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∈δD
if there exists a path γ in D ending at a such that
u(p) tends to b as p tends to a along γ.
The set of all points on δD 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 α>0, let [Mscr ]α be the class of subharmonic functions
u in IRn+1+
≡{(x, y)[ratio ]x∈IRn,
y>0} satisfying the growth condition
formula here
for some constant C(u) depending on u. Denote by
[Fscr ](u) the Fatou set of u, which
consists of points on δIRn+1+
where u has finite vertical limits. For β>0, denote by
Hβ the β-dimensional Hausdorff content.
The following theorem is due to Barth
and Rippon [1], Fernández, Heinonen and Llorente
[5], and Gardiner [6].