Commutators of multiplier operators
type: Item
instance of: scholarly article
title: Commutators of multiplier operators
zbMATH Open document ID: 0617.47036
author: Tao Qian
published in: Chinese Annals of Mathematics. Series B
publication date: 1985-01-01T00:00:00Z
review text: Given functions a and G on \({\mathbb{K}}^ K\) and m in \({\mathbb{N}}\), define the operator \(S^{(m)}\) on \(S({\mathbb{R}}^ K)\) by the formula \[ S^{(m)}f=\sum_{| \beta | <m}(\beta !)^{- 1}(X^{\beta}G)*((D^{\beta}a).f), \] where. and * denotes pointwise multiplication and convolution respectively, \((X^{\beta}G)(x)=x^{\beta}G(x)\), and \((D^{\beta}a)(x)=(\partial /\partial x)^{\beta}a(x)\). Formally, a.(G*f) is equal to \(S^{(\infty)}f.\) The paper under review obtains estimates for \[ \| a.(G*f)- S^{(m)}f\|_ p, \] in terms of \(\| f\|_ p\) and norms of derivatives of a, as well as estimates for iterates and derivatives of this difference. The techniques are generalizations of the work of \textit{R. R. Coifman} and \textit{Y. Meyer} [Au delà des opérateurs pseudodifferentiels, Astérisque 57, 185 p. (1978; Zbl 0483.35082)].
Mathematics Subject Classification ID: 47Gxx
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 47B47
zbMATH DE Number: 4000823
zbMATH Keywords: pseudodifferential operators
zbMATH Keywords: Fourier multiplier
zbMATH Keywords: pointwise multiplier
zbMATH Keywords: pointwise multiplication
zbMATH Keywords: convolution
reviewed by: Michael G. Cowling
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4000823