# Duality of Variable Exponent Triebel-Lizorkin and Besov Spaces

Authors: Takahiro Noi
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

We will prove the duality and reflexivity of variable exponent Triebel-Lizorkin and Besov spaces. It was shown by many authors that variable exponent Triebel-Lizorkin spaces coincide with variable exponent Bessel
potential spaces, Sobolev spaces, and Lebesgue spaces when appropriate indices are chosen. In consequence of the results, these variable exponent function spaces are shown to be reflexive.


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## Body

## 1. Introduction

Recently, variable exponent function spaces have been studied by many authors [1–13], and in particular, the papers about the variable exponent Triebel-Lizorkin and Besov spaces have been published in [14–18]. Diening et al. [15] and Almeida and Hästö [14] studied the spaces Fp(·),q(·)α(·) and Bp(·),q(·)α(·) and Xu studied [16–18] the spaces Fp(·),qs(ℝn) and Bp(·),qs(ℝn).In this paper, we will show the duality of variable exponent Triebel-Lizorkin spacesFp(·),qs(ℝn) and Besov spaces Bp(·),qs(ℝn) under suitable conditions by using the same arguments in Triebel [19]. The duality follows from the fact that Fp(·),qs(ℝn) and Bp(·),qs(ℝn) are reflexive under same conditions.Xu [16] showed that variable exponent Bessel potential space Ls,p(·)(ℝn) coincides with Fp(·),2s(ℝn) if s≥0 and p(·)∈ℬ(ℝn). Diening et al. [15] showed that the variable exponent Lebesgue space Lp(·)(ℝn) coincides with Fp(·),20(ℝn) under suitable assumptions on p(·). Gurka et al. [7] showed that Lk,p(·)(ℝn) coincides with variable exponent Sobolev spaces Wk,p(·)(ℝn) if k∈ℕ and p(·)∈ℬ(ℝn). In consequences of these results, we have the duality and reflexivity of Ls,p(·)(ℝn)(s≥0), although the duality and reflexivity of Lp(·)(ℝn) have been obtained in Kováčik and Rákosník [9] under the assumptions on p(·) which are weaker than ours.


## 2. Definitions of Variable Exponent Function Spaces

We first introduce variable exponent Lebesgue spaces. Letp(·) be a measurable function on ℝn with range in (1,∞). Let Lp(·)(ℝn) denote the set of all complex-valued functions f on ℝn such that, for some λ>0,
(2.1)∫Rn(|f(x)|λ)p(x)dx<∞.
The set becomes a Banach function space when it is equipped with the Luxemburg-Nakano norm:
(2.2)‖f‖Lp(⋅)=inf⁡{λ>0:∫Rn(|f(x)|λ)p(x)dx≤1}.
If p(x)≡p is a constant function, then the above norm coincides with the usual Lp-norm and so the notation is not confusional. In Kováčik and Rákosník [9], variable exponent Lebesgue spaces are defined on arbitrary measurable subset of ℝn. Denote by 𝒫(ℝn) the set of measurable functions p(·) on ℝn with range in (1,∞) such that
(2.3)1<p-=essinf⁡x∈Rnp(x),esssup⁡x∈Rnp(x)=p+<∞.
For a complex-valued locally Lebesgue-integrable function f(·) on ℝn, let
(2.4)(Mf)(x)=sup⁡1|B|∫B|f(y)|dy
denote the Hardy-Littlewood maximal function, where the supremum is taken over all balls B centered at x. There exists p(·)∈𝒫(ℝn) such that the Hardy-Littlewood maximal operator ℳ is not bounded on Lp(·)(ℝn) [11], although the operator ℳis bounded on Lp(ℝn) for p>1. Some sufficient conditions on p(·) for maximal operator ℳ to be bounded on Lp(·)(ℝn) are known. Let ℬ(ℝn) be the set of p(·)∈𝒫(ℝn) such that the Hardy-Littlewood maximal operator ℳ is bounded on Lp(·)(ℝn).Let𝒮(ℝn) be the Schwartz space of all complex-valued rapidly decreasing and infinitely differentiable functions on ℝn. Let 𝒮′(ℝn) be the set of all the tempered distribution on ℝn. For φ∈𝒮(ℝn), let ℱφ denote the Fourier transform of φ and ℱ-1φ the inverse Fourier transform of φ. We write ℱ-1mℱf=ℱ-1[m·ℱf] for the sake of simplicity.Lets≥0 and p(·)∈𝒫(ℝn). The variable exponent Bessel potential space Ls,p(·)(ℝn) is the collection of f∈Lp(·)(ℝn) such that the norm:
(2.5)‖f‖Ls,p(⋅)=‖F-1(1+|⋅|2)s/2Ff(⋅)‖Lp(⋅)<∞.Letk∈ℕ and p∈𝒫(ℝn). The variable exponent Sobolev space Wk,p(·)(ℝn) is the collection of f∈Lp(·)(ℝn) such that the derivatives (in the sense of distribution) up to the order k belong to Lp(·)(ℝn) and the norm:
(2.6)‖f‖Wk,p(⋅)=∑|α|≤k‖Dαf‖Lp(⋅)<∞,
where α is a multi-index and |α|=α1+⋯+αn.LetΦ∈𝒮(ℝn) be a function such that suppℱΦ⊂B(0,1) and ℱΦ=1 on B(0,1/2), where B(0,λ) is an open ball centered at 0 with radius λ. Set Φj(x)=2njΦ(2jx) for x∈ℝn and j∈ℤ. We also set
(2.7)θj(x)=Φj(x)-Φj-1(x).
It follows that
(2.8)∑j=0∞Fθj(ξ)≡1,
where θ0=Φ0.Definition 2.1.
Lets∈ℝ, 0<q<∞, and p(·)∈𝒫(ℝn). Let θj, j∈ℕ0=ℕ∪{0} as above. The variable exponent Triebel-Lizorkin space Fp(·),qs(ℝn) is the collection of f∈𝒮′(ℝn) such that
(2.9)‖f‖Fp(⋅),qs=‖{2sjθj*f}0∞‖Lp(⋅)(lq)<∞.
Let s∈ℝ, 0<q≤∞, and p(·)∈𝒫(ℝn). The variable exponent Besov space Bp(·),qs(ℝn) is the collection of f∈𝒮′(ℝn) such that
(2.10)‖f‖Bp(⋅),qs=‖{2sjθj*f}0∞‖lq(Lp(⋅))<∞.Here,Lp(·)(ℓq) and ℓq(Lp(·)) are the spaces of all sequences {gj} of measurable functions on ℝn such that quasi-norms:
(2.11)‖{gj}0∞‖Lp(⋅)(lq)=‖‖{gj}0∞‖lq‖Lp(⋅)=‖(∑j=0∞|gj(⋅)|q)1/q‖Lp(⋅),‖{gj}0∞‖lq(Lp(⋅))=‖{‖gj‖Lp(⋅)}0∞‖lq=(∑j=0∞‖gj(⋅)‖Lp(⋅)q)1/q
are finite, respectively.Definition 2.2.
Letp(·)∈𝒫(ℝn) and 0<q≤∞. For a sequence of compact subsets Ω={Ωk}k=0∞ of ℝn, Lp(·)Ω(ℓq) denotes the space of all sequences {fk}k=0∞ of 𝒮′(ℝn) such that
(2.12)supp⁡Ffk⊂Ωkfork=0,1,2,…,
and ∥fk∥Lp(·)(ℓq)<∞. For a compact subset Γ of ℝn, Lp(·)Γ(ℝn) denotes the space of all elements f∈𝒮′(ℝn) such that
(2.13)supp⁡Ff⊂Γ,
and ∥f∥Lp(·)<∞.


## 3. Preliminaries

We need the following fundamental properties ofLp(·)(ℝn).Theorem 3.1 (see [5, Theorem 8.1]).
Letp(·)∈𝒫(ℝn). Then, the following conditions are equivalent:
(a)p(·)∈ℬ(ℝn),
(b)p(·)/t∈ℬ(ℝn) for some 1<t<p-,
(c)p′(·)∈ℬ(ℝn), where
(3.1)p′(⋅)=p(⋅)p(⋅)-1.In [1, 5], some other conditions equivalent to the above are given. Let p(·)∈ℬ(ℝn) and 0<q<∞. Then we write
(3.2)rp-,q=sup⁡{r∈R:0<r<min⁡(p-,q),p(⋅)r∈B(Rn)}.
If q=p-, then, we write rp-,p-=rp-.Remark 3.2.
Letp(·)∈ℬ(ℝn) and t as in Theorem 3.1. Then, for any w∈(1,t], we have p(·)/w∈ℬ(ℝn). Furthermore, for any w∈(0,1], we have also p(·)/w∈ℬ(ℝn) by Jensen inequality. The proofs are found in [5]. Consequently, if p(·)∈ℬ(ℝn), then, p(·)/r∈ℬ(ℝn) for any r∈(0,rp-). For any r∈(0,rp-,q), we have also p(·)/r∈ℬ(ℝn).The next theorem gives a generalized Hölder inequality, which is shown in [9, 13].Theorem 3.3 (see [9, 13]).
Letp∈𝒫(ℝn). Then,
(3.3)∫Rn|f(x)g(x)|dx≤(1+1p--1p+)‖f‖Lp(⋅)‖g‖Lp′(⋅),
for every f∈Lp(·)(ℝn) and g∈Lp′(·)(ℝn).The next theorem is shown in [6].Theorem 3.4.
Letp(·)∈𝒫(ℝn) and f∈Lp(·)(ℝn). Then,
(3.4)min⁡{(∫Rn|f(x)|p(x)dx)1/p-,(∫Rn|f(x)|p(x)dx)1/p+}≤‖f‖Lp(⋅)≤max⁡{(∫Rn|f(x)|p(x)dx)1/p-,(∫Rn|f(x)|p(x)dx)1/p+}.The next theorem is shown in [9].Theorem 3.5.
(i) Letp(·)∈𝒫(ℝn). Then, Lp(·)(ℝn) is a Banach space.
(ii) Ifp(·)∈ℬ(ℝn), then C0∞(ℝn) is dense in Lp(·)(ℝn).Letp(·)∈𝒫(ℝn) such that p(·) is not a constant function. Then, according to [9, Example 2.9, Theorem 2.10], for every Lp(·), there exists a function f(·)∈Lp(·)(ℝn) such that its translation f(β+·)∉Lp(·)(ℝn) if β∈ℝn∖{(0,0,…,0)}. However, we can prove that all elements of 𝒮(ℝn) belong to Lp(·)(ℝn) for every p(·)∈𝒫(ℝn). Let f(·)∈𝒮(ℝn). Then, we have
(3.5)sup⁡β∈Rn‖f(β+α⋅)‖Lp(⋅)≤{|α|-n/p+πn/p-sup⁡x∈Rn{(1+|x|2)n|f(x)|}if|α|≥1,|α|-n/p-πn/p-sup⁡x∈Rn{(1+|x|2)n|f(x)|}if|α|<1,
where α∈ℝ∖{0}.Hence, by the same arguments of Triebel [20], we have the following two theorems.Theorem 3.6.
LetΩ be a compact subset of ℝn and α an arbitrary multi-index. Let p(·), q(·)∈ℬ(ℝn) such that 1<p(x)≤q(x)<∞. Then, there exists a positive constant c such that
(3.6)‖Dαf‖Lq(⋅)≤c‖f‖Lp(⋅),(3.7)‖Dαf‖L∞≤c‖f‖Lp(⋅),
for all f∈Lp(·)Ω(ℝn).Letf(·)∈Lp(·)Ω(ℝn) and ℱ-1M∈L1(ℝn). Then,
(3.8)(F-1MFf)(x)=c∫Rn(F-1M)(x-y)f(y)dy
makes sense for any x∈ℝn by the classical Hölder inequality and (3.7).Lets be a real number and H2s(ℝn) the spaces of f∈𝒮′(ℝn) such that
(3.9)‖f‖H2s=‖(1+|⋅|2)s/2(Ff)(⋅)‖L2<∞.Theorem 3.7.
Letp(·)∈ℬ(ℝn) and 0<q<∞. Let Ω={Ωk}k=0∞ be a sequence of compact subsets of ℝn. Let dk>0 be the diameter of Ωk. If v>n/2+n/rp-,q, then there exists a number c such that
(3.10)‖F-1MkFfk‖Lp(⋅)(lq)≤csup⁡l‖Ml(dl⋅)‖H2v‖fk‖Lp(⋅)(lq)
for {fk(x)}k=0∞∈Lp(·)Ω(ℓq) and {Mk(x)}k=0∞∈H2v(ℝn).


## 4. Basic Properties of Variable Exponent Triebel-Lizorkin and Besov Spaces

We will first consider the quasi-norms onFp(·),qs(ℝn) and Bp(·),qs(ℝn).Definition 4.1.
(i) The setΨ(ℝn) is the collection of all systems φ={φj}j=0∞⊂𝒮(ℝn) with ℱφk(ξ)=ℱφ(2-kξ) for k=1,2,…, such that
(4.1)supp⁡⁡Fφ0⊂{x:|x|≤2},supp⁡⁡Fφj⊂{x:2j-1≤|x|≤2j+1}forj=1,2,…,
for every multi-index α, there exists a positive number cα such that
(4.2)2j|α||DαFφj(x)|≤cα,
for j=0,1,…, and x∈ℝn and there exists a positive number c such that
(4.3)c≤∑j=0∞Fφj(x),
for x∈ℝn.
(ii) The setΦ(ℝn) is the collection of all systems φ={φj}j=0∞⊂𝒮(ℝn) such that
(4.4)supp⁡φ0⊂{x:|x|≤2},supp⁡φj⊂{x:2j-1≤|x|≤2j+1}forj=1,2,…,
for every multi-index α, there exists a positive number cα such that
(4.5)2j|α||Dαφj(x)|≤cα,
for j=0,1,…, and x∈ℝn, and
(4.6)∑j=0∞φj(x)=1,
for x∈ℝn.Theorem 4.2.
(i) Letp(·)∈𝒫(ℝn), 0<q≤∞, and s∈ℝ. Let θj be as in Definition 2.1 and φj=ℱθj. Then, the norm ∥f∥Fp(·),qs coincides with ∥2jsℱ-1φjℱf∥Lp(·)(ℓq) and the norm ∥f∥Bp(·),qs with ∥2jsℱ-1φjℱf∥ℓq(Lp(·)).
(ii) Letp(·)∈ℬ(ℝn), 0<q≤∞, and s∈ℝ. Let φ, ψ∈Φ(ℝn). Then, ∥2jsℱ-1φjℱf∥Lp(·)(ℓq) and ∥2jsℱ-1ψjℱf∥Lp(·)(ℓq) are equivalent quasi-norms on Fp(·),qs(ℝn). Similarly, ∥2jsℱ-1φjℱf∥ℓq(Lp(·)) and ∥2jsℱ-1ψjℱf∥ℓq(Lp(·)) are equivalent quasi-norms on Bp(·),qs(ℝn).
(iii) Letp(·)∈ℬ(ℝn), 0<q≤∞, and s∈ℝ. Let φ∈Φ(ℝn) and ψ∈Ψ(ℝn). Then, ∥2jsℱ-1φjℱf∥Lp(·)(ℓq) and ∥2jsψj*f∥Lp(·)(ℓq) are equivalent quasi-norms on Fp(·),qs(ℝn). Similarly, ∥2jsℱ-1φjℱf∥ℓq(Lp(·)) and ∥2jsψj*f∥ℓq(Lp(·)) are equivalent quasi norms on Bp(·),qs(ℝn).Proof.
Letθj be as in Definition 2.1. It is obvious that θj*f=dℱℱ-1(ℱθj·ℱf) for some positive number dℱ, which depends only on the definition of Fourier transform ℱ. Let φj=ℱθj. Then {φj}j=0∞∈Φ(ℝn). Hence, the norm ∥f∥Fp(·),qs coincides with ∥2jsℱ-1φjℱf∥Lp(·)(ℓq). Similarly, the variable exponent Besov norm ∥f∥Bp(·),qs coincides with ∥2jsℱ-1φjℱf∥ℓq(Lp(·)). This proves  (i). The proof of  (ii) is done as in the proof of [20, page 46, Proposition 1] with using Theorem 3.7, the scalar case of Theorem 3.7, and Definition 4.1.
We will prove  (iii). Letf∈Fp(·),qs(ℝn), gj=ℱ-1(φjℱf), and ψ∈Ψ(ℝn). Let ψk≡gk≡0 for k<0. Then,
(4.7)f*ψk(x)=S′∑j=0∞gj*ψk(x)=∑j=k-2k+2(gj*ψk)(x),
because
(4.8)gj*ψk=F-1F(gj*ψk)=dFF-1(Fψk⋅Fgj)=0,
for j<k-2 and j>k+2. Here, f=𝒮′∑0∞aj means ∑0Naj converges to f in 𝒮′(ℝn). It follows that
(4.9)‖{2skf*ψk}0∞‖Lp(⋅)(lq)≤∑r=-22‖{2skgk+r*ψk}0∞‖Lp(⋅)(lq)≤∑r=-22‖Fψ(2r+1⋅)‖H2ϰ‖{2skgk+r}0∞‖Lp(⋅)(lq),
where we use Theorem 3.7 with fk=gk+r, Mk=ℱψk, and dk=2k+1+r. Hence, we have
(4.10)‖{2skf*ψk}0∞‖Lp(⋅)(lq)≤∑r=-22‖Fψ(2r+1⋅)‖H2ϰ‖f‖Fp(⋅),qs.
We will prove the opposite inequality. Let ρ(x) be a real function with ρ(x)=1 for 2-1≤|x|≤2 and
(4.11)ρ(⋅)∈C0∞({ξ:2-2≤|ξ|≤22}).
We set ρk(x)=ρ(2-kx), for k=1,2,…. Furthermore, let ρ0(x) be a real function such that ρ0(x)=1 for |x|≤2 and
(4.12)ρ0(⋅)∈C0∞({ξ:|ξ|≤22}).
We define hk(x) by
(4.13)Fhk(ξ)=(∑l=0∞Fψl(ξ))-1ρk(ξ),
for k=0,1,2,… (hk=0 for k<0). Then, ℱhk(·)∈C0∞(ℝn) and hk(·)∈𝒮(ℝn). Let γk=ψk*hk. It is obvious that {ℱγk/dℱ}∈Φ(ℝn) and {2skψk*f}0∞∈Lp(·)Ω(ℓq) by Theorem 4.8 and (4.10), where Ω is some sequence of compact subsets of ℝn. Then,
(4.14)‖{2ksf*γk}0∞‖Lp(⋅)(lq)=‖{2ksf*ψk*hk}0∞‖Lp(⋅)(lq)≤sup⁡k‖Fhk(2k+1⋅)‖H2ϰ‖{2ksf*ψk}0∞‖Lp(⋅)(lq),Fhk(2k+1x)=ρk(2k+1x)∑l=0∞Fψl(2k+1x)=ρk(2k+1x)∑l=k-3k+3Fψl(2k+1x)=ρ(2x)∑r=-44Fψ(2rx),
for any k=1,2,…. This means that ℱhk(2k+1x) is independent of k∈ℕ and in C0∞(ℝn). Hence, we have supk∥ℱhk(2k+1·)∥H2ϰ<∞. This implies that
(4.15)‖f‖Fp(⋅),qs≤sup⁡k‖Fhk(2k+1⋅)‖H2ϰ‖{2ksf*ψk}0∞‖Lp(⋅)(lq).
This proves the Fp(·),qs case of  (iii). The proof of the Bp(·),qs case of  (iii) is essentially the same as one of the Fp(·),qs case.Corollary 4.3.
Letp(·)∈ℬ(ℝn), s∈ℝ, and 1<q<∞.
(i)f∈Fp(·),qs(ℝn) if and only if there exist continuous functions {aj}j∈ℕ0 such that f=𝒮′∑j=0∞aj, ∥{2jsaj}0∞∥Lp(·)(ℓq)<∞,
(4.16)supp⁡Fa0⊂{ξ:|ξ|≤2},supp⁡Faj⊂{ξ:2j-1≤|ξ|≤2j+1},forj∈N0.
In particular, ∥f∥Fp(·),qs coincides with inff=∑aj∥{2jsaj}0∞∥Lp(·)(ℓq), where the infimum is taken over all representation f=∑aj.
(ii)f∈Bp(·),qs(ℝn) if and only if, there exist continuous functions {aj}j∈ℕ0 such that f=𝒮′∑j=0∞aj, ∥{2jsaj}0∞∥ℓq(Lp(·))<∞, suppℱa0⊂{ξ:|ξ|≤2}, and suppℱaj⊂{ξ:2j-1≤|ξ|≤2j+1} for j∈ℕ0. In particular, the norm ∥f∥Bp(·),qs coincides with inff=∑aj∥{2jsaj}0∞∥ℓq(Lp(·)).Proposition 4.4.
(i) Letp(·)∈𝒫(ℝn), s∈ℝ, and 0<q1≤q2≤∞. Then,
(4.17)Bp(⋅),q1s(Rn)⊂Bp(⋅),q2s(Rn),Fp(⋅),q1s(Rn)⊂Fp(⋅),q2s(Rn).
(ii) Letp(·)∈𝒫(ℝn), s∈ℝ, 0<q1≤∞, 0<q2≤∞, and ϵ>0. Then
(4.18)Bp(⋅),q1s+ϵ(Rn)⊂Bp(⋅),q2s(Rn),Fp(⋅),q1s+ϵ(Rn)⊂Fp(⋅),q2s(Rn).
(iii) Letp(·)∈𝒫(ℝn) and s∈ℝ. If 0<q≤∞, then
(4.19)Bp(⋅),min⁡⁡{p-,q}s(Rn)⊂Fp(⋅),qs(Rn)⊂Bp(⋅),max⁡⁡{p+,q}s(Rn).The proof is almost same as in [20]. Furthermore, Almeida and Hästö [14] proved the above inclusion for Fp(·),q(·)α(·)(ℝn) and Bp(·),q(·)α(·)(ℝn).Theorem 4.5.
Let0<q<∞, s∈ℝ, and p(·)∈ℬ(ℝn). Let Ap(·),qs be either Bp(·),qs(ℝn) or Fp(·),qs(ℝn). Then, 𝒮(ℝn)⊂Ap(·),qs⊂𝒮′(ℝn). Furthermore, S(ℝn) is dense in Ap(·),qs(ℝn) and Ap(·),qs(ℝn) is a quasi Banach space (a Banach space if 1≤q≤∞).Proof.
To prove the inclusion, we may restrict ourselves toBp(·),qs(ℝn) with q=∞ by Proposition 4.4. Let f∈𝒮(ℝn) and φ={φk(x)}k=0∞∈Φ(ℝn). We recall that the topology of the complete locally convex space 𝒮(ℝn) is generated by seminorms:
(4.20)pN(f)=sup⁡x∈Rn∑|α|+k≤N(1+|x|2)k|Dαf(x)|,N∈N,ℱ yields a one-to-one mapping from 𝒮(ℝn) onto itself, and, in particular, pN(ℱφ) with N=1,2,… generates the topology of 𝒮(ℝn). If N is a sufficiently large natural number, then
(4.21)‖f‖Bp(⋅),qs=sup⁡k2ks‖F-1φkFf‖Lp(⋅)≤cpN(Ff),
where c is a positive number. This proves the left-hand side of the inclusion with q=∞. We prove the right-hand side of the inclusion with q=∞. Let φ∈Φ(ℝn) be the above system. It is well known that
(4.22)f(ψ)=∑k=0∞F-1φkFf(ψ)
in 𝒮′(ℝn) for all f∈𝒮′(ℝn) and ψ∈𝒮(ℝn). By the Paley-Wiener-Schwartz Theorem, the definition of φk implies that ℱ-1φkℱf are analytic functions, and that these functions are regular distributions. We put χk(x)=φk-1(x)+φk(x)+φk+1(x) if k=0,1,…, with φ-1=0. If f∈Bp(·),∞s(ℝn) and ψ∈𝒮(ℝn), then f(ψ) denotes the value of the functional f of 𝒮′(ℝn) of the test function ψ. We obtain
(4.23)|f(ψ)|≤|∑k=0∞F-1φkFf(FχkF-1ψ)|≤∑k=0∞‖F-1φkFf‖L∞‖FχkF-1ψ‖L1
by the classical Hölder inequality. Because both ℱ-1φkℱf and ℱχkℱ-1ψ are analytic functions, the last estimate makes sense. By applying (3.7) with α=(0,0,…,0) to ℱ-1φkℱf, we have
(4.24)|f(ψ)|≤c‖f‖Bp(⋅),∞s∑k=0∞2-sk‖FχkF-1ψ‖L1≤c‖f‖Bp(⋅),∞s‖ψ‖B1,1-s.
If N is a sufficiently large natural number, then, for any ψ∈𝒮(ℝn), we have
(4.25)|f(ψ)|≤c‖f‖Bp(⋅),∞spN(ψ)
by the left-hand side of the inclusion for B1,1-s(ℝn). Hence, we have the inclusion for Ap(·),qs.
The proof of the completeness and the density of𝒮(ℝn) is given by the same arguments of step 4 and step 5 of the proof of [20, page 48, Theorem] with p replaced by p(·).As we mentioned in Section1, the next theorem is found in [15, 16].Theorem 4.6.
Lets≥0 and p(·)∈ℬ(ℝn). Then, Fp(·),2s(ℝn) coincides with Ls,p(·)(ℝn). Let k∈ℕ and p(·)∈ℬ(ℝn). Then, Fp(·),2k(ℝn) coincides with Wk,p(·)(ℝn).Definition 4.7.
Letφ∈𝒮(ℝn) such that (ℱφ)(ξ)=1 for 1/2≤|ξ|≤2 and
(4.26)0≤Fφ∈C0∞({ξ∈Rn:12-ϵ<|ξ|<2+ϵ}),
and let ρ∈𝒮(ℝn) such that
(4.27)0≤Fρ∈C0∞({ξ∈Rn:12+δ<|ξ|<2-δ}),
where ϵ and δ are positive numbers. We construct {φk}k=0∞ and {ρk}k=0∞ by
(4.28)(Fφk)(ξ)=(Fφ)(2-kξ),(Fρk)(ξ)=(Fρ)(2-kξ).
We choose ϵ and δ sufficiently small so that
(4.29)Fφk⋅Fρj≡0
for k≠j.Diening [3] shows that Young inequality ∥f*g∥Lp(·)≤∥f∥Lp(·)∥g∥L1 holds if and only if p(·) is a constant function. However, the next theorem, which is a part of Corollary 3.6 of [3], holds.Theorem 4.8.
Letp(·)∈ℬ(ℝn). Let ϕ be an integrable function on ℝn with range in ℝ and set ϕϵ(x)=ϵ-nϕ(x/ϵ) for all ϵ>0. Assume that the least decreasing radial majorant of ϕ is integrable, that is,  A=∫ℝnsup|y|≥|x||ϕ(y)|dx<∞. Then, for any f∈Lp(·)(ℝn), one has
(4.30)sup⁡ϵ>0|(f*ϕϵ)(x)|≤2AMf(x).
Hence, there exist a positive number C(A,p) such that
(4.31)‖f*ϕϵ‖Lp(⋅)≤C(A,p)‖f‖Lp(⋅),
where C(A,p) only depends on A and p(·).The duality pair ofg∈(Fp(·),qs(ℝn))′ and f∈Fp(·),qs(ℝn) is denoted by 〈g,f〉, although, for φ∈𝒮(ℝn)⊂Fp(·),qs(ℝn), we also write g(φ) as 〈g,φ〉. In order to prove the duality of Fp(·),qs(ℝn) and Bp(·),qs(ℝn) (Theorem 5.1), we will first prove the following two lemmas which correspond to [19, Lemmas  7.1.3, and 7.1.5].Lemma 4.9.
Letg∈(Fp(·),qs)′, {ρk} as in Definition 4.7 and
(4.32)g̃=∑k=0∞F-1(Fρk⋅Fg)
in 𝒮′(ℝn). Then, g̃∈(Fp(·),qs)′ and
(4.33)‖g̃‖(Fp(⋅),qs)′≤c‖g‖(Fp(⋅),qs)′,
where c does not depend on g.Proof.
We use the same argument of the proof of [19, Lemma 7.1.3]. Let f∈𝒮(ℝn). Then, it holds that
(4.34)g̃(f)=(Fg̃)(F-1f)=∑k=0∞Fg(FρkF-1f)=c∑k=0∞g(ρk∨*f)=cg(∑k=0∞ρk∨*f),
where ρk∨(x)=ρk(-x). Using g∈(Fp(·),qs)′ and Theorem 3.7 with Mk=ℱ(ρk∨) and dk=2k+1, we find that
(4.35)|g̃(f)|≤‖g‖(Fp(⋅),qs)′‖{2ksρk∨*φk*f}0∞‖Lp(⋅)(lq)≤c‖g‖(Fp(⋅),qs)′‖{2ksφk*f}0∞‖Lp(⋅)(lq),
where {φk}k=0∞ is a system of Definition 4.7. Thus, we get
(4.36)|g̃(f)|≤c‖g‖(Fp(⋅),qs)′‖f‖Fp(⋅),qs,
from Definition 2.1 and Theorem 4.2.Lemma 4.10.
Let{ρk} and {φk} be ones as in Definition 4.7, g∈(Fp(·),qs)′, and g̃ the functional in Lemma 4.9. Let {bk}k=0N be a system of functions such that bk∈Lp(·)(ℝn). Set
(4.37)ak=F-1(Fρk⋅Fg),(4.38)ck=bk*φk,
for k=0,1,2,…,N, and
(4.39)f=∑k=0Nck.
Then,
(4.40)〈g̃,f〉=c∫Rn∑k=0Nakbkdx,
where c depends only on the definition of Fourier transform.Proof.
We use the same argument of the proof of [19, Lemma 7.1.5]. We first prove that ak∈Lp′(·), where p′(·) is defined by (3.1). The consideration of the proof of Lemma 4.9 implies that for f∈𝒮(ℝn),
(4.41)|ak(f)|≤c‖g‖(Fp(⋅),qs)′‖{2sjδj,kφk*f}j‖Lp(⋅)(lq)≤c′‖f‖Lp(⋅),
where δk,k=1 and δj,k=0 if j≠k. This implies ak∈Lp′(·) and the right-hand side of (4.40) has a sense. We assume bk∈𝒮(ℝn). Then, ck∈𝒮(ℝn) and f∈𝒮(ℝn). By (4.32), we have
(4.42)〈g̃,f〉=g̃(f)=∑k=0∞ak(f)
in 𝒮′(ℝn) and
(4.43)ak(f)=∑l=0Nak(cl)=∑l=0NFak(F-1cl)=c∑l=0N(Fak⋅F-1φl)(F-1bl).
Since (ℱ-1h)(ξ)=(ℱh)(-ξ), (4.29), (4.37), and (4.38) imply that
(4.44)ak(f)=cFak(F-1bk)=c∫Rnakbkdx.
From this, (4.40) follows for bk∈𝒮(ℝn). Let bk∈Lp(·)(ℝn), bk,j∈𝒮(ℝn) such that
(4.45)bk,j⟶bk
in Lp(·)(ℝn) as j→∞ and
(4.46)ck,j=bk,j*φk,fj=∑k=0Nck,j.
Thus, (4.40) holds for fj and bk,j. For any h∈𝒮(ℝn), the least decreasing radial majorant of h is integrable. By the definition of φk(x),
(4.47)φk(x)=2knφ(2kx).
Hence, ck,j converges to ck in Lp(·)(ℝn) as j→∞ by Theorem 4.8 and fj converges to f in Fp(·),qs(ℝn) as j→∞. Equation (4.40) follows from the last relation.


## 5. Duality and Reflexivity of Variable Exponent Function Spaces

Theorem 5.1.
Let1<q<∞, s∈ℝ, and p(·)∈ℬ(ℝn). Let Ap(·),qs be either Bp(·),qs(ℝn) or Fp(·),qs(ℝn). Then,
(5.1)(Ap(⋅),qs(Rn))′=Ap′(⋅),q′-s(Rn),
where
(5.2)p′(⋅)=p(⋅)p(⋅)-1,
and 1/q+1/q′=1. The spaces Fp(·),qs(ℝn) and Bp(·),qs(ℝn) are reflexive.Proof.
We will only prove (5.1) because the reflexivity of Ap(·),q(·)s(·) follows from Theorem 3.1.
Step 1.
We will first prove theFp(·),qs case. We use the arguments of the proof of [19, Theorems  7.1.7, and 7.2.2]. Let f∈𝒮(ℝn), g∈Fp′(·),q′-s(ℝn), and {ψk}k=0∞ replaced by {θk}k=0∞ as in Definition 2.1. Then, f=∑k=0∞f*ψk in 𝒮(ℝn), f*ψk∈𝒮(ℝn), and g=∑k=0∞g*ψk in 𝒮′(ℝn). It follows from Definition 2.1 that
(5.3)g(f)=∑k=0∞(g*ψk)(f)=∑k=0∞(∑l=0∞(g*ψk)(f*ψl))=∑k=0∞(∑r=-2N2N(g*ψk)(f*ψk+r))
in 𝒮′(ℝn). Here, ψj=0 for j<0. Using Hölder inequality, we get
(5.4)|g(f)|≤c∫Rn‖{2-ksg*ψk}‖lq′‖{2ksf*ψk}‖lqdx≤c‖{2-ksg*ψk}‖Lp′(⋅)(lq′)‖{2ksf*ψk}‖Lp(⋅)(lq)≤c′‖g‖Fp′(⋅),q′-s‖f‖Fp(⋅),qs,
where c′ does not depend on g, f and ψ. Hence, we see g∈(Fp(·),qs)′ and ∥g∥(Fp(·),qs)′≤c∥g∥Fp′(·),q′-s since 𝒮(ℝn) is dense in Fp(·),qs(ℝn).Step 2.
Letg∈(Fp(·),qs)′. We assume that g̃ and the functions φk and ρk have the same sense as in Definition 4.7, Lemmas 4.9, and 4.10. We show that g̃∈Fp′(·),q′-s(ℝn). For this purpose, we construct functions ak as (4.37) and set
(5.5)bk(x)=sgn⁡ak⋅|ak(x)|q′-12-sk′q′‖{2-jsaj}j=0N‖lq′p′(x)-q′,
for k=0,1,…,N. If ak=0, we set bk=0. We have
(5.6)‖{2ksbk(x)}k=0N‖lq={∑k=0N(|ak|q′-12ks(1-q′)‖{2-jsaj}j=0N‖lq′p′(x)-q′)q}1/q=‖{2-jsaj}j=0N‖lq′p′(x)-q′{∑k=0N|ak|q′2-ksq′}1/q=‖{2-jsaj}j=0N‖lq′q′/q+p′(x)-q′,
where we used (q′-1)q=q′. We set
(5.7)α={essinf⁡x∈Rnp′(x)p(x),if‖{2-ksak}k=0N‖Lp′(⋅)(lq′)≤1,esssup⁡x∈Rnp′(x)p(x),if‖{2-ksak}k=0N‖Lp′(⋅)(lq′)>1,
and λ=∥{2-ksak}k=0N∥Lp′(·)(ℓq′). Then,
(5.8)∫Rn(‖{2ksbk}k=0N‖lqλα)p(x)dx=∫Rn(‖{2-jsaj}j=0N‖lq′q′/q+p′(x)-q′λα)p(x)dx=∫Rn‖{2-jsaj}j=0N‖lq′(q′/q+p′(x)-q′)p(x)λαp(x)dx=∫Rn(‖{2-jsaj}j=0N‖lq′λα(p(x)/p′(x)))p′(x)dx≤∫Rn(‖{2-jsaj}j=0N‖lq′λ)p′(x)dx≤1,
where we used
(5.9)q′qp(x)+p(x)(p′(x)-q′)=p′(x),esssup⁡x∈Rnp′(x)p(x)=(essinf⁡x∈Rnp(x)p′(x))-1,essinf⁡x∈Rnp′(x)p(x)=(esssup⁡x∈Rnp(x)p′(x))-1.
Hence, we have
(5.10)‖{2ksbk}k=0N‖Lp(⋅)(lq)≤‖{2-ksak}k=0N‖Lp′(⋅)(lq′)α.
In particular, bk∈Lp(·)(ℝn). Now, we construct the same function f by (4.39). Using Lemma 4.10, (4.37) and Theorem 4.8, we have
(5.11)∫Rn‖{2-ksak}0N‖lq′p′(x)dx=∫Rn∑k=0N2-skq′|ak|q′‖{2-jsaj}j=0N‖lq′p′(x)-q′dx=∫Rn∑k=0Nakbkdx=〈g̃,f〉≤c‖g‖(Fp(⋅),qs)′‖{2ksbk*φk}0N‖Lp(⋅)(lq)≤c‖g‖(Fp(⋅),qs)′‖{2ksbk}k=0N‖Lp(⋅)(lq).
We set
(5.12)β={esssup⁡x∈Rnp′(x),if‖{2-ksak}k=0N‖Lp′(⋅)(lq′)≤1,essinf⁡x∈Rnp′(x),if‖{2-ksak}k=0N‖Lp′(⋅)(lq′)>1.
Then, it follows from Theorem 3.4, (5.11), and (5.10) that
(5.13)‖{2-ksak}k=0N‖Lp′(⋅)(lq′)β-α≤c‖g‖(Fp(⋅),qs)′.
If p′(x) is not a constant function, then β-α≠1. So, we normalize the ak. We put ak′=ak/λ and
(5.14)bk′=sgn⁡ak′⋅|ak′|q′-12-sk′q′‖{2-jsaj′}j=0N‖lq′p′(⋅)-q′.
Then, we have
(5.15)‖{2ksbk′}k=0N‖Lp(⋅)(lq)≤1,‖{2-ksak′}k=0N‖Lp′(⋅)(lq′)β-α≤c‖gλ‖(Fp(⋅),qs)′.
Hence we have
(5.16)‖{2-ksak}k=0N‖Lp′(⋅)(lq′)≤c‖g‖(Fp(⋅),qs)′.
The last estimate, (4.32), (4.37), and Corollary 4.3 lead us to
(5.17)g̃∈Fp′(⋅),q′-s,‖g̃‖Fp′(⋅),q′-s≤c‖g‖(Fp(⋅),qs)′.
We specialize the function ρ(x) in Definition 4.7 by setting (ℱρ)(ξ)=1 for
(5.18)12+2δ≤|ξ|≤2-2δ,
where f is a sufficiently small positive number. Then, we find a function ρ̃(x)∈𝒮(ℝn) such that
(5.19)0≤Fρ̃∈C0∞({ξ∈Rn:2-2δ<|ξ|<2+5δ}),(ℱρ+ℱρ̃+ℱρ1)=1 for 1/2+2δ<|ξ|≤2(2-2δ). As in (4.28), we construct ρ̃k(x), and, as in (4.32), we put
(5.20)g̃̃=∑k=0∞F-1(Fρ̃kFg)
in 𝒮′(ℝn). The function g̃̃ has properties similar to g̃, notably, (5.17) holds. Changing the function ρ̃0=ρ̃, we obtain g=g̃+g̃̃. Then, (5.17) implies g∈Fp′(·),q′-s(ℝn):(5.21)‖g‖Fp′(⋅),q′-s≤c‖g‖(Fp(⋅),qs)′.
This completes Theorem 5.1 for the Fp(·),qs(ℝn).Step 3.
Finally, we will prove theBp(·),qs case. Let f∈𝒮(ℝn) and g∈Bp′(·),q′-s(ℝn). Then, we have g∈(Bp(·),qs(ℝn))′ and see that there exists a positive number c such that
(5.22)‖g‖(Bp(⋅),qs)′≤c‖g‖Bp′(⋅),q′-s,
by the same manner as in the  Step 1.Step 4.
Letg∈(Bp(·),qs)′ and g̃ as in (4.32). Then, we have g̃∈(Bp(·),qs)′ and
(5.23)‖g̃‖(Bp(⋅),qs)′≤c‖g‖(Bp(⋅),qs)′
by the same argument in the proof of Lemma 4.9. Furthermore, Lemma 4.10 also holds for g∈(Bp(·),qs)′ by the same argument in the proof of Lemma 4.10. Let ak as in (4.37). Then, ak∈Lp′(·)(ℝn). We set
(5.24)bk(x)=sgn⁡ak⋅|ak(x)|p′(x)-12-skq′‖ak‖p′(⋅)q′-p′(x),
for k=0,1,2,…. Then, we have ∥bk∥Lp(·)≤2-skq′∥ak∥Lp′(·)q′/q by the following calculating:
(5.25)∫Rn(|bk(x)|2-ksq′‖ak‖Lp′(⋅)q′/q)p(x)dx=∫Rn(|ak(x)|p′(x)-1‖ak‖Lp′(⋅)q′-p′(x)‖ak‖Lp′(⋅)q′/q)p(x)dx=∫Rn|ak(x)|p′(x)‖ak‖Lp′(⋅)(q′/q)p(x)-p′(x)‖ak‖Lp′(⋅)(q′/q)p(x)dx=∫Rn(|ak(x)|‖ak‖Lp′(⋅))p′(x)dx=1.
Hence, we have
(5.26)‖{2ksbk}0N‖lq(Lp(⋅))≤(∑k=0N2skq(1-q′)‖ak‖Lp′(⋅)q′)1/q=(∑k=0N2-skq′‖ak‖Lp′(⋅)q′)q′/(q′⋅q)=‖{2-ksak}0N‖lq′(Lp′(⋅))q′/q.
We have
(5.27)∫Rn∑k=0Nakbkdx=∑k=0N2-skq′∫Rn|ak(x)|p′(x)‖ak‖Lp′(⋅)q′-p′(x)dx=∑k=0N2-skq′‖ak‖Lp′(⋅)q′=‖{2-ksak}0N‖lq′(Lp′(⋅))q′.
Hence, we have
(5.28)‖{2-ksak}0N‖lq′(Lp′(⋅))q′≤‖g‖(Bp(⋅),qs)′‖{2-ksak}0N‖lq′(Lp′(⋅))q′/q
by the same manner as in Step 2. It follows from this, as in Step 2, that
(5.29)g∈Bp′(⋅),q′-s(Rn),‖g‖Bp′(⋅),q′-s≤c‖g‖(Bp(⋅),qs)′.
This complete the proof.The next corollary is an immediate consequence of Theorems4.6 and 5.1.Corollary 5.2.
Lets≥0 and p(·)∈ℬ(ℝn). Then,
(5.30)(Ls,p(⋅)(Rn))′=Fp′(⋅),2-s(Rn).
In addition, for k∈ℕ0, one has
(5.31)(Wk,p(⋅)(Rn))′=Fp′(⋅),2-k(Rn).
Hence, Ls,p(·)(ℝn) and Wk,p(·)(ℝn) are reflexive.Remark 5.3.
LetΩ be ℝn or any measurable subset of ℝn. As we mentioned in Section 1, Kováčik and Rákosník [9] showed that the dual of Lp(·)(Ω) is Lp′(·)(Ω) if p(·) is a measurable function on Ω with a range in [1,∞] and p(·)∈L∞(Ω). Furthermore, Lp(·)(Ω) is reflexive if 1<p-≤p+<∞. There is no assumption of boundedness of Hardy-Littlewood maximal operator ℳ on Lp(·)(Ω).



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