Cluster Sets of Harmonic Functions at the Boundary of a Half‐Space

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 = R n −1 × (0, + ∞), where n ⩾ 2. Let f : D → [−∞, +∞] and let Z ∈ δ D . 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 δ D . The following results are due to Doob [ 2 , Theorem 6.1 and p. 123]. 1991 Mathematics Subject Classification 31B25.