Extension Theorem for Complex Clifford Algebras-Valued Functions on Fractal Domains
© Ricardo Abreu-Blaya et al. 2010
Received: 1 December 2009
Accepted: 20 March 2010
Published: 30 March 2010
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.
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 algebras-valued functions from Hölder continuous space extending monogenically from the fractal boundary of a domain through the whole domain?
In  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 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  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.
The product of two Clifford vectors splits up into two parts:
Let us introduce the so-called Dirac operator given by
It is a first-order elliptic operator whose fundamental solution is given by
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 .
We recall (see ) that a Whitney extension of , being compact in , is a compactly supported function such that and
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 ) the box dimension of , denoted by , is equal to where stands for the least number of -balls needed to cover .
Fix , assuming that the improper integral converges. Note that this is in agreement with  for to be -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 .
3.1. Monogenic Extension Theorem
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 .
Finally, due to the fact that
3.2. Isotonic Extension Theorem
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 [14–18].
then a primitive idempotent is given by
We have the following conversion relations:
Let us introduce the following real Clifford vectors and their corresponding Dirac operators:
We find ourselves forced to introduce two extra Cauchy kernels, defined by
It is straightforward to deduce that
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.
In this last section, we will briefly discuss two particular cases which arise when considering (3.17).
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.
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.
- Gürlebeck K, Sprössig W: Quaternionic Analysis and Elliptic Boundary Value Problems, International Series of Numerical Mathematics. Volume 89. Birkhäuser, Basel, Switzerland; 1990:253.View ArticleMATHGoogle Scholar
- Gürlebeck K, Sprössig W: Quaternionic and Clifford Calculus for Physicists and Engineers. John Wiley & Sons, New York, NY, USA; 1997.MATHGoogle Scholar
- Gürlebeck K, Habetha K, Sprössig W: Holomorphic Functions in the Plane and n-Dimensional Space. Birkhäuser, Basel, Switzerland; 2007.Google Scholar
- Abreu-Blaya A, Bory-Reyes J, Moreno-García T:Holomorphic extension theorems in Lipschitz domains of . Advances in Applied Clifford Algebras 2010,20(1):1-12. 10.1007/s00006-008-0137-0MathSciNetView ArticleMATHGoogle Scholar
- Abreu-Blaya A, Bory-Reyes J, Peña DP, Sommen F: Biregular extendability via isotonic Clifford analysis. Mathematical Methods in the Applied Sciences 2010,33(4):384-393.MathSciNetMATHGoogle Scholar
- Abreu-Blaya R, Bory-Reyes J, Moreno-García T: Teodorescu transform decomposition of multivector fields on fractal hypersurfaces. In Wavelets, Multiscale Systems and Hypercomplex Analysis, Operator Theory: Advances and Applications. Volume 167. Birkhäuser, Basel, Switzerland; 2006:1-16. 10.1007/3-7643-7588-4_1View ArticleGoogle Scholar
- Abreu-Blaya R, Bory-Reyes J, Moreno-García T:A biregular extension theorem from a fractal surface in . to appear in Georgian Mathematical JournalGoogle Scholar
- Shapiro M, Vasilievski NL, Zhdanov MS: Space analogs of the Cauchy type integrals and the quaternion theory. In Integral Transforms in Geophysics. Edited by: Zhdanov MS. Springer, Heidelberg, Germany; 1988:367. (Appendix A: 344–350)Google Scholar
- Brackx F, Delanghe R, Sommen F: Clifford Analysis, Research Notes in Mathematics. Volume 76. Pitman, Boston, Mass, USA; 1982:x+308.Google Scholar
- Stein EM: Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, no. 30. Princeton University Press, Princeton, NJ, USA; 1970:xiv+290.Google Scholar
- Falconer KJ: The Geometry of Fractal Sets, Cambridge Tracts in Mathematics. Volume 85. Cambridge University Press, Cambridge, UK; 1986:xiv+162.Google Scholar
- Harrison J, Norton A: The Gauss-Green theorem for fractal boundaries. Duke Mathematical Journal 1992,67(3):575-588. 10.1215/S0012-7094-92-06724-XMathSciNetView ArticleMATHGoogle Scholar
- Federer H: Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften. Volume 153. Springer, New York, NY, USA; 1969:xiv+676.Google Scholar
- Abreu-Blaya R, Bory-Reyes J: A Martinelli-Bochner formula on fractal domains. Archiv der Mathematik 2009,92(4):335-343. 10.1007/s00013-009-3049-xMathSciNetView ArticleMATHGoogle Scholar
- Abreu-Blaya R, Bory-Reyes J, Peña Peña D, Sommen F: The isotonic Cauchy transform. Advances in Applied Clifford Algebras 2007,17(2):145-152. 10.1007/s00006-007-0025-zMathSciNetView ArticleMATHGoogle Scholar
- Abreu-Blaya A, Bory-Reyes J, Peña DP, Sommen F:Holomorphic extension theorems in Lipschitz domains of . Advances in Applied Clifford Algebras 2010,20(1):1-12. 10.1007/s00006-008-0137-0MathSciNetView ArticleMATHGoogle Scholar
- Bory-Reyes J, Peña Peña D, Sommen F: A Davydov theorem for the isotonic Cauchy transform. Journal of Analysis and Applications 2007,5(2):109-121.MathSciNetMATHGoogle Scholar
- Sommen F, Peña Peña D: Martinelli-Bochner formula using Clifford analysis. Archiv der Mathematik 2007,88(4):358-363. 10.1007/s00013-006-1932-2MathSciNetView ArticleMATHGoogle Scholar
- Brackx F, Pincket W: A Bochner-Martinelli formula for the biregular functions of Clifford analysis. Complex Variables: Theory & Application 1984,4(1):39-48.MathSciNetView ArticleMATHGoogle Scholar
- Brackx F, Pincket W: The biregular functions of Clifford analysis: some special topics. In Clifford Algebras and Their Applications in Mathematical Physics (Canterbury, 1985), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences. Volume 183. Reidel, Dordrecht, The Netherlands; 1986:159-166.View ArticleGoogle Scholar
- Brackx F, Pincket W: Two Hartogs theorems for nullsolutions of overdetermined systems in Euclidean space. Complex Variables: Theory & Application 1985,4(3):205-222.MathSciNetView ArticleMATHGoogle Scholar
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