On some quaternionic generalized slice regular functions

TL;DR

This paper explores global and local properties of quaternionic generalized slice regular functions, extending the theory through a perturbed differential operator and establishing connections with complex generalized analytic functions.

Contribution

It introduces a perturbed global-type operator and demonstrates how it characterizes a new class of quaternionic functions, linking them to complex analysis on slices.

Findings

Global properties derived from the perturbed operator

Local properties include a version of the Splitting Lemma

Representation Theorem connects quaternionic functions to complex analysis

Abstract

The quaternionic valued functions of a quaternionic variable, often referred to as slice regular functions has been studied extensively due to the large number of generali\-zed results of the theory of one complex variable, see \cite{cgs,CSS,GSC,GS2,gssbook,gp,gpr,GS} and the references given there. Recently, several global properties of these functions has been found of the study of a differential operator, see \cite{GlobalOp,GP_2, G, GG1,GG2}. Particularly, given a structural set $\psi$ the Borel-Pompieu formula induced by the operator ${}^{\psi}G$ and its consequences in the slice regular function theory were studied in \cite{GG1}. The aim of this paper is to present some global and local properties of a kind of quaternionic generalized slice regular functions. We shall see that the global properties are consequences of the study of the perturbed global-type operator: \begin{align*} {}^{\psi}G_v [f] := {}^{\psi} G [f] -\frac{{\bf x}_{\psi}}{ 2} ({\bf x}_{\psi} v + v {\bf x}_{\psi} ) f , \end{align*} where $v$ is a quaternionic constant and $f$ is a quaternionic-valued continuously differentiable function with domain in $\mathbb H$ since our generalized slice regular function space coincides with $\textrm{Ker} {}^{\psi_{\textrm{st}}}G_v$ associated to an axially symmetric s-domain, where the $\psi_{\textrm{st}}$ is standard structural set. Among the local properties studied in this work are the versions of Splitting Lemma and Representation Theorem that show us a deep relationship between this generalized slice regular function space with a complex generalized analytic function space on each slice.