Open Access Research Article

Hermitean Téodorescu Transform Decomposition of Continuous Matrix Functions on Fractal Hypersurfaces

Ricardo Abreu-Blaya1, Juan Bory-Reyes2, Fred Brackx3 and Hennie De Schepper3*

Author Affiliations

1 Facultad de Informática y Matemática, Universidad de Holguín, Holguín 80100, Cuba

2 Departamento de Matemática, Universidad de Oriente, Santiago de Cuba 90500, Cuba

3 Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Galglaan 2, 9000 Gent, Belgium

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Boundary Value Problems 2010, 2010:791358  doi:10.1155/2010/791358

Published: 20 May 2010

Abstract

We consider Hölder continuous circulant () matrix functions defined on the fractal boundary of a domain in . The main goal is to study under which conditions such a function can be decomposed as , where the components are extendable to -monogenic functions in the interior and the exterior of , respectively. -monogenicity are a concept from the framework of Hermitean Clifford analysis, a higher-dimensional function theory centered around the simultaneous null solutions of two first-order vector-valued differential operators, called Hermitean Dirac operators. -monogenic functions then are the null solutions of a () matrix Dirac operator, having these Hermitean Dirac operators as its entries; such matrix functions play an important role in the function theoretic development of Hermitean Clifford analysis. In the present paper a matricial Hermitean Téodorescu transform is the key to solve the problem under consideration. The obtained results are then shown to include the ones where domains with an Ahlfors-David regular boundary were considered.