The Boundedness For Multilinear Commutator Of Multiplier Operator

In this paper, we mainly study the boundedness of multilinear commutators T^? generated by multiplier operator T and locally integrable functions.First, the sharp function estimates for multilinear commutators T^? of multiplier operator are established. By using the sharp inequalities, we obtain the L^p(w)(1?,where (?)=(b₁,...,b_m),b_i∈BMO(Rⁿ),w∈A₁.Second, we prove the multilinear commutators of multiplier operator T^? was bounded form H_?^p(Rⁿ) to L^p(Rⁿ), and weak bounded from H¹(Rⁿ) to L¹(Rⁿ), where(?)=(b₁,...,b_m),b_i∈BMO(Rⁿ),1≤i≤m.Furthermore, we proved the multilinear commutator of multiplier operator T^?, which is generated by multiplier operator and functions in Lipschitz space, is bounded on Lipschitz space, Hardy space and is weak bounded in Hardy space . That is T^? is bounded from L^p(Rⁿ) to F_p^mβ,∞(Rⁿ),L^p(Rⁿ) to L^r(Rⁿ), H_?^p(Rⁿ) to L^q(Rⁿ) and is weak bounded from H^n/(n+mβ)(Rⁿ) to L¹(Rⁿ), when indexes of those spaces satisfy proper conditions, where (?)=(b₁,..., b_m), b_j∈Lip_β(Rⁿ), 1≤j≤m.Finally, the weighted endpoint estimates for multilinear commutators of multiplier operator T^? generated by multiplier operator and BMO functions are studied. That is for w∈A₁, T^? is bounded from L^∞(w) to BMO(w), and T^? is bounded from H¹(w) to L¹(w) and from B_p(w) to CMO(w) on suitable condition.