On some quaternionic generalized slice regular functions

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*} {}^ψG_v [f] := {}^ψ G [f] -\frac{{\bf x}_ψ}{ 2} ({\bf x}_ψ v + v {\bf x}_ψ ) 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 $\text{Ker}{}^{{\psi }_{\text{st}}}{G}_v$ associated to an axially symmetric s-domain, where the ${\psi }_{\text{st}}$ 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.

References in corpus (1)

• The C-property for slice regular functions and applications to the Bergman space