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Multiplicativity Factors for Function Norms

Arens, R. and Goldberg, M. and Luxemburg, W. A. J. (1993) Multiplicativity Factors for Function Norms. Journal of Mathematical Analysis and Applictions, 177 (2). pp. 368-385. ISSN 0022-247X.

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Let (T, Ω, m) be a measure space; let ρ be a function norm on = (T, Ω, m), the algebra of measurable functions on T; and let Lρ be the space {f ∈ : ρ(f) < ∞} modulo the null functions. If Lρ, is an algebra, then we call a constant μ > 0 a multiplicativity factor for ρ if ρ(fg) ≤ μρ(f) ρ(g) for all f, g ∈ Lρ. Similarly, λ > 0 is a quadrativity factor if ρ(f2) ≤ λρ(f)2 for all f. The main purpose of this paper is to give conditions under which Lρ, is indeed an algebra, and to obtain in this case the best (least) multiplicativity and quadrativity factors for ρ. The first of our two principal results is that if ρ is σ-subadditive, then Lρ is an algebra if and only if Lρ is contained in L∞. Our second main result is that if (T, Ω, m) is free of infinite atoms, ρ is σ-subadditive and saturated, and Lρ, is an algebra, then the multiplicativity and quadrativity factors for ρ coincide, and the best such factor is determined by sup{||f||∞: f ∈ Lρ, ρ(f) ≤ 1}.

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Additional Information:© 1993 Academic Press. Under an Elsevier user license.
Issue or Number:2
Record Number:CaltechAUTHORS:20181009-131548532
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Official Citation:R. Arens, M. Goldberg, W.A.J. Luxemburg, Multiplicativity Factors for Function Norms, Journal of Mathematical Analysis and Applications, Volume 177, Issue 2, 1993, Pages 368-385, ISSN 0022-247X, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:90196
Deposited By: Joy Painter
Deposited On:10 Oct 2018 18:02
Last Modified:03 Oct 2019 20:23

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