本文選題：Hardy型平均算子 + Heisenberg群��； 參考：《山東師范大學(xué)》2017年碩士論文

【摘要】：眾所周知,算子在某些空間的有界性理論及其應(yīng)用是調(diào)和分析領(lǐng)域研究的中心內(nèi)容.著名數(shù)學(xué)家、美國科學(xué)院院士、普林斯頓大學(xué)Stein教授[2]把調(diào)和分析中的算子歸結(jié)為三類,分別是以Hardy型算子為代表的平均算子、以Hillbert變換為基本形式的奇異積分算子、以Fourier變換為雛形的震蕩型積分算子.以Hardy型算子為代表的平均算子理論自創(chuàng)立以來,便在調(diào)和分析中處于重要地位,我們將研究Hardy型積分算子的加權(quán)情形在Heisenberg群上的最佳估計(jì)問題.本文主要論述了加權(quán)Hardy算子在Lp(Hn),BMO(Hn), Morrey空間上的有界性,多線性的加權(quán)Hardy算子在乘積型Lp(Hn)空間和乘積型Morrey空間的有界性以及加權(quán)Cesaro算子和多線性的加權(quán)Cesaro算子在Heisenberg群相關(guān)函數(shù)空間上的有界估計(jì).本文的主要內(nèi)容安排如下:在第一章中,首先介紹有關(guān)Hardy型平均算子的研究背景和研究現(xiàn)狀,然后介紹了Heiseuberg群的定義及相關(guān)性質(zhì),從而給出了加權(quán)Hardy算子在Heisenberg群上的定義,并將加權(quán)Hardy算子推廣到多線性的情形,給出明確的定義,接下來主要討論本文中用到的幾類經(jīng)典函數(shù)空間的定義形式和將要用到的一些必要引理,最后簡單的介紹本文的主要研究工作.在第二章中,我們依次給出了加權(quán)Hardy算子在Lp(Hn)、BMO(Hn)和Morrey空間上有界時(shí)對權(quán)函數(shù)的刻畫的充分必要條件,并確定相應(yīng)的范數(shù).在第三章中,我們依次給出了多線性的加權(quán)Hardy算子在乘積型Lp(Hn)和乘積型Morrey空間上有界時(shí)對權(quán)函數(shù)的刻畫的充分必要條件,并確定相應(yīng)的范數(shù).在第四章中,首先給出加權(quán)Cesaro算子和多線性的加權(quán)Cesaro算子在Heisenberg群上的定義,然后給出了加權(quán)Cesaro算子是加權(quán)Hardy算子的伴隨算子及相關(guān)性質(zhì),最后根據(jù)第二章和第三章給出加權(quán)Cesaro算子和多線性的加權(quán)Cesaro算子在Heisenberg群相關(guān)函數(shù)空間上的有界估計(jì)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.
【學(xué)位授予單位】：山東師范大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O177

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本文編號：1941049