##
**Interpolation functors and interpolation spaces. Vol. 1.**
*(English)*
Zbl 0743.46082

North-Holland Mathematical Library. 47. Amsterdam etc.: North-Holland. xv, 718 p. (1991).

The theory of interpolation spaces has its origin in the classical theorems of Riesz, Thorin and Marcinkiewicz. It was first extensively developed and applied as a tool in analysis through the fundamental contributions of, among others, N. Aronszajn, A. P. Calderón, E. Gagliardo, S. G. Krein, J. L. Lions, J. Peetre, during the late fifties and early sixties. Although historically interpolation theory was developed as an abstract tool to solve concrete problems in partial differential equations and harmonic analysis, the theory has evolved over the last 30 years as an independent field of study with its own methods and problems. In particular, the last 15 years or so have witnessed a refocusing of research on its foundations. The theory has also established itself as an important tool in other areas of analysis including partial differential equations, harmonic analysis, approximation theory, operator theory, numerical analysis, geometry of Banach spaces, etc.

The book under review is the first volume, of a projected series of three, dealing with the theory of interpolation spaces and its applications. It aims to present the important theoretical developments that have taken place in the foundations of the theory during the last 15 years. The present volume is devoted to the real method of interpolation, while the second and third volumes will deal, respectively, with the complex method and the applications of the theory to other areas of mathematics. The volume presents a definitive account concentrating mainly in the real method of interpolation, a part of the theory where the authors have made a fundamental contribution through their solution of the so called \(K\) divisibility problem. The material is presented in 4 chapters, the first chapter deals with the classical interpolation theorems, the second with general interpolation functors while the remaining two are devoted to the real method. Since it covers the foundations of the theory in great detail the book should provide useful for beginners. The book should also become an invaluable tool for experts in the field since it contains new results and innovative approaches to the basic aspects of interpolation. Particularly useful features are the inclusion at the end of each chapter of a bibliographical section where credits and references to the literature are provided, and the collection of open problems presented at the end of chapters 2, 3 and 4.

We shall now examine the contents in somewhat more detail. The first chapter deals with the classical interpolation theorems of Riesz, Thorin, and Marcinkiewicz. The second chapter presents an abstract setting for interpolation theory and develops the Aronszajn-Gagliardo theory as it has evolved in the last ten years through the contributions of Janson and Ovchinnikov, among others. A major effect has been placed to use the language of the theory of categories. In particular the authors have included a theory of “computable functors” a natural idea that allows them to formulate a number of new results in interpolation theory. In the last two chapters the authors develop the modern theory of the real method. A prominent role is played here by the theory of \(K\)-divisibility of the \(K\) functional, and the strong version of the “fundamental lemma of interpolation theory” as developed by the authors and M. Cwikel among others. In chapter three the authors also briefly discuss the interpolation theory of some concrete scales of spaces and in chapter four they also discuss the stability of the real method and the important theory of Calderon couples.

The complete table of contents is as follows:

Chapter 1: Classical Interpolation methods; Introduction, The space of measurable functions, The spaces \(L_ p\), M. Riesz “convexity theorem”, Some generalizations, The three circles theorem, The Riesz- Thorin theorem, Generalizations, The spaces \(L_{p,q}\), The Marcinkiewicz theorem, Comments and supplements, References, Supplements: The Riesz constant, The Riesz theorem as a corollary of theorem 1.7.1, The meaning of the Theorems of Riesz and Thorin for \(p_ i<1\), Interpolation of quasilinear operators, Interpolation of spaces \(H_ p\);

Chapter 2: Interpolation spaces and Interpolation functors; Banach couples, Intermediate spaces and Interpolation spaces, Interpolation functors, Duality, Minimal and computable functors, Interpolation methods, Comments and additional remarks, References, Additional remarks, Category language, Further extension of the concept of couple, Density of the set of dual operators for finite dimensional couples, some unsolved problems;

Chapter 3: The real interpolation method, The \(K\) and \(J\) functional, \(K\) divisibility, The \(K\) method, The \(J\) method, Equivalence theorems, Theorems on density and relative completeness, duality theorem, computations, comments and supplements, References, Supplements (Computation of \(K\) functional and Real method spaces), General approach, Couples of Banach lattices, BMO and \(H_ p\), Differentiable and smooth functions, Interpolation of operator spaces, Some unsolved problems;

Chapter 4: Selected questions of the theory of the real interpolation method, Nonlinear interpolation, Real interpolation functors, Stability of real method functors, Calderón couples, Inverse problems of real interpolation, Banach geometry of real method spaces, Comments and supplements, Real method for finite sets of Banach spaces, Calderón couples, Some unsolved problems; References; Subject index.

In conclusion I fully agree with J. Peetre’s statement in the preface of the book asserting that (the book) “will set the mark for all serious work in this area of mathematics for the coming decade, if not longer”.

The book under review is the first volume, of a projected series of three, dealing with the theory of interpolation spaces and its applications. It aims to present the important theoretical developments that have taken place in the foundations of the theory during the last 15 years. The present volume is devoted to the real method of interpolation, while the second and third volumes will deal, respectively, with the complex method and the applications of the theory to other areas of mathematics. The volume presents a definitive account concentrating mainly in the real method of interpolation, a part of the theory where the authors have made a fundamental contribution through their solution of the so called \(K\) divisibility problem. The material is presented in 4 chapters, the first chapter deals with the classical interpolation theorems, the second with general interpolation functors while the remaining two are devoted to the real method. Since it covers the foundations of the theory in great detail the book should provide useful for beginners. The book should also become an invaluable tool for experts in the field since it contains new results and innovative approaches to the basic aspects of interpolation. Particularly useful features are the inclusion at the end of each chapter of a bibliographical section where credits and references to the literature are provided, and the collection of open problems presented at the end of chapters 2, 3 and 4.

We shall now examine the contents in somewhat more detail. The first chapter deals with the classical interpolation theorems of Riesz, Thorin, and Marcinkiewicz. The second chapter presents an abstract setting for interpolation theory and develops the Aronszajn-Gagliardo theory as it has evolved in the last ten years through the contributions of Janson and Ovchinnikov, among others. A major effect has been placed to use the language of the theory of categories. In particular the authors have included a theory of “computable functors” a natural idea that allows them to formulate a number of new results in interpolation theory. In the last two chapters the authors develop the modern theory of the real method. A prominent role is played here by the theory of \(K\)-divisibility of the \(K\) functional, and the strong version of the “fundamental lemma of interpolation theory” as developed by the authors and M. Cwikel among others. In chapter three the authors also briefly discuss the interpolation theory of some concrete scales of spaces and in chapter four they also discuss the stability of the real method and the important theory of Calderon couples.

The complete table of contents is as follows:

Chapter 1: Classical Interpolation methods; Introduction, The space of measurable functions, The spaces \(L_ p\), M. Riesz “convexity theorem”, Some generalizations, The three circles theorem, The Riesz- Thorin theorem, Generalizations, The spaces \(L_{p,q}\), The Marcinkiewicz theorem, Comments and supplements, References, Supplements: The Riesz constant, The Riesz theorem as a corollary of theorem 1.7.1, The meaning of the Theorems of Riesz and Thorin for \(p_ i<1\), Interpolation of quasilinear operators, Interpolation of spaces \(H_ p\);

Chapter 2: Interpolation spaces and Interpolation functors; Banach couples, Intermediate spaces and Interpolation spaces, Interpolation functors, Duality, Minimal and computable functors, Interpolation methods, Comments and additional remarks, References, Additional remarks, Category language, Further extension of the concept of couple, Density of the set of dual operators for finite dimensional couples, some unsolved problems;

Chapter 3: The real interpolation method, The \(K\) and \(J\) functional, \(K\) divisibility, The \(K\) method, The \(J\) method, Equivalence theorems, Theorems on density and relative completeness, duality theorem, computations, comments and supplements, References, Supplements (Computation of \(K\) functional and Real method spaces), General approach, Couples of Banach lattices, BMO and \(H_ p\), Differentiable and smooth functions, Interpolation of operator spaces, Some unsolved problems;

Chapter 4: Selected questions of the theory of the real interpolation method, Nonlinear interpolation, Real interpolation functors, Stability of real method functors, Calderón couples, Inverse problems of real interpolation, Banach geometry of real method spaces, Comments and supplements, Real method for finite sets of Banach spaces, Calderón couples, Some unsolved problems; References; Subject index.

In conclusion I fully agree with J. Peetre’s statement in the preface of the book asserting that (the book) “will set the mark for all serious work in this area of mathematics for the coming decade, if not longer”.

Reviewer: M. Milman (Boca Raton)

### MSC:

46M35 | Abstract interpolation of topological vector spaces |

46B70 | Interpolation between normed linear spaces |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |