现代调和分析的算子理论起源于 20 世纪 50~60 年代 A. Zygmund 和 A Calderon 创立的 Calderon- Zygmund 奇异积分理论．在此后的二三十年内 ， 调和分 析建立起 Hardy空间的实变理论和小波分析等诸多理论 ， 这些理论不仅深刻地影响 着经典调和分析 ， 同时在偏微分方程、多复变函数和概率论等领域中均有着重要的 应用.

在调和分析的理论中 ， 奇异积分算子的弱端点估计一直是核心问题之一 2006 年 ， P. Janakiraman 通过计算 Hardy-Littlewood 极大算子 ， Riesz 变换以及在某种光 滑性条件下的奇异积分算子的弱极限行为 ， 得到了它们的弱 (1 ,1)范数的下界此 后 ， 入们对粗糙核奇异积分算子和强极大算子等重要算子的弱极限行为也进行了研 究.

非标准奇异积分算子是一类与 Calderon 交换子关联紧密的算子 ， 之所以称之 为非标准 ， 是因为它不满足经典的核条件本文致力于推进非标准奇异积分算子的 弱极限行为研究 ， 得到了该算子的弱型范数的下界此外 ， 通过该算子与 Calderon 交换子的联系 ， 本文还得到了一种 Calderon 交换子的新的有界性结果. 全文共分为 三章.

第一章首先介绍了奇异积分算子和非标准奇异积分算子的背景知识 ， 梳理了历 来关于奇异积分算子的弱极限行为的主要结果 ， 提出了本文研究的核心问题．其次 ， 介绍了 Calderon 交换子的基本知识和该算子历来主要的研究结果 ， 并阐述了利用 非标准奇异积分算子研究 Calderon 交换子有界性的思路．最后 ， 给出了本文所得到 的主要结论.

第二章对非标准奇异积分算子的弱端点估计做了更精细的研究 ， 在此基础上得 到了非标准奇异积分算子及其对偶算子的弱极限行为.

第三章应用非标准奇异积分算子的对偶算子的性质得到了一种 Calderon 交换 子的新的有界性证明和结果.

The operator theory of modern harmonic analysis originates from the CalderonZygmund singular integral theory established by A. Zygmund and A. Calderon in the 1950s and 1960s. In the following two or three decades, harmonic analysis established real Hardy space theory, wavelet analysis and many other theories, these theories not only profoundly influenced the classical harmonic analysis, but also have important applications in the field of partial differential equations, functions of several complex variables and probability theory.

In the theory of harmonic analysis, weak type endpoint estimation of singular integral operator has always been one of the core issues. In 2006, P. Janakiraman calculated the limiting weak-type behavior of Hardy-Littlewood maximal operator, riesz transform and the singular integral operator under certain smooth conditions, and the lower bound of their weak (1,1) norm is obtained. Since then, the limiting weak-type behavior of some important operators such as rough kernel singular integral operator and strong maximal opera or has been studied.

The non-standard singular integral operator is closely related to Calderon commutator. It 's called non-standard because it do not satisfy classical kernel conditions. In this paper, the limiting weak-type behavior of the non-standard singular integral operator is studied and the lower bound of the weak type norm of the operator is obtained. In addition, a new bounded result of the Calderon commutator is bt ained by the connection between the operator and the non-standard singular integral operator. This paper is divided into three chapters.

In Chapter 1, we introduce the background knowledge of the singular integral operator and the non-standard singular integral operator, sort out the main results about the limiting weak-type behavior of the singular integral operator, and put forward the core problem of this paper. Secondly, the basic knowledge of the Calderon commutator and the main research results of this operator are introduced, and the idea. of using the non-standard singular integral operator to sttldy the boundedness of the Calderon commutator is expounded. Finally. the main conclusions of this paper are given.

In Chapter 2, the weak type endpoint estimation of the non-standard singular integral operator is studied in det ail, and t hen the limiting weak-type behavior of the non-standard singular integral operator and its dual operator is obtained.

In Chapter 3, a new bounded proof and result of the Calderon commutator are obtained by using the properties of the dual operator of the non-standard singular integral operator.