Uniformly normal structure and uniform non-squareness of Orlicz-Lorentz function spaces endowed with the Orlicz norm. (English) Zbl 1471.46013

Characterizations of uniform normal structure and uniform nonsquareness of Orlicz-Lorentz spaces endowed with the Luxemburg norm appeared in [A. Kamińska et al., Lect. Notes Pure Appl. Math. 175, 229–238 (1996; Zbl 0856.46015)] and [P. Foralewski et al., J. Inequal. Appl. 2013, Paper No. 32, 25 p. (2013; Zbl 1291.46011)]. In the article under review, the authors consider these geometric properties in Orlicz-Lorentz spaces endowed with the Orlicz norm.
Let \(\Lambda^{\text{o}}_{\varphi, w}[0,\gamma)\) denote the Orlicz-Lorentz space on \([0,\gamma)\) endowed with the Orlicz norm where \(\varphi\) is an Orlicz function and \(w\) is a weight function. The authors prove that, if \(w(t)>0\) for \(t\in [0,1)\), the Orlicz-Lorentz space \(\Lambda^{\text{o}}_{\varphi, w}[0,1)\) has uniform normal structure if and only if \(w\) is regular (in the sense that there exists \(K>1\) such that \(\int_0^{2x} w(t)\,dt \geq K \int_0^x w(t)\,dt\) for all \(x\in [0, 1/2)\)) and both \(\varphi\) and its conjugate function \(\psi\) satisfy \(\Delta_2\)-conditions for large values of their arguments. The authors obtain an analogous result for \(\Lambda^{\text{o}}_{\varphi, w}[0,\infty)\). In fact, they prove that the following are equivalent: (a) \(\Lambda^{\text{o}}_{\varphi, w}[0,\infty)\) has uniform normal structure; (b) \(w\) is regular and both \(\varphi\) and \(\psi\) satisfy \(\Delta_2\)-conditions on \(\mathbb{R}\); and (c) \(\Lambda^{\text{o}}_{\varphi, w}[0,\infty)\) is uniformly nonsquare. Finally, similar characterizations of uniformly nonsquare Orlicz-Lorentz spaces on \([0,1)\) are obtained.


46B20 Geometry and structure of normed linear spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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