A Decomposition of the Dual Space of Some Banach Function Spaces

as in Definitions 2.3 and 2.4. If X is a reflexive Banach Function Space, then the dual space X∗ is canonically isometrically isomorphic to the associate space X′ 1, page 23 . On the other hand, for example, if we consider the Orlicz space EXP Ω of exponentially integrable functions, which is not reflexive, the associate space EXP Ω ′ coincides with the Zygmund space L logL Ω , while the dual can be represented by

In this paper, we deal with the following issue.What is the difference between the dual space X * and the associate space X of a Banach Function Space X?
By associate space X of X we mean the space determined by the associate norm ρ : as in Definitions 2.3 and 2.4.If X is a reflexive Banach Function Space, then the dual space X * is canonically isometrically isomorphic to the associate space X 1, page 23 .On the other hand, for example, if we consider the Orlicz space EXP Ω of exponentially integrable functions, which is not reflexive, the associate space EXP Ω coincides with the Zygmund space L log L Ω , while the dual can be represented by where exp Ω is the closure of L ∞ Ω with respect to the EXP norm see 2, Chapter IV , 3 and also Corollary 3.4 .
Our aim is to show that the decomposition for the dual space as in 1.2 holds in a more general setting: namely, if X is a rearrangement invariant Banach Function Space on Ω such that its fundamental function ϕ X verifies where X b denotes the closure of L ∞ Ω in X.We stress that, due to assumption 1.3 , our argument is much shorter than the corresponding one, treated in Zaanen 4, Section 70, Theorem 2, page 467 in the more abstract setting of normed K öethe spaces.See also 2, Chapter IV, Proposition 2.8 and Theorem 2.11 .
In Section 3, we consider our decomposition in the particular case of EXP α spaces, Marcinkiewicz spaces, and the Grand Lebesgue Spaces, specifying case by case the expression of the associate space.
Let us note that in general a Banach Function Space X can be identified with a closed subspace of X * 1 , while the spaces mentioned in our particular cases verify as shown in Theorem 3.7.

Preliminaries
Let Ω be a set of Lebesgue measure |Ω| < ∞ and let M o be the set of all measurable functions, whose values lie in 0, ∞ , finite a.e. in Ω.
Definition 2.1.A mapping ρ : for all constants a ≥ 0, and for all measurable subsets E ⊂ Ω, the following properties hold.
for some constant C E , 0 < C E < ∞, depending on E and ρ, but independent of f.Definition 2.2.If ρ is a Banach function norm, the Banach space Recall that the simple functions are contained in every Banach Function Spaces X 1 .
Definition 2.4.Let ρ be a function norm and let X X ρ be the Banach Function Space determined by ρ.Let ρ be the associate norm of ρ.The Banach Function Space X X ρ determined by ρ is called the associate space of X.
In particular, from the definition of f X , it follows that the norm of a function g in the associate space X is given by The set of all functions in X of absolutely continuous norm is denoted by X a .If X X a , then the space X itself is said to have absolutely continuous norm.Definition 2.6.Let f ∈ M o .The function where here we use the convention inf ∅ ∞.Two functions having the same distribution function are called equimeasurable.Let us recall that a function norm ρ is said to be rearrangement invariant briefly, "r.i." if ρ f ρ g for every couple of equimeasurable functions.The Banach Function Space arising from a r.i.function norm is called a rearrangement-invariant space.
Definition 2.7.Let X be a r.i.Banach Function Space determined by a function norm ρ.For each t ∈ 0, |Ω| , let E t ⊆ Ω be a set of measure t.The fundamental function of X, ϕ X t , is defined by 2.9 Definition 2.8.Let 1 ≤ p ≤ ∞ and α ∈ R, then the Zygmund space L p log L α Ω is the set of all measurable functions f in Ω for which the quantity For p 1 and α 1 we will replace With these notations, the usual space EXP α of the exponentially integrable functions corresponds to the Zygmund space L ∞ / log L 1/α Ω and consists of all measurable functions f in Ω for which the quantity is finite.All these spaces are particular cases of the Orlicz spaces.Let φ : 0, ∞ → 0, ∞ be a right-continuous, increasing function, such that φ 0 0 and lim t → ∞ φ t ∞, then the function defined by Definition 2.9.The Orlicz space L Φ Ω consists of all measurable functions f on Ω for which there exists some λ > 0 such that where − Ω stands for 1/|Ω| Ω .This is a Banach space with respect to the Luxemburg norm:

2.14
The Orlicz spaces are a standard example of rearrangement-invariant Banach Function Space: the associate space of L Φ Ω is given by L Φ Ω , where Φ denotes the complementary function of Φ, defined by Moreover, we notice that, for Φ t t p , Φ t t p log t α , and Φ t e t α − 1, the Orlicz space associated reduces, respectively, to the spaces L p Ω , L p log L α Ω and to EXP α Ω .Definition 2.10.Given 1 ≤ p, q ≤ ∞, the Lorentz space L p,q Ω consists of all measurable functions f in Ω for which is finite.The space L p,∞ Ω Weak-L p Ω is known as the Marcinkiewicz space, and it is another example of r.i.Banach Function Space.
The quantity 2.16 is not a norm since the triangle inequality may fail; however, for p > 1, replacing f * t with f * * t , we obtain a norm equivalent to 2.16 .
In particular, for q ∞, in the case of a nonatomic measure space, 2.16 is equivalent to 2.17 Now, we recall the definitions of Grand and Small Lebesgue Spaces, introduced, respectively, in 5 and in 6 .
Definition 2.11.Let 1 < p < ∞ and θ ≥ 0; the Grand Lebesgue Space L p),θ is the Banach Function Space of all measurable functions f on Ω such that

2.19
If 1 < p < ∞ and p is its H ölder conjugate exponent, according to 7 , the Small Lebesgue Space L p ,θ can be identified as the set of all measurable functions f on Ω such that is finite.The Grand and Small Lebesgue Spaces are r.i.Banach Function Spaces 7 .
Definition 2.12.A vector space V is the direct sum of its subspaces U and W, denoted by V U ⊕ W, if and only if

2.21
Elements v of the direct sum U ⊕ W are representable uniquely in the form u w : u ∈ U , w ∈ W.

2.22
Definition 2.13.Let X be a Banach space and M ⊂ X a vectorial subspace of X.The orthogonal space M ⊥ of M is where ., . is the duality inner product.It is known that M ⊥ is a closed subspace of X * .We conclude this section by recalling some classical results, which will be useful in the sequel.
Theorem 2.14 H ölder's inequality 1 .Let X be a Banach Function Space with associate space X .If f ∈ X and g ∈ X , then fg is integrable and Ω fg dx ≤ f X g X .

2.24
Lemma 2.15 see 1, Lemma 2.6, page 10 .In order that a measurable function g belongs to the associate space X , it is necessary and sufficient that fg is integrable for every f in X.
Theorem 2.16 see 1, Theorem 2.7, page 10 .Every Banach Function Space X coincides with its second associate space X X .
Theorem 2.17 see 1, Theorem 2.9, page 13 .The associate space X of a Banach Function Space X is canonically isometrically isomorphic to a closed norm-fundamental subspace of the Banach space dual X * of X.
where ϕ X t is the fundamental function of X.

Main Results
In this Section, we establish a decomposition for the dual space of a r.i.Banach Function Space.
Theorem 3.1.Let X be a rearrangement-invariant Banach Function Space on Ω.For each t ∈ 0, |Ω| , let E be a subset of Ω with |E| t and let ϕ X t be the fundamental function of X.
then the following decomposition Proof.Let l ∈ X * , for all measurable sets F in Ω, we define the set function which is σ-additive and absolutely continuous with respect to the Lebesgue measure |F|.
Thus, ν has a locally integrable Radon-Nikodym derivative g and where K is a constant.Hence, for all f ∈ L ∞ , Ω fg dx ≤ K f X .

3.6
By Lemma 2.15, it follows that g ∈ X .
To any g ∈ X , we can associate the functional By H ölder's inequality, l g belongs to X * b , which is equivalent to X thanks to Theorem 2.24.Finally, let l s be defined by l s l − l g , then l s f l s , f 0 for all f ∈ X b .Therefore, l s belongs to X b ⊥ .
Hence, l l g l s ∈ X X ⊥ b .

3.8
Since it is easily seen that X and X ⊥ b , subspaces of X * , verify X ∩ X ⊥ b {0}, then the proof is complete.
Remark 3.2.Let us point out that, by Theorem 2.24, the decomposition 3.2 can also be written as

3.11
Proof.If X L Φ Ω is an Orlicz space, then the fundamental function is .12 see 7 .Therefore, lim t → 0 ϕ X t 0 and the claim follows from Theorem 3.1 and Remark 3.2.
where exp α Ω denotes the closure of L ∞ Ω in EXP α Ω .
Corollary 3.5.Let p ∈ 1, ∞ , p be its Hölder conjugate exponent and X L p,∞ Ω , then 3.14 Proof.The Marcinkievicz space L p,∞ Ω is the largest of all rearrangement-invariant spaces having the same fundamental function as L p Ω see 1 , which is ϕ L p t t 1/p .

3.15
Moreover, the associate space of L p,∞ Ω see 1 is, up to equivalence of norms, the Lorentz space L p ,1 Ω .Therefore, the statement easily follows by Theorem 3.1.
A decomposition of the dual of L p,∞ was also given in 8 .

3.16
Proof.Let ϕ X t be the fundamental function of the space L p ,θ Ω , then ϕ X t ≈ t 1/p log 1 t −θ/p 3.17 as t → 0 see 7 .
Therefore the claim easily follows by Theorem 3.1 and by the relation L p),θ Ω L p',θ Ω see 7 .
Theorem 2.19 see 1, Theorem 3.11, page 18 .Let X be a Banach Function Space.Then, X a ⊆ X b ⊆ X.The subspaces X a and X b coincide if and only if the characteristic function χ E has absolutely continuous norm for every set E of finite measure.Corollary 4.3, page 23 .The Banach space dual X * of a Banach Function Space X is canonically isometrically isomorphic to the associate space X if and only if X has absolutely continuous norm.Theorem 2.24 see 1, Theorem 5.5, page 67 .Let Ω, μ be a totally σ-finite nonatomic measure space and let X be an arbitrary rearrangement-invariant space over Ω, μ .The following conditions on X are equivalent: Proposition 2.18 see 1, Proposition 2.10, page 13 .If X and Y are Banach Function Spaces and X ⊂ Y (continuous embedding), then Y ⊂ X (continuous embedding).