双线性Fourier乘子交换子的有界性与紧性

Boundedness and Compactness for Commutators of Bilinear Fourier Multipliers

假定T_σ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sup_(κ∈Z)‖σ_k‖W^s(R^(2m))<∞.对于p_1,p_2,p∈(1,∞)且满足1/p=1/p_1+1/p_2和ω=(ω_1,ω_2)∈A_(p/t)(R^(2n)),建立了T_σ及其与函数b=(b_1,b_2)∈(BMO(R^n))~2生成的交换子T_(σ,b)由L^(p_1,λ)(ω_1)×L^(p_2,λ)(ω_2)到L^(p,λ)(v_w)的有界性;同时,在b_1,b_2∈CMO(R^n)(C_c~∞(R^n)在BMO拓扑下的闭包)的条件下,证明交换子T_(σ,b)是L^(p_1,λ)(ω_1)×L^(p_2,λ)(ω_2)到L^(p,λ)(v_w)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.

Let Ta be the bilinear Fourier multiplier operator associated with multiplier a satisfying the Sobolev regularity that supk∈z｜｜σk｜｜wa（R2n）〈∞ for some s∈（n,2n]. We give the boundedness of Tσ and the commutators Tσ,b generated by Tσ and b = （bl, b2）∈ （BMO（Rn）}2, as well as the compactness of, the BMO-closure of Cc （Rn）） from to Lp for appropriate indices Pl, P2, P c （1,0e） （1/p = 1/pl ＋ l/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 pre- compact set, which are in themselves interesting.