The Study On Boundedness For Maximal Multilinear Commutator Generated By Bochner-Riesz Operator

In this paper,we study the boundedness for the maximal multilinear commutator Bδ,*b generated by the Bochner-Riesz operator on some function spaces,those function spaces include Lebesgue space,Besov space,Morrey space,Weighted Lipschitz space.At first,we prove the Mk-type inequality for the maximal multilinear commutator Bδ,*b. By using the Mk-type inequality,we obtain The Bδ,*b is bounded on Lp(w)for 1<p<∞and w∈Ap.and we obtain the weighted boundedness on the generalized Morrey spaces for the maximal multilinear Commutator,where 1<p<∞,w∈Ap,bj∈BMO(Rn),j=1,…,m.Secondly,a goodλestimate for the maximal multilinear commutator Bδ,*b is obtained.Under this result,we get the Bδ,*b is bounded from Lp(Rn)to Lq(Rn) for 1<p<n/mβand 1/p-1/q=mβ/n,where 0<β<1 and bj∈∧β(Rn)for j=1,…,m.And we get Bδ,*b is bounded on Lp(Rn),J=1,…,m for 1<p<∞.Thirdly,we prove the the boundedness for the maximal multilinear operator on the Besov spaces whenδ>(n-1)/2,O<β<1/m and bj∈∧β(Rn) for j=1…,m Suppose that B*δis bounded on Lp(Rn)for any 1<p<∞.Then Bδ,*b is bounded from Lp(Rn) to∧mβ-n/p(Rn) for any n/(mβ+δ)≤p≤n/δ,and Bδ,*b is bounded from Kq1α,∞(Rn)to CL-α/n-1q2,q2(Rn)with the appropriate condition.Finally,let v∈A1(Rn) and bj∈Lipβ,v(Rn) for j=1,…m,1/q=1/p-mβ/n for 0<β<1,0<ε<1<s<n/β.Then there exists a constant C>0 such that for any smooth function f and a.e.x∈Rn,and where and we get the commutator Bδ,*b is bounded from Lp(v)to Lq(v1-q),where v∈A1(Rn), 1/q=1/p-β/n for 0<β<1 and 1<p<q<∞.