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Nonsquareness and Locally Uniform Nonsquareness in OrliczBochner Function Spaces Endowed with Luxemburg Norm
Journal of Inequalities and Applicationsvolume 2011, Article number: 875649 (2011)
Abstract
Criteria for nonsquareness and locally uniform nonsquareness of OrliczBochner function spaces equipped with Luxemburg norm are given. We also prove that, in OrliczBochner function spaces generated by locally uniform nonsquare Banach space, nonsquareness and locally uniform nonsquareness are equivalent.
1. Introduction
A lot of nonsquareness concepts in Banach spaces are known (see [1]). Nonsquareness are important notions in geometry of Banach space. One of reasons is that these properties are strongly related to the fixed point property (see [2]). The criteria for nonsquareness and locally uniform nonsquareness in the classical Orlicz function spaces have been given in [3, 4] already. However, because of the complicated structure of OrliczBochner function spaces equipped with the Luxemburg norm, the criteria for nonsquareness and locally uniform nonsquareness of them have not been found yet. The aim of this paper is to give criteria for nonsquareness and locally uniform nonsquareness of OrliczBochner function spaces equipped with Luxemburg norm.
Let be a real Banach space. and denote the unit sphere and unit ball, respectively. Let us recall some geometrical notions concerning nonsquareness. A Banach space is said to be nonsquare if for any we have . A Banach space is said to be uniformly nonsquare if there exists such that for any , . A Banach space is said to be locally uniformly nonsquare if for any , there exists such that , where .
Let be set of real numbers. A function is called an function if is convex, even, , and , and .
Let be a nonatomic measurable space. denotes right derivative of . Moreover, for a given Banach space , we denote by the set of all strongly measurable function from to , and for each , we define the modular of by
Put
The linear set endowed with the Luxemburg norm
is a Banach space. We say that an Orlicz function satisfies condition if there exist and such that
First let us recall a known result that will be used in the further part of the paper.
Lemma 1.1 (see [3]).
Suppose . Then
2. Main Results
Theorem 2.1.
is nonsquare if and only if
(a);
(b) is nonsquare.
In order to prove the theorem, we give a lemma.
Lemma 2.2.
If is nonsquare, then for any , we have
Proof.
Case 1.
If , then
or
Case 2.
If , then
or
This implies . This completes the proof.
Proof of Theorem 2.1.

(a)
Necessity. Suppose that , then there exist and such that . Pick such that is not a null set. Since , there exist sequence and disjont subsets of such that
(2.6)
Therefore, if we define , then for any , we have
This yields
Hence, . But , and we deduce that . Moreover, we have and , a contradiction with nonsquareness of .
If (b) is not true, then there exist such that . Pick such that . Put
Then we have
It is easy to see . We know that
Hence, we have
It is easy to see , a contradiction!
Sufficiency. Suppose that there exists such that
We will derive a contradiction for each of the following two cases.
Case 1.
. Let . Hence, we have
Since , we have . Hence, . This implies , a contradiction!
Case 2.
. By Lemma 2.2, without loss of generality, we may assume that there exists such that , and . Therefore,
Since , we have . Hence, . This implies , a contradiction!
Theorem 2.3.
is locally uniformly nonsquare if and only if
(a);
(b) is locally uniformly nonsquare.
In order to prove the theorem, we give a lemma.
Lemma 2.4.
If is locally uniformly nonsquare, then
(a)For any , , we have
(b)If , then , where
Proof.

(a)
Since is locally uniformly nonsquare, we have and , where and
(2.18)
In fact, since is locally uniformly nonsquare, we have
Case 1.
If , then
or
Case 2.
If , then
or
Therefore, we get, the following inequality
holds.
(b1) Suppose that , where . Then there exist and subsequence of , such that . By definition of , there exist such that
We will derive a contradiction for each of the following two cases.
Case 1.
. Since , there exists such that . Therefore,
This implies , a contradiction!
Case 2.
. Without loss of generality, we may assume , where . Since , there exists such that . Therefore, we have
This implies
Similarly, we have
Therefore, we have
This implies , a contradiction! Hence, .
(b2) Suppose that , where . Then there exist and subsequence of , such that . Since , then there exist such that , whenever . By definition of , there exist such that
Therefore, we have
whenever . Since , there exists such that , where
Hence, we have
This implies
which contradict (2.32). Hence, .
Combing (b1) with (b2), we get . This completes the proof.
Proof of Theorem 2.3.
Necessity. By Theorem 2.1, . If (b) is not true, then there exist , such that and as . Pick such that . Put
Then we have
It is easy to see . We know that
Moreover, we have , . By the dominated convergence theorem, we have
It is easy to see , as . By Lemma 1.1, we have and as , a contradiction with locally uniform nonsquareness of .
Sufficiency. Suppose that there exist , such that as . We will derive a contradiction for each of the following two cases.
Case 1.
There exist , such that , where . Put
We have
This implies . Hence, . We define a function
on , where . By Lemma 2.4, we have a.e on . Let a.e on , where is simple function. Hence,
is measurable. By Lemma 2.4, we have a.e on . Then is measurable. Using
we get that there exists such that , where
Let , , . It is easy to see , and . If , by Lemma 2.4, we have
Without loss of generality, we may assume that there exists such that
Moreover, for any , we have
Hence, if , then
Let . Then . Therefore,
By Lemma 1.1, we have , as . This is in contradiction with .
Case 2.
For any , , there exists such that whenever . By the Riesz theorem, without loss of generality, we may assume that a.e on . Using
we get that there exist such that , where
Since is function, we can choose such that . Since a.e on , by the Egorov theorem, there exists such that whenever , where , . Next, we will prove that if , then
In fact, we have
Moreover, we have
By (2.54) and (2.55), we have
This shows that if , then
It is easy to see . Therefore,
for large enough. By Lemma 1.1, we have , as , which contradicts , for large enough. This completes the proof.
Corollary 2.5.
The following statements are equivalent:
(a) is locally uniformly nonsquare if and only if is nonsquare;
(b) is locally uniformly nonsquare.
References
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James RC: Uniform nonsquare Banach space. Annals of Mathematics 1964,80(3):542–550. 10.2307/1970663
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GarcíaFalset J, LlorensFuster E, MazcuñanNavarro EM: Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings. Journal of Functional Analysis 2006,233(2):494–514. 10.1016/j.jfa.2005.09.002
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Chen ST: Geometry of Orlicz spaces. Dissertationes Math 1996, 356: 1–204.
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Wu CX, Wang TF, Chen ST, Wang YW: Geometry Theory of Orlicz Spaces. H.I.T Print House, Harbin, China; 1986.
Acknowledgments
The authors would like to thank the anonymous referee for some suggestions to improve the manuscript. This work was supported by China Natural Science Fund under Grant no. 11061022.
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Keywords
 Banach Space
 Measurable Function
 Function Space
 Convergence Theorem
 Measurable Space