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Journal of Function Spaces and Applications
VolumeΒ 2012, Article IDΒ 737534, 10 pages
Research Article

A Decomposition of the Dual Space of Some Banach Function Spaces

1Istituto per le Applicazioni del Calcolo β€œMauro Picone”, CNR, sez. Napoli, Via P. Castellino 111, 80131 Napoli, Italy
2Dipartimento di Statistica e Matematica per la Ricerca Economica, UniversitΓ  degli Studi di Napoli β€œParthenope”, Via Medina 40, 80133 Napoli, Italy

Received 7 April 2010; Accepted 8 June 2010

Academic Editor: CarloΒ Sbordone

Copyright Β© 2012 Claudia Capone and Maria Rosaria Formica. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We give a decomposition for the dual space of some Banach Function Spaces as the Zygmund space EXP𝛼 of the exponential integrable functions, the Marcinkiewicz space 𝐿𝑝,∞, and the Grand Lebesgue Space 𝐿𝑝),πœƒ.

1. Introduction

Let Ω be a set of Lebesgue measure |Ω|<+∞.

In this paper, we deal with the following issue. What is the difference between the dual space π‘‹βˆ— and the associate space 𝑋′ of a Banach Function Space 𝑋?

By associate space π‘‹ξ…ž of 𝑋 we mean the space determined by the associate norm πœŒξ…ž: πœŒξ…žξ‚»ξ€œ(𝑔)=supΩ𝑓𝑔𝑑π‘₯βˆΆπ‘“βˆˆβ„³+ξ‚Ό,𝜌(𝑓)≀1(1.1) as in Definitions 2.3 and 2.4.

If 𝑋 is a reflexive Banach Function Space, then the dual space π‘‹βˆ— is canonically isometrically isomorphic to the associate space π‘‹ξ…ž [1, page  23]. On the other hand, for example, if we consider the Orlicz space EXP(Ξ©) of exponentially integrable functions, which is not reflexive, the associate space (EXP(Ξ©))β€² coincides with the Zygmund space 𝐿log𝐿(Ξ©), while the dual can be represented by(EXP(Ξ©))βˆ—=𝐿log𝐿(Ξ©)βŠ•(exp(Ξ©))βŸ‚,(1.2) where exp(Ξ©) is the closure of 𝐿∞(Ξ©) 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 Ξ© such that its fundamental function πœ‘π‘‹ verifiesπœ‘π‘‹(0+)=0,(1.3) then,π‘‹βˆ—=π‘‹β€²βŠ•ξ€·π‘‹π‘ξ€ΈβŸ‚,(1.4) where 𝑋𝑏 denotes the closure of 𝐿∞(Ξ©) 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 KΓΆethe 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 EXP𝛼 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′)βˆ— [1], while the spaces mentioned in our particular cases verify𝑋𝑋=ξ…žξ€Έβˆ—(1.5) as shown in Theorem 3.7.

2. Preliminaries

Let Ξ© be a set of Lebesgue measure |Ξ©|<+∞ and let β„³+π‘œ be the set of all measurable functions, whose values lie in [0,+∞], finite a.e. in Ξ©.

Definition 2.1. A mapping πœŒβˆΆβ„³+π‘œβ†’[0,+∞] is called a Banach function norm if, for all 𝑓,𝑔,𝑓𝑛(𝑛=1,2,3,…) in β„³+π‘œ, for all constants π‘Žβ‰₯0, and for all measurable subsets πΈβŠ‚Ξ©, the following properties hold. 𝜌,𝜌(𝑓)=0βŸΊπ‘“=0a.e.inΞ©,(π‘Žπ‘“)=π‘ŽπœŒ(𝑓),𝜌(𝑓+𝑔)β‰€πœŒ(𝑓)+𝜌(𝑔),0≀𝑔≀𝑓a.e.inΩ⟹𝜌(𝑔)β‰€πœŒ(𝑓)0≀𝑓𝑛𝑓↑𝑓a.e.inΞ©βŸΉπœŒπ‘›ξ€Έ||𝐸||ξ€·πœ’β†‘πœŒ(𝑓),<+βˆžβŸΉπœŒπΈξ€Έ||𝐸||ξ€œ<+∞,<+βˆžβŸΉπΈπ‘“π‘‘π‘₯β‰€πΆπΈπœŒ(𝑓)(2.1) for some constant 𝐢𝐸, 0<𝐢𝐸<∞, depending on 𝐸 and 𝜌, but independent of 𝑓.

Definition 2.2. If 𝜌 is a Banach function norm, the Banach space ξ€½ξ€·||𝑓||𝑋=π‘“βˆˆβ„³βˆΆπœŒ<+∞(2.2) is called a Banach Function Space.
For each π‘“βˆˆπ‘‹, define ‖𝑓‖𝑋||𝑓||ξ€Έ=𝜌.(2.3) Recall that the simple functions are contained in every Banach Function Spaces 𝑋 [1].

Definition 2.3. If 𝜌 is a function norm, its associate norm πœŒξ…ž is defined on β„³+π‘œ by πœŒξ…žξ‚»ξ€œ(𝑔)=supΩ𝑓𝑔𝑑π‘₯βˆΆπ‘“βˆˆβ„³+ξ‚Ό,𝜌(𝑓)≀1.(2.4)

Definition 2.4. Let 𝜌 be a function norm and let 𝑋=𝑋(𝜌) be the Banach Function Space determined by 𝜌. Let πœŒβ€² be the associate norm of 𝜌. The Banach Function Space 𝑋′=𝑋(πœŒξ…ž) determined by πœŒβ€² 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 β€–π‘”β€–π‘‹β€²ξ‚»ξ€œ=supΩ𝑓𝑔𝑑π‘₯βˆΆπ‘“βˆˆπ‘‹,‖𝑓‖𝑋≀1.(2.5)

Definition 2.5. A function 𝑓 in a Banach Function Space 𝑋 is said to have absolutely continuous norm in 𝑋 if β€–π‘“πœ’πΈπ‘›β€–π‘‹β†’0 for every sequence {𝐸𝑛}βˆžπ‘›=1 satisfying πΈπ‘›β†’βˆ…a.e. The set of all functions in 𝑋 of absolutely continuous norm is denoted by π‘‹π‘Ž. If 𝑋=π‘‹π‘Ž, then the space 𝑋 itself is said to have absolutely continuous norm.

Definition 2.6. Let π‘“βˆˆβ„³π‘œ. The function πœ‡π‘“||ξ€½||𝑓||ξ€Ύ||(πœ†)=π‘₯∈Ω∢(π‘₯)>πœ†βˆ€πœ†β‰₯0(2.6) is called the distribution function of 𝑓. The decreasing rearrangement of 𝑓,π‘“βˆ—, is defined on [0,|Ξ©|] by π‘“βˆ—ξ€½(𝑑)=infπœ†>0βˆΆπœ‡π‘“ξ€Ύ(πœ†)≀𝑑,(2.7) where here we use the convention infβˆ…=+∞.
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, β€œr.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.
By π‘“βˆ—βˆ—βˆΆ(0,∞)β†’[0,∞], we denote the function given by π‘“βˆ—βˆ—(1𝑑)=π‘‘ξ€œπ‘‘0π‘“βˆ—(𝑠)𝑑𝑠,(𝑑>0).(2.8) The function π‘“βˆ—βˆ— is nonincreasing and verifies π‘“βˆ—(𝑑)β‰€π‘“βˆ—βˆ—(𝑑)(𝑑>0).

Definition 2.7. Let 𝑋 be a r.i. Banach Function Space determined by a function norm 𝜌. For each π‘‘βˆˆ[0,|Ξ©|], let πΈπ‘‘βŠ†Ξ© be a set of measure 𝑑. The fundamental function of 𝑋, πœ‘π‘‹(𝑑), is defined by πœ‘π‘‹ξ€·πœ’(𝑑)=πœŒπΈπ‘‘ξ€Έ=β€–β€–πœ’πΈπ‘‘β€–β€–π‘‹.(2.9)

Definition 2.8. Let 1β‰€π‘β‰€βˆž and π›Όβˆˆπ‘, then the Zygmund space 𝐿𝑝(log𝐿)𝛼(Ξ©) is the set of all measurable functions 𝑓 in Ξ© for which the quantity ‖𝑓‖𝐿𝑝(log𝐿)𝛼(Ξ©)=β€–β€–β€–ξ‚΅ξ‚΅||Ξ©||1+logπ‘‘ξ‚Άξ‚Άπ›Όπ‘“βˆ—β€–β€–β€–(𝑑)𝐿𝑝(0,|Ξ©|)(2.10) is finite.
For 𝑝=1 and 𝛼=1 we will replace 𝐿1(log𝐿)1(Ξ©) by 𝐿(log𝐿)(Ξ©).

With these notations, the usual space EXP𝛼 of the exponentially integrable functions corresponds to the Zygmund space (𝐿∞/(log𝐿)1/𝛼)(Ξ©) and consists of all measurable functions 𝑓 in Ξ© for which the quantity β€–π‘“β€–πΏβˆž/(log𝐿)1/𝛼(Ξ©)=‖𝑓‖EXP𝛼(Ξ©)=sup||Ξ©||0<𝑑<ξ‚΅ξ‚΅||Ξ©||1+logπ‘‘ξ‚Άξ‚Άβˆ’1/π›Όπ‘“βˆ—(𝑑)(2.11) is finite.

All these spaces are particular cases of the Orlicz spaces.

Let πœ™βˆΆ[0,∞)β†’[0,∞)be a right-continuous, increasing function, such that πœ™(0)=0 and limπ‘‘β†’βˆžπœ™(𝑑)=∞, then the function defined by ξ€œΞ¦(𝑑)=𝑑0πœ™(𝑠)𝑑𝑠(2.12) is called N function; it is a continuous, convex, increasing function such that limπ‘‘β†’βˆž(Ξ¦(𝑑)/𝑑)=+∞ and lim𝑑→0(Ξ¦(𝑑)/𝑑)=0.

Definition 2.9. The Orlicz space 𝐿Φ(Ξ©) consists of all measurable functions 𝑓 on Ξ© for which there exists some πœ†>0 such that ΩΦ||𝑓||πœ†ξ‚Ά<∞,(2.13) where ⨍Ω stands for ∫(1/|Ξ©|)Ξ©.
This is a Banach space with respect to the Luxemburg norm: ‖𝑓‖𝐿Φ(Ξ©)=infπœ†>0βˆΆΞ©Ξ¦ξ‚΅||||𝑓(π‘₯)πœ†ξ‚Άξ‚Όπ‘‘π‘₯≀1.(2.14)
The Orlicz spaces are a standard example of rearrangement-invariant Banach Function Space: the associate space of 𝐿Φ(Ξ©) is given by 𝐿Φ(Ξ©), where Φ denotes the complementary function of Ξ¦, defined by Φ(𝑑)=max{π‘ π‘‘βˆ’Ξ¦(𝑠)βˆΆπ‘ β‰₯0}.(2.15) Moreover, we notice that, for Ξ¦(𝑑)=𝑑𝑝, Ξ¦(𝑑)=𝑑𝑝(log𝑑)𝛼, and Ξ¦(𝑑)=π‘’π‘‘π›Όβˆ’1, the Orlicz space associated reduces, respectively, to the spaces 𝐿𝑝(Ξ©),𝐿𝑝(log𝐿)𝛼(Ξ©) and to EXP𝛼(Ξ©).

Definition 2.10. Given 1≀𝑝,π‘žβ‰€βˆž, the Lorentz space 𝐿𝑝,π‘ž(Ξ©) consists of all measurable functions 𝑓 in Ξ© for which ‖𝑓‖𝑝,π‘ž=⎧βŽͺ⎨βŽͺβŽ©ξ€œβˆž0𝑑1/π‘π‘“βˆ—ξ€»(𝑑)π‘žπ‘‘π‘‘π‘‘,0<π‘ž<∞,sup𝑑>0𝑑1/π‘π‘“βˆ—(𝑑),π‘ž=∞(2.16) is finite.
The space 𝐿𝑝,∞(Ξ©)=π‘Šπ‘’π‘Žπ‘˜-𝐿𝑝(Ξ©) 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 π‘“βˆ—(𝑑) with π‘“βˆ—βˆ—(𝑑), we obtain a norm equivalent to (2.16).
In particular, for π‘ž=∞, in the case of a nonatomic measure space, (2.16) is equivalent to ξ‚»||𝐸||sup1/π‘βˆ’1ξ€œπΈ||𝑓||𝑑π‘₯,πΈβŠ‚Ξ©measurable.(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 πœƒβ‰₯0; the Grand Lebesgue Space 𝐿p),πœƒ is the Banach Function Space of all measurable functions 𝑓 on Ξ© such that ‖𝑓‖p),πœƒ=sup0<πœ€<π‘βˆ’1ξ‚΅πœ€πœƒξ€§Ξ©||𝑓||π‘βˆ’πœ€ξ‚Άπ‘‘π‘₯1/(π‘βˆ’πœ€)(2.18) is finite.
Notice that 𝐿𝑝),0(Ξ©)=𝐿𝑝(Ξ©),𝐿p),1(Ξ©)=𝐿p)(Ξ©).(2.19) If 1<𝑝<+∞ and π‘ξ…ž is its HΓΆlder conjugate exponent, according to [7], the Small Lebesgue Space 𝐿(π‘ξ…ž,πœƒ can be identified as the set of all measurable functions 𝑓 on Ξ© such that ‖𝑓‖(𝑝′,πœƒξƒ―||||ξ€œ=supΞ©||||𝑓𝑔𝑑π‘₯βˆΆβ€–π‘“β€–πΏp),πœƒ(Ξ©)≀1(2.20) 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=π‘ˆβŠ•π‘Š, if and only if 𝑉=π‘ˆ+π‘Š={𝑒+π‘€βˆΆπ‘’βˆˆπ‘ˆ,π‘€βˆˆπ‘Š},π‘‰βˆ©π‘Š={0}.(2.21) Elements 𝑣 of the direct sum π‘ˆβŠ•π‘Š are representable uniquely in the form 𝑒+π‘€βˆΆπ‘’βˆˆπ‘ˆ,π‘€βˆˆπ‘Š.(2.22)

Definition 2.13. Let 𝑋 be a Banach space and π‘€βŠ‚π‘‹ a vectorial subspace of 𝑋. The orthogonal space π‘€βŸ‚ of 𝑀 is π‘€βŸ‚=ξ€½π‘“βˆˆπ‘‹βˆ—ξ€ΎβˆΆβŸ¨π‘“,π‘₯⟩=0,βˆ€π‘₯βˆˆπ‘€,(2.23) where ⟨.,.⟩ is the duality inner product.
It is known that π‘€βŸ‚ is a closed subspace of π‘‹βˆ—.
We conclude this section by recalling some classical results, which will be useful in the sequel.

Theorem 2.14 (HΓΆlder's inequality [1]). Let 𝑋 be a Banach Function Space with associate space 𝑋′. If π‘“βˆˆπ‘‹ and π‘”βˆˆπ‘‹β€², then 𝑓𝑔 is integrable and ξ€œΞ©π‘“π‘”π‘‘π‘₯β‰€β€–π‘“β€–π‘‹β€–π‘”β€–π‘‹ξ…ž.(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 π‘‹ξ…žξ…ž=(𝑋′)ξ…ž.

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 π‘‹βˆ— of 𝑋.

Proposition 2.18 (see [1, Proposition  2.10, page 13]). If 𝑋 and π‘Œ are Banach Function Spaces and π‘‹βŠ‚π‘Œ (continuous embedding), then π‘Œβ€²βŠ‚π‘‹β€² (continuous embedding).

Theorem 2.19 (see [1, Theorem  3.11, page  18]). Let 𝑋 be a Banach Function Space. Then, π‘‹π‘ŽβŠ†π‘‹π‘βŠ†π‘‹.

Corollary 2.20. If π‘‹π‘Ž=𝑋, then 𝑋𝑏=𝑋.

Theorem 2.21 (see [1, Theorem  3.13, page  19]). The subspaces π‘‹π‘Ž and 𝑋𝑏 coincide if and only if the characteristic function πœ’πΈ has absolutely continuous norm for every set 𝐸 of finite measure.

Theorem 2.22 (see [1, Corollary  4.2, page  23]). Let 𝑋 be a Banach Function Space. If π‘‹π‘Ž contains the simple functions, then (π‘‹π‘Ž)βˆ—=𝑋′.

Theorem 2.23 (see [1, Corollary  4.3, page  23]). The Banach space dual π‘‹βˆ— 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 (Ξ©,πœ‡) be a totally 𝜎-finite nonatomic measure space and let 𝑋 be an arbitrary rearrangement-invariant space over (Ξ©,πœ‡). The following conditions on 𝑋 are equivalent: (i)lim𝑑→0+πœ‘π‘‹(𝑑)=0; (ii)π‘‹π‘Ž=𝑋𝑏; (iii)(𝑋𝑏)βˆ—=𝑋′,   where πœ‘π‘‹(𝑑) is the fundamental function of 𝑋.

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 𝑋 be a rearrangement-invariant Banach Function Space on Ξ©. For each π‘‘βˆˆ[0,|Ξ©|], let 𝐸 be a subset of Ξ© with |𝐸|=𝑑 and let πœ‘π‘‹(𝑑) be the fundamental function of 𝑋.
If lim𝑑→0+πœ‘π‘‹(𝑑)=0,(3.1) then the following decomposition π‘‹βˆ—=π‘‹ξ…žβŠ•ξ€·π‘‹π‘ξ€ΈβŸ‚(3.2) holds.

Proof. Let π‘™βˆˆπ‘‹βˆ—, for all measurable sets 𝐹 in Ξ©, we define the set function πœˆξ€·πœ’(𝐹)=𝑙𝐹,(3.3) 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 ξ€œπ‘™(𝑓)=Ω𝑓𝑔𝑑π‘₯,foranyπ‘“βˆˆπΏβˆž(Ξ©).(3.4) Since π‘™βˆˆπ‘‹βˆ— for all π‘“βˆˆπ‘‹, it is 𝑙(𝑓)≀𝐾‖𝑓‖𝑋,(3.5) where 𝐾 is a constant. Hence, for all π‘“βˆˆπΏβˆž, ξ€œΞ©π‘“π‘”π‘‘π‘₯≀𝐾‖𝑓‖𝑋.(3.6) By Lemma 2.15, it follows that π‘”βˆˆπ‘‹β€².
To any π‘”βˆˆπ‘‹ξ…ž, we can associate the functional π‘™π‘”βˆΆπ‘“βˆˆπ‘‹π‘βŸΆξ€œΞ©π‘“π‘”π‘‘π‘₯.(3.7) By HΓΆlder's inequality, 𝑙𝑔 belongs to π‘‹βˆ—π‘, which is equivalent to π‘‹ξ…ž thanks to Theorem 2.24.
Finally, let 𝑙𝑠 be defined by 𝑙𝑠=π‘™βˆ’π‘™π‘”, then 𝑙𝑠(𝑓)=βŸ¨π‘™π‘ ,π‘“βŸ©=0 for all π‘“βˆˆπ‘‹π‘. Therefore, 𝑙𝑠 belongs to (𝑋𝑏)βŸ‚.
Hence, 𝑙=𝑙𝑔+π‘™π‘ βˆˆπ‘‹ξ…ž+π‘‹βŸ‚π‘.(3.8) Since it is easily seen that π‘‹ξ…ž and π‘‹βŸ‚π‘, subspaces of π‘‹βˆ—, verify π‘‹ξ…žβˆ©π‘‹βŸ‚π‘={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 π‘‹βˆ—=ξ€·π‘‹π‘ξ€Έβˆ—βŠ•ξ€·π‘‹π‘ξ€ΈβŸ‚,𝑋(3.9)βˆ—=ξ€·π‘‹π‘Žξ€Έβˆ—βŠ•ξ€·π‘‹π‘Žξ€ΈβŸ‚.(3.10)

Corollary 3.3. Let 𝑋 be an Orlicz space, then π‘‹βˆ—=π‘‹ξ…žβŠ•ξ€·π‘‹π‘ξ€ΈβŸ‚=ξ€·π‘‹π‘ξ€Έβˆ—βŠ•ξ€·π‘‹π‘ξ€ΈβŸ‚=ξ€·π‘‹π‘Žξ€Έβˆ—βŠ•ξ€·π‘‹π‘Žξ€ΈβŸ‚.(3.11)

Proof. If 𝑋=𝐿Φ(Ξ©) is an Orlicz space, then the fundamental function is πœ‘π‘‹1(𝑑)=Ξ¦βˆ’1ξ€»||Ξ©||ξ€»(1/𝑑),βˆ€π‘‘βˆˆ0,(3.12) (see [7]). Therefore, lim𝑑→0+πœ‘π‘‹(𝑑)=0 and the claim follows from Theorem 3.1 and Remark 3.2.

Corollary 3.4. Let 𝑋=EXP𝛼(Ξ©),𝛼>0, then ξ€·EXP𝛼(Ξ©)βˆ—=𝐿log1/𝛼𝐿(Ξ©)βŠ•exp𝛼(Ξ©)βŸ‚=ξ€·exp𝛼(Ξ©)βˆ—βŠ•ξ€·exp𝛼(Ξ©)βŸ‚,(3.13) where exp𝛼(Ξ©) denotes the closure of 𝐿∞(Ξ©) in EXP𝛼(Ξ©).

Proof. The result follows by Corollary 3.3, and by (EXP𝛼(Ξ©))β€²=𝐿log1/𝛼𝐿(Ξ©),𝛼>0, (see [1]).

Corollary 3.5. Let π‘βˆˆ]1,∞[, π‘ξ…ž be its HΓΆlder conjugate exponent and 𝑋=𝐿𝑝,∞(Ξ©), then (𝐿𝑝,∞(Ξ©))βˆ—=𝐿𝑝′,1𝐿(Ξ©)βŠ•π‘π‘,βˆžξ€Έ(Ξ©)βŸ‚.(3.14)

Proof. The Marcinkievicz space 𝐿𝑝,∞(Ξ©) is the largest of all rearrangement-invariant spaces having the same fundamental function as 𝐿𝑝(Ξ©) (see [1]), which is πœ‘πΏπ‘(𝑑)=𝑑1/𝑝.(3.15) Moreover, the associate space of 𝐿𝑝,∞(Ξ©) (see [1]) is, up to equivalence of norms, the Lorentz space 𝐿𝑝′,1(Ξ©).
Therefore, the statement easily follows by Theorem 3.1.
A decomposition of the dual of 𝐿𝑝,∞ was also given in [8].

Corollary 3.6. Let π‘βˆˆ]1,∞[, πœƒβ‰₯0 and 𝑋=𝐿p),πœƒ(Ξ©), then 𝐿𝑝),πœƒξ€Έ(Ξ©)βˆ—=𝐿(𝑝′,πœƒξ‚€πΏ(Ξ©)βŠ•π‘π‘),πœƒξ‚(Ξ©)βŸ‚.(3.16)

Proof. Let πœ‘π‘‹(𝑑) be the fundamental function of the space 𝐿𝑝),πœƒ(Ξ©), then πœ‘π‘‹(𝑑)β‰ˆπ‘‘1/𝑝1logπ‘‘ξ‚ξ‚„βˆ’πœƒ/𝑝(3.17) as 𝑑→0+ (see [7]).
Therefore the claim easily follows by Theorem 3.1 and by the relation (𝐿p),πœƒ(Ξ©))ξ…ž=𝐿(p',πœƒ(Ξ©) (see [7]).

In the next theorem, we show the relation between a Banach Function Space 𝑋 and the dual of its associate space (π‘‹ξ…ž)βˆ—.

Theorem 3.7. Let 𝑋 be a Banach Function Space, then the following inclusion ξ€·π‘‹π‘‹βŠ†ξ…žξ€Έβˆ—(3.18) holds, with equality occurring if and only if the associate space π‘‹ξ…ž of 𝑋 has absolutely continuous norm.

Proof. By Theorem 2.17 applied to the Banach Function Space π‘‹ξ…ž, we may identify (π‘‹ξ…ž)ξ…ž with a closed subspace of (π‘‹ξ…ž)βˆ—; hence, Theorem 2.16 implies 𝑋=π‘‹ξ…žξ…ž=ξ€·π‘‹ξ…žξ€Έξ…žβŠ†ξ€·π‘‹ξ…žξ€Έβˆ—.(3.19)
Furthermore, if π‘‹ξ…ž has absolutely continuous norm, that is π‘‹ξ…ž=π‘‹ξ…žπ‘Ž, since every Banach Function Space contains the simple functions, by Theorem 2.22 applied to the space π‘‹ξ…ž and by Theorem 2.16, we have (π‘‹ξ…ž)βˆ—=(π‘‹ξ…žπ‘Ž)βˆ—=(π‘‹ξ…ž)ξ…ž=π‘‹ξ…žξ…ž=𝑋.
On the other hand, if 𝑋=(π‘‹ξ…ž)βˆ—, then (π‘‹ξ…ž)ξ…ž=𝑋=(π‘‹ξ…ž)βˆ—, and Theorem 2.23 yields that π‘‹ξ…ž 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 𝐿1. In fact, if 𝑋=𝐿1, then ξ€·π‘‹ξ…žξ€Έβˆ—=𝐿1ξ€Έξ…žξ‚βˆ—=(𝐿∞)βˆ—βŠƒπΏ1,(3.20) as confirmed by the fact that 𝐿∞ has not absolutely continuous norm.


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