Banach Journal of Mathematical Analysis

Normed Orlicz function spaces which can be quasi-renormed with easily calculable quasinorms

Paweł Foralewski, Henryk Hudzik, Radosław Kaczmarek, and Miroslav Krbec

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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.

Article information

Source
Banach J. Math. Anal., Volume 11, Number 3 (2017), 636-660.

Dates
Received: 7 June 2016
Accepted: 29 September 2016
First available in Project Euclid: 9 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1496973700

Digital Object Identifier
doi:10.1215/17358787-2017-0009

Mathematical Reviews number (MathSciNet)
MR3679899

Zentralblatt MATH identifier
1380.46024

Subjects
Primary: 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Secondary: 46B03: Isomorphic theory (including renorming) of Banach spaces 46B42: Banach lattices [See also 46A40, 46B40]

Keywords
Orlicz spaces quasinorms Simonenko indices regularity conditions for Orlicz functions embeddings into Lebesgue and weighted Lebesgue spaces

Citation

Foralewski, Paweł; Hudzik, Henryk; Kaczmarek, Radosław; Krbec, Miroslav. Normed Orlicz function spaces which can be quasi-renormed with easily calculable quasinorms. Banach J. Math. Anal. 11 (2017), no. 3, 636--660. doi:10.1215/17358787-2017-0009. https://projecteuclid.org/euclid.bjma/1496973700


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