Boundary Behaviour of Harmonic Functions in a Half-Space and Brownian Motion

Abstract

The behaviour of harmonic functions in the half-space $R_ + ^{n + 1}$ has been discussed from two points of view: geometrical and probabilistic. In this paper, we compare these two view points with respect to (1) local convergence at the boundary and (2) the H^p-spaces. The results are as follows: (1) The existence of a nontangential limit for almost all points in a set E of positive Lebesgue measure in ${R^n}\left( { = \partial \,R_ + ^{n + 1}} \right)$ is more restrictive than the existence of a « fine » or probability limit almost everywhere in E when $n \geqslant 2$. When $n=1$, the existence of a nontangential limit almost everywhere in E implies the existence of a « fine » limit almost everywhere in E and conversely.