On harmonicity in a disc and $n$-harmonicity
Bull. Korean Math. Soc. 2010 Vol. 47, No. 4, 815-823
Published online July 1, 2010
Jaesung Lee
Sogang University
Abstract : Let $\tau \neq \delta_0$ be either a power bounded radial measure with compact support on the unit disc $D$ with $\tau(D)=1$ such that there is a $\delta >0$ so that $|\hat{\tau}(s)| \neq 1$ for every $s \in \Sigma (\delta) \setminus \{0,1\}$, or just a radial probability measure on $D$. Here, we provide a decomposition of the set ${\bf{X}} = \{ h \in L^{\infty}(D) \ | \ \lim_{n \rightarrow \infty} h \ast \tau^{n} \mbox{ exists}\}$. Let $\tau_1,\ldots,\tau_n$ be measures on $D$ with above mentioned properties. Here, we prove that if $f \in L^{\infty}(D^n)$ satisfies an invariant volume mean value property with respect to $\tau_1,\ldots,\tau_n$, then $f$ is $n$-harmonic.
Keywords : mean value property, harmonicity, $n$-harmonicity, convolution, spectrum
MSC numbers : Primary 31B30; Secondary 47B38
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