Complex Convexity of Musielak-Orlicz Function Spaces Equipped with the 𝑝-Amemiya Norm

Abstract

The 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 , 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 be a complex Banach space over the complex field , let be the complex number satisfying , and let and be the closed unit ball and the unit sphere of , respectively. In the sequel, and 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, -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 point is said to be a complex extreme point of if for every nonzero there holds . 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 convexity in general complex spaces.

A point is said to be a complex strongly extreme point of if for every we have , where 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 be a nonatomic and complete measure space with . By we denote a Musielak-Orlicz function; that is, satisfies the following:(1)for each , is a -measurable function of on ;(2)for a.e. , , and there exists such that ;(3)for a.e. , is convex on the interval with respect to .

Let be the space of all -equivalence classes of complex and -measurable functions defined on . For each , we define on the convex modular of by

We define supp  and the Musielak-Orlicz space generated by the formula

Set then and are -measurable (see [9]).

The notion of -Amemiya norm has been introduced in [10]; for any and , define

Furthermore, define for all and . Notice that the function is increasing on for ; however, the function is increasing on the interval only.

For , define the -Amemiya norm by the formula

For convenience, from now on, we write ; it is easy to see that is a Banach space. For , , set

2. Main Results

We begin this section from the following useful lemmas.

Lemma 1 (see [9]). For any , there exists such that if and then where

Lemma 2. for a.e. , then for any , where .

Proof. For any , there exists such that . Let and define the subsets for each . It is easy to see that and . Then there exists such that .
It follows from the definition of -Amemiya norm that there is a sequence satisfying For any , we have If , notice that which means the sequence is bounded. Hence, without loss of generality, we assume that as . We can also choose the monotonic increasing or decreasing subsequence of that converges to the number . Applying Levi Theorem and Lebesgue Dominated Convergence Theorem, we obtain which implies .

In order to exclude the case when such sets are empty for , in the sequel we will assume, without any special mention, that for a.e. .

Theorem 3. Assume , . Then the following assertions are equivalent: (1) is a complex strongly extreme point of the unit ball ;(2) is a complex extreme point of the unit ball ;(3)for any , .

Proof. The implication is trivial. Let be a complex extreme point of the unit ball and there exists such that . Then we can find such that . Indeed, if for any , then we set and define the subsets for each . Notice that and ; we can find that which is a contradiction.
Let and define ; we have and for any with , which shows that is not a complex extreme point of the unit ball .
. Suppose that is not a complex strongly extreme point of the unit ball ,; then there exists such that . That is, there exist with and satisfying such that which gives Setting , we have
For the above , by Lemma 1, there exists such that if and then
For each , let
It is easy to see that , . Since when , the following inequalities hold for large enough.
Therefore, which shows that for large enough. Furthermore, for any , we have
To complete the proof, we consider the following two cases.
(I) One has , where .
For each , we get Furthermore, we notice that Since when , we can see that the inequality holds for large enough. Moreover, we find that Then we deduce a contradiction.
(II) One has , where , .
Then for each , Let , we deduce that .
From (I), we obtain that
Now we consider the following two subcases.
(a) Consider for some .
Then we obtain a contradiction.
(b) Consider for any .
For large enough, we observe that Hence, we get Thus, we see the equality holds for large enough. It follows that since for large enough.
On the other hand, by (3), we deduce that Hence, we get a contradiction:

Theorem 4. Assume ; then the following assertions are equivalent: (1) is complex midpoint locally uniformly convex;(2) is complex strictly convex;(3) for a.e. .

Proof. The implication is trivial. Now assume that is complex strictly convex. If , let and it is not difficult to find an element satisfying supp . Take , and define
Obviously, . On the other hand, Thus, . However, for , we find , which implies is not a complex extreme point of from Theorem 3.
. Suppose that is not a complex strongly extreme point of . It follows from Theorem 3 that for some , consequently which is a contradiction.

Remark 5. If then -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].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by Grants from Heilongjiang Provincial Natural Science Foundation for Youths (no. QC2013C001), Natural Science Foundation of Heilongjiang Educational Committee (no. 12531099), Youth Science Fund of Harbin University of Science and Technology (no. 2011YF002), and Tianyuan Funds of the National Natural Science Foundation of China (no. 11226127).