Higher order gradients of monogenic functions

TL;DR

This paper extends classical results on harmonic functions to monogenic functions in quaternionic, Clifford, and octonionic algebras, showing that certain powers of their higher order gradients are subharmonic and identifying optimal parameters.

Contribution

It generalizes Calderon and Zygmund's result to higher order gradients of monogenic functions across various algebraic structures.

Findings

$| abla^m f|^ ext{optimal } eta$ is subharmonic for monogenic functions

Determines the optimal $eta$ for subharmonicity

Extends classical harmonic analysis results to non-commutative algebras

Abstract

Given a monogenic function on the quaternionic algebra $\mathbb{H}$, the Clifford algebra $\mathbb{R}_n$ or the octonionic algebra $\mathbb{O}$ we prove that $|\nabla^m f|^\alpha$ is subharmonic for some $\alpha>0$ where $\nabla^m f$ is the $m$-th order gradient of $f$. We find also the optimal value of $\alpha$. This is generalization of a result of Calderon and Zygmund.