Weighted norm inequalities for the multilinear Calderón-Zygmund operators
type: Item
instance of: scholarly article
title: Weighted norm inequalities for the multilinear Calderón-Zygmund operators
zbMATH Open document ID: 1222.42019
published in: Science China. Mathematics
publication date: 2011-02-25T00:00:00Z
review text: The purpose of this paper is to establish some weighted estimates for the multilinear Calderón-Zygmund operator and its corresponding commutator. Let \(m\geq 1\), let \(K(x;y_1,\dots, y_m)\) be a locally integrable function defined away from the diagonal \(x=y_1=y_2=\cdots=y_m\) in \((\mathbb R^n)^{m+1}\), let \(A>0\) and \(\gamma \in (0,1]\) be two constants. \(K\) is a kernel in \(m\text{-CZK}(A, \gamma)\) if it satisfies the size condition that for all \((x, y_1, \dots, y_m) \in (\mathbb R^n)^{m+1}\) with \(x \neq y_j\) for some \(1 \leq j \leq m\), \[ |K(x;y_1,\dots,y_m)| \leq \frac{A}{(|x-y_1|+\cdots+|x-y_m|)^{mn}}, \] and satisfies the regularity condition that \[ K(x;y_1,\dots,y_m) - K(x';y_1,\dots, y_m)| \leq \frac{A|x-x'|^{\gamma}}{(|x-y_1|+\cdots+|x-y_m|)^{mn+\gamma}} \] whenever \(\max_{1\leq k \leq m}|x-y_k| \geq 2|x-x'|\), and also that for each fixed \(k\) with \(1\leq k \leq m\), \[ |K(x;y_1,\dots,y_k,\dots,y_m)-K(x;y_1,\dots,y'_k,\dots,y_m)| \leq \frac{A|y_k-y'_k|^\gamma}{(|x-y_1|+\cdots+|x-y_m|)^{mn+\gamma}} \] whenever \(\max_{1\leq j \leq m}|x-y_j| \geq 2|y_k-y'_k|\). An operator \(T\) defined on the \(m\)-fold product of Schwartz spaces and taking values in the space of tempered distributions, is said to be an \(m\)-linear Calderón-Zygmund operator with kernel \(K\) if {\parindent=6,5mm \begin{itemize}\item[(a)] \(T\) is \(m\)-linear, \item[(b)] for some \(q_1,\dots, q_m \in [1,\infty]\) and some \(q \in (0,\infty)\) with \(1/q = \sum_{k=1}^{m}1/q_k\), \(T\) can be extended to be a bounded operator from \(L^{q_1}(\mathbb R^n) \times L^{q_2}(\mathbb R^n) \times \cdots \times L^{q_m}(\mathbb R^n)\) to \(L^{q}(\mathbb R^n)\), \item[(c)] for \(f_1,\dots, f_m \in L^2(\mathbb R^n)\) with compact supports, and for \(x \not\in \bigcap_{k=1}^{m}\operatorname{supp}f_k\), \[ T(f_1,\dots, f_m)(x)=\int_{(\mathbb R^n)^m} K(x;y_1,\dots, y_m) \prod_{k=1}^{m}f_k(y_k)\,dy_1\cdots dy_m, \] and \(K\) is in \(m\text{-CZK}(A,\gamma)\) for some constants \(A\) and \(\g…
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 5857640
zbMATH Keywords: multilinear Calderón-Zygmund operator
zbMATH Keywords: commutator
zbMATH Keywords: weighted norm inequality
zbMATH Keywords: interpolation
zbMATH Keywords: extrapolation
author: Guoen Hu
MaRDI profile type: Publication
full work available at URL: s11425-010-4016-5
OpenAlex ID: W1986055030
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DOI: 10.1007/S11425-010-4016-5
description: scientific article; zbMATH DE number 5857640