GROWTH AND ASYMPTOTIC SETS OF SUBHARMONIC FUNCTIONS

Abstract

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 H^β the β-dimensional Hausdorff content. The following theorem is due to Barth and Rippon [1], Fernández, Heinonen and Llorente [5], and Gardiner [6].