Weighted estimates for the Calder\'on commutator

Abstract

In this paper, the authors establish some weighted estimates for the Calder\'on commutator defined by \begin{eqnarray*} &&\mathcal{C}_{m+1,\,A}(a_1,\dots,a_{m};f)(x) &&\quad={\rm p.\,v.}\,\int_{\mathbb{R}}\frac{P_2(A;\,x,\,y)\prod_{j=1}^m(A_j(x)-A_j(y))}{(x-y)^{m+2}}f(y){\rm d}y, \end{eqnarray*} with $P_2(A;\,x,\,y)=A(x)-A(y)-A'(y)(x-y)$. Dominating this operator by multi(sub)linear sparse operators, the authors establish the weighted bounds from $L^{p_1}(\mathbb{R},w_1)$ $\times\dots\times L^{p_m}(\mathbb{R},w_m)$ to $L^{p}(\mathbb{R},\nu_{\vec{w}})$, with $p_1,\dots,p_m \in (1,\,\infty)$, $1/p=1/p_1+\dots+1/p_m$, and $\vec{w}=(w_1,\,\dots,\,w_m)\in A_{\vec{P}}(\mathbb{R}^{m+1})$. The authors also obtain the weighted weak type endpoint estimates for this operator