Some properties of the Kuramochi boundary

F. Maeda
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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
Kuramochi边界的一些性质
已经证明黎曼曲面或格林空间的Kuramochi边界具有许多有用的势论性质(见Q9, [_4~], C H I等)。本文给出了Kuramochi边界的几个性质。我们考虑一个Green空间Ω在Brelot-Choquet[3]的意义上,用i2*表示它的Kuramochi紧化Ω(见[_4Γ\, [9J和[_14Γ\的定义)。设Γ为J = Ω* - Ω上的谐波边界,即谐波测度Ω = Ω xq (x0 e Ω)的支撑。根据定义,Γ是Δ的非空封闭子集。设KQ为Ω中的固定紧实球。对于任意函数φ在J上,设h φ为Ω-Ko上的Dirichlet解,边值为φ在Δ上,边值为0在dK0上(= k的相对边界)。关于Hφ9的存在参见[11H]。如果φ是Γ上的一个函数,并且是Δ上的一个解析函数φ的限制,那么H~ψ是由φ唯一确定的,我们也用Hφ表示。利用这一约定,我们考虑了Γ上的函数φ的空间RD(Γ),这些函数φ是Δ上的解析函数的限制,Hφ e HD0。这里,HD0是所有调和函数u在Ω - Ko上具有有限狄利克雷积分D[_υΓ\在Ω - Ko上并消失在dK0上的空间。确定两个相等的函数ω-几乎在任何地方,我们都可以在RD(Γ)上定义一个范数|| ||
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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