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內容簡介

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.

作者簡介

作者：(美國)施泰恩(SteinE.M.)

內頁插圖

目錄

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

精彩書摘

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.
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前言/序言

奇異積分和函數的可微性（英文） [Singular Integrals and Diffferentiability Properties of Functions] 下載 mobi epub pdf txt 電子書

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本套叢書是數學大師給本科生們寫的分析學係列教材。第一作者E. M. Stein是一位調和分析大師，他是1999年沃爾夫奬獲得者，同時，他也是一位卓越的教師。他的學生，和學生的學生，加起來超過兩百多人，其中有兩位已經獲得瞭菲爾茲奬，2006年的菲爾茲奬獲奬者之一即為他的學生陶哲軒。 　　這套教材在Princeton大學使用，同時其它學校，比如UCLA等名校也在本科生教學中使用。其教學目的是，用統一的、聯係的觀點來把現代分析的核心內容教給本科生們，力圖使本科生的分析學課程能接上現代數學研究的脈絡。這套書共有四本，依次是： 　　傅立葉分析； 　　復分析； 　　實分析； 　　泛函分析。 　　這些課程僅僅假定讀者讀過大一微積分和綫性代數，所以可看作是本科生高年級（大二到大三共四個學期）的必修課程，每學期一門。 　　非常值得注意的是，作者把傅裏葉分析作為學完大一微積分後的第一門高級分析課。同時，在後續課程中，螺鏇式上升，將其貫穿下去。我本人是極為贊同這種做法的。一則，現代數學中傅裏葉分析無處不在，既在純數學，如數論的各個方麵都有深入的應用，又在應用數學中是絕對的基礎工具。二則，傅裏葉分析不光有用，其本身的內容，可以說，就能夠把數學中的幾大主要思想都體現齣來。這樣，學生們先學這門課，對數學就能有鮮活的瞭解，既知道它的用處，又能夠“連續”地欣賞到數學中的各種大思想、大美妙。接下來，是學同樣集理論優美和深刻應用於一體的復分析。學完這兩門課，學生已經有瞭相當多的例子和感覺，既懂得其用又懂得其妙。這樣，再學後麵比較抽象的實分析和泛函分析時，就自然得多，動機也充分得多。 　　這種教法目前在國內還很欠缺，也缺乏相應的教材。這主要是因為我們的教育體製還存在一些問題，比如數學係研究生的入學考試，以往最關鍵的是初試，但初試隻考數學分析和高等代數，也就是本科生低年級的課程。長此以往，中國的大多數本科生，隻用功在這兩門低年級課程上，而在高年級的後續課程，以及現代數學的眼界上就有很大的欠缺。這樣，勢必導緻他們在研究生階段後勁不足，需要補的東西過多，因而疲於奔命。

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好書推薦…

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經典調和分析數目 必讀的 贊贊

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2. 本書應用將近萬次

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書寫的很好 是好書

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很經典的書！

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不錯，不錯。

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