A Decomposition of the Dual Space of Some Banach Function Spaces

Abstract

We give a decomposition for the dual space of some Banach Function Spaces as the Zygmund space EXP_𝛼 of the exponential integrable functions, the Marcinkiewicz space 𝐿^𝑝,∞, and the Grand Lebesgue Space 𝐿^𝑝),𝜃.

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 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^: 𝜌^(𝑔)=sup_Ω𝑓𝑔𝑑𝑥∶𝑓∈ℳ⁺,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 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 𝐿log𝐿(Ω), while the dual can be represented by(EXP(Ω))^∗=𝐿log𝐿(Ω)⊕(exp(Ω))^⟂,(1.2) 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 verifies𝜑_𝑋(0+)=0,(1.3) then,𝑋^∗=𝑋^′⊕𝑋_𝑏^⟂,(1.4) 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 Köethe 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 verify𝑋=𝑋^^∗(1.5) as shown in Theorem 3.7.

2. 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∶ℳ^+𝑜→[0,+∞] is called a Banach function norm if, for all 𝑓,𝑔,𝑓_𝑛(𝑛=1,2,3,…) in ℳ^+𝑜, for all constants 𝑎≥0, and for all measurable subsets 𝐸⊂Ω, the following properties hold. 00≤≤𝑓𝜌𝑔_𝑛(≤𝑓𝜌)(𝑓𝑓=𝜌+a0.(e𝑔𝑎.⟺)𝑓in≤)=Ω𝜌𝑓(𝑎=⟹𝑓𝜌)0(𝑓+a𝜌).𝜌e(,𝑔.(𝑔)in)≤,Ω𝜌,(𝑓),||𝐸↑||<𝑓+a.∞e.i⟹nΩ𝜌⟹𝜒_𝐸𝜌𝑓_𝑛↑𝜌(𝑓),||𝐸||<+∞⟹_𝐸<+∞,𝑓𝑑𝑥≤𝐶_𝐸𝜌(𝑓)(2.1) 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 𝑋=𝑓∈ℳ∶𝜌||𝑓||<+∞(2.2) is called a Banach Function Space.
For each 𝑓∈𝑋, define ‖𝑓‖_𝑋=𝜌||𝑓||.(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 𝜌^ is defined on ℳ^+𝑜 by 𝜌^(𝑔)=sup_Ω𝑓𝑔𝑑𝑥∶𝑓∈ℳ⁺,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 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 ‖𝑔‖_𝑋^′=sup_Ω𝑓𝑔𝑑𝑥∶𝑓∈𝑋,‖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 ‖𝑓𝜒_𝐸𝑛‖_𝑋→0 for every sequence {𝐸_𝑛}^∞𝑛=1 satisfying 𝐸_𝑛→∅a.e. 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 𝜇_𝑓(𝜆)=||𝑥∈Ω∶||𝑓(𝑥)||>𝜆||∀𝜆≥0(2.6) is called the distribution function of 𝑓. The decreasing rearrangement of 𝑓,𝑓^∗, is defined on [0,|Ω|] by 𝑓^∗(𝑡)=inf𝜆>0∶𝜇_𝑓(𝜆)≤𝑡,(2.7) 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.
By 𝑓^∗∗∶(0,∞)→[0,∞], we denote the function given by 𝑓^∗∗(𝑡)=1𝑡^𝑡0𝑓^∗(𝑠)𝑑𝑠,(𝑡>0).(2.8) The function 𝑓^∗∗ is nonincreasing and verifies 𝑓^∗(𝑡)≤𝑓^∗∗(𝑡)(𝑡>0).

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 𝜑_𝑋(𝑡)=𝜌𝜒_𝐸𝑡=‖‖𝜒_𝐸𝑡‖‖_𝑋.(2.9)

Definition 2.8. Let 1≤𝑝≤∞ and 𝛼∈𝐑, then the Zygmund space 𝐿^𝑝(log𝐿)^𝛼(Ω) is the set of all measurable functions 𝑓 in Ω for which the quantity ‖𝑓‖_𝐿^𝑝_(log𝐿)^𝛼_(Ω)=‖‖‖1+log||Ω||𝑡^𝛼𝑓^∗(𝑡)‖‖‖_{𝐿𝑝(0,|Ω|)}(2.10) is finite.
For 𝑝=1 and 𝛼=1 we will replace 𝐿¹(log𝐿)¹(Ω) by 𝐿(log𝐿)(Ω).

With these notations, the usual space EXP_𝛼 of the exponentially integrable functions corresponds to the Zygmund space (𝐿^∞/(log𝐿)^1/𝛼)(Ω) and consists of all measurable functions 𝑓 in Ω for which the quantity ‖𝑓‖_𝐿^∞_/(log𝐿)^1/𝛼_(Ω)=‖𝑓‖_EXP𝛼(Ω)=sup_{||Ω||0<𝑡<}1+log||Ω𝑡||^−1/𝛼𝑓^∗(𝑡)(2.11) 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 Φ(𝑡)=^𝑡0𝜙(𝑠)𝑑𝑠(2.12) is called N function; it is a continuous, convex, increasing function such that lim_𝑡→∞(Φ(𝑡)/𝑡)=+∞ and lim_𝑡→0(Φ(𝑡)/𝑡)=0.

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 _ΩΦ||𝑓||𝜆<∞,(2.13) where ⨍_Ω stands for (1/|Ω|)∫_Ω.
This is a Banach space with respect to the Luxemburg norm: ‖𝑓‖_𝐿^Φ(Ω)=inf𝜆>0∶_ΩΦ||𝑓(𝑥)||𝜆𝑑𝑥≤1.(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 𝐿Φ(Ω), where Φ denotes the complementary function of Φ, defined by Φ(𝑡)=max{𝑠𝑡−Φ(𝑠)∶𝑠≥0}.(2.15) Moreover, we notice that, for Φ(𝑡)=𝑡^𝑝, Φ(𝑡)=𝑡^𝑝(log𝑡)^𝛼, and Φ(𝑡)=𝑒^𝑡𝛼−1, the Orlicz space associated reduces, respectively, to the spaces 𝐿^𝑝(Ω),𝐿^𝑝(log𝐿)^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 ‖𝑓‖_𝑝,𝑞=⎧⎪⎨⎪⎩^∞0𝑡^1/𝑝𝑓^∗(𝑡)^𝑞𝑑𝑡sup_𝑡>0𝑡,0<𝑞<∞,𝑡^1/𝑝𝑓^∗(𝑡),𝑞=∞(2.16) 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 sup||𝐸||^1/𝑝−1_𝐸||𝑓||𝑑𝑥,𝐸⊂Ωmeasurable.(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 ‖𝑓‖_p),𝜃=sup_{0<𝜀<𝑝−1}𝜀^𝜃_Ω||𝑓||^𝑝−𝜀𝑑𝑥^{1/(𝑝−𝜀)}(2.18) is finite.
Notice that 𝐿^𝑝),0(Ω)=𝐿^𝑝(Ω),𝐿^p),1(Ω)=𝐿^p)(Ω).(2.19) If 1<𝑝<+∞ and 𝑝^ is its Hölder 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 ‖𝑓‖_{(𝑝′,𝜃}=sup||||_Ω𝑓𝑔𝑑𝑥||||∶‖𝑓‖_𝐿^p),𝜃_(Ω)≤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=𝑈⊕𝑊, if and only if 𝑉=𝑈+𝑊=𝑉{∩𝑢+𝑊𝑤=∶{𝑢0}∈.𝑈,𝑤∈𝑊},(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 𝑀^⟂=𝑓∈𝑋^∗∶⟨𝑓,𝑥⟩=0,∀𝑥∈𝑀,(2.23) 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 (Hölder'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 _Ω𝑓𝑔𝑑𝑥≤‖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.

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).

Theorem 2.19 (see [1, Theorem  3.11, page  18]). Let 𝑋 be a Banach Function Space. Then, 𝑋_𝑎⊆𝑋_𝑏⊆𝑋.

Corollary 2.20. If 𝑋_𝑎=𝑋, then 𝑋_𝑏=𝑋.

Theorem 2.21 (see [1, Theorem  3.13, page  19]). 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.

Theorem 2.22 (see [1, Corollary  4.2, page  23]). Let 𝑋 be a Banach Function Space. If 𝑋_𝑎 contains the simple functions, then (𝑋_𝑎)^∗=𝑋′.

Theorem 2.23 (see [1, 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: (i)lim_𝑡→0⁺𝜑_𝑋(𝑡)=0; (ii)𝑋_𝑎=𝑋_𝑏; (iii)(𝑋_𝑏)^∗=𝑋′,   where 𝜑_𝑋(𝑡) 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 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.
If lim_𝑡→0+𝜑_𝑋(𝑡)=0,(3.1) then the following decomposition 𝑋^∗=𝑋^⊕𝑋_𝑏^⟂(3.2) holds.

Proof. Let 𝑙∈𝑋^∗, for all measurable sets 𝐹 in Ω, we define the set function 𝜈(𝐹)=𝑙𝜒_𝐹,(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 𝑙(𝑓)=_Ω𝑓𝑔𝑑𝑥,forany𝑓∈𝐿^∞(Ω).(3.4) Since 𝑙∈𝑋^∗ for all 𝑓∈𝑋, it is 𝑙(𝑓)≤𝐾‖𝑓‖_𝑋,(3.5) where 𝐾 is a constant. Hence, for all 𝑓∈𝐿^∞, _Ω𝑓𝑔𝑑𝑥≤𝐾‖𝑓‖_𝑋.(3.6) By Lemma 2.15, it follows that 𝑔∈𝑋^′.
To any 𝑔∈𝑋^, we can associate the functional 𝑙_𝑔∶𝑓∈𝑋_𝑏⟶_Ω𝑓𝑔𝑑𝑥.(3.7) By Hölder'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 𝑋𝑋^∗=𝑋_𝑏^∗⊕𝑋_𝑏^⟂,(3.9)_∗=𝑋_𝑎^∗⊕𝑋_𝑎^⟂.(3.10)

Corollary 3.3. Let 𝑋 be an Orlicz space, then 𝑋^∗=𝑋^⊕𝑋_𝑏^⟂=𝑋_𝑏^∗⊕𝑋_𝑏^⟂=𝑋_𝑎^∗⊕𝑋_𝑎^⟂.(3.11)

Proof. If 𝑋=𝐿^Φ(Ω) is an Orlicz space, then the fundamental function is 𝜑_𝑋(𝑡)=1Φ⁻¹(1/𝑡),∀𝑡∈0,||Ω||(3.12) (see [7]). Therefore, lim_𝑡→0⁺𝜑_𝑋(𝑡)=0 and the claim follows from Theorem 3.1 and Remark 3.2.

Corollary 3.4. Let 𝑋=EXP_𝛼(Ω),𝛼>0, then EXP_𝛼(Ω)^∗=𝐿log^1/𝛼𝐿(Ω)⊕exp_𝛼(Ω)^⟂=exp_𝛼(Ω)^∗⊕exp_𝛼(Ω)^⟂,(3.13) where exp_𝛼(Ω) denotes the closure of 𝐿^∞(Ω) in EXP_𝛼(Ω).

Proof. The result follows by Corollary 3.3, and by (EXP_𝛼(Ω))^′=𝐿log^1/𝛼𝐿(Ω),𝛼>0, (see [1]).

Corollary 3.5. Let 𝑝∈]1,∞[, 𝑝^ be its Hölder conjugate exponent and 𝑋=𝐿^𝑝,∞(Ω), then (𝐿^𝑝,∞(Ω))^∗=𝐿^𝑝′,1(Ω)⊕𝐿^𝑏𝑝,∞(Ω)^⟂.(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].

Corollary 3.6. Let 𝑝∈]1,∞[, 𝜃≥0 and 𝑋=𝐿^p),𝜃(Ω), then 𝐿^𝑝),𝜃(Ω)^∗=𝐿^{(𝑝′,𝜃}(Ω)⊕𝐿^{𝑏𝑝),𝜃}(Ω)^⟂.(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1𝑡^−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]).

In the next theorem, we show the relation between a Banach Function Space 𝑋 and the dual of its associate space (𝑋^)^∗.

Theorem 3.7. Let 𝑋 be a Banach Function Space, then the following inclusion 𝑋⊆𝑋^^∗(3.18) holds, with equality occurring if and only if the associate space 𝑋^ of 𝑋 has absolutely continuous norm.

Proof. By Theorem 2.17 applied to the Banach Function Space 𝑋^, we may identify (𝑋^)^ with a closed subspace of (𝑋^)^∗; hence, Theorem 2.16 implies 𝑋=𝑋^=𝑋^^⊆𝑋^^∗.(3.19)
Furthermore, if 𝑋^ has absolutely continuous norm, that is 𝑋^=𝑋^𝑎, since every Banach Function Space contains the simple functions, by Theorem 2.22 applied to the space 𝑋^ and by Theorem 2.16, we have (𝑋^)^∗=(𝑋^𝑎)^∗=(𝑋^)^=𝑋^=𝑋.
On the other hand, if 𝑋=(𝑋^)^∗, then (𝑋^)^=𝑋=(𝑋^)^∗, and Theorem 2.23 yields that 𝑋^ 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 𝑋^^∗=𝐿¹^^∗=(𝐿^∞)^∗⊃𝐿¹,(3.20) as confirmed by the fact that 𝐿^∞ has not absolutely continuous norm.