The Further Study Of Boundedness For Multilinear Commutators Of Singular Integral Operator With Variable Calderon-Zygmund Kernel

The article mainly study the boundedness for the multilinear commutator relat-ed to the singular integral operator with variable Calderon-Zygmund kernel and some locally integrable functions. In this paper, we research the boundedness of the mul-tilinear commutators Tb generated by the singular operator T and BMO functions or Lipschitz functions in Lp(1<p <∞) spaces、Besov space、Triebel-Lizorkin space、Morrey-Herz space. Moreover, we consider some kind of endpoint estimates for the multilinear commutator.At first, the weighted Lp-boundedness for the multilinear commutator Tr is proved. In this section, we prove a sharp function inequality firstly. By using it, we obtain Tb is bounded from Lp(ω) to Lq(ω) as well as from Lp,φ(ω) to Lp,φ(ω), where1<p <∞and ω∈Ap. Later, we prove the BMO estimate for the multilinear 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 related to the singular integral operator with variable Calderon-Zygmund kernel and the weighted Lipschitz functions. We obtain the boundedness of the multilinear commutator Tb from Lp(ω) to where0<β<1,ω∈A1and bj∈Lipβ(ω) for1≤j≤m,l <p <n/mβ and1/q=1/p-mβ/n. And Tb is bounded from Lp(ω) to where0<β<1,ω∈A1and bj∈Lipβ(ω) for1≤j≤m,1<p <∞.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-1/p(Rn) for any n/mβ≤p≤n, where0<β<1/m and bj∈∧β(Rn) for j=1,…,m. And for another, we prove Tb is bounded from Kqiα,∞(Rn) to CL-α/n-1/q2,q2(Rn), where0<β<1/2m,1<q1<n/mβ,1q2=1/q1-mβ/n,-n/q2-1/2<α≤-n/q2and bj∈∧β(Rn) for j=1,…,m.