Nonmultipliers of the Sobolev spaces \(W^{k,1}({\mathbb{R}}^ n)\)
type: Item
instance of: scholarly article
title: Nonmultipliers of the Sobolev spaces \(W^{k,1}({\mathbb{R}}^ n)\)
zbMATH Open document ID: 0627.46034
DOI: 10.1016/0022-1236(87)90021-8
author: S. Poornima
author: Aline Bonami
published in: Journal of Functional Analysis
publication date: 1987-01-01T00:00:00Z
review text: Let \(W^{k,1}({\mathbb{R}}^ n)\) be the usual Sobolev space of order k on \(L^ 1({\mathbb{R}}^ n)\) and let \(\overset\circ W^{k,1}({\mathbb{R}}^ n)\) denote the corresponding homogeneous Sobolev space. A multiplier is defined as a tempered distribution S such that convolution with S defines a continuous linear operator on \(W^{k,1}({\mathbb{R}}^ n).\) Theorem 1 asserts that if m is a continuous function in \({\mathbb{R}}^ n\) satisfying some limit condition, then, if m is a Fourier multiplier of \(W^{k,1}({\mathbb{R}}^ n)\), the corresponding limit is a constant function. This implies that some natural singular integral operators are not multiplies. This result follows from Theorem 2, which says that if \(\Omega\) is a homogeneous function of degree 0, which is continuous in \(R^ n-\{0\}\), then \(\Omega\) is a Fourier multiplier of \(W^{k,1}\) if and only if it is a constant. The main tool for the proof is Lemma 4, asserting that if m is a bounded function vanishing in a neighborhood of the origin, it is a Fourier multiplier of \(W^{k,1}({\mathbb{R}}^ n)\) if and only if it is a Fourier multipler of \(\overset\circ W^{k,1}({\mathbb{R}}^ n)\).
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 42B15
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 4019754
zbMATH Keywords: homogeneous Sobolev space
zbMATH Keywords: tempered distribution
zbMATH Keywords: Fourier multiplier
zbMATH Keywords: singular integral operators
reviewed by: Jesús Hernández
MaRDI profile type: Publication
full work available at URL: 0022-1236(87)90021-8
OpenAlex ID: W2040576748
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description: scientific article; zbMATH DE number 4019754