The Study On Boundedness For Vector-Valued Multilinear Commutator Of Fractional Area Integral Operator

In this paper, we mainly study the boundedness of the vector-valued multi-linear commutator|Sψ,δb|r generated by the fractional area integral operator Sψ,δ and locally integrable functions b=(b1,…, bm), where0<δ<n,1<r<∞First, the sharp function estimate for the vector-valued multilinear commu-tator|Sψ,δb|r is established. By using the sharp estimate, By using the Holder and Minkowski inequalitiess. we obtain the boundedness of the commutator from Lp(Rn) to Lq(Rn), where b=(b1,…,bm), bj∈BMO(Rn),1≤j≤m.Second, we proved the vector-valued multilinear commutator|Sψ,δb|r, which is generated by the fractional area integral operator Sψ,δ and functions in Lipβ(Rn), is bounded on Triebel-Lizorkin and Lebegue spaces. That is|Sψ,δb|r is bounded from Lp(Rn) to Fqmβ,∞(Rn) and Lp(Rn) to Lq(Rn), when indexes of those spaces satisfy proper conditions, where b=(b1,…,bm), bj∈Lipβ(Rn),1≤j≤m.Finally, the endpoint estimates for the commutator|Sψ,δb|r generated by the fractional area integral operator Sψ,δ and BMO functions are studied. That is, When the parameters satisfy certain conditions,|Sψ,δb|r is bounded from Ln/δ to BMO(Rn) and Bpδ(Rn) to CMO(Rn), where b=(b1,…,bm) with bj∈BMO(Rn) for1≤j≤m.