## Banach Journal of Mathematical Analysis

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

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

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

#### Abstract

We are interested in the widest possible class of Orlicz functions $\Phi $ such that the easily calculable quasinorm $[f{]}_{\Phi ,p}:=\Vert f{\Vert}_{E}\{{I}_{\Phi}\left(\frac{f}{\Vert f{\Vert}_{E}}\right){\}}^{1p}$ if $f\ne 0$ 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 $p\in [1,{p}_{S}^{a}(\Phi \left)\right]$ satisfying ${q}_{S}^{a}(\Phi )-p\le 1$, 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 $r\in [1,{p}_{S}^{l}(\Phi \left)\right]$, when the measure $\mu $ is nonatomic and finite, and the weighted Lebesgue spaces ${L}_{\omega}^{r}(\Omega ,\Sigma ,\mu )$, with $r\in [1,{p}_{S}^{a}(\Phi \left)\right]$ 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 ${p}_{S}^{a}(\Phi )>1$, 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