【作者】 杨杰；

【导师】 陈文艺；

【作者基本信息】 武汉大学 ， 基础数学， 2012， 博士

【摘要】 本文共分四章,主要介绍和讨论了如下几个内容：拟微分算子的交换子在Hardy空间的有界性和紧性；非双倍测度下,带参数的Marcinkiewicz奇异积分算子的交换子在Lebesgue空间,Hardy空间和RBMO空间有界性.全文内容安排如下：第一章首先介绍了奇异积分算子及其交换子的发展历史和现状,特别对拟微分算子和Marcinkiewicz奇异积分算子及其交换子的研究背景和现状做了详细的介绍.然后针对这两个算子的交换子研究现状提出了五个问题.最后简要的介绍了我们得到的主要结果.第二章讨论了象征(?)(χ,ξ)属于象征类S01,δ(0≤δ<1),当b分别在BMO函数空间或BMO∞时,对应于此象征的交换子T(?)b是H1(Rd)到L1(Rd)的有界算子的充分必要条件是b∈LMO或b∈LMO∞.第三章给出了象征(?)(χ,ξ)属于象征类S01,δ5(0≤δ占<1)时,交换子T(?)b是H1(Rd)到L1(Rd)的紧算子的充分条件为b∈CLMO或b∈CLMO∞.第四章证明了在非齐次型空间上,若b属于Lipschitz空间Lipβ(μ)(0<β≤1), Marcinkiewicz积分核满足某种Hormander条件,由b和带参数的Marcinkiewicz积分算子(?)(?)生成的交换子3(?)b在Lp(μ)(1<p<∞)空间、Hardy空间H1(μ)和RBMO(μ)(非齐次型空间上的平均震荡函数空间)上的有界性.更多还原

【Abstract】 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).更多还原