On weighted Compactness of commutators of bilinear maximal Calder\'on-Zygmund singular integral operators

TL;DR

This paper proves the weighted compactness of commutators of bilinear Calderón-Zygmund operators and their maximal truncations when symbols are in CMO and weights are in the multiple weights class.

Contribution

It establishes the weighted compactness of commutators of bilinear maximal Calderón-Zygmund operators with symbols in CMO, extending previous results to the weighted setting.

Findings

Commutators are compact operators on weighted Lebesgue spaces.

Results hold for all p1, p2 in (1, ∞) with 1/p = 1/p1 + 1/p2.

Applicable to weights in the multiple weights class A_{p}.

Abstract

Let $T$ be a bilinear Calder\'on-Zygmund singular integral operator and $T^*$ 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 $T^*_{b,j}$ (j=1,2), $T^*_{\vec{b}}\ $ be the commutators in the j-th entry and the iterated commutators of $T^*$, 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 $T^*_{b,j}$ and $T^*_{\vec{b}}$ are compact operators from $L^{p_1}(w_1)\times L^{p_2}(w_2)$ to $L^p(v_{\vec{w}})$, if $b,b_1,b_2\in{\rm CMO}(\mathbb{R}^n)$ 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 ${\rm CMO}(\mathbb{R}^n)$ denotes the closure of $\mathcal{C}_c^\infty(\mathbb{R}^n)$ in the ${\rm BMO}(\mathbb{R}^n)$ topology and $A_{\vec{p}}$ is the multiple weights class.