ON HARMONICITY IN A DISC AND n-HARMONICITY

초록

Let tau not equal 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 vertical bar(tau) over cap (s)vertical bar not equal 1 for every s is an element of Sigma(delta) \ {0, 1}, or just a radial probability measure on D. Here, we provide a decomposition of the set X = {h is an element of L(infinity) (D) vertical bar lim(n ->infinity) h * tau(n) exists}. Let tau(1), ..., tau(n) be measures on D with above mentioned properties. Here, we prove that if f is an element of L(infinity) (D(n)) satisfies an invariant volume mean value property with respect to tau(1), ..., tau(n), then f is n-harmonic.

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