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Nonsquareness and Locally Uniform Nonsquareness in Orlicz-Bochner Function Spaces Endowed with Luxemburg Norm
Journal of Inequalities and Applications volume 2011, Article number: 875649 (2011)
Abstract
Criteria for nonsquareness and locally uniform nonsquareness of Orlicz-Bochner function spaces equipped with Luxemburg norm are given. We also prove that, in Orlicz-Bochner 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 Orlicz-Bochner 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 Orlicz-Bochner 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ1_HTML.gif)
Put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ2_HTML.gif)
The linear set endowed with the Luxemburg norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ3_HTML.gif)
is a Banach space. We say that an Orlicz function satisfies condition
if there exist
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ4_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ6_HTML.gif)
Proof.
Case 1.
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ7_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ8_HTML.gif)
Case 2.
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ9_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ10_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ12_HTML.gif)
This yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ13_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ14_HTML.gif)
Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ15_HTML.gif)
It is easy to see . We know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ16_HTML.gif)
Hence, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ17_HTML.gif)
It is easy to see , a contradiction!
Sufficiency. Suppose that there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ18_HTML.gif)
We will derive a contradiction for each of the following two cases.
Case 1.
. Let
. Hence, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ19_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ20_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ21_HTML.gif)
(b)If , then
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ22_HTML.gif)
Proof.
-
(a)
Since
is locally uniformly nonsquare, we have
and
, where
and
(2.18)
In fact, since is locally uniformly nonsquare, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ24_HTML.gif)
Case 1.
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ25_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ26_HTML.gif)
Case 2.
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ27_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ28_HTML.gif)
Therefore, we get, the following inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ29_HTML.gif)
holds.
(b1) Suppose that , where
. Then there exist
and subsequence
of
, such that
. By definition of
, there exist
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ30_HTML.gif)
We will derive a contradiction for each of the following two cases.
Case 1.
. Since
, there exists
such that
. Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ31_HTML.gif)
This implies , a contradiction!
Case 2.
. Without loss of generality, we may assume
, where
. Since
, there exists
such that
. Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ32_HTML.gif)
This implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ33_HTML.gif)
Similarly, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ34_HTML.gif)
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ35_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ36_HTML.gif)
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ37_HTML.gif)
whenever . Since
, there exists
such that
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ38_HTML.gif)
Hence, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ39_HTML.gif)
This implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ40_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ41_HTML.gif)
Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ42_HTML.gif)
It is easy to see . We know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ43_HTML.gif)
Moreover, we have ,
. By the dominated convergence theorem, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ44_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ45_HTML.gif)
We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ46_HTML.gif)
This implies . Hence,
. We define a function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ47_HTML.gif)
on , where
. By Lemma 2.4, we have
-a.e on
. Let
-a.e on
, where
is simple function. Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ48_HTML.gif)
is -measurable. By Lemma 2.4, we have
-a.e on
. Then
is
-measurable. Using
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ49_HTML.gif)
we get that there exists such that
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ50_HTML.gif)
Let ,
,
. It is easy to see
,
and
. If
, by Lemma 2.4, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ51_HTML.gif)
Without loss of generality, we may assume that there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ52_HTML.gif)
Moreover, for any , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ53_HTML.gif)
Hence, if , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ54_HTML.gif)
Let . Then
. Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ55_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ56_HTML.gif)
we get that there exist such that
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ57_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ58_HTML.gif)
In fact, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ59_HTML.gif)
Moreover, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ60_HTML.gif)
By (2.54) and (2.55), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ61_HTML.gif)
This shows that if , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ62_HTML.gif)
It is easy to see . Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F875649/MediaObjects/13660_2010_Article_2365_Equ63_HTML.gif)
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.
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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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Shang, S., Cui, Y. & Fu, Y. Nonsquareness and Locally Uniform Nonsquareness in Orlicz-Bochner Function Spaces Endowed with Luxemburg Norm. J Inequal Appl 2011, 875649 (2011). https://doi.org/10.1155/2011/875649
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DOI: https://doi.org/10.1155/2011/875649