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Growth theorems for slice Dirac-regular mappings over Clifford algebras

Ting Yang, Xinyuan Dou

TL;DR

The paper introduces slice Dirac-regular mappings over Clifford algebras using $O(3)$-stem mappings and shows that Dirac-regularity corresponds to a CR-type system on the stem. It proves a representation formula linking $O(3)$-slice mappings to stems and provides a criterion $D_{\mathbb{I}}(f\circ\varphi^{I,J})=0$ for slice Dirac-regularity, establishing both a structural and analytic handle on these functions. Growth-type results are established for slice Dirac-regular mappings in bounded, starlike, and slice circular domains, including refinements for $k$-fold symmetry and for slice convex domains, with sharp inequalities expressed via a defining function $\rho$ and the slice maps $\varphi^{I,J}$. These results advance understanding of multivariable slice analysis on Clifford algebras and yield sharp size bounds for images, contributing to the functional-analytic and geometric theory of slice-regular-like mappings in higher dimensions.

Abstract

In this paper, we define a class of slice Dirac-regular mappings of several variables over Clifford algebras, based on the concept of O(3)-stem mappings. We prove that the slice mappings vanish under the slice Dirac operator, which is equivalent to its O(3)-stem mappings satisfy the generalized version of the Cauchy-Riemann equation. Moreover, we establish the growth theorem for slice Dirac-regular starlike mappings in the slice cones of Clifford algebras, as well as for slice Dirac-regular k-fold symmetric mappings.

Growth theorems for slice Dirac-regular mappings over Clifford algebras

TL;DR

The paper introduces slice Dirac-regular mappings over Clifford algebras using -stem mappings and shows that Dirac-regularity corresponds to a CR-type system on the stem. It proves a representation formula linking -slice mappings to stems and provides a criterion for slice Dirac-regularity, establishing both a structural and analytic handle on these functions. Growth-type results are established for slice Dirac-regular mappings in bounded, starlike, and slice circular domains, including refinements for -fold symmetry and for slice convex domains, with sharp inequalities expressed via a defining function and the slice maps . These results advance understanding of multivariable slice analysis on Clifford algebras and yield sharp size bounds for images, contributing to the functional-analytic and geometric theory of slice-regular-like mappings in higher dimensions.

Abstract

In this paper, we define a class of slice Dirac-regular mappings of several variables over Clifford algebras, based on the concept of O(3)-stem mappings. We prove that the slice mappings vanish under the slice Dirac operator, which is equivalent to its O(3)-stem mappings satisfy the generalized version of the Cauchy-Riemann equation. Moreover, we establish the growth theorem for slice Dirac-regular starlike mappings in the slice cones of Clifford algebras, as well as for slice Dirac-regular k-fold symmetric mappings.
Paper Structure (4 sections, 9 theorems, 91 equations)

This paper contains 4 sections, 9 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Priliminaries
  3. Main results
  4. Growth Theorem In The Starlike OR Convex Domains

Key Result

Proposition 3.3

Let be $O(3)$-stem with $F_\ell:D\rightarrow \mathbb{R}_m^n, \ell=0,1,2,3$. Then where $x_0,x_1\in\mathbb{R}^n\subset\mathbb{R}_m^n$.

Theorems & Definitions (30)

  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Definition 3.4
  • Proposition 3.5
  • proof
  • Remark 3.6
  • Proposition 3.7
  • proof
  • ...and 20 more