Functional inequalities and relative capacities
type: Item
instance of: scholarly article
title: Functional inequalities and relative capacities
zbMATH Open document ID: 1029.46022
DOI: 10.1023/A:1019845927654
author: V. S. Klimov
author: E. S. Panasenko
published in: Mathematical Notes
publication date: 2003-06-15T00:00:00Z
review text: The authors deal with functional inequalities of the form \[ \|f;Q\|\leq C\varphi\bigl( \|\nabla f;P\|,\|f,R\|\bigr), \tag{1} \] where \(P,Q\), and \(R\) are Banach ideal spaces of functions on a domain \(\Omega\subset\mathbb{R}^n\), the constant \(C\) is the same for all compactly supported functions satisfying a Lipschitz condition, \(\nabla f\) is the gradient of a function \(f\), and \(\varphi\) is a continuous degree-one homogeneous function. Compatibility conditions for norms on the spaces \(P,Q\), and \(R\) are given that ensure the equivalence of the functional inequality (1) to a geometric inequality between the norms of indicators and relative capacities of compact subsets of the domain \(\Omega\). The best comprehensible results relate to the case where \(P,Q\) and \(R\) are symmetric spaces.
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 1928277
zbMATH Keywords: Banach ideal spaces
zbMATH Keywords: relative capacity
zbMATH Keywords: symmetric spaces
zbMATH Keywords: relative capacities
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: a:1019845927654
OpenAlex ID: W124825090
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description: scientific article; zbMATH DE number 1928277