Extremal Problems in Interpolation Theory, Whitney-Besicovitch Coverings, and Singular Integrals

  • Sergey Kislyakov
  • Natan Kruglyak

Part of the Monografie Matematyczne book series (MONOGRAFIE, volume 74)

Table of contents

  1. Front Matter
    Pages i-x
  2. Background

    1. Front Matter
      Pages 21-22
    2. Sergey Kislyakov, Natan Kruglyak
      Pages 23-45
    3. Sergey Kislyakov, Natan Kruglyak
      Pages 47-64
    4. Sergey Kislyakov, Natan Kruglyak
      Pages 65-89
    5. Sergey Kislyakov, Natan Kruglyak
      Pages 91-122
    6. Sergey Kislyakov, Natan Kruglyak
      Pages 123-144
    7. Sergey Kislyakov, Natan Kruglyak
      Pages 145-151
    8. Sergey Kislyakov, Natan Kruglyak
      Pages 153-157
  3. Advanced theory

    1. Front Matter
      Pages 159-160
    2. Sergey Kislyakov, Natan Kruglyak
      Pages 161-196
    3. Sergey Kislyakov, Natan Kruglyak
      Pages 197-219
    4. Sergey Kislyakov, Natan Kruglyak
      Pages 221-242
    5. Sergey Kislyakov, Natan Kruglyak
      Pages 243-272
  4. Back Matter
    Pages 273-316

About this book

Introduction

In this book we suggest a unified method of constructing near-minimizers for certain important functionals arising in approximation, harmonic analysis and ill-posed problems and most widely used in interpolation theory. The constructions are based on far-reaching refinements of the classical Calderón–Zygmund decomposition. These new Calderón–Zygmund decompositions in turn are produced with the help of new covering theorems that combine many remarkable features of classical results established by Besicovitch, Whitney and Wiener. In many cases the minimizers constructed in the book are stable (i.e., remain near-minimizers) under the action of Calderón–Zygmund singular integral operators.

The book is divided into two parts. While the new method is presented in great detail in the second part, the first is mainly devoted to the prerequisites needed for a self-contained presentation of the main topic. There we discuss the classical covering results mentioned above, various spectacular applications of the classical Calderón–Zygmund decompositions, and the relationship of all this to real interpolation. It also serves as a quick introduction to such important topics as spaces of smooth functions or singular integrals.

Keywords

Calderón-Zygmund decomposition covering theorem near-minimizer real interpolation singular integral operator

Authors and affiliations

  • Sergey Kislyakov
    • 1
  • Natan Kruglyak
    • 2
  1. 1.St. Petersburg Branch, Department of MathematicsSteklov Mathematical InstituteSt. PetersburgRussia
  2. 2., Department of MathematicsLinköping UniversityLinköpingSweden

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-0348-0469-1
  • Copyright Information Springer Basel 2013
  • Publisher Name Birkhäuser, Basel
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-0348-0468-4
  • Online ISBN 978-3-0348-0469-1
  • About this book