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Algebra of slice regular functions on non-symmetric domains in several quaternionic variables

Xinyuan Dou, Ming Jin, Guangbin Ren, Ting Yang

TL;DR

Addresses the algebraic structure of weakly slice regular functions in several quaternionic variables on non-symmetric domains. Develops an algebra via a $*$-product that preserves the path-slice property and introduces holomorphic stem functions on path spaces, linking holomorphy to weak slice regularity. Proves that the $*$-product of two weakly slice regular functions remains slice regular, enabling the space $\\mathcal{SR}(\\Omega)$ to form an associative unital real algebra under $+$ and $*$ on self-stem-preserving, real-path-connected domains. These results extend slice-regularity theory to non-axially symmetric domains and provide a robust algebraic framework for multi-variable quaternionic analysis.

Abstract

The primary objective of this paper is to establish an algebraic framework for the space of weakly slice regular functions over several quaternionic variables. We recently introduced a $*$-product that maintains the path-slice property within the class of path-slice functions. It is noteworthy that this $*$-product is directly applicable to weakly slice regular functions, as every slice regular function defined on a slice-open set inherently possesses path-slice properties. Building on this foundation, we propose a precise definition of an open neighborhood for a path $γ$ in the path space $\mathscr{P}(\mathbb{C}^n)$. This definition is pivotal in establishing the holomorphism of stem functions. Consequently, we demonstrate that the $*$-product of two weakly slice regular functions retains its weakly slice regular nature. This retention is facilitated by holomorphy of stem functions and their relationship with weakly slice regular functions, providing a comprehensive algebraic structure for this class of functions.

Algebra of slice regular functions on non-symmetric domains in several quaternionic variables

TL;DR

Addresses the algebraic structure of weakly slice regular functions in several quaternionic variables on non-symmetric domains. Develops an algebra via a -product that preserves the path-slice property and introduces holomorphic stem functions on path spaces, linking holomorphy to weak slice regularity. Proves that the -product of two weakly slice regular functions remains slice regular, enabling the space to form an associative unital real algebra under and on self-stem-preserving, real-path-connected domains. These results extend slice-regularity theory to non-axially symmetric domains and provide a robust algebraic framework for multi-variable quaternionic analysis.

Abstract

The primary objective of this paper is to establish an algebraic framework for the space of weakly slice regular functions over several quaternionic variables. We recently introduced a -product that maintains the path-slice property within the class of path-slice functions. It is noteworthy that this -product is directly applicable to weakly slice regular functions, as every slice regular function defined on a slice-open set inherently possesses path-slice properties. Building on this foundation, we propose a precise definition of an open neighborhood for a path in the path space . This definition is pivotal in establishing the holomorphism of stem functions. Consequently, we demonstrate that the -product of two weakly slice regular functions retains its weakly slice regular nature. This retention is facilitated by holomorphy of stem functions and their relationship with weakly slice regular functions, providing a comprehensive algebraic structure for this class of functions.
Paper Structure (4 sections, 12 theorems, 55 equations)

This paper contains 4 sections, 12 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Holomorphic stem functions
  4. Star-product of slice regular functions

Key Result

Proposition 2.6

Dou2023003. Given a domain $\Omega\subset\mathbb{H}_s^n$, a function $f\in\mathcal{PS}(\Omega)$, a path $\gamma\in\mathscr{P}(\mathbb{C}^n,\Omega)$, and a path-slice stem function $F$ associated with $f$, for any distinct $I, J\in\mathbb{S}(\Omega,\gamma)$, it holds that

Theorems & Definitions (30)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Definition 2.7
  • Definition 2.8
  • Lemma 2.9
  • Definition 2.10
  • ...and 20 more