Weighted Compact Commutator of Bilinear FourierMultiplier Operator

Citation:

Guoen HU.Weighted Compact Commutator of Bilinear FourierMultiplier Operator[J].Chinese Annals of Mathematics B,2017,38(3):795~814
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Authors:

Guoen HU;

Foundation:

This work was supported by the National Natural Science Foundation of China (No.11371370).
Abstract: Let $T_{\sigma}$ be the bilinear Fourier multiplier operator with associated multiplier $\sigma$ satisfying the Sobolev regularity that $\sup\limits_{\kappa\in \mathbb{Z}}\|\sigma_{\kappa}\|_{W^{s}(\mathbb{R}^{2n})}<\infty$ for some $s\in (n, 2n]$. In this paper, it is proved that the commutator generated by $T_{\sigma}$ and ${\rm CMO}(\mathbb{R}^n)$ functions is a compact operator from $L^{p_1}(\mathbb{R}^n, w_1)\times L^{p_2}(\mathbb{R}^n, w_2)$ to $L^p(\mathbb{R}^n, \nu_{\vec{w}})$ for appropriate indices $p_1, p_2, p\in (1, \infty)$ with $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$ and weights $w_1, w_2$ such that $\vec{w}=(w_1, w_2)\in A_{\vec{p}/\vec{t}}(\mathbb{R}^{2n})$.

Keywords:

Bilinear Fourier multiplier, Commutator, Bi(sub)linear maximal operator, Compact operator

Classification:

42B15, 47B07, 42B25
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