Sobolev-Orlicz imbeddings, weak compactness, and spectrum
type: Item
instance of: scholarly article
title: Sobolev-Orlicz imbeddings, weak compactness, and spectrum
zbMATH Open document ID: 0974.46036
published in: Journal of Functional Analysis
publication date: 2001-11-30T00:00:00Z
review text: The aim of the paper under consideration is to show how weak compactness in the \(L^1(X,m)\) space may be used to relate the existence of a Sobolev-Orlicz embedding to the \(L^2(X,m)\)-spectral properties of an operator \(H\). First, it is shown that a Sobolev-Orlicz embedding implies that the bottom of the \(L^2\)-spectrum of \(H\) is an eigenvalue of finite multiplicity provided \(m\) is finite. Secondly, it is proved that for a large class of operators, namely those for which Persson's characterisation of the bottom of the essential spectrum is true, a Sobolev-Orlicz embedding always implies the discreteness of the \(L^2\)-spectrum of \(H\) provided \(m\) is finite. Finally, a certain converse of the last result is proved: the discreteness of the \(L^2\)-spectrum of \(H\) always implies the existence of an Orlicz space for which a Sobolev-Orlicz embedding holds true for \(H\). The case of logarithmic Sobolev inequalities is considered and provides the original motivation of the author's investigation.
reviewed by: Walter Farkas
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 1542415
zbMATH Keywords: Sobolev-Orlicz embeddings
zbMATH Keywords: weak compactness
zbMATH Keywords: discrete spectrum
zbMATH Keywords: essential spectrum
zbMATH Keywords: Dirichlet form
zbMATH Keywords: logarithmic Sobolev inequalities
author: Fabio Cipriani
MaRDI profile type: Publication
full work available at URL: jfan.2000.3633
OpenAlex ID: W1983250083
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DOI: 10.1006/JFAN.2000.3633
description: scientific article; zbMATH DE number 1542415