Positive harmonic functions on an end

Mitsuru Ozawa
1960 Kodai Mathematical Seminar Reports  
It is well known that the Martin theory on positive harmonic functions plays an important role in the theory of open Riemann surfaces. Its whole theory depends upon the potential theory and the so-called Martin compactification of the given surface. In the present paper we shall give a proof of it, especially the representation theorem on positive harmonic functions on an end, which seems extremely simple. In our proof we shall introduce a suggestive functional and make use of a variational
more » ... f a variational method. Let W be an open Riemann surface and {PP m } be its exhaustion in the usual sense. Let HP(W-W m ) be a class of positive harmonic functions on W-W m vanishing continuously on ΘW m . Evidently HP(W~-W m ) is a positively linear space. LEMMA 1. The space HP(W-W m ) is a metric space with the metric p(u, v) = 1 --1 uv I ds. jsw m On LEMMA 2. The uniform convergence in the wider sense in W-W m in the class HP (W-Wm) is equivalent to the ^-convergence in the space HP(W-W m ). LEMMA 3. The unit sphere Up in the space HP(W-W m ) is a p-compact convex set. Proof. Let v e U P , then i d --v(p) ds = l. sw m θn L et ω q (p) be the harmonic measure ω(p, ΘW q , W q -W m ) and M = maxv(p), d = mmv(p) on ΘW q . Then we have dω q (p)^v(p) on W q -W m and hence on dW m d Let I denote the value of the integral d
doi:10.2996/kmj/1138844322 fatcat:yg4wakkyf5e35iynjn3k64v3ja