# A Decomposition of the Dual Space of Some Banach Function Spaces

Authors: Claudia Capone; Maria Rosaria Formica
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

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


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## Body

## 1. Introduction

LetΩ be a set of Lebesgue measure |Ω|<+∞.In this paper, we deal with the following issue. What is the difference between the dual spaceX* and the associate space X′ of a Banach Function Space X?By associate spaceX′ of X we mean the space determined by the associate norm ρ′: (1.1)ρ′(g)=sup⁡{∫Ωfgdx:f∈M+,ρ(f)≤1}
as in Definitions 2.3 and 2.4.IfX 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(1.2)(EXP(Ω))*=Llog⁡L(Ω)⊕(exp⁡(Ω))⊥,
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 φX verifies(1.3)φX(0+)=0,
then,(1.4)X*=X′⊕(Xb)⊥,
where Xb 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 Section3, 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 SpaceX can be identified with a closed subspace of (X′)* [1], while the spaces mentioned in our particular cases verify(1.5)X=(X′)*
as shown in Theorem 3.7.


## 2. Preliminaries

LetΩ be a set of Lebesgue measure |Ω|<+∞ and let ℳo+ be the set of all measurable functions, whose values lie in [0,+∞], finite a.e. in Ω.Definition 2.1.
A mappingρ:ℳo+→[0,+∞] is called a Banach function norm if, for all f,g,fn(n=1,2,3,…)  in ℳo+, for all constants a≥0, and for all measurable subsets E⊂Ω, the following properties hold.
(2.1)ρ(f)=0⟺f=0a.e.inΩ,ρ(af)=aρ(f),ρ(f+g)≤ρ(f)+ρ(g),0≤g≤fa.e.inΩ⟹ρ(g)≤ρ(f),0≤fn↑fa.e.inΩ⟹ρ(fn)↑ρ(f),|E|<+∞⟹ρ(χE)<+∞,|E|<+∞⟹∫Efdx≤CEρ(f)
for some constant CE, 0<CE<∞, depending on E and ρ, but independent of f.Definition 2.2.
Ifρ is a Banach function norm, the Banach space
(2.2)X={f∈M:ρ(|f|)<+∞}
is called a Banach Function Space.
For eachf∈X, define
(2.3)‖f‖X=ρ(|f|).
Recall that the simple functions are contained in every Banach Function Spaces X [1].Definition 2.3.
Ifρ is a function norm, its associate normρ′ is defined on ℳo+ by
(2.4)ρ′(g)=sup⁡{∫Ωfgdx:f∈M+,ρ(f)≤1}.Definition 2.4.
Letρ 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 spaceof 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
(2.5)‖g‖X′=sup⁡{∫Ωfgdx:f∈X,‖f‖X≤1}.Definition 2.5.
A functionf in a Banach Function Space X is said to haveabsolutely continuous norm in X if ∥fχEn∥X→0 for every sequence {En}n=1∞ satisfying En→∅a.e. The set of all functions in X of absolutely continuous norm is denoted by Xa. If X=Xa, then the space X itself is said to haveabsolutely continuous norm.Definition 2.6.
Letf∈ℳo. The function
(2.6)μf(λ)=|{x∈Ω:|f(x)|>λ}|∀λ≥0
is called the distribution function of f. The decreasing rearrangement of f,f*, is defined on [0,|Ω|] by
(2.7)f*(t)=inf⁡{λ>0:μf(λ)≤t},
where here we use the convention inf∅=+∞.
Two functions having the same distribution function are called equimeasurable.
Let us recall that a function normρ 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.
Byf**:(0,∞)→[0,∞], we denote the function given by
(2.8)f**(t)=1t∫0tf*(s)ds,(t>0).
The function f** is nonincreasing and verifies  f*(t)≤f**(t)(t>0).Definition 2.7.
LetX be a r.i. Banach Function Space determined by a function norm ρ. For each t∈[0,|Ω|], let Et⊆Ω be a set of measure t. Thefundamental functionof X, φX(t), is defined by
(2.9)φX(t)=ρ(χEt)=‖χEt‖X.Definition 2.8.
Let1≤p≤∞ and α∈R, then theZygmund spaceLp(logL)α(Ω)  is the set of all measurable functions f in Ω for which the quantity
(2.10)‖f‖Lp(log⁡L)α(Ω)=‖(1+log⁡(|Ω|t))αf*(t)‖Lp(0,|Ω|)
is finite.
Forp=1 and α=1  we will replace L1(logL)1(Ω)  by L(logL)(Ω).With these notations, the usual spaceEXPα 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
(2.11)‖f‖L∞/(log⁡L)1/α(Ω)=‖f‖EXPα(Ω)=sup⁡0<t<|Ω|(1+log⁡(|Ω|t))-1/αf*(t)
is finite.All these spaces are particular cases of the Orlicz spaces.Letϕ:[0,∞)→[0,∞)be a right-continuous, increasing function, such that ϕ(0)=0 and limt→∞ϕ(t)=∞, then the function defined by
(2.12)Φ(t)=∫0tϕ(s)ds
is called N function; it is a continuous, convex, increasing function such that limt→∞(Φ(t)/t)=+∞ and limt→0(Φ(t)/t)=0.Definition 2.9.
TheOrlicz spaceLΦ(Ω) consists of all measurable functions f on Ω for which there exists some λ>0 such that
(2.13)⨍ΩΦ(|f|λ)<∞,
where ⨍Ω stands for (1/|Ω|)∫Ω.
This is a Banach space with respect to the Luxemburg norm:(2.14)‖f‖LΦ(Ω)=inf⁡{λ>0:⨍ΩΦ(|f(x)|λ)dx≤1}.
The Orlicz spaces are a standard example of rearrangement-invariant Banach Function Space: the associate space ofLΦ(Ω) is given by LΦ̃(Ω), where Φ̃ denotes the complementary function of Φ, defined by
(2.15)Φ̃(t)=max⁡{st-Φ(s):s≥0}.
Moreover, we notice that, for Φ(t)=tp, Φ(t)=tp(logt)α, and Φ(t)=etα-1, the Orlicz space associated reduces, respectively, to the spaces Lp(Ω),Lp(logL)α(Ω) and to EXPα(Ω).Definition 2.10.
Given1≤p,q≤∞, the Lorentz spaceLp,q(Ω) consists of all measurable functions f in Ω for which
(2.16)‖f‖p,q={∫0∞[t1/pf*(t)]qdtt,0<q<∞,sup⁡t>0t1/pf*(t),q=∞
is finite.
The spaceLp,∞(Ω)=Weak-Lp(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, forq=∞, in the case of a nonatomic measure space, (2.16) is equivalent to
(2.17)sup⁡{|E|1/p-1∫E|f|dx,E⊂Ωmeasurable}.
Now, we recall the definitions of Grand and Small Lebesgue Spaces, introduced, respectively, in [5] and in [6].Definition 2.11.
Let1<p<+∞ and θ≥0; the Grand Lebesgue SpaceLp),θ is the Banach Function Space of all measurable functions f on Ω such that
(2.18)‖f‖p),θ=sup⁡0<ɛ<p-1(ɛθ⨍Ω|f|p-ɛdx)1/(p-ɛ)
is finite.
Notice that(2.19)Lp),0(Ω)=Lp(Ω),Lp),1(Ω)=Lp)(Ω).
If 1<p<+∞ and p′ is its Hölder conjugate exponent, according to [7], the Small Lebesgue SpaceL(p′,θ can be identified as the set of all measurable functions f on Ω such that
(2.20)‖f‖(p′,θ=sup⁡{|∫Ωfgdx|:‖f‖Lp),θ(Ω)≤1}
is finite.
The Grand and Small Lebesgue Spaces are r.i. Banach Function Spaces [7].Definition 2.12.
A vector spaceV is the direct sum of its subspaces U and W, denoted by V=U⊕W, if and only if
(2.21)V=U+W={u+w:u∈U,w∈W},V∩W={0}.
Elements v of the direct sum U⊕W are representable uniquely in the form
(2.22)u+w:u∈U,w∈W.Definition 2.13.
LetX be a Banach space and M⊂X a vectorial subspace of X. The orthogonal spaceM⊥ of M is
(2.23)M⊥={f∈X*:〈f,x〉=0,∀x∈M},
where 〈.,.〉 is the duality inner product.
It is known thatM⊥ 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]).
LetX be a Banach Function Space with associate space X′. If f∈X and g∈X′, then fg is integrable and
(2.24)∫Ωfgdx≤‖f‖X‖g‖X′.Lemma 2.15 (see [1, Lemma 2.6, page  10]).
In order that a measurable functiong 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 SpaceX coincides with its second associate space X′′=(X′)′.Theorem 2.17 (see [1, Theorem  2.9, page  13]).
The associate spaceX′ 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]).
IfX 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]).
LetX be a Banach Function Space. Then, Xa⊆Xb⊆X.Corollary 2.20.
IfXa=X, then Xb=X.Theorem 2.21 (see [1, Theorem  3.13, page  19]).
The subspacesXa and Xb coincide if and only if the characteristic function χE has absolutely continuous norm for every set E of finite measure.Theorem 2.22 (see [1, Corollary  4.2, page  23]).
LetX be a Banach Function Space. If Xa contains the simple functions, then (Xa)*=X′.Theorem 2.23 (see [1, Corollary  4.3, page  23]).
The Banach space dualX* 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)
limt→0+φX(t)=0;(ii)
Xa=Xb;(iii)
(Xb)*=X′,   where φX(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.
LetX 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 φX(t) be the fundamental function of X.
If(3.1)lim⁡t→0+φX(t)=0,
then the following decomposition
(3.2)X*=X′⊕(Xb)⊥
holds.Proof.
Letl∈X*, for all measurable sets F in Ω, we define the set function
(3.3)ν(F)=l(χF),
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
(3.4)l(f)=∫Ωfgdx,foranyf∈L∞(Ω).
Since l∈X* for all f∈X, it is
(3.5)l(f)≤K‖f‖X,
where K is a constant. Hence, for all f∈L∞,
(3.6)∫Ωfgdx≤K‖f‖X.
By Lemma 2.15, it follows that g∈X′.
To anyg∈X′, we can associate the functional
(3.7)lg:f∈Xb⟶∫Ωfgdx.
By Hölder's inequality, lg belongs to Xb*, which is equivalent to X′ thanks to Theorem 2.24.
Finally, letls be defined by ls=l-lg, then ls(f)=〈ls,f〉=0 for all f∈Xb. Therefore, ls belongs to (Xb)⊥.
Hence,(3.8)l=lg+ls∈X′+Xb⊥.
Since it is easily seen that X′ and Xb⊥, subspaces of X*, verify X′∩Xb⊥={0}, then the proof is complete.Remark 3.2.
Let us point out that, by Theorem2.24, the decomposition (3.2) can also be written as
(3.9)X*=(Xb)*⊕(Xb)⊥,(3.10)X*=(Xa)*⊕(Xa)⊥.Corollary 3.3.
LetX be an Orlicz space, then
(3.11)X*=X′⊕(Xb)⊥=(Xb)*⊕(Xb)⊥=(Xa)*⊕(Xa)⊥.Proof.
IfX=LΦ(Ω) is an Orlicz space, then the fundamental function is
(3.12)φX(t)=1Φ-1(1/t),∀t∈]0,|Ω|]
(see [7]). Therefore, limt→0+φX(t)=0 and the claim follows from Theorem 3.1 and Remark 3.2.Corollary 3.4.
LetX=EXPα(Ω),α>0, then
(3.13)(EXPα(Ω))*=Llog⁡1/αL(Ω)⊕(exp⁡α(Ω))⊥=(exp⁡α(Ω))*⊕(exp⁡α(Ω))⊥,
where expα(Ω) denotes the closure of L∞(Ω) in EXPα(Ω).Proof.
The result follows by Corollary3.3, and by (EXPα(Ω))′=Llog1/αL(Ω),α>0, (see [1]).Corollary 3.5.
Letp∈]1,∞[, p′ be its Hölder conjugate exponent and X=Lp,∞(Ω), then
(3.14)(Lp,∞(Ω))*=Lp′,1(Ω)⊕(Lbp,∞(Ω))⊥.Proof.
The Marcinkievicz spaceLp,∞(Ω) is the largest of all rearrangement-invariant spaces having the same fundamental function as Lp(Ω) (see [1]), which is
(3.15)φLp(t)=t1/p.
Moreover, the associate space of Lp,∞(Ω) (see [1]) is, up to equivalence of norms, the Lorentz space Lp′,1(Ω).
Therefore, the statement easily follows by Theorem3.1.
A decomposition of the dual ofLp,∞ was also given in [8].Corollary 3.6.
Letp∈]1,∞[, θ≥0 and X=Lp),θ(Ω), then
(3.16)(Lp),θ(Ω))*=L(p′,θ(Ω)⊕(Lbp),θ(Ω))⊥.Proof.
LetφX(t) be the fundamental function of the space Lp),θ(Ω), then
(3.17)φX(t)≈t1/p[log⁡(1t)]-θ/p
as t→0+ (see [7]).
Therefore the claim easily follows by Theorem3.1 and by the relation (Lp),θ(Ω))′=L(p',θ(Ω) (see [7]).In the next theorem, we show the relation between a Banach Function SpaceX and the dual of its associate space (X′)*.Theorem 3.7.
LetX be a Banach Function Space, then the following inclusion
(3.18)X⊆(X′)*
holds, with equality occurring if and only if the associate space X′ of X has absolutely continuous norm.Proof.
By Theorem2.17 applied to the Banach Function Space X′, we may identify (X′)′ with a closed subspace of (X′)*; hence, Theorem 2.16 implies
(3.19)X=X′′=(X′)′⊆(X′)*.
Furthermore, ifX′ has absolutely continuous norm, that is X′=Xa′, 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′)*=(Xa′)*=(X′)′=X′′=X.
On the other hand, ifX=(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
(3.20)(X′)*=((L1)′)*=(L∞)*⊃L1,
as confirmed by the fact that L∞ has not absolutely continuous norm.



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