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

  • Ricardo Abreu-Blaya1,

    Affiliated with

    • Juan Bory-Reyes2 and

      Affiliated with

      • Paul Bosch3Email author

        Affiliated with

        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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq2_HTML.gif .

        The complex Clifford algebra, denoted by http://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

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

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

        For http://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

        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq11_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq12_HTML.gif with the real Clifford vectors http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq13_HTML.gif , then http://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 http://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:

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

        where

        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq16_HTML.gif -valued functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq17_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq18_HTML.gif of the form

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

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq19_HTML.gif are http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq22_HTML.gif .

        In particular, for bounded set http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq24_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq25_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq26_HTML.gif will be denoted by http://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

        http://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

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

        where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq29_HTML.gif .

        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq30_HTML.gif is open in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq31_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq32_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq33_HTML.gif is said to be monogenic if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq34_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq35_HTML.gif . Denote by http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq38_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq39_HTML.gif being compact in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq40_HTML.gif , is a compactly supported function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq41_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq42_HTML.gif and

        http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq45_HTML.gif whose boundary http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq46_HTML.gif is a compact topological surface. By http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq47_HTML.gif we denote the complement domain of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq49_HTML.gif , denoted by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq50_HTML.gif , is equal to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq51_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq52_HTML.gif stands for the least number of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq53_HTML.gif -balls needed to cover http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq54_HTML.gif .

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

        Fix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq63_HTML.gif , assuming that the improper integral http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq65_HTML.gif to be http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq66_HTML.gif -summable.

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

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

        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq73_HTML.gif such that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq75_HTML.gif and we may choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq76_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq77_HTML.gif . If for such http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq78_HTML.gif we can prove that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq79_HTML.gif then by in [3, Proposition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq80_HTML.gif ] it follows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq81_HTML.gif represents a continuous function in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq82_HTML.gif . Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq83_HTML.gif for any http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq87_HTML.gif is the trace of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq88_HTML.gif , then
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq89_HTML.gif is the trace of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq90_HTML.gif , for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq91_HTML.gif .

        Proof.

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

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

        Note that the boundary of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq94_HTML.gif , denoted by http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq96_HTML.gif ) of some cubes http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq97_HTML.gif . We will denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq98_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq99_HTML.gif the outward pointing unit normal to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq100_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq102_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq103_HTML.gif be so large chosen that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq104_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq105_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq106_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq107_HTML.gif is a cube of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq108_HTML.gif . Here and below http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq109_HTML.gif denotes the diameter of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq110_HTML.gif as a subset of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq111_HTML.gif .

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

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

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ12_HTML.gif
        (3.4)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq119_HTML.gif be an http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq120_HTML.gif -dimensional face of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq121_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq122_HTML.gif the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq123_HTML.gif -cube containing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq124_HTML.gif ; then if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq125_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ13_HTML.gif
        (3.5)
        Each face of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq126_HTML.gif is one of those http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq127_HTML.gif of some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq128_HTML.gif . Therefore, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq129_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ14_HTML.gif
        (3.6)
        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq130_HTML.gif , we get
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ16_HTML.gif
        (3.8)
        Therefore
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq132_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq133_HTML.gif .

        Finally, due to the fact that

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

        we prove that http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq136_HTML.gif -summability of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq137_HTML.gif together with the fact that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq138_HTML.gif .

        For http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq140_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq141_HTML.gif is the trace of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq142_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq143_HTML.gif , then
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq144_HTML.gif is the trace of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq145_HTML.gif , for any http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq147_HTML.gif is even whence we will put http://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

        http://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

        http://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:

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

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

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

        http://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:

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

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

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

        We will denote by http://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 http://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

        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq158_HTML.gif to be

        http://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

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

        Theorem 3.3.

        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq159_HTML.gif , is the trace of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq160_HTML.gif , then
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq161_HTML.gif is the trace of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq162_HTML.gif , for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq163_HTML.gif .

        Proof.

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

        http://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 http://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 ( http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq175_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq176_HTML.gif is isotonic if and only if
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_Equ32_HTML.gif
        (4.1)

        which means that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq178_HTML.gif complex variables http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq179_HTML.gif .

        Case 2.

        If http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq181_HTML.gif , then
        http://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:
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq182_HTML.gif by a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq183_HTML.gif -valued, respectively, http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F513186/MediaObjects/13661_2009_Article_931_IEq186_HTML.gif or http://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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        © Ricardo Abreu-Blaya et al. 2010

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