## 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.

### MSC:

 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.)

### Citations:

Zbl 0856.46015; Zbl 1291.46011
Full Text:

### References:

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