Foundations of symmetric spaces of measurable functions. Lorentz, Marcinkiewicz and Orlicz spaces
type: Item
instance of: scholarly article
title: Foundations of symmetric spaces of measurable functions. Lorentz, Marcinkiewicz and Orlicz spaces
zbMATH Open document ID: 1361.42001
DOI: 10.1007/978-3-319-42758-4
author: Genady Ya. Grabarnik
author: M. A. Muratov
author: Yu. S. Pashkova
author: Ben-Zion Rubshtein
published in: Developments in Mathematics
publication date: 2016-08-10T00:00:00Z
review text: This book represents a consistent reference on the basic theory of symmetric spaces of measurable functions also known as rearrangement invariant function spaces. The theory of these spaces, and all the concepts involved, are introduced, including a detailed study of Lorentz, Marcinkiewicz and Orlicz spaces, at the same time as examples and counterexamples. The basic functional properties for these spaces are studied, such as completeness, separability and duality. The text contains many illustrations in order to clarify concepts and make the theory more comprehensive and understandable. The book is essentially self-contained and it is presented in four parts whose contain is the following: Part I deals with the classical Lebesgue spaces including \(L^1\cap L^{\infty}\) and \(L^1+ L^{\infty}\) which plays an important role in the theory of rearrangement invariant spaces, as it is evidenced in the embedding theorem which is included in Part II, where also the notion of fundamental function relative to a symmetric space \(X\) is introduced. In this second part, the main functional characteristics relative to these spaces are studied such as the dual and the associated spaces and properties such as order continuity in norm, or monotonically completeness in norm as equivalent to separability and reflexivity, respectively. Part III is devoted to the study of Lorentz and Marcinkiewicz spaces. Besides its functional characteristics, and previous to the construction of Marcinkiewicz spaces defined in terms of the maximal function, they are studied properties of fundamental functions as examples of quasiconcave functions and properties that reveal the importance of least concave majorant property in the study of certain functional aspects . Part IV addresses the study of Orlicz spaces and, in particular, they are presented results concerning the equivalence of the fact that the Young function \(\Phi\) satisfies a \(\Delta_2\) condition and the separability or the reflexivity…
reviewed by: Santiago Boza
Mathematics Subject Classification ID: 42-01
Mathematics Subject Classification ID: 46-01
zbMATH DE Number: 6610672
zbMATH Keywords: symmetric spaces
zbMATH Keywords: rearrangement invariant spaces
zbMATH Keywords: Lebesgue spaces
zbMATH Keywords: Lorentz spaces
zbMATH Keywords: Marcinkiewicz spaces
zbMATH Keywords: Orlicz spaces
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-42758-4
OpenAlex ID: W2560822358
description: scientific article; zbMATH DE number 6610672