Research On Multilinear Commutators Of Marcinkiewicz Integral Operator

In this paper,we study the boundedness of multililinear commutator μΩ,δb or generated by Marcinkiewicz integral operator μΩ,δ and locally integrable functions. We study system the boundedness of multilinear commutators μΩ,δb(0<δ<n)generated by Marcinkiewicz operator μΩ,δ and BMO functions or weighted Lipschitz functions on Lebegue space、Besov space、Triebel-Lizorkin space、Morrey-Herz space、CMO space、CBMO space.Moreover,we consiider some kind of endpoint estimates for the multilimear commutator.At first,the Sharp function inequality for the multtilinear commutator μΩ,δb is proved.by using it,we obtain μΩ,δb is bounded on generalized Morrey spaces,by using the(Lp,Lq)-boundedness for the multiliilear commutator μΩ,δb where1<p<q<n/δ,1/q=1/p-δ/n；the Good-λ inequality for the multilinear commutator μΩ,δb is proved.After that,we get that μΩ,δb is bounded from Lp(w)to Lq(w1-q(m/n)),where w∈A1,0<δ<n,1<p<n/(δ+mβ),1/p-1／q=(δ+mβ)/n；Lp(w)to Fqmγ,∞(w1-qm(1-(δ-β)/n),where w∈A1,0<δ<n,1<p<n/mδ,1/p-1/q mδ/n,where μΩ,δb is generated by the Marcinkiewivz integral operator and functions in the weighted Lipschitz space.Then,we prove the the boundedness for the multilinear commutator μΩ,δb on the Besov spaces. Then μΩ,δb is bounded from Lp(Rn)to (?)(δ+mβ)n/p(Rn),and μΩ,δb is bounded from Kq1α,∞(Rn)to CL-α/n-1/q2,q2(Rn),when indaxes of those spaces satisfy proper conditions,where bi∈(?)β(Rn),1≤i≤m,b=(b1,…,bm).Finally,the BMO estimates for the multilinear commutator μΩ,δb are proved.In this chapter,we contain two parts.For one thing,we establish λ-central BMO estimate for μΩ,δb on central Morrey spaces.For another thing,we prove CBMO estimates for the multilinear commutator on Herz and Morrey-Herz spaces.