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Book review

Author(s) Brudnyi, Yu.A.
Krugljak, N.Ya.
Title Interpolation functors and interpolation spaces, Volume 1
Publisher North-Holland
Year of publication 1991
Reviewed by Miroslav Englis

This is an excellent book, which, moreover, comes exactly when the time is ripe for it. It was in the late seventies when, almost simultaneously, three monographs (by now classical) on interpolation theory appeared: Interpolation spaces, by Bergh and Löfström (Springer, 1976); Interpolation of linear operators, by Krein, Petunin, and Semenov (Nauka, 1978); and Interpolation theory, function spaces, differential operators, by Triebel (VEB Deutsche Verlag, 1978). During the next fifteen years, however, an important research has been taking place, and a lot of new results have been obtained. One of the major open questions, the so-called K-divisibility problem, was settled in 1981 by Brudnyi and Krugljak. Apart from presenting the traditional background as well as the new achievements of other people, a large part of the present book does harvest what has come out of this outstanding result.

The book begins with an expose (90 pp.) of the classical interpolation theory: the theorems of Riesz (on convexity), Riesz and Thorin, and Marcinkiewicz; decreasing rearrangements of functions; and a few applications (harmonic conjugation operator, Hp spaces, etc.). The next chapter (199 pp.) begins with the usual notions of interpolation couples and interpolation spaces, and introduces the category language: interpolation functors, orbits and coorbits, duality, computability, etc. The third chapter (204 pp.) discusses the real interpolation method in detail; in particular, K- and J- methods are introduced, K-divisibility is proved, then the connection between the "abstract" theory from the previous chapter and the "concrete" interpolation methods is established, etc. The last chapter (194 pp.) is devoted to selected special topics in the theory of the real interpolation (nonlinear interpolation, other interpolation methods, Calderón's construction, inverse problems, various properties of the interpolation spaces, etc.). There are 27 pages of bibliography, and an (adequate) index. The authors are planning to continue by writing Volume 2, dealing with the complex interpolation method, and Volume 3, devoted to applications.

Each chapter is concluded by comments to the relevant bibliographical references, by various "supplements" mentioning (usually without proofs) some further related material, and by a list of unsolved problems. Both the style of the exposition and the translation from Russian are very good, and so is the graphic appearance of the book.

If the book is not available in your local bookstore, it can be obtained from Elsevier Science Publishers, PO Box 211, 1000 AE Amsterdam, The Netherlands, or (in USA/Canada) from Elsevier Science Publishing, PO Box 882, Madison Square Station, New York, NY 10159, USA.

In my opinion, this book is going to become as classical as the three older monographs mentioned at the beginning of this review, if not to surpass them; and anyone whose interest lies in the interpolation theory should not hesitate to buy it.