Fundaments of Quaternionic Clifford Analysis II: Splitting of Equations

TL;DR

This paper explores the structure of quaternionic Clifford analysis, showing how quaternionic monogenic functions relate to generalized gradients and other Clifford analysis branches, advancing the theoretical framework.

Contribution

It introduces a splitting of equations in quaternionic Clifford analysis and characterizes quaternionic monogenicity via generalized gradients, connecting different Clifford analysis branches.

Findings

Quaternionic monogenic functions characterized by generalized gradients.

Connections established between quaternionic, Hermitian, and Euclidean monogenic functions.

Theoretical framework for splitting equations in quaternionic Clifford analysis.

Abstract

Quaternionic Clifford analysis is a recent new branch of Clifford analysis, a higher dimensional function theory which refines harmonic analysis and generalizes to higher dimension the theory of holomorphic functions in the complex plane. So-called quaternionic monogenic functions satisfy a system of first order linear differential equations expressed in terms of four interrelated Dirac operators. The conceptual significance of quaternionic Clifford analysis is unraveled by showing that quaternionic monogenicity can be characterized by means of generalized gradients in the sense of Stein and Weiss. At the same time, connections between quaternionic monogenic functions and other branches of Clifford analysis, viz Hermitian monogenic and standard or Euclidean monogenic functions are established as well.