Hermitean Téodorescu Transform Decomposition of Continuous Matrix Functions on Fractal Hypersurfaces

• Main Authors: Bory-Reyes Juan, Brackx Fred, De Schepper Hennie, Abreu-Blaya Ricardo
• Format: Article
• Language: English
• Published: SpringerOpen 2010-01-01
• Series: Boundary Value Problems
• Online Access: http://www.boundaryvalueproblems.com/content/2010/791358

<p/> <p>We consider H&#246;lder continuous circulant (<inline-formula> <graphic file="1687-2770-2010-791358-i1.gif"/></inline-formula>) matrix functions <inline-formula> <graphic file="1687-2770-2010-791358-i2.gif"/></inline-formula> defined on the fractal boundary <inline-formula> <graphic file="1687-2770-2010-791358-i3.gif"/></inline-formula> of a domain <inline-formula> <graphic file="1687-2770-2010-791358-i4.gif"/></inline-formula> in <inline-formula> <graphic file="1687-2770-2010-791358-i5.gif"/></inline-formula>. The main goal is to study under which conditions such a function <inline-formula> <graphic file="1687-2770-2010-791358-i6.gif"/></inline-formula> can be decomposed as <inline-formula> <graphic file="1687-2770-2010-791358-i7.gif"/></inline-formula>, where the components <inline-formula> <graphic file="1687-2770-2010-791358-i8.gif"/></inline-formula> are extendable to <inline-formula> <graphic file="1687-2770-2010-791358-i9.gif"/></inline-formula>-monogenic functions in the interior and the exterior of <inline-formula> <graphic file="1687-2770-2010-791358-i10.gif"/></inline-formula>, respectively. <inline-formula> <graphic file="1687-2770-2010-791358-i11.gif"/></inline-formula>-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. <inline-formula> <graphic file="1687-2770-2010-791358-i12.gif"/></inline-formula>-monogenic functions then are the null solutions of a (<inline-formula> <graphic file="1687-2770-2010-791358-i13.gif"/></inline-formula>) 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 T&#233;odorescu 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.</p>