The Boundedness Of Multilinear Littlewood-Paley Commutator

In this paper, we mainly study the boundedness of multilinear commutators S_ψ^? generated by Littlewood- Paley operator S_ψand locally integrable functions.First, the sharp estimates for multilinear Littlewood- Paley commutators S_ψ^? are es- tablished. By using the sharp inequalities, we obtain the L^p(w)(1＜p＜∞) boundedness and Llog L estimates of S_ψ^?, where w∈A₁.Second, we proved the multilinear Littlewood - Paley commutators S_ψ^? is bounded form H_?^p to L^p(Rⁿ), HK_q,(?)^α,p to K_q^α,p(Rⁿ) and HK_q,(?)^α,p to K_q^α,p(Rⁿ), where HK_q,(?)^α,p,K_q^α,p(Rⁿ) is nonhomogeneous Herz-Hardy space and nonhomogeneous Herz space, (?) = (b₁,…, b_m), b_i∈BMO(Rⁿ), 1≤i≤m.Furthermore, we proved the multilinear commutator S_ψ^?, which is generated by Littlewood- Paley operator and functions in Lip_β, is bounded on Triebel - Lizorkin space, Hardy space and Herz - Hardy space. That is S_ψ^? is bounded from L^P(Rⁿ) to F_p^mβ,∞(Rⁿ),L^p(Rⁿ) to L^p(Rⁿ) H^P(Rⁿ)to L^q(Rⁿ) and HK_q1^α,p(Rⁿ) to K_q2^α,p(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 S_ψ^? generated by Littlewood - Paley operator and BMO functions are studied. That is for w∈A₁, S_ψ^? is bounded from L^∞(w) to BMO(w) and B_p(w) to CMO(w). Also s_ψ^? is bounded from H¹(w) to weak L¹(w), further more, S_ψ^? is bounded from H¹(w) to L¹(w) on suitable condition.