Extrapolation of weighted norm inequalities for multivariable operators and applications
type: Item
instance of: scholarly article
title: Extrapolation of weighted norm inequalities for multivariable operators and applications
zbMATH Open document ID: 1049.42007
DOI: 10.1007/BF02921864
author: Loukas Grafakos
author: José María Martell
published in: The Journal of Geometric Analysis
publication date: 2004-06-08T00:00:00Z
review text: The celebrated Rubio de Francia extrapolation theorem states that if a given operator is bounded on \(L^{p_0}(w)\) for some \(1\leq p_0<\infty\) and all \(A_p\) weights \(w\in A_{p_0}\), then it is bounded on all spaces \(L^p(\mathbb R^n)\) for \(1<p<\infty\). The authors extend this to the case of many functions. More precisely, let \(1\leq q_1, \ldots, q_m<\infty\) with \(1/q=1/q_1+\cdots+1/q_m\leq m\). Suppose that for all \(B>1\) there is a constant \(C_0(B)\) such that \(\| T(f_1,\ldots,f_m)\| _{L^q(w_1^{q}\cdots w_m^q)}\leq C_0 \prod_{j=1}^{m}\| f_j\| _{L^{q_j}(w_j^{q_j})}\) for all tuples of weights \(w_1^{q_1},\ldots, w_m^{q_m}\in (A_{q_1},\ldots, A_{q_m})\) with \([w_j^{q_j}]_{A_{q_j}}\leq B\). Then for all indices \(1<p_1,\ldots, p_m<\infty\) with \(1/p=1/p_1+\cdots+1/p_m<m\), all \(B>1\), and all weights \(w_1^{p_1},\ldots, w_m^{p_m}\in (A_{p_1},\ldots, A_{p_m})\) with \([w_j^{p_j}]_{A_{p_j}}\leq B\), there exists \(C=C(B)\) such that \(\| T(f_1,\ldots,f_m)\| _{L^p(w_1^{p}\cdots w_m^p)}\leq C \prod_{j=1}^{m}\| f_j\| _{L^{p_j}(w_j^{p_j})}\). They give a version of this for which there is only one weight. They also consider the situation for which the initial estimates are of weak type. As applications, they treat multilinear Calderón-Zygmund operators, vector-valued inequalities, a multilinear extension of the Marcinkiewicz and Zygmund theorem on \(\ell^2\)-valued extensions of bounded linear operators.
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 47H60
zbMATH DE Number: 2070478
zbMATH Keywords: extrapolation
zbMATH Keywords: multilinear operators
zbMATH Keywords: \(A_p\) weights
zbMATH Keywords: vector-valued inequalities
zbMATH Keywords: Calderón-Zygmund singular integral operators
reviewed by: Kôzô Yabuta
MaRDI profile type: Publication
cites work: CONVOLUTION OPERATORS ON BANACH SPACE VALUED FUNCTIONS
cites work: On Commutators of Singular Integrals and Bilinear Singular Integrals
cites work: Commutateurs d'intégrales singulières et opérateurs multilinéaires
cites work: An Extrapolation Theorem in the Theory of A p Weights
cites work: Q3700300
cites work: Some remarks on multilinear maps and interpolation
cites work: The disc as a bilinear multiplier
cites work: Multilinear interpolation between adjoint operators
cites work: Multilinear Calderón-Zygmund theory
cites work: Maximal operator and weighted norm inequalities for multilinear singular integrals
cites work: Multilinear estimates and fractional integration
cites work: On Calderón's conjecture
cites work: Q4462261
cites work: Factorization Theory and A P Weights
cites work: Calderón-Zygmund theory for operator-valued kernels
cites work: Q5607484
cites work: Interpolation of Operators with Change of Measures
cites work: Q3254471
description: scientific article; zbMATH DE number 2070478