Abstract
Let (Ω, A, μ) be a measure space, ϱ a function seminorm on M, the space of measurable functions on Ω, and M ϱ 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 M ϱ 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.
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Cuenya, H.H. φ-Factors for Function Seminorms. Analysis Mathematica 26, 175–186 (2000). https://doi.org/10.1023/A:1005601118086
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DOI: https://doi.org/10.1023/A:1005601118086