算子的紧性和正则性是调和分析中的两个基本问题.本学位论文主要研究几类重要算子的交换子的紧性以及局部双线性极大算子的正则性和连续性问题.交换子的紧性主要涉及的算子有：振荡奇异积分算子、双线性极大Calderón-Zygmund奇异积分算子以及与薛定谔算子相关的各类算子.这几类算子在调和分析以及其他领域中起着重要的作用,如：振荡积分与Fourier变换、Bochner-Riesz平均以及Radon变换密切相关.近年来,具有多项式相位函数的振荡奇异积分算子的有界性研究取得了许多重要成果.多线性Calderón-Zygmund奇异积分及其相关算子,包括极大奇异积分算子以及分数次奇异积分算子是调和分析中研究的重要算子.这方面的研究始于70年代Coifman和Meyer的著名工作,随后Grafakos和Torres对其做了系统的研究.具有非负位势的薛定谔算子在某些次椭圆算子的研究中起着非常重要的作用.自Fefferman, Shen以及Zhong等人在薛定谔算子的基本解估计及其有界性方面作出了奠基性的工作后,薛定谔型算子以及与薛定谔算子相关的各类算子引起了众多知名学者的关注.本学位论文致力于研究上述三类算子的交换子的紧性以及基于Lacey和Thiele研究的双线性Hilbert变换背景下的局部双线性极大算子的Sobolev正则性.本学位论文共分为四章.

    在第一章中,主要介绍几类重要算子的交换子的紧性以及极大算子的正则性的研究背景,并给出本学位论文的主要结果.

    在第二章中,我们致力于研究三类重要算子的交换子的紧性问题.  交换子的紧性研究最早源于1975年, Cordes在线性Fourier乘子和拟微分算子的交换子的紧性方面的工作.奇异积分算子交换子的紧性研究始于1978年Uchiyama的工作.此后, Torres、Hyt\"onen、Clop和Cruz等人也在各类经典算子的交换子的紧性方面做出了一系列的重要工作.这一章中我们的主要工作如下：

    1.振荡奇异积分算子的交换子的加权紧性.

    我们给出了两类核条件下,具有多项式相位函数的振荡奇异积分算子的交换子在加权L^p空间上的紧性.首先,我们证明了当多项式P(x,y)的次数满足条件\deg_x(P)\leq 1或者\deg_y(P)\leq1时,非卷积核满足H\"older核条件的振荡奇异积分算子的交换子在加权L^p空间上的紧性(其有界性我们将在第三章中给出).其次,在比H\"older核条件稍强的条件下,我们证明了对任意的实值多项式,满足标准核条件的振荡奇异积分算子的交换子在加权L^p空间上的紧性.

    2.双线性极大Calderón-Zygmund奇异积分算子的交换子的加权紧性.

    我们证明了双线性极大Calderón-Zygmund奇异积分算子T^*的交换子的加权紧性,包括单通道交换子T^*_{b,1}, T^*_{b,2}以及迭代交换子T^*_{(b_1,b_2)}.我们将Ding, Mei和Xue的结果推广到了加权的情形,这里的权属于多重权类A_{\vec{p}}.由于T^*是次线性的,并且权不具有平移不变性,所以研究它的交换子的加权紧性会更加困难,也更有意义.同时,我们也得到了当权属于A_{\vec{p}}时,双线性Calderón-Zygmund奇异积分算子交换子的加权紧性. Bu和Chen将P\'{e}rez和Torres对于双线性Calderón-Zygmund奇异积分算子交换子的加权(权属于A_p\times A_p)紧性结果推广到了多重权类A_{\vec{p}},但他们需要假设v_{\vec{w}}\in A_p且p>1.我们去掉了他们的这一假设条件,并且将指标做到了最好的范围p>1/2.

    3.与薛定谔算子相关的算子的交换子的加权紧性.

    我们得到了当b\in{CMO}_\theta(\rho)(\mathbb{R}^n)且权w\in A_p^{\rho,\theta}(或w\in A^{\rho}_{(p,q)})时,与薛定谔算子-\Delta+V(x)相关的半群极大算子以及分数次积分算子的交换子的加权紧性.这里的空间\CMO_\theta(\rho)(\mathbb{R}^n)比经典的\CMO(\mathbb{R}^n)更大,权类A_p^{\rho,\theta}要比经典的Muckenhoupt权类A_p更广.这是首次考虑与薛定谔算子相关的算子的紧性问题.

    在第三章中,我们将Al-Qassem, Cheng以及Pan对于双线性相位函数B(x,y)的振荡奇异积分算子的有界性结果推广到了多项式相位函数的情形,证明了当多项式P(x,y)的次数满足条件\deg_x(P)\leq1或者\deg_y(P)\leq1时,非卷积核满足H\"{o}lder核条件的振荡奇异积分算子及其交换子的加权强有界性以及端点估计.而Al-Qassem等人的结果要求\deg_x(B)\leq1且\deg_y(B)\leq1.更具有挑战性的是证明这一结论对任意的实值多项式都成立,目前仍是未解决的问题.

    在第四章中,我们致力于研究基于Lacey和Thiele研究的双线性Hilbert变换背景下的局部双线性极大算子的Sobolev正则性.我们首先建立了局部双线性极大算子以及局部双线性分数次极大算子导数的一些新的界.然后,利用这些估计我们得到了这两类算子在Sobolev空间中上的有界性和连续性.

    Compactness and regularity of operators are two basic problems in Harmonic analysis. The purpose of this dissertation is to study the compactness of commutators of some important operators and regularity of locally bilinear maximal operators. The compactness of commutators mainly involve operators such as oscillatory singular integral operators, bilinear maximal Calder\'on-Zygmund singular integral operators and some operators related to Schr\"odinger operator. These operators play important roles in Harmonic analysis and other fields. For example, oscillatory integral  has a close relationship with  Fourier transform, Bochner Riesz and Radon transform. Recently, the boundedness of oscillatory singular integral operators with polynomial phase has been widely concerned and many important results have been obtained. Multilinear Calder\'on-Zygmund singular integral and its related operators, including maximal singular integral operator and fractional singular integral operator, are important operators in Harmonic analysis. The research of this subject originated from the celebrated works of Coifman and Meyer in the 1970s. Later on, Grafakos and Torres did a systematic research on it. The Schr\"odinger operator with nonnegative potentials is very useful in the study of certain subelliptic operators. Fefferman, Shen and Zhong have done important works on the boundedness and the  fundamental solutions of Schr\"odinger operators. Since then, Schr\"odinger type operators and various operators related to Schrodinger operators have attracted much attention of many famous scholars. In this dissertation, we are devoted to study the compactness of commutators and the Sobolev regularity properties of the local bilinear maximal operator based on the bilinear Hilbert transform studied by Lacey and Thiele. This dissertation including four chapters.

    In Chapter 1, we recall the background of compactness of commutators of several important operators and regularity of maximal operators. We also briefly state the main results of this dissertation.

    In Chapter 2, we focus on the compactness of commutators of three kinds of important operators. The compactness of commutator of singular integral operator originated from the work of Uchiyama in 1978. After that, Cordes, Hyt\"onen, Torres, Clop and Cruz et al. have done a series of works on this issue. In this chapter, our main works are as follows.

    1. The weighted compactness of the commutator of oscillatory singular integral operator.

     We establish the compactness of the commutator of oscillatory singular integral operator with polynomial phase on weighted L^p spaces.  We prove that the commutator of oscillatory singular integral operator is a compact operator on L^p(w) with the kernel K(x,y) satisfies a H\"{o}lder condition and P(x,y) satisfies the condition \deg_x(P)\leq1 or \deg_y(P)\leq1 (the boundedness will be given in Chapter 3). We also show the compactness of commutators of oscillatory singular integral operators with stronger kernel condition and any real-valued polynomials.

    2. The weighted compactness of the commutators of the bilinear maximal Calder\'on-Zygmund singular integral operators.

    We show that the commutators of the bilinear maximal Calder\'on-Zygmund singular integral operators, including the commutators in the j-th entry T^*_{b,j}(j=1,2) and the iterated commutator T^*_{\vec{b}} are compact operators on L^p(v_{\vec{w}}).  We generalize the results of Ding, Mei and Xue to the weighted case, where the weights belong to the multiple weights class A_{\vec{p}}.  At the same time, we also obtain the weighted compactness for the commutators of the bilinear Calder\'on-Zygmund singular integral operators with the weight \vec{w}\in A_{\vec{p}} and p>1/2. Recently, Bu and Chen generalized the results of P\'{e}rez and Torres. They proved that the commutators of the bilinear Calder\'on-zygmund singular integral operators are compact operators with \vec{w}\in A_{\vec{p}} and p>1. However, they need to assume v_{\vec{w}}\in A_p. Moreover, it seems that the condition p>1 is not natural, since the strong boundedness holds for p>1/2.

    3. The weighted compactness of the commutators of some operators associated to the Schr\"odinger operator.

    We give the weighted compactness of the commutators of the semi-group maximal function and fractional integral associated to the Schr\"odinger operator -\Delta+V(x)when b\in{CMO}_\theta(\rho)(\mathbb{R}^n) and w\in A_p^{\rho,\theta}(or w\in A^{\rho}_{(p,q)}). It should be pointed out that the space where b belongs to and the weight w belongs to are more larger than the usual {CMO}(\mathbb{R}^n) space and the Muckenhoupt weights class A_p, respectively. It is the first paper to consider the compactness of operators related to Schr\"odinger operator.

     In Chapter 3, we extend and generalize some known results for oscillatory singular integrals with polynomial phases. We establish the weighted strong boundedness and endpoint estimate for oscillatory singular integral operators and its commutators with the kernel satisfying H\"older condition and P(x,y) satisfies the condition \deg_x(P)\leq1 or \deg_y(P)\leq1. Al-Qassem, Cheng and Pan gave the boundedeness of oscillatory singular integrals with bilinear phase B(x,y), where B satisfies \deg_x(B)\leq1 and \deg_y(B)\leq1.  Our results are valid when \deg_x(P)\leq1 or \deg_y(P)\leq1. However, a more challenging question is to prove the result with P being a general real-valued polynomial. It is still an open problem.

    In Chapter 4, we are devoted to study the Sobolev regularity properties of the local bilinear maximal operator and its fractional variant. Some new bounds for the derivatives of the above maximal functions are established. These estimates can be used to obtain boundedness and continuity for these operators in Sobolev spaces.