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
Categories: math.CA