The Boundedness Of Multilinear Littlewood-Paley Commutator

In this paper, we study the boundedness of multilinear Littlewood ? Paleycommutators gψb generated by Littlewood ? Paley operator gψand locally inte-grable functions.At first, the sharp inequalities for multilinear Littlewood?Paley commuta-tors gψb are proved. By using the sharp inequalities, we obtain gψb is bounded onLp(ω), whereω∈A1, 1 < p <∞.After that, the boundedness for multilinear Littlewood?Paley commutatorsgψbon Hbp , H K˙qα,,bp and HKqα,,bp are proved, where b = (b1,···,bm), bi∈BMO,1≤i≤m.Then, the boundedness for the multilinear Littlewood ? Paley commutatorgψb on Triebel?Lizorkin space, Hardy space and Herz?Hardy space are proved,which generated by Littlewood ? Paley operator and functions in Lipβ, that isgψb is bounded from Lp(Rn) to F˙pmβ,∞(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 weighted endpoint estimates for multilinear Littlewood ? Paleycommutatorsgψb are studied. That is gψb is bounded from L∞(ω) to BMO(ω) andgψb is bounded from H1(ω) to weak L1(ω), further more, iffor all cube Q, then gψb is bounded from H1(ω) to L1(ω). Also we get that gψb isbounded from Bp(ω) to CMO(ω), whereω(x)∈A1.