Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-23T05:32:21.518Z Has data issue: false hasContentIssue false

SHARP WEIGHTED ESTIMATES FOR MULTILINEAR COMMUTATORS

Published online by Cambridge University Press:  24 March 2003

C. PÉREZ
Affiliation:
Departmento de Análisis Matematico, Facultad de Matemáticas, Universidad de Sevilla, 41080 Sevilla, [email protected]
R. TRUJILLO-GONZÁLEZ
Affiliation:
Departmento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna, S/C de Tenerife, [email protected]
Get access

Abstract

Multilinear commutators with vector symbol $\vec{b}=(b_1,\ldots,b_m)$ defined by \[ T_{\vec{b}}(f)(x)=\int_{{\bb R}^n}\Bigg[\prod\limits^m_{j=1}(b_j(x)-b_j(y))\Bigg]K(x,y)f(y)dy \] are considered, where $K$ is a Calderón–Zygmund kernel. The following a priori estimates are proved for $w\in A_\infty$ . For $0 < p < \infty$ , there exists a constant $C$ such that \[ \|\dot{T}_{{\vec{b}}}(f)\|_{L^P(w)}\le C\|\vec{b}\|\|M_{L(\log\,L)^{1/r}}(f)\|_{L^P(w)} \] and \[ \sup_{t>0}\frac{1}{\Phi(\frac{1}{t})}w(\{y\in{\bb R}^n:|T_{\vec{b}}f(y)|>t\})\le C\sup_{t>0}\frac{1}{\Phi(\frac{1}{t})}w(\{y\in{\bb R}^n:M_{L(\log\,L)^{1/r}}(\|\vec{b}\|f)(y)>t\}), \] where \begin{eqnarray*} &\|\vec{b}\|=\prod\limits^m_{j=1}\|b_j\|_{osc_{\exp L}^r j},\\ &\Phi(t)=t\log^{1/r}(e+t),\quad \frac{1}{r}=\frac{1}{r_1}+\cdots+\frac{1}{r_m}, \end{eqnarray*} and $M_{L(\log L)^{\alpha}}$ is an Orlicz type maximal operator. This extends, with a different approach, classical results by Coifman.

As a corollary, it is deduced that the operators $T_{\vec{b}}$ are bounded on $L^p(w)$ when $w\in A_p$ , and that they satisfy corresponding weighted $L(\log\,L)^{1/r}$ -type estimates with $w\in A_1$ .

Type
Research Article
Copyright
The London Mathematical Society, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)