LiliChen,YunanCui,andYanfengZhaoΒ Β· Research Article Complex Convexity of Musielak-Orlicz Function...

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Research Article Complex Convexity of Musielak-Orlicz Function Spaces Equipped with the -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 Β© 2014 Lili Chen et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e complex convexity of Musielak-Orlicz function spaces equipped with the -Amemiya norm is mainly discussed. It is obtained that, for any Musielak-Orlicz function space equipped with the -Amemiya norm when 1≀<∞, complex strongly extreme points 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 (,β€– β‹… β€–) be a complex Banach space over the complex field C, let be the complex number satisfying 2 = βˆ’1, and let () and () be the closed unit ball and the unit sphere of , respectively. In the sequel, N and R denote the set of natural numbers and the set of real numbers, respectively. In the early 1980s, a huge number of papers in the area of the geometry of Banach spaces were directed to the complex geometry of complex Banach spaces. It is well known that the complex geometric properties of complex Banach spaces have applications in various branches, among others in Harmonic Analysis eory, Operator eory, Banach Algebras, βˆ— - Algebras, Differential Equation eory, Quantum Mechanics eory, and Hydrodynamics eory. It is also known that extreme points which are connected with strict convexity of the whole spaces are the most basic and important geometric points in geometric theory of Banach spaces (see [1–6]). In [7], orp and Whitley first introduced the concepts of complex extreme point and complex strict convexity when they studied the conditions under which the Strong Maximum Modulus eorem for analytic functions always holds in a complex Banach space. A point ∈ () is said to be a complex extreme point of () if for every nonzero ∈ there holds sup ||≀1 β€–+β€– > 1. A complex Banach space is said to be complex strictly convex if every element of () is a complex extreme point of (). In [8], we further studied the notions of complex strongly extreme point and complex midpoint locally uniform con- vexity in general complex spaces. A point ∈ () is said to be a complex strongly extreme point of () if for every >0 we have Ξ” (, ) > 0, where Ξ” (, ) = inf {1 βˆ’ || : βˆƒ ∈ s.t. Β± ≀ 1, Β± ≀ 1, β‰₯ } . (1) A complex Banach space is said to be complex midpoint locally uniformly convex if every element of () is a complex strongly extreme point of (). Let (, Ξ£, ) be a nonatomic and complete measure space with () < ∞. By Ξ¦ we denote a Musielak-Orlicz function; that is, Ξ¦ : Γ— [0, +∞) β†’ [0, +∞] satisfies the following: (1) for each ∈ [0, ∞], Ξ¦(, ) is a -measurable function of on ; (2) for -a.e. ∈, Ξ¦(, 0) = 0, lim β†’βˆž Ξ¦(, ) = ∞ and there exists >0 such that Ξ¦(, )<∞; (3) for -a.e. ∈ , Ξ¦(, ) is convex on the interval [0, ∞) with respect to . Let () be the space of all -equivalence classes of complex and Ξ£-measurable functions defined on . For each ∈ (), we define on () the convex modular of by Ξ¦ () = ∫ Ξ¦ (, | ()|) . (2) Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 190203, 6 pages http://dx.doi.org/10.1155/2014/190203

Transcript of LiliChen,YunanCui,andYanfengZhaoΒ Β· Research Article Complex Convexity of Musielak-Orlicz Function...

  • Research ArticleComplex Convexity of Musielak-Orlicz Function SpacesEquipped with the 𝑝-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 Β© 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 𝑝-Amemiya norm is mainly discussed. It is obtainedthat, for any Musielak-Orlicz function space equipped with the 𝑝-Amemiya norm when 1 ≀ 𝑝 < ∞, 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 (𝑋, β€– β‹… β€–) be a complex Banach space over the complexfield C, let 𝑖 be the complex number satisfying 𝑖2 = βˆ’1, andlet 𝐡(𝑋) and 𝑆(𝑋) be the closed unit ball and the unit sphereof 𝑋, 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, πΆβˆ—-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 [1–6]).

    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 π‘₯ ∈ 𝑆(𝑋) is said to be a complex extreme point of𝐡(𝑋) if for every nonzero𝑦 ∈ 𝑋 there holds sup

    |πœ†|≀1β€–π‘₯+πœ†π‘¦β€– >

    1. A complex Banach space 𝑋 is said to be complex strictlyconvex if every element of 𝑆(𝑋) is a complex extreme pointof 𝐡(𝑋).

    In [8], we further studied the notions of complex stronglyextreme point and complex midpoint locally uniform con-vexity in general complex spaces.

    A point π‘₯ ∈ 𝑆(𝑋) is said to be a complex strongly extremepoint of 𝐡(𝑋) if for every πœ€ > 0 we have Ξ”

    𝑐(π‘₯, πœ€) > 0, where

    Δ𝑐(π‘₯, πœ€) = inf {1 βˆ’ |πœ†| : βˆƒπ‘¦ ∈ 𝑋s.t. πœ†π‘₯ Β± 𝑦

    ≀ 1,

    πœ†π‘₯ Β± 𝑖𝑦 ≀ 1,

    𝑦 β‰₯ πœ€} .

    (1)

    A complex Banach space 𝑋 is said to be complex midpointlocally uniformly convex if every element of 𝑆(𝑋) is a complexstrongly extreme point of 𝐡(𝑋).

    Let (𝑇, Ξ£, πœ‡) be a nonatomic and complete measure spacewith πœ‡(𝑇) < ∞. By Ξ¦ we denote a Musielak-Orlicz function;that is, Ξ¦ : 𝑇 Γ— [0, +∞) β†’ [0, +∞] satisfies the following:

    (1) for each 𝑒 ∈ [0,∞], Ξ¦(𝑑, 𝑒) is a πœ‡-measurablefunction of 𝑑 on 𝑇;

    (2) for πœ‡-a.e. 𝑑 ∈ 𝑇,Ξ¦(𝑑, 0) = 0, limπ‘’β†’βˆž

    Ξ¦(𝑑, 𝑒) = ∞ andthere exists 𝑒 > 0 such thatΞ¦(𝑑, 𝑒) < ∞;

    (3) for πœ‡-a.e. 𝑑 ∈ 𝑇, Ξ¦(𝑑, 𝑒) is convex on the interval[0,∞) with respect to 𝑒.

    Let 𝐿𝑐(πœ‡) be the space of all πœ‡-equivalence classes ofcomplex and Ξ£-measurable functions defined on 𝑇. For eachπ‘₯ ∈ 𝐿𝑐(πœ‡), we define on 𝐿𝑐(πœ‡) the convex modular of π‘₯ by

    𝐼Φ(π‘₯) = ∫

    𝑇

    Ξ¦ (𝑑, |π‘₯ (𝑑)|) 𝑑𝑑. (2)

    Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 190203, 6 pageshttp://dx.doi.org/10.1155/2014/190203

  • 2 Abstract and Applied Analysis

    We define supp π‘₯ = {𝑑 ∈ 𝑇 : |π‘₯(𝑑)| ΜΈ= 0} and the Musielak-Orlicz space 𝐿

    Ξ¦generated by the formula

    𝐿Φ= {π‘₯ ∈ 𝐿

    𝑐(πœ‡) : 𝐼

    Ξ¦(πœ†π‘₯) < ∞ for some πœ† > 0} . (3)

    Set

    𝑒 (𝑑) = sup {𝑒 β‰₯ 0 : Ξ¦ (𝑑, 𝑒) = 0} ,

    𝐸 (𝑑) = sup {𝑒 β‰₯ 0 : Ξ¦ (𝑑, 𝑒) < ∞} ;(4)

    then 𝑒(𝑑) and 𝐸(𝑑) are πœ‡-measurable (see [9]).The notion of 𝑝-Amemiya norm has been introduced in

    [10]; for any 1 ≀ 𝑝 ≀ ∞ and 𝑒 > 0, define

    𝑠𝑝(𝑒) = {

    (1 + 𝑒𝑝)1/𝑝

    , for 1 ≀ 𝑝 < ∞,max {1, 𝑒} , for 𝑝 = ∞.

    (5)

    Furthermore, define 𝑠Φ,𝑝(π‘₯) = 𝑠

    π‘βˆ˜πΌΞ¦(π‘₯) for all 1 ≀ 𝑝 ≀ ∞

    and π‘₯ ∈ 𝐿𝑐(πœ‡). Notice that the function 𝑠𝑝is increasing on 𝑅

    +

    for 1 ≀ 𝑝 < ∞; however, the function π‘ βˆžis increasing on the

    interval [1,∞) only.For π‘₯ ∈ 𝐿𝑐(πœ‡), define the 𝑝-Amemiya norm by the

    formula

    β€–π‘₯β€–Ξ¦,𝑝 = infπ‘˜>0

    1

    π‘˜π‘ Ξ¦,𝑝(π‘˜π‘₯) , 1 ≀ 𝑝 ≀ ∞. (6)

    For convenience, from now on, we write 𝐿Φ,𝑝

    =

    (𝐿Φ, β€– β‹… β€–Ξ¦,𝑝); it is easy to see that 𝐿

    Ξ¦,𝑝is a Banach space. For

    1 ≀ 𝑝 < ∞, π‘₯ ∈ 𝐿Φ,𝑝

    , set

    𝐾𝑝(π‘₯) = {π‘˜ > 0 : β€–π‘₯β€–Ξ¦,𝑝 =

    1

    π‘˜{1 + 𝐼

    𝑝

    Ξ¦(π‘˜π‘₯)}1/𝑝

    } . (7)

    2. Main Results

    We begin this section from the following useful lemmas.

    Lemma 1 (see [9]). For any πœ€ > 0, there exists 𝛿 ∈ (0, 1/2)such that if 𝑒, V ∈ C and

    |V| β‰₯πœ€

    8max𝑒|𝑒 + 𝑒V| , (8)

    then

    |𝑒| ≀1 βˆ’ 2𝛿

    4Σ𝑒 |𝑒 + 𝑒V| , (9)

    where

    max𝑒|𝑒 + 𝑒V| = max {|𝑒 + V| , |𝑒 βˆ’ V| , |𝑒 + 𝑖V| , |𝑒 βˆ’ 𝑖V|} ,

    Σ𝑒 |𝑒 + 𝑒V| = |𝑒 + V| + |𝑒 βˆ’ V| + |𝑒 + 𝑖V| + |𝑒 βˆ’ 𝑖V| .

    (10)

    Lemma 2. If limπ‘’β†’βˆž

    (Ξ¦(𝑑, 𝑒)/𝑒) = ∞ for πœ‡-a.e. 𝑑 ∈ 𝑇, then𝐾𝑝(π‘₯) ΜΈ= 0 for any π‘₯ ∈ 𝐿

    Ξ¦,𝑝\ {0}, where 1 ≀ 𝑝 < ∞.

    Proof. For any π‘₯ ∈ 𝐿Φ,𝑝

    \ {0}, there exists 𝛼 > 0 such thatπœ‡{𝑑 ∈ 𝑇 : |π‘₯(𝑑)| β‰₯ 𝛼} > 0. Let 𝑇

    1= {𝑑 ∈ 𝑇 : |π‘₯(𝑑)| β‰₯ 𝛼} and

    define the subsets

    𝐹𝑛= {𝑑 ∈ 𝑇

    1:Ξ¦ (𝑑, 𝑒)

    𝑒β‰₯4β€–π‘₯β€–Ξ¦,𝑝

    𝛼 β‹… πœ‡π‘‡1

    , 𝑒 β‰₯ 𝑛} (11)

    for each 𝑛 ∈ N. It is easy to see that 𝐹1βŠ† F2βŠ† β‹… β‹… β‹… βŠ† 𝐹

    π‘›βŠ† β‹… β‹… β‹…

    and limπ‘›β†’βˆž

    πœ‡πΉπ‘›= πœ‡π‘‡1. Then there exists 𝑛

    0∈ N such that

    πœ‡πΉπ‘›0> (1/2)πœ‡π‘‡

    1.

    It follows from the definition of 𝑝-Amemiya norm thatthere is a sequence {π‘˜

    𝑛} satisfying

    β€–π‘₯β€–Ξ¦,𝑝 = limπ‘›β†’βˆž

    1

    π‘˜π‘›

    {1 + 𝐼𝑝

    Ξ¦(π‘˜π‘›π‘₯)}1/𝑝

    . (12)

    For any π‘˜ > 0, we have

    1

    π‘˜{1 + 𝐼

    𝑝

    Ξ¦(π‘˜π‘₯)}1/𝑝

    =1

    π‘˜{1 + [∫

    𝑇

    Ξ¦ (𝑑, π‘˜ |π‘₯ (𝑑)|) 𝑑𝑑]

    𝑝

    }

    1/𝑝

    β‰₯

    βˆ«πΉπ‘›0

    Ξ¦ (𝑑, π‘˜ |π‘₯ (𝑑)|) 𝑑𝑑

    π‘˜

    β‰₯ βˆ«πΉπ‘›0

    Ξ¦ (𝑑, π‘˜π›Ό)

    π‘˜π‘‘π‘‘

    = π›Όβˆ«πΉπ‘›0

    Ξ¦ (𝑑, π‘˜π›Ό)

    π‘˜π›Όπ‘‘π‘‘.

    (13)

    If π‘˜ > 𝑛0/𝛼, notice that

    1

    π‘˜{1 + 𝐼

    𝑝

    Ξ¦(π‘˜π‘₯)}1/𝑝

    β‰₯ 𝛼 β‹…4β€–π‘₯β€–Ξ¦,𝑝

    𝛼 β‹… πœ‡π‘‡1

    β‹… πœ‡πΉπ‘›0> 2β€–π‘₯β€–Ξ¦,𝑝, (14)

    which means the sequence {π‘˜π‘›} is bounded. Hence, without

    loss of generality, we assume that π‘˜π‘›β†’ π‘˜0as 𝑛 β†’ ∞.

    We can also choose the monotonic increasing or decreasingsubsequence of {π‘˜

    𝑛} that converges to the number π‘˜

    0. Apply-

    ing Levi Theorem and Lebesgue Dominated ConvergenceTheorem, we obtain

    β€–π‘₯β€–Ξ¦,𝑝 = limπ‘›β†’βˆž

    1

    π‘˜π‘›

    {1 + 𝐼𝑝

    Ξ¦(π‘˜π‘›π‘₯)}1/𝑝

    =1

    π‘˜0

    {1 + 𝐼𝑝

    Ξ¦(π‘˜0π‘₯)}1/𝑝

    (15)

    which implies π‘˜0∈ 𝐾𝑝(π‘₯).

    In order to exclude the case when such sets 𝐾𝑝(π‘₯) are

    empty for π‘₯ ∈ 𝐿Φ,𝑝\{0}, in the sequel we will assume, without

    any special mention, that limπ‘’β†’βˆž

    (Ξ¦(𝑑, 𝑒)/𝑒) = ∞ for πœ‡-a.e.𝑑 ∈ 𝑇.

    Theorem 3. Assume 1 ≀ 𝑝 < ∞, π‘₯ ∈ 𝑆(𝐿Φ,𝑝). Then the

    following assertions are equivalent:

    (1) π‘₯ is a complex strongly extreme point of the unit ball𝐡(𝐿Φ,𝑝);

    (2) π‘₯ is a complex extreme point of the unit ball 𝐡(𝐿Φ,𝑝);

    (3) for any π‘˜ ∈ 𝐾𝑝(π‘₯), πœ‡{𝑑 ∈ 𝑇 : π‘˜|π‘₯(𝑑)| < 𝑒(𝑑)} = 0.

    Proof. The implication (1) β‡’ (2) is trivial. Let π‘₯ be a complexextreme point of the unit ball 𝐡(𝐿

    Ξ¦,𝑝) and there exists π‘˜

    0∈

    𝐾𝑝(π‘₯) such that πœ‡{𝑑 ∈ 𝑇 : π‘˜

    0|π‘₯(𝑑)| < 𝑒(𝑑)} > 0. Then we can

  • Abstract and Applied Analysis 3

    find 𝑑 > 0 such that πœ‡{𝑑 ∈ 𝑇 : π‘˜0|π‘₯(𝑑)| + 𝑑 < 𝑒(𝑑)} > 0.

    Indeed, if πœ‡{𝑑 ∈ 𝑇 : π‘˜0|π‘₯(𝑑)| + 𝑑 < 𝑒(𝑑)} = 0 for any 𝑑 > 0,

    then we set 𝑑𝑛= 1/𝑛 and define the subsets

    𝑇𝑛= {𝑑 ∈ 𝑇 : π‘˜

    0 |π‘₯ (𝑑)| + 𝑑𝑛 < 𝑒 (𝑑)} (16)

    for each 𝑛 ∈ N. Notice that 𝑇1βŠ† 𝑇2βŠ† β‹… β‹… β‹… βŠ† 𝑇

    π‘›βŠ† β‹… β‹… β‹… and

    {𝑑 ∈ 𝑇 : π‘˜0|π‘₯(𝑑)| ≀ 𝑒(𝑑)} = ⋃

    ∞

    𝑛=1𝑇𝑛; we can find that

    πœ‡ {𝑑 ∈ 𝑇 : π‘˜0 |π‘₯ (𝑑)| ≀ 𝑒 (𝑑)} = 0 (17)

    which is a contradiction.Let 𝑇0= {𝑑 ∈ 𝑇 : π‘˜

    0|π‘₯(𝑑)| + 𝑑 < 𝑒(𝑑)} and define 𝑦 =

    (𝑑/π‘˜0)πœ’π‘‡0; we have 𝑦 ΜΈ= 0 and for any πœ† ∈ C with |πœ†| ≀ 1,

    π‘₯ + πœ†π‘¦Ξ¦,𝑝 ≀

    1

    π‘˜0

    {1 + 𝐼𝑝

    Ξ¦(π‘˜0(π‘₯ + πœ†π‘¦))}

    1/𝑝

    =1

    π‘˜0

    {1 + [𝐼Φ(π‘˜0π‘₯πœ’π‘‡\𝑇0

    )

    +𝐼Φ(π‘˜0π‘₯πœ’π‘‡0+ πœ†π‘‘πœ’

    𝑇0)]𝑝

    }1/𝑝

    ≀1

    π‘˜0

    {1 + [𝐼Φ(π‘˜0π‘₯πœ’π‘‡\𝑇0

    )

    +𝐼Φ((π‘˜0 |π‘₯| + 𝑑) πœ’π‘‡0

    )]𝑝

    }1/𝑝

    =1

    π‘˜0

    {1 + [𝐼Φ(π‘˜0π‘₯πœ’π‘‡\𝑇0

    )]𝑝

    }1/𝑝

    =1

    π‘˜0

    (1 + 𝐼𝑝

    Ξ¦(π‘˜0π‘₯))1/𝑝

    = 1,

    (18)

    which shows that π‘₯ is not a complex extreme point of the unitball 𝐡(𝐿

    Ξ¦,𝑝).

    (3) β‡’ (1). Suppose that π‘₯ ∈ 𝑆(𝐿Φ,𝑝) is not a complex

    strongly extreme point of the unit ball 𝐡(𝐿Φ,𝑝),; then there

    exists πœ€0> 0 such thatΞ”

    𝑐(π‘₯, πœ€0) = 0.That is, there existπœ†

    π‘›βˆˆ C

    with |πœ†π‘›| β†’ 1 and𝑦

    π‘›βˆˆ 𝐿Φ,𝑝

    satisfying ‖𝑦𝑛‖Φ,𝑝

    β‰₯ πœ€0such that

    πœ†π‘›π‘₯0 Β± 𝑦𝑛Φ,𝑝 ≀ 1,

    πœ†π‘›π‘₯0 Β± 𝑖𝑦𝑛Φ,𝑝 ≀ 1, (19)

    which givesπ‘₯0±𝑦𝑛

    πœ†π‘›

    Ξ¦,𝑝

    ≀1πœ†π‘›

    ,

    π‘₯0Β± 𝑖𝑦𝑛

    πœ†π‘›

    Ξ¦,𝑝

    ≀1πœ†π‘›

    . (20)

    Setting 𝑧𝑛= 𝑦𝑛/πœ†π‘›, we have𝑧𝑛Φ,𝑝 β‰₯

    𝑦𝑛Φ,𝑝 β‰₯ πœ€0,

    π‘₯0 Β± 𝑧𝑛Φ,𝑝 ≀

    1πœ†π‘›

    ,

    π‘₯0 Β± 𝑖𝑧𝑛Φ,𝑝 ≀

    1πœ†π‘›

    .

    (21)

    For the above πœ€0> 0, by Lemma 1, there exists 𝛿

    0∈

    (0, 1/2) such that if 𝑒, V ∈ C and

    |V| β‰₯πœ€0

    8max𝑒|𝑒 + 𝑒V| , (22)

    then

    |𝑒| ≀1 βˆ’ 2𝛿

    0

    4Σ𝑒 |𝑒 + 𝑒V| . (23)

    For each 𝑛 ∈ N, let

    𝐴𝑛= {𝑑 ∈ 𝑇 :

    𝑧𝑛 (𝑑) β‰₯

    πœ€0

    8max𝑒

    π‘₯0 (𝑑) + 𝑒𝑧𝑛 (𝑑)} ,

    𝑧(1)

    𝑛: 𝑧(1)

    𝑛(𝑑) = 𝑧

    𝑛(𝑑) (𝑑 βˆ‰ 𝐴

    𝑛) , 𝑧

    (1)

    𝑛(𝑑) = 0 (𝑑 ∈ 𝐴

    𝑛) ,

    𝑧(2)

    𝑛: 𝑧(2)

    𝑛(𝑑) = 0 (𝑑 βˆ‰ 𝐴

    𝑛) , 𝑧

    (2)

    𝑛(𝑑) = 𝑧

    𝑛(𝑑) (𝑑 ∈ 𝐴

    𝑛) .

    (24)

    It is easy to see that 𝑧𝑛= 𝑧(1)

    𝑛+𝑧(2)

    𝑛,βˆ€π‘› ∈ N. Since |πœ†

    𝑛| β†’ 1

    when 𝑛 β†’ ∞, the following inequalities

    𝑧(1)

    𝑛

    Ξ¦,𝑝

    = infπ‘˜>0

    1

    π‘˜{1 + [∫

    π‘‘βˆ‰π΄π‘›

    Ξ¦(𝑑, π‘˜π‘§π‘› (𝑑)

    ) 𝑑𝑑]

    𝑝

    }

    1/𝑝

    ≀ infπ‘˜>0

    1

    π‘˜{1+

    [βˆ«π‘‘βˆ‰π΄π‘›

    Ξ¦(𝑑, π‘˜πœ€0

    8max𝑒

    π‘₯0 (𝑑) + 𝑒𝑧𝑛 (𝑑)) 𝑑𝑑]

    𝑝

    }

    1/𝑝

    β‰€πœ€0

    8

    max𝑒

    π‘₯0 + 𝑒𝑧𝑛

    Ξ¦,𝑝

    β‰€πœ€0

    8

    Σ𝑒π‘₯0 + 𝑒𝑧𝑛

    Ξ¦,𝑝

    β‰€πœ€0

    2πœ†π‘›

    <3πœ€0

    4

    (25)

    hold for 𝑛 large enough.Therefore, ‖𝑧(2)

    𝑛‖Φ,𝑝

    > πœ€0/4 which shows that πœ‡(𝐴

    𝑛) > 0

    for 𝑛 large enough. Furthermore, for any 𝑑 ∈ 𝐴𝑛, we have

    π‘₯0 (𝑑) ≀

    1 βˆ’ 2𝛿0

    4Σ𝑒

    π‘₯0 (𝑑) + 𝑒𝑧𝑛 (𝑑) .

    (26)

    To complete the proof, we consider the following twocases.

    (I) One has π‘˜π‘›β†’ ∞(𝑛 β†’ ∞), where π‘˜

    π‘›βˆˆ 𝐾𝑝((1/4)

    Σ𝑒|π‘₯0+ 𝑒𝑧𝑛|).

  • 4 Abstract and Applied Analysis

    For each 𝑛 ∈ N, we get

    1 =π‘₯0

    Ξ¦,𝑝

    ≀1

    π‘˜π‘›

    {1 + 𝐼𝑝

    Ξ¦(π‘˜π‘›π‘₯0)}1/𝑝

    =1

    π‘˜π‘›

    {1 + [βˆ«π‘‘βˆˆπ΄π‘›

    Ξ¦(𝑑, π‘˜π‘›

    π‘₯0 (𝑑)) 𝑑𝑑

    +βˆ«π‘‘βˆ‰π΄π‘›

    Ξ¦(𝑑, π‘˜π‘›

    π‘₯0 (𝑑)) 𝑑𝑑]

    𝑝

    }

    1/𝑝

    ≀1

    π‘˜π‘›

    {1 + [βˆ«π‘‘βˆˆπ΄π‘›

    Ξ¦(𝑑, π‘˜π‘›(1 βˆ’ 2𝛿

    0

    4

    ×Σ𝑒

    π‘₯0 (𝑑) + 𝑒𝑧𝑛 (𝑑) )) 𝑑𝑑

    + βˆ«π‘‘βˆ‰π΄π‘›

    Ξ¦(𝑑, π‘˜π‘›

    Γ— (1

    4Σ𝑒

    π‘₯0 (𝑑) + 𝑒𝑧𝑛 (𝑑))) 𝑑𝑑]

    𝑝

    }

    1/𝑝

    ≀1

    π‘˜π‘›

    {1

    + [ (1 βˆ’ 2𝛿0)

    Γ— βˆ«π‘‘βˆˆπ΄π‘›

    Ξ¦(𝑑, π‘˜π‘›(1

    4Σ𝑒

    π‘₯0 (𝑑) + 𝑒𝑧𝑛 (𝑑))) 𝑑𝑑

    + βˆ«π‘‘βˆ‰π΄π‘›

    Ξ¦(𝑑, π‘˜π‘›

    Γ—(1

    4Σ𝑒

    π‘₯0 (𝑑) + 𝑒𝑧𝑛 (𝑑))) 𝑑𝑑]

    𝑝

    }

    1/𝑝

    =1

    π‘˜π‘›

    {1

    + [𝐼Φ(π‘˜π‘›(1

    4Σ𝑒

    π‘₯0 + 𝑒𝑧𝑛)) βˆ’ 2𝛿0

    Γ— βˆ«π‘‘βˆˆπ΄π‘›

    Ξ¦(𝑑, π‘˜π‘›

    Γ—(1

    4Σ𝑒

    π‘₯0 (𝑑) + 𝑒𝑧𝑛 (𝑑))) 𝑑𝑑]

    𝑝

    }

    1/𝑝

    .

    (27)

    Furthermore, we notice that

    βˆ«π‘‘βˆˆπ΄π‘›

    Ξ¦(𝑑, π‘˜π‘›(1

    4Σ𝑒

    π‘₯0 (𝑑) + 𝑒𝑧𝑛 (𝑑))) 𝑑𝑑

    β‰₯ βˆ«π‘‘βˆˆπ΄π‘›

    Ξ¦(𝑑, π‘˜π‘›

    𝑧𝑛 (𝑑)) 𝑑𝑑

    β‰₯ [π‘˜π‘

    𝑛

    𝑧(2)

    𝑛

    𝑝

    Ξ¦,π‘βˆ’ 1]1/𝑝

    β‰₯ [(πœ€0

    4π‘˜π‘›)

    𝑝

    βˆ’ 1]

    1/𝑝

    .

    (28)

    Since π‘˜π‘›β†’ ∞when 𝑛 β†’ ∞, we can see that the inequality

    ((πœ€0/4)π‘˜π‘›)π‘βˆ’ 1 > 0 holds for 𝑛 large enough. Moreover, we

    find that

    𝐼Φ(π‘˜π‘›(1

    4Σ𝑒

    π‘₯0 + 𝑒𝑧𝑛))

    β‰₯ 𝐼Φ(π‘˜π‘›π‘₯0)

    β‰₯ (π‘˜π‘

    π‘›βˆ’ 1)1/𝑝

    β‰₯ 2𝛿0((πœ€0

    4π‘˜π‘›)

    𝑝

    βˆ’ 1)

    1/𝑝

    .

    (29)

    Then we deduce

    1 =π‘₯0

    Ξ¦,𝑝

    ≀1

    π‘˜π‘›

    {1 + [𝐼Φ(π‘˜π‘›(1

    4Σ𝑒

    π‘₯0 + 𝑒𝑧𝑛)) βˆ’ 2𝛿0

    Γ— βˆ«π‘‘βˆˆπ΄π‘›

    Ξ¦(𝑑, π‘˜π‘›

    Γ—(1

    4Σ𝑒

    π‘₯0 (𝑑) + 𝑒𝑧𝑛 (𝑑))) 𝑑𝑑]

    𝑝

    }

    1/𝑝

    ≀1

    π‘˜π‘›

    {1 + [𝐼Φ(π‘˜π‘›(1

    4Σ𝑒

    π‘₯0 + 𝑒𝑧𝑛))

    βˆ’2𝛿0((πœ€0

    4π‘˜π‘›)

    𝑝

    βˆ’ 1)

    1/𝑝

    ]

    𝑝

    }

    1/𝑝

    <1

    π‘˜π‘›

    {1 + 𝐼𝑝

    Ξ¦(π‘˜π‘›(1

    4Σ𝑒

    π‘₯0 + 𝑒𝑧𝑛))}

    1/𝑝

    = 1,

    (30)

    a contradiction.(II) One has π‘˜

    𝑛→ π‘˜

    0(𝑛 β†’ ∞), where π‘˜

    π‘›βˆˆ

    𝐾𝑝((1/4)Ξ£

    𝑒|π‘₯0+ 𝑒𝑧𝑛|), π‘˜0∈ R.

    Then for each 𝑛 ∈ N,

    1

    πœ†π‘›

    β‰₯

    1

    4Σ𝑒

    π‘₯0 + 𝑒𝑧𝑛

    Ξ¦,𝑝

    =1

    π‘˜π‘›

    {1 + 𝐼𝑝

    Ξ¦(π‘˜π‘›

    4Σ𝑒

    π‘₯0 + 𝑒𝑧𝑛)}

    1/𝑝

    β‰₯1

    π‘˜π‘›

    {1 + 𝐼𝑝

    Ξ¦(π‘˜π‘›π‘₯0)}1/𝑝

    β‰₯π‘₯0

    Ξ¦,𝑝 = 1.

    (31)

    Let 𝑛 β†’ ∞, we deduce that 1 = (1/π‘˜0){1 + 𝐼

    𝑝

    Ξ¦(π‘˜0π‘₯0)}1/𝑝

    =

    β€–π‘₯0β€–Ξ¦,𝑝

    .

  • Abstract and Applied Analysis 5

    From (I), we obtain that

    1 =β€– π‘₯0β€–Ξ¦,𝑝

    ≀1

    π‘˜π‘›

    {1+ [𝐼Φ(π‘˜π‘›(1

    4Σ𝑒

    π‘₯0 + 𝑒𝑧𝑛)) βˆ’ 2𝛿0

    Γ— βˆ«π‘‘βˆˆπ΄π‘›

    Ξ¦(𝑑, π‘˜π‘›

    Γ— (1

    4Σ𝑒

    π‘₯0 (𝑑) + 𝑒𝑧𝑛 (𝑑))) 𝑑𝑑]

    𝑝

    }

    1/𝑝

    .

    (32)

    Now we consider the following two subcases.(a) Consider ∫

    π‘‘βˆˆπ΄π‘›

    Ξ¦(𝑑, π‘˜π‘›((1/4)Ξ£

    𝑒|π‘₯0(𝑑) + 𝑒𝑧

    𝑛(𝑑)|))𝑑𝑑 > 0

    for some 𝑛 ∈ N.Then we obtain

    1 =β€– π‘₯0β€–Ξ¦,𝑝

    ≀1

    π‘˜π‘›

    {1 + [𝐼Φ(π‘˜π‘›(1

    4Σ𝑒

    π‘₯0 + 𝑒𝑧𝑛)) βˆ’ 2𝛿0

    Γ— βˆ«π‘‘βˆˆπ΄π‘›

    Ξ¦(𝑑, π‘˜π‘›

    Γ—(1

    4Σ𝑒

    π‘₯0 (𝑑) + 𝑒𝑧𝑛 (𝑑))) 𝑑𝑑]

    𝑝

    }

    1/𝑝

    <1

    π‘˜π‘›

    {1 + 𝐼𝑝

    Ξ¦(π‘˜π‘›(1

    4Σ𝑒

    π‘₯0 + 𝑒𝑧𝑛))}

    1/𝑝

    = 1,

    (33)

    a contradiction.(b) Consider ∫

    π‘‘βˆˆπ΄π‘›

    Ξ¦(𝑑, π‘˜π‘›((1/4)Ξ£

    𝑒|π‘₯0(𝑑) + 𝑒𝑧

    𝑛(𝑑)|))𝑑𝑑 = 0

    for any 𝑛 ∈ N.For 𝑛 large enough, we observe that

    π‘˜π‘›>1 βˆ’ 2𝛿

    0

    1 βˆ’ 𝛿0

    π‘˜0. (34)

    Hence, we get

    0 = βˆ«π‘‘βˆˆπ΄π‘›

    Ξ¦(𝑑, π‘˜π‘›(1

    4Σ𝑒

    π‘₯0 (𝑑) + 𝑒𝑧𝑛 (𝑑))) 𝑑𝑑

    β‰₯ βˆ«π‘‘βˆˆπ΄π‘›

    Ξ¦(𝑑,

    π‘˜π‘›

    1 βˆ’ 2𝛿0

    π‘₯0(𝑑)

    ) 𝑑𝑑

    β‰₯ βˆ«π‘‘βˆˆπ΄π‘›

    Ξ¦(𝑑,

    π‘˜0

    1 βˆ’ 𝛿0

    π‘₯0(𝑑)

    ) 𝑑𝑑 β‰₯ 0.

    (35)

    Thus, we see the equalityβˆ«π‘‘βˆˆπ΄π‘›

    Ξ¦(𝑑, |(π‘˜0/(1βˆ’π›Ώ

    0))π‘₯0(𝑑)|) 𝑑𝑑 = 0

    holds for 𝑛 large enough. It follows that

    π‘˜0

    1 βˆ’ 𝛿0

    π‘₯0(𝑑)

    ≀ 𝑒 (𝑑) , πœ‡-a.e. 𝑑 ∈ 𝐴

    𝑛 (36)

    since πœ‡(𝐴𝑛) > 0 for 𝑛 large enough.

    On the other hand, by (3), we deduce that

    π‘˜0π‘₯0 (𝑑) β‰₯ 𝑒 (𝑑) , πœ‡-a.e. 𝑑 ∈ 𝐴𝑛. (37)

    Hence, we get a contradiction:

    π‘˜0

    1 βˆ’ 𝛿0

    π‘₯0(𝑑)

    β‰₯

    𝑒 (𝑑)

    1 βˆ’ 𝛿0

    > 𝑒 (𝑑) , πœ‡-a.e. 𝑑 ∈ 𝐴𝑛. (38)

    Theorem 4. Assume 1 ≀ 𝑝 < ∞; then the following assertionsare equivalent:

    (1) 𝐿Φ,𝑝

    is complex midpoint locally uniformly convex;

    (2) 𝐿Φ,𝑝

    is complex strictly convex;

    (3) 𝑒(𝑑) = 0 for πœ‡-a.e. 𝑑 ∈ 𝑇.

    Proof. The implication (1) β‡’ (2) is trivial. Now assume that𝐿Φ,𝑝

    is complex strictly convex. If πœ‡{𝑑 ∈ 𝑇 : 𝑒(𝑑) > 0} > 0, let𝑇0= {𝑑 ∈ 𝑇 : 𝑒(𝑑) > 0} and it is not difficult to find an element

    π‘₯ ∈ 𝑆(𝐿Φ,𝑝) satisfying supp π‘₯ = 𝑇 \ 𝑇

    0. Take π‘˜ ∈ 𝐾

    𝑝(π‘₯), and

    define

    𝑦 (𝑑) ={

    {

    {

    𝑒 (𝑑)

    2π‘˜for 𝑑 ∈ 𝑇

    0,

    π‘₯ (𝑑) for 𝑑 ∈ 𝑇 \ 𝑇0.

    (39)

    Obviously, ‖𝑦‖Φ,𝑝

    β‰₯ β€–π‘₯β€–Ξ¦,𝑝

    = 1. On the other hand,

    𝑦Φ,𝑝 ≀

    1

    π‘˜{1 + [∫

    π‘‘βˆˆπ‘‡0

    Ξ¦(𝑑,𝑒 (𝑑)

    2) 𝑑𝑑

    +βˆ«π‘‡\𝑇0

    Ξ¦ (𝑑, π‘˜π‘₯ (𝑑)) 𝑑𝑑]

    𝑝

    }

    1/𝑝

    =1

    π‘˜(1 + 𝐼

    𝑝

    Ξ¦(π‘˜π‘₯))1/𝑝

    = 1.

    (40)

    Thus, ‖𝑦‖Φ,𝑝

    = (1/π‘˜)(1 + 𝐼𝑝

    Ξ¦(π‘˜π‘¦))1/𝑝

    = 1. However, for 𝑑 βˆˆπ‘‡0, we find π‘˜|𝑦(𝑑)| = 𝑒(𝑑)/2 < 𝑒(𝑑), which implies 𝑦 ∈ 𝑆(𝐿

    Ξ¦,𝑝)

    is not a complex extreme point of 𝐡(𝐿Φ,𝑝) fromTheorem 3.

    (3) β‡’ (1). Suppose that π‘₯ ∈ 𝑆(𝐿Φ,𝑝) is not a complex

    strongly extreme point of 𝐡(𝐿Φ,𝑝). It follows fromTheorem 3

    that πœ‡{𝑑 ∈ 𝑇 : π‘˜0|π‘₯(𝑑)| < 𝑒(𝑑)} > 0 for some π‘˜

    0∈ 𝐾𝑝(π‘₯),

    consequently πœ‡{𝑑 ∈ 𝑇 : 𝑒(𝑑) > 0} > 0which is a contradiction.

    Remark 5. If 𝑝 = ∞ then 𝑝-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. Pavlović, β€œComplex convexity and vector-valued Littlewood-Paley inequalities,” Bulletin of the LondonMathematical Society, vol. 35, no. 6, pp. 749–758, 2003.

    [2] C. Choi, A. Kamińska, and H. J. Lee, β€œComplex convexity ofOrlicz-Lorentz spaces and its applications,” Bulletin of the PolishAcademy of Sciences:Mathematics, vol. 52, no. 1, pp. 19–38, 2004.

    [3] H. Hudzik and A. Narloch, β€œRelationships betweenmonotonic-ity and complex rotundity properties with some consequences,”Mathematica Scandinavica, vol. 96, no. 2, pp. 289–306, 2005.

    [4] H. J. Lee, β€œMonotonicity and complex convexity in Banachlattices,” Journal of Mathematical Analysis and Applications, vol.307, no. 1, pp. 86–101, 2005.

    [5] H. J. Lee, β€œComplex convexity and monotonicity in Quasi-Banach lattices,” Israel Journal of Mathematics, vol. 159, no. 1, pp.57–91, 2007.

    [6] M. M. Czerwińska and A. Kamińska, β€œComplex rotunditiesand midpoint local uniform rotundity in symmetric spaces ofmeasurable operators,” Studia Mathematica, vol. 201, no. 3, pp.253–285, 2010.

    [7] E. Thorp and R. Whitley, β€œThe strong maximum modulustheorem for analytic functions into a Banach space,” Proceedingsof the AmericanMathematical Society, vol. 18, pp. 640–646, 1967.

    [8] L. Chen, Y. Cui, and H. Hudzik, β€œCriteria for complex stronglyextreme points of Musielak-Orlicz function spaces,” NonlinearAnalysis: Theory, Methods & Applications, vol. 70, no. 6, pp.2270–2276, 2009.

    [9] S. Chen, Geometry of Orlicz Spaces, Dissertationes Mathemati-cae, 1996.

    [10] Y. Cui, L. Duan, H. Hudzik, and M. WisΕ‚a, β€œBasic theory of p-Amemiya norm in Orlicz spaces (1 ≀ 𝑝 ≀ ∞): extreme pointsand rotundity in Orlicz spaces endowed with these norms,”Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no.5-6, pp. 1796–1816, 2008.

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