The theory of interpolation spaces has its origin in the classical work of Riesz and Marcinkiewicz but had its first flowering in the years around 1960 with the pioneering work of Aronszajn, Calderón, Gagliardo, Krein, Lions and a few others. It is interesting to note that what originally triggered off this avalanche were concrete problems in the theory of elliptic boundary value problems related to the scale of Sobolev spaces. Later on, applications were found in many other areas of mathematics: harmonic analysis, approximation theory, theoretical numerical analysis, geometry of Banach spaces, nonlinear functional analysis, etc. Besides this the theory has a considerable internal beauty and must by now be regarded as an independent branch of analysis, with its own problems and methods. Further development in the 1970s and 1980s included the solution by the authors of this book of one of the outstanding questions in the theory of the real method, the K -divisibility problem. In a way, this book harvests the results of that solution, as well as drawing heavily on a classic paper by Aronszajn and Gagliardo, which appeared in 1965 but whose real importance was not realized until a decade later. This includes a systematic use of the language, if not the theory, of categories. In this way the book also opens up many new vistas which still have to be explored.
This volume is the first of three planned books. Volume II will deal with the complex method, while Volume III will deal with applications.
Author(s): Unknown Author
Series: North-Holland Mathematical Library 1
Edition: 1
Publisher: North Holland
Year: 1991
Language: English
Pages: 735
Interpolation Functors and Interpolation Spaces, Volume I......Page 4
Copyright Page......Page 5
PREFACE......Page 6
PREFACE TO THE ENGLISH TRANSLATION......Page 12
Contents......Page 14
1.1. Introduction......Page 18
1.2. The Space of Measurable Functions......Page 20
1.3. The Spaces Lp......Page 25
1.4. M. Riesz’s “Convexity Theorem”......Page 30
1.5. Some Generalizations......Page 40
1.6. The Three Circles Theorem......Page 48
1.7. The Riesz-Thorin Theorem......Page 51
1.8. Generalizations......Page 56
1.9. The Spaces Lpq......Page 65
1.10. The Marcinkiewicz Theorem......Page 83
1.11. Comments and Supplements......Page 101
2.1. Banach Couples......Page 108
2.2. Intermediate and Interpolation Spaces......Page 130
2.3. Interpolation Functors......Page 157
2.4. Duality......Page 191
2.5. Minimal and Computable Functors......Page 228
2.6. Interpolation Methods......Page 262
2.7. Comments and Additional Remarks......Page 271
3.1. The K- and J-functionals......Page 306
3.2. K-divisibility......Page 332
3.3. The K-method......Page 355
3.4. The J-method......Page 377
3.5. Equivalence Theorems......Page 404
3.6. Theorems on Density and Relative Completeness......Page 426
3.7. Duality Theorem......Page 439
3.8. Computations......Page 455
3.9. Comments and Supplements......Page 476
4.1. Nonlinear Interpolation......Page 510
4.2. Real Interpolation Functors......Page 522
4.3. Stability of Real Method Functors......Page 570
4.4. Calderón Couples......Page 595
4.5. Inverse Problems of Real Interpolation......Page 627
4.6. Banach Geometry of Real-Method Spaces......Page 651
4.7. Comments and Supplements......Page 684
REFERENCES......Page 704
SUBJECT INDEX......Page 732
