Extremal Problems in Interpolation Theory Whitney Besicovitch Coverings and Singular Integrals Preface Introduction Definitions notation and some standard facts Part 1. Background Chapter 1. Classical Calderón–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. 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.