Local uniform non-squareness of Orlicz–Lorentz function spaces

We give criteria for local uniform non-squareness of Orlicz–Lorentz function spaces Λφ,ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varLambda _{\varphi ,\omega }$$\end{document} 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 Lφ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\varphi }$$\end{document}, which complete the already known results, are deduced.

Let L 0 = L 0 ([0, γ )) be the space of all (equivalence classes of) Lebesgue measurable real-valued functions defined on the interval [0, γ ), where γ ≤ ∞. For any x, y ∈ L 0 , we write x ≤ y if x(t) ≤ y(t) almost everywhere with respect to the Lebesgue measure m on the interval [0, γ ).
Recall that the Köthe space E is called a symmetric space if E is rearrangement invariant in the sense that if x ∈ E, y ∈ L 0 and x * = y * , then y ∈ E and x = y . For basic properties of symmetric spaces we refer to [1,14,15].
Recall that an Orlicz function ϕ satisfies the condition Δ 2 for all u ∈ R (ϕ ∈ Δ 2 (R), for short) if there exists a constant K > 0 such that the inequality holds for any u ∈ R (then we have a ϕ = 0 and b ϕ = ∞). Analogously, we say that an Orlicz function ϕ satisfies the condition Δ 2 at infinity (ϕ ∈ Δ 2 (∞), for short) if there exist a constant K > 0 and a constant u 0 ≥ 0 such that ϕ(u 0 ) < ∞ and inequality (2) holds for any u ≥ u 0 (then we obtain b ϕ = ∞). For any Orlicz function ϕ we define its complementary function in the sense of Young by the formula for all u ∈ R. It is easy to show that ψ is also an Orlicz function.
Let ω : [0, γ ) → R + be a non-increasing and locally integrable function, which will be called a weight function. Define Recall that Orlicz-Lorentz spaces-a natural generalization of Orlicz and Lorentz spaces-were introduced at the beginning of the 1990s [11,12,[16][17][18]. These investigations spurred further investigations of Orlicz-Lorentz spaces.
Given an Orlicz function ϕ and a weight function ω, we define a convex modular I ϕ,ω : The modular space generated by the modular I ϕ,ω , is called the Orlicz-Lorentz function space. Since the modular unit ball {x ∈ L 0 : I ϕ,ω (x) ≤ 1} is an absolutely convex and absorbing subset of Λ ϕ,ω ([0, γ )), its Minkowski functional · ϕ,ω , defined by It is easy to check that, thanks to the condition ϕ(u) → ∞ as u → ∞, it is a norm in Λ ϕ,ω ([0, γ )), called the Luxemburg norm. In our investigations we apply the results concerning the monotonicity properties of Lorentz function spaces first presented in [8] and [4]. Let us recall that a Lorentz function space Λ ω is defined by the formula A Banach lattice E = (E, ≤, · ) is said to be locally uniformly monotone, whenever for any x ∈ (E) + (the positive cone of E) with x = 1 and any ε ∈ (0, 1) there is δ(x, ε) ∈ (0, 1) such that the conditions 0 ≤ y ≤ x and y ≥ ε imply that x − y ≤ 1 − δ(x, ε). In our investigations we will also apply Lemma 1. It is proved analogously as in the case of Orlicz spaces using the convexity of the modular I ϕ,ω (cf. also [13] for a more general case).

Lemma 1 Suppose that an Orlicz function ϕ satisfies a suitable condition
In particular, for any x ∈ Λ ϕ,ω , we then have x = 1 if and only if I ϕ,ω (x) = 1.
Proof. Sufficiency. Let us fix x ∈ S(Λ ϕ,ω ). Since ϕ ∈ Δ 2 (R), by Lemma 1 it is enough to show that there exists δ ∈ (0, 1) such that min I ϕ,ω for some ξ ∈ (0, 1]. Without loss of generality we may assume that and w > t 2 , otherwise. By [14], property 7 • , page 64 there exist sets e t 1 and e t 2 with m(e t 1 ) = t 1 , m(e t 2 ) = t 2 , Moreover, by the properties of the function ϕ, we get By the inequality (5), we obtain I ϕ,ω xχ e t 2 \e t 1 ≥ ξ , whence I ϕ,ω xχ B 1 ≥ ξ 2 or I ϕ,ω xχ B 2 ≥ ξ 2 . Let us assume that the first inequality holds. Then we have By the local uniform monotonicity of the Lorentz space Λ ω we get is a constant from the definition of local uniform monotonicity of the Lorentz space Λ ω corresponding to ϕ • x and ε = ξ ·η x 2 . In the second case, that is if I ϕ,ω xχ B 2 ≥ ξ 2 , proceeding analogously we can prove that I ϕ,ω , then Λ ϕ,ω contains an order isometric copy of l ∞ (see [11,Theorem 2.4]). Suppose now that and which means that the space Λ ϕ,ω is not locally uniformly non-square.
Then the following conditions are equivalent: Proof The implication (iii)⇒(ii) is obvious. The implication (ii)⇒(i) has been proved in Theorem 2.2 in [3]. We shall show (i)⇒(iii). Let us take an arbitrary x ∈ S Λ ϕ,ω . By the assumption that Proceeding analogously as in the proof of sufficiency in Theorem 2, we can find δ > 0 depending only on x such that min x−y Proof Sufficiency. Since ϕ ∈ Δ 2 (∞), by Lemma 1, it is sufficient to prove the corresponding inequalities for the modular. Case 1. Suppose that γ 0 > 0. Let us take an arbitrary x ∈ S Λ ϕ,ω . If then we proceed analogously as in the proof of the implication (i)⇒(iii) in Theorem 3.