Local Boundedness Of Singular Integral Commutators

In this paper,we mainly study the boundedness of commutator plus local weight of singular integral operator and boundedness of local weight under the nondoubling measure.Firstly,the classical singular integral theory and the theory of commutator of singular integral operator are introduced in this paper.Then,the definition of local BMO space and the definition of the corresponding singular integral commutator are introduced.The local weak boundedness of the singular singular integral commutator is proved by means of the Young function and the local maximal operator and the Vitali covering lemma.And then it will be extend to the nondoubling measure,by using the local Ap right under nondou-bling measure and Cauchy theorem,the singular integral commutator is exchanged into a differentiable function.Lastly,Minkowski inequality could prove the local boundedness of singular integral commutator under nondoubling measures.This paper proves the local boundedness of the singular integral commuta-tor,enriches the theory of the boundedness of the singular integral commutator,and provides a direction for the future study of other commutators and their local boundedness in other Spaces.