© 1998 by London Mathematical Society
© The London Mathematical Society
Growth and Asymptotic Sets of Subharmonic Functions
Department of Mathematics, University of Illinois 1409 West Green Street, Urbana, IL 61801, USA
Received 12 May 1996. Revision received 21 January 1997.
We study the relation between the growth of a subharmonic function in the half space and the size of its asymptotic set.
A function f defined on a domain D has an asymptotic value b
[–
, ] at a
D if there exists a path 
in D ending at a such that u(p) tends to b as p tends to a along . The set of all points on D at which f has an asymptotic value b is denoted by A(f, b).
G. R. MacLane [10, 11] studied the class of analytic functions in the unit disk having asymptotic values at a dense subset of the unit circle. Hornblower [8, 9] studied the analogous class for subharmonic functions. Many theorems have since been proved having the following character: for a function f of a given growth, if A(f, +) is a small set then f has nice boundary behavior on a large set. See [1, 3–7] and the references therein.
For 
>0, let M be the class of subharmonic functions u in 
{(x, y):xRn, y>0} satisfying the growth condition
u(x, y) 
C(u)y– for 0 < y < 1
for some constant C(u) depending on u. Denote by F(u) the Fatou set of u, which consists of points on 
where u has finite vertical limits. For ß>0, denote by Hß the ß-dimensional Hausdorff content. The following theorem is due to Barth and Rippon [1], Fernández, Heinonen and Llorente [5], and Gardiner [6].