Harmonic functions on finitely sheeted unlimited covering surfaces.
type: Item
instance of: scholarly article
title: Harmonic functions on finitely sheeted unlimited covering surfaces.
zbMATH Open document ID: 1046.30026
author: Hiroaki Masaoka
author: Shigeo Segawa
published in: Journal of the Mathematical Society of Japan
publication date: 2003-06-30T00:00:00Z
review text: It is considered the following objects: the classes \(HP(R)\) and \(HB(R)\) of positive and bounded harmonic functions on a Riemann surface \(R\), open Riemann surface \(W\) possessing Green's functions, p-sheeted unlimited covering surface \(\widetilde{W}\) of \(W\) with projection map \(\varphi\). Let \(X\) be one of the symbols \(P\) or \(B\). Evidently, that \[ HX(W)\circ \varphi :=\{h\circ\varphi: h\in HX(W)\subset HX(\widetilde{W})\} \] The author give a necessary and suffisient condition, in terms of Martin boundary, for \(HX(W)\circ\varphi=HX(\widetilde{W})\). As example they give the following proposition. Let \(A=(1-2^{-n-1}\exp{\frac{2\pi i k}{2^{n+2}}}\), \(n=1,2,\dots\), \(k=1,2,\dots,2^{n+2})\) and let \(\widetilde{D}\) be p-sheets unlimited covering surface of the unit disk \(D\) with branch points in \(\widetilde{D}\) of multiplicity \(p\) for every \(z\in A\). Then \(HP(D)\circ \varphi=HP(\widetilde{D})\).
Mathematics Subject Classification ID: 30F15
Mathematics Subject Classification ID: 30F25
Mathematics Subject Classification ID: 31C25
Mathematics Subject Classification ID: 31C35
zbMATH DE Number: 1943313
zbMATH Keywords: unlimited covering surface
zbMATH Keywords: positive harmonic function
zbMATH Keywords: bounded harmonic function
zbMATH Keywords: Martin boundary
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/1191419119
OpenAlex ID: W2040491323
DOI: 10.2969/JMSJ/1191419119
description: scientific article; zbMATH DE number 1943313