φ-Factors for Function Seminorms

Abstract

Let (Ω, A, μ) be a measure space, ϱ a function seminorm on M, the space of measurable functions on Ω, and Mϱ the space {f ∈ M : ϱ(f) < ∞}. Every Borel measurable function φ : [0, ∞) → [0, ∞) induces a function Φ : M → M by Φ(f)(x) = φ(|f(x)|). We introduce the concepts of φ-factor and Φ-invariant space. If ϱ is a σ-subadditive seminorm function, we give, under suitable conditions over φ, necessary and sufficient conditions in order that Mϱ be invariant and prove the existence of φ-factors for ϱ. We also give a characterization of the best φ-factor for a σ-subadditive function seminorm when μ is σ-finite. All these results generalize those about multiplicativity factors for function seminorms proved earlier.