Quasi-nearly subharmonic functions in locally uniformly homogeneous spaces
type: Item
instance of: scholarly article
title: Quasi-nearly subharmonic functions in locally uniformly homogeneous spaces
zbMATH Open document ID: 1225.31001
author: Miroslav Pavlović
author: Juhani Riihentaus
published in: Positivity
publication date: 2011-05-27T00:00:00Z
review text: The authors define non-negative quasi-nearly subharmonic functions on so-called locally uniformly homogeneous spaces. This function class is rather general and includes quasi-nearly subharmonic (thus also subharmonic, quasi-subharmonic and nearly subharmonic) functions on domains of Euclidean spaces \(\mathbb{R}^n\), \(n\geq2\). In addition, quasi-nearly subharmonic functions with respect to various measures on domains of \(\mathbb{R}^n\), \(n\geq2\), are included. As examples the authors list the cases of the hyperbolic measure on the unit ball \(B^n\) of \(\mathbb{R}^n\), the \(\mathcal{M}\)-invariant measure on the unit ball \(B^{2n}\) of \(\mathbb{C}^n\), \(n\geq1\), and the hyperbolic measure on any domain \(D\subset\mathbb{R}^n\), \(D\neq\mathbb{R}^n\). For \(Y\subset X\) and \(x\in X\), let \[ \delta^Y_K(x):=\inf\big\{d_K(x,y):y\in Y^c\big\}, \] where \(d_K(x,y)\) is a quasi-distance on \(X\) and \(Y^c\) is the complement of \(Y\) taken in \(X\). The main result of the paper is the following theorem. Theorem 2.5. Let \(X\) be a locally uniformly homogeneous space satisfying the following additional condition: if \(B:=B_K(x_0,r_0)\) is a ball in \(X\), \(0\leq r_0\leq\rho_0\), and \(x\in B\), then \[ \delta^B_K(x)=\sup\big\{\varrho>0:B_K(x,\varrho)\subset B\big\}\,. \] Let \(u:X\rightarrow[0,+\infty)\) be \(C\)-quasi-nearly subharmonic. If \(\psi:[0,+\infty)\rightarrow[0,+\infty)\) is permissible, then \(\psi\circ u\) is \(C_1\)-quasi-nearly subharmonic on \(X\), where \(C_1=C_1(C,K,\psi,\varepsilon_0,\rho_0)\).
Mathematics Subject Classification ID: 31-02
Mathematics Subject Classification ID: 31C45
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 31C05
zbMATH DE Number: 5900667
zbMATH Keywords: locally uniformly homogeneous space
zbMATH Keywords: hyperbolic measure
zbMATH Keywords: \(\mathcal{M}\)-invariant measure
zbMATH Keywords: subharmonic function
MaRDI profile type: Publication
full work available at URL: s11117-009-0037-0
OpenAlex ID: W2140010878
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DOI: 10.1007/S11117-009-0037-0
description: scientific article; zbMATH DE number 5900667