Local uniform non-squareness of Orlicz–Lorentz function spaces

Abstract

We give criteria for local uniform non-squareness of Orlicz-Lorentz function spaces Λ ϕ,ω equipped with the Luxemburg norm, where the widest possible class of convex Orlicz functions is admitted. As immediate consequences, criteria for local uniform non-squareness of Orlicz function spaces L ϕ , which complete the already known results, are deduced.

Kadec–Klee properties of Orlicz–Lorentz sequence spaces equipped with the Orlicz norm

“…Moreover, by (13) we have • (k n(s(t)) x n(s(t)) − k 0 x) * • ω → 0 uniformly as t → ∞. Hence, by the Lebesgue dominated convergence theorem,…”

“…Recall that Orlicz-Lorentz spaces, which are a natural generalization of Orlicz and Lorentz spaces, were introduced in the early nineties of the twentieth century [18,24,27]. These investigations gave an impulse to further study of the spaces, results of which have been published among others in papers [1,3,7,[10][11][12][13][14][15]19,22,33].…”

Non-squareness and local uniform non-squareness properties of Orlicz-Lorentz function spaces endowed with the Orlicz norm

“…Therefore we have Lemma 11,17]). The Lorentz function space L 1,ω is strictly monotone if and only if ω is positive on [0, γ) and 6]). The Lorentz function space L 1,ω is lower locally uniformly monotone if and only if ω is positive on [0, γ) and…”

“…In 2013 Foralewski, Hudzik and Kolwicz show the sufficient and necessary conditions for Orlicz-Lorentz space with the Luxemburg norm being non-square and uniformly non-square [4,5] . Recently Foralewski and Kończak got the criterion for Orlicz-Lorentz space with the Luxemburg norm being locally uniformly non-square [6] . In this paper we will discuss the characteristic of Orlicz-Lorentz function space with the Orlicz norm being non-square and locally uniformly non-square.…”

Kadec–Klee property with respect to the local convergence in measure of Orlicz–Lorentz spaces

“…At first, studies on their structure concerned the case of the Luxemburg norm (see, e.g., [4, 7, 8, 11–13, 17, 19, 21]). Recently there have been some papers addressing the case of the Orlicz norm (see [10, 23, 24]), but only in articles [5] and [9] it was assumed for the first time that $$\varphi$$ is any Orlicz function and not so‐called $$N$$‐function. In this paper, we continue those studies by finding criteria for the Kadec–Klee property with respect to the local convergence in measure in Orlicz–Lorentz spaces and their subspaces of order continuous elements.…”