Boundedness and compactness of convolution operators with unbounded coefficients on locally compact groups
type: Item
instance of: scholarly article
title: Boundedness and compactness of convolution operators with unbounded coefficients on locally compact groups
zbMATH Open document ID: 0591.43005
DOI: 10.1007/BF01156250
author: B. Ya. Shtejnberg
published in: Mathematical Notes
publication date: 1985-01-01T00:00:00Z
review text: This paper considers the operators arising when convolution operators (with bounded kernels with compact support) and multiplication operators are composed and commuted. The author finds necessary and sufficient conditions that all operators \(A_ fA\), or all operators \(AA_ f\), or all operators \(AA_ f-A_ fA\), etc. be compact, when \(A_ f\) is multiplication by a fixed function f and A varies over the set of all convolution operators with bounded compactly supported kernels. To give the flavour of the results obtained, we mention one interesting application: the Fourier multiplier operator \(A: g\to F^{-1}(a(Fg))\), F denoting the Fourier transformation on \(L^ 2({\mathbb{R}}^ n)\), is of local type, meaning \(AA_ f-A_ fA\) is compact for any continuous f vanishing at \(\infty\), iff a is in \(L^{\infty}({\mathbb{R}}^ n)\) and also \[ \lim_{t\to \infty}\int_{W}| \int_{D}[a(t+x)-a(t+y)] dx|^ 2 dy=0 \] for all relatively compact open sets W and D in \({\mathbb{R}}^ n\).
Mathematics Subject Classification ID: 43A22
Mathematics Subject Classification ID: 45P05
Mathematics Subject Classification ID: 44A35
zbMATH DE Number: 3948964
zbMATH Keywords: composition of operators
zbMATH Keywords: commutator
zbMATH Keywords: compact operator
zbMATH Keywords: convolution operators
zbMATH Keywords: multiplication operators
zbMATH Keywords: Fourier multiplier
reviewed by: Michael G. Cowling
MaRDI profile type: Publication
cites work: Q5610581
cites work: On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators
full work available at URL: https://doi.org/10.1007/bf01156250
OpenAlex ID: W1967059285
description: scientific article; zbMATH DE number 3948964