Lattice homomorphisms between Sobolev spaces
type: Item
instance of: scholarly article
title: Lattice homomorphisms between Sobolev spaces
zbMATH Open document ID: 1222.47064
author: M. Biegert
published in: Positivity
publication date: 2010-08-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/0805.4740
review text: The author shows that every vector lattice homomorphism \(T\) from \(W^{1,p}_{0}(\Omega_{1})\) into \(W^{1,q}(\Omega_{2})\) has the representation \(Tu=(u \circ \xi)g\) (\(\text{Cap}_{q}\)-quasi everywhere on \(\Omega_{2}\)). Here, \(p,q\in (1,\infty)\), \(\Omega_{1}\) and \(\Omega_{2}\) are open sets in \(\mathbb R^{n}\), \(\xi\) maps \(\Omega_{1}\) to \(\Omega_{2}\), and \(g\) maps \(\Omega_{2}\) to \([0,\infty)\). This is a corollary of a more general and abstract representation theorem, which the author also proves. Some other applications of the theorem and examples are given as well.
reviewed by: Riikka Korte
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 31C15
Mathematics Subject Classification ID: 28A12
zbMATH DE Number: 5770266
zbMATH Keywords: lattice homomorphisms
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: weighted composition operators
MaRDI profile type: Publication
OpenAlex ID: W2065876983
arXiv ID: 0805.4740
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DOI: 10.1007/S11117-009-0022-7
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description: scientific article