The Boundedness Of Multilinear Commutators Of Marcinkiewicz Operator

In this paper, we study the boundedness of multilinear Marcinkiewicz commutators generated by Marcinkiewicz operator and locally integrable functions,i.e., we study system the boundedness of multilinear commutators generated by Marcinkiewicz operator and BMO functions or Lipschitz functions on L^p(1＜p＜∞) space,Hardy space,Herz—Hardy space,Triebel—Lizorkin space. Moreover, we consider some kind of endpoint estimates for multilinear Marcinkiewicz commutators.At first, the sharp inequalities for multilinear Marcinkiewicz commutators are proved. By using it, we obtain are bounded on L^p space,where 1＜p＜∞.After that, the boundedness for multilinear Marcinkiewicz commutators on (Rⁿ), (Rⁿ) and (Rⁿ) are proved, where b)i∈BMO(Rⁿ),1≤i≤m,=(b₁,…,b_m).Then, we get they are bounded from L^p(Rⁿ) to(Rⁿ),L^p(Rⁿ) to L^q(Rⁿ), where 1/p-1/q=mβ/n and 1/p＞mβ/n, H^p(Rⁿ) to L^q(Rⁿ), (Rⁿ) to, (Rⁿ) to (Rⁿ), which generated by Marcinkiewicz operator and functions in Lipschitz space.Finally, the weighted endpoint estimates for multilinear Marcinkiewicz commutators are studied. They are bounded from L^∞to BMO(w). Moreover, if w(Q)≥1 for any cube Q and t＞max{p, s},λ≤0,then are bounded from B^t,λ(w) to CMO^s,λ(w), where w∈A_p, (p＞1).