The Research On Multilinear Commutators Of Singular Integral Operators With A Variant Of H(?)rmander's Kernel

The singular integral theory has important applications in the study of differential equation,complex analysis and operator theory.In this paper,we focus on study about the boundedness of the multilinear commutators Tb generated by the singular integral operators T with a variant of H(?)rmander's condition and the locally integrable function b.First of all,in this thesis we obtain the sharp maximal function estimate for the multilinear commutators related to the singular integral operators satisfying a variant of H(?)rmander's condiction.Further we obtain Tb is bounded in LP(Rn)(1<p<?).At the same time it is bounded in Lp,?(Rn)(1<p<?),where b=(b1,...,bm),bj ? BMO(Rn),1 ? j?m.Secondly,in the thesis we study the weighted endpoint estimates for the mul-tilinear commutators related to the singular integral operators satisfying a variant of H(?)rmander's condition.If ??A1,then Tb is bounded from L?(w)to BMOA(w).And Tb is also bounded from Bp(w)to CMO(w),where b=(b1,...,bm)bj?BMO(Rn),1?j?m.Furthermore,in the thesis the weighted boundedness for the multilinear com-mutators of singular integral operators satisfying a variant of H(?)rmander's condic-tion associated to the weighted BMO and Lipschitz functions are obtained.One case,if ??A1,bj ? Lip?(w)(1? j?m),and indexes of those spaces satisfy poper conditions,then Tb is bounded from LP(w)to Lq(w1-m+(q-1)m?/n).Case two,if ??A1,bj? BMO(?)(1?j ?m),then Tb is bounded from Lp(?)to Lp(?1-m).Finally,in the thesis,the M?-type inequality for the multilinear commutators related to singular integral operators satisfying a variant of H(?)rmander's condition is obtained.By using the M?-type inequality,we obtain Tb is bounded in Lp(?)(1<p<?).At the same time it is bounded in Lp,?(?)(1<p<?),where b =(b1,...,bm),bj ? BMO(Rn),1 ? j?m.