Symmetric Spaces of Measurable Functions: Some New and Old Advances

The article is an extensive review in the theory of symmetric spaces of measurable functions. It contains a number of new (recent) and old (known) results in this field. For the most of the results, we give their proofs or exact references, where they can be found. The symmetric spaces under consideration are Banach (or quasi-Banach) latices of measurable functions equipped with symmetric (rearrangement invariant) norm (or quasinorm). We consider symmetric spaces E = E(Ω, F_μ, μ) (Formula presented.)L₀(Ω, F_μ, μ) on general measure spaces (Ω, F_μ, μ), where the measures μ are assumed to be finite or infinite σ-finite and nonatomic, while there are no assumptions that (Ω, F_μ, μ) is a separable or Lebesgue space. In the first section of the review, we describe main classes and basic properties of symmetric spaces, consider minimal, maximal and associate spaces, properties (A), (B), (C) and Fatou’s property. The list of specific symmetric spaces we use includes Orlicz L_Φ(Ω, F_μ, μ), Lorentz Λ_W(Ω, F_μ, μ), Marcinkiewicz M_V (Ω, F_μ, μ) and Orlicz–Lorentz L_W,Φ(Ω, F_μ, μ) spaces, and, in particular, the spaces L_p(w), M_p(w), L_p,q and L_∞(U). In the second section, we deal with the dilation (Boyd) indexes of symmetric spaces and some applications of classical Hardy–Littlewood operator H. One of the main problem here is: when H acts as a bounded operator on a given symmetric space E(Ω, F_μ, μ)? A special attention is paid to symmetric spaces, which have Hardy–Littlewood property (HLP) or weak Hardy–Littlewood property (WHLP). In the third section, we consider some interpolation theorems for the pair of spaces (L₁, L_∞) including the classical Calderon–Mityagin theorem. As an application of general theory, in the last section of review, we prove ergodic theorems for Cesaro averages of positive contractions in symmetric spaces. Studying the various types of convergence, we are interested in dominant ergodic theorem (Formula presented.), individual (pointwise) ergodic theorem (Formula presented.), order ergodic theorem (Formula presented.) and also mean (statistical) ergodic theorem (Formula presented.).