Symmetric Spaces of Measurable Functions: Some New and Old Advances

The article is an extensive review in the theory of symmetric spaces measurable functions. It contains a number new (recent) and old (known) results this field. For most results, we give their proofs or exact references, where they can be found. under consideration are Banach (or quasi-Banach) latices functions equipped with (rearrangement invariant) norm quasinorm). We consider E = E(Ω, ℱμ, μ) ⊂ L0(Ω, on general measure (Ω, μ), measures μ assumed to finite infinite σ-finite nonatomic, while there no assumptions that separable Lebesgue space. In first section review, describe main classes basic properties spaces, minimal, maximal associate (A), (B), (C) Fatou’s property. list specific use includes Orlicz LΦ(Ω, Lorentz ΛW(Ω, Marcinkiewicz MV Orlicz–Lorentz LW,Φ(Ω, and, particular, Lp(w), Mp(w), Lp,q L∞(U). second section, deal dilation (Boyd) indexes some applications classical Hardy–Littlewood operator H. One problem here is: when H acts as bounded given space μ)? A special attention paid which have property (ℋℒ