Open Access

Extension Theorem for Complex Clifford Algebras-Valued Functions on Fractal Domains

Boundary Value Problems20102010:513186

DOI: 10.1155/2010/513186

Received: 1 December 2009

Accepted: 20 March 2010

Published: 30 March 2010

Abstract

Monogenic extension theorem of complex Clifford algebras-valued functions over a bounded domain with fractal boundary is obtained. The paper is dealing with the class of Hölder continuous functions. Applications to holomorphic functions theory of several complex variables as well as to that of the so-called biregular functions will be deduced directly from the isotonic approach.

1. Introduction

It is well known that methods of Clifford analysis, which is a successful generalization to higher dimension of the theory of holomorphic functions in the complex plane, are a powerful tool for study boundary value problems of mathematical physics over bounded domains with sufficiently smooth boundaries; see [13].

One of the most important parts of this development is the particular feature of the existence of a Cauchy type integral whose properties are similar to its famous complex prototype. However, if domains with boundaries of highly less smoothness (even nonrectifiable or fractal) are allowed, then customary definition of the Cauchy integral falls, but the boundary value problems keep their interest and applicability. A natural question arises as follows.

Can we describe the class of complex Clifford algebras-valued functions from Hölder continuous space extending monogenically from the fractal boundary of a domain through the whole domain?

In [4] for the quaternionic case and in [57] for general complex Clifford algebra valued functions some preliminaries results are given. However, in all these cases the condition ensures that extendability is given in terms of box dimension and Hölder exponent of the functions space considered.

In this paper we will show that there is a rich source of material on the roughness of the boundaries permitted for a positive answer of the question which has not yet been exploited, and indeed hardly touched.

At the end, applications to holomorphic functions theory of several complex variables as well as to the so-called biregular functions (to be defined later) will be deduced directly from the isotonic approach.

The above motivation of our work is of more or less theoretical mathematical nature but it is not difficult to give arguments based on an ample gamma of applications.

Indeed, the M. S. Zhdanov book cited in [8] is a translation from Russian and the original title means literally "The analogues of the Cauchy-type integral in the Theory of Geophysics Fields". In this book is considered, as the author writes, one of the most interesting questions of the Potential plane field theory, a possibility of an analytic extension of the field into the domain occupied by sources.

He gives representations of both a gravitational and a constant magnetic field as such analogues in order to solve now the spatial problems of the separation of field as well as analytic extension through the surface and into the domain with sources.

Our results can be applied to the study of the above problems in the more general context of domains with fractal boundaries, but the detailed discussion of this technical point is beyond the scope of this paper.

2. Preliminaries

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq1_HTML.gif be an orthonormal basis of the Euclidean space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq2_HTML.gif .

The complex Clifford algebra, denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq3_HTML.gif , is generated additively by elements of the form

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ1_HTML.gif
(2.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq4_HTML.gif is such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq5_HTML.gif , and so the complex dimension of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq6_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq7_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq8_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq9_HTML.gif is the identity element.

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq10_HTML.gif , the conjugation and the main involution are defined, respectively, as

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ2_HTML.gif
(2.2)

If we identify the vectors https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq11_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq12_HTML.gif with the real Clifford vectors https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq13_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq14_HTML.gif may be considered as a subspace of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq15_HTML.gif .

The product of two Clifford vectors splits up into two parts:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ3_HTML.gif
(2.3)

where

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ4_HTML.gif
(2.4)

Generally speaking, we will consider https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq16_HTML.gif -valued functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq17_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq18_HTML.gif of the form

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ5_HTML.gif
(2.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq19_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq20_HTML.gif -valued functions. Notions of continuity and differentiability of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq21_HTML.gif are introduced by means of the corresponding notions for its complex components https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq22_HTML.gif .

In particular, for bounded set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq23_HTML.gif , the class of continuous functions which satisfy the Hölder condition of order https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq24_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq25_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq26_HTML.gif will be denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq27_HTML.gif .

Let us introduce the so-called Dirac operator given by

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ6_HTML.gif
(2.6)

It is a first-order elliptic operator whose fundamental solution is given by

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ7_HTML.gif
(2.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq28_HTML.gif is the area of the unit sphere in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq29_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq30_HTML.gif is open in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq31_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq32_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq33_HTML.gif is said to be monogenic if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq34_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq35_HTML.gif . Denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq36_HTML.gif the set of all monogenic functions in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq37_HTML.gif . The best general reference here is [9].

We recall (see [10]) that a Whitney extension of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq38_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq39_HTML.gif being compact in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq40_HTML.gif , is a compactly supported function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq41_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq42_HTML.gif and

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ8_HTML.gif
(2.8)

Here and in the sequel, we will denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq43_HTML.gif certain generic positive constant not necessarily the same in different occurrences.

The following assumption will be needed through the paper. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq44_HTML.gif be a Jordan domain, that is, a bounded oriented connected open subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq45_HTML.gif whose boundary https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq46_HTML.gif is a compact topological surface. By https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq47_HTML.gif we denote the complement domain of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq48_HTML.gif .

By definition (see [11]) the box dimension of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq49_HTML.gif , denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq50_HTML.gif , is equal to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq51_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq52_HTML.gif stands for the least number of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq53_HTML.gif -balls needed to cover https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq54_HTML.gif .

The limit above is unchanged if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq55_HTML.gif is thinking as the number of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq56_HTML.gif -cubes with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq57_HTML.gif intersecting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq58_HTML.gif . A cube https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq59_HTML.gif is called a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq60_HTML.gif -cube if it is of the form: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq61_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq62_HTML.gif are integers.

Fix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq63_HTML.gif , assuming that the improper integral https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq64_HTML.gif converges. Note that this is in agreement with [12] for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq65_HTML.gif to be https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq66_HTML.gif -summable.

Observe that a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq67_HTML.gif -summable surface has box dimension https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq68_HTML.gif . Meanwhile, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq69_HTML.gif has box dimension less than https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq70_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq71_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq72_HTML.gif -summable.

3. Extension Theorems

We begin this section with a basic result on the usual Cliffordian Théodoresco operator defined by

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ9_HTML.gif
(3.1)

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq73_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq74_HTML.gif , which we may assume, then it follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq75_HTML.gif and we may choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq76_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq77_HTML.gif . If for such https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq78_HTML.gif we can prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq79_HTML.gif then by in [3, Proposition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq80_HTML.gif ] it follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq81_HTML.gif represents a continuous function in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq82_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq83_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq84_HTML.gif , which is due to the fact that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq85_HTML.gif .

In the remainder of this section we assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq86_HTML.gif .

3.1. Monogenic Extension Theorem

Theorem 3.1.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq87_HTML.gif is the trace of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq88_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ10_HTML.gif
(3.2)

Conversely, assuming that (3.2) holds, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq89_HTML.gif is the trace of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq90_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq91_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq92_HTML.gif and define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ11_HTML.gif
(3.3)

and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq93_HTML.gif .

Note that the boundary of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq94_HTML.gif , denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq95_HTML.gif , is actually composed by certain faces (denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq96_HTML.gif ) of some cubes https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq97_HTML.gif . We will denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq98_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq99_HTML.gif the outward pointing unit normal to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq100_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq101_HTML.gif , respectively, in the sense introduced in [13].

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq102_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq103_HTML.gif be so large chosen that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq104_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq105_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq106_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq107_HTML.gif is a cube of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq108_HTML.gif . Here and below https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq109_HTML.gif denotes the diameter of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq110_HTML.gif as a subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq111_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq112_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq113_HTML.gif a cube containing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq114_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq115_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq116_HTML.gif .

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq117_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq118_HTML.gif , we have

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ12_HTML.gif
(3.4)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq119_HTML.gif be an https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq120_HTML.gif -dimensional face of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq121_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq122_HTML.gif the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq123_HTML.gif -cube containing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq124_HTML.gif ; then if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq125_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ13_HTML.gif
(3.5)
Each face of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq126_HTML.gif is one of those https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq127_HTML.gif of some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq128_HTML.gif . Therefore, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq129_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ14_HTML.gif
(3.6)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq130_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ15_HTML.gif
(3.7)
By Stokes formula we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ16_HTML.gif
(3.8)
Therefore
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ17_HTML.gif
(3.9)

The same conclusion can be drawn for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq131_HTML.gif . The only point now is to note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq132_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq133_HTML.gif .

Finally, due to the fact that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ18_HTML.gif
(3.10)

we prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq134_HTML.gif , and the second assertion follows directly by taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq135_HTML.gif .

The finiteness of the last sum follows from the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq136_HTML.gif -summability of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq137_HTML.gif together with the fact that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq138_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq139_HTML.gif the following analogous result can be obtained.

Theorem 3.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq140_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq141_HTML.gif is the trace of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq142_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq143_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ19_HTML.gif
(3.11)

Conversely, assuming that (3.11) holds, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq144_HTML.gif is the trace of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq145_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq146_HTML.gif .

3.2. Isotonic Extension Theorem

For our purpose we will assume that the dimension of the Euclidean space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq147_HTML.gif is even whence we will put https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq148_HTML.gif from now on.

In a series of recent papers, so-called isotonic Clifford analysis has emerged as yet a refinement of the standard case but also has strong connections with the theory of holomorphic functions of several complex variables and biregular ones, even encompassing some of its results; see [1418].

Put

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ20_HTML.gif
(3.12)

then a primitive idempotent is given by

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ21_HTML.gif
(3.13)

We have the following conversion relations:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ22_HTML.gif
(3.14)

with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq149_HTML.gif (complex Clifford algebra generated by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq150_HTML.gif ).

Note that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq151_HTML.gif one also has that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ23_HTML.gif
(3.15)

Let us introduce the following real Clifford vectors and their corresponding Dirac operators:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ24_HTML.gif
(3.16)

The function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq152_HTML.gif is said to be isotonic in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq153_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq154_HTML.gif is continuously differentiable in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq155_HTML.gif and moreover satisfies the equation

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ25_HTML.gif
(3.17)

We will denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq156_HTML.gif the set of all isotonic functions in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq157_HTML.gif .

We find ourselves forced to introduce two extra Cauchy kernels, defined by

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ26_HTML.gif
(3.18)

Now we may introduce the isotonic Théodoresco transform of a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq158_HTML.gif to be

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ27_HTML.gif
(3.19)

It is straightforward to deduce that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ28_HTML.gif
(3.20)

Theorem 3.3.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq159_HTML.gif , is the trace of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq160_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ29_HTML.gif
(3.21)

Conversely, assuming that (3.21) holds, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq161_HTML.gif is the trace of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq162_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq163_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq164_HTML.gif be an isotonic extension of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq165_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq166_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq167_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq168_HTML.gif is a monogenic extension of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq169_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq170_HTML.gif , which obviously belongs to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq171_HTML.gif . Therefore
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ30_HTML.gif
(3.22)

by Theorem 3.1.

We thus get

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ31_HTML.gif
(3.23)

the first equality being a direct consequence of (3.20). According to (3.15) we have (3.21), which is the desired conclusion.

On account of Theorem 3.1 again, the converse assertion follows directly by taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq172_HTML.gif , and the proof is complete.

Remark 3.4.

Theorems 3.1 and 3.3 extend the results in [47], since the restriction putted there ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq173_HTML.gif ) implies that of this paper.

4. Applications

In this last section, we will briefly discuss two particular cases which arise when considering (3.17).

Case 1.

It is easily seen that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq174_HTML.gif takes values in the space of scalars https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq175_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq176_HTML.gif is isotonic if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ32_HTML.gif
(4.1)

which means that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq177_HTML.gif is a holomorphic function with respect to the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq178_HTML.gif complex variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq179_HTML.gif .

Case 2.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq180_HTML.gif , isotonic function, takes values in the real Clifford algebra https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq181_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ33_HTML.gif
(4.2)
or, equivalently, by the action of the main involution on the second equation we arrive to the overdetermined system:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ34_HTML.gif
(4.3)

whose solutions are called biregular functions. For a detailed study we refer the reader to [1921].

The proof of Theorem 3.3 may readily be adapted to establish analogous results for both holomorphic and biregular functions context. Clearly, we prove that if we replace https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq182_HTML.gif by a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq183_HTML.gif -valued, respectively, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq184_HTML.gif -valued function, such that (3.21) holds, then there exists an isotonic extension https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq185_HTML.gif , which, by using the classical Dirichlet problem, takes values precisely in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq186_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq187_HTML.gif , respectively. On the other direction the proof is immediate. The corresponding statements are left to the reader.

Declarations

Acknowledgments

The topic covered here has been initiated while the first two authors were visiting IMPA, Rio de Janeiro, in July of 2009. Ricardo Abreu and Juan Bory wish to thank CNPq for financial support. Ricardo Abreu wishes to thank the Faculty of Ingeneering, Universidad Diego Portales, Santiago de Chile, for the kind hospitality during the period in which the final version of the paper was eventually completed. This work has been partially supported by CONICYT (Chile) under FONDECYT Grant 1090063.

Authors’ Affiliations

(1)
Departamento de Matemática, Universidad de Holguín
(2)
Departamento de Matemática, niversidad de Oriente
(3)
Facultad de Ingeniería, Universidad Diego Portales

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Copyright

© Ricardo Abreu-Blaya et al. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.