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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.
......

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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歡迎您撰寫這本書的原創書評

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評分☆☆☆☆☆

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調和分析入門的好書

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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。

評分☆☆☆☆☆

這個小鎮的每個人，都在經曆內心激烈的衝擊。他們一方麵到處打聽那些勇敢邁進舞廳的人，打聽那白白的大腿和金色的牆麵，另一方麵又馬上擺齣一種道貌岸然的神情，嚴肅地加以批評。但誰都知道，隨著沸騰的財富，每個人內心的各種欲求在湧動。財富消除瞭飢餓感和貧窮感，放鬆瞭人心。以前，貧窮像個設置在內心的安全閥門，讓每個人都對隱藏在其中的各種欲望不聞不問。然而現在，每個人就要直接麵對自己瞭。

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