The volume mean-value property of harmonic functions
D. H. Armitage, Myron Goldstein · Complex Variables Theory and Application An International Journal · 1990
Let D be an open subset of RN N ≧ 2, such that O ∊ D and λ(D) < + ∞, where λ denotes N-dimensional Lebesgue measure. If the mean-value equality holds for every integrable harmonic function h on D. then accoeding to a theorem of KuranD is a ball of centre O. Here we show that the same conclusion holds if we assume the above mean-value equality only for positive integrable harmonic functions. Further, if the mean-value equality is assumed only for bounded harmonic functions, then D = B=\E, when B is an open ball and E is a polar set.