CLUSTER SETS OF HARMONIC FUNCTIONS AT THE BOUNDARY OF A HALF-SPACE

Abstract

The purpose of this paper is to answer some questions posed by Doob [2] in 1965 concerning the boundary cluster sets of harmonic and superharmonic functions on the half-space D given by D = ℝⁿ⁻¹ × (0, +∞), where n[ges ]2. Let f[ratio ]D→[−∞, + ∞] and let Z∈δD. Following Doob, we write B_Z (respectively C_Z) for the non-tangential (respectively minimal fine) cluster set of f at Z. Thus l∈B_Z if and only if there is a sequence (X_m) of points in D which approaches Z non-tangentially and satisfies f(X_m)→l. Also, l∈C_Z if and only if there is a subset E of D which is not minimally thin at Z with respect to D, and which satisfies f(X)→l as X→Z along E. (We refer to the book by Doob [3, 1.XII] for an account of the minimal fine topology. In particular, the latter equivalence may be found in [3, 1.XII.16].) If f is superharmonic on D, then (see [2, §6]) both sets B_Z and C_Z are subintervals of [−∞, + ∞]. Let λ denote (n−1)-dimensional measure on δD. The following results are due to Doob [2, Theorem 6.1 and p. 123].