Extremal Problems in Interpolation Theory, Whitney-Besicovitch Coverings, and Singular Integrals
By: by Sergey Kislyakov, Natan Kruglyak
Contributors: Kruglyak, Natan; SpringerLink (Online service)
Series: Monografie Matematyczne -- 74
Publisher: Springer Basel; Imprint: Birkhäuser
Published: 2013
Place of publication: Basel
Language: eng
Format: [electronic resource] /
Number of pages: 316
ISBN-13: 9783034804691
Subjects: Functional analysis; Mathematics; Approximations and Expansions; Real Functions
Description: <p>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.</p><p>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.</p>
Table of contents: title: <p>Preface; title: Introduction; title: Definitions, notation, and some standard facts; title: Part 1. Background; title: Chapter 1. Classical Calderón–Zygmund decomposition and real interpolation; title: Chapter 2. Singular integrals; title: Chapter 3. Classical covering theorems; title: Chapter 4. Spaces of smooth functions and operators on them; title: Chapter 5. Some topics in interpolation; title: Chapter 6. Regularization for Banach spaces; title: Chapter 7. Stability for analytic Hardy spaces; title: Part 2. Advanced theory; title: Chapter 8. Controlled coverings; title: Chapter 9. Construction of near-minimizers; title: Chapter 10. Stability of near-minimizers; title: Chapter 11. The omitted case of a limit exponent; title: Chapter A. Appendix. Near-minimizers for Brudnyi and Triebel–Lizorkin spaces; title: Notes and remarks; title: Bibliography; title: Index.</p>.