Spectral theory of some degenerate elliptic operators with local singularities
type: Item
instance of: scholarly article
title: Spectral theory of some degenerate elliptic operators with local singularities
zbMATH Open document ID: 1198.47067
author: Leszek Skrzypczak
author: Dorothee D. Haroske
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-08-11T00:00:00Z
review text: The present paper extends the results from [\textit{D.\,D.\thinspace Haroske} and \textit{L.\,Skrzypczak}, Rev.\ Mat.\ Complut.\ 21, No.\,1, 135--177 (2008; Zbl 1202.46039)] and [\textit{D.\,Haroske} and \textit{H.\,Triebel}, Math.\ Nachr.\ 168, 109--137 (1994; Zbl 0829.46020)] regarding compact embeddings of Muckenhoupt weighted function spaces of Besov and Triebel-Lizorkin type with example weights of polynomial growth near infinity and near some local singularity. The authors obtain eigenvalue estimates for degenerate pseudodifferential operators of type \(b_{2}\circ p(x,D)\circ b_{1}\), where \(b_i \in L_{r_i}(\mathbb R^n, w_i),\) \(w_i \in \mathcal A_{\infty },\) \(i=1,2,\) and \(p(x, D) \in \psi ^{-x}_{1,0},\) \(x > 0.\) Moreover, by means of the Birman-Schwinger principle, the ``negative spectrum'' of operators like \(H_{\gamma }=A - \gamma V\) for \(\gamma \rightarrow \infty \) is studied, where the potential \(V\) may have singularities (in terms of Muckenhoupt weights) and \(A\) is a positive elliptic pseudodifferential operator of order \(x>0,\) which is selfadjoint in \(L_2(\mathbb R^n)\). Various examples and comparisons to already existing results are given as well.
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 47G30
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35S05
zbMATH DE Number: 5768948
zbMATH Keywords: Muckenhoupt weighted function spaces
zbMATH Keywords: compact embeddings
zbMATH Keywords: distribution of eigenvalues
zbMATH Keywords: degenerate pseudodifferential operators
zbMATH Keywords: Birman-Schwinger principle
zbMATH Keywords: negative spectrum
Wikidata QID: Q57339758
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.05.026
OpenAlex ID: W1963781882
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DOI: 10.1016/J.JMAA.2010.05.026
description: scientific article; zbMATH DE number 5768948