lilichen,yunancui,andyanfengzhaoย ยท research article complex convexity of musielak-orlicz function...
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Research ArticleComplex Convexity of Musielak-Orlicz Function SpacesEquipped with the ๐-Amemiya Norm
Lili Chen, Yunan Cui, and Yanfeng Zhao
Department of Mathematics, Harbin University of Science and Technology, Harbin 150080, China
Correspondence should be addressed to Lili Chen; [email protected]
Received 27 November 2013; Revised 5 April 2014; Accepted 13 April 2014; Published 8 May 2014
Academic Editor: Angelo Favini
Copyright ยฉ 2014 Lili Chen et al.This is an open access article distributed under the Creative CommonsAttribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The complex convexity of Musielak-Orlicz function spaces equipped with the ๐-Amemiya norm is mainly discussed. It is obtainedthat, for any Musielak-Orlicz function space equipped with the ๐-Amemiya norm when 1 โค ๐ < โ, complex strongly extremepoints of the unit ball coincide with complex extreme points of the unit ball. Moreover, criteria for them in above spaces are given.Criteria for complex strict convexity and complex midpoint locally uniform convexity of above spaces are also deduced.
1. Introduction
Let (๐, โ โ โ) be a complex Banach space over the complexfield C, let ๐ be the complex number satisfying ๐2 = โ1, andlet ๐ต(๐) and ๐(๐) be the closed unit ball and the unit sphereof ๐, respectively. In the sequel, N and R denote the set ofnatural numbers and the set of real numbers, respectively.
In the early 1980s, a huge number of papers in the area ofthe geometry of Banach spaces were directed to the complexgeometry of complex Banach spaces. It is well known that thecomplex geometric properties of complex Banach spaces haveapplications in various branches, among others in HarmonicAnalysis Theory, Operator Theory, Banach Algebras, ๐ถโ-Algebras, Differential EquationTheory, QuantumMechanicsTheory, and Hydrodynamics Theory. It is also known thatextreme points which are connected with strict convexity ofthe whole spaces are the most basic and important geometricpoints in geometric theory of Banach spaces (see [1โ6]).
In [7], Thorp and Whitley first introduced the conceptsof complex extreme point and complex strict convexitywhen they studied the conditions under which the StrongMaximum Modulus Theorem for analytic functions alwaysholds in a complex Banach space.
A point ๐ฅ โ ๐(๐) is said to be a complex extreme point of๐ต(๐) if for every nonzero๐ฆ โ ๐ there holds sup
|๐|โค1โ๐ฅ+๐๐ฆโ >
1. A complex Banach space ๐ is said to be complex strictlyconvex if every element of ๐(๐) is a complex extreme pointof ๐ต(๐).
In [8], we further studied the notions of complex stronglyextreme point and complex midpoint locally uniform con-vexity in general complex spaces.
A point ๐ฅ โ ๐(๐) is said to be a complex strongly extremepoint of ๐ต(๐) if for every ๐ > 0 we have ฮ
๐(๐ฅ, ๐) > 0, where
ฮ๐(๐ฅ, ๐) = inf {1 โ |๐| : โ๐ฆ โ ๐s.t. ๐๐ฅ ยฑ ๐ฆ
โค 1,
๐๐ฅ ยฑ ๐๐ฆ โค 1,
๐ฆ โฅ ๐} .
(1)
A complex Banach space ๐ is said to be complex midpointlocally uniformly convex if every element of ๐(๐) is a complexstrongly extreme point of ๐ต(๐).
Let (๐, ฮฃ, ๐) be a nonatomic and complete measure spacewith ๐(๐) < โ. By ฮฆ we denote a Musielak-Orlicz function;that is, ฮฆ : ๐ ร [0, +โ) โ [0, +โ] satisfies the following:
(1) for each ๐ข โ [0,โ], ฮฆ(๐ก, ๐ข) is a ๐-measurablefunction of ๐ก on ๐;
(2) for ๐-a.e. ๐ก โ ๐,ฮฆ(๐ก, 0) = 0, lim๐ขโโ
ฮฆ(๐ก, ๐ข) = โ andthere exists ๐ข > 0 such thatฮฆ(๐ก, ๐ข) < โ;
(3) for ๐-a.e. ๐ก โ ๐, ฮฆ(๐ก, ๐ข) is convex on the interval[0,โ) with respect to ๐ข.
Let ๐ฟ๐(๐) be the space of all ๐-equivalence classes ofcomplex and ฮฃ-measurable functions defined on ๐. For each๐ฅ โ ๐ฟ๐(๐), we define on ๐ฟ๐(๐) the convex modular of ๐ฅ by
๐ผฮฆ(๐ฅ) = โซ
๐
ฮฆ (๐ก, |๐ฅ (๐ก)|) ๐๐ก. (2)
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 190203, 6 pageshttp://dx.doi.org/10.1155/2014/190203
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2 Abstract and Applied Analysis
We define supp ๐ฅ = {๐ก โ ๐ : |๐ฅ(๐ก)| ฬธ= 0} and the Musielak-Orlicz space ๐ฟ
ฮฆgenerated by the formula
๐ฟฮฆ= {๐ฅ โ ๐ฟ
๐(๐) : ๐ผ
ฮฆ(๐๐ฅ) < โ for some ๐ > 0} . (3)
Set
๐ (๐ก) = sup {๐ข โฅ 0 : ฮฆ (๐ก, ๐ข) = 0} ,
๐ธ (๐ก) = sup {๐ข โฅ 0 : ฮฆ (๐ก, ๐ข) < โ} ;(4)
then ๐(๐ก) and ๐ธ(๐ก) are ๐-measurable (see [9]).The notion of ๐-Amemiya norm has been introduced in
[10]; for any 1 โค ๐ โค โ and ๐ข > 0, define
๐ ๐(๐ข) = {
(1 + ๐ข๐)1/๐
, for 1 โค ๐ < โ,max {1, ๐ข} , for ๐ = โ.
(5)
Furthermore, define ๐ ฮฆ,๐(๐ฅ) = ๐
๐โ๐ผฮฆ(๐ฅ) for all 1 โค ๐ โค โ
and ๐ฅ โ ๐ฟ๐(๐). Notice that the function ๐ ๐is increasing on ๐
+
for 1 โค ๐ < โ; however, the function ๐ โis increasing on the
interval [1,โ) only.For ๐ฅ โ ๐ฟ๐(๐), define the ๐-Amemiya norm by the
formula
โ๐ฅโฮฆ,๐ = inf๐>0
1
๐๐ ฮฆ,๐(๐๐ฅ) , 1 โค ๐ โค โ. (6)
For convenience, from now on, we write ๐ฟฮฆ,๐
=
(๐ฟฮฆ, โ โ โฮฆ,๐); it is easy to see that ๐ฟ
ฮฆ,๐is a Banach space. For
1 โค ๐ < โ, ๐ฅ โ ๐ฟฮฆ,๐
, set
๐พ๐(๐ฅ) = {๐ > 0 : โ๐ฅโฮฆ,๐ =
1
๐{1 + ๐ผ
๐
ฮฆ(๐๐ฅ)}1/๐
} . (7)
2. Main Results
We begin this section from the following useful lemmas.
Lemma 1 (see [9]). For any ๐ > 0, there exists ๐ฟ โ (0, 1/2)such that if ๐ข, V โ C and
|V| โฅ๐
8max๐|๐ข + ๐V| , (8)
then
|๐ข| โค1 โ 2๐ฟ
4ฮฃ๐ |๐ข + ๐V| , (9)
where
max๐|๐ข + ๐V| = max {|๐ข + V| , |๐ข โ V| , |๐ข + ๐V| , |๐ข โ ๐V|} ,
ฮฃ๐ |๐ข + ๐V| = |๐ข + V| + |๐ข โ V| + |๐ข + ๐V| + |๐ข โ ๐V| .
(10)
Lemma 2. If lim๐ขโโ
(ฮฆ(๐ก, ๐ข)/๐ข) = โ for ๐-a.e. ๐ก โ ๐, then๐พ๐(๐ฅ) ฬธ= 0 for any ๐ฅ โ ๐ฟ
ฮฆ,๐\ {0}, where 1 โค ๐ < โ.
Proof. For any ๐ฅ โ ๐ฟฮฆ,๐
\ {0}, there exists ๐ผ > 0 such that๐{๐ก โ ๐ : |๐ฅ(๐ก)| โฅ ๐ผ} > 0. Let ๐
1= {๐ก โ ๐ : |๐ฅ(๐ก)| โฅ ๐ผ} and
define the subsets
๐น๐= {๐ก โ ๐
1:ฮฆ (๐ก, ๐ข)
๐ขโฅ4โ๐ฅโฮฆ,๐
๐ผ โ ๐๐1
, ๐ข โฅ ๐} (11)
for each ๐ โ N. It is easy to see that ๐น1โ F2โ โ โ โ โ ๐น
๐โ โ โ โ
and lim๐โโ
๐๐น๐= ๐๐1. Then there exists ๐
0โ N such that
๐๐น๐0> (1/2)๐๐
1.
It follows from the definition of ๐-Amemiya norm thatthere is a sequence {๐
๐} satisfying
โ๐ฅโฮฆ,๐ = lim๐โโ
1
๐๐
{1 + ๐ผ๐
ฮฆ(๐๐๐ฅ)}1/๐
. (12)
For any ๐ > 0, we have
1
๐{1 + ๐ผ
๐
ฮฆ(๐๐ฅ)}1/๐
=1
๐{1 + [โซ
๐
ฮฆ (๐ก, ๐ |๐ฅ (๐ก)|) ๐๐ก]
๐
}
1/๐
โฅ
โซ๐น๐0
ฮฆ (๐ก, ๐ |๐ฅ (๐ก)|) ๐๐ก
๐
โฅ โซ๐น๐0
ฮฆ (๐ก, ๐๐ผ)
๐๐๐ก
= ๐ผโซ๐น๐0
ฮฆ (๐ก, ๐๐ผ)
๐๐ผ๐๐ก.
(13)
If ๐ > ๐0/๐ผ, notice that
1
๐{1 + ๐ผ
๐
ฮฆ(๐๐ฅ)}1/๐
โฅ ๐ผ โ 4โ๐ฅโฮฆ,๐
๐ผ โ ๐๐1
โ ๐๐น๐0> 2โ๐ฅโฮฆ,๐, (14)
which means the sequence {๐๐} is bounded. Hence, without
loss of generality, we assume that ๐๐โ ๐0as ๐ โ โ.
We can also choose the monotonic increasing or decreasingsubsequence of {๐
๐} that converges to the number ๐
0. Apply-
ing Levi Theorem and Lebesgue Dominated ConvergenceTheorem, we obtain
โ๐ฅโฮฆ,๐ = lim๐โโ
1
๐๐
{1 + ๐ผ๐
ฮฆ(๐๐๐ฅ)}1/๐
=1
๐0
{1 + ๐ผ๐
ฮฆ(๐0๐ฅ)}1/๐
(15)
which implies ๐0โ ๐พ๐(๐ฅ).
In order to exclude the case when such sets ๐พ๐(๐ฅ) are
empty for ๐ฅ โ ๐ฟฮฆ,๐\{0}, in the sequel we will assume, without
any special mention, that lim๐ขโโ
(ฮฆ(๐ก, ๐ข)/๐ข) = โ for ๐-a.e.๐ก โ ๐.
Theorem 3. Assume 1 โค ๐ < โ, ๐ฅ โ ๐(๐ฟฮฆ,๐). Then the
following assertions are equivalent:
(1) ๐ฅ is a complex strongly extreme point of the unit ball๐ต(๐ฟฮฆ,๐);
(2) ๐ฅ is a complex extreme point of the unit ball ๐ต(๐ฟฮฆ,๐);
(3) for any ๐ โ ๐พ๐(๐ฅ), ๐{๐ก โ ๐ : ๐|๐ฅ(๐ก)| < ๐(๐ก)} = 0.
Proof. The implication (1) โ (2) is trivial. Let ๐ฅ be a complexextreme point of the unit ball ๐ต(๐ฟ
ฮฆ,๐) and there exists ๐
0โ
๐พ๐(๐ฅ) such that ๐{๐ก โ ๐ : ๐
0|๐ฅ(๐ก)| < ๐(๐ก)} > 0. Then we can
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Abstract and Applied Analysis 3
find ๐ > 0 such that ๐{๐ก โ ๐ : ๐0|๐ฅ(๐ก)| + ๐ < ๐(๐ก)} > 0.
Indeed, if ๐{๐ก โ ๐ : ๐0|๐ฅ(๐ก)| + ๐ < ๐(๐ก)} = 0 for any ๐ > 0,
then we set ๐๐= 1/๐ and define the subsets
๐๐= {๐ก โ ๐ : ๐
0 |๐ฅ (๐ก)| + ๐๐ < ๐ (๐ก)} (16)
for each ๐ โ N. Notice that ๐1โ ๐2โ โ โ โ โ ๐
๐โ โ โ โ and
{๐ก โ ๐ : ๐0|๐ฅ(๐ก)| โค ๐(๐ก)} = โ
โ
๐=1๐๐; we can find that
๐ {๐ก โ ๐ : ๐0 |๐ฅ (๐ก)| โค ๐ (๐ก)} = 0 (17)
which is a contradiction.Let ๐0= {๐ก โ ๐ : ๐
0|๐ฅ(๐ก)| + ๐ < ๐(๐ก)} and define ๐ฆ =
(๐/๐0)๐๐0; we have ๐ฆ ฬธ= 0 and for any ๐ โ C with |๐| โค 1,
๐ฅ + ๐๐ฆฮฆ,๐ โค
1
๐0
{1 + ๐ผ๐
ฮฆ(๐0(๐ฅ + ๐๐ฆ))}
1/๐
=1
๐0
{1 + [๐ผฮฆ(๐0๐ฅ๐๐\๐0
)
+๐ผฮฆ(๐0๐ฅ๐๐0+ ๐๐๐
๐0)]๐
}1/๐
โค1
๐0
{1 + [๐ผฮฆ(๐0๐ฅ๐๐\๐0
)
+๐ผฮฆ((๐0 |๐ฅ| + ๐) ๐๐0
)]๐
}1/๐
=1
๐0
{1 + [๐ผฮฆ(๐0๐ฅ๐๐\๐0
)]๐
}1/๐
=1
๐0
(1 + ๐ผ๐
ฮฆ(๐0๐ฅ))1/๐
= 1,
(18)
which shows that ๐ฅ is not a complex extreme point of the unitball ๐ต(๐ฟ
ฮฆ,๐).
(3) โ (1). Suppose that ๐ฅ โ ๐(๐ฟฮฆ,๐) is not a complex
strongly extreme point of the unit ball ๐ต(๐ฟฮฆ,๐),; then there
exists ๐0> 0 such thatฮ
๐(๐ฅ, ๐0) = 0.That is, there exist๐
๐โ C
with |๐๐| โ 1 and๐ฆ
๐โ ๐ฟฮฆ,๐
satisfying โ๐ฆ๐โฮฆ,๐
โฅ ๐0such that
๐๐๐ฅ0 ยฑ ๐ฆ๐ฮฆ,๐ โค 1,
๐๐๐ฅ0 ยฑ ๐๐ฆ๐ฮฆ,๐ โค 1, (19)
which gives๐ฅ0ยฑ๐ฆ๐
๐๐
ฮฆ,๐
โค1๐๐
,
๐ฅ0ยฑ ๐๐ฆ๐
๐๐
ฮฆ,๐
โค1๐๐
. (20)
Setting ๐ง๐= ๐ฆ๐/๐๐, we have๐ง๐ฮฆ,๐ โฅ
๐ฆ๐ฮฆ,๐ โฅ ๐0,
๐ฅ0 ยฑ ๐ง๐ฮฆ,๐ โค
1๐๐
,
๐ฅ0 ยฑ ๐๐ง๐ฮฆ,๐ โค
1๐๐
.
(21)
For the above ๐0> 0, by Lemma 1, there exists ๐ฟ
0โ
(0, 1/2) such that if ๐ข, V โ C and
|V| โฅ๐0
8max๐|๐ข + ๐V| , (22)
then
|๐ข| โค1 โ 2๐ฟ
0
4ฮฃ๐ |๐ข + ๐V| . (23)
For each ๐ โ N, let
๐ด๐= {๐ก โ ๐ :
๐ง๐ (๐ก) โฅ
๐0
8max๐
๐ฅ0 (๐ก) + ๐๐ง๐ (๐ก)} ,
๐ง(1)
๐: ๐ง(1)
๐(๐ก) = ๐ง
๐(๐ก) (๐ก โ ๐ด
๐) , ๐ง
(1)
๐(๐ก) = 0 (๐ก โ ๐ด
๐) ,
๐ง(2)
๐: ๐ง(2)
๐(๐ก) = 0 (๐ก โ ๐ด
๐) , ๐ง
(2)
๐(๐ก) = ๐ง
๐(๐ก) (๐ก โ ๐ด
๐) .
(24)
It is easy to see that ๐ง๐= ๐ง(1)
๐+๐ง(2)
๐,โ๐ โ N. Since |๐
๐| โ 1
when ๐ โ โ, the following inequalities
๐ง(1)
๐
ฮฆ,๐
= inf๐>0
1
๐{1 + [โซ
๐กโ๐ด๐
ฮฆ(๐ก, ๐๐ง๐ (๐ก)
) ๐๐ก]
๐
}
1/๐
โค inf๐>0
1
๐{1+
[โซ๐กโ๐ด๐
ฮฆ(๐ก, ๐๐0
8max๐
๐ฅ0 (๐ก) + ๐๐ง๐ (๐ก)) ๐๐ก]
๐
}
1/๐
โค๐0
8
max๐
๐ฅ0 + ๐๐ง๐
ฮฆ,๐
โค๐0
8
ฮฃ๐๐ฅ0 + ๐๐ง๐
ฮฆ,๐
โค๐0
2๐๐
<3๐0
4
(25)
hold for ๐ large enough.Therefore, โ๐ง(2)
๐โฮฆ,๐
> ๐0/4 which shows that ๐(๐ด
๐) > 0
for ๐ large enough. Furthermore, for any ๐ก โ ๐ด๐, we have
๐ฅ0 (๐ก) โค
1 โ 2๐ฟ0
4ฮฃ๐
๐ฅ0 (๐ก) + ๐๐ง๐ (๐ก) .
(26)
To complete the proof, we consider the following twocases.
(I) One has ๐๐โ โ(๐ โ โ), where ๐
๐โ ๐พ๐((1/4)
ฮฃ๐|๐ฅ0+ ๐๐ง๐|).
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4 Abstract and Applied Analysis
For each ๐ โ N, we get
1 =๐ฅ0
ฮฆ,๐
โค1
๐๐
{1 + ๐ผ๐
ฮฆ(๐๐๐ฅ0)}1/๐
=1
๐๐
{1 + [โซ๐กโ๐ด๐
ฮฆ(๐ก, ๐๐
๐ฅ0 (๐ก)) ๐๐ก
+โซ๐กโ๐ด๐
ฮฆ(๐ก, ๐๐
๐ฅ0 (๐ก)) ๐๐ก]
๐
}
1/๐
โค1
๐๐
{1 + [โซ๐กโ๐ด๐
ฮฆ(๐ก, ๐๐(1 โ 2๐ฟ
0
4
รฮฃ๐
๐ฅ0 (๐ก) + ๐๐ง๐ (๐ก) )) ๐๐ก
+ โซ๐กโ๐ด๐
ฮฆ(๐ก, ๐๐
ร (1
4ฮฃ๐
๐ฅ0 (๐ก) + ๐๐ง๐ (๐ก))) ๐๐ก]
๐
}
1/๐
โค1
๐๐
{1
+ [ (1 โ 2๐ฟ0)
ร โซ๐กโ๐ด๐
ฮฆ(๐ก, ๐๐(1
4ฮฃ๐
๐ฅ0 (๐ก) + ๐๐ง๐ (๐ก))) ๐๐ก
+ โซ๐กโ๐ด๐
ฮฆ(๐ก, ๐๐
ร(1
4ฮฃ๐
๐ฅ0 (๐ก) + ๐๐ง๐ (๐ก))) ๐๐ก]
๐
}
1/๐
=1
๐๐
{1
+ [๐ผฮฆ(๐๐(1
4ฮฃ๐
๐ฅ0 + ๐๐ง๐)) โ 2๐ฟ0
ร โซ๐กโ๐ด๐
ฮฆ(๐ก, ๐๐
ร(1
4ฮฃ๐
๐ฅ0 (๐ก) + ๐๐ง๐ (๐ก))) ๐๐ก]
๐
}
1/๐
.
(27)
Furthermore, we notice that
โซ๐กโ๐ด๐
ฮฆ(๐ก, ๐๐(1
4ฮฃ๐
๐ฅ0 (๐ก) + ๐๐ง๐ (๐ก))) ๐๐ก
โฅ โซ๐กโ๐ด๐
ฮฆ(๐ก, ๐๐
๐ง๐ (๐ก)) ๐๐ก
โฅ [๐๐
๐
๐ง(2)
๐
๐
ฮฆ,๐โ 1]1/๐
โฅ [(๐0
4๐๐)
๐
โ 1]
1/๐
.
(28)
Since ๐๐โ โwhen ๐ โ โ, we can see that the inequality
((๐0/4)๐๐)๐โ 1 > 0 holds for ๐ large enough. Moreover, we
find that
๐ผฮฆ(๐๐(1
4ฮฃ๐
๐ฅ0 + ๐๐ง๐))
โฅ ๐ผฮฆ(๐๐๐ฅ0)
โฅ (๐๐
๐โ 1)1/๐
โฅ 2๐ฟ0((๐0
4๐๐)
๐
โ 1)
1/๐
.
(29)
Then we deduce
1 =๐ฅ0
ฮฆ,๐
โค1
๐๐
{1 + [๐ผฮฆ(๐๐(1
4ฮฃ๐
๐ฅ0 + ๐๐ง๐)) โ 2๐ฟ0
ร โซ๐กโ๐ด๐
ฮฆ(๐ก, ๐๐
ร(1
4ฮฃ๐
๐ฅ0 (๐ก) + ๐๐ง๐ (๐ก))) ๐๐ก]
๐
}
1/๐
โค1
๐๐
{1 + [๐ผฮฆ(๐๐(1
4ฮฃ๐
๐ฅ0 + ๐๐ง๐))
โ2๐ฟ0((๐0
4๐๐)
๐
โ 1)
1/๐
]
๐
}
1/๐
<1
๐๐
{1 + ๐ผ๐
ฮฆ(๐๐(1
4ฮฃ๐
๐ฅ0 + ๐๐ง๐))}
1/๐
= 1,
(30)
a contradiction.(II) One has ๐
๐โ ๐
0(๐ โ โ), where ๐
๐โ
๐พ๐((1/4)ฮฃ
๐|๐ฅ0+ ๐๐ง๐|), ๐0โ R.
Then for each ๐ โ N,
1
๐๐
โฅ
1
4ฮฃ๐
๐ฅ0 + ๐๐ง๐
ฮฆ,๐
=1
๐๐
{1 + ๐ผ๐
ฮฆ(๐๐
4ฮฃ๐
๐ฅ0 + ๐๐ง๐)}
1/๐
โฅ1
๐๐
{1 + ๐ผ๐
ฮฆ(๐๐๐ฅ0)}1/๐
โฅ๐ฅ0
ฮฆ,๐ = 1.
(31)
Let ๐ โ โ, we deduce that 1 = (1/๐0){1 + ๐ผ
๐
ฮฆ(๐0๐ฅ0)}1/๐
=
โ๐ฅ0โฮฆ,๐
.
-
Abstract and Applied Analysis 5
From (I), we obtain that
1 =โ ๐ฅ0โฮฆ,๐
โค1
๐๐
{1+ [๐ผฮฆ(๐๐(1
4ฮฃ๐
๐ฅ0 + ๐๐ง๐)) โ 2๐ฟ0
ร โซ๐กโ๐ด๐
ฮฆ(๐ก, ๐๐
ร (1
4ฮฃ๐
๐ฅ0 (๐ก) + ๐๐ง๐ (๐ก))) ๐๐ก]
๐
}
1/๐
.
(32)
Now we consider the following two subcases.(a) Consider โซ
๐กโ๐ด๐
ฮฆ(๐ก, ๐๐((1/4)ฮฃ
๐|๐ฅ0(๐ก) + ๐๐ง
๐(๐ก)|))๐๐ก > 0
for some ๐ โ N.Then we obtain
1 =โ ๐ฅ0โฮฆ,๐
โค1
๐๐
{1 + [๐ผฮฆ(๐๐(1
4ฮฃ๐
๐ฅ0 + ๐๐ง๐)) โ 2๐ฟ0
ร โซ๐กโ๐ด๐
ฮฆ(๐ก, ๐๐
ร(1
4ฮฃ๐
๐ฅ0 (๐ก) + ๐๐ง๐ (๐ก))) ๐๐ก]
๐
}
1/๐
<1
๐๐
{1 + ๐ผ๐
ฮฆ(๐๐(1
4ฮฃ๐
๐ฅ0 + ๐๐ง๐))}
1/๐
= 1,
(33)
a contradiction.(b) Consider โซ
๐กโ๐ด๐
ฮฆ(๐ก, ๐๐((1/4)ฮฃ
๐|๐ฅ0(๐ก) + ๐๐ง
๐(๐ก)|))๐๐ก = 0
for any ๐ โ N.For ๐ large enough, we observe that
๐๐>1 โ 2๐ฟ
0
1 โ ๐ฟ0
๐0. (34)
Hence, we get
0 = โซ๐กโ๐ด๐
ฮฆ(๐ก, ๐๐(1
4ฮฃ๐
๐ฅ0 (๐ก) + ๐๐ง๐ (๐ก))) ๐๐ก
โฅ โซ๐กโ๐ด๐
ฮฆ(๐ก,
๐๐
1 โ 2๐ฟ0
๐ฅ0(๐ก)
) ๐๐ก
โฅ โซ๐กโ๐ด๐
ฮฆ(๐ก,
๐0
1 โ ๐ฟ0
๐ฅ0(๐ก)
) ๐๐ก โฅ 0.
(35)
Thus, we see the equalityโซ๐กโ๐ด๐
ฮฆ(๐ก, |(๐0/(1โ๐ฟ
0))๐ฅ0(๐ก)|) ๐๐ก = 0
holds for ๐ large enough. It follows that
๐0
1 โ ๐ฟ0
๐ฅ0(๐ก)
โค ๐ (๐ก) , ๐-a.e. ๐ก โ ๐ด
๐ (36)
since ๐(๐ด๐) > 0 for ๐ large enough.
On the other hand, by (3), we deduce that
๐0๐ฅ0 (๐ก) โฅ ๐ (๐ก) , ๐-a.e. ๐ก โ ๐ด๐. (37)
Hence, we get a contradiction:
๐0
1 โ ๐ฟ0
๐ฅ0(๐ก)
โฅ
๐ (๐ก)
1 โ ๐ฟ0
> ๐ (๐ก) , ๐-a.e. ๐ก โ ๐ด๐. (38)
Theorem 4. Assume 1 โค ๐ < โ; then the following assertionsare equivalent:
(1) ๐ฟฮฆ,๐
is complex midpoint locally uniformly convex;
(2) ๐ฟฮฆ,๐
is complex strictly convex;
(3) ๐(๐ก) = 0 for ๐-a.e. ๐ก โ ๐.
Proof. The implication (1) โ (2) is trivial. Now assume that๐ฟฮฆ,๐
is complex strictly convex. If ๐{๐ก โ ๐ : ๐(๐ก) > 0} > 0, let๐0= {๐ก โ ๐ : ๐(๐ก) > 0} and it is not difficult to find an element
๐ฅ โ ๐(๐ฟฮฆ,๐) satisfying supp ๐ฅ = ๐ \ ๐
0. Take ๐ โ ๐พ
๐(๐ฅ), and
define
๐ฆ (๐ก) ={
{
{
๐ (๐ก)
2๐for ๐ก โ ๐
0,
๐ฅ (๐ก) for ๐ก โ ๐ \ ๐0.
(39)
Obviously, โ๐ฆโฮฆ,๐
โฅ โ๐ฅโฮฆ,๐
= 1. On the other hand,
๐ฆฮฆ,๐ โค
1
๐{1 + [โซ
๐กโ๐0
ฮฆ(๐ก,๐ (๐ก)
2) ๐๐ก
+โซ๐\๐0
ฮฆ (๐ก, ๐๐ฅ (๐ก)) ๐๐ก]
๐
}
1/๐
=1
๐(1 + ๐ผ
๐
ฮฆ(๐๐ฅ))1/๐
= 1.
(40)
Thus, โ๐ฆโฮฆ,๐
= (1/๐)(1 + ๐ผ๐
ฮฆ(๐๐ฆ))1/๐
= 1. However, for ๐ก โ๐0, we find ๐|๐ฆ(๐ก)| = ๐(๐ก)/2 < ๐(๐ก), which implies ๐ฆ โ ๐(๐ฟ
ฮฆ,๐)
is not a complex extreme point of ๐ต(๐ฟฮฆ,๐) fromTheorem 3.
(3) โ (1). Suppose that ๐ฅ โ ๐(๐ฟฮฆ,๐) is not a complex
strongly extreme point of ๐ต(๐ฟฮฆ,๐). It follows fromTheorem 3
that ๐{๐ก โ ๐ : ๐0|๐ฅ(๐ก)| < ๐(๐ก)} > 0 for some ๐
0โ ๐พ๐(๐ฅ),
consequently ๐{๐ก โ ๐ : ๐(๐ก) > 0} > 0which is a contradiction.
Remark 5. If ๐ = โ then ๐-Amemiya norm equals Luxem-burg norm, the problem of complex convexity of Musielak-Orlicz function spaces equipped with the Luxemburg normhas been investigated in [8].
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
-
6 Abstract and Applied Analysis
Acknowledgments
This work is supported by Grants from HeilongjiangProvincial Natural Science Foundation for Youths (no.QC2013C001), Natural Science Foundation of HeilongjiangEducational Committee (no. 12531099), Youth Science Fundof Harbin University of Science and Technology (no.2011YF002), and Tianyuan Funds of the National NaturalScience Foundation of China (no. 11226127).
References
[1] O. Blasco and M. Pavlovicฬ, โComplex convexity and vector-valued Littlewood-Paley inequalities,โ Bulletin of the LondonMathematical Society, vol. 35, no. 6, pp. 749โ758, 2003.
[2] C. Choi, A. Kaminฬska, and H. J. Lee, โComplex convexity ofOrlicz-Lorentz spaces and its applications,โ Bulletin of the PolishAcademy of Sciences:Mathematics, vol. 52, no. 1, pp. 19โ38, 2004.
[3] H. Hudzik and A. Narloch, โRelationships betweenmonotonic-ity and complex rotundity properties with some consequences,โMathematica Scandinavica, vol. 96, no. 2, pp. 289โ306, 2005.
[4] H. J. Lee, โMonotonicity and complex convexity in Banachlattices,โ Journal of Mathematical Analysis and Applications, vol.307, no. 1, pp. 86โ101, 2005.
[5] H. J. Lee, โComplex convexity and monotonicity in Quasi-Banach lattices,โ Israel Journal of Mathematics, vol. 159, no. 1, pp.57โ91, 2007.
[6] M. M. Czerwinฬska and A. Kaminฬska, โComplex rotunditiesand midpoint local uniform rotundity in symmetric spaces ofmeasurable operators,โ Studia Mathematica, vol. 201, no. 3, pp.253โ285, 2010.
[7] E. Thorp and R. Whitley, โThe strong maximum modulustheorem for analytic functions into a Banach space,โ Proceedingsof the AmericanMathematical Society, vol. 18, pp. 640โ646, 1967.
[8] L. Chen, Y. Cui, and H. Hudzik, โCriteria for complex stronglyextreme points of Musielak-Orlicz function spaces,โ NonlinearAnalysis: Theory, Methods & Applications, vol. 70, no. 6, pp.2270โ2276, 2009.
[9] S. Chen, Geometry of Orlicz Spaces, Dissertationes Mathemati-cae, 1996.
[10] Y. Cui, L. Duan, H. Hudzik, and M. Wisลa, โBasic theory of p-Amemiya norm in Orlicz spaces (1 โค ๐ โค โ): extreme pointsand rotundity in Orlicz spaces endowed with these norms,โNonlinear Analysis: Theory, Methods & Applications, vol. 69, no.5-6, pp. 1796โ1816, 2008.
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