The Boundedness For Multilinear Commutators Of Integral Operator

The article mainly st udy the boundedness for the multiltilinear commutator related to some integral operator and some locally integrable functions. In this paper, we research the boundedness of the multilinear commutators Tb generated by integral operator T and BMO functions or Lipschitz functions in Lp(1<p<∞)spaces.Besov space.Triebel-Lizorkin space、Morrey-Herz space.At first,the weighted Lp-boundedness for the multilinear commutator Tb is proved. In this section,we prove a sharp function inequality firstly.By using it,we obtain Tb is bounded from Lp(w)to Lq(w) as well as from Lp,φ(w)to Lp,φ(w),where1<p<∞and w∈Ap.Later,we prove the BMO estimate for the muItilinear commutator Tb. Respectively,we give the conclusions of λ-central BMO estimates for Tb on central Morrey spaces and CBMO estimates for the multilinear commutator on Herz and Morrey-Herz spaces.Then,we get the weighted estimates for the multilinear commutator Tb relat-ed to the integral operator and the weighted Lipschitz functions. We obtain the boundedness of the multilinear commutator Tb from Lp(w)to Lq(w1-m+(q-1)mβ/n) and from Lp(w) to Fpmβ,∞(w1-m-mβ/n).In the discussion of boundedness from Lp(w) to Lq(w1-m+(q-1)mβ/n),the corresponding indicators should satisfies0<β<1,w∈A1and1<p<n/mβ,1/q=1/p-mβ/n,and in the situation from Lp(w)to Fpmβ,∞(w1-m-mβ/n),they should satisfy1<p<n/m、0<β<1、w∈A1.Finally, we prove the the boundedness for the multilinear commutator Tb on the Besov spaces. In this chapter,we also give the related proof according two aspects. For one thing,we obtain that Tb is bounded from Lp(Rn) to∧mβ-n/p(Rn),where0<β<1/m,n/mβ≤p<∞and bj∈λβ(Rn):And for anther, we prove Tb is bounded Kq1α，∞(Rn) to CL-α/n-1/q2,q2(Rn),where0<β<1/m,1<q1<n/mβ,1/q2=1/q1-mβ/n,-n/q2-1<α≤-n/q2,,and bj∈λβ(Rn),j=1,…,m。...