On the mean value property of harmonic functions and best harmonic $L\sp 1$-approximation

Abstract

The present paper deals with the inverse mean value property of harmonic functions and with the existence, uniqueness, and characterization of a best harmonic L1-approximant to strictly subharmonic functions. The main theorem concerning the inverse mean value property of harmonic functions is based on a generalization of a theorem due to U. Kuran as well as on an approximation theorem proved by J. C. Polking and also by L. I. Hedberg. The inverse mean value property will be applied in order to prove necessary and sufficient conditions for the existence of a best harmonic L'-apprcocimant to a subharmonic function s satisfying As > 0 a.e. in the unit ball.