Weighted compact commutator of bilinear Fourier multiplier operator

Abstract

Let T _σ be the bilinear Fourier multiplier operator with associated multiplier σ satisfying the Sobolev regularity that $\mathop {\sup }\limits_{k \in \mathbb{Z}} {\left\| {{\sigma _k}} \right\|_{{W^s}\left( {{\mathbb{R}^{2n}}} \right)}} < \infty$ for some s ∈ (n, 2n]. In this paper, it is proved that the commutator generated by T _σ and CMO(ℝⁿ) functions is a compact operator from ${L^{{p_1}}}\left( {{\mathbb{R}^n},{\omega _1}} \right) \times {L^{{p_2}}}\left( {{\mathbb{R}^n},{\omega _2}} \right)$ to ${L^p}\left( {{\mathbb{R}^n},{\nu _{\vec \omega }}} \right)$ for appropriate indices p ₁, p ₂, p ∈ (1,∞) with $\frac{1}{p} = \frac{1}{{{p_1}}} + \frac{1}{{{p_2}}}$ and weights ω ₁,ω ₂ such that $\vec \omega = \left( {{\omega _1},{\omega _2}} \right) \in {A_{\vec p/\vec t}}\left( {{\mathbb{R}^{2n}}} \right)$ .