On weighted compactness of commutators of bilinear maximal Calderón–Zygmund singular integral operators

Let T be a bilinear Calderón–Zygmund singular integral operator and let T * {T^{*}} be its corresponding truncated maximal operator. For any b ∈ BMO ⁡ ( ℝ n ) {b\in\operatorname{BMO}({\mathbb{R}^{n}})} and b → = ( b 1 , b 2 ) ∈ BMO ⁡ ( ℝ n ) × BMO ⁡ ( ℝ n ) {\vec{b}=(b_{1},b_{2})\in\operatorname{BMO}({\mathbb{R}^{n}})\times% \operatorname{BMO}({\mathbb{R}^{n}})} , let T b , j * {T^{*}_{b,j}} ( j = 1 , 2 {j=1,2} ) and T b → * {T^{*}_{\vec{b}}} be the commutators in the j-th entry and the iterated commutators of T * {T^{*}} , respectively. In this paper, for all 1 < p 1 , p 2 < ∞ {1<p_{1},p_{2}<\infty} , 1 p = 1 p 1 + 1 p 2 {\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}} , we show that T b , j * {T^{*}_{b,j}} and T b → * {T^{*}_{\vec{b}}} are compact operators from L p 1 ⁢ ( w 1 ) × L p 2 ⁢ ( w 2 ) {L^{p_{1}}(w_{1})\times L^{p_{2}}(w_{2})} to L p ⁢ ( v w → ) {L^{p}(v_{\vec{w}})} if b , b 1 , b 2 ∈ CMO ⁡ ( ℝ n ) {b,b_{1},b_{2}\in\operatorname{CMO}(\mathbb{R}^{n})} and w → = ( w 1 , w 2 ) ∈ A p → {\vec{w}=(w_{1},w_{2})\in A_{\vec{p}}} , v w → = w 1 p / p 1 ⁢ w 2 p / p 2 {v_{\vec{w}}=w_{1}^{p/p_{1}}w_{2}^{p/p_{2}}} . Here CMO ⁡ ( ℝ n ) {\operatorname{CMO}(\mathbb{R}^{n})} denotes the closure of 𝒞 c ∞ ⁢ ( ℝ n ) {\mathcal{C}_{c}^{\infty}(\mathbb{R}^{n})} in the BMO ⁡ ( ℝ n ) {\operatorname{BMO}(\mathbb{R}^{n})} topology and A p → {A_{\vec{p}}} is the multiple weights class.