| In this paper, we mainly study the boundedness of multilinear commutators gμ,δ(?) generated by Littlewood-Paley operator gμ,δ and locally integrable functions.First, the sharp estimates for multilinear Littlewood-Paley commutators gμ,δ(?) are established. By using the sharp inequalities, we obtain the boundedness of the commutator from Lp(Rn) to Lq(Rn),where (?)= (b1,…,bm),bj∈BMO(Rn),1≤j≤m.Second, we proved the multilinear Littlewood-Paley commutators gμ,δ(?) is bound ed from H(?)p to Lq(Rn),HKq1,(?)α,p to Kq2α,p(Rn) and HKq1,(?)α,p to Kq2α,p(Rn),where HKq1,(?)α,p Kq2α,p(Rn) is nonhomogeneous Herz-Hardy space and nonhomogeneous Herz space, (?) = (b1,…,bm),bj∈BMO(Rn),1≤j≤m.Furthermore, we proved the multilinear commutator gμ,δ(?),which is generated by Littlewood-Paley operator and functions in Lipβ(Rn),is bounded on Triebel-Lizorkin space,Hardy space and Herz-Hardy space. That is gμ,δ(?) is bounded from Lp(Rn) to Fqmβ,∞(Rn),Lp(Rn) to Lq(Rn),Hp(Rn) to Lq(Rn),HKq1,(?)α,p to Kq2α,p(Rn),HKq1n(1-1/q1)+ε,p(Rn) to WKq2n(1-1/q1)+ε,p(Rn),when indexes of those spaces satisfy proper conditions, where (?)=(b1,…,bm),bj∈Lipβ(Rn),1≤j≤m.Finally, the endpoint estimates for multilinear commutators gμ,δ(?) generated by Littlewood-Paley operator and BMO functions are studied. They are bounded from Ln/δ to BMO(Rn).Moreover,Iet 0 <δ< n,1 < p < n/δand (?) = (b1,…,bm) with bj∈BMO(Rn) for 1≤j≤m.Then gμ,δ(?) is bounded from Bpδ(Rn) to CMO(Rn).Last, Let 0 <δ< n and (?) = (b1,…,bm) with bj∈BMO(Rn) for 1≤j≤m.If for any H1(Rn)-atom a supported on certain cube Q and u∈Q,then gμ,δ(?) is bounded from H1(Rn) to Ln/(n-δ)(Rn).
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