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ISBN：9780691080796

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简介

简介

Summary: Publisher Summary 1 Singular integrals are among the most interesting and important objects of study in analysis, one of the three main branches of mathematics. They deal with real and complex numbers and their functions. In this book, Princeton professor Elias Stein, a leading mathematical innovator as well as a gifted expositor, produced what has been called the most influential mathematics text in the last thirty-five years. One reason for its success as a text is its almost legendary presentation: Stein takes arcane material, previously understood only by specialists, and makes it accessible even to beginning graduate students. Readers have reflected that when you read this book, not only do you see that the greats of the past have done exciting work, but you also feel inspired that you can master the subject and contribute to it yourself.Singular integrals were known to only a few specialists when Stein's book was first published. Over time, however, the book has inspired a whole generation of researchers to apply its methods to a broad range of problems in many disciplines, including engineering, biology, and finance.Stein has received numerous awards for his research, including the Wolf Prize of Israel, the Steele Prize, and the National Medal of Science. He has published eight books with Princeton, including Real Analysisin 2005.

Table Of Contents:
Preface vii
Notation ix

Some Fundamental Notions of Real-Variable Theory 3(23)

The maximal function 4(8)

Behavior near general points of measurable sets 12(4)

Decomposition in cubes of open sets in Rn 16(4)

An interpolation theorem for Lp 20(2)

Further results 22(4)

Singular Integrals 26(28)

Review of certain aspects of harmonic analysis in Rn 27(1)

Singular integrals: the heart of the matter 28(6)

Singular integrals: some extensions and variants of the preceding 34(4)

Singular integral operators which commute with dilations 38(7)

Vector-valued analogues 45(3)

Further results 48(6)

Riesz Transforms, Poisson Integrals, and Spherical harmonics 54(27)

The Riesz transforms 54(6)

Poisson integrals and approximations to the identity 60(8)

Higher Riesz transforms and spherical harmonics 68(9)

Further results 77(4)

The Littlewood-Paley Theory and Multipliers 81(35)

The Littlewood-Paley g-function 82(4)

The function gλ* 86(8)

Multipliers (first version) 94(5)

Application of the partial sums operators 99(4)

The dyadic decomposition 103(5)

The Marcinkiewicz multiplier theorem 108(4)

Further results 112(4)

Differentiability Properties in Terms of Function Spaces 116(50)

Riesz potentials 117(4)

The Sobolev spaces, Lpk(Rn) 121(9)

Bessel potentials 130(11)

The spaces Λα of Lipschitz continuous functions 141(9)

The spaces Λα p,q 150(9)

Further results 159(7)

Extensions and Restrictions 166(30)

Decomposition of open sets into cubes 167(3)

Extension theorems of Whitney type 170(10)

Extension theorem for a domain with minimally smooth boundary 180(12)

Further results 192(4)

Return to the Theory of Harmonic Functions 196(44)

Non-tangential convergence and Fatou's theorem 196(9)

The area integral 205(12)

Application of the theory of Hp spaces 217(18)

Further results 235(5)

Differentiation of Functions 240(39)

Several notions of pointwise differentiability 241(5)

The splitting of functions 246(4)

A characterization of differentiability 250(7)

Desymmetrization principle 257(5)

Another characterization of differentiability 262(4)

Further results 266(5)
Appendices

A. Some Inequalities 271(1)

B. The Marcinkiewicz Interpolation Theorem 272(2)

C. Some Elementary Properties of Harmonic Functions 274(2)

D. Inequalities for Rademacher Functions 276(3)
Bibliography 279(10)
Index 289

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