On conjugate harmonic pairs $(U_r, V_{r-1})$ of multi-vector valued functions

Abstract

Let $\mathbb{R}_{0,m}$ be the real Clifford algebra constructed over the real quadratic space $\mathbb{R}^{0,m}$ with signature $(0,m)$ and let $U_r$ be an $\mathbb{R}^+_{0,m}$-valued harmonic function in an appropriate open domain $\Omega$ of $\mathbb{R}^{m+1}$ $(0 < r \leq m; m \geq 2)$. Then a necessary and sufficient condition is given upon $U_r$ for the existence of an $\mathbb{R}^{r-1}_{0,m}$-valued harmonic function in $\Omega$ which is conjugate to $U_r$.