本文主要研究积分算子交换子的有界性及紧性特征以及交换子在~Morrey~空间上有界的特征刻画. 全文分为五章. 第一章介绍文章的研究背景及本文的结构;第二章研究了~Calder\'{o}n-Zygmund~奇异积分算子的交换子在~Morrey~空间上的紧性特征;第三章给出了分数次积分算子的交换子在~Morrey~空间上有界的特征刻画并讨论了~Riesz~位势算子的交换子在~Morrey~空间上的紧性特征;第四章给出了Littlewood-Paley~算子交换子在~Morrey~空间上有界的特征刻画并讨论了~Marcinkiewicz~积分算子的交换子在~Morrey~空间上的紧性特征;第五章研究了粗糙核算子的交换子在~Sobolev~空间上的有界性问题.行文结构安排如下：第一章介绍文章的研究背景及本文的结构. 第二章主要讨论 ~Calder\'{o}n-Zygmund~奇异积分算子的交换子在~Morrey~ 空间上的紧性特征,实现了对已有结果的改进和推广, \,并且关于奇异积分算子的结果是在核\;$\Omega$ 很弱的条件下得到的.第三章主要通过分数次积分算子的交换子在~Morrey~空间上有界来刻画~BMO~空间,其结果是在核~$\Omega$~很弱的条件下得到的; 并讨论了~Riesz~位势算子的交换子在~Morrey~空间上的紧性特征.第四章通过~Littlewood-Paley算子的交换子在~Morrey~空间上的有界性给出了~BMO~空间的刻画,并进一步研究了~Marcinkiewicz~积分交换子在~Morrey~空间上的紧性特征,实现了对已有结果的改进和推广, 并且这些结果是在核\;$\Omega$很弱的条件下得到的.第五章研究了带粗糙核的超奇异积分及超奇性~Marcinkiecicz~积分的交换子从齐次~Sobolev~空间~$\dotL^p_\gamma$ 到~$L^p$~空间上的有界性.

There are five chapters in this dissertation, which focuseson four contents: the compactness of commutators ofCalder\'on-Zygmund singular integral operator on Morrey space; theresults about the commutators of fractional integrals on Morrey space; the theories of thecommutators of Littlewood-Paley operators on Morrey space; theboundedness of the commutator of hypersingular integral operator andMarcinkiewicz integral from homogeneous Sobolev space to Lebesguespace.The outline of the paper is arranged as follows: Chapter 1 presents the background and the outline of thisdissertation. Chapter 2 is devoted to study the compactness of the commutator which is formed by BMO functions with Calder\'on-Zygmund singular integral operatoron Morrey space, which extends and improves on previous results.In Chapter 3, we proceed to give a characterization of BMO space bystudying the boundedness of the commutators of fractional integralwith the kernel function $\Omega$ on the unit sphere and thecompactness of the commutator which is formed by BMO functions withRiesz potential operator on Morrey space, which extends and improveson previous results. The purposes of Chapter 4 are to give a characterization of BMO space bystudying the boundedness of the commutators of Littlewood-Paleyoperator with the kernel function $\Omega$ on the unit sphere anddiscuss the compactness of the commutator which is formed by BMO functions withMarcinkiewicz integral operator. The results extend and improve on previous results.In Chapter 5, we give the boundedness of the commutator$[b,T_{\gamma}]$ and $[b,\mu_{\gamma,\Omega}]$ from the homogeneousSobolev space $\dot L^p_\gamma(\mathbb R^n)$ to the Lebesgue space$L^p(\mathbb R^n)$ for $1 ﹀