Symmetric Spaces of Measurable Functions: Old and New 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, Fμ, μ) ⊂ L0 (Ω, Fμ, μ) on general measure spaces (Ω, Fμ, μ), where the measures μ are assumed to be finite or infinite σ-finite and nonatomic, while there are no assumptions that (Ω, Fμ, μ) is 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, the properties (A), (B), and (C), and Fatou’s property. The list of specific symmetric spaces we use includes Orlicz LΦ(Ω, Fμ, μ), Lorentz ΛW (Ω, Fμ, μ), Marcinkiewicz MV (Ω, Fμ, μ), and Orlicz-Lorentz LW,Φ (Ω, Fμ, μ) spaces, and, in particular, the spaces Lp (w), Mp(w), Lp,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 problems here is: when H acts as a bounded operator on a given symmetric space E(Ω, Fμ, μ)? A spacial 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, we prove in the last section of review Ergodic Theorems for Cesaro averages of positive contractions in symmetric spaces. Studying various types of convergence, we are interested in Dominant Ergodic Theorem (DET ), Individual (Pointwise) Ergodic Theorem (IET ), Order Ergodic Theorem (OET ), and also Mean (Statistical) Ergodic Theorem (MET ).