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**Theory of Orlicz spaces.**
*(English)*
Zbl 0724.46032

The first monograph on Orlicz spaces is the classical book of M. A. Krasnosel’skiĭ and Ya. B. Rutitskiĭ [Convex functions and Orlicz spaces. Moscow: Fizmatgiz (1958; Zbl 0095.09103)]. As every book of Krasnosel’skiĭ, this work was pioneering, opened the way to a new branch of linear and nonlinear analysis, and entailed a wealth of papers on both the theory and applications of these spaces. In spite of the vast literature concerned with special aspects, there were only very few papers of survey type in the sequel; a good example is the well-written brief overview by J. J. Grobler [From A to Z, Proc. Symp. in Honour of A. C. Zaanen, Leiden 1982, Math. Cent. Tracts 149, 1–12 (1982; Zbl 0492.46024)].

This is the most complete, best readable, and most systematic account of the theory of Orlicz spaces available at present. It consists of seven chapters containing the basic notions and results, and three chapters on further developments and related questions.

The first chapter is introductory, the second chapter is concerned with Young functions, their growth properties, and their comparison. Interestingly, this chapter contains solutions of many problems raised in Krasnosel’skiĭ’s above-mentioned monograph. In the third chapter, Orlicz spaces are introduced, equipped with both the Orlicz norm and the Luxemburg norm (“gauge norm”), and the usual closure of \(L_{\infty}\) with respect to these norms is discussed. Linear functionals and weak topologies are studied in the fourth chapter; here also important notions as reflexivity and weak compactness come in. A rather important and advanced part of the theory concerns comparison and imbedding theorems, which are dealt with in the fifth chapter. A crucial point is here the “interplay” between the analytical properties of the imbedding operator between two Orlicz spaces, on the one hand, and the rate of growth between the corresponding Young functions, on the other. While the authors emphasize, in this connection, the role of various types of products (pointwise, tensor, tc.) of Orlicz spaces, they do not mention, surprisingly, the interesting theory of multiplicators [see e.g. P. P. Zabreiko’s and the reviewer’s book on nonlinear superposition operators. Cambridge: Cambridge University Press (1990; Zbl 0701.47041)]. Such constructions are useful in the analysis of linear (in particular, integral) operators between Orlicz spaces, which is the topic of the sixth chapter. Here the authors show how the classical Riesz-Thorin and Marcinkiewicz-Stein-Weiss interpolation theorems for Lebesgue spaces carry over to Orlicz spaces. The seventh chapter, finally, is concerned with the geometry of Orlicz spaces (such as rotundity, uniform convexity, etc.) which is much more involved than that of Lebesgue spaces. This chapter closes the main part of the book.

The last 100 pages of the book consist of three additional chapters, with the headings “Orlicz spaces based on sets of measures”, “Some related function spaces” (e.g., Hardy-Orlicz or Orlicz-Sobolev spaces), and “Generalized Orlicz spaces”.

The book is very clear and readable throughout. Each chapter starts with a brief summary of the contents and closes with interesting bibliographical notes. It can be strongly recommended to all specialists in functional analysis and operator theory.

This is the most complete, best readable, and most systematic account of the theory of Orlicz spaces available at present. It consists of seven chapters containing the basic notions and results, and three chapters on further developments and related questions.

The first chapter is introductory, the second chapter is concerned with Young functions, their growth properties, and their comparison. Interestingly, this chapter contains solutions of many problems raised in Krasnosel’skiĭ’s above-mentioned monograph. In the third chapter, Orlicz spaces are introduced, equipped with both the Orlicz norm and the Luxemburg norm (“gauge norm”), and the usual closure of \(L_{\infty}\) with respect to these norms is discussed. Linear functionals and weak topologies are studied in the fourth chapter; here also important notions as reflexivity and weak compactness come in. A rather important and advanced part of the theory concerns comparison and imbedding theorems, which are dealt with in the fifth chapter. A crucial point is here the “interplay” between the analytical properties of the imbedding operator between two Orlicz spaces, on the one hand, and the rate of growth between the corresponding Young functions, on the other. While the authors emphasize, in this connection, the role of various types of products (pointwise, tensor, tc.) of Orlicz spaces, they do not mention, surprisingly, the interesting theory of multiplicators [see e.g. P. P. Zabreiko’s and the reviewer’s book on nonlinear superposition operators. Cambridge: Cambridge University Press (1990; Zbl 0701.47041)]. Such constructions are useful in the analysis of linear (in particular, integral) operators between Orlicz spaces, which is the topic of the sixth chapter. Here the authors show how the classical Riesz-Thorin and Marcinkiewicz-Stein-Weiss interpolation theorems for Lebesgue spaces carry over to Orlicz spaces. The seventh chapter, finally, is concerned with the geometry of Orlicz spaces (such as rotundity, uniform convexity, etc.) which is much more involved than that of Lebesgue spaces. This chapter closes the main part of the book.

The last 100 pages of the book consist of three additional chapters, with the headings “Orlicz spaces based on sets of measures”, “Some related function spaces” (e.g., Hardy-Orlicz or Orlicz-Sobolev spaces), and “Generalized Orlicz spaces”.

The book is very clear and readable throughout. Each chapter starts with a brief summary of the contents and closes with interesting bibliographical notes. It can be strongly recommended to all specialists in functional analysis and operator theory.

Reviewer: Jürgen Appell (Würzburg)

### MSC:

46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |

47G10 | Integral operators |

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

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |

46M35 | Abstract interpolation of topological vector spaces |