海森堡群上加权Hardy算子的最佳估计

发布时间：2018-05-27 07:27

本文选题：Hardy型平均算子 + Heisenberg群　；参考：《山东师范大学》2017年硕士论文

【摘要】：众所周知,算子在某些空间的有界性理论及其应用是调和分析领域研究的中心内容.著名数学家、美国科学院院士、普林斯顿大学Stein教授[2]把调和分析中的算子归结为三类,分别是以Hardy型算子为代表的平均算子、以Hillbert变换为基本形式的奇异积分算子、以Fourier变换为雏形的震荡型积分算子.以Hardy型算子为代表的平均算子理论自创立以来,便在调和分析中处于重要地位,我们将研究Hardy型积分算子的加权情形在Heisenberg群上的最佳估计问题.本文主要论述了加权Hardy算子在Lp(Hn),BMO(Hn), Morrey空间上的有界性,多线性的加权Hardy算子在乘积型Lp(Hn)空间和乘积型Morrey空间的有界性以及加权Cesaro算子和多线性的加权Cesaro算子在Heisenberg群相关函数空间上的有界估计.本文的主要内容安排如下:在第一章中,首先介绍有关Hardy型平均算子的研究背景和研究现状,然后介绍了Heiseuberg群的定义及相关性质,从而给出了加权Hardy算子在Heisenberg群上的定义,并将加权Hardy算子推广到多线性的情形,给出明确的定义,接下来主要讨论本文中用到的几类经典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,我们依次给出了加权Hardy算子在Lp(Hn)、BMO(Hn)和Morrey空间上有界时对权函数的刻画的充分必要条件,并确定相应的范数.在第三章中,我们依次给出了多线性的加权Hardy算子在乘积型Lp(Hn)和乘积型Morrey空间上有界时对权函数的刻画的充分必要条件,并确定相应的范数.在第四章中,首先给出加权Cesaro算子和多线性的加权Cesaro算子在Heisenberg群上的定义,然后给出了加权Cesaro算子是加权Hardy算子的伴随算子及相关性质,最后根据第二章和第三章给出加权Cesaro算子和多线性的加权Cesaro算子在Heisenberg群相关函数空间上的有界估计定理.
[Abstract]:It is well known that the boundedness theory of operators in some spaces and its applications are the central contents of harmonic analysis. The famous mathematician, academician of the American Academy of Sciences, Professor Stein of Princeton University [2] reduced the operators in harmonic analysis to three categories, which are the averaging operators represented by Hardy type operators and the singular integral operators in which the Hillbert transformation is the basic form. Oscillation integral operator based on Fourier transform. The averaging operator theory, represented by Hardy type operators, has played an important role in harmonic analysis since its inception. We will study the optimal estimation of the weighted case of Hardy type integral operators on Heisenberg groups. In this paper, we discuss the boundedness of weighted Hardy operator on Morrey space. The boundedness of multilinear weighted Hardy operators in product type Morrey spaces and product type Morrey spaces, and the bounded estimates of weighted Cesaro operators and multilinear weighted Cesaro operators on Heisenberg group correlation function spaces. The main contents of this paper are as follows: in the first chapter, the research background and research status of Hardy type averaging operators are introduced, then the definition and related properties of Heiseuberg groups are introduced, and the definition of weighted Hardy operators on Heisenberg groups is given. The weighted Hardy operator is extended to the multilinear case, and a clear definition is given. Then, the definition forms of some classical function spaces used in this paper and some necessary lemmas to be used are discussed. Finally, the main research work of this paper is briefly introduced. In the second chapter, we give the sufficient and necessary conditions for the weighted Hardy operator to characterize the weight function when the weighted Hardy operator is bounded on the LpHN and Morrey spaces, and determine the corresponding norms. In Chapter 3, we give a sufficient and necessary condition for the characterization of weighted Hardy operators on product type LpHn) and product type Morrey spaces, and determine the corresponding norms. In chapter 4, we first give the definitions of weighted Cesaro operator and multilinear weighted Cesaro operator on Heisenberg group, then we give the adjoint operator of weighted Cesaro operator and its related properties. Finally, according to the second and third chapters, the bounded estimate theorems of weighted Cesaro operator and multilinear weighted Cesaro operator on the space of Heisenberg group correlation function are given.
【学位授予单位】：山东师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O177

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