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 $[f{]}_{\Phi ,p}:=\Vert f{\Vert }_E\{I_{\Phi }\left(\frac{f}{\Vert f{\Vert }_E}\right){\}}^{1p}$ 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.