Growth and Asymptotic Sets of Subharmonic Functions Ii

We study the relation between the growth of a subharmonic function in the half space R + and the size of its asymptotic set. In particular, we prove that for any n ≥ 1 and 0 < α ≤ n, there exists a subharmonic function u in the R + satisfying the growth condition of order α : u(x) ≤ x−α n+1 for 0 < xn+1 < 1, such that the Hausdorff dimension of the asymptotic set ⋃ λ6=−∞ A(λ) is exactly n−α. Here A(λ) is the set of boundary points at which f tends to λ along some curve. This proves the sharpness of a theorem due to Berman, Barth, Rippon, Sons, Fernández, Heinonen, Llorente and Gardiner cumulatively. A function f defined in a domain D is said to have an asymptotic value b ∈ [−∞,∞] at a point a ∈ ∂D provided that there exists a path γ in D ending at a so 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) and called the asymptotic set for the value b. G. R. MacLane [M1], [M2] studied the class of analytic functions in the unit disk having asymptotic values at a dense subset of the unit circle. Hornblower studied the analogous class of subharmonic functions. Since then, many have worked on problems of the following nature: for a subharmonic function u of a certain growth, if A(u,+∞) is a small set, then u has nice boundary behavior on a large set. Denote by R + = {(x, y) : x = (x1, . . . , xn) ∈ R, y > 0} the upper half space in R. For α > 0, denote by Mα the class of subharmonic functions u in R + which satisfy the growth condition: u(x, y) ≤ C(u)y−α for 0 < y < 1 for some constant C(u) > 0. Research partially supported by the National Science Foundation.