The Study On Boundedness Of Vector-valued Multilinear Commutators Of Littlewood P Aley Operator

In this paper, we system study the boundedness of the vector-valued multilin-ear commutators generated by Littlewood-Paley operator and BMO functions or Lipschitz functions on Lp(1 < p <∞) space and Triebel - Lizorkin space. More-over, we consider some kind of weighted endpoint estimates for the vector-valued multilinear Littlewood - Paley commutators.At first, the sharp inequalities for the vector-valued multilinear Littlewood-Paley commutators |gψb|r are proved, i.e., when 1 < r <∞, bj∈BMO(Rn) (j=1,...,m), for each fixed f∈C0∞(Rn) and x∈Rn, then By using it, we obtain that |gψb|r are bounded on Lp space, where 1 < p <∞.Secondly, when 1 < r <∞, 0 <β< min(1,ε/m), 1 < p <∞, b= (b1,...,bm), where bj∈Lipβ(Rn), 1≤j≤m, we get |gψb|r is bounded from Lp(Rn) to Fpmβ,∞(Rn), Lp(Rn) to Lq(Rn), where 1/p- 1/q=mβ/n and 1/p>mβ/n, which is generated by Littlewood- Paley operator and the functions in Lipschitz space.Finally, the weighted endpoint estimates for the vector-valued multilinear Littlewood - Paley commutators |gψb|r are discussed. When 1 < r <∞, w∈A1, b = (b1,...,bm), where bj∈BMO(Rn), |gψb|r is bounded from L∞(w) to BMO(w). When 1 < r <∞, 1 < p <∞, w∈A1, b = (b1,...,bm), bj∈BMO(Rn), |gψb|r is bounded from Bp(w) to CMO(w).