奇異積分和函數(shù)的可微性

內(nèi)容概要

　　本書內(nèi)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.

書籍目錄

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

章節(jié)摘錄

　　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.　?。?）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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