Harmonic Functions in Upper Half Space

Abstract

In this paper, we prove that if the positive part $u^{+}(x)$ of a harmonic function $u(x)$ in the upper half space satisfies a fast growing condition, then its negative part $u^{-}(x)$ can also be dominated by a similar growing condition. Meanwhile, $u(x)$ can be represented in terms of the modified Poisson integral and a harmonic function vanishing on the boundary.