Research Article
Received 18 June 2009 Published online 11 August 2009 in Wiley InterScience
(www.interscience.wiley.com) DOI: 10.1002/mma.1209
MOS subject classification: 30 G 35
Biregular extendability via isotonic Clifford
analysis
Ricardo Abreu Blaya,
a
Juan Bory Reyes,
b
Dixan Peña Peña
c ∗ †
and Frank Sommen
c
We use the so-called isotonic functions to obtain extension theorems in the framework of biregular functions of Clifford
analysis. In this context we also prove the Plemelj–Sokhotski formulae for the Bochner–Martinelli integral and an
expression for the square of its singular version. Copyright © 2009 John Wiley & Sons, Ltd.
Keywords: Clifford algebras; isotonic functions; biregular extension theorem
1. Introduction
The biregular functions were introduced by Brackx and Pincket as a natural generalization to higher dimension of the theory
of holomorphic functions in C
2
(see [1--4]). For other overdetermined systems in Clifford analysis we refer the reader to [5--11],
and for an introduction to the general theory of overdetermined systems we refer the reader to [12] and the references
therein.
The aim of this paper is to study the question under which conditions a function defined and continuous on the boundary of
a bounded domain
+
⊂ R
m
has a biregular extension to
+
. The results obtained may be viewed as generalizations to the case
of biregular functions of those obtained by Aronov, Kytmanov and A˘ ızenberg (among others) for holomorphic functions of several
complex variables (see [13--15]). That is, extension theorems in terms of an analog of the Bochner–Martinelli integral (denoted by
M
1
). It should be noted that results of this nature have been worked out for more general overdetermined elliptic systems in [12].
It is important to remark that our approach not only enables us to get simplified and elegant proofs, but also to obtain alternative
characterizations using a second integral operator and its singular version (denoted by M
2
and N
2
, respectively). Finally, in our
results we also deal with quite general surfaces (Ahlfors–David-regular (AD-regular) surfaces).
The outline of the paper is as follows. For the reader who is not familiar with Clifford analysis, we recall some of its basics in
Section 2. In Section 3 we introduce our main techniques of work: the isotonic functions and the isotonic Cauchy-type integral. In
the last two sections our results are stated and proved.
2. Some basic notions of Clifford analysis
Clifford analysis (see e.g. [16--18]) offers a function theory that is a higher-dimensional analogue of the theory of the holomorphic
functions of one complex variable.
The functions considered are defined in the Euclidean space R
m
(m>1) and take their values in the complex Clifford
algebra C
m
. Let (e
1
,... ,e
m
) be an orthonormal basis of R
m
, then a basis for the Clifford algebra C
m
is given by (e
A
: A ⊂
{1,... ,m}), where e
∅
= 1 is the identity element, e
{j}
= e
j
,j = 1,... ,m and e
A
= e
j
1
... e
j
k
,A ={j
1
,... ,j
k
} being ordered such that
j
1
< ··· <j
k
.
a
Facultad de Informática y Matemática, Universidad de Holguín, Holguín 80100, Cuba
b
Departamento de Matemática, Universidad de Oriente, Santiago de Cuba 90500, Cuba
c
Department of Mathematical Analysis, Ghent University, Galglaan 2, B-9000 Gent, Belgium
∗
Correspondence to: Dixan Peña Peña, Department of Mathematical Analysis, Ghent University, Galglaan 2, B-9000 Gent, Belgium.
†
E-mail: dixan@cage.UGent.be, dixanpena@gmail.com
Contract/grant sponsor: CAPES-MES project; contract/grant number: 028/07
Contract/grant sponsor: Ghent university
384
Copyright © 2009 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2010, 33 384–393