Asymptotic Behavior for a Class of Modified -Potentials in a Half Space

A class of α-potentials represented as the sum of modified Green potential and modified Poisson integral are proved to have the growth estimates Rα,l,l x o x β n|x| h |x| −1 at infinity in the upper-half space of the n-dimensional Euclidean space, where the function h |x| is a positive nondecreasing function on the interval 0,∞ satisfying certain conditions. This result generalizes the growth properties of analytic functions, harmonic functions, and superharmonic functions.