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 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.

From the reviews:

The monograph is a good source of information on spaces of smooth functions and real interpolation methods. the monograph could be a good reference on methods and results in the area. The book is highly recommended for specialists in approximation theory, harmonic analysis, functional analysis, and related areas. the monograph should be interesting for specialists in applied mathematics and computer science. Graduate students specializing in analysis should find many interesting topics here, as well as motivation for further research. (Alexander V. Tovstolis, Mathematical Reviews, August, 2013) This book consists of two parts. Each chapter closes with comments, historical remarks and further references. Everybody who is interested in this topic should consult this important book. Without doubt, this monograph will stimulate further research. (Manfred Tasche, zbMATH, Vol. 1270, 2013)

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.