectively. They are continuous in the corresponding closed domains, vanish at infinity, and show jump https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq295_HTML.gif over the boundary https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq296_HTML.gif .

Proof.

For simplicity we restrict ourselves to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq297_HTML.gif , the proof for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq298_HTML.gif running along similar lines. The continuity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq299_HTML.gif on the closed domains follows from the fact that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq300_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq301_HTML.gif satisfying (3.8). On the other hand, a direct calculation shows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq302_HTML.gif and that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ29_HTML.gif
(3.10)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ30_HTML.gif
(3.11)

Finally, the monogenicity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq303_HTML.gif is a direct consequence of the well-known fact that the Téodorescu transform constitutes a right inverse of the Dirac operator.

Summarizing, any function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq304_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq305_HTML.gif as in (3.7), can be decomposed as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ31_HTML.gif
(3.12)

where the components https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq306_HTML.gif are extendable to monogenic functions in the interior and the exterior of the domain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq307_HTML.gif , with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq308_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq309_HTML.gif , respectively. Note that a decomposition of type (3.12) is said to be of class https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq310_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq311_HTML.gif . The remaining question is whether the decomposition (3.12) is unique. In order to investigate this, we will need the following version of the Dolzhenko theorem, as proved in [22].

Theorem 3.3.

Let the compact set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq312_HTML.gif be such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq313_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq314_HTML.gif . Then, a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq315_HTML.gif which is monogenic in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq316_HTML.gif is monogenic in the whole of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq317_HTML.gif .

We then arrive at the following result.

Theorem 3.4.

Suppose that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ32_HTML.gif
(3.13)

Then, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq318_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq319_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq320_HTML.gif , there exists a unique decomposition (3.12) of class https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq321_HTML.gif .

Proof.

The existence being shown above, it remains to prove the uniqueness. To this end, assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq322_HTML.gif admits two decompositions of class https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq323_HTML.gif , denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq324_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq325_HTML.gif , respectively. Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ33_HTML.gif
(3.14)
implying that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ34_HTML.gif
(3.15)

Consequently, the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq326_HTML.gif is monogenic in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq327_HTML.gif and belongs to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq328_HTML.gif , whence it is monogenic in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq329_HTML.gif on account of Theorem 3.3, while it also vanishes at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq330_HTML.gif . By Liouville's Theorem we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq331_HTML.gif .

4. A Matricial Hermitean Téodorescu Transform

A first step in the solution of the Hermitean matrix decomposition problem is the introduction of the matricial Hermitean Téodorescu transform:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ35_HTML.gif
(4.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq332_HTML.gif is the associated volume element given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ36_HTML.gif
(4.2)
and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq333_HTML.gif denote the Hermitean counterparts of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq334_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ37_HTML.gif
(4.3)

It clearly holds that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq335_HTML.gif .

A direct calculation reveals that the Hermitean Téodorescu transform https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq336_HTML.gif can be expresed in terms of the Euclidean Téodorescu transforms https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq337_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq338_HTML.gif as follows (see [15]):
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ38_HTML.gif
(4.4)
In particular, for the special case of the matrix function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq339_HTML.gif (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq340_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq341_HTML.gif ) this expression reduces to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ39_HTML.gif
(4.5)
In what follows we will denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq342_HTML.gif the Whitney extension of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq343_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ40_HTML.gif
(4.6)

The following theorem then contains some of its basic properties of the matricial Hermitean Téodorescu transform. They can be proven using standard techniques applied to the present matrix context.

Theorem 4.1.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq344_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq345_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq346_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq347_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq348_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq349_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ41_HTML.gif
(4.7)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq350_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq351_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq352_HTML.gif as in (3.7), then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ42_HTML.gif
(4.8)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq353_HTML.gif satisfying (3.8).

5. The H-Monogenic Decomposition Problem

We are now in the possibility to treat the H-monogenic decomposition problem; it means to study under which conditions a given matrix function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq354_HTML.gif can be decomposed as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ43_HTML.gif
(5.1)

where the components https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq355_HTML.gif , are Hölder continuous matrix functions on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq356_HTML.gif , which are Hermitean monogenically extendable to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq357_HTML.gif respectively, and moreover https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq358_HTML.gif . The following theorem provides an answer to that question.

Theorem 5.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq359_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq360_HTML.gif as in (3.7). Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq361_HTML.gif admits the Hermitean monogenic decomposition (5.1), where the components are explicitly given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ44_HTML.gif
(5.2)

Proof.

On account of the assumption on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq362_HTML.gif , it follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq363_HTML.gif belongs to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq364_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq365_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq366_HTML.gif , simultaneously. Then, the Hölder continuity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq367_HTML.gif directly follows from Theorem 4.1, (i) and (iii). Next, the matrix inversion formula (ii) in Theorem 4.1 yields
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ45_HTML.gif
(5.3)

showing the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq368_HTML.gif -monogenicity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq369_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq370_HTML.gif , respectively.

In order to rephrase Theorem 5.1 in the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq371_HTML.gif -monogenic setting, as studied in [23], we only need to ensure that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ46_HTML.gif
(5.4)
or, equivalently, that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ47_HTML.gif
(5.5)
which, by means of some direct calculations, can be rewritten as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ48_HTML.gif
(5.6)

This is precisely the condition under which a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq372_HTML.gif has been found to admit an https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq373_HTML.gif -monogenic decomposition (3.12); see [23, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq374_HTML.gif ]. This result may be reformulated into the present setting as follows.

Theorem 5.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq375_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq376_HTML.gif as in (3.7), and consider the corresponding matrix function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq377_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq378_HTML.gif admits the decomposition (5.1) in terms of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq379_HTML.gif -monogenic functions if and only if (5.4) holds.

Remark 5.3.

Even though the decomposition (5.1) is not unique in general, it will be so in the corresponding class:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ49_HTML.gif
(5.7)

for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq380_HTML.gif ; see Theorem 3.4.

Remark 5.4.

When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq381_HTML.gif is sufficiently regular, for example, Ahlfors-David regular, the expressions (5.2) reduce to the ones obtained in [17] in terms of the matricial Hermitean Cauchy integral, the latter being easily obtained using the Hermitean Borel-Pompeiu formula, as proved in [16]. Indeed, applying this Borel-Pompeiu formula to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq382_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ50_HTML.gif
(5.8)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq383_HTML.gif is the Hermitean Cauchy integral given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ51_HTML.gif
(5.9)
Here, the additional circulant matrix
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ52_HTML.gif
(5.10)
contains (up to a factor) the Hermitean projections https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq384_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq385_HTML.gif of the outward unit normal vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq386_HTML.gif at the point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq387_HTML.gif , while the matrix Hausdorff measure https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq388_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ53_HTML.gif
(5.11)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq389_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ54_HTML.gif
(5.12)
or, using the notations of (5.2),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_Equ55_HTML.gif
(5.13)

which coincides with [17, equation ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F791358/MediaObjects/13661_2009_Article_955_IEq390_HTML.gif )].

Declarations

Acknowledgments

This paper was written during a scientific stay of the first author at the Clifford Research Group of the Department of Mathematical Analysis of Ghent University, supported by a "Visiting Postdoctoral Fellowship" of the Flemish Research Foundation. He wishes to thank the members of the Clifford Research Group for their kind hospitality during this stay.

Authors’ Affiliations

(1)
Facultad de Informática y Matemática, Universidad de Holguín
(2)
Departamento de Matemática, Universidad de Oriente
(3)
Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering, Ghent University

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© Ricardo Abreu-Blaya et al. 2010

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