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

This book proposes a unified method for the construction of near-minimizers for certain important functions, which arise in approximation, harmonic analysis and ill-posed problems and are most widely used in interpolation theory.

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 CalderónZygmund decomposition. These new CalderónZygmund 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 CalderónZygmund 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 CalderónZygmund 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.

Quick and concise introduction to several important classical topics of real analysis Exposition of powerful results of recent research in a self-contained manner, making them accessible to beginners Presents results not yet available in existing literature Contains descriptions of new techniques which may be useful in other research problems ? Includes supplementary material: sn.pub/extras

Contenu

Preface.- Introduction.- Definitions, notation, and some standard facts.- Part 1. Background.- Chapter 1. Classical CalderónZygmund 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 TriebelLizorkin spaces.- Notes and remarks.- Bibliography.- Index.