Endpoint Estimates For Two Classes Of The Commutators

In this dissertation, we focus on the boundedness or compactness for commutator of cer-tain pseudo-differential operators and parametrized Marcinkiewicz singular integral operators. It comprises four chapters.In chapter1, we give a survey on the background and recent development of general sin-gular integral operators and their commutator. In particular, we gather some notations about pseudo-differential operators, Marcinkiewicz singular integral operators and their commuta-tors. Finally, we simply list our work.In chapter2, we discuss endpoint estimates for the commutator of certain pseudo-differential operators. Let σ(x,ξ)∈S01δ(0≤δ<1),b∈BMO(b∈BMO∞), then the commutator [b, Tσ] is bounded from H1(Rn) into L1(Rn) if and only if b∈LMO(b∈LMO∞).In chapter3, we show two sufficient condition which the commutator [b,Tσ] is com-pacted operators. If b∈CLMO or b∈CLMO∞, then the commutator [b,Tσ] is compacted from H1(Rn) into L1(Rn) for any σ{x,ξ)∈01δ^(0≤δ<1).In chapter4, we deal with the endpoint estimates for the commutator of parametrized Marcinkiewicz singular integral operators. On non-double measure space, if b∈Lipβ(μ)(0<β≤1) and Marcinkiewicz integral kernel satisfy some condition, then the parametrized Marcinkiewicz commutators M(?)b is bounded on Lebesgue spaces L/?(μ)(1<p<∞), Hardy space H1(μ) and RBMOμO).