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July 2017 Normed Orlicz function spaces which can be quasi-renormed with easily calculable quasinorms
Paweł Foralewski, Henryk Hudzik, Radosław Kaczmarek, Miroslav Krbec
Banach J. Math. Anal. 11(3): 636-660 (July 2017). DOI: 10.1215/17358787-2017-0009

Abstract

We are interested in the widest possible class of Orlicz functions Φ such that the easily calculable quasinorm [f]Φ,p:=fE{IΦ(ffE)}1p if f0 and [f]Φ,p=0 if f=0, on the Orlicz space LΦ(Ω,Σ,μ) generated by Φ, is equivalent to the Luxemburg norm Φ. To do this, we use a suitable Δ2-condition, lower and upper Simonenko indices pSa(Φ) and qSa(Φ) for the generating function Φ, numbers p[1,pSa(Φ)] satisfying qSa(Φ)p1, and an embedding of LΦ(Ω,Σ,μ) into a suitable Köthe function space E=E(Ω,Σ,μ). We take as E the Lebesgue spaces Lr(Ω,Σ,μ) with r[1,pSl(Φ)], when the measure μ is nonatomic and finite, and the weighted Lebesgue spaces Lωr(Ω,Σ,μ), with r[1,pSa(Φ)] and a suitable weight function ω, when the measure μ is nonatomic infinite but σ-finite. We also use condition 3 if pSa(Φ)=1 and condition 2 if pSa(Φ)>1, proving their necessity in most of the considered cases. Our results seem important for applications of Orlicz function spaces.

Citation

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Paweł Foralewski. Henryk Hudzik. Radosław Kaczmarek. Miroslav Krbec. "Normed Orlicz function spaces which can be quasi-renormed with easily calculable quasinorms." Banach J. Math. Anal. 11 (3) 636 - 660, July 2017. https://doi.org/10.1215/17358787-2017-0009

Information

Received: 7 June 2016; Accepted: 29 September 2016; Published: July 2017
First available in Project Euclid: 9 June 2017

zbMATH: 1380.46024
MathSciNet: MR3679899
Digital Object Identifier: 10.1215/17358787-2017-0009

Subjects:
Primary: 46E30
Secondary: 46B03, 46B42

Rights: Copyright © 2017 Tusi Mathematical Research Group

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Vol.11 • No. 3 • July 2017
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