Growth and Asymptotic Sets of Subharmonic Functions

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 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 ∂ 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 α>0, let M α be the class of subharmonic functions u inR + n + 1≡{( x , y ): x ∈ ℝ n , y >0} satisfying the growth condition u ( x , y ) ⩽ C ( u ) y −α 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 ∂ R + n + 1where 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 ].