Invariant Mean-Value Property and M-Harmonicity in the Unit Ball of ℝⁿ

Abstract

In 1993, Ahern, Flores and Rudin showed that, if f is integrable over the unit ball $B^{n}_{\mathbb{C}}$ of ℂⁿ and satisfies

$$ {\int_{B^{n}_{\mathbb{C}} } {f \circ \psi dv = f{\left( {\psi {\left( 0 \right)}} \right)}} } $$

for every $\psi \in {\text{Aut}}{\left( {B^{n}_{\mathbb{C}} } \right)}$, then f is M-harmonic if and only if n ≤ 11. The present paper is about an analogous question in the context of the unit ball B _n of ℝⁿ as well as in the weighted setting.