Wiener’s compactification andΦ-bounded harmonic functions in the classification of harmonic spaces

Wellington Ham Ow

Vol. 38 (1971), No. 3, 759–769

DOI: 10.2140/pjm.1971.38.759

Abstract

The class HΦ of Φ-bounded harmonic functions on Riemann surfaces first investigated by Parreau for the special case where Φ is increasing and convex, was later characterized by Nakai in its complete generality by assuming only that Φ was a nonnegative real-valued function on [0,∞). In this paper we show that Nakai’s theory can be presented in the axiomatic setting of Brelot. The theory of Wiener compactifications which is indispensable in the study of potential theory on Riemann surfaces is extended to harmonic spaces and shown to be equally useful in the potential theory there.

Mathematical Subject Classification 2000

Primary: 31D05

Milestones

Received: 18 September 1970

Revised: 18 March 1971

Published: 1 September 1971