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

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 Calder?nZygmund decomposition. These new Calder?nZygmund 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 Calder?nZygmund 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 Calder?nZygmund 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.

Preface.- Introduction.- Definitions, notation, and some standard facts.- Part 1. Background.- Chapter 1. Classical Calder?nZygmund 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.- NolS†