ON HARMONICITY IN A DISC AND n-HARMONICITY

Title & Authors
ON HARMONICITY IN A DISC AND n-HARMONICITY
Lee, Jae-Sung;

Abstract
Let $\small{{\tau}\;{\neq}\;\delta_0}$ be either a power bounded radial measure with compact support on the unit disc D with $\small{\tau(D)\;=\;1}$ such that there is a $\small{\delta}$ > 0 so that $\small{{\mid}\hat{\tau}(s){\mid}\;{\neq}\;1}$ for every $\small{s\;{\in}\;\Sigma(\delta)}$ \ {0,1}, or just a radial probability measure on D. Here, we provide a decomposition of the set X = {$\small{h\;{\in}\;L^{\infty}(D)\;{\mid}\;lim_{n{\rightarrow}{\infty}}\;h\;*\;\tau^n}$ exists}. Let $\small{\tau_1}$, ..., $\small{\tau_n}$ be measures on D with above mentioned properties. Here, we prove that if $\small{f\;{in}\;L^{\infty}(D^n)}$ satisfies an invariant volume mean value property with respect to $\small{\tau_1}$, ..., $\small{\tau_n}$, then f is n-harmonic.
Keywords
mean value property;harmonicity;n-harmonicity;convolution;spectrum;
Language
English
Cited by
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