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.
Categories: math.CV