Geometry of Orlicz spaces equipped with norms generated by some lattice norms in $\mathbb{R}^{2}$

In Orlicz spaces generated by convex Orlicz functions a family of norms generated by some lattice norms in $\mathbb{R}^{2}$ are defined and studied. This family of norms includes the family of the p-Amemiya norms ($1\leq p\leq\infty$) studied in [10-11], [14-15] and [20]. Criteria for strict monotonicity, lower and upper local uniform monotonicities and uniform monotonicities of Orlicz spaces and their subspaces of order continuous elements, equipped with these norms, are given in terms of the generating Orlicz functions, and the lattice norm in $\mathbb{R}^{2}$. The problems of strict convexity and of the existence of order almost isometric as well as of order isometric copies in these spaces are also discussed.

Let us demote by A Φ (1) the modular unit ball, that is, Since the Orlicz function Φ is absolutely convex, that is, for all u, ν ∈ R and all α, β ∈ R with |α| + |β| ≤ 1, we obtain absolute convexity of the functional I Φ and, in consequence, also the absolute convexity of the set A Φ (1). The Minkovski functional generated by the set A Φ (1) can be defined for these elements from L 0 (Ω, Σ, µ) which are absorbing by A Φ (1). It is easy to see that the biggest subspace of L 0 (Ω, Σ, µ), the elements of which are absorbed by A Φ (1), is just the Orlicz space L Φ (µ). Namely, ∃λ > 0 s.t. : x λ ∈ A Φ (1)) ⇔ (x ∈ L Φ (µ)).
The Minkovski functional of the set A Φ (1) is called in the literature the Luxemburg norm, it is denoted by · Φ and defined by the formula (see [5], [36], [37]and [39]) The following family of the norms, called p-Amemiya norms, was already defined and used in Orlicz spaces for 0 ≤ p ≤ ∞: where 1 ≤ p ≤ ∞ (see [10][11], [14][15], [20]). The norm · Φ,1 is the Orlicz norm · 0 Φ which was defined by Orlicz in [41] by the formula x(t)y(t)dµ| : y ∈ L 0 (Ω, Σ, µ) and I Φ * (y) ≤ 1}, where Φ * is the function complementary to Φ in the sense of Young, that is, For p = ∞, we have (see [24]) It is well known that all the norms from the family { · Φ,p } p∈ [1,∞] are equivalent and that the Luxemburg norm · Φ = x Φ,∞ is the smallest norm and the Orlicz norm · 0 Φ = x Φ,1 is the biggest one. The Orlicz space L Φ (µ) equipped with every norm from this family of norms is a Banach space, which is even the Banach function lattice, called also the Kőthe space (see [31][32], [35]and [41]), which means that for any p ∈ [1, +∞], the space (L Φ (µ), · Φ,p ) has the following properties: The same properties has the space (E Φ (µ), · Φ,p ) defined below. Let us recall that an element x of a Kőthe space (E, · E ) is said to be order continuous if for any sequence {x n } ∞ n=1 in E such that 0 ≤ x n (t) ≤ |x(t)| for all n ∈ N and µ a.e.t ∈ T , the condition x n (t) → 0 as n → ∞ for µ a.e. t ∈ Ω implies that x n E → 0 as n → ∞. The set of all order continuous elements in E is denoted by E a , and the space (E a , · E ) is again a Kőthe space. It is obvious that equivalent norms keep the order continuity property. It is well known that (see [35]and [41]) In this paper we will introduce a new family of norms in the Orlicz space L Φ (µ). Namely, given any lattice norm p(·) in R 2 such that p((1, 0)) = 1, we define the following functional in L Φ (µ): We will prove that such functionals are norms in L Φ (µ). Of course, these norms are equivalent each others. We will work on criteria for strict convexity and various their monotonicity properties (strict monotonicity, lower and upper local uniform monotonicity and uniform monotonicity) as well as on order almost isometric copies of l ∞ and order isometric copies of l ∞ . We need to define all others notions that will be used in this paper. A Banach lattice X = (X, ≤, · ), for the definition of which we refer to [2], [31], [35] and [41] is said to be strictly monotone if for any x, y ∈ X such that 0 ≤ x ≤ y and x = y we have x < y . By the homogenity of the norm · , we can restrict ourselves in this definition to y ≥ 0 satisfying y = 1. Let us denote by X + the positive cone in X that is the set of all x ∈ X such that x ≥ 0. In our definitions below X always denotes a Banach lattice (X, ≤, · ). X is said to be uniformly monotone (see [2] and [33]) if for any ε ∈ (0, 1) there exists δ(ε) ∈ (0, 1) such that if x, y ∈ X, 0 ≤ x ≤ y; x ≥ ε and y = 1, then y − x ≤ 1 − δ(ε). The biggest function δ X : (0, 1) → (0, 1) with this property, that is, the function is called the modulus of monotonicity of X (see [2]) and for the properties of δ X (·) also ([23]). It is known (see [33]) that X is uniformly monotone if and only if for any ε > 0 there exists σ(ε) > 0 such that for any x, y ∈ X + such that x ≥ ε and y = 1 there holds y + x ≥ 1 + σ(ε). X is said to be lower (upper) locally uniformly monotone if for any y ∈ X + with y = 1 and any ε ∈ (0, 1) (resp. any ε > 0) there exists δ(y, ε) ∈ (0, 1)(resp. σ(y, ε) > 0 ) such that for any x ∈ X satisfying 0 ≤ x ≤ y and x ≥ ε (resp. x ≥ 0 with x ≥ ε), we have y − x ≤ 1 − δ(y, ε) (resp. y + x ≥ 1 + σ(y, ε)). For the definition of these two properties see [2,4,25].
If Ω a is finite and µ(Ω n−a ) > 0, then the suitable condition ∆ 2 for Φ is the ∆ 2 -condition for the non-atomic measure space defined above. If µ(Ω n−a ) = 0 and Ω a = N, Σ = 2 N and µ is the counting measure on 2 N , then the suitable condition ∆ 2 is the condition ∆ 2 (0). If µ(Ω n−a ) > 0, Ω a = N, Σ = 2 N and µ is the counting measure on 2 N , then the suitable ∆ 2 -condition for Φ is the conjunction of the suitable ∆ 2 -condition for the non-atomic measure space and of the condition △ 2 (0). In our paper, we always assume that all atoms have the measure 1 and we identify the atoms with the singletons {n}, where n ∈ N (the set of all natural numbers).
Problems on estimates or calculations of the characteristic of monotonicity in Orlicz spaces and Orlicz-Lorentz spaces were studied in [16], [19] and [23]. Applications of the monotonicity properties and the ergodic theory in Banach lattices were studied in [1].

Theorem 2 For any Orlicz function Φ and any lattice norm p(·) in the Eu
Now, we will show that the functional · Φ,p(·) is absolutely homogeneous. Let us take any x ∈ L Φ (µ) and any λ > 0. If x = 0, then λx = 0 for any λ ∈ R, whence So let us assume that x = 0. Then Finally, we will show that the functional · Φ,p(·) satisfies the triangle inequality. Let us take arbitrarily x, y ∈ L Φ (µ). If at least one element among x and y is equal to zero function, then the triangle inequality is obvious. So assume that x = 0 and y = 0. Let us take any ε > 0. There exists constants λ > 0 and l > 0 such that By the arbitrariness of ε > 0, we obtain the inequality which finishes the proof of the theorem.
Proof. Since p(·) is a lattice norm in R 2 , we have for any u > 0 that Hence, and from condition (1) as well as by the fact that there exist positive constants k 0 and k 1 such that k 0 < k 1 < +∞, I Φ (k 1 x) < ∞, and .
The function f (k) := I Φ (kx)) is convex and it has finite values on the compact interval [k 0 , k 1 ], so it is continuous on this interval. In consequence, by continuity of the norm p(·), the function g : [k 0 , k 1 ] → R + defined by is also continuous. Therefore, the desired number l ∈ (0, +∞) exists.

Lemma 2 For any Orlicz function Φ, any lattice norm on
Proof. We have On the other hand Theorem 3 Let p(·) be a norm on R 2 as in Lemma 2. If Φ is an Orlicz function Φ which does not satisfy suitable ∆ 2 -condition, then (L Φ (µ), · Φ,p(·) ) contains an order linearly almost isometric copy of l ∞ , that is, for any ε > 0 there exists a linear nonnegative operator P ε : l ∞ → L Φ (µ) such that Proof. Under the assumptions on Φ, given any ε > 0, there exists a sequence {x n } ∞ n=1 in L Φ (µ) with pairwise disjoint supports and such that I Φ (x n ) ≤ ε/2 n , x n ≥ 0 and I Φ (λx n ) = ∞ for any n ∈ N and λ > 1. Let us define the operator P ε on L Φ (µ) by the formula where the series is defined pointwisely for t ∈ Ω. It is obvious that P ε is linear and nonnegative. There is no problem with the pointwise convergence of the series by pairwise disjointness of the supports of the element x n ∈ L Φ (µ), whence for any t ∈ Ω there exists at least one n ∈ N such that t ∈ suupx n .
Let us note first that for any k > 0 and z ∈ l ∞ , we have In consequence, by Lemma 2, we have for any z = {z n } ∞ n=1 ∈ l ∞ , x n ))) On the other hand, since for any λ > 0 there exist n λ ∈ N such that λ|z n l |/ z ∞ > 1,we have whence, by Lemma 2, By the arbitrariness of λ > 1, we have P ε z Φ,p(·) ≥ z ∞ for any z ∈ l ∞ , which finishes the proof. Proof. Under the assumptions on the measure space, there exists a sequence {A n } ∞ n=1 of pairwise disjoint set with µ(A n ) = +∞ for any n ∈ N . Let us define where the series is defined pointwisely (no problem with it's pointwise convergence because of pairwise disjointness of the sets A n ). It is obvious that I Φ (x) = 0 and I Φ (x n ) = 0 as well as that I Φ (λx) = I Φ (λx n ) = +∞ for any n ∈ N and λ > 1. Moreover, x n Φ,p(·) = min( inf In the same way, we can prove that x Φ,p(·) = 1. Let us define the following operator on l ∞ : Let us first note that P : l ∞ −→ L Φ (µ). Namely, On the other hand, given any λ > 1, one can find n λ ∈ N such that λ | z n λ |> z ∞ , consequently, By the arbitrariness of λ > 1, we obtain that P z Φ,p(·) ≥ z ∞ , which together with the opposite inequality proved already gives the equality P z Φ,p(·) = z ∞ for any z ∈ l ∞ , which means that P is an isometry. It is obvious that the operator P is linear. Since the functions x n are non-negative, so P is also non-negative, that is, P z ≥ 0 for any z ∈ l ∞ , z ≥ 0. In consequence, the operator P is a linear order isometry, which finishes the proof.
Hence, and by the equality we obtain , Taking in place of ε a sequence {ε n } ∞ n=1 such that ε n ց 0 as n ր ∞ and applying the Beppo Levi theorem, we obtain that (1 + ε)I Φ ( x 1+ε ) ր I Φ (x) as n ր ∞. Since the modulus of monotonicity is continuous on the interval [0, 1), we obtain the desired inequality.
Assume now that x and y are as above, but they belong to the space E Φ (µ). Let us note that the space (E Φ (µ), Φ,p(·) ) has the following property.