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Acta Mathematica Vietnamica

Compactness for the Commutator of the Multilinear Fourier Multiplier on the Morrey Space

icon-email Peng Li , Jiang Zhou

Abstract

Given s1,,sm(n/2,n], let Tσ be a multilinear Fourier multiplier operator associated with a multilinear multiplier σ satisfying a Sobolev regularity condition supZσWs1,,sm(Rmn)<. By the strongly precompactness of Banach space, the authors prove that if b1,,bmCMO(Rn), then the commutator Tσ,Σb is a compact operator from the product Morrey space Lp1,λ(Rn)××Lpm,λ(Rn) to the Morrey space Lp,λ(Rn). As an application, the compactness of the commutator Tσ,Σb from the product Morrey space Lp1,λ(Rn)××Lpm,λ(Rn) to the Morrey space Lp,λ(Rn) is also obtained under the Sobolev regularity condition supZσWs(Rmn)< for s(mn/2,mn].