A characterization of $mathcal{M}$-harmonicity

摘要

If $f$ is $mathcal{M}$-harmonic and integrable with respect to a weighted radial measure $u_{lpha}$ over the unit ball $B_n$ of $mathbb{C}^n$, then $int_{B_n} (fcircpsi) du_{lpha}=f(psi(0))$ for every $psi in {hbox{Aut}(B_n)}$. Equivalently $f$ is fixed by the weighted Berezin transform; $T_{lpha}f=f$. 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 ${T_{lpha}}'s$, then $f$ is $mathcal{M}$-harmonic.