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E-raamat: Singular Integrals and Differentiability Properties of Functions

  • Formaat: 304 pages
  • Sari: Princeton Mathematical Series
  • Ilmumisaeg: 02-Jun-2016
  • Kirjastus: Princeton University Press
  • Keel: eng
  • ISBN-13: 9781400883882
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  • Formaat: 304 pages
  • Sari: Princeton Mathematical Series
  • Ilmumisaeg: 02-Jun-2016
  • Kirjastus: Princeton University Press
  • Keel: eng
  • ISBN-13: 9781400883882
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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 Analysis in 2005.

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