Weighted estimates for multilinear commutators of generalized Calderón–Zygmund operators

Abstract

Let \(T\) be the generalized Calderon–Zygmund operator introduced by Chang et al. (Contemp Math 445:61–70, 2007) and \(\mathbf {b}=(b_1, \ldots ,b_m)\) with \(b_i\in {BMO}({\mathbb R}^n)\). The multilinear commutator \(T_{\mathbf {b}}\) generated by \(T\) and \(\mathbf {b}\) is formally defined by $$\begin{aligned} T_{\mathbf {b}}f(x)=\int _{{\mathbb R}^n}\left[ \prod _{i=1}^m \big (b_i(x)-b_i(y)\big )\right] K(x,y)f(y)dy. \end{aligned}$$ In this paper, the weighted \(L^p\)-boundedness and the weighted weak-\(L{\log }L\) type estimate for the multilinear commutator \(T_{\mathbf {b}}\) are established.