Surfaces carrying sufficiently many Dirichlet finite harmonic functions that are automatically bounded
type: Item
instance of: scholarly article
title: Surfaces carrying sufficiently many Dirichlet finite harmonic functions that are automatically bounded
zbMATH Open document ID: 1264.30027
author: Mitsuru Nakai
published in: Journal of the Mathematical Society of Japan
publication date: 2012-04-03T00:00:00Z
review text: Let \(H(R)\) be the linear space of harmonic functions on a Riemann surface \(R\). Let \(HB(R)\) and \(HD(R)\) be the linear subspaces of \(H(R)\) consisting of the bounded harmonic functions on \(R\) and the harmonic functions on \(R\) with finite Dirichlet integral, respectively. In general, there is no inclusion relation between these two subspaces, but there are a lot of Riemann surfaces satisfying \(HB(R)\subset HD(R)\) with the linear dimension \(\dim HB(R)<\infty\) or \(HD(R)\subset HB(R)\) with \(\dim HD(R)<\infty\). In this paper the author studies the problem whether the relation \(HD(R)\subset HB(R)\) always implies the finiteness of \(\dim HD(R)\) or not, and shows the existence of a Riemann surface \(R\) such that \(HD(R)\subset HB(R)\) with \(\dim HD(R)=\infty\). For the construction of such a Riemann surface, the author makes use of the Sario-Tôki disc. With respect to the converse inclusion relation, it is already shown by the author that there exists a Riemann surface \(R\) such that \(HB(R)\subset HD(R)\) with \(\dim HB(R)=\infty\).
reviewed by: Gou Nakamura
Mathematics Subject Classification ID: 30F20
Mathematics Subject Classification ID: 30F25
Mathematics Subject Classification ID: 30F15
Mathematics Subject Classification ID: 31A05
zbMATH DE Number: 6020914
zbMATH Keywords: Riemann surface
zbMATH Keywords: bounded harmonic functions
zbMATH Keywords: harmonic functions with finite Dirichlet integral
zbMATH Keywords: Sario-Tôki disc
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/06410201
OpenAlex ID: W2068142946
cites work: Q5593095
cites work: Q5719551
cites work: Q4693180
cites work: Dirichlet integrals on harmonic spaces
cites work: Extremal functions for capacities
cites work: Welding of polygons and the type of Riemann surfaces
cites work: Q5597712
cites work: Classification theory of Riemannian manifolds. Harmonic, quasiharmonic and biharmonic functions
DOI: 10.2969/JMSJ/06410201
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description: scientific article; zbMATH DE number 6020914