Iterated Integral Operators in Clifford Analysis

Abstract

Integral representation formulas of Cauchy-Pompeiu type expressing Clifford-algebra-valued functions in domains of ${\mathbb{R}}^m$ through its boundary values and its first order derivatives in form of the Dirac operator are iterated in order to get higher order Cauchy-Pompeiu formulas. In the most general representation formulas obtained the Dirac operator is replaced by products of powers of the Dirac and the Laplace operator. Boundary values of lower order operators are involved too. In particular the integral operators provide particular solutions to the inhomogeneous equations ${\partial }^{k}w=f,{\Delta }^{k}w=g$ and $\partial {\Delta }^{k}w=h$. The main subject of this paper is to develop the representation formulas. Properties of the integral operators are not studied here.