Advances in Applied Clifford Algebras ( IF 1.1 ) Pub Date : 2021-01-01 , DOI: 10.1007/s00006-020-01103-6 J. Oscar González Cervantes , Daniel González Campos
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.
中文翻译：
保形映射与全局算子G
切片规则函数的一些重要全局属性已从全局算符获得
$$ \ begin {aligned} G：= \ Vert \ mathbf {x} \ Vert ^ 2 \ partial _0 + \ mathbf {x} \ sum _ {i = 1} ^ 3 x_i \ partial _i，\ end {aligned} $$例如全局特征，全局柯西积分定理和全局Borel–Pompeiu公式，请参阅Colombo等。（Trans Am Math Soc 365：303–318，2013），冈萨雷斯·塞万提斯（复杂的肛门操作理论13：2527–2539、2019）和冈萨雷斯·塞万提斯和冈萨雷斯·坎波斯（复杂变量椭圆方程65：1–10，2020，https ：//doi.org/10.1080/17476933.2020.1738410）[4，13，14]。这项工作的目的是显示：G与构形算子之间的一些关系与共形映射，G的共形协方差性质及其对协变函子的解释，这些事实对切片正则函数的所有后果，与算子G相关的Leibnitz规则 并根据非恒定系数二阶微分方程对切片正则函数的实数分量进行表征。
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