2021/11/10 by José Óscar González-Cervantes, González-Cervantes, José Oscar
Mathematics · #30G35 #Advanced Topics in Algebra #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2111.05520
openalex publication_date 2021/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \citecgs,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 \citeGlobalOp,GP2, G, GG1,GG2. Particularly, given a structural set ψ the Borel-Pompieu formula induced by the operator ψG and its consequences in the slice regular function theory were studied in \citeGG1. 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: ψGv [f] := ψ G [f] -\frac\bf xψ 2 (\bf xψ v + v \bf xψ ) f , 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 \textrmKer ^ψ_\textrmstGv associated to an axially symmetric s-domain, where the ψ_\textrmst 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.