A CHARACTERIZATION OF M-HARMONICITY

Title & Authors
A CHARACTERIZATION OF M-HARMONICITY
Lee, Jae-Sung;

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