The Study Of Boundedness For Multilinear Commutators Of Singular Integral Operator On Spaces Of Homogeneous Type

The article mainly study the boundedness fbr the multilinear commutator related to the singular integral Operator and some locally integrable functions On the spaces of homogeneous type. In this paper,we research the boundedness of the multilinear commutators Tb�'generated by the singular operator T and BMO functions or Lipschitz functions in Lp(1<p<∞)spaces.Besou space、Triebel—Lizorkin space、Morrey-Herz space on the spaces of homogeneous type X.At first,the weighted Lp-boundedness for the multilinear commuator Tb�'is proved. In this section,we prove a sharp function inequality fistly By using it,we obtainTb�'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 multilinear commuator 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 estiates for the multilinear commutator Tb�'related to the singular 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) and from Lp(w) to Fpmβ,∞(w1-m-mβ). In the discussion of boundedness from Lp(w) to Lq(w1-m+(q-1)mβ),the corresponding indicators should satisfies0<β<1and w∈A1,and in the situation from Lp(w) to Fpmβ,∞(w1-m-mβ),they should satisfy0<β<1.w∈A1and w-1∈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(X) to(?)mβ-1/p(X),where0<β<1/m,1<p<∞and bj∈(?)β(X) for j=1,…,m；And for another,we prove Tb�'is bounded Kq1α,∞(X) to CL-α-1/q2,q2(X),where0<β<1/m,1<q1<∞,1/q2=1/q1-mβ,-1/q2-1<α≤-1/q2and bj∈(?)β(X) for j=1,…,m.