双线性Fourier乘子交换子的有界性与紧性

毛素珍, 孙丽静, 伍火熊

数学学报 ›› 2016, Vol. 59 ›› Issue (3) : 317-334.

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数学学报 ›› 2016, Vol. 59 ›› Issue (3) : 317-334. DOI: 10.12386/A2016sxxb0030
论文

双线性Fourier乘子交换子的有界性与紧性

    毛素珍1, 孙丽静2, 伍火熊1
作者信息 +

Boundedness and Compactness for Commutators of Bilinear Fourier Multipliers

    Su Zhen MAO1, Li Jing SUN2, Huo Xiong WU1
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文章历史 +

摘要

假定Tσ是关于乘子σ的双线性Fourier乘子算子,其中σ满足如下Sobolev正则条件:对某个s∈(n,2n],有supk∈Z||σk||Ws (R2n).对于p1,p2,p∈(1,∞)且满足1/p=1/p1+1/p2ω=(ω1,ω2)∈Ap/t(R2n),建立了Tσ及其与函数b=(b1,b2)∈(BMO (Rn))2生成的交换子Tσ,bLp1,λ (ω1Lp2,λ(ω2)到Lp,λ(νw)的有界性;同时,在b1,b2∈CMO (Rn)(Cc(Rn)在BMO拓扑下的闭包)的条件下,证明交换子Tσ,bLp1,λ(ω1Lp2,λ(ω2)到Lp,λ(νw)的紧算子.为了得到主要结果,我们先后建立了几个双(次)线性极大函数在加多权Morrey空间上的有界性以及该空间中准紧集的判定.

Abstract

Let Tσ be the bilinear Fourier multiplier operator associated with multiplier σ satisfying the Sobolev regularity that supk∈Z ||σk||Ws (R2n)<∞ for some s∈(n,2n].We give the boundedness of Tσ and the commutators Tσ,b generated by Tσ and b=(b1,b2)∈(BMO (Rn))2,as well as the compactness of Tσ,b (if b1,b2∈CMO (Rn),the BMO-closure of Cc (Rn)) from Lp1,λ(ω1Lp2,λ(ω2) to Lp,λ(νw) for appropriate indices p1,p2,p∈(1,∞)(1/p=1/p1+1/p2) and multiple weights ω=(ω1,ω2)∈Ap/t (R2n).The main ingredient is to establish the multiple weighted estimates for the variants of certain multi (sub) linear maximal operators on the weighted Morrey spaces,and a sufficient condition for a subset in the weighted Morrey spaces to be a strongly precompact set,which are in themselves interesting.

关键词

双线性Fourier乘子 / 交换子 / 双(次)线性极大算子 / 紧性 / 加权Morrey空间

Key words

bilinear Fourier multipliers / commutators / bi (sub) linear maximal operators / compactness / weighted Morrey spaces

引用本文

导出引用
毛素珍, 孙丽静, 伍火熊. 双线性Fourier乘子交换子的有界性与紧性. 数学学报, 2016, 59(3): 317-334 https://doi.org/10.12386/A2016sxxb0030
Su Zhen MAO, Li Jing SUN, Huo Xiong WU. Boundedness and Compactness for Commutators of Bilinear Fourier Multipliers. Acta Mathematica Sinica, Chinese Series, 2016, 59(3): 317-334 https://doi.org/10.12386/A2016sxxb0030

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基金

国家自然科学基金资助项目(11371295, 11471041);福建省自然科学基金项目(2015J01025)

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