vol. 55, no. 2 (2015), 183–209
Geometric properties of Orlicz spaces equipped with
p-Amemiya norms – results and open questions
Marek Wisła
Summary. The classical Orlicz and Luxemburg norms generated by an Orlicz function Φ can be defined with the use of the Amemiya formula [H. Hudzik and L. Maligranda, Amemiya norm equals Orlicz norm in general, Indag. Math. 11 (2000), no. 4, 573–585]. Moreover, in this article Hudzik and Maligranda suggested investigating a family of p-Amemiya norms defined by the formula ∥x ∥Φ, p= infk >0 1
k(1 + I
p
Φ(k x ))1/p, where 1 ⩽ p ⩽ ∞ (under the convention: (1 + u∞)1/∞= limp→∞(1 + up)1/p= max {1, u} for all u ⩾ 0). Based on this idea, a number of papers have been published in the past few years. In this paper, we present some major results concerning the geometric properties of Orlicz spaces equipped with p-Amemiya norms. In the last section, a more general case of Ame- miya type norms is investigated. A few open questions concerning this theory will be stated as well.
Keywords rotundity;
non-squareness;
uniform monotonicity;
dominated best approximation problem;
Amemiya type norm
MSC 2010 46B20; 46E30
Received: 2016-03-10, Accepted: 2016-05-05
The paper is dedicated to Professor Henryk Hudzik on his 70th birthday.
1. Introduction
The main idea of Władysław Orlicz when introducing in 1932 (see [44]) a new class of Banach spaces, known today as Orlicz spaces, was to produce a wide class of Banach spaces of measurable functions constructed in a similar way to the classical Lebesgue spaces Lp.
Marek Wisła, Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Umultowska 87, 61-614 Poznań, Poland (e-mail:Marek.Wisla@amu.edu.pl)
DOI 10.14708/cm.v55i2.1104 © 2015Polish Mathematical Society
The new subspaces of measurable functions were defined by a simple generalization of the power function up to a convex, even, vanishing at 0 function Φ(u), and by appeal to the notion of modular space: an Orlicz space consists of those measurable functions x for which the modular IΦ(λx ) = ∫TΦ(λx (t))d µ is finite for some λ > 0. But the key point that decided of the importance and wide applicability of Orlicz spaces was the norm defined for these spaces and a precise description of the set of continuous linear functionals generated by this norm. To do this, the concept of a complementary function in the sense of Young was used. The norm, known today as the Orlicz norm, was defined as follows
∥x ∥oΦ= sup {∫
T
∣x (t) y(t)∣d µ ∶ y ∈ LΨ, IΨ( y) ⩽ 1} , where the complementary function Ψ is defined by
Ψ(u) = sup{∣u∣v − Φ(v) ∶ v ⩾ 0}.
Although the space conjugate to the Orlicz space was described already in the thirties, a precise description of the conjugate norm is attributed to Nakano (1950), Morse–Transue (1950), and W. A. Luxemburg (1955) (see [37]). They investigated the norm, known today as the Luxemburg norm, defined with the use of the notion of the Minkowski functional with respect to the unit modular ball as follows
∥x ∥Φ= inf {λ > 0 ∶ IΦ( x λ
) ⩽ 1} .
It was proved, under some minor assumptions, that Orlicz space generated by the function Φ and equipped with the Luxemburg norm ∥⋅∥Φis conjugate to the Orlicz space generated by the function Ψ (conjugate to Φ in the sense of Young) and equipped with the Orlicz norm ∥ ⋅ ∥oΨ.
At first sight the Orlicz and Luxemburg norms seem far from similar. In fact, in many cases the geometric properties of Orlicz space under each of these norms differ from each other. But in the fifties, I. Amemiya (see [42] p. 218) considered a norm defined by the formula
∥x ∥AΦ= inf
k >0
1 k
(1 + IΦ(k x )).
Krasnoselskii–Rutickii [34], Nakano [42], Luxemburg and Zaanen [38] proved, under ad- ditional assumptions on the function Φ, that the Orlicz norm can be expressed exactly by the Amemiya formula, i.e. ∥⋅∥oΦ = ∥⋅∥AΦ. In the most general case of the Orlicz function Φ, a similar result was obtained by Hudzik and Maligranda [29]. Moreover, it is not dif- ficult to verify that the Luxemburg norm can be expressed by an Amemiya-like formula (see [6,43]), namely
∥x ∥Φ = inf
k >0
1 k
max{1, IΦ(k x )}.
The only difference between the two Amemyia formulas is the function under the infimum operation (we will call it the outer function): for all u ⩾ 0, s(u) = 1 + u (for the Orlicz norm) and s(u) = max {1, u} (for the Luxemburg norm). In [29], Hudzik and Maligranda suggested investigating the Amemiya formula generated by outer functions of the type sp(u) = (1 + up)1/p, where 1 ⩽ p ⩽ ∞. We obtain a family of topologically equivalent norms (called the p-Amemiya norms and denoted by ∥⋅∥Φ, p), indexed by 1 ⩽ p ⩽ ∞ and satisfying the inequalities
∥x ∥Φ = ∥x ∥Φ,∞⩽ ∥x ∥Φ, p⩽ ∥x ∥Φ,q⩽ ∥x ∥Φ,1= ∥x ∥oΦ ⩽ 2
1
p∥x ∥Φ, p,
for all 1 ⩽ q ⩽ p ⩽ ∞. There is a natural Köthe duality between these norms, that is, the Köthe dual of ∥⋅∥Φ, pis ∥⋅∥Ψ,q, where 1/ p + 1/q = 1 and Ψ is the conjugate function to Φ in the sense of Young.
The p-Amemiya norms can be used in renorming-type theorems, when a norm topo- logically equivalent to an original one, with better or worse properties, is sought. Indeed, the outer function s1 is affine, but it is not strictly convex at any point u > 0. The outer function s∞ is linear for u > 1 (that is good), but it is constant on [0, 1] and possesses only one point of strict convexity, namely u = 1. On the other hand, the outer functions sp, 1 < p < ∞, are increasing, strictly convex, and differentiable at any u > 0. Therefore, we can expect that geometric properties of the p-Amemiya norm for 1 < p < ∞ will be better (i.e. easier archieved) than in the case of the Orlicz or Luxemburg norm.
Another purpose in investigating the p-Amemiya norms is to look at the theory of Orlicz spaces from a common point of view. This encourages the search for universal pro- ofs that cover all norms ∥⋅∥Φ, p, 1 ⩽ p ⩽ ∞, in contrast to the current practice consisting in developing the theory of Orlicz spaces in two separate cases – for the Orlicz norm and for the Luxemburg norm.
In this paper we present some major results on Orlicz spaces equipped with the p- -Amemiya norm obtained in the last few years. A few open questions will be raised as
well.
Further details on Orlicz spaces equipped with the Luxemburg or the Orlicz norm, can be found in [4,34,39–41,45].
2. Preliminaries
In the sequel, by N and R we denote the sets of natural and real numbers. S(X) and B(X) denote the unit sphere and the unit ball of the Banach space X , respectively.
For any map Φ∶ R → [0, ∞] define
aΦ = sup{u ⩾ 0 ∶ Φ(u) = 0}, bΦ= sup{u > 0 ∶ Φ(u) < ∞}.
A map Φ∶ R → [0, ∞] is said to be an Orlicz function if Φ(0) = 0, Φ is not identical- ly zero, Φ is even, convex on the interval (−bΦ, bΦ), and Φ is left-continuous at bΦ, i.e.
limu→b− Φ
Φ(u) = Φ(bΦ). We say that w ∈ R is a point of strict convexity of Φ (we wri- te w ∈ S CΦ) if Φ(u+v
2 ) < 1
2(Φ(u) + Φ(v)) for every u, v ∈ R such that u ≠ v and w = 1
2(u + v). By S C+Φwe denote the set of all nonnegative points of strict convexity of Φ, i.e. S C+Φ= S CΦ∩ [0, ∞). Evidently, Φ(w) < ∞ for every w ∈ S CΦ.
An Orlicz function Φ satisfies the ∆2-condition for all u ∈ R (resp., at infinity) [resp., at zero] if there is a constant K > 0 (resp., and a constant u0 > 0 with Φ(u0) < ∞) [resp., and a constant u0 > 0 with Φ(u0) > 0] such that Φ(2u) ⩽ K Φ(u) for all u ∈ R (resp., for every ∣u∣ ⩾ u0) [resp., for every ∣u∣ ⩽ u0]. We will shortly write Φ ∈ ∆2(R) (resp., Φ ∈ ∆2(∞)) [resp., Φ ∈ ∆2(0)]. Evidently, Φ ∈ ∆2(R) if and only if Φ ∈ ∆2(∞) and Φ ∈ ∆2(0).
Let(T , Σ, µ) be a measure space with a σ -finite, nonatomic and complete measure µ and Lo(µ) be the set of all µ-equivalence classes of Σ-measurable real functions defined on T .
We say that an Orlicz function Φ satisfies the ∆2(µ)-condition if Φ ∈ ∆2(∞) provi- ded µ(T ) < ∞ and Φ ∈ ∆2(R) in the other case.
For a given Orlicz function Φ, we define a convex functional on the space L0(µ) (called a pseudomodular [41]) by
IΦ(x ) = ∫
T
Φ(x (t))d µ.
The Orlicz space LΦ generated by an Orlicz function Φ is a linear space of measurable functions defined by the formula
LΦ= {x ∈ L0(µ) ∶ IΦ(λx ) < ∞ for some λ > 0 depending on x }.
By EΦwe denote the linear space of all measurable functions such that IΦ(λx ) < ∞ for all λ > 0. It may happen that the space EΦconsists of only one element – the zero function.
In the following, by p+ (resp. p−) we denote the right-hand-side (resp., left-hand- -side) derivative of Φ on [0, ∞], with the following conventions p−(0) = 0, p+(∞) =
limu→∞p+(u), p−(∞) = limu→∞p−(u), and, if bΦ < ∞, p+(u) = ∞ for all u ⩾ bΦ p−(u) = ∞ for all u > bΦ. Let us emphasize that by p+, p−we always mean functions, while the single p always denotes a number from the interval [1, ∞].
For any 1 ⩽ p ⩽ ∞ and u ⩾ 0, define the outer function by
sp(u) =
⎧⎪
⎪
⎨
⎪⎪
⎩ (1 + up)
1
p for 1 ⩽ p < ∞, max {1, u} for p = ∞.
Note that the functions spare convex. Moreover, for 1 ⩽ p < ∞ the function spis increasing on R+, but the function s∞is increasing on the interval [1, ∞) only. By the p-Amemiya norm of an element x ∈ L0we mean the norm defined by the formula (see [6,29])
∥x ∥Φ, p= inf
k >0
1 k
sp(IΦ(k x )) , 1 ⩽ p ⩽ ∞.
Let us emphasize that for p = 1 the p-Amemiya norm coincides with the Orlicz norm and for p = ∞ with the Luxemburg norm.
3. Norm attainability
In each case when it is necessary to calculate the least upper bound (or the greatest lower bound) of a partially ordered set, it is welcome to have a LUB or a GLB that is realized by an element of that set. It simplifies calculations and often gives a lot of additional informa- tion. In the case of the p-Amemiya norm, many geometric local (pointwise) properties strongly depend on whether the norm is attained at some k > 0 or not. In [6], a detailed investigation of the p-Amemiya norm attainability was done. To present these results here, we need a few definitions.
Let Ψ be the complementary Orlicz function to Φ in the sense of Young. By αp∶ LΦ, p→ [−1, ∞] we denote a function defined by
αp(x ) =
⎧⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎩
IΦp−1(x )IΨ( p+(∣x ∣)) − 1, 1 ⩽ p < ∞,
−1, p = ∞, IΦ(x ) ⩽ 1,
IΨ( p+(∣x ∣)), p = ∞, IΦ(x ) > 1,
(it will be called the norm attainability indicator), and let k∗p∶ LΦ, p→ [0, ∞), k∗∗p ∶ LΦ, p→ (0, ∞] be defined by
k∗
p(x ) = inf {k ⩾ 0 ∶ αp(k x ) ⩾ 0} (with inf ∅ = ∞), k∗∗p (x ) = sup{k ⩾ 0 ∶ αp(k x ) ⩽ 0}.
It is evident that k∗p(x ) ⩽ k∗∗p (x ) for every 1 ⩽ p ⩽ ∞ and x ∈ LΦ, p. Further, let the set Kp(x ) be defined by
Kp(x ) = {0 < k < ∞ ∶ k∗p(x ) ⩽ k ⩽ k∗∗p (x )} .
Let us note that for any k from the interior of Kp(x ), the norm attainability indicator at k x is equal to 0, i.e. αp(k x ) = 0. But this fact does not hold for all k ∈ Kp(x ). In Theorem 3.7 in [6] it was proved that the set of those k′s at which the norm of an element x ∈ LΦ, p∖ {0} is attained forms an interval, and this interval was completely described. It
is worth noticing that the description of the norm attainability interval is universal for all p-Amemiya norms. Below a slight reformulation of that theorem is presented.
3.1. Theorem ([6]). For any 1 ⩽ p ⩽ ∞ the set of those k′s at which the p-Amemiya norm of an element x ∈ LΦ, p∖ {0} is attained is equal to the interval Kp(x ).
Thus, the above theorem gives a good reason for introducing the following defini- tions. An Orlicz function Φ is said to be k∗p-finite if k∗p(x ) < ∞ for all x ∈ LΦ, p∖ {0}
(i.e. the interval of norm attainability of x is non-empty). An Orlicz function Φ is said to be k∗∗p -finite if k∗∗p (x ) < ∞ for all x ∈ LΦ, p∖ {0} (i.e. the interval of norm attaina- bility of x is closed and bounded). Finally, an Orlicz function Φ is said to be kp-unique, if k∗p(x ) = k∗∗p (x ) < ∞ for all x ∈ LΦ, p ∖ {0} (i.e., the interval of norm attainability of x consists of only one element). Just how important is the k∗p-finiteness of the Orlicz function Φ shows Lemma 2.5 in [12].
3.2. Theorem ([12]). If the Orlicz function Φ is not k∗p-finite, then the Orlicz space LΦ, p contains a linearly isometric copy of L1.
From the geometric point of view, it is important to know when the Orlicz func- tion Φ is k∗p-finite, k∗∗p -finite, or kp-unique. The relevant theorems were given in [6, The- orem 4.3, 4.4 and 4.5]. Below a slight reformulation of these theorems in terms of the norm attainability interval is presented.
3.3. Theorem ([6]). Let Φ be an Orlicz function and let 1 ⩽ p ⩽ ∞.
(i) The interval of norm attainability of all x ∈ LΦ, p∖ {0} is empty if and only if one of the following conditions is satisfied:
(a) p = 1 and Φ admits an asymptote at infinity.
(b) Φ is linear on [0, ∞).
(ii) The interval of norm attainability of all x ∈ LΦ, p∖ {0} is nonempty, closed and boun- ded (in fact, equal to [k∗p(x ), k∗∗p (x )]) if and only if one of the following conditions is satisfied:
(a) p = 1 and Φ does not admit an asymptote at infinity.
(b) 1 < p ⩽ ∞ and Φ is not linear on [0, ∞).
(c) Φ takes infinite values.
(iii) The interval of norm attainability of all x ∈ LΦ, p∖{0} is a one-point set (in fact, k∗p(x ) = k∗∗p (x )) if and only if one of the following conditions is satisfied:
(a) p = 1, Φ does not admit an asymptote at infinity, and either a = bΦor Φ is strictly convex on the interval (a, bΦ), where a = sup {u > 0 ∶ Ψ( p+(u))µ(T ) < 1}.
(b) 1 < p < ∞ and Φ is not linear on [0, ∞).
(c) p = ∞ and Φ is not linear on any interval [0, b + ε) with ε > 0, where b = inf {u ⩾ 0 ∶ Φ(u)µ(T ) > 1}.
Let us emphasize the simplicity of the above conditions in the case 1 < p < ∞. If the Orlicz function Φ is not linear on [0, ∞), then the interval of norm attainability always consists of a single point, i.e. the (generally set-valued) function x ↦ [k∗p(x ), k∗∗p (x )] is
a single valued function. In [5, Proposition 1], it was proved, under additional assumptions, that in the case of the Orlicz norm the above set-valued function is upper semicontinuous.
3.4. Corollary. Let Φ be an Orlicz function with Φ(u)/u → 0 as u → 0 and Φ(u)/u → ∞ as u → ∞. Then the set-valued function K1∶ LΦ,1 → R, K1(x ) = [k∗1(x ), k∗∗1 (x )] is single valued and continuous.
Let us note that the assumption Φ(u)/u → ∞ as u → ∞ implies that Φ is k∗1-finite, whence the sets K1(x ) are nonempty for all x ∈ LΦ∖ {0}. But this assumption is stronger than k∗1- finiteness. Moreover, let us emphasize that the existence of a continuous selector of the set-valued function K1(⋅) is an exceptional case.
3.5. Theorem ([5]). Let Φ be an Orlicz function with Φ(u)/u → 0 as u → 0 and Φ(u)/u →
∞ as u → ∞. Then the set-valued function K1(⋅) admits a continuous selector if and only if it is single valued.
Thus we can ask the following questions: Can the assumption Φ(u)/u → ∞ as u → ∞ be weakened in the Corollary3.4and Theorem3.5? And can this assumption be omitted in the case 1 < p < ∞ or p = ∞?
4. Extreme points and rotundity
In the following, by χ{∞}( p) we denote the characteristic function defined on the set 1 ⩽ p ⩽ ∞. This function gets nonzero value (in fact, is equal to 1) if and only if p = ∞.
As it was mentioned in the introduction, the outer function sp in the p-Amemiya formula has better properties for 1 < p < ∞ than for p = 1 or p = ∞; for example, sp is strictly convex for 1 < p < ∞. Thus one can suspect that this fact should have big impact on the rotundity properties of Orlicz spaces with the p-Amemiya norms. We start with a description of the extreme points of the unit ball B(LΦ, p).
4.1. Theorem ([6]). For 1 ⩽ p ⩽ ∞, an element x ∈ S(LΦ, p) is an extreme point of the unit ball B(LΦ, p) if and only if
(i) k∗p(x ) = k∗∗p (x ) < ∞,
(ii) k∗p(x )x (t) ∈ S CΦfor µ-a.e. t ∈ T ,
(iii) IΦ(k∗p(x )x ) ⩾ χ{∞}( p) or ∣x (t)∣ = bΦfor µ-a.e. t ∈ T . As an immediate consequence, we get the following corollary.
4.2. Corollary. For 1 < p < ∞, an element x ∈ S(LΦ, p) is an extreme point of the unit ball B(LΦ, p) if and only if
(i) Φ is not linear on [0, ∞),
(ii) k x (t) ∈ S CΦfor µ-a.e. t ∈ T , where Kp(x ) = {k}.
Proof. Necessity. Let x be an extreme point. Suppose that Φ(u) = cu for some c > 0 and every u ∈ R. Then
∥x ∥Φ, p= inf
k >0
1 k
sp(IΦ(k x )) = inf
k >0
1 k
sp(c k∥x ∥1) = c∥x ∥1,
so the Orlicz space LΦ, p is isometric to the Lebesgue space L1(µ) for every 1 ⩽ p ⩽ ∞.
Thus Ext B(LΦ, p) = Ext(B(L1(µ))) = ∅, a contradiction. The rest of the proof follows from Theorem4.1.
Looking at the extreme points globally, we get a theorem on rotundity.
4.3. Theorem ([6]). The Orlicz space LΦ, pis rotund if and only if (i) Φ is kp-unique,
(ii) Φ is strictly convex on (−bΦ, bΦ) and (a) 1 ⩽ p < ∞ or
(b) p = ∞ and Φ ∈ ∆2(µ).
Thus, for every 1 < p < 1 and every Orlicz function Φ which is strictly convex on (−bΦ, bΦ), the Orlicz space LΦ, p is rotund. The Theorem4.3 can serve as a renorming theorem as well.
4.4. Corollary. Let Φ be a strictly convex Orlicz function. The Orlicz space LΦcan be equ- ipped with a norm ∥⋅∥ topologically equivalent to the Orlicz and Luxemburg norm such that the Orlicz space (LΦ, ∥⋅∥) is rotund.
We can also look at Theorem4.3from a chain of norms point of view with two boun- daries: the Orlicz norm ( p = 1) and the Luxemburg norm ( p = ∞). If an Orlicz space is rotund under one of the boundary norm, then the Orlicz spaces equipped with any norm from the interior of the chain is rotund as well. More precisely:
4.5. Corollary. If the Orlicz space LΦ, pis rotund for p = 1 or p = ∞, then it is rotund for all 1 < p < ∞ as well.
Proof. If the Orlicz space LΦ,1 is rotund, then Φ is strictly convex on (−bΦ, bΦ) and k1-unique. By Theorem3.3, the function Φ does not admit an asymptote at infinity and either a = bΦor Φ is strictly convex on the interval (a, bΦ), where
a = sup {u > 0 ∶ Ψ( p+(u))µ(T ) < 1} .
Thus Φ cannot be a linear function on [0, ∞), whence Φ is kp-unique for all 1 < p < ∞.
By Theorem4.3, the Orlicz space LΦ, pis rotund for all 1 < p < ∞.
Further, if the Orlicz space LΦ,∞is rotund, then Φ is strictly convex on (−bΦ, bΦ), k∞-unique and Φ ∈ ∆2(µ). By Theorem3.3, the function Φ is not linear at any interval
[0, ε) (ε > 0), whence Φ cannot be a linear function on [0, ∞). Thus Φ is kp-unique for all 1 < p < ∞ and, by Theorem4.3, the Orlicz space LΦ, pis rotund for all 1 < p < ∞.
Let us note that rotundity of LΦ, pfor any (and even for all) 1 < p < ∞ does not imply the rotundity of either of the boundary norms. For example, take the function Φ(u) =
∣u∣e1/∣u∣for u ≠ 0 and Φ(0) = 0 and a measure space of infinite measure (see [6]). Then Φ is strictly convex, admits an asymptote at infinity and does not satisfy the ∆2condition (at 0). Thus the Orlicz space LΦ, pis rotund for all 1 < p < ∞ but is not rotund for p = 1 or p = ∞.
For more results on extreme points in Orlicz spaces, we refer to [10,14,20,22,23,28, 30,48,49].
5. Kadec–Klee properties
For a Banach space X , a point x ∈ S(X) is said to be a strongly extreme point of the unit ball B(X) if for any sequences ( yn) and (zn) in X such that ∥ yn∥X, ∥zn∥X → 1 the condition yn+ zn = 2x for any n ∈ N implies that ∥ yn− x ∥X → 0. The set of all extreme (resp. strongly extreme) points of the unit ball B(X) will be denoted by Ext B(X) (resp.
SExt B(X)).
We say that a point x ∈ S(X) is an Hµ-point, if for every sequence (xn) such that xn
µ
Ð→ x and ∥xn∥ → ∥x ∥ we have ∥xn− x ∥ → 0. We say that a Banach function space X possesses the Kadec–Klee property (with respect to convergence in measure), if every point of its unit sphere is an Hµ-point. In [13], the notion of p-spherical convergence was introduced: a sequence (xn) is called p-spherically convergent to x (denoted by xn
p−s
Ð→ x ), if there exists a bounded sequence (kn) of positive numbers such that
1 kn
sp(IΦ(knxn)) → ∥x ∥Φ, p.
Let us note that, in general, convergence in measure together with convergence of norms do not imply p-spherical convergence. Just consider the Lebesgue space L1. No sequence in L1is p-spherically convergent but there are plenty of sequences that are convergent in measure and have convergent norms (in fact, look at the norm convergent sequences).
5.1. Lemma ([13]). Let 1 ⩽ p ⩽ ∞, xn, x ∈ LΦ, p, and let x ≠ 0.
(i) If ∥xn∥Φ,∞→ ∥x ∥Φ,∞≠ 0, then xn
∞−s
Ð→ x .
(ii) If limu→∞Φ(u)/u = ∞, ∥xn∥Φ, p→ ∥x ∥Φ, pand xn
µ
Ð→ x , then (xn) is p-spherically convergent to x .
Thus, ∞-spherical convergence is weaker than convergence of norms, but for 1 ⩽ p <
∞ we need additional assumptions. In spite of Lemma5.1, we can consider a bit more
general geometric property of Kadec–Klee type – by replacing the convergence of norms with p-spherical convergence.
5.2. Theorem ([13]). Let Φ be an Orlicz function such that either Φ ∈ ∆2(µ) or aΦ > 0 and Φ ∈ ∆2(∞). Further, let xn, x ∈ LΦ, p ∖ {0} be such that xn
µ
Ð→ x . Moreover, as- sume that IΦ(x / ∥x ∥Φ,∞) = 1 for p = ∞. If (xn) is p-spherically convergent to x , then
∥xn− x ∥Φ, p→ 0.
As a corollary we get a description of the Kadec–Klee property in Orlicz spaces with p-Amemiya norms.
5.3. Theorem ([13]). Let Φ be an Orlicz function such that either Φ ∈ ∆2(µ) or aΦ > 0 and Φ ∈ ∆2(∞). If 1 ⩽ p < ∞, then each point of x ∈ LΦ, p∖ {0} with Kp(x ) ≠ ∅ is an Hµ-point, i.e. the Orlicz space LΦ, phas the Kadec–Klee property as long as the function Φ is k∗p-finite. If p = ∞, then each point with IΦ(x / ∥x ∥Φ,∞) = 1 is an Hµ-point.
Considering the Kadec–Klee property it is also interesting to consider two sequences with a fixed midpoint instead of just one sequence. More precisely ([13]): we say that a po- int x of a Banach space is an H2,µ-point, if for every pair of sequences (xn) and ( yn) such that xn+ yn= 2x , xn− yn
µ
Ð→ 0, ∥xn∥ → ∥x ∥ and ∥ yn∥ → ∥x ∥ we have ∥xn− yn∥ → 0. We say that a Banach space X has the Kadec–Klee 2 property (with respect to convergence in measure), if each point of its unit sphere is an H2,µ-point. Evidently, every strongly extreme point of the unit ball B(X) is an H2,µ-point. Moreover, every Hµ-point is also an H2,µ-point, so the Kadec–Klee property with respect to convergence in measure implies the Kadec–Klee 2 property with respect to convergence in measure. But the converse im- plication is not true.
5.4. Proposition ([13]). Let Φ be an Orlicz function such that Φ(bΦ)µ(T ) ⩽ 1 (with 0 ⋅ ∞ =
∞). Then the function x = ±bΦχTis an H2,µ-point of the unit ball B(LΦ,∞) but is not an Hµ-point. In particular, the functions x = ± χTare H2,µ-points of the unit ball B(L∞(µ)) but are not Hµ-points.
The following theorem gives sufficient conditions for a point in LΦ, pto be an H2,µ- -point. Note that those conditions coincide with the sufficient conditions for Hµ-points.
However, those conditions need not be necessary.
5.5. Theorem ([13]). Let Φ be an Orlicz function such that either Φ ∈ ∆2(µ) or aΦ >
0 and Φ ∈ ∆2(∞). If 1 ⩽ p < ∞ then every point x ∈ LΦ, p ∖ {0} with Kp(x ) ≠ ∅ is an H2,µ-point, i.e. the Orlicz space LΦ, p has the Kadec–Klee 2 property with respect to convergence in measure as long as the function Φ is k∗p-finite. If p = ∞, then every point x with IΦ(x / ∥x ∥Φ,∞) = 1 is an H2,µ-point.
6. Strongly extreme points and midpoint local uniform rotundity
The class of Orlicz functions that is considered in this paper contains the functions that can take infinite values (in the other words: “can jump to infinity”), i.e. the coefficient bΦ can be finite. Orlicz spaces generated by this kind of functions are subspaces of the Banach space L∞. There are two types of such Orlicz functions: one when the function Φ(u) tends asymptotically to infinity as u tends to bΦfrom the left and, the other, when Φ(bΦ) < ∞.
6.1. Theorem ([13]). Let 1 ⩽ p ⩽ ∞ and let Φ be an Orlicz function with bΦ< ∞.
(i) If Φ(bΦ)µ(T ) < ∞ then the Orlicz space LΦ, pis linearly isomorphic to the space L∞. We have
b−Φ1∥x ∥∞⩽ ∥x ∥Φ, p⩽ b−Φ1sp(IΦ(bΦχT)) ∥x ∥∞,
so the mapping P∶ LΦ, p→ L∞defined by P(x ) = b−Φ1x is a linear isomorphism between the spaces LΦ, pand L∞.
(ii) The Orlicz space LΦ, pis linearly isometric to the space L∞if and only if either Φ(bΦ) = 0 or p = ∞ and Φ(bΦ)µ(T ) ⩽ 1.
Part (ii) of the following Lemma states that the set of strongly extreme points of the unit ball B(LΦ, p) is empty if the function Φ(u) tends asymptotically to infinity as u tends to bΦfrom the left and bΦ is finite. Thus, the set SExt B(LΦ, p) is not empty if and only if Φ(bΦ) < ∞ or bΦ = ∞. Moreover, the absolute value of a function that is a strongly extreme point is always an almost everywhere constant function on T .
6.2. Theorem ([13]). Let Φ be an Orlicz function with bΦ < ∞.
(i) SExt B(LΦ, p) ⊂ {x ∈ S(LΦ, p) ∶ k∗p(x )∣x ∣ = bΦχT}.
(ii) If Φ(bΦ) = ∞ then SExt B(LΦ, p) = ∅.
(iii) If Φ(bΦ) < ∞ then SExt B(LΦ, p) = {x ∈ S(LΦ, p) ∶ k∗p(x )∣x ∣ = bΦχT}.
Let us note that Theorem6.2does not give criteria for the set SExt B(LΦ, p) to be nonempty. It only gives necessary and sufficient conditions for a point x ∈ S(LΦ, p) to be strongly extreme. The next theorem is more precise and gives a complete description of the set SExt B(LΦ, p) in the case when bΦ < ∞. In the theorem the norm attainability indicator has been used for the function x = bΦµ(T ) with a slight change – the left-hand- -side derivative p−is used instead of the derivative p+. Further, q is such a number that
1/ p + 1/q = 1 (with 1
∞ = 0) and (S CΦ)ddenotes the set of all accumulation points of the set of strict convexity points of the function Φ.
6.3. Theorem ([13]). Let Φ be an Orlicz function such that bΦ < ∞. Then SExt B(LΦ, p) ≠
∅ if and only if Φ(bΦ)µ(T ) < ∞ and one of the following conditions holds true (with conventions: 0 ⋅ ∞ = 0 and 1
∞ = 0):
(i) Φ
1
q(bΦ)Ψ
1
p( p−(bΦ))µ(T ) < 1;
(ii) Φ
1
q(bΦ)Ψ
1
p( p−(bΦ))µ(T ) = 1 and either (a) 1 < p ⩽ ∞ or
(b) p = 1 and bΦ ∈ (S CΦ)d. If the set SExt B(LΦ, p) is not empty then
SExt B(LΦ, p) = {x ∈ LΦ, p ∶ ∣x ∣ = bΦ⋅ (sp(IΦ(bΦχT)))−1χT} .
For example, take the Orlicz function Φ(u) = ∣u∣ for ∣u∣ ⩽ 1, Φ(u) = ∞ for ∣u∣ > 1. It is known that the Orlicz space LΦ,1(equipped with the Orlicz norm) coincides with the interpolation space L1∩ L∞(see [26,30]). For all 1 ⩽ p < ∞ we have
SExt B(LΦ, p) = {x ∈ LΦ, p ∶ ∣x ∣ = (1 + µp(T ))−
1
pχT} ≠ ∅ ⇐⇒ µ(T ) < ∞, SExt B(LΦ, p) ≠ Ext B(LΦ, p).
In the case p = ∞ we have
SExt B(LΦ,∞) = {x ∈ LΦ,∞ ∶ ∣x ∣ = χT} = Ext B(LΦ,∞) ≠ ∅ ⇐⇒ µ(T ) ⩽ 1.
The next example is very interesting since it gives a geometric property (non-emptiness of the set of strongly extreme points) that holds true only if p ⩽ p0, where p0is a number such that 1 < p0 < ∞. This shows that some geometric properties distinguish the spaces LΦ, pnot only when p is just one of the three cases: equal to 1, equal to ∞, or from the inte- rval (1, ∞). This is a direct consequence of the fact that in the case bΦ< ∞ the description of strongly extreme points depends not only on the Orlicz function Φ, but on the value of the measure µ(T ) as well.
6.4. Example ([13]). Let Φ(u) = 1
β∣u∣βfor ∣u∣ ⩽ 1, Φ(u) = ∞ for ∣u∣ > 1, where 1 < β < ∞.
Let γ ⩾ 1 be such that 1
β+ 1
γ = 1.
(i) For all 1 ⩽ p ⩽ ∞,
∅ ≠ SExt B(LΦ, p) =
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
{x ∈ LΦ, p ∶ ∣x ∣ = (1 + 1
βpµp(T ))
−1
pχT} , 1 ⩽ p < ∞, {x ∈ LΦ, p ∶ ∣x ∣ = χT} , p = ∞ if and only if µ(T ) ⩽ β
1 qγ
1 p.
(ii) If β = 2, then SExt B(LΦ, p) ≠ ∅ for every 1 ⩽ p ⩽ ∞ if and only if µ(T ) ⩽ 2.
(iii) If µ(T ) ⩽ min {β, β
β−1}, then SExt B(LΦ, p) ≠ ∅ for every 1 ⩽ p ⩽ ∞.
(iv) If µ(T ) > max {β, β
β−1}, then SExt B(LΦ, p) = ∅ for every 1 ⩽ p ⩽ ∞.
(v) Let min {β, β
β−1} < µ(T ) ⩽ max {β, β
β−1} and let p0 = −ln(β−1)
ln µ(T)−ln β (with 1
0 = ∞).
Then SExt B(LΦ, p) ≠ ∅ if and only if (a) 1 < β < 2 and 1 ⩽ p ⩽ p0, or (b) 2 < β < ∞ and p0⩽ p ⩽ ∞.
p y
β = 4
β = 2 β =4/3
1
The graph of the function y = β
1 qγ
1
p = β
(β−1)1/p
The following theorem gives a general description of the points of strict convexity of a unit ball in Orlicz spaces.
6.5. Theorem ([13]). Let Φ be an Orlicz function and x ∈ S(LΦ, p). The function x is a stron- gly extreme point of the unit ball B(LΦ, p) if and only if the following conditions are satisfied:
(i) the set Kp(x ) is a singleton, that is, Kp(x ) = {k} for some k ⩾ 1, (ii) k x (t) ∈ S CΦfor µ-a.e. t ∈ T ,
(iii) either ∣x ∣ = bΦ(sΦ, p(bΦχT))−1χTor
(iv) IΦ(k x ) ⩾ χ{∞}( p) and at least one of the conditions:
(a) aΦ> 0 and Φ ∈ ∆2(∞), (b) Φ ∈ ∆2(µ)
is satisfied.
The next corollary states that in the case of Orlicz functions taking only finite values, if one of the extreme points of the unit ball B(LΦ, p) is strongly extreme, then every extreme point of B(LΦ, p) is strongly extreme.
6.6. Corollary ([13]). Let Φ be an Orlicz function such that bΦ= ∞ and SExt B(LΦ, p) ≠ ∅.
Then SExt B(LΦ, p) = Ext B(LΦ, p) for every 1 ⩽ p ⩽ ∞.
Recall that a Banach space X is called midpoint locally uniformly rotund (MLUR) if every point of the unit sphere S(X) is strongly extreme. Applying the previous theorem we get the following criteria for the MLUR property of Orlicz spaces.
6.7. Theorem ([13]). For every 1 ⩽ p ⩽ ∞, the Orlicz space LΦ, pis MLUR if and only if (i) Φ is kp-unique,
(ii) Φ is finite and strictly convex on R, (iii) Φ ∈ ∆2(µ).
6.8. Corollary ([13]).
(i) The Orlicz space LΦ, pis rotund if and only if the space LΦ, p′ is rotund for every p′ ∈ (1, ∞) ∪ { p}.
(ii) The Orlicz space LΦ, p is MLUR if and only if the space LΦ, p′ is MLUR for every p′ ∈ (1, ∞] ∪ { p}.
Taking into account that Φ ∈ ∆2(µ) implies the Orlicz function Φ takes only finite values, we get the following renorming theorem.
6.9. Theorem ([13]). Let Φ be a strictly convex Orlicz function on (−bΦ, bΦ). If ∥ ⋅ ∥ is any norm on the Orlicz space LΦ that is equivalent to either the Orlicz or the Luxemburg norm, then the space (LΦ, ∥ ⋅ ∥) can be renormed by an equivalent norm ∥ ⋅ ∥′such that the space (LΦ, ∥ ⋅ ∥′) becomes rotund. If, moreover, the function Φ satisfies the ∆2(µ)-condition, then the space (LΦ, ∥ ⋅ ∥′) becomes MLUR.
7. Non-squareness
R. C. James in [31] introduced the concepts of uniformly non-square, locally uniformly non-square and non-square Banach spaces and proved that uniformly non-square Banach spaces are reflexive. Recall that a Banach space X is said to be uniformly non-square if there are no sequences (xn) and ( yn) in S(X ) such that
∥xn+ yn∥ → 1 and ∥xn− yn∥ → 1. (1) If there is no x0∈ S (X ) and no sequence ( yn) in S(X ) such that (1) holds with xn= x0for n = 1, 2, . . ., then X is said to be locally uniformly non-square. If there are no x , y ∈ S(X) such that
∥x + y∥ = ∥x − y∥ = 1, then X is said to be non-square.
In this section, linearity of the Orlicz function at 0 plays a crucial role. Define dΦ= sup{u ⩾ 0 ∶ Φ is linear on the interval [0, u]}.
Using Young’s equality it is not difficult to prove that for any Orlicz function Φ Φ(dΦ)Ψ(dΨ) = 0,
where Ψ is the Orlicz function complementary to Φ in the sense of Young ([12]).
7.1. Theorem ([12]). The Orlicz space LΦ, p is non-square if and only if the following three conditions hold true (with 0 ⋅ ∞ = 0):
(i) Φ is k∗p-finite, (ii) aΦ< bΦ, (iii) aΦµ(T ) < ∞,
and moreover one of the following conditions is satisfied:
(a) p = ∞, Φ(dΦ)µ(T ) < 2 and Φ ∈ ∆2(µ), (b) 1 < p < ∞,
(c) p = 1 and Ψ(dΨ)µ(T ) < 2.
Since in the case 1 < p < ∞ no condition other than (i), (ii), and (iii) is required, the following corollary is evident.
7.2. Corollary. If the Orlicz space LΦ, pis non-square for p = 1 or p = ∞, then it is non-square for all 1 < p < ∞.
As in the case of rotundity, non-squareness of LΦ, pfor any (or for all) 1 < p < ∞ does not imply non-squareness in the boundary cases p = 1 or p = ∞. Let Φ(u) = max {0, eu− e} and let the measure of T be such that 2e−1⩽ µ(T ) < ∞ (see [12]). Eviden- tly aΦ= 1. Then the Orlicz space LΦ, pis not non-square for p = 1 (because Ψ(dΨ)µ(T ) >
2), and for p = ∞ (because Φ ∉ ∆2(∞)), but it is non-square for all 1 < p < ∞ (because Φ is k∗p-finite, bΦ= ∞, and aΦµ(T ) < ∞).
The following theorem provides the conditions that guarantee local uniform non- -squareness of Orlicz spaces.
7.3. Theorem ([12]). The Orlicz space LΦ, pis locally uniformly non-square if and only if it is non-square and either
(i) 1 < p ⩽ ∞,
(ii) p = 1 and Ψ(dΨ)µ(T ) < 1, or
(iii) p = 1, 1 ⩽ Ψ(dΨ)µ(T ) < 2, and Φ ∈ ∆2(∞).
In consequence, if the Orlicz space LΦ, p is locally non-square for p = 1 or p = ∞, then it is locally non-square for all 1 < p < ∞. Evidently, local non-squareness of LΦ, pfor any (or for all) 1 < p < ∞ does not imply local non-squareness in the boundary cases p = 1 or p = ∞.
The most important property of the three non-squareness properties is the uniform non-squareness. Uniform non-square Banach spaces are reflexive and a Banach space X is uniformly non-square if and only if its dual space X∗is uniformly non-square [18,33,47].
Moreover, J. Garcia-Falset, E. Llorens-Fuster and E. Mazcuñan-Navarro [21] proved that uniformly non-square Banach spaces have the fixed point property.
7.4. Theorem ([12]). The Orlicz space LΦ, pis uniformly non-square if and only if (i) (aΦ+ aΨ)µ(T ) < ∞,
(ii) Φ, Ψ ∈ ∆2(µ) and
(iii) 1 < p < ∞ or
(iv) p ∈ {1, ∞} and (Φ(dΦ) + Ψ(dΨ))µ(T ) < 2,
where Ψ is the Orlicz function complementary to Φ in the sense of Young, and 1/ p + 1/q = 1.
Some other results on non-squareness of Orlicz spaces can be found in [2,3,8,9,24,46].
8. Monotonicity properties
In order to present the results, we need to recall same definitions. Let X be a Banach lattice with a lattice norm ∥⋅∥.
– X is said to be strictly monotone (SM) if
(0 ⩽ y ⩽ x , y ≠ 0) Ô⇒ ∥x − y∥ < ∥x ∥ .
– X is said to be uniformly monotone (UM) ([1]) if for every ε > 0 there exists δ(ε) > 0 such that
∥ y∥ ⩾ ε Ô⇒ ∥x + y∥ > 1 + δ(ε)
for all y ∈ X+(the positive cone of X ) and all x ∈ S+(X ) (the unit sphere of positive cone of X). It is known [35] that X is UM if and only if for every ε ∈ (0, 1] there exists η(ε) ∈ (0, 1] such that
(0 ⩽ y ⩽ x , ∥ y∥ ⩾ ε) Ô⇒ ∥x − y∥ ⩽ 1 − η(ε) for all y ∈ X+and all x ∈ S+(X ).
– X is said to be lower locally uniformly monotone (LLUM) if for every ε ∈ (0, 1] and every x ∈ S+(X ) there exists δ(x , ε) > 0 such that
(0 ⩽ y ⩽ x , ∥ y∥ ⩾ ε) Ô⇒ ∥x − y∥ ⩽ 1 − δ(x , ε)
for all y ∈ X+. It is known that a Banach lattice X is LLUM if and only if for every x ∈ S+(X ) and every sequence (xn) with 0 ⩽ xn⩽ x the implication
∥xn∥ → 1 Ô⇒ ∥x − xn∥ → 0
holds true.
– X is said to be upper locally uniformly monotone (ULUM) if for every ε > 0 and every x ∈ S+(X ) there exists σ (x , ε) > 0 such that
( y ⩾ 0, ∥ y∥ ⩾ ε) Ô⇒ ∥x + y∥ ⩾ 1 + σ (x , ε).
Equivalently (see [15]), the ULUM property can be expressed in sequence terms: A Ba- nach lattice X is ULUM if and only if for every x ∈ S+(X ) and every sequence (xn) with x ⩽ xn, we have
∥xn∥ → 1 Ô⇒ ∥xn− x ∥ → 0.
In [11], monotonicity properties of Orlicz spaces equipped with p-Amemiya norms are investigated. One of the most important problems is to get a precise estimation (from above) of the distance ∥ y − x ∥. It is interesting that the distance ∥ y − x ∥Φ, pcan be estima- ted by the modulus of monotonicity of the sequence space ℓp. Namely, Lemma 2.3 in [11]
states that for 1 ⩽ p < ∞ and all x , y ∈ LΦ, pwith ∥ y∥Φ, p= 1 and 0 ⩽ x ⩽ y we have
∥ y − x ∥Φ, p⩽ 1 − δm, p(IΦ(x )) ,
where δm, pis the modulus of monotonicity of the space ℓpdefined by
δm, p(ε) = inf {1 − ∥x − y∥p ∶ x , y ∈ ℓp, 0 ⩽ y ⩽ x , ∥x ∥p= 1, ∥ y∥p⩾ ε} . 8.1. Theorem ([11]).
(i) The Orlicz space LΦ, pis strictly monotone if and only if one of the following two condi- tions is satisfied:
(a) Φ vanishes only at 0 whenever 1 ⩽ p < ∞.
(b) Φ vanishes only at 0 and Φ ∈ ∆2(µ) whenever p = ∞.
(ii) For any 1 ⩽ p ⩽ ∞, the space EΦ, pis strictly monotone if and only if either Φ vanishes only at 0 or EΦ, p= {0}.
The criteria for UM of the Orlicz space are more restricting. In fact, these criteria describe not only the UM property, but LLUM, ULUM properties as well (and some other monotonicity properties – see [11]). It is also interesting that the criteria do not depend on p, i.e. they are valid for all 1 ⩽ p ⩽ ∞.
8.2. Theorem ([11]). Let 1 ⩽ p ⩽ ∞. The following conditions are equivalent.
(i) LΦ, pis UM.
(ii) LΦ, pis LLUM.
(iii) LΦ, pis ULUM.
(iv) EΦ, p≠ {0} and EΦ, pis ULUM.
(v) aΦ= 0 and Φ ∈ ∆2(µ).
For more results on the monotonicity properties of Orlicz spaces, we refer to [19,25,27].
9. Best approximation properties
For a nonempty subset A of a Banach space X and any x ∈ X, the number d (x , A) = inf {∥x − y∥ ∶ y ∈ A}
is called the distance of x from A, and the sequence ( yn) of elements of A is said to be an x -minimizing sequence whenever limn→∞∥x − yn∥ = d (x , A). Moreover, the set-valued function PA(x ) = X → 2Xdefined by
PA(x ) = {z ∈ A ∶ d (x , A) = ∥x − z∥}
is called the projection from X onto A, and for any x ∈ X the set PA(x ) is called the projection of x onto A.
The problem of describing the elements of the set PA(x ) is called a best approxima- tion problem. If PA(x ) ≠ ∅ (resp., card PA(x ) = 1), then the best approximation problem is called solvable (resp., uniquely solvable) for x ∈ X. Moreover, the best approximation problem is said to be stable for x ∈ A, if for every x -minimizing sequence ( yn) in A we have d ( yn, PA(x )) → 0 as n → ∞. Finally, the best approximation problem is called strongly solvable if it is uniquely solvable and stable.
A set A is called a sublattice of the Banach lattice X if A ⊂ X and for any x , y ∈ A there exist x ∧ y ∈ A and x ∨ y ∈ A. The best approximation problem restricted to a sublattice A and x being a (lower or upper) boundary of A is called a dominated best approximation problem.
Using order continuity and the ULUM property of Orlicz spaces equipped with the p-Amemiya norm, we get the following theorems.
9.1. Theorem ([11]). Let Φ be an Orlicz function that vanishes only at 0 and satisfies the
∆2(µ)-condition if p = ∞. For every p-Amemiya norm ∥⋅∥Φ, p, 1 ⩽ p ⩽ ∞, and every closed sublattice A of the Orlicz space LΦ, p, the dominated best approximation problem is uniquely solvable for any (lower or upper) boundary of A as far as it is solvable.
9.2. Theorem ([11]). Let Φ be an Orlicz function that vanishes only at 0 and satisfies the
∆2(µ)-condition. For every p-Amemiya norm ∥⋅∥Φ, p, 1 ⩽ p ⩽ ∞, and every closed sublattice A of the Orlicz space LΦ, pthe dominated best approximation problem is strongly solvable for any (lower or upper) boundary of the sublattice A.
Although the dominated best approximation problem is more specific than the best approximation problem, the disadvantage of the first is that it is restricted to sublattices while the general one is formulated for convex sets. In [7], the authors concern themse- lves with the dominated best approximation problem on the order-bounded convex set A, where the boundaries of A are elements of the set
D(A) = {z ∈ X ∶ A − z is an absolutely directed set}
x y
w
x ∨ y
x ∧ y B
A closed convex absolutely directed set B which is not a sublattice
(we call it the modified best approximation problem). Recall that a subset B ⊂ X is said to be absolutely directed if for any x , y ∈ B there exists w ∈ B such that ∣w∣ ⩽ ∣x ∣ ∧ ∣ y∣.
9.3. Theorem ([11]). Let Φ be an Orlicz function that vanishes only at 0 and satisfies the
∆2(µ)-condition. Then, for every p-Amemiya norm ∥⋅∥Φ, p, 1 ⩽ p ⩽ ∞, and every closed convex subset A of the Orlicz space LΦ, p, we have:
(i) The modified best approximation problem is uniquely solvable for every x ∈ D(A).
(ii) The best approximation operator PA∶ D(A) → A is continuous.
10. Amemiya-type norms generated by outer functions
In the previous sections we presented a number of results on geometric properties of Orlicz spaces equipped with the p-Amemiya norm. As stated in the preliminaries, the p-Amemiya norm is defined with the help of two functions: the (inner) Orlicz function Φ and the outer function
sp(u) =
⎧⎪
⎪
⎨
⎪⎪
⎩ (1 + up)
1
p for 1 ⩽ p < ∞, max {1, u} for p = ∞.
The family {sp ∶ 1 ⩽ p ⩽ ∞} consists of functions that are convex and nondecreasing on (0, ∞) and that have exactly one common point (knot) at 0 (i.e. sp(0) = 1 for all 1 ⩽ p ⩽ ∞).
Moreover, the functions sp on the interval (0, ∞) are strictly increasing for 1 ⩽ p < ∞, strictly convex for 1 < p < ∞, and sp(u) < sp′(u) for every 1 ⩽ p < p′⩽ ∞.
In fact, to define an Amemiya-type norm on an Orlicz space we do not need all of the above properties of the outer function.
10.1. Proposition. Let s∶ R+→ [1, ∞) be a convex function such that max {1, u} ⩽ s(u) ⩽ 1 + u. The functional
∥x ∥Φ,s= inf
k >0
1 k
s(IΦ(k x )) is a norm on the Orlicz space LΦ.
Proof. Evidently ∥0∥Φ,s = 0. Further, ∥x ∥Φ,s⩾ ∥x ∥Φ,∞> 0 for all x ∈ LΦ∖ {0}. Since the Orlicz function Φ is even, the norm ∥⋅∥Φ,sis absolutely homogeneous. Finally, let x , y ∈ LΦ ∖ {0} and let ε > 0. We can find k , l > 0 such that 1
ks(IΦ(l x )) ⩽ ∥x ∥Φ,s + ε and
1
ls(IΦ(k y)) ⩽ ∥ y∥Φ,s+ ε. Hence, by convexity of Φ and s,
∥x + y∥Φ,s⩽ k + l
k l
s (IΦ( k l k + l
(x + y))) = k + l
k l
s (IΦ( l k + l
k x + k k + l
l y))
⩽ k + l
k l (
l k + l
s(IΦ(k x )) + k k + l
s(IΦ(l y)))
= 1 k
s(IΦ(k x )) + 1 l
s(IΦ(l y)) ⩽ ∥x ∥Φ,s+ ∥ y∥Φ,s+ 2ε.
Thus, by the arbitrariness of ε > 0, ∥⋅∥Φ,ssatisfies the triangle inequality.
For example, let us consider the family of outer functions sc(u) = max {1, u + c} for 0 ⩽ c ⩽ 1.
The norm ∥⋅∥Φ,s
1
coincides with the Orlicz norm, while ∥⋅∥Φ,s
0
is equal to the Luxemburg norm. The graphs of the family sccover the area
Ω = {(u, v) ∶ u ⩾ 0, max {1, u} ⩽ v ⩽ u + 1}
in the sense that for all (u, v) ∈ Ω there exists 0 ⩽ c ⩽ 1 such that sc(u) = v. The solution c of this equation is unique if and only if 1 < v ⩽ u + 1.
Another family of outer functions can be defined as follows
̂sc(u) =
⎧⎪
⎪
⎨
⎪⎪
⎩
(c − 1)u + 1 for 0 ⩽ u ⩽ 1, u + c − 1 for u > 1, where 1 ⩽ c ⩽ 2. Then, as above, ∥⋅∥Φ,̂s
2coincides with the Orlicz norm and ∥⋅∥Φ,̂s
1is equal to the Luxemburg norm. The graphs of the family ̂sccover the area Ω and, moreover, for each (u, v) ∈ Ω there exists unique 1 ⩽ c ⩽ 2 such that ̂sc(u) = v.
The family of p-Amemiya norms ∥⋅∥Φ, p, 1 ⩽ p ⩽ ∞, admits yet another but very important property: the Köthe dual norm to the p-Amemiya norm ∥⋅∥Φ, pis also a p-Ame- miya norm. Namely, the norm ∥⋅∥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 conjugate to the p-Amemiya norm ∥⋅∥Φ, p.
We will say that two outer functions s, σ are conjugate in the sense of Hölder if
∣∫
T
x (t) y(t)d µ∣ ⩽ ∥x ∥Φ,s⋅ ∥ y∥Ψ,σ
for all Orlicz functions Φ, Ψ complementary to each other in the sense of Young and for all measurable functions x , y.
10.2. Theorem. All outer functions s, σ satisfying the condition
u + v ⩽ s(u) ⋅ σ (v) for all u, v ⩾ 0 (2) are conjugate in the sense of Hölder.
Proof. Let Φ and Ψ be Orlicz functions complementary in the sense of Young. For all measurable functions x , y and all numbers k , l > 0, applying Young’s inequality, we obtain
∣∫
T
x (t) y(t)d µ∣ ⩽ 1 k l ∫
T
∣k x (t)∣ ∣l y(t)∣ d µ
⩽ 1 k l
(IΦ(k x ) + IΨ(l y)) ⩽ 1 k
s(IΦ(k x )) ⋅ 1 l
σ (IΨ(l y)).
Taking infimum over k and l we get Hölder’s inequality
∣∫
T
x (t) y(t)d µ∣ ⩽ ∥x ∥Φ,s∥ y∥Ψ,σ.
As immediate consequences of the above theorem we get the following corollaries.
10.3. Corollary. If s(u) = u + 1, then every outer function σ is conjugate to s in the sense of Hölder.
Proof. For every u > 0 and v ⩾ 1 we have
s(u) ⋅ σ (v) ⩾ (u + 1)v = uv + v ⩾ u + v . Further, since σ (v) ⩾ 1,
s(u) ⋅ σ (v) ⩾ u + 1 ⩾ u + v . for every u > 0 and 0 < v ⩽ 1.
10.4. Corollary. If the outer functions s, σ are conjugate in the sense of Hölder and the Orlicz functions Φ, Ψ are complementary in the sense of Young, then for every y ∈ LΨ the linear functional
fy∶ (LΦ, ∥⋅∥Φ,s) → R, fy(x ) = ∫
T
x (t) y(t)d µ is continuous. Moreover,
∥ fy∥ = sup
∥x ∥Φ,
s⩽1
∣ fy(x )∣ ⩽ inf
σ ∈Hs
∥ y∥Ψ,σ,
where Hsdenotes the family of all outer functions σ that are conjugate to s in the sense of Hölder.
So we can pose the following question: Does for any outer function s there exist an outer function σ such that
∥ fy∥ = ∥ y∥Ψ,σ
for every y ∈ EΨ? The answer is affirmative for sp(u) = (1 + up)1/p, 1 ⩽ p ⩽ ∞. In that case σ (u) = sq(u) = (1 + uq)1/q, where 1/ p + 1/q = 1.
Let us denote by Zs,σthe set of those (u, v) for which inequality (2) becomes equality, i.e.
Zs,σ = {(u, v) ∶ u, v ⩾ 0 and u + v = s(u) ⋅ σ (v)} . Note that u + v ⩾ 1 for every (u, v) ∈ Zs,σ.
10.5. Theorem. Let s, σ be outer functions conjugate in the sense of Hölder such that
(IΦ(q+(k ∣ y∣)), IΨ(k y)) ∈ Zs,σ, (3) where y ∈ LΨ∖ {0}, k ∈ {k > 0 ∶ 1
kσ (IΨ(k y)) = ∥ y∥Ψ,σ} ≠ ∅, and the Orlicz functions Φ, Ψ are complementary in the sense of Young. Then the linear functional
fy∶ (LΦ, ∥⋅∥Φ,s) → R, fy(x ) = ∫
T
x (t) y(t)d µ is continuous and ∥ fy∥ = ∥ y∥Ψ,σ.
Proof. Since (IΦ(q+(k ∣ y∣)), IΨ(k y)) ∈ Zs,σfor some k > 0 with 1
kσ (IΦ(k y)) = ∥ y∥Ψ,σ, we have IΦ(q+(k ∣ y∣)) < ∞, whence q+(k ∣ y∣) ∈ LΦ. Further, by Young’s inequality (3),
∫
T
q+(k ∣ y(t)∣)k ∣ y(t)∣ d µ = IΦ(q+(k ∣ y∣)) + IΨ(k y) > 0,
so q+(k ∣ y∣) ≠ 0, whence ∥q+(k ∣ y∣)∥Φ,s > 0. Applying again the Young inequality we obtain
∣∫
T
q+(k ∣ y(t)∣) y(t)d µ∣ = 1 k∫
T
q+(k ∣ y(t)∣)k ∣ y(t)∣ d µ = 1 k
(IΦ(q+(k ∣ y∣)) + IΨ(k y))
= 1 k
s(IΦ(q+(k ∣ y∣)) ⋅ 1 k
σ (IΨ(k y)) ⩾ ∥q+(k ∣ y∣)∥Φ,s⋅ ∥ y∥Ψ,σ.
