Sharp weighted estimates for multilinear commutators

Abstract

Multilinear commutators with vector symbol Formula=(b1,…,bm) defined by

Formula

are considered, where K is a Calderón–Zygmund kernel. The following a priori estimates are proved for w ∈ A∞. For 0 < p < ∞, there exists a constant C such that

Formula

and

Formula

where

Formula

Formula

and ML(log L)α 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 Formula are bounded on Lp(w) when w ∈ Ap, and that they satisfy corresponding weighted L(log L)1/r-type estimates with w ∈ A1.