Generalizations of harmonic functions in $\mathbb{R}^m$

TL;DR

This paper explores the relationship between generalized harmonic functions and inframonogenic functions within Clifford analysis, focusing on their properties and differences in the context of elliptic PDEs in ^m.

Contribution

It clarifies the relationship between ^m -harmonic and inframonogenic functions, extending the understanding of their structural properties in Clifford analysis.

Findings

Established connections between ^m -harmonic and inframonogenic functions.

Analyzed the properties of solutions to elliptic PDEs in Clifford analysis.

Highlighted differences in structural properties of generalized harmonic functions.

Abstract

In recent works, arbitrary structural sets in the non-commutative Clifford analysis context have been used to introduce non-trivial generalizations of harmonic Clifford algebra valued functions in $\mathbb{R}^m$. Being defined as the solutions of elliptic (generally non-strongly elliptic) partial differential equations, $(\varphi,\psi)$-inframonogenic and $(\varphi,\psi)$-harmonic functions do not share the good structure and properties of the harmonic ones. The aim of this paper it to show and clarified the relationship between these classes of functions.