Foundations of symmetric spaces of measurable functions. Lorentz, Marcinkiewicz and Orlicz spaces
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 of the Orlicz space generated by \(\Phi\). The text also contain several complements, such as the study of symmetric spaces on the unit interval, or in sequence spaces, the Lebesgue spaces \(L^p\), \(0<p<1\), and the classical Lorentz spaces \(L^{p,q}\), among others. The book is structured in such a way that it should correspond to a one-semester special course of lectures, and it is presented with only previous requirements in basic concepts and theorems of measure theory and functional analysis.
scientific article; zbMATH DE number 6610672