COMPACTNESS OF THE COMMUTATOR OF BILINEAR FOURIER MULTIPLIER OPERATOR

Abstract

Let $b_1, b_2 \in {\rm CMO}(\mathbb{R}^n)$ and $T_{\sigma}$ be the bilinear Fourier multiplier operator with associated multiplier $\sigma$ satisfies the Sobolev regularity that $\sup_{\kappa\in \mathbb{Z}}\|\sigma_{\kappa}\|_{W^{s_1,\,s_2}(\mathbb{R}^{2n})}\lt \infty$ for some $s_1,\,s_2\in (n/2,\,n]$. In this paper, it is proved that the commutator defined by $$T_{\sigma,\,\vec{b}}(f_1,\,f_2)(x) = b_1(x) T_{\sigma}(f_1,\,f_2)(x) - T_{\sigma}(b_1 f_1,\,f_2)(x) + b_2(x) T_{\sigma}(f_1,\,f_2)(x) - T_{\sigma}(f_1,\,b_2 f_2)(x)$$ is a compact operator from $L^{p_1}(\mathbb{R}^n) \times L^{p_2}(\mathbb{R}^n)$ to $L^p(\mathbb{R}^n)$ when $p_k \in (n/s_k,\,\infty)$ $(k=1,\,2)$, $p \in (1,\,\infty)$ with $1/p = 1/p_1 + 1/p_2$.