Extremal problems in interpolation theory, Whitney-Besicovitch coverings, and singular integrals
QA281 .K57 2013
- Unknown QA281 .K57 2013
- Kruglyak, Natan.
- Includes bibliographical references (p. 305-311) and index.
- Preface.- Introduction.- Definitions, notation, and some standard facts.- Part 1. Background.- Chapter 1. Classical Calderon-Zygmund decomposition and real interpolation.- Chapter 2. Singular integrals.- Chapter 3. Classical covering theorems.- Chapter 4. Spaces of smooth functions and operators on them.- Chapter 5. Some topics in interpolation.- Chapter 6. Regularization for Banach spaces.- Chapter 7. Stability for analytic Hardy spaces.- Part 2. Advanced theory.- Chapter 8. Controlled coverings.- Chapter 9. Construction of near-minimizers.- Chapter 10. Stability of near-minimizers.- Chapter 11. The omitted case of a limit exponent.- Chapter A. Appendix. Near-minimizers for Brudnyi and Triebel-Lizorkin spaces.- Notes and remarks.- Bibliography.- Index.
- (source: Nielsen Book Data)
- Publisher's Summary
- 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 Calderon-Zygmund decomposition. These new Calderon-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 Calderon-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 Calderon-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.
(source: Nielsen Book Data)
- Publication date
- Sergey Kislyakov, Natan Kruglyak.
- Monografie matematyczne ; new ser., v. 74