On the Conformal Mappings and the Global Operator G

Abstract

Some important global properties of the slice regular functions have been obtained from the global operator $$\begin{aligned}G:= \Vert \mathbf {x}\Vert ^2 \partial _0 + \mathbf {x} \sum _{i=1}^3 x_i \partial _i , \end{aligned}$$ such as a global characterization, a global Cauchy integral theorem and a global Borel–Pompeiu formula, see Colombo et al. (Trans Am Math Soc 365:303–318, 2013), Gonzalez Cervantes (Complex Anal Oper Theory 13:2527–2539, 2019) and Gonzalez Cervantes and Gonzalez-Campos (Complex Var Elliptic Equ 65:1–10, 2020, https://doi.org/10.1080/17476933.2020.1738410) [4, 13, 14], respectively. The aim of this work is to show: some relationships between G and the composition operator with the conformal mappings, a conformal covariance property of G along with its interpretations in terms of a covariant functor, all consequences of these facts for the slice regular functions, a Leibnitz rule associated to the operator G and a characterization of the real Components of slice regular functions in terms of a Non-constant Coefficient second order differential equation.