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Book review

Author(s) Rao, M.M.
Ren, Z.D.
Title Theory of Orlicz Spaces
Publisher Marcel Dekker
Year of publication 1991
Reviewed by S. Cobzas

The Orlicz spaces, defined first in 1931 by Z.W. Birnbaum and W. Orlicz and immediately developed by Orlicz himself in a series of papers, are natural extensions of Lebesgue L^p-spaces. Originally an Orlicz space was defined as a space of measurable functions f satisfying the condition <formula>, for some <formula>, where <formula> is a convex function increasing to infinity for t increasing to infinity, called Young function. For <formula> one obtains the usual L^p-spaces . Later on the convexity assumption on <formula> was frequently ommited.

The development of the theory is motivated by the wealth of properties of these spaces studied from a functional analyst viewpoint (as Banach or, more general, as metric linear spaces) as well by their numerous applications to differential and integral equations with kernels of nonpower type. Recently a new interest in the study of these spaces emerged in connection to their convexity and smoothness properties (especially in the vector-valued case), Boyd indices and the theory of rearrangement invariant function spaces.

The first systematic presentation of the theory, in the framework of bounded Lebesgue measurable sets in R^n and nice Young functions (N-functions), was done by M.A. Krasnosel'skii and Yu. B. Rutickii in a monograph published in Russian in 1958. The English version of this book - "Convex Functions and Orlicz Spaces" , P. Noordhoff Ltd., Groningen 1961, raised the interest of a larger audience. Beside a systematic account of basic facts of Orlicz space theory, Krasnosel'skii and Rutickii's book contains also a number of open problems. Since then the theory has progressed in several directions and most of these problems were solved.

The book under review concentrates on the developments since the appearance of the above mentioned book. The treatement is more general including abstract measure spaces. The book is divided into ten chapters. The basic result of the theory are presented in the first seven chapters and the last three chapters contain further developments. The chapters are headed as follows: I. Introduction and Preliminaries; II. Some classes of Young's Functions; III. Orlicz Function Spaces; IV. Linear Functionals and Weak Topologies; V. Comparison of Orlicz Spaces; VI. Analysis of Linear Operators; VII. Geometry and Smoothness; VIII. Orlicz Spaces Based on Sets of Measures; IX. Some Related Function Spaces; X. Generalized Orlicz Spaces.

The book is clearly written, contains a wealth of results presented in a form accessible to a graduate student, bringing him in a short time to the frontier of current mathematical research. It is printed in excellent typographical conditions and, undoubtely, will remain a fundamental reference work in the field.

A good companion in reading it will be the book by N.M. Kozlowski, Modular Function Spaces, Monographs and Textbooks in Pure and Applied Mathematics v. 122, Marcel Dekker 1988, where it is presented the theory of a class of function spaces containing Orlicz spaces and having also many applications in complex analysis, approximation theory, nonlinear operators.