Higher order gradients of monogenic functions

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.

16 pages. Comments are welcome