Hermitean Cauchy Integral Decomposition of Continuous Functions on Hypersurfaces

• Main Authors: Brackx Fred, De Knock Bram, De Schepper Hennie, Peña DixanPeña, Sommen Frank, Reyes JuanBory, Blaya RicardoAbreu
• Format: Article
• Language: English
• Published: SpringerOpen 2008-01-01
• Series: Boundary Value Problems
• Online Access: http://www.boundaryvalueproblems.com/content/2008/425256

<p/> <p>We consider H&#246;lder continuous circulant <inline-formula> <graphic file="1687-2770-2008-425256-i1.gif"/></inline-formula> matrix functions <inline-formula> <graphic file="1687-2770-2008-425256-i2.gif"/></inline-formula> defined on the Ahlfors-David regular boundary <inline-formula> <graphic file="1687-2770-2008-425256-i3.gif"/></inline-formula> of a domain <inline-formula> <graphic file="1687-2770-2008-425256-i4.gif"/></inline-formula> in <inline-formula> <graphic file="1687-2770-2008-425256-i5.gif"/></inline-formula>. The main goal is to study under which conditions such a function <inline-formula> <graphic file="1687-2770-2008-425256-i6.gif"/></inline-formula> can be decomposed as <inline-formula> <graphic file="1687-2770-2008-425256-i7.gif"/></inline-formula>, where the components <inline-formula> <graphic file="1687-2770-2008-425256-i8.gif"/></inline-formula> are extendable to two-sided <inline-formula> <graphic file="1687-2770-2008-425256-i9.gif"/></inline-formula>-monogenic functions in the interior and the exterior of <inline-formula> <graphic file="1687-2770-2008-425256-i10.gif"/></inline-formula>, respectively. <inline-formula> <graphic file="1687-2770-2008-425256-i11.gif"/></inline-formula>-monogenicity is 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="1687-2770-2008-425256-i12.gif"/></inline-formula>-monogenic functions then are the null solutions of a <inline-formula> <graphic file="1687-2770-2008-425256-i13.gif"/></inline-formula> matrix Dirac operator, having these Hermitean Dirac operators as its entries; such functions have been crucial for the development of function theoretic results in the Hermitean Clifford context.</p>