Multilinear Calderón-Zygmund theory
type: Item
instance of: scholarly article
title: Multilinear Calderón-Zygmund theory
zbMATH Open document ID: 1032.42020
author: Loukas Grafakos
author: Rodolfo H. Torres
published in: Advances in Mathematics
publication date: 2002-07-04T00:00:00Z
review text: In this paper the authors consider a systematic treatment of multilinear Calderón-Zygmund operators introduced earlier in the papers of Coifman and Meyer and of \textit{M. Lacey} and \textit{C. Thiele} [Ann. Math. (2) 146, 693-724 (1997; Zbl 0914.46034); ibid. 149, 475-496 (1999; Zbl 0934.42012)]. The first main result reads as follows: Let \(m\)-linear operators be \(T: [{\mathcal S}(\mathbb{R}^n)]^m\to{\mathcal S}'(\mathbb{R}^n)\) for which there is a function \(K\) defined away from the diagonal \(x= y_1=\cdots= y_m\) in \((\mathbb{R}^n)^{m+1}\) satisfying \[ |K(y_0,y_1,\dots, y_m)|\leq {c_{n,m}A\over (\sum^m_{k,l=0}|y_k- y_l|)^{nm}} \] and \[ |K(y_0,\dots, y_j,\dots,y_m)- K(y_0,\dots, y_j',\dots, y_m)|\leq {c_{n,m}A|y_j- y_j'|^\varepsilon\over (\sum^m_{k,l=0}|y_k- y_l|)^{nm+ \varepsilon}}, \] whenever \(0\leq j\leq m\) and \(|y_j- y_j'|\leq{1\over 2}\max_{0\leq k\leq m}|y_j- y_k|\). Let \(q_j\in [1,\infty)\) be given numbers with \(1/q= \sum^m_{j=1} 1/q_j\). Suppose that \(T\) maps \(L^{q_1,1}\times\cdots\times L^{q_m,1}\) into \(L^{q,\infty}\) if \(q> 1\) or \(L^1\) if \(q= 1\). Then for any \(p_j\in [1,\infty]\) such that \(1/m\leq p< \infty\), \(T\) extends to a bounded map from \(L^{p_1}\times\cdots\times L^{p_m}\) into \(L^p\) if all \(p_j> 1\) and into \(L^{p,\infty}\) if some \(p_j= 1\). If some \(p_k= \infty\), \(L^{p_k}\) should be replaced by \(L^\infty_c\). Moreover, \(T\) extends to a bounded map from \(L^\infty\times\cdots\times L^\infty\) to BMO. Next, the authors obtain the version of the multilinear T1 theorem by \textit{G. David} and \textit{J.-L. Journé} [Ann. Math. (2) 120, 371-397 (1984; Zbl 0567.47025)]. It is proved that if \(T(e_{\xi_1},\dots, e_{\xi_m})\) and \(T^{*j}(e_{\xi_1},\dots, e_{\xi_m})\) \((\xi_1,\dots, \xi_m\in \mathbb{R}^n\), \(1\leq j\leq m)\) are bounded subsets of BMO, then \(T\) has a bounded extension from \(L^{q_1}\times\cdots\times L^{q_m}\) into \(L^q\) if \(1< q,q_j<\infty\). Here \(j\)th transpose \(T^{*j}\) of \(T\)…
reviewed by: Sunggeum Hong
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 1763525
zbMATH Keywords: multilinear Calderón-Zygmund operators
zbMATH Keywords: BMO
zbMATH Keywords: multilinear T1 theorem
zbMATH Keywords: multilinear translation invariant operators
zbMATH Keywords: multlinear pseudodifferential operators
zbMATH Keywords: multilinear multipliers
MaRDI profile type: Publication
full work available at URL: aima.2001.2028
OpenAlex ID: W2048755684
Wikidata QID: Q61611370
cites work: CONVOLUTION OPERATORS ON BANACH SPACE VALUED FUNCTIONS
cites work: Une algebre maximale d'operateurs pseudo-differentiels d'operateurs pseudo-differentiels
cites work: On the existence of certain singular integrals
cites work: On Singular Integrals
cites work: Polynomial growth estimates for multilinear singular integral operators
cites work: On Commutators of Singular Integrals and Bilinear Singular Integrals
cites work: Commutateurs d'intégrales singulières et opérateurs multilinéaires
cites work: Q3941791
cites work: Q3761164
cites work: A boundedness criterion for generalized Calderón-Zygmund operators
cites work: Some remarks on multilinear maps and interpolation
cites work: Uniform bounds for the bilinear Hilbert transforms. I
cites work: Estimates for translation invariant operators in \(L^p\) spaces
cites work: Q3996146
cites work: Q3977301
cites work: Multilinear estimates and fractional integration
cites work: \(L^p\) estimates on the bilinear Hilbert transform for \(2<p<\infty\)
cites work: On Calderón's conjecture
cites work: Q4338833
cites work: On convolution operators leaving \(L^{p,\lambda}\) spaces invariant
cites work: Q5601699
cites work: Q5607484
cites work: Q3142876
cites work: A Multilinear Version of the Marcinkiewicz Interpolation Theorem
DOI: 10.1006/AIMA.2001.2028
Recommended article: Q4945135
Recommended article: Multilinear operator-valued Calderón-Zygmund theory
Recommended article: Multilinear Calderón–Zygmund Singular Integrals
Recommended article: Multilinear theory of strongly singular Calderón-Zygmund operators and applications
Recommended article: Multilinear Calderón-Zygmund operator on Morrey type spaces
Recommended article: Multilinear Calderón-Zygmund operators on Hardy spaces
Recommended article: Multilinear pseudodifferential operators beyond Calderón-Zygmund theory
Recommended article: Multilinear Calderón-Zygmund theory with geometric mean Hörmander conditions
Recommended article: Multilinear Calderón–Zygmund operators on weighted Hardy spaces
Recommended article: Maximal multilinear Calderón-Zygmund operators with non-doubling measures
description: scientific article; zbMATH DE number 1763525