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Order Continuous Seminorms and Weak Compactness in Orlicz Spaces

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Order Continuous Seminorms and Weak Compactness in Orlicz Spaces Collect. Math. 44 (1993), 217–236 c 1994 Universitat de Barcelona Order continuous seminorms and weak compactness in Orlicz spaces Marian Nowak Institute of Mathematics, Adam Mickiewicz University, Matejki 48/49, 60-769 Poznan,´ Poland Abstract Let Lϕ be an Orlicz space defined by a Youngfunction ϕ over a σ-finite measure ∗ space, and let ϕ denote the complementary function in the sense of Young. We ∗ ∗ give a characterization of the Mackey topology τ(L ,Lϕ ) in terms of some family of norms defined by some regular Young functions. Next, we describe order continuous (= absolutely continuous) Riesz seminorms on Lϕ, and obtain ∗ a criterion for relative σ(Lϕ,Lϕ )-compactness in Lϕ. As an application we get a representation of Lϕ as the union of some family of other Orlicz spaces. Finally, we apply the above results to the theory of Lebesgue spaces. 0. Introduction and preliminaries 1 In 1915 de la Vall´ee Poussin (see [12]) showed that a set Z of L for a 1 finite measure space (Ω, Σ,µ) has uniformly absolutely continuous L -norms (i.e., lim (sup |x(t)|dµ) = 0) iff there exists a Young function ψ such that → µ(E) 0 x∈Z E lim ψ(u)/u = ∞ in terms of which sup ψ(|x(t)|)dµ < ∞. u→∞ x∈Z Ω On the other hand, in view of the Dunford-Pettis criterion (on relatively com- pact sets in L1)(see [3, p. 294]) the set Z ⊂ L1 has uniformly absolutely continuous L1-norms iff it is relatively σ(L1,L∞)-compact. Thus we have the following criterion for relative weak compactness in L1 (for 1 1 ∞ finite measures): a set Z of L is relatively σ(L ,L )-compact iff there exists a Young function ψ such that lim ψ(u)/u = ∞ and sup ψ(|x(t)|)dµ < ∞. u→∞ x∈Z Ω 217 218 Nowak ∗ In 1962 T. Ando [2, Theorem 2] found similar criterion for relative σ(Lϕ,Lϕ )- compactness in Lϕ for ϕ being an N-function and a finite measure. This criterion was extended by the present author to the case of σ-finite measures ([15, Theorem 1.2]). In this paper, using a different method, we extend the Ando’s criterion to the case of ϕ belonging to a much wider class of Young functions and σ-finite measures. We can include Lϕ being equal to L1 + L∞ (so L1 if µ(Ω) < ∞), L1 + Lp,Lp + L∞ (p>1). In section 1, making use of the author’s results concerning the so-called modular T ∧ ϕ topology ϕ on L (see [13], [14], [18], [19]), we obtain a characterization of the ∗ Mackey topology τ(Lϕ,Lϕ ) in terms of some family of norms defined by some regular Young functions, dependent on ϕ (see Theorem 1.5). As an application we have a description of absolutely continuous (= order continuous) Riesz seminorms on Lϕ (see Corollary 1.6). ∗ In section 2, in view of the close connection between relative σ(Lϕ,Lϕ )- ∗ compactness in Lϕ and the absolute continuity of some seminorm in Lϕ , we can ∗ describe relatively σ(Lϕ,Lϕ )-compact sets in Lϕ as norm bounded subsets of an Orlicz space Lψ for some regular Young function ψ (see Theorem 2.4). As an appli- cation we get a representation of the Orlicz space Lϕ as the union of some family of ∗ other Orlicz spaces. At last, we examine the absolute weak topology |σ|(Lϕ,Lϕ ). In section 3 we apply the results of sections 1 and 2 to the theory of Lebesgue spaces. For notation and terminology concerning Riesz spaces we refer to [1], [21]. As usual, N stands for the set of all natural numbers. Let (Ω, Σ,µ)beσ-finite measure space, and let L0 denote the set of equivalence classes of all real valued measurable functions defined and a.e. finite on Ω. Then L0 is a super Dedekind complete Riesz space under the ordering x ≤ y whenever x(t) ≤ y(t) a.e. on Ω. The Riesz F -norm | | x(t) 0 x0 = f(t)dµ for x ∈ L , Ω 1+|x(t)| where f:Ω → (0, ∞) is measurable and f(t)dµ = 1, determines the Lebesgue Ω 0 topology T0 on L , which generates the convergence in measure on subsets of Ω 0 of finite measure. For a sequence (xn)inL we will write xn → x(µ) whenever xn − x0 → 0. 0 For a subset A of Ωand x ∈ L we will write xA = x · χA, where χA stands for the characteristic function of A. We will write En ∅ if (En) is a decreasing Order continuous seminorms and weak compactness in Orlicz spaces 219 sequence of measurable subsets of Ωsuch that µ(En ∩ E) → 0 for every set E ⊂ Ω of finite measure. Now we recall some notation and terminology concerning Orlicz spaces (see [6], [8], [10], [20] for more details). By an Orlicz function we mean a function ϕ:[0, ∞) → [0, ∞] which is non- decreasing, left continuous, continuous at 0 with ϕ(0) = 0, not identically equal to 0. An Orlicz function ϕ is called convex, whenever ϕ(αu + βv) ≤ αϕ(u)+βϕ(v) for α, β ≥ 0,α+ β = 1 and u, v ≥ 0. A convex Orlicz function is usually called a Young function. For a Young function ϕ we denote by ϕ∗ the function complementary to ϕ in the sense of Young, i.e., ϕ∗(v) = sup uv − ϕ(u): u ≥ 0 for v ≥ 0. For a set Ψ of Young functions we will write Ψ∗ = {ψ∗: ψ ∈ Ψ} . We shall say that an Orlicz function ψ is completely weaker than another ϕ a s for all u (resp. for small u; resp. for large u), in symbols ψ ϕ(resp. ψ ϕ; l resp. ψ ϕ), if for an arbitrary c>1 there exists a constant d>0 such that ψ(cu) ≤ dϕ(u) for u ≥ 0 (resp. for 0 ≤ u ≤ u0; resp. for u ≥ u0 ≥ 0). (See [2], [20, Ch. II]). It is seen that ϕ satisfies the so called ∆2-condition for all u (resp. for small u; a s l resp. for large u) if and only if ϕ ϕ(resp. ϕ ϕ; resp. ϕ ϕ). We shall say that an Orlicz function ϕ increases more rapidly than another ψ a s for all u (resp. for small u; resp. for large u) in symbols ψ ≺− ϕ (resp. ψ ≺− ϕ; resp. l ≺− ≤ 1 ψ ϕ), if for an arbitrary c>0 there exists d>0 such that cψ(u) d ϕ(du) for all u ≥ 0 (resp. for 0 ≤ u ≤ u0; resp. for u ≥ uo ≥ 0). Note that ϕ satisfies the so called ∇2-condition for all u (resp. for small u; resp. a s l for large u) if and only if ϕ ≺− ϕ (resp. ϕ ≺− ϕ; resp. ϕ ≺− ϕ). a One can verify that for given Young functions ψ and ϕ the relation ψ ϕ(resp. s l a s l ψ ϕ; resp. ψ ϕ) holds iff ϕ∗ ≺− ψ∗ (resp. ϕ∗ ≺− ϕ∗, resp. ϕ∗ ≺− ψ∗) holds (see [2], [20, Proposition 2.2.4]). 220 Nowak An Orlicz function ϕ determines the functional m : L0 → [0, ∞]by ϕ mϕ(x)= ϕ(|x(t)|)dµ. Ω The Orlicz space generated by ϕ is the ideal of L0 defined by ϕ 0 L = {x ∈ L : mϕ(λx) < ∞ for some λ>0}. ϕ The functional mϕ restricted to L is an orthogonally additive semimodular (see [10], [11]). ϕ L can be equipped with the complete metrizable linear topology Tϕ of the Riesz F -norm x x = inf λ>0: m ≤ λ . ϕ ϕ λ Moreover, when ϕ is a Young function, the topology Tϕ can be generated by two Riesz norms (called the Orlicz and the Luxemburg norms resp.) defined as follows: ϕ∗ xϕ = sup |x(t)y(t)|dµ: y ∈ L ,mϕ∗ (y) ≤ 1 Ω x |||x||| = inf λ>0: m ( ) ≤ 1 . ϕ ϕ λ For an Orlicz function ϕ let ϕ 0 E = {x ∈ L : mϕ(λx) < ∞ for all λ>0} and ϕ { ∈ ϕ → ∅} La = x L : xEn ϕ 0asEn . It is well known that for ϕ taking only finite values these spaces coincide, i.e., ϕ ϕ E = La . ∗ 1. The Mackey topology τ(Lϕ,Lϕ ) First we recall the definition and the basic properties of the so-called modular topol- ogy on Orlicz spaces (see [13], [14]). Let ϕ be an Orlicz function vanishing only at 0. For given ε>0, let Uϕ(ε)= ϕ {x ∈ L : mϕ(x) ≤ ε}. Then the family of all sets of the form ∞ N Uϕ(εn) N=1 n=1 where (εn) is a sequence of positive numbers, forms a base of neighborhoods of 0 for a linear topology on Lϕ, that will be called the modular topology on Lϕ and will be T ∧ denoted by ϕ . T ∧ The basic properties of ϕ are included in the following theorem (see [14, The- orem 1.1], [18, Theorem 2.2], [19. Theorem 4.2]): Order continuous seminorms and weak compactness in Orlicz spaces 221 Theorem 1.1 Let ϕ be an Orlicz function vanishing only at 0. Then the following statements hold: T ∧ ϕ (i) ϕ is the finest σ-Lebesgue topology on L . T ∧ ⊂T T ∧ T ∈ (ii) ϕ ϕ and the equality ϕ = ϕ holds whenever ϕ ∆2. T ∧ ϕ ϕ∗ (iii) ϕ coincides with the Mackey topology τ(L ,L ), whenever ϕ is a Young function. To present the crucial for this paper characterization of the modular topology T ∧ ϕ we will distinguish some classes of Orlicz functions.
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