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

1 Introduction and preliminaries

This paper is devoted to studying the abstract-valued Orlicz space of range-varying type. Orlicz space was firstly introduced in [1]. Due to the power in dealing with the nonstandard growing phenomena, it has wide applications in many fields 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 ϱ_φ 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 L⁰(I, μ) with L⁰(I, X) the collection of all strongly measurable X-valued functions, and replace |f(t)| with ∥f(t)∥_X for f ∈ L⁰(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).

Here we focus on a special type of the abstract-valued Orlicz space, whose members have a varying range. This new type of function space was firstly introduced in [10], and later studied in [11, 12, 13]. As the value space varies as t changes, it is crucial to give a suitable description of measurability for the functions of this type. In [10], the authors introduced the notions of totally bounded topological lattice 𝓐, regular Banach space net {X_α : α ∈ 𝓐}, and order-continuous map θ : I → 𝓐. Based on these notions, they introduced a suitably measurable X_θ(⋅)-valued function space L⁰(I, X_θ(⋅)) on an interval I. Functions in this space have a common property, that is the norm function t ↦ ∥f(t)∥_Xθ(⋅) is measurable. There are three types of subspaces of L⁰(I, X_θ(⋅)) according to their constructions: continuous type C⁻(I, X_θ(⋅)) (cf. [10]), norm-modular type L^p(⋅)(I, X_θ(⋅)), L+p(⋅) (I, X_θ(⋅)) (cf. [11]), and pure modular type L^ϱ_θ(⋅)(I, X_θ(⋅)) (cf. [12]). Each of which owns useful examples, such as C⁻(I, (X, D(A))_{1−1/p(⋅),p(⋅)}), trace of the maximal regular space W01,p(⋅) (I, X) ∩ L^1,p(⋅)(I, D(A)) associated with a sectorial operator A, raised in [10] and treated in [13], Lebesgue space L^p(x,t)(I × Ω) with the double variable exponent, and the anisotropic space {u ∈ L²(I, L²(Ω)) : ∂_xiu(t) ∈ L^{p_i(⋅, t)}(Ω)}, used in [14, 15, 16, 17] to deal with the anisotropic parabolic equations.

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

when Ω is an unbounded domain in ℝ^N, because of the restriction

1. X_β ↪ X_α provided α ≺ β

in its definition. In order to incorporate (1.1) into our framework, in this paper, the above hypothesis is replaced by

1. X_α ∩ X_γ ↪ X_β ↪ X_α + X_γ whenever α ≺ β ≺ γ.

Adapted to this change, continuity and successive assumption are revised slightly. After these modifications, except for (1.1), a lot of space families including the complex interpolation series {[X₀, X₁]_s : s ∈ [0, 1]} and the real interpolation series {(X₀, X₁)_p,s : 0 < a ≤ s ≤ b < 1} 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 defined 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 {ϱ_α : α ∈ 𝓐}. We will prove that, if the norms of X_α ∩ X_γ, X_α + X_γ and X_β are produced by ϱ_α ∧ ϱ_γ, ϱ_α ∨ ϱ_γ and 2ϱ(⋅/2) respectively, then under some reasonable hypotheses, every partially continuous semimodular net generates a regular Banach space net (II). This gives a beneficial supplement of that in [11].

This paper is organised as follows. In section 1, we make some reviews on pre-semimodular and semimodular, including ϱ_α ∧ ϱ_γ, a semimodular, producing an equivalent norm of X_ϱα ∩ X_ϱγ, and ϱ_α ∨ ϱ_γ, a pre-semimodular, producing an equivalent norm of X_ϱα + X_ϱγ. 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 scale {[X₀, X₁]_s : s ∈ [0, 1]}, real interpolation space scale {(X₀, X₁)_s,q : s ∈ [a, b]} (0 < a < b < 1, 1 < q < ∞), and {L^p(⋅)(Ω) : p ∈ 𝓟₀(Ω)} on a unbounded open set Ω ⊆ ℝ^N.

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^* satisfies 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 {X_α : 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 Xα∗ 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 different intermediate spaces (X₀, X₁)_s,ϕ,θ,K, (X₀, X₁)_s,ϕ,θ,J, (X₀, X₁)_s,ϕ,ϑ,K and (X₀, X₁)_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 index p̄_φ satisfy 1 < p_φ ≤ p̄_φ < ∞, then the four intermediate spaces are mutually equivalent, i.e.

Moreover, for the dual space, we have

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

for all u ∈ X₀ ∩ X₀. In this sense, the four intermediate spaces can also be viewed as the interpolation spaces between X₀ and X₁. Finally, in concrete applications, the Φ-function φ can take the form τ^p(t), τ^pw(t) or τ(log(1 + τ))^p(t) etc.

Before the main parts of this paper, as preliminaries, let us firstly make some reviews and arguments on the semimodular and semimodular space. Let X be a complex or real linear space and ϱ : X → [0, ∞] be a convex functional with ϱ(0) = 0. If ϱ(λ u) = ϱ(u) whenever |λ| = 1, and ϱ(λ u) = 0 for all λ > 0 leads to u = 0, 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) = 0 implies u = 0, 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 X_ϱ is called a semimodular space on which ϱ is lower semicontinuous, and the unit ball property

holds (refer to [5, § 2.1]).

Recall that (refer to [5, § 2.2]), given a semimodular ϱ on X with the semimodular space X_ϱ, the dual functional

is also a semimodular on the dual space Xϱ∗ , and the induced space (Xϱ∗)ϱ∗ is equivalent to Xϱ∗ . Furthermore, for the double dual ϱ^**, we have

2 Regular Banach space net of type (II)

Taking any f ∈ 𝓕, define

Since log M(⋅) is convex (cf. [19, § VI. 10]), for every s ∈ (0, 1) and arbitrary δ > 0, we have

Then by the arbitrariness of δ, we can conclude that,

for all u ∈ [X₀, X₁]_s.

As for the dual space, we know that (cf. [18, § 1.11.3] or [20, § 4.5]) if one of X_j (j = 0, 1) is reflexive, then

Moreover, by the density of X₀ ∩ X₁ in X₀ and X₁, we can deduce the density of X0∗ ∩ X1∗ in X0∗ and X1∗ . Consequently, for all α ∈ [0, 1], X₀ ∩ X₁ and X0∗ ∩ X1∗ are dense in X_α and Xα∗ respectively ([20, § 4.1, § 4.5]).

Let 𝓐 = [0, 1] with the natural topology and the general order ≤, and let X_s = [X₀, X₁]_s for s ∈ (0, 1), then we obtain a family of Banach space {X_s : s ∈ [0, 1]}. Here spaces X_j, j = 0, 1 can be replaced by the interpolation spaces [X₀, X₁]_j, j = 0, 1, since the latter ones are the closed subspaces of the former ones respectively, and (refer to [21])

for all s ∈ [0, 1]. Suppose that both of X₀ and X₁ are reflexive, then {X_s : s ∈ [0, 1]} is a regular BSN of type (II).

Firstly, for all α, s, γ ∈ [0, 1], α ≤ s ≤ γ, we have

with the equal norms (cf. [20, § 4.6]). This fact, together with (2.5), leads to the inlusion X_s ↪ X_α + X_γ, and

In addition, since [⋅, ⋅]_s is an exact interpolation functor (cf. [21]), we have X_α ∩ X_γ ↪ X_s, and

where λ ∈ [0, 1] making s = (1 − λ)α + λγ. Therefore {X_s} is a uniformly bounded BSN, and

if α_k ↑ β and γ_k ↓ β.

In order to show the norm-continuous and successive properties of {X_s}, we need to prove the inverse inequalities of (2.6), (2.7). To this end, let s ⊆ [0, 1], and u ∈ X₀ ∩ X₁. If 0 < α < s, then

Letting α ↑ s, we obtain

Similarly, if s < γ < 1, then

letting γ ↓ s, we obtain

Therefore, for the sequences α_k ↑ β and γ_k ↓ β, we have

which, combined with (2.7), leads to the equality

Now we can prove the norm-continuity of {X_α}. If u ∈ X₀ ∩ X₁, then using (2.4) and (2.8) for the dual spaces, we have

This inequality, together with (2.6), produces

If u ∈ X_s, then for arbitrary ε > 0, there is a u_ε ∈ X₀ ∩ X₁ such that ∥u_ε − u∥_s < ε. Thus

which yields (2.9) for u ∈ X_s.

Finally, suppose that u ∈ ⋂k=1∞ (X_αk + X_γk) satisfying (2.3) for some K > 0. Then for all ξ ∈ X0∗ ∩ X1∗ , we have

Let k → ∞, using (2.8) for the dual spaces, we get

Since X0∗ ∩ X1∗ is dense in Xs∗ , we have u ∈ X_s, and ∥u∥_s ≤ K. This proves the successive property of {X_α}. Putting all the properties together, we conclude that {X_α} is a regular BSN (II).

Here 𝓢(u) is the collection of all X₀ ∩ X₁-valued functions strongly measurable in the sum space X₀ + X₁ and satisfying

and

is the equivalent norm of w ∈ X₀ ∩ X₁.

By [18, § 1.6.1, 1.11.2] or [20, § 3.3, 3.4, 3.7], we know that X₀ ∩ X₁ is dense in (X₀, X₁)_s,q. Moreover, if X₀ ∩ X₁ is dense in both X₀ and X₁, then

where 1/q + 1/q′ = 1.

Given 0 < a < b < 1 and 1 < q < ∞, let 𝓐 = [a, b] be the BTL as above, X_s,q = (X₀, X₁)_s,q for a ≤ s ≤ b, then under all the assumptions in the previous example, the real interpolation space family {X_s,q : s ∈ [a, b]} is a regular BSN (II). Firstly, for all a ≤ α < s < γ ≤ b and 1 < p, q, r < ∞, we have (cf. [18, § 1.10.2], [20, § 3.5]) or [22, Theorem 7.21])

where 0 < λ < 1, and (1 − λ)α + λγ = s. This infers that {X_s,q : s ∈ [a, b]} is a BSN (II).

Notice that the equivalent constant in (2.10) is proportional to (γ − α)^−1/q′ and consequently blows up as α ↑ s and γ ↓ s, hence we could not get the unform boundedness of {X_s,q : s ∈ [a, b], 1 < q < ∞} from (2.10). By this reason, we fix the second exponent q in this example, and use the splitting method to derive the unform boundedness of {X_s,q : s ∈ [a, b]}. More precisely, for all a ≤ α ≤ s ≤ γ ≤ b, u ∈ X_s,q and f ∈ 𝓢(u), let f₁ = fχ_(0,1], f₂ = fχ_(1,∞), and

Obviously, f_i ∈ 𝓢(u_i) and u = u₁ + u₂ in X₀ + X₁. Since

and

we can deduce that u₁ ∈ X_α,q, u₂ ∈ X_γ,q, and ∥u₁∥_α,q + ∥u₂∥_α,q ≤ ∥u∥_s,q, which in turn yields

On the other hand, if u ∈ X_α,q ∩ X_γ,q, then

which implies that

Inequalities (2.11) and (2.12) jointly show the uniform boundedness of {X_s,q : s ∈ [a, b]} and

as α_k ↑ s and γ_k ↓ s for u ∈ X_s,q. Moreover, using Lebesgue’s convergence theorem, we have

provided u ∈ X₀ ∩ X₁. Then similar to the previews example, using the reflexivity of the dual interpolation spaces, and the density of X₀ ∩ X₁ in X₀ and X₁, we can derive the norm-continuity and the successive property of {X_s,q : s ∈ [a, b]}.

In the sequel, we will use ∥u∥_ϱα∨γ as the norm of X_ϱα + X_ϱγ, and use 2ϱ_β(u/2) to produce the norm of X_ϱβ. We also assume that the space nets {X_ϱα} and { Xϱα∗ } are compatible, i.e. for all α, β, γ ∈ 𝓐, α ≺ β ≺ γ, and all ξ ∈ Xϱα∗ ∩ Xϱγ∗ , all u ∈ X_β, the dual products 〈ξ,u〉(Xϱα∗∩Xϱγ∗)×(Xϱα+Xϱγ) and 〈ξ,u〉Xϱβ∗×Xϱβ are equal. Due to these conventions, in the following arguments, we will omit the subscript and only use 〈ξ, u〉 to denote the dual product between ξ and u.