On the Bergmann Kernel Function in the Clifford Analysis

Abstract

We define and study the Bergmann kernel function and corresponding integral operator for Clifford algebra-valued hyperholomorphic functions. For any bounded domain Ω with a smooth enough boundary Г = ∂Ω we have established the following facts:

The formula which gives an independent definition of _ψ Β by means of the Green function g of a domain Ω; _ψ Β is left-ψ-h.h. with respect to the first variable, is right-anti-ψ-h.h. with respect to the second one, is hermitian conjugate; Bergmannh.h. operator^ψ B (i.e. the integral operator with the kernel _ψ B) reproduces left-ψ-h.h. functions; the integral representation of _ψ Β containing the integral over the boundary with the Clifford Cauchy kernel and the integral over Ω giving a continuous function.