Orlicz–Lorentz function spaces equipped with the Orlicz norm

We investigate Orlicz–Lorentz function spaces equipped with the Orlicz norm generated by any Orlicz function and any non-increasing weight function. As far as we know, this is the first time such a general research is conducted. First we show some basic properties of the Orlicz norm, including its equality to the Amemiya norm, the problem of attainability of infimum in the definition of the Amemiya norm and a formula for the norm of a characteristic function. Then we find criteria for the order continuity and strict monotonicity and we study the problem of existence of order linearly isometric copies of l∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l^{\infty }$$\end{document}.


Introduction
Orlicz-Lorentz spaces, a natural generalization of Orlicz spaces as well as Lorentz spaces, have been intensely studied for the last 30 years. Attention has focused primarily on the Luxemburg norm (see e.g. [5, 8-11, 15, 16]). In recent years some papers concerning Orlicz-Lorentz spaces equipped with the Orlicz norm have been published (see [12,26,27] and also [5]). However, in the case of these papers, as well as in the case of classical Orlicz spaces equipped with the Orlicz norm, it was assumed that the Orlicz function ϕ is an N -function, that means ϕ(u) ∈ (0, ∞) for u ∈ (0, ∞), lim u→0 ϕ(u) u = 0 and lim u→∞ ϕ(u) u = ∞ (see [3,17,21,22,25]). In the case of Orlicz spaces, there was an exception to that rule in the paper [13], where the authors proved the equality of the Orlicz and Amemiya norms in Orlicz In memory of Henryk Hudzik, a great man, our teacher and colleague. spaces defined for any Orlicz function. In this context it is worth mentioning the paper [4], where the problem of attaining infimum in the definition of the Orlicz-Amemiya norm was considered, provided that lim u→∞ ϕ(u) u = B < ∞ and lim u→∞ (Bu − ϕ(u)) = ∞ (that means the function ϕ does not have an oblique asymptote at infinity).
Herein we begin a study of Orlicz-Lorentz function spaces equipped with the Orlicz norm generated by any Orlicz function and any non-increasing weight function.
This paper is organized as follows. In Sect. 2 we recall some basic definitions. In Sect. 3, after proving the equality of the Orlicz and Amemiya norms, we show some basic properties of these spaces, among others the problem of attaining infimum in the definition of the Amemiya norm and a formula for the norm of a characteristic function. According to our knowledge, the above-mentioned problems have not been considered so far by any other authors in such a general way, not even in Orlicz spaces. Hence, we hope that the presented results will provide an impetus toward further investigation of not only Orlicz-Lorentz spaces but also Orlicz spaces equipped with the Orlicz norm in full generality. Finally, in Sects. 4 and 5 we give criteria for the order continuity and strict monotonicity of the considered spaces and we study inclusion of respective copies of l ∞ .
Analogous results for the case of sequence spaces can be found in the paper [7] (see also [6]); however, in the case of function spaces some different techniques had to be implemented in many components of the proofs.
Note that the left continuity of ϕ on (0, ∞) is equivalent to the fact that lim u→(b ϕ ) − ϕ (u) = ϕ b ϕ .
Recall that an Orlicz function ϕ satisfies the condition 2 for all u ∈ R (ϕ ∈ 2 (R) for short) if there exists a constant K > 0 such that the inequality ϕ(2u) ≤ K ϕ(u) (1) 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 2 at infinity [at zero] (ϕ ∈ 2 (∞) [ϕ ∈ 2 (0)] for short) if there exist constants K , u 0 ∈ (0, ∞) such that ϕ(u 0 ) < ∞ [ϕ(u 0 ) > 0] and inequality (1) holds for any u ≥ u 0 [u ≤ u 0 ] (then we have b ϕ = ∞ [a ϕ = 0]). For any Orlicz function ϕ we define its complementary function in the sense of Young by the formula It is easy to show that ψ is also an Orlicz function and that ϕ and ψ satisfy the In some proofs in this paper the case when the inequality (2) becomes an equality will be important. Let us define a subdifferential ∂ϕ (u) of ϕ at u ≥ 0 as follows: By l and p we denote the left and right derivative of ϕ respectively. Then we have Let ω : [0, γ ) → R + be a non-increasing and locally integrable function, called a weight function. We will assume that ω is a non-zero function.
Similarly as in the case of Orlicz spaces, we can define in ϕ,ω ([0, γ )) two additional norms, namely the Orlicz norm and the Amemiya norm Remark 2.1 Let ϕ be any Orlicz function and ω any non-negative, locally integrable and nontrivial weight function (not necessarily non-increasing).
. If a ϕ < b ϕ , then the following conditions are equivalent: the weight function ω is nonincreasing, the second equality in (4) holds true and the semimodular I ϕ,ω is convex (see [8,Theorem 1.2]). The convexity of the semimodular I ϕ,ω is in turn an essential condition to prove that the functionals defined by (5) and (7) are norms. The assumption about the non-increasing weight function is also important while proving that the functional defined by (6) is a norm.
Assume to the contrary that I ψ,ω p kx * = ∞.
Without loss of generality we can assume that ω(t 0 ) > 0. We will consider two cases.
Let us assume first that a ϕ = b ϕ . By the assumption x * (0) > a ϕ and by the right continuity of the non-increasing rearrangement, we conclude that there exists Defining y = u 0 χ [0,t 0 ) , we get y * = y, I ψ,ω (y) = 1 and which gives a contradiction with the assumption Defining y = u 0 χ [0,t 0 ) again, we have y * = y and I ψ,ω (y) = 1. Simultaneously which again leads to a contradiction.
Suppose now that b ϕ < ∞ and define for any t ∈ A and n ≥ n 2 , where n 2 is the smallest natural number such that ϕ,ω ≤ 1 for the same n. Moreover, defining t n := m{t ∈ A : |y(t)| > x * (0) − 1 n } for n ≥ n 2 , we get lim n→∞ t n = 0 and z * n (t) = y * (t) for any n ≥ n 2 and any t ∈ [t n , t 0 ). In consequence, I ψ,ω p z * n I ψ,ω ( p (y * )), whence 1 < I ψ,ω p z * n < ∞ starting from a certain n 3 ≥ n 2 . Using again (14) and (15), we obtain z n O ϕ,ω > 1 for n ≥ n 3 , which again leads to a contradiction.
Finally assume that ). Therefore, we can find n 4 ∈ N such that 1 < I ψ,ω p w * n < ∞ for n ≥ n 4 . Hence, by (14) and (15), we have w n O ϕ,ω > 1 for n ≥ n 4 , which gives again a contradiction.
Then by the proof of (ii), we have I ψ,ω ( p (x * )) ≤ 1, whence by the equality in the Young inequality (see (3)), we get (iv) By (iii) the inequality I ϕ,ω On the other hand, by the Young inequality and left-hand continuity of the modular, we have Remark 3.1 By Lemma 3.1(iv), we conclude that the space ϕ,ω 0, γ ) , · O ϕ,ω is complete, therefore, it is a Banach lattice as well as a Banach symmetric space. By Beppo Levi's theorem, it also has the Fatou property. Analogous conclusions for the space ϕ,ω 0, γ ) , · A ϕ,ω follow from Lemma 3.1(v).

Remark 3.2
In the example below we will show that in the case when is not necessarily finite. We will also prove that if does not need to be true.
If γ = ∞ and x = χ [0,∞) , then λ ∞ = λ ∞ (x) = 1 for both functions ϕ 1 and ϕ 2 and In the first theorem we will show that the Orlicz and Amemiya norms are equal for any Orlicz function ϕ and any weight function ω. Let us recall that Hudzik and Maligranda proved the above-mentioned equality in Orlicz spaces generated by any Orlicz function in [13]. In that paper the reader will also find many important facts concerning all three norms in Orlicz spaces, including the history of research on relations between them.

Theorem 3.1 The Orlicz and Amemiya norms in an Orlicz
Proof It is easy to show that the Orlicz norm is not bigger than Amemiya norm for any x ∈ ϕ,ω ([0, γ )). In fact, by the Young inequality (2), for any y ∈ ψ,ω 0, γ ) , I ψ,ω (y) ≤ 1, and for any k > 0 we have Hence, x O ϕ,ω ≤ 1 k 1 + I ϕ,ω (kx) for any k > 0 and it follows that x O ϕ,ω ≤ x A ϕ,ω . The converse inequality will be proved for the non-negative simple functions with finite supports first. Set We will consider three cases.
0 ω(t)dt > 1. Moreover, Hence, there exist k 3 > 0 and a non-increasing function y 3 such that y 3 (t) ∈ l (k 3 x * (t)) , p (k 3 x * (t)) for any t ∈ 0, m (supp x)) and I ψ,ω (y 3 ) = 1. Proceeding in the same way as in (17), we get ω (l (k x x)) ≥ 1, we proceed analogously as in Case 3.1. Assume now that I ψ,ω (l (k x x)) < 1. Since ψ assumes finite values on the whole R, we will find β > l b ϕ such that we get z * = z, I ψ,ω (z) = 1 and for any t ∈ 0, m (supp x)). Hence, Finally we take any x ∈ ϕ,ω ([0, γ )). Since x O ϕ,ω = |x| O ϕ,ω and x A ϕ,ω = |x| A ϕ,ω , we can assume without loss of generality that x is non-negative. Then there exists a sequence (x m ) of non-negative simple functions such that x m ⊂ 0, min (γ, m)) for m ∈ N and x m x m-a.e. By the previous part of the proof and the Fatou property for both norms (see Remark 3.1), we get and this finishes the proof.

Remark 3.3
In the paper [13] the reader will find many examples of Orlicz functions suitable for any case considered in the Theorem above.
For our further consideration it will be important to answer the question if the infimum in the formula (7) is attained for any x ∈ ϕ,ω ([0, γ )) \ {θ }; equivalently, if there exists k = k (x) > 0 such that In order to answer this question, for any x ∈ ϕ,ω ([0, γ )) we define two constants First we present two lemmas which short proofs will be shown for the sake of completeness.

Lemma 3.2 For any x
where λ ∞ is defined by the formula (9).

Theorem 3.2 Let
for any k ∈ K (x), while in the case when k * = ∞ we get The proof of this theorem can be obtained immediately from Lemma 3.3.

Order continuity and separability
Let E ⊂ L 0 be a Banach ideal space. An element x ∈ E is said to be order continuous if for any sequence (x n ) in E + (the positive cone of E) with x n ≤ |x| and x n → 0 m-a.e. we have x n E → 0. The subspace E a of all order continuous elements in E is an order ideal in E. The space E is called order continuous if E a = E (see [19]). Since the Lebesgue measure space ([0, γ ), , m) is separable, the Banach ideal space E is separable if and only if it is order continuous.
By [15,Theorem 2.4] and the equivalence of the Luxemburg and Orlicz norms, we obtain the following

Copies of l ∞ and strict monotonicity
It is well known that Banach ideal space (E, · E ) is not order continuous if and only if it contains an order isomorphic copy of l ∞ (see [20]). Moreover, if (E, · E ) is an Orlicz-Lorentz space equipped with the Luxemburg norm, then we can obtain a stronger result, namely ϕ,ω 0, γ ) , · ϕ,ω is not order continuous if and only if it contains an order linearly isometric copy of l ∞ (see [15,Theorem 2.4]). The last statement does not hold if we consider the Orlicz norm instead of the Luxemburg norm. Furthermore, the problem of existence of an order linearly isometric copy of l ∞ is in the case of the Orlicz norm connected to the strict monotonicity of this space (especially for γ = ∞). Recall that Banach lattice E is strictly monotone if y E < x E whenever x, y ∈ E + (the positive cone of E) and y x (see [2]). Analogous results for Orlicz spaces equipped with the Orlicz norm were obtained by Chen, Cui and Hudzik in [4]. Proof (i) ⇒ (ii). Let x, y ∈ ϕ,ω ([0, ∞)) be such that x ≥ 0 and y 0. We have x x + y and in consequence x * ≤ (x + y) * . Moreover, in virtue of Lemma 3.2 in [15], there exists t 0 ∈ [0, ∞) such that x * (t 0 ) < (x + y) * (t 0 ). By the right continuity of (x + y) * , we conclude that there exists t 1 > t 0 such that x * (t 1 ) ≤ x * (t 0 ) < (x + y) * (t 1 ). If K (x + y) = ∅, then for any k 0 ∈ K (x + y) we have I ϕ,ω (k 0 x) < I ϕ,ω (k 0 (x + y)) and consequently Let now K (x + y) = ∅. By Remark 3.4, we have lim n→∞ ω(t)dt = 1 and p (u) < B for any u > 0). Therefore, K (x) = ∅ and by Lemma 3.4 By the arbitrariness of x and y, we conclude that the space ϕ,ω ([0, ∞)) , · O ϕ,ω is strictly monotone.
Proof Sufficiency. If a ϕ = 0, the proof is obtained in the similar way as the proof of the implication (i) ⇒ (ii) in Theorem 5.1. Suppose now that a ϕ > 0; hence, by assumption, we have K (x t ) = ∅ for any t ∈ 0, γ ). By virtue of Remark 3.4, we get lim u→∞ ϕ(u) u = B < ∞ and ψ (B) γ 0 ω(t)dt ≤ 1. Therefore, by Lemma 3.4, we have for any z ∈ ϕ,ω 0, γ ) . Take any x, y ∈ ϕ,ω 0, γ ) such that x ≥ 0 and y 0. We have x x + y, whence x * ≤ (x + y) * . Furthermore, by Lemma 3.2 in [15] and by the right continuity of the rearrangement function, there exist 0 ≤ t 0 < t 1 ≤ γ such that . Proceeding analogously as in (26), we get x O ϕ,ω < x + y O ϕ,ω , which establishes the sufficiency. Necessity. Suppose that ω(t 0 ) = 0 for some t 0 ∈ (0, γ ). Without loss of generality we can assume that ω(t) > 0 for any t < t 0 . Then for x = χ [0,t 0 ) and y = χ [t0,γ ) we have I ϕ,ω (kx) = I ϕ,ω (k (x + y)) for any k > 0, whence x O ϕ,ω = x + y O ϕ,ω . Now suppose that ω(t) > 0 for any t ∈ 0, γ ), a ϕ > 0 and K x t 0 = ∅ for some t 0 ∈ 0, γ ). Let k 0 ∈ K x t 0 and y := min 1, a ϕ k 0 χ [t0,γ ) . We have  ([0, ∞)) , · O ϕ,ω is order continuous, then it is strictly monotone as well. The converse implication does not hold as will be shown in Example 5.1. In the case when γ < ∞, these properties are not comparable (see Example 5.1). (ii) Let us notice that if the space ϕ,ω 0, γ ) is strictly monotone in the case of Luxemburg norm · ϕ,ω , then it is also strictly monotone in the case of Orlicz norm · O ϕ,ω (see [10,Corollary 4.4] and Theorems 5.1 and 5.2). As we will show in Example 5.1, there exist Orlicz-Lorentz spaces that are strictly monotone when equipped with the Orlicz norm and do not have that property when equipped with the Luxemburg norm. Example 5.1 (i) Since the function ϕ(u) = e u − 1 for u ≥ 0 does not satisfy the condition 2 at infinity, the space ϕ,ω 0, γ ) , γ ≤ ∞ generated by ϕ is not order continuous for both norms (that means for the Orlicz and Luxemburg norms). It is also not strictly monotone in the case of Luxembug norm. However, it is strictly monotone in the case of Orlicz norm, whenever ω(t) > 0 for t ∈ 0, γ ) if γ < ∞ or ∞ 0 ω(t)dt = ∞ if γ = ∞.

Remark 5.2 (i) The last theorem is a generalization of results obtained by Hudzik and
Mastyło in [14]. (ii) Let us notice that in the case of isometries defined in the proof of implication (iii) ⇒ (i) in Theorem 5.1 we have P(z) / ∈ ϕ,ω ([0, ∞)) a for any z ∈ c 0 \{0}. The same situation takes place for an isometry defined in the case of an Orlicz-Lorentz space equipped with the Luxemburg norm (see [15,Theorem 2.4]).