# Complex Convexity of Musielak-Orlicz Function Spaces Equipped with thep-Amemiya Norm

Authors: Lili Chen; Yunan Cui; Yanfeng Zhao
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The complex convexity of Musielak-Orlicz function spaces equipped with thep-Amemiya norm is mainly discussed. It is obtained that, for any Musielak-Orlicz function space equipped with the p-Amemiya norm when 1≤p<∞, 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.


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## Body

## 1. Introduction

Let(X,∥·∥) be a complex Banach space over the complex field ℂ, let i be the complex number satisfying i2=-1, and let B(X) and S(X) be the closed unit ball and the unit sphere of X, 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 Theory, Operator Theory, Banach Algebras,C*-Algebras, Differential Equation Theory, Quantum Mechanics Theory, and Hydrodynamics Theory. 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], Thorp 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 Theorem for analytic functions always holds in a complex Banach space.A pointx∈S(X) is said to be a complex extreme point of B(X) if for every nonzero y∈X there holds sup⁡|λ|≤1∥x+λy∥>1. A complex Banach space X is said to be complex strictly convex if every element of S(X) is a complex extreme point of B(X).In [8], we further studied the notions of complex strongly extreme point and complex midpoint locally uniform convexity in general complex spaces.A pointx∈S(X) is said to be a complex strongly extreme point of B(X) if for every ε>0 we have Δc(x,ε)>0, where
(1)Δc(x,ε)=inf⁡{1-|λ|:∃y∈Xs.t.∥λx±y∥≤1,∥λx±iy∥≤1,∥y∥≥ε}.
A complex Banach space X is said to be complex midpoint locally uniformly convex if every element of S(X) is a complex strongly extreme point of B(X).Let(T,Σ,μ) be a nonatomic and complete measure space with μ(T)<∞. By Φ we denote a Musielak-Orlicz function; that is, Φ:T×[0,+∞)→[0,+∞] satisfies the following:(1)
for eachu∈[0,∞], Φ(t,u) is a μ-measurable function of t on T;(2)
forμ-a.e. t∈T, Φ(t,0)=0, lim⁡u→∞Φ(t,u)=∞ and there exists u′>0 such that Φ(t,u′)<∞;(3)
forμ-a.e. t∈T, Φ(t,u) is convex on the interval [0,∞) with respect to u.LetLc(μ) be the space of all μ-equivalence classes of complex and Σ-measurable functions defined on T. For each x∈Lc(μ), we define on Lc(μ) the convex modular of x by
(2)IΦ(x)=∫T‍Φ(t,|x(t)|)dt.We define suppx={t∈T:|x(t)|≠0} and the Musielak-Orlicz space LΦ generated by the formula
(3)LΦ={x∈Lc(μ):IΦ(λx)<∞forsomeλ>0}.Set(4)e(t)=sup⁡{u≥0:Φ(t,u)=0},E(t)=sup⁡{u≥0:Φ(t,u)<∞};
then e(t) and E(t) are μ-measurable (see [9]).The notion ofp-Amemiya norm has been introduced in [10]; for any 1≤p≤∞ and u>0, define
(5)sp(u)={(1+up)1/p,for1≤p<∞,max⁡{1,u},forp=∞.Furthermore, definesΦ,p(x)=sp∘IΦ(x) for all 1≤p≤∞ and x∈Lc(μ). Notice that the function sp is increasing on R+ for 1≤p<∞; however, the function s∞ is increasing on the interval [1,∞) only.Forx∈Lc(μ), define the p-Amemiya norm by the formula
(6)∥x∥Φ,p=inf⁡k>01ksΦ,p(kx),1≤p≤∞.For convenience, from now on, we writeLΦ,p=(LΦ,∥·∥Φ,p); it is easy to see that LΦ,p is a Banach space. For 1≤p<∞, x∈LΦ,p, set
(7)Kp(x)={k>0:∥x∥Φ,p=1k{1+IΦp(kx)}1/p}.


## 2. Main Results

We begin this section from the following useful lemmas.Lemma 1 (see [9]).
For anyε>0, there exists δ∈(0,1/2) such that if u,v∈ℂ and
(8)|v|≥ε8max⁡e|u+ev|,
then
(9)|u|≤1-2δ4Σe|u+ev|,
where
(10)max⁡e|u+ev|=max⁡{|u+v|,|u-v|,|u+iv|,|u-iv|},Σe|u+ev|=|u+v|+|u-v|+|u+iv|+|u-iv|.Lemma 2.
If
lim
⁡
u
→
∞
(
Φ
(
t
,
u
)
/
u
)
=
∞ for μ-a.e. t∈T, then Kp(x)≠∅ for any x∈LΦ,p∖{0}, where 1≤p<∞.Proof.
For anyx∈LΦ,p∖{0}, there exists α>0 such that μ{t∈T:|x(t)|≥α}>0. Let T1={t∈T:|x(t)|≥α} and define the subsets
(11)Fn={t∈T1:Φ(t,u)u≥4∥x∥Φ,pα·μT1,u≥n}
for each n∈ℕ. It is easy to see that F1⊆F2⊆⋯⊆Fn⊆⋯ and lim⁡n→∞μFn=μT1. Then there exists n0∈ℕ such that μFn0>(1/2)μT1.
It follows from the definition ofp-Amemiya norm that there is a sequence {kn} satisfying
(12)∥x∥Φ,p=lim⁡n→∞1kn{1+IΦp(knx)}1/p.
For any k>0, we have
(13)1k{1+IΦp(kx)}1/p=1k{1+[∫T‍Φ(t,k|x(t)|)dt]p}1/p≥∫Fn0‍Φ(t,k|x(t)|)dtk≥∫Fn0‍Φ(t,kα)kdt=α∫Fn0‍Φ(t,kα)kαdt.
If k>n0/α, notice that
(14)1k{1+IΦp(kx)}1/p≥α·4∥x∥Φ,pα·μT1·μFn0>2∥x∥Φ,p,
which means the sequence {kn} is bounded. Hence, without loss of generality, we assume that kn→k0 as n→∞. We can also choose the monotonic increasing or decreasing subsequence of {kn} that converges to the number k0. Applying Levi Theorem and Lebesgue Dominated Convergence Theorem, we obtain
(15)∥x∥Φ,p=lim⁡n→∞1kn{1+IΦp(knx)}1/p=1k0{1+IΦp(k0x)}1/p
which implies k0∈Kp(x).In order to exclude the case when such setsKp(x) are empty for x∈LΦ,p∖{0}, in the sequel we will assume, without any special mention, that lim⁡u→∞(Φ(t,u)/u)=∞ for μ-a.e. t∈T.Theorem 3.
Assume1≤p<∞, x∈S(LΦ,p). Then the following assertions are equivalent: (1)
x is a complex strongly extreme point of the unit ball B(LΦ,p);(2)
x is a complex extreme point of the unit ball B(LΦ,p);(3)
for anyk∈Kp(x), μ{t∈T:k|x(t)|<e(t)}=0.Proof.
The implication(1)⇒(2) is trivial. Let x be a complex extreme point of the unit ball B(LΦ,p) and there exists k0∈Kp(x) such that μ{t∈T:k0|x(t)|<e(t)}>0. Then we can find d>0 such that μ{t∈T:k0|x(t)|+d<e(t)}>0. Indeed, if μ{t∈T:k0|x(t)|+d<e(t)}=0 for any d>0, then we set dn=1/n and define the subsets
(16)Tn={t∈T:k0|x(t)|+dn<e(t)}
for each n∈ℕ. Notice that T1⊆T2⊆⋯⊆Tn⊆⋯ and {t∈T:k0|x(t)|≤e(t)}=⋃n=1∞‍Tn; we can find that
(17)μ{t∈T:k0|x(t)|≤e(t)}=0
which is a contradiction.
LetT0={t∈T:k0|x(t)|+d<e(t)} and define y=(d/k0)χT0; we have y≠0 and for any λ∈ℂ with |λ|≤1,
(18)∥x+λy∥Φ,p≤1k0{1+IΦp(k0(x+λy))}1/p=1k0{+IΦ(k0xχT0+λdχT0)]p1+[IΦ(k0xχT∖T0)+IΦ(k0xχT0+λdχT0)]phhhhhhhhhh+IΦ(k0xχT0+λdχT0)]p}1/p≤1k0{1+[IΦ(k0xχT∖T0)((k0|x|+d)χT0)hhhhhhhhh+IΦ((k0|x|+d)χT0)]p}1/p=1k0{1+[IΦ(k0xχT∖T0)]p}1/p=1k0(1+IΦp(k0x))1/p=1,
which shows that x is not a complex extreme point of the unit ball B(LΦ,p).
(
3
)
⇒
(
1
). Suppose that x∈S(LΦ,p) is not a complex strongly extreme point of the unit ball B(LΦ,p),; then there exists ε0>0 such that Δc(x,ε0)=0. That is, there exist λn∈ℂ with |λn|→1 and yn∈LΦ,p satisfying ∥yn∥Φ,p≥ε0 such that
(19)∥λnx0±yn∥Φ,p≤1,∥λnx0±iyn∥Φ,p≤1,
which gives
(20)∥x0±ynλn∥Φ,p≤1|λn|,∥x0±iynλn∥Φ,p≤1|λn|.
Setting zn=yn/λn, we have
(21)∥zn∥Φ,p≥∥yn∥Φ,p≥ε0,∥x0±zn∥Φ,p≤1|λn|,∥x0±izn∥Φ,p≤1|λn|.
For the aboveε0>0, by Lemma 1, there exists δ0∈(0,1/2) such that if u,v∈ℂ and
(22)|v|≥ε08max⁡e|u+ev|,
then
(23)|u|≤1-2δ04Σe|u+ev|.
For eachn∈ℕ, let
(24)An={t∈T:|zn(t)|≥ε08max⁡e|x0(t)+ezn(t)|},zn(1):zn(1)(t)=zn(t)(t∉An),zn(1)(t)=0(t∈An),zn(2):zn(2)(t)=0(t∉An),zn(2)(t)=zn(t)(t∈An).
It is easy to see thatzn=zn(1)+zn(2), ∀n∈ℕ. Since |λn|→1 when n→∞, the following inequalities
(25)∥zn(1)∥Φ,p=inf⁡k>01k{1+[∫t∉An‍Φ(t,k|zn(t)|)dt]p}1/p≤inf⁡k>01k{[∫t∉An‍Φ(t,kε08max⁡e|x0(t)+ezn(t)|)dt]p}1/p1+hhhhhhhhhh[∫t∉An‍Φ(t,kε08max⁡e|x0(t)+ezn(t)|)dt]p{[∫t∉An‍Φ(t,kε08max⁡e|x0(t)+ezn(t)|)dt]p}1/p1+}1/p≤ε08∥max⁡e|x0+ezn|∥Φ,p≤ε08∥Σe|x0+ezn|∥Φ,p≤ε02|λn|<3ε04
hold for n large enough.
Therefore,∥zn(2)∥Φ,p>ε0/4 which shows that μ(An)>0 for n large enough. Furthermore, for any t∈An, we have
(26)|x0(t)|≤1-2δ04Σe|x0(t)+ezn(t)|.
To complete the proof, we consider the following two cases.
(I) One haskn→∞(n→∞), where kn∈Kp((1/4)Σe|x0+ezn|).
For eachn∈ℕ, we get
(27)1=∥x0∥Φ,p≤1kn{1+IΦp(knx0)}1/p=1kn{∫1+[∫t∈An‍Φ(t,kn|x0(t)|)dthhhhhhhhhh+∫t∉An‍Φ(t,kn|x0(t)|)dt]p}1/p≤1kn{1+[∫t∈An‍Φ(t,kn(1-2δ04hhhhhhhhhhhhhhhhh×Σe|x0(t)+ezn(t)|(1-2δ04))dthhhhhhhhhhhhhhhhhh+∫t∉An‍Φ((14Σe|x0(t)+ezn(t)|))t,knhhhhhhhhhhhhhhh×(14Σe|x0(t)+ezn(t)|))dt[∫t∈An‍Φ(t,kn(1-2δ04]p}1/p≤1kn{+∫t∉An‍Φ(t,kn(14Σe|x0(t)+ezn(t)|))dt]p1+[+∫t∉An‍Φ(t,kn(14Σe|x0(t)+ezn(t)|))dt]p(1-2δ0)hhhhhhhh×∫t∈An‍Φ(t,kn(14Σe|x0(t)+ezn(t)|))dthhhhhhhh+∫t∉An‍Φ(×(14Σe|x0(t)+ezn(t)|))t,knhhhhhhhhhhhhhhh×(14Σe|x0(t)+ezn(t)|))dt[+∫t∉An‍Φ(t,kn(14Σe|x0(t)+ezn(t)|))dt]p(1-2δ0)]p}1/p=1kn{×(14Σe|x0(t)+ezn(t)|))dt]p1+[IΦ(kn(14Σe|x0+ezn|))-2δ0hhhhhhhh×∫t∈An‍Φ(×(14Σe|x0(t)+ezn(t)|))t,knhhhhhhhhhhhhhhh×(14Σe|x0(t)+ezn(t)|))dt]p}1/p.
Furthermore, we notice that
(28)∫t∈An‍Φ(t,kn(14Σe|x0(t)+ezn(t)|))dt≥∫t∈An‍Φ(t,kn|zn(t)|)dt≥[knp∥zn(2)∥Φ,pp-1]1/p≥[(ε04kn)p-1]1/p.
Since kn→∞ when n→∞, we can see that the inequality ((ε0/4)kn)p-1>0 holds for n large enough. Moreover, we find that
(29)IΦ(kn(14Σe|x0+ezn|))≥IΦ(knx0)≥(knp-1)1/p≥2δ0((ε04kn)p-1)1/p.
Then we deduce
(30)1=∥x0∥Φ,p≤1kn{14]p1+[IΦ(kn(14Σe|x0+ezn|))-2δ0hhhhhhhhhh×∫t∈An‍Φ(×(14Σe|x0(t)+ezn(t)|))t,knhhhhhhhhhhhhhhh×(14Σe|x0(t)+ezn(t)|))dt]p}1/p≤1kn{1+[-2δ0((ε04kn)p-1)1/p]pIΦ(kn(14Σe|x0+ezn|))hhhhhhhhhh-2δ0((ε04kn)p-1)1/p]p}1/p<1kn{1+IΦp(kn(14Σe|x0+ezn|))}1/p=1,
a contradiction.
(II) One haskn→k0(n→∞), where kn∈Kp((1/4)Σe|x0+ezn|), k0∈ℝ.
Then for eachn∈ℕ,
(31)1λn≥∥14Σe|x0+ezn|∥Φ,p=1kn{1+IΦp(kn4Σe|x0+ezn|)}1/p≥1kn{1+IΦp(knx0)}1/p≥∥x0∥Φ,p=1.
Let n→∞, we deduce that 1=(1/k0){1+IΦp(k0x0)}1/p=∥x0∥Φ,p.
From (I), we obtain that(32)1=∥x0∥Φ,p≤1kn{∫t∈An‍Φ(t,kn(14Σe|x0(t)+ezn(t)|))dt]p}1/p1+[IΦ(kn(14Σe|x0+ezn|))-2δ0hhhhhhhhhh×∫t∈An‍Φ(×(14Σe|x0(t)+ezn(t)|))t,knhhhhhhhhhhhhhhhhh×(14Σe|x0(t)+ezn(t)|))dt]p{∫t∈An‍Φ(t,kn(14Σe|x0(t)+ezn(t)|))dt]p}1/p1+}1/p.
Now we consider the following two subcases.
(a) Consider∫t∈An‍Φ(t,kn((1/4)Σe|x0(t)+ezn(t)|))dt>0 for some n∈ℕ.
Then we obtain(33)1=∥x0∥Φ,p≤1kn{×(14Σe|x0(t)+ezn(t)|))dt]p}1/p1+[IΦ(kn(14Σe|x0+ezn|))-2δ0hhhhhhhhhh×∫t∈An‍Φ(×(14Σe|x0(t)+ezn(t)|))t,knhhhhhhhhhhhhhhh×(14Σe|x0(t)+ezn(t)|))dt]p}1/p×(14Σe|x0(t)+ezn(t)|))dt]p}1/p<1kn{1+IΦp(kn(14Σe|x0+ezn|))}1/p=1,
a contradiction.
(b) Consider∫t∈An‍Φ(t,kn((1/4)Σe|x0(t)+ezn(t)|))dt=0 for any n∈ℕ.
Forn large enough, we observe that
(34)kn>1-2δ01-δ0k0.
Hence, we get
(35)0=∫t∈An‍Φ(t,kn(14Σe|x0(t)+ezn(t)|))dt≥∫t∈An‍Φ(t,|kn1-2δ0x0(t)|)dt≥∫t∈An‍Φ(t,|k01-δ0x0(t)|)dt≥0.
Thus, we see the equality ∫t∈An‍Φ(t,|(k0/(1-δ0))x0(t)|)dt=0 holds for n large enough. It follows that
(36)|k01-δ0x0(t)|≤e(t),μ-a.e.t∈An
since μ(An)>0 for n large enough.
On the other hand, by (3), we deduce that(37)|k0x0(t)|≥e(t),μ-a.e.t∈An.
Hence, we get a contradiction:
(38)|k01-δ0x0(t)|≥e(t)1-δ0>e(t),μ-a.e.t∈An.Theorem 4.
Assume1≤p<∞; then the following assertions are equivalent: (1)
LΦ,p is complex midpoint locally uniformly convex;(2)
LΦ,p is complex strictly convex;(3)
e(t)=0 for μ-a.e. t∈T.Proof.
The implication(1)⇒(2) is trivial. Now assume that LΦ,p is complex strictly convex. If μ{t∈T:e(t)>0}>0, let T0={t∈T:e(t)>0} and it is not difficult to find an element x∈S(LΦ,p) satisfying supp x=T∖T0. Take k∈Kp(x), and define
(39)y(t)={e(t)2kfort∈T0,x(t)fort∈T∖T0.
Obviously,∥y∥Φ,p≥∥x∥Φ,p=1. On the other hand,
(40)∥y∥Φ,p≤1k{1+[∫t∈T0‍Φ(t,e(t)2)dthhhhhhhhhhh+∫T∖T0‍Φ(t,kx(t))dt]p}1/p=1k(1+IΦp(kx))1/p=1.
Thus, ∥y∥Φ,p=(1/k)(1+IΦp(ky))1/p=1. However, for t∈T0, we find k|y(t)|=e(t)/2<e(t), which implies y∈S(LΦ,p) is not a complex extreme point of B(LΦ,p) from Theorem 3.
(
3
)
⇒
(
1
). Suppose that x∈S(LΦ,p) is not a complex strongly extreme point of B(LΦ,p). It follows from Theorem 3 that μ{t∈T:k0|x(t)|<e(t)}>0 for some k0∈Kp(x), consequently μ{t∈T:e(t)>0}>0 which is a contradiction.Remark 5.
Ifp=∞ then p-Amemiya norm equals Luxemburg norm, the problem of complex convexity of Musielak-Orlicz function spaces equipped with the Luxemburg norm has been investigated in [8].



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