Hermitean Téodorescu Transform Decomposition of Continuous Matrix Functions on Fractal Hypersurfaces. (20th May 2010) - Abreu-Blaya Abreu-Blaya, Ricardo Ricardo; Bory-Reyes Bory-Reyes, Juan Juan; Brackx Brackx, Fred Fred; De Schepper De Schepper, Hennie Hennie
Abstract : We consider Hölder continuous circulant (2 × 2 ) matrix functions G 2 1 defined on the fractal boundary Γ of a domain Ω in ℝ 2 n . The main goal is to study under which conditions such a function G 2 1 can be decomposed as G 2 1 = G 2 1 + − G 2 1 −, where the components G 2 1 ± 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.