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On the Conformal Mappings and the Global Operator G

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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), González Cervantes (Complex Anal Oper Theory 13:2527–2539, 2019) and González Cervantes and González-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.

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Authors and Affiliations

  1. Departamento de Matemáticas, E.S.F.M. del I.P.N., 07338, Mexico City, Mexico

    J. Oscar González Cervantes & Daniel González Campos

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  1. J. Oscar González Cervantes
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  2. Daniel González Campos
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Correspondence to Daniel González Campos.

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Communicated by Vladislav Kravchenko

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Cervantes, J.O.G., Campos, D.G. On the Conformal Mappings and the Global Operator G. Adv. Appl. Clifford Algebras 31, 6 (2021). https://doi.org/10.1007/s00006-020-01103-6

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