In the past few years, the theory of slice monogenic functions has been developed rapidly mainly motivated by the applications to an elegant functional calculus for non-commuting operators. In this article, we introduce the Teodorescu transform in the theory of slice monogenic functions, which turns out to be the right inverse of a slice Cauchy-Riemann operator. The boundednesses of the Teodorescu transform and its derivatives are investigated as well. A Hodge decomposition of the $\mathcal{L}^p$ space and a generalized Bergman projection are introduced at the end as applications.
The Radon transform between monogenic and generalized slice monogenic functions
在[J. Bures，R. Lavicka，V. Soucek，Elements of quaternionic analysis and Radon transform，Textos de Matematica 42，Departamento de Matematica，Universidade de Coimbra，2009]，作者描述了依赖于参数的全纯函数与使用Radon和对偶Radon变换在R^（n+1）上定义的单演函数之间的联系。本文的主要目的是进一步发展这种方法。实际上，值在Clifford代数R_n中的函数的Radon变换是广义Cauchy-Riemann方程的映射解，即，单演函数映射到R_n中的参数全纯函数族，类似地，对偶Radon变换也是将上述参数全纯函数族映射到单演函数。本文所考虑的参数全纯函数族可以看作是所谓切片单演函数的推广。本文所解决的问题的一个重要部分是找到一个合适的定义函数空间作为域和目标的积分变换。
Integral formulas and Hodge decomposition in the theory of generalized partial-slice mo-nogenic functions
School of Mathematical Sciences, Anhui University（安徽大学数学科学学院）；Center for Pure Mathematics, School of Mathematical Sciences, Anhui University（安徽大学数学科学学院纯数学中心）
Slice Fueter- regular functions , originally called slice Dirac- regular functions , are generalized holomorphic functions defined over the octonion algebra $\mathbb{O}$ , recently introduced by M. Jin, G. Ren and I. Sabadini. A function $f:Ω_D\subset\mathbb{O}\to\mathbb{O}$ is called (quaternionic) slice Fueter-regular if, given any quaternionic subalgebra $\mathbb{H}_\mathbb{I}$ of $\mathbb{O}$ generated by a pair $\mathbb{I}=(I,J)$ of orthogonal imaginary units $I$ and $J$ ( $\mathbb{H}_\mathbb{I}$ is a `quaternionic slice' of $\mathbb{O}$ ), the restriction of $f$ to $Ω_D\cap\mathbb{H}_\ma…
Singularities of slice regular functions over real alternative *-algebras
The main goal of this work is classifying the singularities of slice regular functions over a real alternative *-algebra A. This function theory has been introduced in 2011 as a higher-dimensional generalization of the classical theory of holomorphic complex functions, of the theory of slice regular quaternionic functions launched by Gentili and Struppa in 2006 and of the theory of slice monogenic functions constructed by Colombo, Sabadini and Struppa since 2009. Along with this generalization step, the larger class of slice functions over A has been defined. We introduce here a new type of se…
