双线性Fourier乘子交换子的有界性与紧性

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.

Boundedness and Compactness for Commutators of Bilinear Fourier Multipliers

Let T_σ be the bilinear Fourier multiplier operator associated with multiplier σ satisfying the Sobolev regularity that sup_k∈Z ||σk||W^s (R²ⁿ)<∞ for some s∈(n,2n].We give the boundedness of T_σ and the commutators T_σ,b generated by T_σ and b=(b₁,b₂)∈(BMO (Rⁿ))²,as well as the compactness of T_σ,b (if b₁,b₂∈CMO (Rⁿ),the BMO-closure of C_c^∞ (Rⁿ)) from L^p1,λ(ω₁)×L^p2,λ(ω₂) to L^p,λ(ν_w) for appropriate indices p₁,p₂,p∈(1,∞)(1/p=1/p₁+1/p₂) and multiple weights ω=(ω₁,ω₂)∈A_p/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.