Bulletin of the
Korean Mathematical Society BKMS

If $f$ is $\mathcal{M}$-harmonic and integrable with respect to a weighted radial measure $\nu_{\alpha}$ over the unit ball $B_n$ of $\mathbb{C}^n$, then $\int_{B_n} (f\circ\psi)\ d\nu_{\alpha}=f(\psi(0))$ for every $\psi \in {\hbox{Aut}(B_n)}$. Equivalently $f$ is fixed by the weighted Berezin transform; $T_{\alpha}f=f$. In this paper, we show that if a function $f$ defined on $B_n$ satisfies $R(f \circ \phi) \in L^{\infty}(B_{n})$ for every $\phi \in {\hbox{Aut}(B_n)}$ and $Sf=rf$ for some $|r|=1$, where $S$ is any convex combination of the iterations of ${T_{\alpha}}'s$, then $f$ is $\mathcal{M}$-harmonic.

Keywords: $\mathcal{M}$-harmonic function, weighted Berezin transform, Gelfand transform

MSC numbers: Primary 42B35, 31B05; Secondary 31B10