On Cauchy Type Integrals Related to the Cimmino System of Partial Differential Equations

Abstract

In this paper, we established a one-to-one correspondence between quaternionic hyperholomorphic functions in \(\mathbb{R}^{4}\;\cong\;\mathbb{C}^{2}\) and solutions (pairs of complex-valued functions) for Cimmino system of partial differential equations written in complex form. This leads to a pair of Cauchy type integrals associated with Cimmino system. The topics of the paper concern theorems which cover basic properties of those Cauchy type integrals: the Sokhotski– Plemelj and Plemelj–Privalov type theorems for it as well as the necessary and sufficient condition for the possibility to extend a given pair of complexvalued Hölder-continuous functions from such a surface up to a solution of Cimmino system in a Jordan domain. Formulae for the square of the corresponding singular Cauchy type integrals are given. The proofs of all these facts are based on intimate relations between the theory of Cimmino system and some version of quaternionic analysis.KeywordsCimmino systemquaternionic analysishyperholomorphic functions