逆Fueter定理的一个变体与二阶广义多解析Cauchy-Kovalevskaya延拓
A variation of the inverse Fueter theorem and the generalized polyanalytic Cauchy-Kovalevskaya extension of order 2
中文总结 AI 辅助
本文研究二阶轴向多解析函数的广义Cauchy-Kovalevskaya延拓，建立其与Fueter定理的联系，推导Fueter映射分解的可逆性及相关积分表示公式，是超复分析的重要成果。
中文摘要 AI 辅助
本文首先建立二阶轴向多解析函数的广义Cauchy-Kovalevskaya（GCK）延拓，证明该延拓可表示为作用于两个初始函数的微分算子构成的幂级数。还研究了该GCK延拓在二维球面$\boldsymbol{\rm S}$上基于平面波型函数的积分分解。进一步建立二阶多解析函数的GCK延拓与Fueter定理的联系，这是超复分析中最重要的结果之一，可分为两步：第一步，从单复变量全纯函数出发，应用合适算子得到切片超全纯函数；第二步，应用四个实变量的拉普拉斯算子（即Fueter映射）得到轴向单解析函数，即Fueter算子核中的函数。Fueter映射在Fueter算子及其共轭下的合适分解，产生了介于切片超全纯函数与轴向单解析函数之间的两类中间函数：轴向调和函数与二阶轴向多解析函数。本文另一目标是研究Fueter映射在合适开集上对调和函数与二阶多解析函数的分解的可逆性，并推导分解后的Fueter映射逆的积分表示公式，这些积分表示基于二阶多解析函数的Cauchy公式与调和函数的泊松积分公式。
英文摘要
In this paper, we first establish a generalized Cauchy--Kovalevskaya (GCK) extension for axially polyanalytic functions of order $2$. We prove that the extension can be written as a power series involving differential operators acting on two initial functions. We also study the decomposition of the GCK extension in terms of integrals over the sphere 2-sphere $\mathbb{S}$ involving plane-wave type functions. We further establish a connection between the GCK extension for polyanalytic functions of order $2$ and the Fueter theorem. This is one of the most important results in hypercomplex analysis and can be described in two steps. In the first step, starting from holomorphic functions of one complex variable, the application of a suitable operator yields slice hyperholomorphic functions. In the second step, applying the Laplace operator in four real variables (called Fueter map) one obtains axially monogenic functions, i.e. functions in the kernel of the Fueter operator. A suitable factorization of the Fueter map in terms of the Fueter operator and its conjugate gives rise to two intermediate classes of functions between slice hyperholomorphic functions and axially monogenic functions: axially harmonic functions and axially polyanalytic functions of order $2$. Another goal of this paper is to study the invertibility of the factorizations of the Fueter map for harmonic and polyanalytic functions of order 2 on suitable open sets, and to derive integral representation formulas for the inverse of the factorized Fueter map. These integral representations are based on the Cauchy formula for polyanalytic functions of order $2$ and the Poisson integral formula for harmonic functions.
