Sets of Determination for Harmonic Functions
Stephen J. Gardiner · Transactions of the American Mathematical Society · 1993
Let $h$ denote a positive harmonic function on the open unit ball $B$ of Euclidean space ${{\mathbf {R}}^n}\;(n \geq 2)$. 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 $\partial B$ as sums of Poisson kernels corresponding to points in $E$.