奇异积分和函数的可微性（英文） [Singular Integrals and Diffferentiability Properties of Functions] 下载 mobi epub pdf 电子书 2025

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内容简介

This book is an outgrowth of a course which I gave at Orsay duringthe academic year 1 966.67 MY purpose in those lectures was to pre-sent some of the required background and at the same time clarify theessential unity that exists between several related areas of analysis.These areas are：the existence and boundedness of singular integral op-erators；the boundary behavior of harmonic functions；and differentia-bility properties of functions of several variables.AS such the commoncore of these topics may be said to represent one of the central develop-ments in n.dimensional Fourier analysis during the last twenty years，and it can be expected to have equal influence in the future.These pos.

作者简介

作者：(美国)施泰恩(SteinE.M.)

内页插图

目录

PREFACE
NOTATION
I．SOME FUNDAMENTAL NOTIONS OF REAL．VARIABLE THEORY
The maximal function
Behavior near general points of measurable sets
Decomposition in cubes of open sets in R”
An interpolation theorem for L
Further results

II．SINGULAR INTEGRALS
Review of certain aspects of harmonic analysis in R”
Singular integrals：the heart of the matter
Singular integrals：some extensions and variants of the
preceding
Singular integral operaters which commute with dilations
Vector．valued analogues
Further results

III．RIESZ TRANSFORMS，POLSSON INTEGRALS，AND SPHERICAI HARMONICS
The Riesz transforms
Poisson integrals and approximations to the identity
Higher Riesz transforms and spherical harmonics
Further results

IV．THE LITTLEWOOD．PALEY THEORY AND MULTIPLIERS
The Littlewood-Paley g-function
The functiong
Multipliers（first version）
Application of the partial sums operators
The dyadic decomposition
The Marcinkiewicz multiplier theorem
Further results

V．DIFFERENTIABlLITY PROPERTIES IN TERMS OF FUNCTION SPACES
Riesz potentials
The Sobolev spaces
BesseI potentials
The spaces of Lipschitz continuous functions
The spaces
Further results

VI．EXTENSIONS AND RESTRICTIONS
Decomposition of open sets into cubes
Extension theorems of Whitney type
Extension theorem for a domain with minimally smooth
boundary
Further results

VII．RETURN TO THE THEORY OF HARMONIC FUNCTIONS
Non-tangential convergence and Fatou'S theorem
The area integral
Application of the theory of H”spaces
Further results

VIII．DIFFERENTIATION OF FUNCTIONS
Several qotions of pointwise difierentiability
The splitting of functions
A characterization 0f difrerentiability
Desymmetrization principle
Another characterization of difirerentiabiliW
Further results
APPENDICES
Some Inequalities
The Marcinkiewicz Interpolation Theorem
Some Elementary Properties of Harmonic Functions
Inequalities for Rademacher Functions
BlBLl0GRAPHY
INDEX

精彩书摘

The basic ideas of the theory of reaI variables are connected with theconcepts of sets and ftmctions，together with the processes of integrationand difirerentiation applied to them.WhiIe the essential aspects of theseideas were brought to light in the early part of our century，some of theirfurther applications were developed only more recently.It iS from thislatter perspective that we shall approach that part of the theory thatinterests US.In doing SO，we distinguish several main features： The theorem of Lebesgue about the differentiation of the integral.The study of properties related to this process iS best done in terms of a“maximal function”to which it gives rise：the basic features of the latterare expressed in terms of a“weak-type”inequality which iS characteristicof this situation. Certain covering lemmas.In general the idea iS to cover an arbitraryopen set in terms of a disioint union ofcubes or balls，chosen in a mannerdepending on the problem at hand.ORe such example iS a lemma ofWhitney，fTheorem 3）.Sometimes，however，it SHffices to cover only aportion of the set。as in the simple covering lemma，which iS used to provethe weak-type inequality mentioned above. f31 Behavior near a‘'general”point of an arbitrary set.The simplest notion here iS that of point of density.More refined properties are bestexpressed in terms of certain integrals first studied systematically by Marcinkiewicz.
（4）The splitting of functions into their large and small parts.Thisfeature which iS more of a technique than an end in itself，recurs often.ItiS especially useful in proving Linequalities，as in the first theorem ofthis chapter.That part of the proof of the first theorem iS systematizedin the Marcinkiewicz interpolation theorem discussed in§4 of this chapter and also in Appendix B.
......

前言/序言

奇异积分和函数的可微性（英文） [Singular Integrals and Diffferentiability Properties of Functions] 下载 mobi epub pdf txt 电子书 格式

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是一本好书是一本好书

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调和分析的经典名著，绝不容错过

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1. 作者是T. Tao的老师

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本套丛书是数学大师给本科生们写的分析学系列教材。第一作者E. M. Stein是一位调和分析大师，他是1999年沃尔夫奖获得者，同时，他也是一位卓越的教师。他的学生，和学生的学生，加起来超过两百多人，其中有两位已经获得了菲尔兹奖，2006年的菲尔兹奖获奖者之一即为他的学生陶哲轩。 　　这套教材在Princeton大学使用，同时其它学校，比如UCLA等名校也在本科生教学中使用。其教学目的是，用统一的、联系的观点来把现代分析的核心内容教给本科生们，力图使本科生的分析学课程能接上现代数学研究的脉络。这套书共有四本，依次是： 　　傅立叶分析； 　　复分析； 　　实分析； 　　泛函分析。 　　这些课程仅仅假定读者读过大一微积分和线性代数，所以可看作是本科生高年级（大二到大三共四个学期）的必修课程，每学期一门。 　　非常值得注意的是，作者把傅里叶分析作为学完大一微积分后的第一门高级分析课。同时，在后续课程中，螺旋式上升，将其贯穿下去。我本人是极为赞同这种做法的。一则，现代数学中傅里叶分析无处不在，既在纯数学，如数论的各个方面都有深入的应用，又在应用数学中是绝对的基础工具。二则，傅里叶分析不光有用，其本身的内容，可以说，就能够把数学中的几大主要思想都体现出来。这样，学生们先学这门课，对数学就能有鲜活的了解，既知道它的用处，又能够“连续”地欣赏到数学中的各种大思想、大美妙。接下来，是学同样集理论优美和深刻应用于一体的复分析。学完这两门课，学生已经有了相当多的例子和感觉，既懂得其用又懂得其妙。这样，再学后面比较抽象的实分析和泛函分析时，就自然得多，动机也充分得多。 　　这种教法目前在国内还很欠缺，也缺乏相应的教材。这主要是因为我们的教育体制还存在一些问题，比如数学系研究生的入学考试，以往最关键的是初试，但初试只考数学分析和高等代数，也就是本科生低年级的课程。长此以往，中国的大多数本科生，只用功在这两门低年级课程上，而在高年级的后续课程，以及现代数学的眼界上就有很大的欠缺。这样，势必导致他们在研究生阶段后劲不足，需要补的东西过多，因而疲于奔命。

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据说，她本来是个乖巧美丽的女人；据说，她喜欢上一个跟着轮船来这里进货的外地男人：据说，那男人长得身材魁梧，好打抱不平。在这个小镇，结婚前女人不能破身，她却私自把自己给了那男人。他们曾想私奔，最终被拦下，张美丽因而自杀。

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幸亏有张美丽。张美丽作为一个沦陷的标志，牢牢地立在欲望的悬崖边，被反复强化、反复讲述。关于她的细节，成了这个小镇用来教育孩子的最好典型：不准和外地人讲话，不要和男同学私下见面，不能靠近那种漂染头发的发廊……说完不准，大人们会用这样的话收尾：要不你就会像张美丽那样，名声臭遍整个小镇。

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Stein大师著名的调和三部曲之一，买来学习观摩

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幸亏有张美丽。张美丽作为一个沦陷的标志，牢牢地立在欲望的悬崖边，被反复强化、反复讲述。关于她的细节，成了这个小镇用来教育孩子的最好典型：不准和外地人讲话，不要和男同学私下见面，不能靠近那种漂染头发的发廊……说完不准，大人们会用这样的话收尾：要不你就会像张美丽那样，名声臭遍整个小镇。

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那时候，一股莫名的冲动开始在我们这群男同学的内心涌动。而张美丽，一个性感如摩托车女郎的女鬼，总让我们在夜晚提到的时候，血脉偾张。

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