On weighted Compactness of commutators of bilinear maximal Calderón-Zygmund singular integral operators

Abstract

Let $T$ be a bilinear Calderón-Zygmund singular integral operator and $T^{\ast }$ be its corresponding truncated maximal operator. For any $b\in \text{BMO}({\mathbb{R}}^n)$ and $\vec{b}=(b_1,b_2)\in \text{BMO}({\mathbb{R}}^n)\times \text{BMO}({\mathbb{R}}^n)$, let $T_{b,j}^{\ast }$ (j=1,2), $T_{\vec{b}}^{\ast }$ be the commutators in the j-th entry and the iterated commutators of $T^{\ast }$, respectively. In this paper, for all $1<p_1,p_2<\infty$, $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$, we show that $T_{b,j}^{\ast }$ and $T_{\vec{b}}^{\ast }$ are compact operators from $L^{p_1}(w_1)\times L^{p_2}(w_2)$ to $L^p(v_{\vec{w}})$, if $b,b_1,b_2\in \mathrm{CMO}({\mathbb{R}}^n)$ and $\vec{w}=(w_1,w_2)\in A_{\vec{p}}$, $v_{\vec{w}}=w_1^{p/p_1}{w}_2^{p/p_2}$. Here $\mathrm{CMO}({\mathbb{R}}^n)$ denotes the closure of ${\mathcal{C}}_c^{\infty }({\mathbb{R}}^n)$ in the $\mathrm{BMO}({\mathbb{R}}^n)$ topology and $A_{\vec{p}}$ is the multiple weights class.

corrected the author name