On Weighted Compactness of Commutators of Bilinear Vector-valued Singular Integral Operators and Applications

Abstract

Let T be a bilinear vector-valued singular integral operator satisfies some mild regularity conditions, which may not fall under the scope of the theory of standard Calderón–Zygmund classes. For any \(\vec{b}=(b_{1},b_{2})\in (\text{CMO}(\mathbb{R}^{n}))^{2}\), let \([T,b_{j}]_{e_{j}}\ (j=1,2),\ [T,\vec{b}]_{\alpha}\) be the commutators in the j-th entry and the iterated commutators of T, respectively. In this paper, for all p₀ > 1, \({p_{0}\over 2} < p < \infty\), and p₀ ≤ p₁, p₂ < ∞ with 1/p = 1/p₁ + 1/p₂, we prove that \([T,b_{j}]_{e_{j}}\) and \([T,\vec{b}]_{\alpha}\) are weighted compact operators from \(L^{p_{1}}(w_{1})\times L^{p_{2}}(w_{2})\) to \(L^{p}(\nu_{\vec{w}})\), where \(\vec{w}=(w_{1},w_{2})\in A_{\vec{p}/p_{0}}\) and \(\nu_{\vec{w}}=w_{1}^{p/p_{1}}w_{2}^{p/p_{2}}\). As applications, we obtain the weighted compactness of commutators in the j-th entry and the iterated commutators of several kinds of bilinear Littlewood–Paley square operators with some mild kernel regularity, including bilinear g function, bilinear g*_λ function and bilinear Lusin’s area integral. In addition, we also get the weighted compactness of commutators in the j-th entry and the iterated commutators of bilinear Fourier multiplier operators, and bilinear square Fourier multiplier operators associated with bilinear g function, bilinear g*_λ function and bilinear Lusin’s area integral, respectively.