Smoothness of Orlicz function spaces equipped with the p-Amemiya norm

In this paper, we will use the convex modular ρ∗(f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho ^{*}(f)$$\end{document} to investigate ‖f‖Ψ,q∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert f\Vert _{\Psi ,q}^{*}$$\end{document} on (LΦ)∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(L_{\Phi })^{*}$$\end{document} defined by the formula ‖f‖Ψ,q∗=infk>01ksq(ρ∗(kf))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert f\Vert _{\Psi ,q}^{*}=\inf _{k>0}\frac{1}{k}s_{q}(\rho ^{*}(kf))$$\end{document}, which is the norm formula in Orlicz dual spaces equipped with p-Amemiya norm. The attainable points of dual norm ‖f‖Ψ,q∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert f\Vert _{\Psi ,q}^{*}$$\end{document} are discussed, the interval for dual norm ‖f‖Ψ,q∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert f\Vert _{\Psi ,q}^{*}$$\end{document} attainability is described. By presenting the explicit form of supporting functional, we get sufficient and necessary conditions for smooth points. As a result, criteria for smoothness of LΦ,p(1≤p≤∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{\Phi ,p}~(1\le p\le \infty )$$\end{document} is also obtained. The obtained results unify, complete and extended as well the results presented by a number of paper devoted to studying the smoothness of Orlicz spaces endowed with the Luxemburg norm and the Orlicz norm separately.


Introduction
It is well known that smooth points and smoothness are basic concepts in geometric theory of Banach spaces. Smoothness of Orlicz spaces are of importance in applications of the approximation theory, the conditional expectation theory, probability limit theorems and the nonlinear prediction theory as well as in other applications. Criteria for smooth points and smoothness of Orlicz function and sequence spaces equipped with Luxemburg norm were given in [4,13,28]. Criteria for smooth points and smoothness of Orlicz function and sequence spaces equipped with Orlicz norm were given in [5,7,26]. But up to now, the smoothness of Orlicz function spaces equipped with p-Amemiya norm has not been solved. The aim of this paper is to present criteria for smooth points and smoothness of Orlicz function spaces equipped with the p-Amemiya norm.
The rest of the paper is organized as follows. In the first part of the paper some basic notions, terminology and original results are reviewed, which will be used throughout the paper. We also recalled some properties of outer function which were introduced by Wisla in [30] and Köthe predual, i.e., (E Φ,p ) * = L Ψ,q where 1 p + 1 q = 1 and Ψ is the function complementary to the Orlicz function Φ in the sense of Young. In the next part of the paper, we will use the convex modular * (f ) to investigate ‖f ‖ * Ψ,q on (L Φ ) * defined by the formula ‖f ‖ * Ψ,q = inf k>0 1 k s q ( * (kf )) , which is the norm formula in Orlicz dual spaces equipped with p-Amemiya norm. The attainable points of dual norm ‖f ‖ * Ψ,q are discussed, the interval for dual norm ‖f ‖ * Ψ,q attainability is described. In the last part of the paper, we present the explicit form of supporting functional and get sufficient and necessary conditions for smooth points. As a result, criteria for smoothness of L Φ,p (1 ≤ p ≤ ∞) are obtained.
Let X be a real Banach space, and S(X) be the unit sphere of X. By X * we denote the dual space of X. In the sequel N and R denote the set of natural numbers and the set of real numbers, respectively.
For every u, v ∈ R , we have the following Young Inequality: which reduces to an equality when v ∈ [p − (u), p + (u)] if u is given, or when u ∈ [q − (v), q + (v)] if v is given (see [6]). Let us underline that p + , p − , q + , q − will always mean functions, while letters p, q will always refer to numbers.
Let (G, Σ, ) be a measure space with a -finite, nonatomic and complete measure and L 0 ( ) be the set of all -equivalence classes of real and Σ-measurable functions defined on G. To simplify notations, by a characteristic function A of a subset A ⊂ G we will mean the function defined by For a given Orlicz function Φ we define on L 0 ( ) a convex functional (called a pseudomodular [21]) by The Orlicz space L Φ generated by an Orlicz function Φ is a linear space of measurable functions defined by the formula By E Φ we denote the linear space of all measurable functions such that I Φ (cx) < ∞ for all c > 0 . It may happen that the space E Φ consists of only one element-the zero function. For instance, this happens if the measure is atomless and the function Φ jumps to infinity (i.e., b Φ < ∞).
The Orlicz space L Φ is a Banach space when it is endowed with any of the norms: and which are called the Luxemburg norm, Orlicz norm and Amemiya norm, respectively. Krasnoslskii and Rutickii [18], Nakano [23], Luxemburg and Zaanen [20] |uv| ≤ Φ(u) + Ψ(v) proved, under additional assumptions on the function Φ , that the Orlicz norm can be expressed exactly by the Amemiya formula, i.e., ‖x‖ • Φ = ‖x‖ A Φ . In the most general case of Orlicz function Φ , the similar result was obtained by Hudzik and Maligranda ([15]). Moreover, it is not difficult to verify that Luxemburg norm can also be expressed by an Amemiya-like formula (see [9,24]), namely In the paper [15], Hudzik and Maligranda proposed to investigate another class of norms given by the Amemiya formula-norms generated by the functions of the type and In that case we obtain a family of topologically equivalent norms (called p-Amemiya norms and denoted by ‖.‖ Φ,p ), indexed by 1 ≤ p ≤ ∞ and satisfying the inequalities Since that time, an intensive development of research connected with Orlicz spaces equipped with p-Amemiya norms have taken place, many important results broaden the knowledge about the geometry of these spaces (see [3,8,9,11,12,14,17,19]) and some open questions were put (see [29]).
To simplify notation, the Orlicz spaces equipped with the p-Amemiya norms are denoted by L Φ,p = (L Φ , ‖ ⋅ ‖ Φ,p ) . Further, for any function u ∈ L 0 the essential supremum of |u| over G, i.e. sup ees t∈G |u(t)| , no matter whether this number is finite or not, will be denoted by ‖u‖ ∞ .
Lemma 2.1 [9] For every 1 ≤ p ≤ ∞ and u ∈ L Φ,p ⧵{0} , the following conditions hold: Lemma 2.2 [9] Let Φ be an Orlicz function and let 1 ≤ p ≤ ∞ . The set K p (u) is nonempty if and only if one of the following conditions is satisfied: (i) If p = 1 then Φ does not admit an asymptote at infinity.
Remark 2.3 By Lemma 2.2, we know for every 1 ≤ p < ∞, if K p (u) = � , then there exists G 0 ⊂ G such that L Φ (G 0 ) is linearly isometric to L 1 . We know that L ∞ is the dual space of L 1 and L 1 is not a smooth space. For this reason we will assume K p (u) ≠ � in the following whenever smooth points and smoothness are considered.
In the paper [30], Wisla introduced outer functions and presented basic properties of outer functions. We recall them here. A function s ∶ [0, ∞) → [1, ∞) will be called an outer function, if it is convex and To simplify notations, we extend the domain and range of s to the interval [0, ∞] by setting s(∞) = ∞.
Evidently, for every 1 ≤ p ≤ ∞ , s p (⋅) is an outer function.We will say that two outer functions s, are conjugate (to each other) in the Hölder sense, Lemma 2.4 [30] The outer function (v) = 1 + v is conjugate in the Hölder sense to any outer function s(⋅).

Lemma 2.7 [30] Let Φ, Ψ be the Orlicz functions complementary in the sense of Young that take finite values only. If the p-Amemiya norm
Orlicz spaces are endowed with the structure of Banach lattices [1]. This property can be used in a more refined analysis of the (topological) dual space of L Φ , which is denoted by (L Φ ) * . (L Φ ) * is represented in the following way (see [21]): where is singular functional, i.e., (u) = 0 for any u ∈ E Φ,p and v ∈ L Ψ,q where 1 p + 1 q = 1 and Ψ is the function complementary to the Orlicz function Φ in the sense of Young, is the regular functional by the formula: Let us define for each f ∈ (L Φ ) * : ( Proofs of the next three lemmas can be found for N-functions Φ in [16], but they are also true for arbitrary Orlicz functions Φ (see [27], even in the more general case of Musielak-Orlicz functions).

q and norm attainability
Let f ∈ (L Φ ) * be as in (2). Define Cui et al. proved that * (f ) is a convex modular in (L Φ ) * (see [10]). Now, for 1 ≤ p ≤ ∞ , on (L Φ,p ) * we introduce new functionals as follows In the next section we will prove that ‖f ‖ Ψ = ‖f ‖ * Ψ,∞ . We will also prove there that for any 1 ≤ q ≤ ∞ the functional ‖f ‖ * Ψ,q is a norm on (L Φ,p ) * and all the norms ‖f ‖ * Ψ,q are equivalent to each other.
Further, by Theorem 3.1, we have Thus (4) holds true and ‖f ‖ * In the following by the determinant function we shall mean the function

Further, define
The support of a measurable function v ∈ L Ψ,q is defined by In the sequel, together with a measurable function v, we shall often consider a sequence (v n ) of bounded measurable functions with support of finite measure defined by
5 For every f ∈ (L Φ,p ) * ⧵{0} and every 1 ≤ q < ∞ the following conditions hold: Proof Condition (i) follows immediately from the fact that both functions The condition (iii) follows directly from the Lebesgue dominated convergence theorem.
Proof We need to prove conditions (iv)-(vii) only.
If = 0 . The proof is similar to Theorem 4.3 in [9], so we omit it here.
Smoothness of Orlicz function spaces equipped with the… q for all u, v ≥ 0 and every 1 ≤ q < ∞ .

Proof
The part (i) follows directly from the Minkowski Inequality. If we put 1 = 1 , 2 = v , then we get the condition (ii) for 1 ≤ q < ∞ . Similarly, we . So, we will prove the cases For any f ∈ (L Φ,p ) * , if = 0 then Orlicz space L Φ,p is order continuous, i.e, L Φ,p = E Φ,p . By Lemma 2.7, We have (E Φ,p ) * = L Ψ,q . The case has been discussed. So we assume ≠ 0 . By Theorem 3.9, we know l s q ( * (lf )) . Since u and l are arbitrary, we deduce that Take l ∈ K q (f ) , for any > 0 , take k 0 ∈ K p (q + (lv)) and choose y ∈ S(L Φ,p ) such that ). Select > 0 such that then pick k > 0 such that H < and that where H = {t ∈ G ∶ |y(t)| > k} . Define
Conversely, let f = v + ∈ (L Φ,p ) * be norm attainable at u ∈ S(L Φ,p ) . We have Then we obtain the condition (i). By the Young Inequality, the condition (ii) holds. By Lemma 2.6, we have the condition (iii). ◻
Given v 0 is norm attainable at u, take k ∈ K p (u) , l ∈ K q (v 0 ) , by Theorem 4.4(b-ii) and the Young Inequality, we have By Theorem 4.4(b-iii) and = 0 , we have I Φ (ku) ⋅ I q−1 Then by the Young Inequality and the definition of p-Amemiya norm, Necessity. If condition (i) fails, i.e., lv 0 (t) = w(t) ∉ [p − (ku(t)), p + (ku(t))] , by Theo- ⋅ sign u is not a supporting functional of u.
If condition (ii) fails, i.e., I Φ (ku) ⋅ I q−1 Ψ (w) ≠ 1 . In an analogous way as above, by Theorem 4.4(b-iii) and = 0 , we have v 0 is not norm attainable at u. Hence, v = w ‖w‖ Ψ,q ⋅ sign u is not a supporting functional of u. ◻

Smoothness
Let X be a Banach space. u ∈ X is called a smooth point if it has a unique supporting functional f u . If every u ≠ 0 is a smooth point, then X is called a smooth space. Criteria for smooth points of Orlicz function (sequence) spaces equipped with the Orlicz norm and Luxemburg norm were given in [5,7,13,28]. Criteria for smoothness of Orlicz function (sequence) spaces equipped with the Orlicz norm and Luxemburg norm were given in [4,16,26,27]. In this section, we provide a characterization of smooth points in L Φ,p (1 ≤ p ≤ ∞) and as a result, we give necessary and sufficient conditions for the smoothness of L Φ,p . For any u ∈ L Φ,p (1 ≤ p ≤ ∞) , for each n ∈ N , set Lemma 5.1 [6] For any u ∈ L Φ , where u n is defined as in (6) and (u) = inf{ > 0, I Φ u < ∞}.
Proof Let u n be defined as in (6), Then u n ∈ E Φ,p . We have Then the supporting functional of u must be in L Ψ,q where 1 p + 1 q = 1 and Ψ is the function complementary to the Orlicz function Φ in the sense of Young.
Proof If p = 1 or p = ∞ , then [31] has given the proofs. We need to prove the cases of 1 < p < ∞.
Let f be the supporting functional of u. Then f has the unique decomposition f = v + where v ∈ L Ψ,q (1 < q < ∞), ∈ F . Assuming ≠ 0 , then For any k ∈ K p (u), l ∈ K q (f ), by Theorem 5.3, the Young Inequality and the definition of conjugate outer functions, we get a contradiction.
Hence, w 1 and w 2 are both supporting functionals of u ‖u‖ Φ,1 . Thus, is not a smooth point of L Φ,1 .
Let 1 < p < ∞ and v i = w i ‖w i ‖ Ψ,q , by the Young inequality and Lemma 2.6, we have is not a smooth point of L Φ,p .
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