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Hermitean téodorescu transform decomposition of continuous matrix functions on fractal hypersurfaces

Abstract:

We consider Hölder continuous circulant (2 × 2) matrix functions G 21 defined on the fractal boundary Γ of a domain ω in ℝ2n. The main goal is to study under which conditions such a function G 21 can be decomposed as G 21 = G 21+ - G 21-, where the componentsG 21± are extendable to H -monogenic functions in the interior and the exterior of ω,respectively. H -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. H -monogenic functions then are the null solutions of a (2 × 2) 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. Copyright © 2010 Ricardo Abreu-Blaya et al.