Cauchy-Riemann operators and local slice analysis over real alternative algebras

TL;DR

This paper establishes formulas connecting Cauchy-Riemann operators, slice-regularity, and spherical Dirac operators in hypercomplex subspaces of alternative *-algebras, enabling local slice analysis and harmonicity results.

Contribution

It introduces a new framework linking Cauchy-Riemann operators to slice-regular functions and harmonicity in hypercomplex algebras, expanding the analytical tools available.

Findings

Formulas relating Cauchy-Riemann operators to slice-regularity and spherical Dirac operators

Definition of locally slice-regular functions

Results on harmonicity and polyharmonicity of slice-regular functions

Abstract

We prove some formulas relating Cauchy-Riemann operators defined on hypercomplex subspaces of an alternative *-algebra to a differential operator associated with the concept of slice-regularity and to the spherical Dirac operator. These results in particular allow to introduce a definition of locally slice-regular function and open the path for local slice analysis. Since Cauchy-Riemann operators factor the corresponding Laplacian operators, the proven formulas let us also obtain several results about the harmonicity and polyharmonicity properties of slice-regular functions.