Slice Fueter-regular functions

TL;DR

This paper explores the properties of slice Fueter-regular functions over octonions, showing they are standard slice functions with holomorphic-like features, and introduces operators to characterize them through differential systems.

Contribution

It demonstrates that slice Fueter-regular functions are standard slice functions with holomorphic properties and introduces global operators to characterize them via differential systems.

Findings

Slice Fueter-regular functions are standard slice functions with holomorphic properties.

They possess Cauchy integral formulas, series expansions, and maximum modulus principles.

The paper introduces operators to characterize these functions through second order differential systems.

Abstract

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 $f:\Omega_D\subset\mathbb{O}\to\mathbb{O}$ 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 $\Omega_D\cap\mathbb{H}_\mathbb{I}$ belongs to the kernel of the corresponding Cauchy-Riemann-Fueter operator $\frac{\partial}{\partial x_0}+I\frac{\partial}{\partial x_1}+J\frac{\partial}{\partial x_2}+(IJ)\frac{\partial}{\partial x_3}$. The goal of this paper is to show that slice Fueter-regular functions are standard (complex) slice functions, whose stem functions satisfy a Vekua system having exactly the same form of the one characterizing axially monogenic functions of degree zero. The mentioned standard sliceness of slice Fueter-regular functions is able to reveal their `holomorphic nature': slice Fueter-regular functions have Cauchy integral formulas, Taylor and Laurent series expansions, and a version of Maximum Modulus Principle, and each of these properties is global in the sense that it is true on genuine $8$-dimesional domains of $\mathbb{O}$. Slice Fueter-regular functions are real analytic. Furthermore, we introduce the global concepts of spherical Dirac operator $\Gamma$ and of slice Fueter operator $\bar{\vartheta}_F$ over octonions, which allow to characterize slice Fueter-regular functions as the $\mathscr{C}^2$-functions in the kernel of $\bar{\vartheta}_F$ satisfying a second order differential system associated with $\Gamma$. The paper contains eight open problems.