#### Abstract

We give a decomposition for the dual space of some Banach Function Spaces as the Zygmund space of the exponential integrable functions, the Marcinkiewicz space , and the Grand Lebesgue Space .

#### 1. Introduction

Let be a set of Lebesgue measure .

In this paper, we deal with the following issue. What is the difference between the dual space and the associate space of a Banach Function Space

By associate space of we mean the space determined by the associate norm : as in Definitions 2.3 and 2.4.

If is a reflexive Banach Function Space, then the dual space is canonically isometrically isomorphic to the associate space [1, pageββ23]. On the other hand, for example, if we consider the Orlicz space of exponentially integrable functions, which is not reflexive, the associate space coincides with the Zygmund space , while the dual can be represented by where is the closure of with respect to the EXP norm (see [2, ChapterββIV], [3] and also Corollaryββ3.4).

Our aim is to show that the decomposition for the dual space as in (1.2) holds in a more general setting: namely, if is a rearrangement invariant Banach Function Space on such that its fundamental function verifies then, where denotes the closure of in . We stress that, due to assumption (1.3), our argument is much shorter than the corresponding one, treated in Zaanen ([4, Sectionββ70, Theoremββ2, pageββ467]) in the more abstract setting of normed KΓΆethe spaces. (See also [2, ChapterββIV, Propositionββ2.8 and Theoremββ2.11]).

In Section 3, we consider our decomposition in the particular case of 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 can be identified with a closed subspace of [1], while the spaces mentioned in our particular cases verify as shown in Theorem 3.7.

#### 2. Preliminaries

Let be a set of Lebesgue measure and let be the set of all measurable functions, whose values lie in , finite a.e. in .

*Definition 2.1. *A mapping is called a *Banach function norm* if, for all in , for all constants , and for all measurable subsets , the following properties hold.
for some constant , , depending on and , but independent of .

*Definition 2.2. *If is a Banach function norm, the Banach space
is called a *Banach Function Space*.

For each , define
Recall that the simple functions are contained in every Banach Function Spaces [1].

*Definition 2.3. *If is a function norm, its *associate norm * is defined on by

*Definition 2.4. *Let be a function norm and let be the Banach Function Space determined by . Let be the associate norm of . The Banach Function Space determined by is called the *associate space *of .

In particular, from the definition of , it follows that the norm of a function in the associate space is given by

*Definition 2.5. *A function in a Banach Function Space is said to have* absolutely continuous norm * in if for every sequence satisfying a.e. The set of all functions in of absolutely continuous norm is denoted by . If , then the space itself is said to have* absolutely continuous norm. *

*Definition 2.6. *Let . The function
is called the *distribution function* of . The *decreasing rearrangement* of , is defined on by
where here we use the convention .

Two functions having the same distribution function are called equimeasurable.

Let us recall that a function norm is said to be *rearrangement invariant* (briefly, βr.i.β) if for every couple of equimeasurable functions. The Banach Function Space arising from a r.i. function norm is called a *rearrangement-invariant space*.

By , we denote the function given by
The function is nonincreasing and verifies .

*Definition 2.7. *Let be a r.i. Banach Function Space determined by a function norm . For each , let be a set of measure . The* fundamental function *of , , is defined by

*Definition 2.8. *Let and , then the* Zygmund space * is the set of all measurable functions in for which the quantity
is finite.

For and we will replace by .

With these notations, the usual space of the *exponentially integrable functions* corresponds to the Zygmund space and consists of all measurable functions in for which the quantity
is finite.

All these spaces are particular cases of the Orlicz spaces.

Let be a right-continuous, increasing function, such that and , then the function defined by
is called *N function*; it is a continuous, convex, increasing function such that and .

*Definition 2.9. *The* Orlicz space * consists of all measurable functions on for which there exists some such that
where stands for .

This is a Banach space with respect to the Luxemburg norm:

The Orlicz spaces are a standard example of rearrangement-invariant Banach Function Space: the associate space of is given by , where denotes the complementary function of , defined by
Moreover, we notice that, for , , and , the Orlicz space associated reduces, respectively, to the spaces and to .

*Definition 2.10. *Given , the *Lorentz space * consists of all measurable functions in for which
is finite.

The space - 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 , replacing with , we obtain a norm equivalent to (2.16).

In particular, for , in the case of a nonatomic measure space, (2.16) is equivalent to

Now, we recall the definitions of Grand and Small Lebesgue Spaces, introduced, respectively, in [5] and in [6].

*Definition 2.11. *Let and ; the *Grand Lebesgue Space * is the Banach Function Space of all measurable functions on such that
is finite.

Notice that
If and is its HΓΆlder conjugate exponent, according to [7], the *Small Lebesgue Space * can be identified as the set of all measurable functions on 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 , if and only if
Elements of the direct sum are representable uniquely in the form

*Definition 2.13. *Let be a Banach space and a vectorial subspace of . The *orthogonal space * of is
where is the duality inner product.

It is known that is a closed subspace of .

We conclude this section by recalling some classical results, which will be useful in the sequel.

Theorem 2.14 (HΓΆlder's inequality [1]). *Let be a Banach Function Space with associate space . If and , then is integrable and
*

Lemma 2.15 (see [1, Lemmaβ2.6, pageββ10]). *In order that a measurable function belongs to the associate space , it is necessary and sufficient that is integrable for every in .*

Theorem 2.16 (see [1, Theoremββ2.7, pageββ10]). *Every Banach Function Space coincides with its second associate space . *

Theorem 2.17 (see [1, Theoremββ2.9, pageββ13]). *The associate space of a Banach Function Space is canonically isometrically isomorphic to a closed norm-fundamental subspace of the Banach space dual of .*

Proposition 2.18 (see [1, Propositionββ2.10, pageβ13]). *If and are Banach Function Spaces and (continuous embedding), then (continuous embedding).*

Theorem 2.19 (see [1, Theoremββ3.11, pageββ18]). *Let be a Banach Function Space. Then, .*

Corollary 2.20. *If , then .*

Theorem 2.21 (see [1, Theoremββ3.13, pageββ19]). *The subspaces and coincide if and only if the characteristic function has absolutely continuous norm for every set of finite measure.*

Theorem 2.22 (see [1, Corollaryββ4.2, pageββ23]). *Let be a Banach Function Space. If contains the simple functions, then .*

Theorem 2.23 (see [1, Corollaryββ4.3, pageββ23]). *The Banach space dual of a Banach Function Space is canonically isometrically isomorphic to the associate space if and only if has absolutely continuous norm.*

Theorem 2.24 (see [1, Theoremββ5.5, pageββ67]). *Let be a totally -finite nonatomic measure space and let be an arbitrary rearrangement-invariant space over . The following conditions on are equivalent: *(i)*;
*(ii)*;
*(iii)*,
** ββwhere is the fundamental function of .*

#### 3. Main Results

In this Section, we establish a decomposition for the dual space of a r.i. Banach Function Space.

Theorem 3.1. *Let be a rearrangement-invariant Banach Function Space on . For each , let be a subset of with and let be the fundamental function of .**If
**
then the following decomposition
**
holds.*

*Proof. *Let , for all measurable sets in , we define the set function
which is -additive and absolutely continuous with respect to the Lebesgue measure . Thus, has a locally integrable Radon-Nikodym derivative and
Since for all , it is
where is a constant. Hence, for all ,
By Lemma 2.15, it follows that .

To any , we can associate the functional
By HΓΆlder's inequality, belongs to , which is equivalent to thanks to Theorem 2.24.

Finally, let be defined by , then for all . Therefore, belongs to .

Hence,
Since it is easily seen that and , subspaces of , verify , 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

Corollary 3.3. *Let be an Orlicz space, then
*

*Proof. *If is an Orlicz space, then the fundamental function is
(see [7]). Therefore, and the claim follows from Theorem 3.1 and Remark 3.2.

Corollary 3.4. *Let , then
**
where denotes the closure of in .*

*Proof. *The result follows by Corollary 3.3, and by , (see [1]).

Corollary 3.5. *Let , be its HΓΆlder conjugate exponent and , then
*

*Proof. *The Marcinkievicz space is the largest of all rearrangement-invariant spaces having the same fundamental function as (see [1]), which is
Moreover, the associate space of (see [1]) is, up to equivalence of norms, the Lorentz space .

Therefore, the statement easily follows by Theorem 3.1.

A decomposition of the dual of was also given in [8].

Corollary 3.6. *Let , and , then
*

* Proof. *Let be the fundamental function of the space , then
as (see [7]).

Therefore the claim easily follows by Theorem 3.1 and by the relation (see [7]).

In the next theorem, we show the relation between a Banach Function Space and the dual of its associate space .

Theorem 3.7. *Let be a Banach Function Space, then the following inclusion
**
holds, with equality occurring if and only if the associate space of has absolutely continuous norm.*

*Proof. * By Theorem 2.17 applied to the Banach Function Space , we may identify with a closed subspace of ; hence, Theorem 2.16 implies

Furthermore, if has absolutely continuous norm, that is , since every Banach Function Space contains the simple functions, by Theorem 2.22 applied to the space and by Theorem 2.16, we have .

On the other hand, if , then , and Theorem 2.23 yields that has absolutely continuous norm.

*Remark 3.8. *An example of a Banach Function Space verifying the proper inclusion in (3.18) is given by the Lebesgue space . In fact, if , then
as confirmed by the fact that has not absolutely continuous norm.