The Study On Boundedness Of Multilinear Commutators Of Multiplier Operator

In this paper, we mainly study the boundedness of the multilinear commuta-tors Tmb generated by the multiplier operator Tm and the locally integrable func-tions b=(b1,…,bm).Firstly, we prove the Mk inequality for multilinear commutator generated by multiplier operator and BMO functions. By using the Mk inequality, we obtain this multilinear commutator is (p, p) bounded on the Morrey space.Secondly, by using the Mmβ,r inequality, we prove that the multilinear com-mutator generated by multiplier operator and Lipschitz functions is bounded from Lp(w) to Lq(w1-m+(q-1)mβ/n) and Lp(w) to Famβ,∞(w1-m-mβ/n), where Fpmβ,∞(w1-m-mβ/n) is the weighted homogeneous Triebel-Lizorkin space.Furthermore, we discuss the boundedness of the multilinear commutator gen-erated by multiplier operator and Besov functions from Lebesgue space to Besov space and from homogeneous Herz space to central Campanato space. That is, this multilinear commutator is bounded from LP(Rn) to Λmβ-n/P(Rn), Kq1α,∞(Rn) to CL-α/n-1/q2,q2(Rn), when indexes of those spaces satisfy proper conditions.Finally, by proving the multiplier operator is bounded to itself on central Morrey space, we obtain the (Bp,λ(Rn), Bq,λ(Rn)) boundedness of the multilinear commutator generated by multiplier operator and CBMO functions, when indexes of those spaces satisfy proper conditions. Here, Bp,λ(Rn) is the central Morrey-space.