Orlicz spaces and interpolation.

*(English)*Zbl 0874.46022
Seminários de Matemática. 5. Campinas, SP: Univ. Estadual de Campinas, Dep. de Matemática, 206 p. (1989).

The author is known as one of the leading specialists in interpolation theory. The present monograph gives a detailed account of the state-of-the-art of some aspects of the theory of Orlicz spaces, as well as interpolation theory for operators in Orlicz spaces. It consists of 15 chapters with the following headings: 1. Modular spaces; 2. Examples of modulars; 3. Orlicz spaces and Orlicz classes; 4. Separability of Orlicz spaces; 5. Bounded and convex sets in \(L_\varphi\); 6. Existence and nonexistence of continuous linear functionals; 7. Nonexistence of nontrivial compact operators; 8. Complementary function and the Orlicz norm; 9. General form of a continuous linear functional; 10. The product of functions and the Landau theorem; 11. Indices of Orlicz spaces; 12. Orlicz spaces generated by Young’s functions; 13. Orlicz’s nonlinear interpolation theorem and applications; 14. Interpolation of Orlicz spaces; 15. Calderón-Lozanovskij spaces and interpolation of operators.

After the pioneering book by M. A. Krasnosel’skij and Ya. B. Rutitskij [Convex functions and Orlicz spaces, Moscow (1958; Zbl 0084.10104); for the review of the English translation from 1954 see Zbl 0095.09103], the monograph by J. Musielak [Orlicz spaces and modular spaces, Berlin, Springer (1983; Zbl 0557.46020)], and the more recent book by M. M. Rao and Z. D. Ren [Theory of Orlicz spaces, New York, Dekker (1991; Zbl 0724.46032)], this is another monograph which is very suitable for the reader who wants to learn some basic facts without getting drowned in technical details. The material is carefully chosen, the presentation is clear and understandable throughout, and the abstract theorems are illustrated by many remarks and illuminating examples. This book may be warmly recommended to anybody who is interested in functional analysis and operator theory. The more specialized reader may find new material on the subject in some new papers of the author and his collaborators [Stud. Math. 95, No. 1, 43-58 (1989; Zbl 0713.46047), Mat. Z. 208, No. 1, 57-63 (1991; Zbl 0738.46013), Bull. Pol. Acad. Sci., Math. 37, No. 7-12, 453-457 (1989; Zbl 0758.46043), Bull. Pol. Acad. Sci., Math. 38, No. 1-12, 125-134 (1990; Zbl 0758.46043), Proc. Orlicz Mem. Conf., Oxford/MS (USA), Exp. No. 5, 1-12 (1991; Zbl 0760.46058), Boll. Unione Mat. Ital., VII. Ser. B 8, No. 1, 37-55 (1994; Zbl 0821.46024), Indag. Math. New Ser. 3, No. 3, 313-321 (1992; Zbl 0802.46042), Math. Nachr. 178, 81-101 (1996; Zbl 0848.46013)].

After the pioneering book by M. A. Krasnosel’skij and Ya. B. Rutitskij [Convex functions and Orlicz spaces, Moscow (1958; Zbl 0084.10104); for the review of the English translation from 1954 see Zbl 0095.09103], the monograph by J. Musielak [Orlicz spaces and modular spaces, Berlin, Springer (1983; Zbl 0557.46020)], and the more recent book by M. M. Rao and Z. D. Ren [Theory of Orlicz spaces, New York, Dekker (1991; Zbl 0724.46032)], this is another monograph which is very suitable for the reader who wants to learn some basic facts without getting drowned in technical details. The material is carefully chosen, the presentation is clear and understandable throughout, and the abstract theorems are illustrated by many remarks and illuminating examples. This book may be warmly recommended to anybody who is interested in functional analysis and operator theory. The more specialized reader may find new material on the subject in some new papers of the author and his collaborators [Stud. Math. 95, No. 1, 43-58 (1989; Zbl 0713.46047), Mat. Z. 208, No. 1, 57-63 (1991; Zbl 0738.46013), Bull. Pol. Acad. Sci., Math. 37, No. 7-12, 453-457 (1989; Zbl 0758.46043), Bull. Pol. Acad. Sci., Math. 38, No. 1-12, 125-134 (1990; Zbl 0758.46043), Proc. Orlicz Mem. Conf., Oxford/MS (USA), Exp. No. 5, 1-12 (1991; Zbl 0760.46058), Boll. Unione Mat. Ital., VII. Ser. B 8, No. 1, 37-55 (1994; Zbl 0821.46024), Indag. Math. New Ser. 3, No. 3, 313-321 (1992; Zbl 0802.46042), Math. Nachr. 178, 81-101 (1996; Zbl 0848.46013)].

Reviewer: J.Appell (Würzburg)

##### MSC:

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

46M35 | Abstract interpolation of topological vector spaces |

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

46B70 | Interpolation between normed linear spaces |