Boundedness Of Multilinear Commutators Of Littlewood-Paley Operator

In this paper, we study the boundedness of multilinear Littlewood - Paley commutators generated by Littlewood - Paley operator and locally integrable functions. We study system the boundedness of multilinear commutators g_ψ,δ^?(0 <δ< n), generated by Littlewood-Paley operator g_ψ,δ and BMO functions or Lipschitz functions on L^p(1ψ,δ^? are proved. By using it, we obtain g_ψ,δ^? are bounded on L^p space,where b_i∈BMO(Rⁿ),1≤i≤m,(?) = (b₁,…,b_m),1ψ,δ^?on H(?)(Rⁿ),HK₍q,(?)^α,p(Rⁿ) and HK_q,(?)^α,p(Rⁿ) are proved, where b_i∈BMO(Rⁿ),1≤i≤m,(?)= (b₁,…,b_m).Then, we get they are bounded from L^p(Rⁿ) to F_q^mβ,∞(Rⁿ),LP(Rⁿ) to L^q(Rⁿ), where 1/p - 1/q = mβ+δ/n and 1/p > mβ+δ/n,H^p(Rⁿ) to L^q(Rⁿ),HK_q1^α,p(Rⁿ) to K_q2^α,p,HK_q1^{n(1-1/q₁)+ε,p}(Rⁿ) to WK_q2^{n(1-1/q₁)+ε,p}(Rⁿ),which generated by Littlewood-Paley operator and functions in Lipschitz space.Finally, the endpoint estimates for multilinear Littlewood-Paley commutators g_ψ,δ^? are studied. They are bounded from L^n/δ to BMO(Rⁿ).Moreover,let 1 < p < n/δand (?) = (b₁,…,b_m) with b_j∈BMO(Rⁿ) for 1≤j≤m.Then g_ψ,δ^? is bounded from B_p^δ(Rⁿ) to CMO(Rⁿ).Last,let (?)=(b₁,…,b_m) with b_j∈BMO(Rⁿ) for 1≤j≤m. If for any H¹(Rⁿ)-atom a supported on certain cube Q and u∈Q,there isthen g_ψ,δ^? is bounded from H¹(Rⁿ) to L^n/(n-δ)(Rⁿ).