Holomorphic functions on the lie ball and their monogenic counterparts

TL;DR

This paper explores the relationship between holomorphic functions on the Lie ball and their monogenic counterparts in Clifford analysis, using integral formulas to establish a correspondence and constructing an inverse map via the Cauchy-Hua formula.

Contribution

It introduces a method to explicitly construct the inverse map from holomorphic functions on the Lie ball to monogenic functions, extending Morimoto's work with the Cauchy-Hua formula.

Findings

Established a formula linking holomorphic and monogenic functions

Constructed an explicit inverse map using the Cauchy-Hua formula

Extended the theoretical framework of Clifford analysis on the Lie ball

Abstract

The Cauchy integral formula in Clifford analysis allows us to associate a holomorphic function $\tilde f:L_n\to \C$ on the Lie ball $L_n$ in $\C^n$ with its monogenic counterpart $f:B_1(0)\to \C^{n+1}$ via the formula $\tilde f(z) = \int_{S^n}G_\om(z)\bs n(\om)f(\om)\,d\mu(\om)$, $z\in L_n.$ The inverse map $\tilde f\mapsto f$ is constructed here using the Cauchy-Hua formula for the Lie ball following the work of M. Morimoto \cite{Mori2}.