A Cauchy kernel for the Hermitian submonogenic system

TL;DR

This paper introduces a Cauchy kernel for the Hermitian submonogenic system, enabling integral representations and extending properties similar to the standard Dirac operator in Euclidean space.

Contribution

It constructs a Cauchy kernel for the Hermitian submonogenic system and establishes an integral representation formula, advancing the analysis of these functions.

Findings

Constructed a Cauchy kernel for the Hermitian submonogenic system

Derived a Stokes type formula for integral representation

Identified properties similar to the standard Dirac operator

Abstract

Hermitian monogenic functions are the null solutions of two complex Dirac type operators. The system of these complex Dirac operators is overdetermined and may be reduced to constraints for the Cauchy datum together with what we called the Hermitian submonogenic system (see [8],[9]). This last system is no longer overdetermined and it has properties that are similar to those of the standard Dirac operator in Euclidean space, such as a Cauchy-Kowalevski extension theorem and Vekua type solutions. In this paper, we investigate plane wave solutions of the Hermitian submonogenic system, leading to the construction of a Cauchy kernel. We also establish a Stokes type formula that, when applied to the Cauchy kernel provides an integral representation formula for Hermitian submonogenic functions.