Rubshtein B.-Z.A., Grabarnik G.Y., Muratov M.A., Pashkova Y.S. Foundations of Symmetric Spaces of Measurable Functions: Lorentz, Marcinkiewicz and Orlicz Spaces

Springer International Publishing, Switzerland, 2017. — 262 p. — (Developments in Mathematics. Volume 45) — ISBN: 3319427563Key definitions and results in symmetric spaces, particularly Lp, Lorentz, Marcinkiewicz and Orlicz spaces are emphasized in this textbook. A comprehensive overview of the Lorentz, Marcinkiewicz and Orlicz spaces is presented based on concepts and results of symmetric spaces. Scientists and researchers will find the application of linear operators, ergodic theory, harmonic analysis and mathematical physics noteworthy and useful.
This book is intended for graduate students and researchers in mathematics and may be used as a general reference for the theory of functions, measure theory, and functional analysis. This self-contained text is presented in four parts totaling seventeen chapters to correspond with a one-semester lecture course. Each of the four parts begins with an overview and is subsequently divided into chapters, each of which concludes with exercises and notes. A chapter called “Complements” is included at the end of the text as supplementary material to assist students with independent work.Symmetric Spaces. The Spaces L p
Definition of Symmetric Spaces
Spaces L
The Space L
The Space L
Symmetric Spaces. The Embedding Theorem. Properties (A), (B), (C)
Embeddings L
Embeddings. Minimality and Separability. Property (A)
Associate Spaces
Maximality. Properties (B) and (C)
Lorentz and Marcinkiewicz Spaces
Lorentz Spaces
Quasiconcave Functions
Marcinkiewicz Spaces
Embedding
Orlicz Spaces
Definition and Examples of Orlicz Spaces
Separable Orlicz Spaces
Duality for Orlicz Spaces
Comparison of Orlicz Spaces
Intersections and Sums of Orlicz Spaces
Complements