The Boundedness Of Multilinear Littlewood-Paley Commutator

In this paper, we mainly study the boundedness of multilinear commutators g_μ^bgenerated by Littlewood - Paley operator g_μand locally integrable functions.First, the sharp estimates for multilinear Littlewood - Paley commutators g_μ^bare established. By using the sharp inequalities, we obtain the L^p(w)(1 < p <∞)boundedness and LlogL estimates of g_μ^b, where w∈A₁.Second, we proved the multilinear Littlewood-Paley commutators g_μ^b is boundedfrom Hbp to L^p(Rⁿ), H K_q,b^α,p to K_q^α,p(Rⁿ) and HK_q,b^α,p to K_q^α,p(Rⁿ), where HK_q,b^α,p ,K_q^α,p(Rⁿ) is nonhomogeneous Herz - Hardy space and nonhomogeneous Herz space, b =(b₁,···,b_m), b_i∈BMO, 1≤i≤m.Furthermore, we proved the multilinear commutator g_μ^b, which is generated byLittlewood-Paley operator and functions in Lip_β, is bounded on Triebel -Lizorkinspace, Hardy space and Herz - Hardy space. That is g_μ^b is bounded from L^p(Rⁿ)to F_p^mβ,∞(Rⁿ), L^p(Rⁿ) to L^q(Rⁿ), H^p(Rⁿ)to L^q(Rⁿ) and H K_q1^{α, p}(Rⁿ) to K_q2^{α, p}(Rⁿ)when indexes of those spaces satisfy proper conditions, where b = (b₁,···,b_m), b_j∈Lip_β(Rⁿ), 1≤j≤m.Finally, the weighted endpoint estimates for multilinear commutators g_μ^b gener-ated by Littlewood - Paley operator and BMO functions are studied. That is forw∈A₁, g_μ^b is bounded from L^∞(w) to BMO(w) and B_p(w) to CMO(w). Also g_μ^b isbounded from H¹(w) to weak L¹(w), furthermore, g_μ^b is bounded from H1(w) to L¹(w)on suitable condition.