Asymptotic Values of Subharmonic Functions

In this paper we study the boundary behaviour of subharmonic functions in B", the unit ball of U", or in the upper half space of U\ U+ +l = {(x,y): x E R", y>0}. A function u denned in B" (respectively, U+) has asymptotic value A G [-oo, oo] at the point x e dU (respectively, U") if there is a path y(t), Q^tx as t-+1 and u(y(t))-+ A as t—> 1. The set of x e dB (respectively, W) at which u has an asymptotic limit is denoted by A(u). We will be particularly interested in the case A = °°, and the corresponding subset of dB (respectively, U), consisting of the points at which u has asymptotic value + °°, will be denoted by A(u, +<»). If y is restricted to be a radius (respectively, a vertical line) and u has a finite radial (vertical) limit at x then x e dB" (respectively, R") is said to be a Fatou point of u. We write F(u) to denote the set of all Fatou points of u. These definitions have their origin in the case of the unit disk