Some global operators and the material derivative

TL;DR

This paper extends the study of a class of global operators related to slice monogenic functions, exploring their structure within Clifford and quaternionic analysis, and investigates properties of the material derivative in this context.

Contribution

It introduces a new operator \\mathcal{H}_a that generalizes the operator G, extending function theory in Clifford analysis and quaternionic analysis, and examines properties of the material derivative.

Findings

Extended the operator G to a more general operator \\mathcal{H}_a.

Connected the operator theory to properties of the material derivative.

Analyzed the structure within quaternionic analysis for n=3.

Abstract

The theory of the operator $$G(x) = |\underline{x}|^2 \frac{\partial }{\partial x_0} + \underline{x} \sum_{j=1}^n x_j \frac{\partial }{\partial x_j} $$ is deeply associated with the slice monogenic function theory and has grown in recent years. In particular, for $n=3$ the quaternionic version of $G$ has been recently used to study the quaternionic slice regular function theory. This work extends the study of the $G$ operator in two senses: a) Clifford's analysis structure. The function theory induced by the operator \begin{align*}\mathcal H_a (x) = {\underline a} ( {x}) \frac{\partial }{\partial x_0} - \sum_{i=1}^n \left( \sum_{j=1}^n a_j ( {x}) \frac{\partial (a^{-1})_i}{\partial y_j}\circ a ( {x}) \right) \frac{\partial}{\partial x_i}, \end{align*} where $a$ is a function with certain properties with domain in $\mathbb R^{n+1}$ is presented extending the already known results of the $G$. Also some properties of the material derivative are presented as consequences of function theory induced by $\mathcal H_a$. b) Structure of quaternionic analysis. In particular, the case $n=3$ is approached from the point of view of quaternionic analysis.