Rotundity and monotonicity properties of selected Cesàro function spaces

We study rotundity, strict monotonicity, lower local uniform monotonicity and upper local uniform monotonicity in some classes of Cesàro function spaces. We present full criteria of these properties in the Cesà ro–Orlicz function spaces $$Ces_{\varphi }$$Cesφ (under some mild assumptions on the Orlicz function $$\varphi $$φ). Next, we prove a characterization of strict monotonicity, lower local uniform monotonicity and upper local uniform monotonicity in the Cesàro–Lorentz function spaces $$C\Lambda _{\phi }$$CΛϕ. We also show that the space $$C\Lambda _{\phi }$$CΛϕ is never rotund. Finally, we will prove that Cesàro–Marcinkiewicz function space $$CM_{\phi }^{(*)}$$CMϕ(∗) is neither strictly monotone nor order continuous for any quasi-concave function $$\phi $$ϕ.


Introduction
The structure of different types of spaces defined by a Cesàro operator has been intensively investigated during the last decades from the isomorphic as well as isometric point of view. The classical Cesàro sequence ces p and function Ces p spaces have been studied by many authors (see [1,2] also for further references). It is worth to mention that some properties are fulfilled in the sequence case and are not in function case. Furthermore, sometimes the cases Ces p [0, 1] and Ces p [0, ∞) are essentially different (see an isomorphic description of the Köthe dual of Ces àro spaces in [1,34], see also [23] for the respective isometric description). The spaces generated by the Cesàro operator (including abstract Cesàro spaces) have been considered by Curbera, Delgado, Soria, Ricker, Leśnik and Maligranda in several papers (see [13][14][15][16][34][35][36]).
We are going to discuss several properties of some Cesàro function spaces C X. The Cesàro-Orlicz function spaces Ces ϕ have been studied in [26,27] where the authors, among others, prove the criteria for the existence of order isomorphic (order isomorphically isometric) copy of l ∞ in these spaces. We prove a characterization of strict monotonicity, lower local uniform monotonicity, upper local uniform monotonicity and rotundity in the spaces Ces ϕ . We admit the largest possible class of Orlicz functions giving the maximal generality of spaces under consideration. Finally, we study rotundity, monotonicity properties and order continuity of Cesàro-Lorentz function spaces C φ and Cesàro-Marcinkiewicz function spaces C M ( * ) φ .

Preliminaries
Let R, R + and N be the sets of real, nonnegative real and natural numbers, respectively. Denote by μ the Lebesgue measure on I and by L 0 = L 0 (I ) the space of all classes of real-valued Lebesgue measurable functions defined on I , where I = [0, 1] or I = [0, ∞).
A Banach lattice X = (X, · ) is said to be a Banach ideal space on I if X is a linear subspace of L 0 (I ) which satisfies the ideal property: if g ∈ X , f ∈ L 0 and | f | ≤ |g| a.e. on I then f ∈ X and f ≤ g .
Unless it is stated otherwise then we understand that a Banach ideal space has the so-called weak unit that is there is an element f ∈ X that is positive on whole I . Sometimes we write · X to be sure in which space the norm has been taken. By X + we denote the positive cone of X , that is, X + = {x ∈ X : x ≥ 0}.
For two Banach ideal spaces X and Y on I the symbol X → Y means that the embedding X ⊂ Y is continuous, i.e., there exists constant a C > 0 such that x Y ≤ C x X for all x ∈ X . Moreover, X = Y means that the spaces are the same as the sets and the norms are equivalent.
A Banach ideal space X is called order continuous (X ∈ (OC) shortly) if every element of X is order continuous, that is, for each f ∈ X and for each sequence ( f n ) ⊂ X satisfying 0 ≤ f n ≤ | f | and f n → 0 a.e. on I , we have f n → 0. By X a we denote the subspace of all order continuous elements of X . It is worth to notice that x ∈ X a if and only if xχ A n → 0 for any decreasing sequence of Lebesgue measurable sets A n ⊂ I with empty intersection (see [4,Proposition 3.5,p. 15]).
A Banach lattice X is strictly monotone (X ∈ (SM) for short) if for any x, y ∈ X + such that x ≤ y and y = x, we have x < y , see [5,21]. Moreover, we say that X is upper (lower) locally uniformly monotone, writing shortly X ∈ (U LU M) (X ∈ (L LU M)), if x n − x → 0 for any x ∈ X + and any sequence ( We say a normed space X is rotund (X ∈ (R) for short) if x + y < 2 whenever x and y are distinct points on the unit sphere of X . It is well known that rotundity (strict monotonicity) is a useful tool in the theory of Banach spaces (Banach lattices), e.g. in the best approximation problems in Banach spaces (in the best dominated approximation problems in Banach lattices), see [6] for a local approach and further references. Moreover, rotundity and strict monotonicity are closely related in Banach lattices (see [21]). Note that a Banach space (Banach lattice) is rotund (strictly monotone) if and only if the unit sphere contains no nontrivial line segment (nontrivial order interval).
The continuous Cesàro operator C : with the norm f C X = C| f | X (see [34][35][36]). We always assume that C X = {0} . If C : X → X is bounded and X = {0} then C X = {0} . The reader is referred to [32] for more informations about the boundedness of the operator C.
The same implication is true if we replace the property U LU M by L LU M (or by SM property). The proof follows just from the definition (see also Fact in [26]).

The Cesàro-Orlicz function spaces Ces ϕ
If we denote a ϕ = sup{u ≥ 0 : ϕ(u) = 0}, then 0 ≤ a ϕ ≤ b ϕ ≤ ∞. Moreover, a ϕ < ∞ and b ϕ > 0, since an Orlicz function is neither identically equal to zero nor infinity on (0, ∞). The function ϕ is continuous and non-decreasing on [0, b ϕ ) and is strictly increasing on [a ϕ , b ϕ ). We use notations ϕ > 0, ϕ < ∞ when a ϕ = 0, b ϕ = ∞, respectively. We say an Orlicz function ϕ satisfies the condition 2 for large arguments (ϕ ∈ 2 (∞) for short) if there exists K > 0 and u 0 > 0 such that ϕ(u 0 ) < ∞ and for all u ≥ 0 then we say that ϕ satisfies the condition 2 for all arguments (ϕ ∈ 2 (R + )). These conditions play a crucial role in the theory of Orlicz spaces, see [8,31,38] and [41]. We will write ϕ ∈ 2 in two cases: Recall that an Orlicz function ϕ is strictly convex whenever ϕ u+v The Orlicz function space L ϕ = L ϕ (I ) generated by an Orlicz function ϕ is defined by where dt is a convex modular (for the theory of Orlicz spaces and modular spaces see [38,41]). The space L ϕ is a Banach ideal space with the Luxemburg-Nakano norm It is well known that f ϕ ≤ 1 if and only if I ϕ ( f ) ≤ 1.
The following Lemma is formulated for any Banach ideal space. We will apply it later in the case X = L ϕ .

Lemma 2 Let X be a Banach ideal space on I . If X is rotund then C X is rotund.
Proof Suppose X is rotund. It is well known that X ∈ (R) is equivalent to X + ∈ (R) (see [21,Theorem 2]). Take f, g ∈ C X, f, g ≥ 0, f C X = g C X = 1 and ( f + g)/2 C X = 1. We will show that f = g. Indeed, therefore C f = Cg by our assumption and C( f − g) is identically 0 on I , i.e., x for all 0 < x ∈ I . It is well known (by the Fundamental Theorem of Calculus) that if λ ∈ L 1 [a, b], r ∈ R and we define the function : ]. Therefore f − g = 0 a.e. and so C X ∈ (R). However, we can also use a direct and easy argument. Indeed, put h = f − g and suppose for the contrary that h = 0. We can assume that h(t) > 0 for all 0 < t ∈ A ⊂ I where m(A) > 0. There is a closed set B ⊂ A with m(B) > 0. We have  Proof (⇐). If ϕ is strictly convex and ϕ ∈ 2 then L ϕ is rotund (see [8]) and Ces ϕ is rotund by Lemma 2.
(⇒). We have to consider two cases. I. Suppose I = [0, 1], ϕ ∈ 2 (∞) and ϕ is not strictly convex. Then there is an interval (a, b), a, b ∈ [0, ∞) on which ϕ is affine, i.e., ϕ(t) = αt + β for some α, β ∈ R and all t ∈ (a, b). Note that the case a ϕ > 0 is included in the below proof and is even much simpler. Take a, b ∈ (a, b), a < b such that Let us define functions for n ∈ N and t ∈ [0, 1]. Since F n and G n are absolutely continuous functions for all n ∈ N and product of absolutely continuous functions is also absolutely continuous function, we can find elements f n , For all n ∈ N we have: Properties (i) and (ii) follow from the definition of functions F n and G n and from equalities (3.3). Now we prove (iii). We have We claim that functions t F n (t) and tG n (t) are nondecreasing on the interval 0, 1 2n . We have where the last inequality follows from condition (3.2). The case of the function t F n (t) is analogous. This proves the claim and consequently functions f n and g n are nonnegative on the interval 0, 1 2n . Thus Applying (ii), (3.4) and (3.5) it is enough to prove that Observe that Consequently, and the proof of condition (iii) is finished. Note that ρ ϕ ( f n χ (0,1/2n) ) → 0 as n → ∞. Indeed, from condition (ii) and the proof of part (iii) we have for n ∈ N and c ∈ R. We define the function for c ∈ [0, ∞). Note that h(0) < 1 and h(c) ≤ ϕ(max{b, c}) < ∞, which means that h takes finite values because ϕ < ∞. Moreover, for 0 < λ < 1 and c 1 , because ρ ϕ is a convex modular. This means that h is convex and therefore continuous function on [0, ∞). Applying the Darboux property we find a number c 0 satisfying Then applying the equality (iii) we obtain ρ ϕ (x) = ρ ϕ (y) = 1. Consequently, The proof is the same. We need only to notice that all constructed elements are well defined (the respective modulars are finite) by Proposition 3 from [26].

Remark 4
In the proof of Theorem 3 above, the assumption ϕ ∈ 2 has been used only in the proof of implication: if ϕ is strictly convex then Ces ϕ ∈ (R). In the proof of reverse implication we use only the assumption ϕ < ∞. Obviously, if ϕ ∈ 2 (∞) or ϕ ∈ 2 (R + ), then ϕ < ∞.
Now we present criteria for several properties in the spaces Ces ϕ . The following well known notion will be useful.
Suppose ψ, γ : [0, ∞) → [0, ∞). We say ψ and γ are equivalent (weak equivalent) for "all arguments" if there are constants A, B > 0 such that , for simplicity. If the above inequalities are satisfied for u ∈ [u 0 , ∞) with u 0 > 0 such that ψ(u 0 ) > 0 then we say that ψ, γ are equivalent (weak equivalent) for "large arguments" and we write shortly ψ ∼ l γ (ψ w ∼ l γ ). The proof of the next fact is just an easy exercise -it is enough to apply equivalent formulation for condition 2 , see [8].
Let ϕ be an Orlicz function with b ϕ < ∞ and ϕ(b ϕ ) = ∞, γ be a convex function and p > 1. We will write ϕ 1/ p ∼ a γ (ϕ 1/ p ∼ l γ ) provided there are constants Recall that we have the following implications R ⇒ SM, U LU M ⇒ SM and L LU M ⇒ SM.
Easy proof follows, in fact, from the definitions of these properties, cf. [21].

Corollary 6
Let ϕ be an Orlicz function.

Remark 7
The criteria of uniform monotonicity and property LLUM have been proved in [26,Theorem 11] using the theorem about the existence of isomorphic copy of l ∞ . If we consider properties SM and ULUM this argument is not enough and we need [27,Th. 3 and Th. 4].
(ii) ⇒ (iii) . By Theorem 1.1 from [18], there exists an Orlicz function ϕ w ∼ a ϕ such that ϕ > 0, ϕ (1) = 1 and p S ( ϕ) > 1, where p S (ψ) = inf t>0 tψ (t) ψ(t) is the lower Simonenko index. Take 1 < p < p S ( ϕ) . Consequently, by Lemma 2.3 from [19], the function ϕ(t) t p is non-decreasing (the condition ϕ ∈ 2 (R + ) used in the proof of Lemma 2.3 in [19] is not needed in this part). Thus ϕ 1/ p (t) t is non-decreasing. Denote Then γ is convex. Moreover, it is easy to see that γ It is enough to apply the condition 2 in equivalent form (see [8]). Consequently, as far as we know, it is not possible to replace the condition (+) in the Corollary 6 (3) by the condition α ϕ > 1, because we use the condition (+) exactly in proving the necessity of ϕ ∈ 2 .
(3) Clearly, if α ϕ > 1 then we have only the weak equivalence ϕ 1/ p w ∼ a γ. Unfortunately, in the proofs of results concerning the existence of isometric copy of l ∞ it is not enough (see [27]). (4) The implication (ii) ⇒ (i) under additional assumption that ϕ ∈ 2 has been proved in [24, Theorem 1.7].

Question 10
Is the condition ϕ ∈ 2 "really" needed to demonstrate the equivalence (ii) ⇒ (i) in Proposition 8?
Question 11 Suppose X = L ϕ and ϕ 1/ p is equivalent to a convex function for some p > 1. Then C X is rotund if and only if X is rotund. Indeed, L ϕ is rotund if and only if ϕ ∈ 2 and ϕ is strictly convex (see [8]

C X construction for (quasi-)symmetric spaces X
We also present some results concerning the spaces C X in the case of symmetric spaces X different than L ϕ . Given a vector space X the functional x → x is called a quasi-norm if the following three conditions are satisfied: We call · a p-norm where 0 < p ≤ 1 if, in addition, it is p-subadditive, that is, x + y p ≤ x p + y p for all x, y ∈ X .
Recall the important Aoki-Rolewicz theorem (cf. [ [40, pp. 6-8]): if 0 < p ≤ 1 is given by C = 2 1/ p−1 , then there exists a p-norm · 1 equivalent to the quasi-norm · so that for all x, y ∈ X . The quasi-norm · induces a metric topology on X : in fact a metric can be defined by d(x, y) = x − y p 1 , when the quasi-norm · 1 is p-subadditive. We say that X = (X, · ) is a quasi-Banach space if it is complete for this metric.
To find out more about quasi-normed spaces (and about even more general -normed spaces) the reader is referred to [25].
Let φ be a quasi-concave function on I , that is, φ(0) = 0, φ is positive, nondecreasing and φ(t)/t is non-increasing for t ∈ (0, m(I )). Then the Marcinkiewicz space M Moreover, if φ is a concave function on I , φ(0) = 0, φ is positive and non-decreasing then the Lorentz function space φ is given by the norm Recall that Marcinkiewicz spaces M ( * ) φ and Lorentz spaces φ are symmetric quasi-Banach function spaces, symmetric Banach function spaces on I, respectively (see [4,33]). The spaces M ( * ) φ and φ are examples symmetrizations X ( * ) of some Banach ideal spaces X (see [29] for some properties and more references). For more details about symmetric (quasi-)Banach function spaces see [4,30,33,39,40].
Any non-trivial symmetric normed function space X on I (X is non-trivial if X = {0}) is intermediate space between the spaces L 1 (I ) and L ∞ (I ). More precisely, (1) and f X is the fundamental function of X , i.e., f X (t) := χ [0,t] X for t ∈ I (see [33], Theorem 4.1). In particular, supp(X ) = I . The situation is different if (X, · ) is non-trivial symmetric quasi-Banach function space. Suppose the constant C comes from the triangle inequality for quasi-norm · and the number p satisfies the equality C = 2 1/ p−1 (see ( 4.1)). Applying Theorem 1 and 2 from [3] we conclude that where (X, · 1 ), Remark 12 (i) If X is a quasi-Banach ideal space on I then the space C X (defined as in Sect. 2) is a quasi-Banach ideal space on I. Note that C X need not have a weak unit even if X has (see Example 2 in [34]). However, if C : X → X is bounded and X has a weak unit then C X has it also.
(ii) Suppose X is a symmetric (quasi-)Banach ideal space on I = [0, 1] . Then C X = {0} . Indeed, we apply the conditions (4.2) and (4.3) for symmetric Banach function space and for symmetric quasi-Banach function space, respectively. Note also that if X → Y then C X → CY (see Remark after Corollary 1 in [26]). Clearly,
Proof The first fact follows from Remark 12 (ii) . In the case of I = [0, ∞) we will use the direct computations. Set . We have and using [34, Theorem 1 (a)] we finish the proof.
Then 0 ≤ y ≤ x and x = y. Clearly, C x = x. Moreover, The implications (b) ⇒ (a) . By the assumptions we conclude that X = φ is strictly monotone (see Lemma 3.1 in [28]). Although we use a little different than in [28] definition of the Lorentz space φ (they coincide if φ(0+) = 0), the proof of required implication is the same. Since X ∈ (SM) so C X ∈ (SM) (see Remark 1),whence the proof is finished. Proof Necessity. We divide the proof into two parts: 1] . Since in the view of [34, Theorem 1 (a)], we conclude that f ∈ φ and f ∈ C φ . Let A n = [0, 1 n ] for n ∈ N. We have which means that C φ is not order continuous.
Remark 16 Note that the above result for I = [0, 1] follows from Proposition 2 in [26] under the assumption that C : φ → φ . We present the direct proof because we need it for I = [0, ∞] and it does not require the assumption C : φ → φ . Note also that we may reformulate theorem above as: C φ is order continuous iff φ is order continuous.
Recall that a Banach ideal space X has the Kadec-Klee property with respect to global convergence in measure, we write X ∈ H g whenever for any sequence (x n ) ⊂ X such that x n → x globally in measure and x n X → x X , we have x n − x X → 0. The implication (c) ⇒ (a) . By the assumptions we conclude that X = φ is lower locally uniformly monotone (see Proposition 3 from [20]). Although we use a little different definition of the Lorentz space φ than in [20], they coincide if φ(0+) = 0. By Remark 1, we finish the proof.

Remark 18
The natural is the question of example of a Banach lattice which is strictly monotone but not lower (upper) locally uniformly monotone. There are not so many such examples. However, by the above two Theorems, taking X = φ on I = [0, 1] such that φ (t) > 0 for all t ∈ (0, μ (I )) and φ (0+) > 0 we get C X ∈ (SM) and

Lemma 19
The Cesàro-Lorentz function space C φ is not rotund for any fundamental function φ.

The
for any a > 0, because φ(t)/t is non-increasing. Therefore, for a = 1 we have Proof The idea of this proof comes from [7,Lemma 2.4]. The elements constructed below are in fact the same but because of slightly different character of the norm in Cesàro-Marcinkiewicz space and different monotonicity property considered in [7] we present the details. Moreover, we use part of this construction again in Proposition 23. Let for t ∈ [0, μ(I )). From [4,Theorem 5.2] we conclude that function f is a fundamental function of some Köthe dual space, so this function must be quasi-concave. Thus, the function defined as λ(t) = d dt f (t) is a non-increasing function defined almost everywhere. Put 1 2 ) (t) and h(t) = λ(t)χ [0,μ(I )) (t).
We have for any 0 ≤ t < 1/2, so and Of course, (i) g ≤ h and g = h, (ii) (Cg) * (t) = Cg(t) and Proof It is enough to take (we use the notation from the proof of Therefore, an element u is not order continuous. Since φ is the arbitrary quasi-concave function we finish the proof.