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Normed Orlicz function spaces which can be quasi-renormed with easily calculable quasinorms. (English) Zbl 1380.46024

Summary: We are interested in the widest possible class of Orlicz functions \(\Phi\) such that the easily calculable quasinorm \([f]_{\Phi,p}:=\| f\|_{E}\{I_{\Phi}(\frac{f}{\| f\|_{E}})\}^{1/ p}\) 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 \(\|\cdot\|_{\Phi}\). 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 \(p_{S}^{a}(\Phi)> 1\), proving their necessity in most of the considered cases. Our results seem important for applications of Orlicz function spaces.

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B03 Isomorphic theory (including renorming) of Banach spaces
46B42 Banach lattices
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