Weak compactness and representation in variable exponent Lebesgue spaces on infinite measure spaces

Relative weakly compact sets and weak convergence in variable exponent Lebesgue spaces Lp(·)(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L^{p(\cdot )}(\Omega )}$$\end{document} for infinite measure spaces (Ω,μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\Omega ,\mu )$$\end{document} are characterized. Criteria recently obtained in [14] for finite measures are here extended to the infinite measure case. In particular, it is showed that the inclusions between variable exponent Lebesgue spaces for infinite measures are never L-weakly compact. A lattice isometric representation of Lp(·)(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L^{p(\cdot )}(\Omega )}$$\end{document} as a variable exponent space Lq(·)(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{q(\cdot )}(0,1)$$\end{document} is given.

compactness criteria were given by Andô in [1] (cf. [21] chapter 4). Later on, many extensions have been given for symmetric function spaces (see f.i. [8] and references within) and also for the vectorial case of Orlicz-Bochner spaces ( [20]). In the non-symmetric setting, the weak compactness in variable exponent Lebesgue spaces L p(·) ( ) for finite measure spaces ( , μ) has been studied recently in [14], obtaining suitable criteria of weak compactness and weak convergence.
In this paper we broaden the study of weak compactness in variable exponent Lebesgue spaces L p(·) ( ). One of our goals is to find suitable weak compactness criteria in L p(·) ( ) for infinite measure spaces ( , μ) extending results obtained in [14] for the case of finite measure. We analyze possible versions of Andô weak compactness characterizations given for Orlicz spaces L ϕ ( ) ( [1]). Another goal in this paper is to find lattice isometric representations of variable exponent Lebesgue spaces L p(·) ( ) over infinite measure spaces ( , μ) as a suitable variable exponent Lebesgue space L q(·) (0, 1) on the probability measure space (0, 1) with the Lebesgue measure. This kind of representations are a useful tool in the study of the structure of general variable exponent Lebesgue spaces.
Often, to lift properties from function spaces over probability measure spaces to the framework of function spaces on infinite measure spaces requires a careful study and additional conditions are usually needed. For example, while for finite measures the L p(·)equi-integrability is equivalent to the uniform integrability, in the case of infinite measures the additional condition of uniform decay at infinity (i.e. for every ε > 0 there exists A ⊂ with μ(A) < ∞ such that sup f ∈S f χ \A < ε) is required for the equivalence (see Proposition 3.1). Thus, the Dunford-Pettis Theorem in L 1 ( ) for σ -finite measure spaces ( , μ) is stated as follows (cf. [11] Theorem IV.8.9):

Theorem 1.1 ([10]) A bounded subset S ⊂ L 1 ( ) is relatively weakly compact if and only if it is equi-integrable if and only if it is uniformly integrable and decays uniformly at infinity.
The paper is organized as follows. After a preliminary section, in Sect. 3 we give a characterization of L p(·) ( )-equi-integrability for infinite measure spaces ( , μ). It is proved that all inclusion operators between L p(·) ( ) spaces for infinite measures are not L-weakly compact (Theorem 3.4). Section 4 is devoted to the study of weak compactness in variable exponent Lebesgue spaces L p(·) (0, ∞) on the real interval (0, ∞). We find suitable weak compactness criteria extending those given in [14] for finite measures. To prove it, we make use of an isometric lattice representation of the spaces L p(·) (0, ∞) as a suitable L q(·) (0, 1) space on the interval (0, 1). From this it follows easily that L p(·) (0, ∞) is weakly Banach-Saks if and only if p + < ∞.
Section 5 is devoted to variable exponent Lebesgue spaces L p(·) ( ) for abstract σ -finite measure spaces ( , μ). Firstly, we give a lattice isometric representation of L p(·) ( ) over non-atomic separable σ -finite spaces as a space L p(·) (0, ∞) for a suitable exponent function p(·) (see Theorem 5.4). This useful representation theorem is obtained using the classical Carathéodory Theorem on algebras of measure (cf. [22] page 399). In particular, it follows that every space L p(·) ( ) for a non-atomic separable σ -finite measure space can be lattice represented as a certain variable exponent space L q(·) (0, 1) on the interval (0, 1). This remarkable fact is not true for the class of Orlicz spaces (see Remark 5.9). Finally, as a consequence of the above representation and previous results in Sect. 4, suitable criteria for weak compactness and weak convergence in L p(·) ( ) are obtained. Namely, S ⊂ L p(·) ( ) is relatively weakly compact if and only if it is bounded, We point out that no extra conditions on the regularity of the exponent functions (like the log-Hölder continuous conditions) will be required along the paper.

Preliminaries
Throughout the paper, ( , , μ) is a non-atomic σ -finite measurable space (unless specified otherwise) and L 0 ( ) is the space of all real μ-measurable function classes Given a μ-measurable function p : → [1, ∞) (called exponent function), the variable exponent Lebesgue space (or Nakano space) L p(·) ( ) is defined to be the set of all real measurable function classes such that the modular ρ p(·) ( f /r ) is finite for some r > 0, where The associated Luxemburg norm is defined as With the usual pointwise order, L p(·) ( ), · p(·) is a Banach function space. The case of p(·) = p constant will be denoted by L p ( ). Let p − := ess inf{ p(t) : t ∈ } and p + := ess sup{ p(t) : t ∈ }. By p + |A and p − |A we denote the essential supremum and infimum of the function p(·) over a measurable subset A ⊂ . The conjugate function p * (·) of p(·) is defined by the equation 1 p(t) + 1 p * (t) = 1 almost everywhere t ∈ . Thus, for p + < ∞ the topological dual of the space L p(·) ( ) is the variable exponent space L p * (·) ( ).
A L p(·) ( ) space is separable if and only if p + < ∞ or, equivalently, if and only if L p(·) ( ) contains no isomorphic copy of ∞ . In the sequel, only separable variable exponent Lebesgue spaces L p(·) ( ) will be considered (note that p * (·) is bounded if and only if p − > 1, so we can consider non separable dual spaces L p * (·) ( )). An space L p(·) ( ) is reflexive if and only if 1 < p − ≤ p + < ∞. This is also equivalent to L p(·) ( ) being uniformly convex ( [17] Theorem 3.3). Notice that, for p + < ∞, f p(·) = 1 if and only if the modular ρ p(·) ( f ) = 1. Also for p + < ∞, a sequence ( f n ) ⊂ L p(·) ( ) satisfies lim n→∞ || f n || p(·) = 0 if and only if lim n→∞ ρ p(·) ( f n ) = 0 (cf. [7]). By B L p(·) we denote the closed unit ball of L p(·) ( ). The essential range of the exponent function p(·) is defined as It is a closed subset of [1, ∞) and it is compact when p(·) is essentially bounded. Fixed an exponent function p(·), we denote p(·) 1 (≡ 1 ) the subset of defined by 1 := p −1 ({1}). Given a sequence ( p n ) in [1, ∞), the Nakano sequence space p n is the Banach lattice of sequences (x n ) such that ∞ n=1 x n r p n < ∞ for some r > 0, equipped with the Luxemburg We refer to [4,6,7,16] for other definitions and basic facts regarding variable exponent Lebesgue spaces and Banach function spaces.

L p(·) equi-integrability
In Banach function spaces on finite measure spaces, both properties are equivalent (cf. [15] Lemma 2.1) However, for infinite measure spaces, uniform integrability is not enough for getting equi-integrability and an additional condition is required.

Proposition 3.1 Let E( ) be an order continuous Banach function space on an infinite measure space ( , μ) and let S ⊂ E( ) bounded. Then, S is equi-integrable if and only if it is uniformly integrable and it decays uniformly at infinity, i.e. for every ε > 0 there exists
Let us first characterize equi-integrability by a more suitable property:

Lemma 3.2 Let E( ) be an order continuous Banach function space and S ⊂ E( ). Then, S is equi-integrable if and only if, for every disjoint sequence of measurable sets (A n )
, Conversely, if S is not equi-integrable, there exist ε > 0, a decreasing sequence A n ∅ in and a sequence ( f n ) in S such that, for all n ∈ N, Now, using that E( ) is order continuous, given n 1 ∈ N we can find n 2 > n 1 large enough so that, for Then, we can find n 3 large enough so that, for Thus, we can define the disjoint sequence (B m ) by B m := A n m \ A n m+1 from the subsequence (A n m ) and, taking the subsequence ( f n m ), for every m ∈ N we get ending the proof.

Proof of Proposition 3.1 If lim
Fixed some n 1 , take n 2 large enough so that Then, we can take n 3 large enough so that Thus, we construct the disjoint sequence (B m ), where B m := A n m \ ∞ k=n m+1 A k , which satisfies, for every natural m, Hence, by Lemma 3.2, S is not equi-integrable. Assume now that there exists some ε > 0 such that, for every A with μ(A) < ∞, it holds sup f ∈S f χ \A ≥ ε. Then we can take an increasing sequence of sets (A n ) of finite measure and ( f n ) in S such that, for all n ∈ N, Now, as before, fixed n 1 take n 2 large enough so that In the same way, we define the disjoint sequence so S is not equi-integrable. Conversely, assume that, given ε > 0, by ( * ) there exists δ > 0 so that, for every A with μ(A) < δ, And by ( * * ), take Then, for every disjoint sequence Thus, by Lemma 3.2, S is equi-integrable.
Inclusions between variable exponent Lebesgue spaces for infinite measure spaces are more restricted but nonetheless they exist (in contrast with the behavior of classical Lebesgue spaces L p ). The following is known (cf. [7] Theorem 3.3.1. See also [6] and [2]): Let us remark that if the inclusion L q(·) ( ) ⊂ L p(·) ( ) holds and μ( d ) = ∞, then for every ε > 0 the set D ε := {t ∈ d : q(t) < p(t) + ε} has infinite measure.
Indeed, assume that there exists ε > 0 such that which, using the above Proposition, gives a contradiction. Proof To prove this theorem, it will be enough to find a disjoint sequence ( f n ) which be seminormalized in L p(·) ( ) as well as in L q(·) ( ). To do so, we distinguish two cases. If μ( d ) < ∞, then μ( \ d ) > 0. Hence, as p(t) ≤ q(t) for almost every t ∈ for the inclusion to hold, L q(·) ( \ d ) = L p(·) ( \ d ) and every disjoint sequence of functions ( f n ) supported in \ d will have equal norm in L p(·) ( ) and in L q(·) ( ). If, on the contrary, μ( d ) = ∞, we take ( f n := χ A n ), where (A n ) is a disjoint sequence satisfying that μ(A n ) = 1 and, for every natural n, Indeed, in that case we have ) = ∞ for every natural n. We take a finite for every i. We can ensure that, for some j, , it is also true that Thus, we deduce that there exists an infinite measure set E n (the set above) such that Finally, since μ(E n ) = ∞ for each natural n, we take the subsets which satisfy the required properties, namely

) is relatively weakly compact if and only if it is norm bounded and
Proof We just have to prove that this condition is equivalent to: But, as the isometry T is also isomodular, it is clear that If μ( 1 ) > 0, combining the above result and Dunford-Pettis Theorem 1.1 we get:

relatively weakly compact if and only if it is norm bounded,
We pass to characterize weakly convergent sequences in L p(·) (0, ∞): and we can take a simple function g s such that (g − g s )χ A ∞ < ε 3 . Thus, and hence ∞ 0 | f n g|dμ ≤ ε for n ∈ N large enough. Now, for an arbitrary g ∈ L p * (·) (0, ∞), by condition (ii), there exists δ > 0 such that Since gχ G m is bounded, we have ∞ 0 | f n g|dμ ≤ ε from n ∈ N large enough.
Proof Clearly, (i) is equivalent to condition (i) in the above Proposition 4.4 and, using Therorem 4.2, we get also condition (ii). Conversely, let g ∈ L p * (·) (0, ∞) and r > 0 such that Take δ > 0 such that, for every measurable set E with μ(E) < δ, Thus, using Young inequality ( [7] Lemma 3.2.20), we have: as well as and so the conditions (iii) and (ii) of the above Proposition 4.4 are respectively satisfied. Thus, we conclude that ( f n ) is weakly convergent to f .

A representation of L p(·) (Ä)
In this section ( , , μ) will denote an abstract non-atomic separable σ -finite measure space (i.e. endowed with the usual metric d( A)), for A, B ∈ , is separable). The real interval (0, ∞) with the Lebesgue measure will be denoted by ((0, ∞), B, λ). In the previous Sect. 4, we have obtained weak compactness criteria in L p(·) (0, ∞) by using the lattice isometry T τ between L p(·) (0, ∞) and a certain L q(·) (0, 1) space. We would like now to formulate this result for general separable σ -finite measure spaces ( , μ). In order to do it, we are going to give a lattice isomorphic representation of the space L p(·) ( ) as a certain variable exponent space L p(·) (0, ∞) in the interval (0, ∞). For that, we will use a version of the classical Carathéodory Theorem on algebras of measure for σ -finite measure spaces. Recall that N μ and N λ are the classes of subsets of measure zero in and (0, ∞) respectively.  Proof Since is σ -finite, consider a sequence (A n ) of sets with μ(A n ) < ∞ such that = ∞ n=1 A n . Taking B n := A n \ i<n A i , we have = ∞ n=1 B n with (B n ) mutually disjoint and μ(B n ) < ∞. Thus, if we take an increasing sequence (x n ) ∞ in (0, ∞) with x 1 = 0 and, for all n ∈ N, x n+1 − x n = μ(B n ), then the above Carathéodory theorem says that, for every natural n, the measure spaces (B n , | B n , μ| B n ) and ((x n , x n+1 ], B| (x n ,x n+1 ] , λ| (x n ,x n+1 ] ) are isomorphic (modulo sets of zero measure). Also, . In consequence, /N μ and B/N λ are isomorphic too.
We can now pursue a representation Theorem. To begin with, given the measure preserving isomorphism φ from /N μ to B/N λ , take any simple function S = n∈ a n χ A n in L 0 ( ) (we assume simple functions in this section with the canonical decomposition, i.e. (A n ) is a disjoint finite sequence such that = n∈ A n ). Define the transformation S := n∈ a n χ φ(A n ) into a simple function in L 0 (0, ∞). Recall that given a positive measurable function f , there exists an increasing sequence of simple functions (S n ) such that S n f pointwise (cf. [5]). Thus, for a positive function f ∈ L 0 ( ), we define the transformation is an increasing sequence of positive simple functions such that S n f , since the sequence ( S n ) is also increasing a.e-λ. The next proposition shows that f is well defined.

Proposition 5.3
Let (S n ) and (R n ) be increasing sequences of positive simple functions in L 0 ( ) such that both S n f , R n f pointwise. Then, both S n f and R n f .
Proof As ( S n ) and ( R n ) are increasing sequences, they have a λ-pointwise limit, noted S n f and R n g. We have to prove that f (s) = g(s) a.e-λ. Assume there is ε > 0 such that, Then, we can take C ⊂ D ε with 0 < μ(C) < ∞. Let A = φ −1 (C) ∈ so μ(A) = λ(C). By Egorov's theorem, we can suppose w.l.o.g. that the convergence on C (and on φ −1 (C) = A) is uniform, i.e, S n | A f | A , R n | A f | A , S n | C f | C and R n | C g| C uniformly. So, and then | S n − R n | → 0 uniformly on C. Finally, f | C − g| C ≤ f | C − S n | C + S n | C − R n | C + R n | C − g| C → 0 uniformly in C, which is a contradiction.
Similarly, it holds that if (S n ) is a decreasing sequence of positive simple functions in L 0 ( ) such that S n f pointwise, then, S n f a.e-λ. We pass to extend this map to general measurable functions, getting an isometry between the spaces L p(·) ( ) and L p(·) (0, ∞) for non-atomic separable σ -finite measure spaces ( , μ). Before we prove this theorem we state several easy preliminary lemmas. Lemma 5.5 Let f , g ∈ L 0 ( ) with f , g ≥ 0. Then, f + g = f + g.
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