Regular Banach space net and abstract-valued Orlicz space of range-varying type

Abstract This paper investigates the abstract-valued Orlicz space of range-varying type. We firstly give the notions and examples of partially continuous modular net and regular Banach space net of type (II), then deal with the definitions, constructions, and geometrical properties of the range-varying Orlicz spaces, including representation of the dual L+φ $\begin{array}{} L_{+}^{\varphi} \end{array}$(I, Xθ(⋅))*, and reflexivity of Lφ(I, Xθ(⋅)), under some reasonable conditions. As an application, we finally make another approach to the real interpolation spaces constructed by a generalized Φ-function.


Introduction and preliminaries
This paper is devoted to studying the abstract-valued Orlicz space of range-varying type. Orlicz space was rstly introduced in [1]. Due to the power in dealing with the nonstandard growing phenomena, it has wide applications in many elds of applied mathematics, such as the model porus medium problem (see [2]), compressible Navier-Stokes equation (see [3]) and nonlinear obstacle problem (see [4]) etc. Roughly speaking, Orlicz space is a special type of semimodular space, where the semimodular ϱφ is commonly constructed by a generalized Φ−function φ (refer to [5, §2.3]), namely where (I, µ) is a complete measure space. Given a Banach space X, if we replace L (I, µ) with L (I, X) the collection of all strongly measurable X−valued functions, and replace |f (t)| with f (t) X for f ∈ L (I, X), then we obtain the abstract-valued Orlicz space, which was receiving a growing interesting in recent decades (cf. [6][7][8][9] etc).
We should admit that the notion Banach space net introduced in [11] has a limitation in application: It does not incorporates the following space family {L p(·) (Ω) : < p ≤ essinf x∈Ω p(x) ≤ esssup x∈Ω p(x) ≤p < ∞} (1.1) when Ω is an unbounded domain in R N , because of the restriction -X β → Xα provided α ≺ β in its de nition. In order to incorporate (1.1) into our framework, in this paper, the above hypothesis is replaced by Adapted to this change, continuity and successive assumption are revised slightly. After these modi cations, except for (1.1), a lot of space families including the complex interpolation series {[X , X ]s : s ∈ [ , ]} and the real interpolation series {(X , X )p,s : < a ≤ s ≤ b < } become regular Banach space nets. This makes the application of the range-varying function spaces much wider. In order to distinguish the notions of Banach space net de ned in [10,11] and this paper, herein after we name them type (I) and type (II) respectively.
Analogous to [11], here we also pay attention to the partially continuous semimodular net {ϱα : α ∈ A}. We will prove that, if the norms of Xα ∩ Xγ, Xα + Xγ and X β are produced by ϱα ∧ ϱγ, ϱα ∨ ϱγ and ϱ(·/ ) respectively, then under some reasonable hypotheses, every partially continuous semimodular net generates a regular Banach space net (II). This gives a bene cial supplement of that in [11].
This paper is organised as follows. In section 1, we make some reviews on pre-semimodular and semimodular, including ϱα ∧ ϱγ, a semimodular, producing an equivalent norm of Xϱ α ∩ Xϱ γ , and ϱα ∨ ϱγ, a presemimodular, producing an equivalent norm of Xϱ α +Xϱ γ . In section 2, we give notions of regular Banach space net (II) and partially continuous semimodular net, together with three useful examples, namely complex  interpolation space Section 3 is devoted to investigating the construction and geometrical properties of abstract-valued Orlicz spaces. With the aid of the associate space L φ (I, X) , we show the equivalence between the dual space L φ (I, X) * and the X * −valued function space L φ (I, X * ), i.e.
under the assumptions that φ is a locally integrable generalized Φ−function, and the dual space X * satis es the Radon-Nikodym's property. Equivalence (1.2) is a natural but not trivial extension of the corresponding result from the scalar case to the vector-valued case. Based on this extension, representation of the dual space of the range-varying Orlicz space constructed by the regular Banach space net (II) is derived, that is It is worth remarking that, representation (1.3) also holds in case that {Xα : α ∈ A} is a regular Banach space net (I). Taking into account that φ is only a locally integral generalized Φ−function, and the extra assumption that X * α is norm-attainable is dropped here, (1.2) can be viewed as an improvement of that in [11].
To illustrate the application of the range-varying Orlicz spaces, in the last section, we make another approach to the real interpolation space, where the usual p−power τ p is replaced by a generalized Φ−function φ, from which, four di erent intermediate spaces (X , X ) s,ϕ,θ,K , (X , X ) s,ϕ,θ,J , (X , X ) s,ϕ,ϑ,K and (X , X ) s,ϕ,ϑ,J are constructed. All of them are produced naturally from the range-varying Orlicz spaces, two ones are the quotient spaces, and the other two are closed subspaces. We will show that if the lower index p φ and the upper indexpφ satisfy < p φ ≤pφ < ∞, then the four intermediate spaces are mutually equivalent, i.e.
In spite that the general interpolated property of the four intermediate spaces linear operators does not remain any more, we have a weak version of the interpolation, that is max{ u s,ϕ,θ,K , u s,ϕ,θ,J , u s,ϕ,ϑ,K , u s,ϕ,ϑ,J } ≤ Cs,φ u −s u s for all u ∈ X ∩ X . In this sense, the four intermediate spaces can also be viewed as the interpolation spaces between X and X . Finally, in concrete applications, the Φ−function φ can take the form τ p(t) , τ p w(t) or τ(log( + τ)) p(t) etc.
Before the main parts of this paper, as preliminaries, let us rstly make some reviews and arguments on the semimodular and semimodular space. Let X be a complex or real linear space and ϱ : X → [ , ∞] be a convex functional with ϱ( ) = . If ϱ(λu) = ϱ(u) whenever |λ| = , and ϱ(λu) = for all λ > leads to u = , then ϱ is called a pre-semimodular. In addition, if ϱ is left-continuous, i.e.
then ϱ is called a semimodular. Furthermore, if additionally ϱ(u) = implies u = , then ϱ is said to be a modular.
Similar to the semimodular, for a pre-semimodular ϱ, the induced space is a normed linear space with the Luxemmburg norm If ϱ is a semimodular, then Xϱ is called a semimodular space on which ϱ is lower semicontinuous, and the unit ball property Proof : Taking two points u, v ∈ X and arbitrary λ ∈ [ , ], notice that for any decompositions u = u + u and v = v + v , (( − λ)u + λv ) + (( − λ)u + λv ) is a decomposition of the convex combination ( − λ)u + λv. Thus using the convexity of ϱα and ϱγ, we get the convexity of ϱα∨γ.
The above inequality shows that both u ,k ϱα and u ,k ϱγ are no more than /k. Let k → ∞, we have i.e. u = .
Recall that (refer to [5, §2.2]), given a semimodular ϱ on X with the semimodular space Xϱ, the dual functional is also a semimodular on the dual space X * ϱ , and the induced space (X * ϱ ) ϱ * is equivalent to X * ϱ . Furthermore, for the double dual ϱ ** , we have

Regular Banach space net of type (II)
De nition 2.1. Suppose that A is a topological space on which there is also de ned an order ≺. We say that the order ≺ is compatible with the topology, if for any net {α i : i ∈ I} convergent to α in A according to the topology, and α i ≺ β for all i ∈ I, one has α ≺ β de nitely. In a word, the order can be preserved through the process of convergence. Under this situation, A is called an ordered topological space. Furthermore, if for every order-bounded subset of A, its order-supremum and order-in mum are both existing, then A is called an topological lattice. For the convenience of use, throughout this paper, we always assume that A is a totally order-bounded topological lattice, or BTL in symbol. Its total order-supremum and total order-in mum are denoted by α + and α − respectively. Given a sequence and lim k→∞ α k = β are ful lled at the same time.

De nition 2.2.
Attached to the BTL A, let {Xα : α ∈ A} be a family of Banach spaces.
-We say {Xα} is a Banach space net of type (II), or BSN (II) for short, provided for all α, β, γ ∈ A, -{Xα} is called uniformly bounded if the imbedding constants of (2.1) are independent of α, β and γ.
Suppose that {α k } and {γ k } are two sequence upper-and lower-approaching β ∈ A respectively.
-If for every u ∈ X β , the limit lim holds, then {Xα} is called norm-continuous, where · β denotes the norm in X β .
means that u ∈ X β , and u β ≤ K, then {Xα} is called successive.
Finally, we say {Xα} is a regular BSN (II), if it is uniformly bounded, norm-continuous and successive simultaneously.
Remark 2.3. Previous notion of Banach space net de ned in [10,11] is called BSN (I) here.
Remark 2.4. For the sake of convenience in applications, in the coming arguments, we always assume that Thus we can de ne the dual product in (X * α ∩ X * γ ) × (Xα + Xγ) as follows Xα ∩ Xγ is densely imbedded in both Xα and Xγ, then we have (refer to [18, p. 69 Example 2.1. Let X and X be two Banach spaces embedded continuously into a topological linear space where i denotes the unit imaginary number. We know that, endowed with the norm then we de ne a norm in [X , X ]s making it be a Banach space. Taking any f ∈ F, de ne Since log M(·) is convex (cf. [19, §VI. 10]), for every s ∈ ( , ) and arbitrary δ > , we have Then by the arbitrariness of δ, we can conclude that, for all u ∈ [X , X ]s. As for the dual space, we know that (cf. [18, §1.11.3] or [20, §4.5]) if one of X j (j = , ) is re exive, then Moreover, by the density of X ∩ X in X and X , we can deduce the density of X * ∩ X * in X * and X * . Consequently, for all α ∈ [ , ] , X ∩ X and X * ∩ X * are dense in Xα and X * α respectively ([20, §4.1, §4.5]).
with the natural topology and the general order ≤, and let Xs = [X , X ]s for s ∈ ( , ), then we obtain a family of Banach space {Xs : s ∈ [ , ]}. Here spaces X j , j = , can be replaced by the interpolation spaces [X , X ] j , j = , , since the latter ones are the closed subspaces of the former ones respectively, and (refer to [21]) [ In addition, since [·, ·]s is an exact interpolation functor (cf. [21]), we have Xα ∩ Xγ → Xs, and In order to show the norm-continuous and successive properties of {Xs}, we need to prove the inverse inequalities of (2.6), (2.7). To this end, let s ⊆ [ , ], and u ∈ X ∩ X . If < α < s, then Letting α ↑ s, we obtain lim sup letting γ ↓ s, we obtain lim sup γ↓s u γ ≤ u s .
This inequality, together with (2.6), produces If u ∈ Xs, then for arbitrary ε > , there is a uε ∈ X ∩ X such that uε − u s < ε. Thus Let k → ∞, using (2.8) for the dual spaces, we get Let (X , X ) be an interpolation couple as above, and (X , X )s,q be the real interpolation space between X and X for s ∈ ( , ) and < q < ∞, i.e.
(X , X )s,q = u ∈ X + X : with the norm Here S(u) is the collection of all X ∩ X −valued functions strongly measurable in the sum space X + X and satisfying and is the equivalent norm of w ∈ X ∩ X . By [18, §1.6.1, 1.11.2] or [20, §3.3, 3.4, 3.7], we know that X ∩ X is dense in (X , X )s,q. Moreover, if X ∩ X is dense in both X and X , then (X , X ) * s,q ∼ = (X * , X * ) s,q , where /q + /q = .
Given < a < b < and < q < ∞, let A = [a, b] be the BTL as above, Xs,q = (X , X )s,q for a ≤ s ≤ b, then under all the assumptions in the previous example, the real interpolation space family {Xs,q : s ∈ [a, b]} is a regular BSN (II). Firstly, for all a ≤ α < s < γ ≤ b and < p, q, r < ∞, we have (cf. [ Notice that the equivalent constant in (2.10) is proportional to (γ − α) − /q and consequently blows up as α ↑ s and γ ↓ s, hence we could not get the unform boundedness of {Xs,q : s ∈ [a, b], < q < ∞} from (2.10). By this reason, we x the second exponent q in this example, and use the splitting method to derive the unform boundedness of {Xs,q : s ∈ [a, b]}. More precisely, for all a ≤ α ≤ s ≤ γ ≤ b, u ∈ Xs,q and f ∈ S(u), let f = fχ ( , ] , f = fχ ( ,∞) , and Obviously, f i ∈ S(u i ) and u = u + u in X + X . Since we can deduce that u ∈ Xα,q, u ∈ Xγ,q, and u α,q + u α,q ≤ u s,q, which in turn yields u Xα,q+Xγ,q ≤ u s,q . (2.11) On the other hand, if u ∈ Xα,q ∩ Xγ,q, then which implies that u s,q ≤ u Xα,q∩Xγ,q . as α k ↑ s and γ k ↓ s for u ∈ Xs,q. Moreover, using Lebesgue's convergence theorem, we have lim β→s u β,q = u s,q provided u ∈ X ∩ X . Then similar to the previews example, using the re exivity of the dual interpolation spaces, and the density of X ∩X in X and X , we can derive the norm-continuity and the successive property of {Xs,q : s ∈ [a, b]}.
Remark 2.7. Unlike the continuous semimodular net, the dual semimodular family of a partially semimodular net is no longer a semimodular net in general.
Proof : For each splitting u = u + u of u ∈ Xϱ β , by the de nition of dual semimodular, we have Taking in mum over the set of all the splitting u = u + u , we obtain (2.15). Equality (2.16) is a straight consequence of (2.15).

Orlicz space of range-varying type
Let I = ( , b] for some < b < ∞ or I = ( , ∞), on which there is a complete and regular Borel measure µ, and let Λ(I) be the collection of all the bounded and closed subinterval of I. Suppose that A is a BTL, and θ : I → A is an order-continuous map, that is for any nest of intervals {J k ∈ Λ(I) : k = , , · · · } shrinking to t, limits lim always hold simultaneously, where θ − J = inf t∈J θ(t) and θ + J = sup t∈J θ(t) according to the order. Given a regular BSN (II) {Xα : α ∈ A}, de ne and L (I, X θ(·) ) = {f ∈ L − (I, X θ(·) ) : f (t) ∈ X θ(t) for a.e. t ∈ I}.
Obviously, both L − (I, X θ(·) ) and L (I, X θ(·) ) are linear spaces according to the sum and scalar multiplication of abstract valued functions.
Using the norm-continuity of {Xα : α ∈ A} and the order-continuity of θ, we can prove that (cf. [10] for a proof of the similar result) Proposition 3.1. For all u ∈ L (I, X θ(·) ), the norm function t → u(t) θ(t) is measurable.
For each k ∈ N, divide I into k equal parts if I = ( , b], or in nite many equal parts with the length / k of each part if I = ( , ∞). Denote by t k,j = jb/ k and J k,j = (t k,j , t k,j+ ] for j = , · · · , k − , if I = ( , b], or t k,j = j/ k and J k,j = (t k,j , t k,j+ ] for j = , , · · · , if I = ( , ∞). Let for t ∈ J k,j and j = , · · · , k − , or j = , , · · · , then we obtain two step functions. Obviously for all t ∈ I since θ is order-continuous. Set then we can de ne two function spaces L φ (I k ,X θ k (·) ) and L φ (I k ,X θ k (·) ) as the semimodular spaces derived from L (I k ,X θ k (·) ) and L (I k ,X θ k (·) ) by the semimodular (3.1) with X θ (t) replaced byX θ k (t) andX θ k (t) respectively. By the uniform boundedness and successive property of {Xα}, adjoint with the monotonicity of φ, we can derive that Theorem 3.3. For all k ∈ N, the following imbeddings hold. Moreover, if f ∈ L φ (I k ,X θ k (·) ) for all k ∈ N, and then f ∈ L φ (I, X θ(·) ), and f L φ (I,X θ(·) ) ≤ C.
The above two results are much similar to those obtained in [11], and here we omit the whole proofs.
Let X be a Banach space, and L φ (I, X) be the abstract-valued Orlicz space of range-xed type. De ne the associate function space with the norm One can easily check that, according to · L φ (I,X) , L φ (I, X) becomes a Banach spaces, and the following equality de nes a linear imbedding map T : L φ (I, X) → L φ (I, X) * with where L φ (I, X) * represents the dual space of L φ (I, X). For the relation between L φ (I, X) and L φ (I, X * ), we have L φ (I, X * ) → L φ (I, X) with the estimates Here φ (t, ·) stands for the conjugate function of φ(t, ·), i.e.
Evidently, {ω j } is an increasing sequence of nonnegative and measurable functions, and for a.e. t ∈ I. Moreover for every j ∈ N + , there is correspondingly a nonnegative scalar simple function s j with supps j ⊂ I j , such that ω j (t) = s j (t) ξ (t) X * − φ(t, s j (t)).
Since φ is locally integrable, for every λ > , we have I φ(t, λ − s j (t))dµ < ∞. Fix < λ < / , then by the absolute convergence of the integral, there is δ j > , such that D φ(t, λ − s j (t))dµ < /j for all measurable subsets D of I j with µ(D) < δ j . For each j ∈ N + , by the Egrov's theorem, there is a measurable set E j ⊆ I j with µ(I j \E j ) < δ j such that η k (t) → ξ (t) in X * uniformly on E j as k → ∞. Thus for su ciently large integer k j , we have Notice that η k j (t) takes only nite many values in X * , so there is another function w j ∈ Sφ(I, X) satisfying w j (t) X = and Let f j = s j w j , then we obtain a member of S(I j , X) satisfying Consequently, Let j → ∞, and use (3.4), we obtain Thus ξ ∈ L φ (I, X * ) and Φ φ (η) ≤ by the arbitrariness of λ ∈ ( , / ). Finally by the scaling arguments, we reach the desired conclusion.
Remark 3.6. Di erent to the scalar case (please compare to [5,Theorem 2.7.4]), for the function ξ ∈ L φ (I, X) , we could not nd a sequence of X−valued simple functions, say {h k }, verifying unless X is separable. To derive the inclusion ξ ∈ L φ (I, X * ), we introduce the multiplier λ ∈ ( , ), along which, the absolute convergence of the integral, and the Egrov's theorem are applied together. Due to these di erences, Theorem 3.5 is not a parallel extension of [5,Theorem 2.7.4] from the scalar case to the vectorvalued case. Proof : It su ces to show that for every Ξ ∈ L φ (I, X) * , there is only one function ξ ∈ L φ (I, X) such that Tξ = Ξ in the sense of (3.3). If Ξ = , then take ξ = and there is nothing to do. If Ξ ≠ , then the proof can be made by the scaling arguments. So we can assume Ξ L φ (I,X) * = . Since Sφ(I) ⊆ L φ (I), for every compact subset E of I and u ∈ X, the function χ E u belongs to L φ (I, X), and Fix k ∈ N, and consider the X * −valued function µ k : acting on the collection of all measurable subsets of I k . By (3.5), we can easily show that µ k is a totally bounded X * −valued measure on I k with the total variation no more than Ξ L φ (I,X) * χ I k L φ (I) . Hence under the Radon-Nikodym's assumption of X * , we can nd a unique function ξ k ∈ L (I k , X * ) satisfying for all measurable subsets E of I k . By the uniqueness of ξ k , it is easy to check that ξ k+ (t) = ξ k (t) a.e. on I k . So if we let ξ (t) = ξn(t) for t ∈ In, then we obtain a strongly measurable X * −valued function on I satisfying for the function f = uχ E with E compact and u ∈ X. By the linearity of Ξ and the integration, we can easily check that (3.6) is also satis ed for all f ∈ Sφ(I, X). As for f ∈ L φ (I, X), there exits a sequence {w k } in S(I, X) such that w k (t) → f (t) in X and w k (t) X ≤ f (t) X a.e. on I. Letw k = w k χ I k sgn( ξ , w k ), thenw k is also a simple function satisfyingw k ∈ Sφ(I, X), w k L φ (I,X) ≤ f L φ (I,X) , and ξ (t),w k (t) → | ξ (t), f | a.e. on I as k → ∞. Thus usingw k to replace f in (3.6), and letting k → ∞, we have Therefore ξ (t) ∈ L φ (I, X) , Tξ = Ξ and ξ L φ (I,X) ≤ Ξ L φ (I,X) * .
Then the function space L φ (I, X θ(·) ) is re exive. Remark 3.12. It is easy to check that Theorem 3.10 and 3.11 still hold respectively for the space L φ + (I, X θ(·) ) * and L φ (I, X θ(·) ) in case that {Xα : α ∈ A} is a dense, regular BSN of type (I). Compare to [11,Theorem 3.12], here φ is merely a general generalized Φ−function with local integrability assumption, and the extra hypothesis that X * α is norm-attainable for every α ∈ A is no longer needed. In this sense, Theorem 3.10 and 3.11 can be viewed as the improvements of Theorem 3.12 and Corollary 3.15 in [11] respectively.

Application in real interpolation spaces
Given an interpolation couple (X , X ) as in Example 2.2, suppose that X ∩ X is dense in X i , i = , . For each t ∈ ( , ∞), letX t = X ∩ X ,X * t = X * ∩ X * endowed with the norms u X t = J(t, u), ξ X* t = J * (t, ξ ), and let X t = X + X ,X * t = X * + X * endowed with the norm u X t = K(t, u), ξ X* , and the latter two ones are dual nets of the former two ones respectively. Let I = ( , ∞), dµ = dt/t and θ(t) = t, ϑ(t) = t − , then we obtain a complete and regular Borel measure, two order-continuous maps, attached to which, four function spaces: L (I,X θ(·) ), L (I,X ϑ(·) ), L (I,X θ(·) ), L (I,X ϑ(·) ), and other four ones for the dual couple (X * , X * ) are well de ned. All of them are equal to the strongly measurable ones, for example, L (I,X θ(·) ) = L + (I,X θ(·) ).
Based on these properties, we can deduce that wherep φ and p φ are the conjugate exponents ofpφ and p φ respectively.