Some properties of the Kuramochi boundary

Abstract

It has been shown that the Kuramochi boundary of a Riemann surface or of a Green space has many useful potential-theoretic properties (see Q9], [_4~], C H I etc.). In this paper, we shall give a few more properties of the Kuramochi boundary. We consider a Green space Ω in the sense of Brelot-Choquet [3] and denote by i2* its Kuramochi compactification of Ω (see [_4Γ\, [9J and [_14Γ\ for the definition). Let Γ be the harmonic boundary on J = Ω* — Ω, i.e., the support of a harmonic measure ω = ωXQ (x0 e Ω). By definition, Γ is a non-empty closed subset of Δ. Let KQ be a fixed compact ball in Ω. For any resolutive function φ on J, let Hφ be the Dirichlet solution on Ω—Ko with boundary values φ on Δ and 0 on dK0 ( = t h e relative boundary of Ko). For the existence of Hφ9 see e.g. [11H. If φ is a function on Γ and is the restriction of a resolutive function φ on Δ, then H~ψ is uniquely determined by φ we denote it also by Hφ. With this convention, we consider the space RD(Γ) of functions φ on Γ which are restrictions of resolutive functions on Δ and for which Hφ e HD0. Here, HD0 is the space of all harmonic functions u on Ω — Ko having finite Dirichlet integral D[_υΓ\ on Ω — Ko and vanishing on dK0. Identifying two functions which are equal ω-almost everywhere, we can define a norm || || on RD(Γ) by