Cluster Sets of Harmonic Functions at the Boundary of a Half-Space |
| |
Authors: | Gardiner Stephen J. |
| |
Affiliation: | Department of Mathematics, University College Dublin Dublin 4, Ireland |
| |
Abstract: | The purpose of this paper is to answer some questions posedby Doob [2] in 1965 concerning the boundary cluster sets ofharmonic and superharmonic functions on the half-space D givenby D = Rn1 x (0, + ), where n 2. Let f: D [,+] and let Z D. Following Doob, we write BZ (respectively CZ)for the non-tangential (respectively minimal fine) cluster setof f at Z. Thus l BZ if and only if there is a sequence (Xm)of points in D which approaches Z non-tangentially and satisfiesf(Xm) l. Also, l CZ if and only if there is a subset E ofD which is not minimally thin at Z with respect to D, and whichsatisfies f(X) l as X Z along E. (We refer to the book byDoob [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 BZ andCZ are subintervals of [, +]. Let denote (n 1)-dimensional measure on D. The following results are due toDoob [2, Theorem 6.1 and p. 123]. 1991 Mathematics Subject Classification31B25. |
| |
Keywords: | |
本文献已被 Oxford 等数据库收录! |
|