Multilinear Littlewood-Paley Commutator

In this paper, we study the boundedness of multilinear Littlewood ? Paleycommutators S_δ~b generated by Littlewood-Paley operator S_δand locally integrablefunctions.At first, the sharp inequalities for multilinear Littlewood?Paley commutatorsSδb are proved. By using the sharp inequalities, we obtain S_δ~b is bounded on Lebesguespaces.After that, the boundedness for multilinear Littlewood?Paley commutatorsSδbon Hbp , H K˙qα,,bp and HKqα,,bp are proved, where b = (b1,···,bm), bi∈BMO(Rn),1≤i≤m.Then, the boundedness for the multilinear Littlewood?Paley commutator Sδbon Triebel - Lizorkin space, Hardy space and Herz - Hardy space are proved,which generated by Littlewood-Paley operator Sδand the functions in Lipβ, thatis S_δ~b is bounded from Lp(Rn) to F˙qmβ,∞(Rn), Lp(Rn) to Lq(Rn), Hp(Rn)to Lq(Rn)and H K˙qα1, p(Rn) to K˙qα2, p(Rn), where b = (b1,···,bm), bj∈Lipβ(Rn),1≤j≤m,indexes of those spaces satisfy proper conditions.Finally, the endpoint estimates for multilinear Littlewood ? Paley commuta-tors S_δ~b are studied. That is S_δ~b is bounded from Ln/δ(Rn) to BMO(Rn), furthermore, if...