Non-squareness properties of Orlicz-Lorentz function spaces
© Foralewski et al.; licensee Springer 2013
Received: 2 October 2012
Accepted: 8 January 2013
Published: 25 January 2013
In this paper, criteria for non-squareness and uniform non-squareness of Orlicz-Lorentz function spaces are given. Since degenerated Orlicz functions φ and degenerated weight functions ω are also admitted, this investigation concerns the most possible wide class of Orlicz-Lorentz function spaces.
It is worth recalling that uniform non-squareness is an important property, because it implies super-reflexivity as well as the fixed point property (see James in Ann. Math. 80:542-550, 1964; Pacific J. Math. 41:409-419, 1972 and García-Falset et al. in J. Funct. Anal. 233:494-514, 2006).
MSC:46B20, 46B42, 46A80, 46E30.
Uniform non-squareness of Banach spaces has been defined by James as the geometric property which implies super-reflexivity (see [1, 2]). So, after proving this property for a Banach space, we know, without any characterization of the dual space, that it is super-reflexive, so reflexive as well. Recently, García-Falset, Llorens-Fuster and Mazcuñan-Navarro have shown that uniformly non-square Banach spaces have the fixed point property (see ).
Therefore, it was natural and interesting to look for criteria of non-squareness properties in various well-known classes of Banach spaces. Among a great number of papers concerning this topic, we list here [4–12].
The problem of uniform non-squareness of Calderón-Lozanovskiĭ spaces was initiated by Cerdà, Hudzik and Mastyło in . Since the class of Orlicz-Lorentz spaces is a subclass of Calderón-Lozanovskiĭ spaces, we can say that also the problem of uniform non-squareness of Orlicz-Lorentz spaces was initiated in . However, the results of our paper show that those results were only some sufficient conditions for uniform non-squareness which were very far from being necessary and sufficient. Analogous results for Orlicz-Lorentz sequence spaces were presented in , but the techniques of the proofs in the function case are different (in some parts completely different) than in the sequence case.
We say that a Banach space is non-square if for any x and y from (the unit sphere of X). A Banach space X is said to be uniformly non-square if there exists such that for any (the unit ball of X). In the last definition, the unit ball can be replaced, equivalently, by the unit sphere .
Let be the space of all (equivalence classes of) Lebesgue measurable real-valued functions defined on the interval , where . For any , we write if almost everywhere with respect to the Lebesgue measure m on the interval .
(under the convention ). We say that two functions are equimeasurable if for all . Then we obviously have .
Let and be totally σ-finite measure spaces. A map σ from into is called a measure preserving transformation if for each -measurable subset A from , the set is a -measurable subset of and (see ). It is well known that a measure preserving transformation induces equimeasurability, that is, if σ is a measure preserving transformation, then x and are equimeasurable functions. The converse is false (see ).
if , and , then and ,
there exists a function x in E that is strictly positive on the whole .
Recall that a Köthe space E is called a symmetric space if E is rearrangement invariant which means that if , and , then and (see ). For basic properties of symmetric spaces, we refer to [15, 16] and .
Let us note that if , then , while left continuity of φ on is equivalent to the fact that .
holds for any (then we have and ). Analogously, we say that an Orlicz function φ satisfies the condition at infinity ( for short) if there exist a constant and a constant such that and inequality (1) holds for any (then we obtain ).
for all . It is easy to show that ψ is also an Orlicz function.
Now we recall the definition of Orlicz-Lorentz spaces. These spaces were introduced by Kamińska (see [26, 27] and ) at the beginning of 1990s. Her investigations gave an impulse to further investigations of the spaces, results of which have been published, among others, in the papers [14, 28–42].
A Banach lattice is said to be strictly monotone if , and imply that . We say that E is uniformly monotone if for any , there is such that whenever , , and (see ). Recall (see ) that in Banach lattices E, strict monotonicity and uniform monotonicity are restrictions of rotundity and uniform rotundity (respectively) to couples of comparable elements in the positive cone only.
The Lorentz function space is strictly monotone if and only if ω is positive on and whenever .
Theorem 1.2 The Lorentz function space is uniformly monotone if and only if the weight function ω is regular and ω is positive on whenever .
In our further investigations, we will also apply Lemma 1.1 and Remark 1.1. By convexity of the modular , Lemma 1.1 can be proved analogously as in the case of Orlicz spaces (cf. also  for considering a more general case).
Lemma 1.1 Suppose that the Orlicz function φ satisfies a suitable condition , that is, if and , and otherwise. Then, for any , there exists such that for any whenever .
In particular, for any , we then have that if and only if .
We start with the following
Theorem 2.1 Let . Then the Orlicz-Lorentz function space is non-square if and only if , and .
where is such that , we get and, consequently, . Thus, is not non-square.
Therefore, if , we have .
In order to do this, we will consider two cases.
Applying inequalities (5), (6), (7) and (8), we obtain (4).
Case 2. Let now . Then there exists v such that and for any t and s satisfying . Proceeding similarly as in the above Case 1, but with v instead of , we get again inequality (4). □
Theorem 2.2 If , then the Orlicz-Lorentz function space is non-square if and only if , and .
which means that is not non-square.
Sufficiency. Let . Analogously as in the proof of Theorem 2.1, it is enough to show that . We divide the proof into several parts.
Consequently, in the remaining part of the proof, we will assume that for any in formula (11), we have equality for both the sum and the difference.
whence we get .
If , then we can assume without loss of generality that , whence we get the equality . Proceeding analogously as in Case 22.214.171.124, we obtain .
So, we get .
Applying convexity of the Orlicz function and the equality in formula (11), we get the conditions , , and . Since , the set has positive measure. If , we can assume that (where is defined analogously as in (19)); in the opposite case, we can assume that . Proceeding analogously as in Case 2.2.1, we obtain . □
Theorem 2.3 In the case when , the Orlicz-Lorentz function space is uniformly non-square if and only if , and ω is regular.
whence we have .
Since , we have . Suppose that
where depends only on . □
Theorem 2.4 If , then the Orlicz-Lorentz function space is uniformly non-square if and only if , , ω is regular and .
Proof Necessity. The necessity of the conditions and has been shown in Theorem 2.2, whereas the necessity of the conditions and regularity of ω can be shown analogously as in Theorem 2.3.
with some depending only on q, and the proof will be finished. In order to show (20), we consider three cases.
where is the constant from the definition of uniform monotonicity of the Lorentz space corresponding to . If , then we get similarly that . Therefore, if , we obtain inequality (20) with .
Now we divide the proof of this case into several parts.
where is the constant from the definition of uniform monotonicity of the Lorentz space corresponding to .
By the definition of and the inequality , we have and , respectively.
Indeed, by the equalities and the definition of , we have and , whence by and the definition of , we get (25).
where is the constant from the definition of uniform monotonicity of the Lorentz space corresponding to .
Case 3. Finally, assume that and . For arbitrary fixed , we define the sets and by formulas (22) and (23). If , then proceeding analogously as in Case 2, we get inequality (24) with the constant .
If or , where the sets and are defined by formulas (28) and (29), then analogously as in Case 2, we obtain inequality (30) with the constant .
Defining and repeating the procedure from Case 2, putting in place of , we get inequality (39) with the constant .
Theorem 2.5 Let and . Then the Orlicz-Lorentz function space is uniformly non-square if and only if , , ω is regular and .
Proof Necessity. Condition follows from Theorem 2.2, while the necessity of remaining conditions can be proved as in Theorem 2.4.
Sufficiency. Analogously as in Theorem 2.4, it is enough to show that there exists such that inequality (20) holds for any .
In order to show (20), we will consider two cases.
The remaining part of the proof of Case 2 will be divided into three subcases.
which is a contradiction.
Summarizing both cases, we get inequality (20) with . □
The authors thank the referees for valuable comments and suggestions. All authors are supported partially by the State Committee for Scientific Research, Poland, Grant No. N N201 362236.
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