SETS OF DETERMINATION FOR HARMONIC-FUNCTIONS

Let h denote a positive harmonic function on the open unit ball B of Euclidean space R(n) (n > 2). This paper characterizes those subsets E of B for which sup(E) H/h = sup(B) H/h or inf(E) H/h = inf(B) H/h for all harmonic functions H belonging to a specified class. In this regard we consider the classes of positive harmonic functions, differences of positive harmonic functions, and harmonic functions with a one-sided quasi-boundedness condition. We also consider the closely related question of representing functions on the sphere partial derivative B as sums of Poisson kernels corresponding to points in E.