The Study On Boundedness Of Multilinear Commutators Of Integral Operator With Parameter Kernel

In this paper,we mainly study the boundedness of the multilinear commutator Tδb generated by the integral operator Tδand the locally integrable functions.The integral operator include the Littlewood-Paley operator,Marcinkiewicz operator and Bochner-Riesz operator.First,the sharp function estimates for the multilinear integral commutators Tδb are established.By using the sharp inequalities,we obtain the boundedness of the commutator from Lp(Rn) to Lq(Rn),where b=(b1,…,bm),bj∈BEO(Rn), 1≤j≤m.Second,we proved the multilinear integral commutators Tδb is bounded from Hbp to Lq(Rn),H Kq1,bα,p to Kq2α,p(Rn) and H Kq1,bα,p to Kq2,α,p(Rn),where H Kq1,bα,p and Kq2α,p(Rn) are the nonhomogeneous Herz-Hardy and nonhomogeneous Herz spaces,b=(b1,…,bm),bj∈BMO(Rn),1≤j≤m.Furthermore,we proved the multilinear commutator Tδb,which is generated by the integral operator and the functions in Lipβ(Rn),is bounded on Triebel-Lizorkin space,Hardy space and Herz-Hardy space.That is Tδb is bounded from Lp(Rn)to Fqmβ,∞(Rn),Lp(Rn)to Lq(Rn),Hp(Rn)to Lq(Rn),H Kq1α,p(Rn) to Kq2α,p(Rn),H Kq1n(1-1/q1)+ε,p(Rn)to W Kq2n(1-1/q1)+ε,p(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 multilinear commutators Tδb generated by the integral operator and BMO functions are studied.That is Tδb is bounded from Ln/δto BMO(Rn),from Bpδ(Rn) to CMO(Rn),where 0<δ