## Banach Journal of Mathematical Analysis

### 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 $\Phi$ such that the easily calculable quasinorm $[f]_{\Phi,p}:=\Vert f\Vert_{E}\{I_{\Phi}(\frac{f}{\Vertf\Vert_{E}})\}^{1\slash p}$ if $f\neq0$ and $[f]_{\Phi,p}=0$ if $f=0$, on the Orlicz space $L^{\Phi}(\Omega,\Sigma,\mu)$ generated by $\Phi$, is equivalent to the Luxemburg norm $\Vert\cdot\Vert_{\Phi}$. To do this, we use a suitable $\Delta_{2}$-condition, lower and upper Simonenko indices $p_{S}^{a}(\Phi)$ and $q_{S}^{a}(\Phi)$ for the generating function $\Phi$, numbers $p\in[1,p_{S}^{a}(\Phi)]$ satisfying $q_{S}^{a}(\Phi)-p\leq1$, and an embedding of $L^{\Phi}(\Omega,\Sigma,\mu)$ into a suitable Köthe function space $E=E(\Omega,\Sigma,\mu)$. We take as $E$ the Lebesgue spaces $L^{r}(\Omega,\Sigma,\mu)$ with $r\in[1,p_{S}^{l}(\Phi)]$, when the measure $\mu$ is nonatomic and finite, and the weighted Lebesgue spaces $L^{r}_{\omega}(\Omega,\Sigma,\mu)$, with $r\in[1,p_{S}^{a}(\Phi)]$ and a suitable weight function $\omega$, when the measure $\mu$ is nonatomic infinite but $\sigma$-finite. We also use condition $\nabla_{3}$ if $p_{S}^{a}(\Phi)=1$ and condition $\nabla^{2}$ if $p_{S}^{a}(\Phi)\gt 1$, 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
Accepted: 29 September 2016
First available in Project Euclid: 9 June 2017

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

#### 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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