The Boundedness Of Multilinear Marcinkiewicz Commutator

In this paper, we mainly study the boundedness of multilinear Marcinkiewicz com-mutatorsμ_S^? generated by Marcinkiewicz operatorμ_S and locally integrable functions.The content of this paper is mainly divided into four parts.In the first part of this paper,we briefly introduce the background of the paper andtheir significance in theory and practice, we also introduce some signs, definitions andlemmas.In the second part, the sharp inequalities for multilinear Marcinkiewicz commuta-torsμ_S^? are proved. By using the sharp inequalities, we obtainμ_S^? is bounded on LP(w),where w∈A₁, 1＜p＜∞.In the third part, the boundedness for multilinear Marcinkiewicz commutatorsμ_S^?(Rⁿ)on H_?^p(Rⁿ), H(?)_q,(?)^α,p(Rⁿ) and HK_q,(?)^α,p(Rⁿ) are proved, where (?)=(b₁,…,b_m),b_i∈BMO(Rⁿ), 1≤i≤m.In the fourth part of this paper, the weighted endpoint estimates for multilinearMarcinkiewicz commutatorsμ_S^? are studied. That isμ_S^? is bounded from L^∞(w) toBMO(w), furthermore,μ_S^? is bounded from H¹(w) to L¹(w) on sutiable condition. Alsowe get thatμ_S^? is bounded from B_p(w) to CMO(w), where w∈A₁.