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

Abstract

We are interested in the widest possible class of Orlicz functions Φ such that the easily calculable quasinorm [f]Φ,p:=‖f‖E{IΦ(f‖f‖E)}1p if f≠0 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)−p≤1, 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.