Monogenic extension theorem of complex Clifford algebrasvalued 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 socalled 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 [1–3].
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 algebrasvalued 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 [5–7] 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 socalled 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 Cauchytype 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 be an orthonormal basis of the Euclidean space .
The complex Clifford algebra, denoted by , is generated additively by elements of the form
where is such that , and so the complex dimension of is . For , is the identity element.
For , the conjugation and the main involution are defined, respectively, as
If we identify the vectors of with the real Clifford vectors , then may be considered as a subspace of .
The product of two Clifford vectors splits up into two parts:
where
Generally speaking, we will consider valued functions on of the form
where are valued functions. Notions of continuity and differentiability of are introduced by means of the corresponding notions for its complex components .
In particular, for bounded set , the class of continuous functions which satisfy the Hölder condition of order in will be denoted by .
Let us introduce the socalled Dirac operator given by
It is a firstorder elliptic operator whose fundamental solution is given by
where is the area of the unit sphere in .
If is open in and , then is said to be monogenic if in . Denote by the set of all monogenic functions in . The best general reference here is [9].
We recall (see [10]) that a Whitney extension of , being compact in , is a compactly supported function such that and
Here and in the sequel, we will denote by certain generic positive constant not necessarily the same in different occurrences.
The following assumption will be needed through the paper. Let be a Jordan domain, that is, a bounded oriented connected open subset of whose boundary is a compact topological surface. By we denote the complement domain of .
By definition (see [11]) the box dimension of , denoted by , is equal to where stands for the least number of balls needed to cover .
The limit above is unchanged if is thinking as the number of cubes with intersecting . A cube is called a cube if it is of the form: where are integers.
Fix , assuming that the improper integral converges. Note that this is in agreement with [12] for to be summable.
Observe that a summable surface has box dimension . Meanwhile, if has box dimension less than , then is summable.
3. Extension Theorems
We begin this section with a basic result on the usual Cliffordian Théodoresco operator defined by
If such that , which we may assume, then it follows that and we may choose such that . If for such we can prove that then by in [3, Proposition ] it follows that represents a continuous function in . Moreover, for any , which is due to the fact that .
In the remainder of this section we assume that .
3.1. Monogenic Extension Theorem
Theorem 3.1.
If is the trace of , then
Conversely, assuming that (3.2) holds, then is the trace of , for any .
Proof.
Let and define
and .
Note that the boundary of , denoted by , is actually composed by certain faces (denoted by ) of some cubes . We will denote by , the outward pointing unit normal to and , respectively, in the sense introduced in [13].
Let and let be so large chosen that and for , where is a cube of . Here and below denotes the diameter of as a subset of .
Let , a cube containing , and such that .
Since , , we have
Let be an dimensional face of and the cube containing ; then if , we have
Each face of is one of those of some . Therefore, for
Since , we get
By Stokes formula we have
Therefore
The same conclusion can be drawn for . The only point now is to note that for .
Finally, due to the fact that
we prove that , and the second assertion follows directly by taking .
The finiteness of the last sum follows from the summability of together with the fact that .
For the following analogous result can be obtained.
Theorem 3.2.
Let . If is the trace of and , then
Conversely, assuming that (3.11) holds, then is the trace of , for any .
3.2. Isotonic Extension Theorem
For our purpose we will assume that the dimension of the Euclidean space is even whence we will put from now on.
In a series of recent papers, socalled 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 [14–18].
Put
then a primitive idempotent is given by
We have the following conversion relations:
with (complex Clifford algebra generated by ).
Note that for one also has that
Let us introduce the following real Clifford vectors and their corresponding Dirac operators:
The function is said to be isotonic in if and only if is continuously differentiable in and moreover satisfies the equation
We will denote by the set of all isotonic functions in .
We find ourselves forced to introduce two extra Cauchy kernels, defined by
Now we may introduce the isotonic Théodoresco transform of a function to be
It is straightforward to deduce that
Theorem 3.3.
If , is the trace of , then
Conversely, assuming that (3.21) holds, then is the trace of , for any .
Proof.
Let be an isotonic extension of to such that . Then, is a monogenic extension of to , which obviously belongs to . Therefore
by Theorem 3.1.
We thus get
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 , and the proof is complete.
Remark 3.4.
Theorems 3.1 and 3.3 extend the results in [4–7], since the restriction putted there () 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 takes values in the space of scalars , then is isotonic if and only if
which means that is a holomorphic function with respect to the complex variables .
Case 2.
If , isotonic function, takes values in the real Clifford algebra , then
or, equivalently, by the action of the main involution on the second equation we arrive to the overdetermined system:
whose solutions are called biregular functions. For a detailed study we refer the reader to [19–21].
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 by a valued, respectively, valued function, such that (3.21) holds, then there exists an isotonic extension , which, by using the classical Dirichlet problem, takes values precisely in or , respectively. On the other direction the proof is immediate. The corresponding statements are left to the reader.
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.
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