Estimates For Commutators Of Some Operators

In this thesis, the author studies the boundedness of a certain kind of Caldero′n -Zygmund signal commutator and Marcinkiewicz order commutator , their correspondingendpoint estimates are also discussed.In chapter 1, we introduce the background of knowledge involved in this paper andsome necessary concepts and the definition of spaces.In chapter 2, we consider the boundedness of the generalized Caldero′n-Zygmund sig-nal commutators with the Lipschitz functions on Lebesgue spaces and Hardy spaces,andthe properties of the commutators in endpoint cases are also discussed.In chapter 3, we consider the boundedness of the order commutatorμ-m,b with BMOfunctions on weighted Herz-type Hardy spaces and the properties of the commutators inendpoint cases.where,ω∈A1 .The definition of Calder′on-Zygmund singular integral operator T is as follows:Let b be a locally integrable function on Rn, then the commutator [T,b] generated byb and T is defined by[T,b]f(x) = b(x)Tf(x) - T(bf)(x),for suitable functions f.The Marcinkiewicz integral commutator is defined byWhereRemark: when m = 1,μ-m,b =μ-,b.