Weighted estimates for the Calderón commutator

Abstract

In this paper, the authors establish some weighted estimates for the Calderón 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 $P_2(A;x,y)=A(x)-A(y)-A^{\prime }(y)(x-y)$. Dominating this operator by multi(sub)linear sparse operators, the authors establish the weighted bounds from $L^{p_1}(\mathbb{R},w_1)$ $\times \cdots \times L^{p_m}(\mathbb{R},w_m)$ to $L^p(\mathbb{R},{\nu }_{\vec{w}})$, with $p_1,\dots ,p_m\in (1,\infty )$, $1/p=1/p_1+\cdots +1/p_m$, 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

21 pages

References in corpus (1)

• Sparse domination theorem for multilinear singular integral operators with $L^r$-Hörmander condition