The Cauchy transform in the slice hyperholomorphic setting and related topics

TL;DR

This paper explores the quaternionic Cauchy transform and its additive splitting in slice hyperholomorphic functions, introducing a fundamental solution and extending results to Clifford algebra contexts.

Contribution

It introduces the quaternionic Cauchy transform's additive splitting and the fundamental solution for the slice hyperholomorphic operator, extending to Clifford algebra settings.

Findings

Additive splitting of the quaternionic Cauchy transform

Introduction of the fundamental solution for the slice hyperholomorphic operator

Results applicable to Clifford algebra-valued functions

Abstract

In this paper we study the additive splitting associated to the quaternionic Cauchy transform defined by the Cauchy formula of slice hyperholomorphic functions. Moreover, we introduce and study the analogue of the fundamental solution of the global operator of slice hyperholomorphic functions. We state our results in the quaternionic setting but several results hold for Clifford algebra-valued function with minor changes in the proofs.