关于核函数满足一类变形的LIPSCHITZ条件的奇异积分算子的单边估计

发布时间：2018-06-05 18:14

本文选题：单边C-Z奇异积分 + 单边权　；参考：《山东师范大学》2015年硕士论文

【摘要】：近半个世纪以来,现代调和分析理论取得了许多重大进展,其思想、方法和技巧在很多数学领域中得到广泛的应用.以Calderon-Zygmund(C-Z)奇异积分算子为代表的算子理论自创立以来,便在调和分析中处于中心地位.本文主要研究有关变形核的单边C-Z奇异积分算子的加权估计,单边Cohen型奇异积分算子交换子在加权Triebel-Lizorkin空间的Lipschitz估计,以及在单边加权Morrey空间中满足一定尺寸条件的单边次线性算子的有界性.本文的主要内容安排如下：在第一章中,我们将内容分为六个小节.首先主要介绍有关核函数的奇异积分算子的研究背景和研究现状.然后给出了经典的C-Z理论和Ap权函数的定义和双边情形下变形的Hormander条件.其次引入单边权函数和单边C-Z奇异积分算子及其交换子的定义及性质.接下来主要讨论本文中用到的几类单边函数空间的定义形式和将要用到的一些必要引理.最后简单的介绍本文的主要研究工作.在第二章中,我们首先给出满足变形的Lipschitz条件的单边C-Z奇异积分算子T+的定义.然后,以单边sharp极大函数为桥梁来求得此类单边奇异积分算子的加权Lp(p1)有界性.对于p=1时,运用单边C-Z分解来完成单边奇异积分算子T+的弱(1,1)有界性估计.第三章分成两个主要的部分.我们主要利用单边权的外推方法,来分别讨论单边Cohen型奇异积分算子交换子和分数次积分算子交换子在单边加权Triebel-Lizorkin空间的Lipschitz的有界性估计.在第四章中,首先介绍了满足一定尺寸条件的单边次线性算子和单边分数次积分算子.然后在单边加权Morrey空间中讨论这两类单边算子的有界性.
[Abstract]:In the last half century, modern harmonic analysis theory has made great progress, and its ideas, methods and techniques have been widely used in many fields of mathematics. The operator theory, represented by Calderon-Zygmund C-Z) singular integral operator, has been at the center of harmonic analysis since it was founded. In this paper, we mainly study the weighted estimates of one-sided C-Z singular integral operators with deformed kernels and the Lipschitz estimates of commutators of one-sided Cohen type singular integral operators in weighted Triebel-Lizorkin spaces. And the boundedness of unilateral sublinear operators satisfying certain size conditions in one-sided weighted Morrey spaces. The main contents of this paper are as follows: in the first chapter, we divide the content into six sections. Firstly, the research background and present situation of singular integral operators for kernel functions are introduced. Then, the classical C-Z theory and the definition of AP weight function and the Hormander condition of deformation in bilateral case are given. Secondly, the definition and properties of one-sided weight function, one-sided C-Z singular integral operator and its commutator are introduced. Then we mainly discuss the definition form and necessary Lemma of several kinds of unilateral function spaces used in this paper. Finally, the main research work of this paper is briefly introduced. In chapter 2, we first give the definition of one-sided C-Z singular integral operator T which satisfies the Lipschitz condition of deformation. Then, we use the one-sided sharp maximal function as the bridge to obtain the weighted LpP-1) boundedness of this singular integral operator. For p = 1, the one-sided C-Z decomposition is used to complete the weakly bound estimate of one-sided singular integral operator T. Chapter three is divided into two main parts. We mainly use the one-sided weight extrapolation method to discuss the boundedness estimates of the Lipschitz of singular integral operator commutators of one-sided Cohen type and fractional integral operator commutators in one-sided weighted Triebel-Lizorkin spaces, respectively. In the fourth chapter, we first introduce the unilateral sublinear operator and the unilateral fractional integral operator which satisfy certain size conditions. Then we discuss the boundedness of these two kinds of one-sided operators in one-sided weighted Morrey spaces.
【学位授予单位】：山东师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O177

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