The Boundedness And Compactness Of Commutator Of Multilinear Fourier Multiplier Operator On Morrey Space

Multilinear Fourier multiplier operator was introduced by Coifman and Meyer in 1978.They obtained that if multiplier symbol σ satisfies Mihlin-Hormander condition, then the multilinear Fourier multiplier operator Tσ is a bounded operator from LP1(Rn)× ...×LPm(Rn)to Lp(Rn). In 2010,Tomita weakened the condition of multiplier symbol σ through a unity decomposition. Let σι(ξ1,...,ξm):=σ(2-ιξ1,...,2-ιξm)Φ(ξ1,...,ξm), where Φ is a unity decomposition, they proved that if multiplier symbol σ satisfies the Sobolev regularity condition sup‖σι‖Ws(Rmn)<∞,S∈(n,∞), then Tσ also is a bounded operator from Lp1(Rn)×…×Lpm(Rn) to Lp(Rn). Then in 2013, Miyachi and Tomita fur-ther improved this result and got that if the multiplier symbol δ satisfies a weaker Sobolev regularity condition sup‖δι‖Ws1,...,sm(Rmn)<∞,S1...,Sm ∈(n/2,n],then the mapping prop-erty of Tσ on Lp(Rn) also holds. Since then, many properties of the multilinear Fourier multiplier operator Tσ was studied by many scholars, including the weighted norm esti-mate for Tσ, the boundedness and compactness for its commutators Tσ,Σb. In this article, we prepare to study the boundedness and compactness of multilinear Fourier multiplier operator Tσ and its commutator Tσ,Σb on Morrey space.In first section, we introduce the developing background and current research situa-tion of multilinear Fourier multiplier operator Tσ and state the frame of the paper.In second section, under the Sobolev regularity condition sup‖σι‖Ws1,...,sm(Rmn)<∞ and condition b=(b1,..., bm) ∈ (BMO(Rn))m, we discuss the boundedness of multilinear Fourier multiplier operator Tσ and its multilinear commutator Tσ,Σb on weighted Morrey space Lp,k(vω). Be as specific as possible, via the sharp maximal theorem on weighted Morrey space, Cotlar inequalities for multilinear Fourier multiplier operator Tσ and its multilinear commutator Tσ,Σb and the boundedness of multilinear Hardy-Littlewood op-erator M on weighted Morrey space Lp,λ(vω), we obtain that Tσ and Tσ,Σb are bounded operators from Lp1,λ(ω1)×…×Lpm,λ(ωm) to Lp,λ(vω), where vω=Πj=1mωjp/pj.In the third section, under the Sobolev regularity condition sup‖σι‖Ws1,...,sm(Rmn)<∞ and condition b=(b1,...,bm)∈(CMO(Rn))m, we obtain the compactness of commu-tator of multilinear Fourier multiplier operator Tσ,Σb on Morrey space Lp,λ(Rn). In fact, via verifying the three conditions of pro-compactness on Morrey space:uniform norm boundedness, uniform translation continuity and uniform control away from the origin, we get that Tσ,Σb is a compact operator from Lp1,λ(Rn)×…×LPm,λ(Rn) to Lp,λ(Rn).In the last section, we pose two problems obout multilinear Fourier multiplier oper-ator and its commutator which can be considered further.