Commutators of bilinear pseudodifferential operators and Lipschitz functions
type: Item
instance of: scholarly article
title: Commutators of bilinear pseudodifferential operators and Lipschitz functions
zbMATH Open document ID: 1401.35363
author: Árpád Bényi
author: Virginia Naibo
published in: The Journal of Fourier Analysis and Applications
publication date: 2018-06-28T00:00:00Z
review text: For a given pseudodifferential operator \(T\) and a function \(a\), the commutator \([T,a]\) is given by \[ [T,a] (f) = T (af) - a T(f). \] Continuity properties of such commutators are intensively studied for different symbol classes of the operator \(T\) and different function spaces as their domains (\(a\) usually being of a Lipschitz class). Some of the main references are [\textit{R. R. Coifman} and \textit{Y. Meyer}, Au dela des opérateurs pseudo-différentiels. Société Mathématique de France, Paris (1978; Zbl 0483.35082)], [\textit{R. R. Coifman} and \textit{Y. Meyer}, Ann. Inst. Fourier 28, No. 3, 177--202 (1978; Zbl 0368.47031)], Zbl [\textit{P. Auscher} and \textit{M. E. Taylor}, Commun. Partial Differ. Equations 20, No. 9--10, 1743--1775 (1995; Zbl 0844.35149)], [\textit{M. Taylor}, Proc. Am. Math. Soc. 131, No. 5, 1501--1507 (2003; Zbl 1022.35096)], [\textit{M. E. Taylor}, Proc. Am. Math. Soc. 143, No. 12, 5265--5274 (2015; Zbl 1334.42034)]. An extension of the classical results to the bilinear case can be found in [\textit{Á. Bényi} and \textit{T. Oh}, J. Fourier Anal. Appl. 20, No. 2, 282--300 (2014; Zbl 1327.35471)]. The authors introduce bilinear versions of different classes of operators considered in [\textit{P. Auscher} and \textit{M. E. Taylor}, Commun. Partial Differ. Equations 20, No. 9--10, 1743--1775 (1995; Zbl 0844.35149)], and show that, modulo smoothing operator, those classes strictly contain the corresponding Hörmander class. In such situations bilinear versions of the main results from [loc. cit.] are given. Moreover, the results from [\textit{Á. Bényi} and \textit{T. Oh}, J. Fourier Anal. Appl. 20, No. 2, 282--300 (2014; Zbl 1327.35471)] are extended to a broader class of symbols. The paper is an important contribution to the theory of commutators for pseudodifferential operators.
reviewed by: Nenad Teofanov
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 47G30
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B15
zbMATH DE Number: 6897260
zbMATH Keywords: commutators
zbMATH Keywords: bilinear pseudodifferential operators
zbMATH Keywords: bilinear Calderón-Zygmund theory
MaRDI profile type: Publication
full work available at URL: s00041-016-9519-1
OpenAlex ID: W2560352466
cites work: Paradifferential operators and commutator estimates
cites work: Bilinear pseudodifferential operators with forbidden symbols on Lipschitz and Besov spaces.
cites work: Smoothing of commutators for a Hörmander class of bilinear pseudodifferential operators
cites work: Symbolic Calculus and the Transposes of Bilinear Pseudodifferential Operators
cites work: Almost orthogonality and a class of bounded bilinear pseudodifferential operators
cites work: On the Hörmander classes of bilinear pseudodifferential operators
cites work: On the Hormander classes of bilinear pseudodifferential operators II
cites work: Une algebre maximale d'operateurs pseudo-differentiels d'operateurs pseudo-differentiels
cites work: COMMUTATORS OF SINGULAR INTEGRAL OPERATORS
cites work: Polynomial growth estimates for multilinear singular integral operators
cites work: Q3941791
cites work: Commutateurs d'intégrales singulières et opérateurs multilinéaires
cites work: Multilinear Calderón-Zygmund theory
cites work: A new proof of the bilinear T(1) Theorem
cites work: Bilinear pseudodifferential operators with symbols in Besov spaces
cites work: Pseudo-differential operaors of type 1,1.
cites work: Multilinear pseudodifferential operators beyond Calderón-Zygmund theory
cites work: Calderon-Vaillancourt--type theorem for bilinear operators
cites work: On the bilinear Hörmander classes in the scales of Triebel-Lizorkin and Besov spaces
cites work: On the $L^\infty \times L^\infty \rightarrow BMO$ mapping property for certain bilinear pseudodifferential operators
cites work: Estimates for rough Fourier integral and pseudodifferential operators and applications to the boundedness of multilinear operators
cites work: Commutator estimates
cites work: Commutator estimates for Hölder continuous and bmo-Sobolev multipliers
DOI: 10.1007/S00041-016-9519-1
description: scientific article; zbMATH DE number 6897260