Area inequality and \(Q_{p}\) norm
type: Item
instance of: scholarly article
title: Area inequality and \(Q_{p}\) norm
zbMATH Open document ID: 1078.30028
author: Rauno Aulaskari
author: Huaihui Chen
published in: Journal of Functional Analysis
publication date: 2005-04-21T00:00:00Z
review text: Let \(R\) be a hyperbolic Riemann surface, and let \(f\) be a function defined and analytic on \(R\). The main result of the paper under review is the following theorem: for any \(p>q\geq 0\) and any \(a\in R\), we have \[ \int\int_R | f'(z)| ^2 g^p(z,a)\,dx\,dy \leq \frac{2^{q-p}\Gamma(p+1)}{\Gamma(q+1)} \int\int_R | f'(z)| ^2 g^q(z,a)\,dx\,dy, \] where \(g(z,a)\) denotes a Green's function of \(R\) with logarithmic singularity at \(a\). The theorem is a generalization of results of [\textit{S. Kobayashi}, Can. J. Math. 36, 747--755 (1984; Zbl 0553.30022)] and of [\textit{T. A. Metzger}, Can. J. Math. 33, 1255--1260 (1981; Zbl 0442.30030)].
Mathematics Subject Classification ID: 30D45
Mathematics Subject Classification ID: 30D50
Mathematics Subject Classification ID: 30F35
zbMATH DE Number: 2160152
zbMATH Keywords: \(Q_p\) space
zbMATH Keywords: Hyperbolic Riemann surface
reviewed by: Zengjian Lou
MaRDI profile type: Publication
full work available at URL: j.jfa.2004.12.007
OpenAlex ID: W2075221615
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DOI: 10.1016/J.JFA.2004.12.007
description: scientific article; zbMATH DE number 2160152