Duality of Variable Exponent Triebel-Lizorkin and Besov Spaces

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.

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 𝐹^{𝛼(⋅)𝑝(⋅),𝑞(⋅)} and 𝐵^{𝛼(⋅)𝑝(⋅),𝑞(⋅)} and Xu studied [16–18] the spaces 𝐹^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛) and 𝐵^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛).

In this paper, we will show the duality of variable exponent Triebel-Lizorkin spaces 𝐹^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛) and Besov spaces 𝐵^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛) under suitable conditions by using the same arguments in Triebel [19]. The duality follows from the fact that 𝐹^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛) and 𝐵^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛) are reflexive under same conditions.

Xu [16] showed that variable exponent Bessel potential space 𝐿^{𝑠,𝑝(⋅)}(ℝ^𝑛) coincides with 𝐹^{𝑠𝑝(⋅),2}(ℝ^𝑛) if 𝑠≥0 and 𝑝(⋅)∈ℬ(ℝ^𝑛). Diening et al. [15] showed that the variable exponent Lebesgue space 𝐿^𝑝(⋅)(ℝ^𝑛) coincides with 𝐹^0𝑝(⋅),2(ℝ^𝑛) under suitable assumptions on 𝑝(⋅). Gurka et al. [7] showed that 𝐿^{𝑘,𝑝(⋅)}(ℝ^𝑛) coincides with variable exponent Sobolev spaces 𝑊^{𝑘,𝑝(⋅)}(ℝ^𝑛) if 𝑘∈ℕ and 𝑝(⋅)∈ℬ(ℝ^𝑛). In consequences of these results, we have the duality and reflexivity of 𝐿^{𝑠,𝑝(⋅)}(ℝ^𝑛)(𝑠≥0), although the duality and reflexivity of 𝐿^𝑝(⋅)(ℝ^𝑛) 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. Let 𝑝(⋅) be a measurable function on ℝ^𝑛 with range in (1,∞). Let 𝐿^𝑝(⋅)(ℝ^𝑛) denote the set of all complex-valued functions 𝑓 on ℝ^𝑛 such that, for some 𝜆>0, _ℝ𝑛|𝑓(𝜆𝑥)|^𝑝(𝑥)d𝑥<∞.(2.1) The set becomes a Banach function space when it is equipped with the Luxemburg-Nakano norm: ‖𝑓‖_𝐿^𝑝(⋅)=inf𝜆>0∶_ℝ𝑛|𝑓(𝜆𝑥)|^𝑝(𝑥)d𝑥≤1.(2.2) If 𝑝(𝑥)≡𝑝 is a constant function, then the above norm coincides with the usual 𝐿^𝑝-norm and so the notation is not confusional. In Kováčik and Rákosník [9], variable exponent Lebesgue spaces are defined on arbitrary measurable subset of ℝ^𝑛. Denote by 𝒫(ℝ^𝑛) the set of measurable functions 𝑝(⋅) on ℝ^𝑛 with range in (1,∞) such that 1<𝑝₋=essinf_{𝑥∈ℝ𝑛}𝑝(𝑥),esssup_{𝑥∈ℝ𝑛}𝑝(𝑥)=𝑝₊<∞.(2.3) For a complex-valued locally Lebesgue-integrable function 𝑓(⋅) on ℝ^𝑛, let (ℳ𝑓)(𝑥)=sup1||𝐵||_𝐵||𝑓(𝑦)||d𝑦(2.4) denote the Hardy-Littlewood maximal function, where the supremum is taken over all balls 𝐵 centered at 𝑥. There exists 𝑝(⋅)∈𝒫(ℝ^𝑛) such that the Hardy-Littlewood maximal operator ℳ is not bounded on 𝐿^𝑝(⋅)(ℝ^𝑛) [11], although the operator ℳis bounded on 𝐿^𝑝(ℝ^𝑛) for 𝑝>1. Some sufficient conditions on 𝑝(⋅) for maximal operator ℳ to be bounded on 𝐿^𝑝(⋅)(ℝ^𝑛) are known. Let ℬ(ℝ^𝑛) be the set of 𝑝(⋅)∈𝒫(ℝ^𝑛) such that the Hardy-Littlewood maximal operator ℳ is bounded on 𝐿^𝑝(⋅)(ℝ^𝑛).

Let 𝒮(ℝ^𝑛) be the Schwartz space of all complex-valued rapidly decreasing and infinitely differentiable functions on ℝ^𝑛. Let 𝒮^(ℝ^𝑛) be the set of all the tempered distribution on ℝ^𝑛. For 𝜑∈𝒮(ℝ^𝑛), 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.

Let 𝑠≥0 and 𝑝(⋅)∈𝒫(ℝ^𝑛). The variable exponent Bessel potential space 𝐿^{𝑠,𝑝(⋅)}(ℝ^𝑛) is the collection of 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛) such that the norm: ‖𝑓‖_𝐿^{𝑠,𝑝(⋅)}=‖‖ℱ⁻¹1+|⋅|²^𝑠/2ℱ𝑓(⋅)‖‖_{𝐿𝑝(⋅)}<∞.(2.5)

Let 𝑘∈ℕ and 𝑝∈𝒫(ℝ^𝑛). The variable exponent Sobolev space 𝑊^{𝑘,𝑝(⋅)}(ℝ^𝑛) is the collection of 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛) such that the derivatives (in the sense of distribution) up to the order 𝑘 belong to 𝐿^𝑝(⋅)(ℝ^𝑛) and the norm: ‖𝑓‖_𝑊^{𝑘,𝑝(⋅)}=_{|𝛼|≤𝑘}‖𝐷^𝛼𝑓‖_𝐿^𝑝(⋅)<∞,(2.6) where 𝛼 is a multi-index and |𝛼|=𝛼₁+⋯+𝛼_𝑛.

Let Φ∈𝒮(ℝ^𝑛) 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 Φ_𝑗(𝑥)=2^𝑛𝑗Φ(2^𝑗𝑥) for 𝑥∈ℝ^𝑛 and 𝑗∈ℤ. We also set 𝜃_𝑗(𝑥)=Φ_𝑗(𝑥)−Φ_𝑗−1(𝑥).(2.7) It follows that ^∞_𝑗=0ℱ𝜃_𝑗(𝜉)≡1,(2.8) where 𝜃₀=Φ₀.

Definition 2.1. Let 𝑠∈ℝ, 0<𝑞<∞, and 𝑝(⋅)∈𝒫(ℝ^𝑛). Let 𝜃_𝑗, 𝑗∈ℕ₀=ℕ∪{0} as above. The variable exponent Triebel-Lizorkin space 𝐹^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛) is the collection of 𝑓∈𝒮′(ℝ^𝑛) such that ‖𝑓‖_𝐹^{𝑠𝑝(⋅),𝑞}=‖‖2^𝑠𝑗𝜃_𝑗∗𝑓^∞0‖‖_{𝐿𝑝(⋅)(ℓ𝑞)}<∞.(2.9) Let 𝑠∈ℝ, 0<𝑞≤∞, and 𝑝(⋅)∈𝒫(ℝ^𝑛). The variable exponent Besov space 𝐵^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛) is the collection of 𝑓∈𝒮′(ℝ^𝑛) such that ‖𝑓‖_𝐵^{𝑠𝑝(⋅),𝑞}=‖‖2^𝑠𝑗𝜃_𝑗∗𝑓^∞0‖‖_{ℓ𝑞(𝐿𝑝(⋅))}<∞.(2.10)

Here, 𝐿^𝑝(⋅)(ℓ^𝑞) and ℓ^𝑞(𝐿^𝑝(⋅)) are the spaces of all sequences {𝑔_𝑗} of measurable functions on ℝ^𝑛 such that quasi-norms: ‖‖𝑔_𝑗^∞0‖‖_{𝐿𝑝(⋅)(ℓ𝑞)}=‖‖‖‖{𝑔_𝑗}^∞0‖‖_ℓ𝑞‖‖_{𝐿𝑝(⋅)}=‖‖‖‖^∞_𝑗=0||𝑔_𝑗(⋅)||^𝑞^1/𝑞‖‖‖‖_{𝐿𝑝(⋅)}‖‖𝑔_𝑗,^∞0‖‖_{ℓ𝑞(𝐿𝑝(⋅))}=‖‖‖‖𝑔_𝑗‖‖_{𝐿𝑝(⋅)}^∞0‖‖_ℓ𝑞=^∞_𝑗=0‖‖𝑔_𝑗(⋅)‖‖^𝑞𝐿_𝑝(⋅)^1/𝑞(2.11) are finite, respectively.

Definition 2.2. Let 𝑝(⋅)∈𝒫(ℝ^𝑛) and 0<𝑞≤∞. For a sequence of compact subsets Ω={Ω_𝑘}^∞𝑘=0 of ℝ^𝑛, 𝐿^Ω𝑝(⋅)(ℓ^𝑞) denotes the space of all sequences {𝑓_𝑘}^∞𝑘=0 of 𝒮^(ℝ^𝑛) such that suppℱ𝑓_𝑘⊂Ω_𝑘for𝑘=0,1,2,…,(2.12) and ‖𝑓_𝑘‖_{𝐿𝑝(⋅)(ℓ}^𝑞₎<∞. For a compact subset Γ of ℝ^𝑛, 𝐿^Γ𝑝(⋅)(ℝ^𝑛) denotes the space of all elements 𝑓∈𝒮′(ℝ^𝑛) such that suppℱ𝑓⊂Γ,(2.13) and ‖𝑓‖_𝐿^𝑝(⋅)<∞.

3. Preliminaries

We need the following fundamental properties of 𝐿^𝑝(⋅)(ℝ^𝑛).

Theorem 3.1 (see [5, Theorem 8.1]). Let 𝑝(⋅)∈𝒫(ℝ^𝑛). Then, the following conditions are equivalent:
(a)𝑝(⋅)∈ℬ(ℝ^𝑛),
(b)𝑝(⋅)/𝑡∈ℬ(ℝ^𝑛) for some 1<𝑡<𝑝₋,
(c)𝑝′(⋅)∈ℬ(ℝ^𝑛), where 𝑝^(⋅)=𝑝(⋅)𝑝(⋅)−1.(3.1)

In [1, 5], some other conditions equivalent to the above are given. Let 𝑝(⋅)∈ℬ(ℝ^𝑛) and 0<𝑞<∞. Then we write 𝑟_𝑝−,𝑞=sup𝑟∈ℝ∶0<𝑟<min𝑝₋,𝑞,𝑝(⋅)𝑟∈ℬ(ℝ^𝑛).(3.2) If 𝑞=𝑝₋, then, we write 𝑟_{𝑝−,𝑝−}=𝑟_𝑝−.

Remark 3.2. Let 𝑝(⋅)∈ℬ(ℝ^𝑛) and 𝑡 as in Theorem 3.1. Then, for any 𝑤∈(1,𝑡], we have 𝑝(⋅)/𝑤∈ℬ(ℝ^𝑛). Furthermore, for any 𝑤∈(0,1], we have also 𝑝(⋅)/𝑤∈ℬ(ℝ^𝑛) by Jensen inequality. The proofs are found in [5]. Consequently, if 𝑝(⋅)∈ℬ(ℝ^𝑛), then, 𝑝(⋅)/𝑟∈ℬ(ℝ^𝑛) for any 𝑟∈(0,𝑟_𝑝−). For any 𝑟∈(0,𝑟_𝑝−,𝑞), we have also 𝑝(⋅)/𝑟∈ℬ(ℝ^𝑛).

The next theorem gives a generalized Hölder inequality, which is shown in [9, 13].

Theorem 3.3 (see [9, 13]). Let 𝑝∈𝒫(ℝ^𝑛). Then, _ℝ𝑛||𝑓(𝑥)𝑔(𝑥)||d𝑥≤1+1𝑝₋−1𝑝₊‖𝑓‖^{𝐿𝑝(⋅)}‖𝑔‖_{𝐿𝑝′(⋅)},(3.3) for every 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛) and 𝑔∈𝐿^𝑝′(⋅)(ℝ^𝑛).

The next theorem is shown in [6].

Theorem 3.4. Let 𝑝(⋅)∈𝒫(ℝ^𝑛) and 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛). Then, 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+.(3.4)

The next theorem is shown in [9].

Theorem 3.5. (i) Let 𝑝(⋅)∈𝒫(ℝ^𝑛). Then, 𝐿^𝑝(⋅)(ℝ^𝑛) is a Banach space.
(ii) If 𝑝(⋅)∈ℬ(ℝ^𝑛), then 𝐶^∞0(ℝ^𝑛) is dense in 𝐿^𝑝(⋅)(ℝ^𝑛).

Let 𝑝(⋅)∈𝒫(ℝ^𝑛) such that 𝑝(⋅) is not a constant function. Then, according to [9, Example 2.9, Theorem 2.10], for every 𝐿^𝑝(⋅), there exists a function 𝑓(⋅)∈𝐿^𝑝(⋅)(ℝ^𝑛) such that its translation 𝑓(𝛽+⋅)∉𝐿^𝑝(⋅)(ℝ^𝑛) if 𝛽∈ℝ^𝑛⧵{(0,0,…,0)}. However, we can prove that all elements of 𝒮(ℝ^𝑛) belong to 𝐿^𝑝(⋅)(ℝ^𝑛) for every 𝑝(⋅)∈𝒫(ℝ^𝑛). Let 𝑓(⋅)∈𝒮(ℝ^𝑛). Then, we have sup_{𝛽∈ℝ𝑛}‖𝑓(𝛽+𝛼⋅)‖_𝐿^𝑝(⋅)≤⎧⎪⎨⎪⎩|𝛼|^{−𝑛/𝑝+}|𝛼|_{−𝑛/𝑝−}𝜋^𝑛/𝑝−sup_{𝑥∈ℝ𝑛}1+|𝑥|²^𝑛||𝑓(𝑥)||if|𝛼|≥1,𝜋^𝑛/𝑝−sup_{𝑥∈ℝ𝑛}1+|𝑥|²^𝑛||𝑓(𝑥)||if|𝛼|<1,(3.5) where 𝛼∈ℝ⧵{0}.

Hence, by the same arguments of Triebel [20], we have the following two theorems.

Theorem 3.6. Let Ω be a compact subset of ℝ^𝑛 and 𝛼 an arbitrary multi-index. Let 𝑝(⋅), 𝑞(⋅)∈ℬ(ℝ^𝑛) such that 1<𝑝(𝑥)≤𝑞(𝑥)<∞. Then, there exists a positive constant 𝑐 such that ‖‖𝐷𝐷^𝛼𝑓‖_𝐿^𝑞(⋅)≤𝑐‖𝑓‖_𝐿^𝑝(⋅),(3.6)_𝛼𝑓‖_𝐿^∞≤𝑐‖𝑓‖_𝐿^𝑝(⋅),(3.7) for all 𝑓∈𝐿^Ω𝑝(⋅)(ℝ^𝑛).

Let 𝑓(⋅)∈𝐿^Ω𝑝(⋅)(ℝ^𝑛) and ℱ⁻¹𝑀∈𝐿¹(ℝ^𝑛). Then, ℱ⁻¹𝑀ℱ𝑓(𝑥)=𝑐_ℝ𝑛ℱ⁻¹𝑀(𝑥−𝑦)𝑓(𝑦)d𝑦(3.8) makes sense for any 𝑥∈ℝ^𝑛 by the classical Hölder inequality and (3.7).

Let 𝑠 be a real number and 𝐻^𝑠2(ℝ^𝑛) the spaces of 𝑓∈𝒮^(ℝ^𝑛) such that ‖𝑓‖_𝐻^𝑠2=‖‖1+|⋅|²^𝑠/2(ℱ𝑓)(⋅)‖‖_𝐿2<∞.(3.9)

Theorem 3.7. Let 𝑝(⋅)∈ℬ(ℝ^𝑛) and 0<𝑞<∞. Let Ω={Ω_𝑘}^∞𝑘=0 be a sequence of compact subsets of ℝ^𝑛. Let 𝑑_𝑘>0 be the diameter of Ω_𝑘. If 𝑣>𝑛/2+𝑛/𝑟_𝑝−,𝑞, then there exists a number 𝑐 such that ‖‖ℱ⁻¹𝑀_𝑘ℱ𝑓_𝑘‖‖_{𝐿𝑝(⋅)(ℓ𝑞)}≤𝑐sup_𝑙‖‖𝑀_𝑙𝑑_𝑙⋅‖‖_𝐻𝑣2‖‖𝑓_𝑘‖‖_{𝐿𝑝(⋅)(ℓ𝑞)}(3.10) for {𝑓_𝑘(𝑥)}^∞𝑘=0∈𝐿^Ω𝑝(⋅)(ℓ^𝑞) and {𝑀_𝑘(𝑥)}^∞𝑘=0∈𝐻^𝑣2(ℝ^𝑛).

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

We will first consider the quasi-norms on 𝐹^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛) and 𝐵^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛).

Definition 4.1. (i) The set Ψ(ℝ^𝑛) is the collection of all systems 𝜑={𝜑_𝑗}^∞𝑗=0⊂𝒮(ℝ^𝑛) with ℱ𝜑_𝑘(𝜉)=ℱ𝜑(2^−𝑘𝜉) for 𝑘=1,2,…, such that suppℱ𝜑_𝑗suppℱ𝜑₀⊂{𝑥∶|𝑥|≤2},⊂𝑥∶2^𝑗−1≤|𝑥|≤2^𝑗+1for𝑗=1,2,…,(4.1) for every multi-index 𝛼, there exists a positive number 𝑐_𝛼 such that 2^𝑗|𝛼|||𝐷^𝛼ℱ𝜑_𝑗(𝑥)||≤𝑐_𝛼,(4.2) for 𝑗=0,1,…, and 𝑥∈ℝ^𝑛 and there exists a positive number 𝑐 such that 𝑐≤^∞_𝑗=0ℱ𝜑_𝑗(𝑥),(4.3) for 𝑥∈ℝ^𝑛.
(ii) The set Φ(ℝ^𝑛) is the collection of all systems 𝜑={𝜑_𝑗}^∞𝑗=0⊂𝒮(ℝ^𝑛) such that supp𝜑_𝑗supp𝜑₀⊂{𝑥∶|𝑥|≤2},⊂𝑥∶2^𝑗−1≤|𝑥|≤2^𝑗+1for𝑗=1,2,…,(4.4) for every multi-index 𝛼, there exists a positive number 𝑐_𝛼 such that 2^𝑗|𝛼|||𝐷^𝛼𝜑_𝑗(𝑥)||≤𝑐_𝛼,(4.5) for 𝑗=0,1,…, and 𝑥∈ℝ^𝑛, and ^∞_𝑗=0𝜑_𝑗(𝑥)=1,(4.6) for 𝑥∈ℝ^𝑛.

Theorem 4.2. (i) Let 𝑝(⋅)∈𝒫(ℝ^𝑛), 0<𝑞≤∞, and 𝑠∈ℝ. Let 𝜃_𝑗 be as in Definition 2.1 and 𝜑_𝑗=ℱ𝜃_𝑗. Then, the norm ‖𝑓‖_𝐹^{𝑠𝑝(⋅),𝑞} coincides with ‖2^𝑗𝑠ℱ⁻¹𝜑_𝑗ℱ𝑓‖_{𝐿𝑝(⋅)(ℓ}^𝑞₎ and the norm ‖𝑓‖_𝐵^{𝑠𝑝(⋅),𝑞} with ‖2^𝑗𝑠ℱ⁻¹𝜑_𝑗ℱ𝑓‖_ℓ^𝑞_{(𝐿𝑝(⋅))}.
(ii) Let 𝑝(⋅)∈ℬ(ℝ^𝑛), 0<𝑞≤∞, and 𝑠∈ℝ. Let 𝜑, 𝜓∈Φ(ℝ^𝑛). Then, ‖2^𝑗𝑠ℱ⁻¹𝜑_𝑗ℱ𝑓‖_{𝐿𝑝(⋅)(ℓ}^𝑞₎ and ‖2^𝑗𝑠ℱ⁻¹𝜓_𝑗ℱ𝑓‖_{𝐿𝑝(⋅)(ℓ}^𝑞₎ are equivalent quasi-norms on 𝐹^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛). Similarly, ‖2^𝑗𝑠ℱ⁻¹𝜑_𝑗ℱ𝑓‖_ℓ^𝑞_{(𝐿𝑝(⋅))} and ‖2^𝑗𝑠ℱ⁻¹𝜓_𝑗ℱ𝑓‖_ℓ^𝑞_{(𝐿𝑝(⋅))} are equivalent quasi-norms on 𝐵^{𝑠𝑝(⋅),𝑞}(ℝⁿ).
(iii) Let 𝑝(⋅)∈ℬ(ℝ^𝑛), 0<𝑞≤∞, and 𝑠∈ℝ. Let 𝜑∈Φ(ℝ^𝑛) and 𝜓∈Ψ(ℝ^𝑛). Then, ‖2^𝑗𝑠ℱ⁻¹𝜑_𝑗ℱ𝑓‖_{𝐿𝑝(⋅)(ℓ}^𝑞₎ and ‖2^𝑗𝑠𝜓_𝑗∗𝑓‖_{𝐿𝑝(⋅)(ℓ}^𝑞₎ are equivalent quasi-norms on 𝐹^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛). Similarly, ‖2^𝑗𝑠ℱ⁻¹𝜑_𝑗ℱ𝑓‖_ℓ^𝑞_{(𝐿𝑝(⋅))} and ‖2^𝑗𝑠𝜓_𝑗∗𝑓‖_ℓ^𝑞_(L𝑝(⋅)) are equivalent quasi norms on 𝐵^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛).

Proof. Let 𝜃_𝑗 be as in Definition 2.1. It is obvious that 𝜃_𝑗∗𝑓=𝑑_ℱℱ⁻¹(ℱ𝜃_𝑗⋅ℱ𝑓) for some positive number 𝑑_ℱ, which depends only on the definition of Fourier transform ℱ. Let 𝜑_𝑗=ℱ𝜃_𝑗. Then {𝜑_𝑗}^∞𝑗=0∈Φ(ℝ^𝑛). Hence, the norm ‖𝑓‖_𝐹^{𝑠𝑝(⋅),𝑞} coincides with ‖2^𝑗𝑠ℱ⁻¹𝜑_𝑗ℱ𝑓‖_{𝐿𝑝(⋅)(ℓ}^𝑞₎. Similarly, the variable exponent Besov norm ‖𝑓‖_𝐵^{𝑠𝑝(⋅),𝑞} coincides with ‖2^𝑗𝑠ℱ⁻¹𝜑_𝑗ℱ𝑓‖_ℓ^𝑞_{(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). Let 𝑓∈𝐹^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛), 𝑔_𝑗=ℱ⁻¹(𝜑_𝑗ℱ𝑓), and 𝜓∈Ψ(ℝ^𝑛). Let 𝜓_𝑘≡𝑔_𝑘≡0 for 𝑘<0. Then, 𝑓∗𝜓_𝑘(𝑥)=_𝒮′^∞_𝑗=0𝑔_𝑗∗𝜓_𝑘(𝑥)=^𝑘+2_𝑗=_𝑘−2𝑔_𝑗∗𝜓_𝑘(𝑥),(4.7) because 𝑔_𝑗∗𝜓_𝑘=ℱ⁻¹ℱ𝑔_𝑗∗𝜓_𝑘=𝑑_ℱℱ⁻¹ℱ𝜓_𝑘⋅ℱ𝑔_𝑗=0,(4.8) for 𝑗<𝑘−2 and 𝑗>𝑘+2. Here, 𝑓=_𝒮^′∑^∞0𝑎_𝑗 means ∑^𝑁0𝑎_𝑗 converges to 𝑓 in 𝒮^(ℝ^𝑛). It follows that ‖‖2^𝑠𝑘𝑓∗𝜓_𝑘^∞0‖‖_{𝐿𝑝(⋅)(ℓ𝑞)}≤²_𝑟₌₋₂‖‖2^𝑠𝑘𝑔_𝑘+𝑟∗𝜓_𝑘^∞0‖‖_{𝐿𝑝(⋅)(ℓ𝑞)}≤²_𝑟₌₋₂‖‖ℱ𝜓2^𝑟+1⋅‖‖_𝐻𝜘2‖‖2^𝑠𝑘𝑔_𝑘+𝑟^∞0‖‖_{𝐿𝑝(⋅)(ℓ𝑞)},(4.9) where we use Theorem 3.7 with 𝑓_𝑘=𝑔_𝑘+𝑟, 𝑀_𝑘=ℱ𝜓_𝑘, and 𝑑_𝑘=2^𝑘+1+𝑟. Hence, we have ‖‖2^𝑠𝑘𝑓∗𝜓_𝑘^∞0‖‖_{𝐿𝑝(⋅)(ℓ𝑞)}≤²_𝑟₌₋₂‖‖ℱ𝜓2^𝑟+1⋅‖‖_𝐻𝜘2‖𝑓‖_𝐹^{𝑠𝑝(⋅),𝑞}.(4.10) We will prove the opposite inequality. Let 𝜌(𝑥) be a real function with 𝜌(𝑥)=1 for 2⁻¹≤|𝑥|≤2 and 𝜌(⋅)∈𝐶^∞0𝜉∶2⁻²≤||𝜉||≤2².(4.11) We set 𝜌_𝑘(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 𝜌₀(⋅)∈𝐶^∞0𝜉∶||𝜉||≤2².(4.12) We define ℎ_𝑘(𝑥) by ℱℎ_𝑘(𝜉)=^∞_𝑙=0ℱ𝜓_𝑙(𝜉)⁻¹𝜌_𝑘(𝜉),(4.13) for 𝑘=0,1,2,… (ℎ_𝑘=0 for 𝑘<0). Then, ℱℎ_𝑘(⋅)∈𝐶^∞0(ℝ^𝑛) and ℎ_𝑘(⋅)∈𝒮(ℝ^𝑛). Let 𝛾_𝑘=𝜓_𝑘∗ℎ_𝑘. It is obvious that {ℱ𝛾_𝑘/𝑑_ℱ}∈Φ(ℝ^𝑛) and {2^𝑠𝑘𝜓_𝑘∗𝑓}^∞0∈𝐿^Ω𝑝(⋅)(ℓ^𝑞) by Theorem 4.8 and (4.10), where Ω is some sequence of compact subsets of ℝ^𝑛. Then, ‖‖2^𝑘𝑠𝑓∗𝛾_𝑘^∞0‖‖_{𝐿𝑝(⋅)(ℓ𝑞)}=‖‖2^𝑘𝑠𝑓∗𝜓_𝑘∗ℎ_𝑘^∞0‖‖_{𝐿𝑝(⋅)(ℓ𝑞)}ℱℎ_𝑘≤sup_𝑘‖‖ℱℎ_𝑘2^𝑘+1⋅‖‖_𝐻𝜘2‖‖2^𝑘𝑠𝑓∗𝜓_𝑘^∞0‖‖_{𝐿𝑝(⋅)(ℓ𝑞)},2^𝑘+1𝑥=𝜌^𝑘2^𝑘+1𝑥∑^∞𝑙=0ℱ𝜓_𝑙2^𝑘+1𝑥=𝜌^𝑘2^𝑘+1𝑥∑^{𝑘+3𝑙=𝑘−3}ℱ𝜓_𝑙2^𝑘+1𝑥=∑𝜌^4𝑟=−4ℱ(2𝑥𝜓)(2^𝑟𝑥),(4.14) for any 𝑘=1,2,…. This means that ℱℎ_𝑘(2^𝑘+1𝑥) is independent of 𝑘∈ℕ and in 𝐶^∞0(ℝ^𝑛). Hence, we have sup_𝑘‖ℱℎ_𝑘(2^𝑘+1⋅)‖_𝐻𝜘2<∞. This implies that ‖𝑓‖_𝐹^{𝑠𝑝(⋅),𝑞}≤sup_𝑘‖‖ℱℎ_𝑘2^𝑘+1⋅‖‖_𝐻𝜘2‖‖2^𝑘𝑠𝑓∗𝜓_𝑘^∞0‖‖_{𝐿𝑝(⋅)(ℓ𝑞)}.(4.15) This proves the 𝐹^{𝑠𝑝(⋅),𝑞} case of (iii). The proof of the 𝐵^{𝑠𝑝(⋅),𝑞} case of (iii) is essentially the same as one of the 𝐹^{𝑠𝑝(⋅),𝑞} case.

Corollary 4.3. Let 𝑝(⋅)∈ℬ(ℝ^𝑛), 𝑠∈ℝ, and 1<𝑞<∞.
(i)𝑓∈𝐹^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛) if and only if there exist continuous functions {𝑎_𝑗}_𝑗∈ℕ0 such that 𝑓=_𝒮∑^∞𝑗=0𝑎_𝑗, ‖{2^𝑗𝑠𝑎_𝑗}^∞0‖_𝐿^𝑝(⋅)_(ℓ^𝑞₎<∞, suppℱ𝑎₀⊂𝜉∶||𝜉||≤2,suppℱ𝑎_𝑗⊂𝜉∶2^𝑗−1≤||𝜉||≤2^𝑗+1,for𝑗∈ℕ₀.(4.16) In particular, ‖𝑓‖_𝐹^{𝑠𝑝(⋅),𝑞} coincides with inf_{∑𝑎𝑓=j}‖{2^𝑗𝑠𝑎_𝑗}^∞0‖_𝐿^𝑝(⋅)_(ℓ^𝑞₎, where the infimum is taken over all representation 𝑓=∑𝑎_𝑗.
(ii)𝑓∈𝐵^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛) if and only if, there exist continuous functions {𝑎_𝑗}_𝑗∈ℕ0 such that 𝑓=_𝒮∑^∞𝑗=0𝑎_𝑗, ‖{2^𝑗𝑠𝑎_𝑗}^∞0‖_ℓ^𝑞_(𝐿^𝑝(⋅)₎<∞, suppℱ𝑎₀⊂{𝜉∶|𝜉|≤2}, and suppℱ𝑎_𝑗⊂{𝜉∶2^𝑗−1≤|𝜉|≤2^𝑗+1} for 𝑗∈ℕ₀. In particular, the norm ‖𝑓‖_𝐵^{𝑠𝑝(⋅),𝑞} coincides with inf_{∑𝑎𝑓=𝑗}‖{2^𝑗𝑠𝑎_𝑗}^∞0‖_ℓ^𝑞_(𝐿^𝑝(⋅)₎.

Proposition 4.4. (i) Let 𝑝(⋅)∈𝒫(ℝ^𝑛), 𝑠∈ℝ, and 0<𝑞₁≤𝑞₂≤∞. Then, 𝐵𝐹^{𝑠𝑝(⋅),𝑞}_{1𝑠𝑝(⋅),𝑞1}(ℝ^𝑛)⊂𝐵^{𝑠𝑝(⋅),𝑞}₂(ℝ^𝑛),(ℝ^𝑛)⊂𝐹^{𝑠𝑝(⋅),𝑞}₂(ℝ^𝑛).(4.17)
(ii) Let 𝑝(⋅)∈𝒫(ℝ^𝑛), 𝑠∈ℝ, 0<𝑞₁≤∞, 0<𝑞₂≤∞, and 𝜖>0. Then 𝐵𝐹^{𝑠+𝜖𝑝(⋅),𝑞}₁(ℝ^𝑛)⊂𝐵^{𝑠𝑝(⋅),𝑞}₂(ℝ^𝑛),_{𝑠+𝜖𝑝(⋅),𝑞1}(ℝ^𝑛)⊂𝐹^{𝑠𝑝(⋅),𝑞}₂(ℝ^𝑛).(4.18)
(iii) Let 𝑝(⋅)∈𝒫(ℝ^𝑛) and 𝑠∈ℝ. If 0<𝑞≤∞, then 𝐵^{𝑠𝑝(⋅),min{𝑝}_−,𝑞}(ℝ^𝑛)⊂𝐹^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛)⊂𝐵^{𝑠𝑝(⋅),max{𝑝}_+,𝑞}(ℝ^𝑛).(4.19)

The proof is almost same as in [20]. Furthermore, Almeida and Hästö [14] proved the above inclusion for 𝐹^{𝛼(⋅)𝑝(⋅),𝑞(⋅)}(ℝ^𝑛) and 𝐵^{𝛼(⋅)𝑝(⋅),𝑞(⋅)}(ℝ^𝑛).

Theorem 4.5. Let 0<𝑞<∞, 𝑠∈ℝ, and 𝑝(⋅)∈ℬ(ℝ^𝑛). Let 𝐴^{𝑠𝑝(⋅),𝑞} be either 𝐵^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛) or 𝐹^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛). Then, 𝒮(ℝ^𝑛)⊂𝐴^{𝑠𝑝(⋅),𝑞}⊂𝒮^(ℝ^𝑛). Furthermore, 𝑆(ℝ^𝑛) is dense in 𝐴^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛) and 𝐴^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛) is a quasi Banach space (a Banach space if 1≤𝑞≤∞).

Proof. To prove the inclusion, we may restrict ourselves to 𝐵^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛) with 𝑞=∞ by Proposition 4.4. Let 𝑓∈𝒮(ℝ^𝑛) and 𝜑={𝜑_𝑘(𝑥)}^∞𝑘=0∈Φ(ℝ^𝑛). We recall that the topology of the complete locally convex space 𝒮(ℝ^𝑛) is generated by seminorms: 𝑝_𝑁(𝑓)=sup_{𝑥∈ℝ𝑛|𝛼|+}_𝑘≤𝑁1+|𝑥|²^𝑘||𝐷^𝛼𝑓(𝑥)||,𝑁∈ℕ,(4.20)ℱ yields a one-to-one mapping from 𝒮(ℝ^𝑛) onto itself, and, in particular, 𝑝_𝑁(ℱ𝜑) with 𝑁=1,2,… generates the topology of 𝒮(ℝ^𝑛). If 𝑁 is a sufficiently large natural number, then ‖𝑓‖_𝐵^{𝑠𝑝(⋅),𝑞}=sup_𝑘2^𝑘𝑠‖‖ℱ⁻¹𝜑_𝑘ℱ𝑓‖‖_{𝐿𝑝(⋅)}≤𝑐𝑝_𝑁(ℱ𝑓),(4.21) 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(ℝ^𝑛) be the above system. It is well known that 𝑓(𝜓)=^∞_𝑘=0ℱ⁻¹𝜑_𝑘ℱ𝑓(𝜓)(4.22) in 𝒮^(ℝ^𝑛) for all 𝑓∈𝒮′(ℝ^𝑛) and 𝜓∈𝒮(ℝ^𝑛). By the Paley-Wiener-Schwartz Theorem, the definition of 𝜑_𝑘 implies that ℱ⁻¹𝜑_𝑘ℱ𝑓 are analytic functions, and that these functions are regular distributions. We put 𝜒_𝑘(𝑥)=𝜑_𝑘−1(𝑥)+𝜑_𝑘(𝑥)+𝜑_𝑘+1(𝑥) if 𝑘=0,1,…, with 𝜑₋₁=0. If 𝑓∈𝐵^{𝑠𝑝(⋅),∞}(ℝ^𝑛) and 𝜓∈𝒮(ℝ^𝑛), then 𝑓(𝜓) denotes the value of the functional 𝑓 of 𝒮′(ℝ^𝑛) of the test function 𝜓. We obtain ||𝑓(𝜓)||≤|||||^∞_𝑘=0ℱ⁻¹𝜑_𝑘ℱ𝑓ℱ𝜒_𝑘ℱ⁻¹𝜓|||||≤^∞_𝑘=0‖‖ℱ⁻¹𝜑_𝑘ℱ𝑓‖‖_𝐿∞‖‖ℱ𝜒_𝑘ℱ⁻¹𝜓‖‖_𝐿1(4.23) by the classical Hölder inequality. Because both ℱ⁻¹𝜑_𝑘ℱ𝑓 and ℱ𝜒_𝑘ℱ⁻¹𝜓 are analytic functions, the last estimate makes sense. By applying (3.7) with 𝛼=(0,0,…,0) to ℱ⁻¹𝜑_𝑘ℱ𝑓, we have ||𝑓(𝜓)||≤𝑐‖𝑓‖_{𝐵𝑠𝑝(⋅),∞}^∞_𝑘=02^−𝑠𝑘‖‖ℱ𝜒_𝑘ℱ⁻¹𝜓‖‖_𝐿1≤𝑐‖𝑓‖_𝐵^{𝑠𝑝(⋅),∞}‖𝜓‖_𝐵^−𝑠1,1.(4.24) If 𝑁 is a sufficiently large natural number, then, for any 𝜓∈𝒮(ℝ^𝑛), we have ||𝑓(𝜓)||≤𝑐‖𝑓‖_{𝐵𝑠𝑝(⋅),∞}𝑝_𝑁(𝜓)(4.25) by the left-hand side of the inclusion for 𝐵^−𝑠1,1(ℝ^𝑛). Hence, we have the inclusion for 𝐴^{𝑠𝑝(⋅),𝑞}.
The proof of the completeness and the density of 𝒮(ℝ^𝑛) 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 Section 1, the next theorem is found in [15, 16].

Theorem 4.6. Let 𝑠≥0 and 𝑝(⋅)∈ℬ(ℝ^𝑛). Then, 𝐹^{𝑠𝑝(⋅),2}(ℝ^𝑛) coincides with 𝐿^{𝑠,𝑝(⋅)}(ℝ^𝑛). Let 𝑘∈ℕ and 𝑝(⋅)∈ℬ(ℝ^𝑛). Then, 𝐹^{𝑘𝑝(⋅),2}(ℝ^𝑛) coincides with 𝑊^{𝑘,𝑝(⋅)}(ℝ^𝑛).

Definition 4.7. Let 𝜑∈𝒮(ℝ^𝑛) such that (ℱ𝜑)(𝜉)=1 for 1/√2≤|𝜉|≤√2 and 0≤ℱ𝜑∈𝐶^∞0𝜉∈ℝ^𝑛∶1√2−𝜖<||𝜉||<√2+𝜖,(4.26) and let 𝜌∈𝒮(ℝ^𝑛) such that 0≤ℱ𝜌∈𝐶^∞0𝜉∈ℝ^𝑛∶1√2+𝛿<||𝜉||<√2−𝛿,(4.27) where 𝜖 and 𝛿 are positive numbers. We construct {𝜑_𝑘}^∞𝑘=0 and {𝜌_𝑘}^∞𝑘=0 by ℱ𝜑_𝑘(𝜉)=(ℱ𝜑)2^−𝑘𝜉,ℱ𝜌_𝑘(𝜉)=(ℱ𝜌)2^−𝑘𝜉.(4.28) We choose 𝜖 and 𝛿 sufficiently small so that ℱ𝜑_𝑘⋅ℱ𝜌_𝑗≡0(4.29) for 𝑘≠𝑗.

Diening [3] shows that Young inequality ‖𝑓∗𝑔‖_𝐿^𝑝(⋅)≤‖𝑓‖_𝐿^𝑝(⋅)‖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. Let 𝑝(⋅)∈ℬ(ℝ^𝑛). Let 𝜙 be an integrable function on ℝ^𝑛 with range in ℝ and set 𝜙_𝜖(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∈𝐿^𝑝(⋅)(ℝ^𝑛), one has sup_𝜖>0||𝑓∗𝜙_𝜖(𝑥)||≤2𝐴ℳ𝑓(𝑥).(4.30) Hence, there exist a positive number 𝐶(𝐴,𝑝) such that ‖‖𝑓∗𝜙_𝜖‖‖_{𝐿𝑝(⋅)}≤𝐶(𝐴,𝑝)‖𝑓‖_𝐿^𝑝(⋅),(4.31) where 𝐶(𝐴,𝑝) only depends on 𝐴 and 𝑝(⋅).

The duality pair of 𝑔∈(𝐹^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛))′ and 𝑓∈𝐹^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛) is denoted by ⟨𝑔,𝑓⟩, although, for 𝜑∈𝒮(ℝ^𝑛)⊂𝐹^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛), we also write 𝑔(𝜑) as ⟨𝑔,𝜑⟩. In order to prove the duality of 𝐹^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛) and 𝐵^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛) (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. Let 𝑔∈(𝐹^{𝑠𝑝(⋅),𝑞})′, {𝜌_𝑘} as in Definition 4.7 and 𝑔̃=^∞_𝑘=0ℱ⁻¹ℱ𝜌_𝑘⋅ℱ𝑔(4.32) in 𝒮^(ℝ^𝑛). Then, 𝑔̃∈(𝐹^{𝑠𝑝(⋅),𝑞})^ and ‖𝑔̃‖_{(𝐹𝑠𝑝(⋅),𝑞)}^′≤𝑐‖𝑔‖_{(𝐹𝑠𝑝(⋅),𝑞)}^′,(4.33) 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(ℝ^𝑛). Then, it holds that 𝑔̃(𝑓)=(ℱ𝑔̃)ℱ⁻¹𝑓=^∞_𝑘=0ℱ𝑔ℱ𝜌_𝑘ℱ⁻¹𝑓=𝑐^∞_𝑘=0𝑔𝜌^∨𝑘∗𝑓=𝑐𝑔^∞_𝑘=0𝜌^∨𝑘∗𝑓,(4.34) where 𝜌^∨𝑘(𝑥)=𝜌_𝑘(−𝑥). Using 𝑔∈(𝐹^{𝑠𝑝(⋅),𝑞})′ and Theorem 3.7 with 𝑀_𝑘=ℱ(𝜌^∨𝑘) and 𝑑_𝑘=2^𝑘+1, we find that ||𝑔̃(𝑓)||≤‖𝑔‖_{(𝐹𝑠𝑝(⋅),𝑞)′}‖‖2^𝑘𝑠𝜌^∨𝑘∗𝜑_𝑘∗𝑓^∞0‖‖_{𝐿𝑝(⋅)(ℓ𝑞)}≤𝑐‖𝑔‖_{(𝐹𝑠𝑝(⋅),𝑞)}^′‖‖2^𝑘𝑠𝜑_𝑘∗𝑓^∞0‖‖_{𝐿𝑝(⋅)(ℓ𝑞)},(4.35) where {𝜑_𝑘}^∞𝑘=0 is a system of Definition 4.7. Thus, we get ||𝑔̃(𝑓)||≤𝑐‖𝑔‖_{(𝐹𝑠𝑝(⋅),𝑞)′}‖𝑓‖_𝐹^{𝑠𝑝(⋅),𝑞},(4.36) from Definition 2.1 and Theorem 4.2.

Lemma 4.10. Let {𝜌_𝑘} and {𝜑_𝑘} be ones as in Definition 4.7, 𝑔∈(𝐹^{𝑠𝑝(⋅),𝑞})′, and 𝑔̃ the functional in Lemma 4.9. Let {𝑏_𝑘}^𝑁𝑘=0 be a system of functions such that 𝑏_𝑘∈𝐿^𝑝(⋅)(ℝ^𝑛). Set 𝑎_𝑘=𝑐ℱ⁻¹ℱ𝜌_𝑘⋅ℱ𝑔,(4.37)_𝑘=𝑏_𝑘∗𝜑_𝑘,(4.38) for 𝑘=0,1,2,…,𝑁, and 𝑓=^𝑁_𝑘=0𝑐_𝑘.(4.39) Then, ⟨𝑔̃,𝑓⟩=𝑐_ℝ𝑛^𝑁_𝑘=0𝑎_𝑘𝑏_𝑘d𝑥,(4.40) 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 𝑎_𝑘∈𝐿^𝑝(⋅), where 𝑝^(⋅) is defined by (3.1). The consideration of the proof of Lemma 4.9 implies that for 𝑓∈𝒮(ℝ^𝑛), ||𝑎_𝑘(𝑓)||≤𝑐‖𝑔‖_{(𝐹𝑠𝑝(⋅),𝑞)}^′‖‖2^𝑠𝑗𝛿_𝑗,𝑘𝜑_𝑘∗𝑓_𝑗‖‖_{𝐿𝑝(⋅)(ℓ𝑞)}≤𝑐^‖𝑓‖_𝐿^𝑝(⋅),(4.41) where 𝛿_𝑘,𝑘=1 and 𝛿_𝑗,𝑘=0 if 𝑗≠𝑘. This implies 𝑎_𝑘∈𝐿^𝑝′(⋅) and the right-hand side of (4.40) has a sense. We assume 𝑏_𝑘∈𝒮(ℝ^𝑛). Then, 𝑐_𝑘∈𝒮(ℝ^𝑛) and 𝑓∈𝒮(ℝ^𝑛). By (4.32), we have ⟨𝑔̃,𝑓⟩=𝑔̃(𝑓)=^∞_𝑘=0𝑎_𝑘(𝑓)(4.42) in 𝒮^(ℝ^𝑛) and 𝑎_𝑘(𝑓)=^𝑁_𝑙=0𝑎_𝑘𝑐_𝑙=^𝑁_𝑙=0ℱ𝑎_𝑘ℱ⁻¹𝑐_𝑙=𝑐^𝑁_𝑙=0ℱ𝑎_𝑘⋅ℱ⁻¹𝜑_𝑙ℱ⁻¹𝑏_𝑙.(4.43) Since (ℱ⁻¹ℎ)(𝜉)=(ℱℎ)(−𝜉), (4.29), (4.37), and (4.38) imply that 𝑎_𝑘(𝑓)=𝑐ℱ𝑎_𝑘ℱ⁻¹𝑏_𝑘=𝑐_ℝ𝑛𝑎_𝑘𝑏_𝑘d𝑥.(4.44) From this, (4.40) follows for 𝑏_𝑘∈𝒮(ℝ^𝑛). Let 𝑏_𝑘∈𝐿^𝑝(⋅)(ℝ^𝑛), 𝑏_𝑘,𝑗∈𝒮(ℝ^𝑛) such that 𝑏_𝑘,𝑗⟶𝑏_𝑘(4.45) in 𝐿^𝑝(⋅)(ℝ^𝑛) as 𝑗→∞ and 𝑐_𝑘,𝑗=𝑏_𝑘,𝑗∗𝜑_𝑘,𝑓_𝑗=^𝑁_𝑘=0𝑐_𝑘,𝑗.(4.46) Thus, (4.40) holds for 𝑓_𝑗 and 𝑏_𝑘,𝑗. For any ℎ∈𝒮(ℝ^𝑛), the least decreasing radial majorant of ℎ is integrable. By the definition of 𝜑_𝑘(𝑥), 𝜑_𝑘(𝑥)=2^𝑘𝑛𝜑2^𝑘𝑥.(4.47) Hence, 𝑐_𝑘,𝑗 converges to 𝑐_𝑘 in 𝐿^𝑝(⋅)(ℝ^𝑛) as 𝑗→∞ by Theorem 4.8 and 𝑓_𝑗 converges to 𝑓 in 𝐹^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛) as 𝑗→∞. Equation (4.40) follows from the last relation.

5. Duality and Reflexivity of Variable Exponent Function Spaces

Theorem 5.1. Let 1<𝑞<∞, 𝑠∈ℝ, and 𝑝(⋅)∈ℬ(ℝ^𝑛). Let 𝐴^{𝑠𝑝(⋅),𝑞} be either 𝐵^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛) or 𝐹^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛). Then, 𝐴^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛)^=𝐴^𝑝−𝑠_{′(⋅),𝑞′}(ℝ^𝑛),(5.1) where 𝑝^(⋅)=𝑝(⋅)𝑝(⋅)−1,(5.2) and 1/𝑞+1/𝑞^=1. The spaces 𝐹^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛) and 𝐵^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛) are reflexive.

Proof. We will only prove (5.1) because the reflexivity of 𝐴^{𝑠(⋅)𝑝(⋅),𝑞(⋅)} follows from Theorem 3.1.
Step 1. We will first prove the 𝐹^{𝑠𝑝(⋅),𝑞} case. We use the arguments of the proof of [19, Theorems  7.1.7, and 7.2.2]. Let 𝑓∈𝒮(ℝ^𝑛), 𝑔∈𝐹^𝑝−𝑠_{′(⋅),𝑞′}(ℝ^𝑛), and {𝜓_𝑘}^∞𝑘=0 replaced by {𝜃_𝑘}^∞𝑘=0 as in Definition 2.1. Then, 𝑓=∑^∞𝑘=0𝑓∗𝜓_𝑘 in 𝒮(ℝ^𝑛), 𝑓∗𝜓_𝑘∈𝒮(ℝ^𝑛), and 𝑔=∑^∞𝑘=0𝑔∗𝜓_𝑘 in 𝒮^(ℝ^𝑛). It follows from Definition 2.1 that 𝑔(𝑓)=^∞_𝑘=0𝑔∗𝜓_𝑘(𝑓)=^∞_𝑘=0^∞_𝑙=0𝑔∗𝜓_𝑘𝑓∗𝜓_𝑙=^∞_𝑘=0^2𝑁_{𝑟=−2𝑁}𝑔∗𝜓_𝑘𝑓∗𝜓_𝑘+𝑟(5.3) in 𝒮^(ℝ^𝑛). Here, 𝜓_𝑗=0 for 𝑗<0. Using Hölder inequality, we get ||𝑔(𝑓)||≤≤𝑐𝑐‖‖_ℝ𝑛2‖‖_−𝑘𝑠2^−𝑘𝑠𝑔∗𝜓_𝑘‖‖_ℓ^𝑞′‖‖2^𝑘𝑠𝑓∗𝜓_𝑘‖‖_ℓ^𝑞d𝑥𝑔∗𝜓_𝑘‖‖_𝐿^𝑝′(⋅)_(ℓ^𝑞′₎‖‖2^𝑘𝑠𝑓∗𝜓_𝑘‖‖_𝐿^𝑝(⋅)_(ℓ^𝑞₎≤𝑐^‖𝑔‖_𝐹^{𝑝′−𝑠(⋅),𝑞′}‖𝑓‖_𝐹^{𝑠𝑝(⋅),𝑞},(5.4) where 𝑐^ does not depend on 𝑔, 𝑓 and 𝜓. Hence, we see 𝑔∈(𝐹^{𝑠𝑝(⋅),𝑞})^ and ‖𝑔‖_(𝐹^{𝑠𝑝(⋅),𝑞}_)≤𝑐‖𝑔‖_𝐹^{−𝑠𝑝′(⋅),𝑞′} since 𝒮(ℝ^𝑛) is dense in 𝐹^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛).Step 2. Let 𝑔∈(𝐹^{𝑠𝑝(⋅),𝑞})^. We assume that 𝑔̃ and the functions 𝜑_𝑘 and 𝜌_𝑘 have the same sense as in Definition 4.7, Lemmas 4.9, and 4.10. We show that 𝑔̃∈𝐹^{−𝑠𝑝(⋅),𝑞}(ℝ^𝑛). For this purpose, we construct functions 𝑎_𝑘 as (4.37) and set 𝑏_𝑘(𝑥)=sgn𝑎_𝑘⋅||𝑎_𝑘(𝑥)||^𝑞′−12^{−𝑠𝑘′𝑞′}‖‖2^−𝑗𝑠𝑎_𝑗^𝑁𝑗=0‖‖^{𝑝′(𝑥)−𝑞′}_ℓ𝑞′,(5.5) for 𝑘=0,1,…,𝑁. If 𝑎_𝑘=0, we set 𝑏_𝑘=0. We have ‖‖2^𝑘𝑠𝑏_𝑘(𝑥)^𝑁𝑘=0‖‖_ℓ𝑞=^𝑁_𝑘=0||𝑎^𝑘||^𝑞′−12^{𝑘𝑠(1−𝑞′)}‖‖2^−𝑗𝑠𝑎_𝑗^𝑁𝑗=0‖‖^{𝑝′(𝑥)−𝑞′}_ℓ𝑞′^𝑞^1/𝑞=‖‖2^−𝑗𝑠𝑎_𝑗^𝑁𝑗=0‖‖^{𝑝′(𝑥)−𝑞′}_ℓ𝑞′^𝑁_𝑘=0||𝑎_𝑘||^𝑞′2^{−𝑘𝑠𝑞′}^1/𝑞=‖‖2^−𝑗𝑠𝑎_𝑗^𝑁𝑗=0‖‖^{𝑞′/𝑞+𝑝′(𝑥)−𝑞′}_ℓ𝑞′,(5.6) where we used (𝑞^−1)𝑞=𝑞^. We set 𝛼=⎧⎪⎨⎪⎩essinf^{𝑥∈ℝ𝑛}𝑝^(𝑥)𝑝(𝑥),if‖‖2^−𝑘𝑠esssup_{𝑥∈ℝ𝑛}𝑎_𝑘^𝑁𝑘=0‖‖_{𝐿𝑝′(⋅)(ℓ𝑞′)}≤1,𝑝^(𝑥)𝑝(𝑥),if‖‖2^−𝑘𝑠𝑎_𝑘^𝑁𝑘=0‖‖_{𝐿𝑝′(⋅)(ℓ𝑞′)}>1,(5.7) and 𝜆=‖{2^−𝑘𝑠𝑎_𝑘}^𝑁𝑘=0‖_{𝐿𝑝′(⋅)(ℓ𝑞′)}. Then, _ℝ𝑛⎛⎜⎜⎝‖‖2^𝑘𝑠𝑏_𝑘^𝑁𝑘=0‖‖_ℓ𝑞𝜆^𝛼⎞⎟⎟⎠^𝑝(𝑥)d𝑥=_ℝ𝑛⎛⎜⎜⎜⎝‖‖2^−𝑗𝑠𝑎_𝑗^𝑁𝑗=0‖‖^{𝑞′/𝑞+𝑝′(𝑥)−𝑞′}_ℓ𝑞′𝜆^𝛼⎞⎟⎟⎟⎠^𝑝(𝑥)=d𝑥_ℝ𝑛‖‖2^−𝑗𝑠𝑎_𝑗^𝑁𝑗=0‖‖^{(𝑞′/𝑞+𝑝′(𝑥)−𝑞′ℓ)𝑝(𝑥)}_𝑞′𝜆^{𝛼𝑝(𝑥)}=d𝑥_ℝ𝑛‖‖{2^−𝑗𝑠𝑎_j}^𝑁𝑗=0‖‖_ℓ𝑞′𝜆^{𝛼(𝑝(𝑥)/𝑝′(𝑥))}^{𝑝′(𝑥)}≤d𝑥_ℝ𝑛⎛⎜⎜⎝‖‖2^−𝑗𝑠𝑎_𝑗^𝑁𝑗=0‖‖_ℓ𝑞′𝜆⎞⎟⎟⎠^{𝑝′(𝑥)}d𝑥≤1,(5.8) where we used 𝑞^𝑞𝑝(𝑥)+𝑝(𝑥)𝑝^esssup_{𝑥∈ℝ𝑛}(𝑥)−𝑞^=𝑝^(𝑥),𝑝^(𝑥)𝑝(𝑥)=essinf^{𝑥∈ℝ𝑛}𝑝(𝑥)𝑝^(𝑥)⁻¹essinf_{𝑥∈ℝ𝑛},𝑝^(𝑥)𝑝(𝑥)=esssup_𝑥∈ℝ^𝑛𝑝(𝑥)𝑝^(𝑥)⁻¹.(5.9) Hence, we have ‖‖2^𝑘𝑠𝑏_𝑘^𝑁𝑘=0‖‖_{𝐿𝑝(⋅)(ℓ𝑞)}≤‖‖2^−𝑘𝑠𝑎_𝑘^𝑁𝑘=0‖‖^𝛼𝐿_{𝑝′(⋅)ℓ𝑞′}.(5.10) In particular, 𝑏_𝑘∈𝐿^𝑝(⋅)(ℝ^𝑛). Now, we construct the same function 𝑓 by (4.39). Using Lemma 4.10, (4.37) and Theorem 4.8, we have _ℝ𝑛‖‖2^−𝑘𝑠𝑎_𝑘^𝑁0‖‖^{𝑝′ℓ(𝑥)}_𝑞′d𝑥=_ℝ𝑛^𝑁_𝑘=0=2^{−𝑠𝑘𝑞′}||𝑎_𝑘||^𝑞′‖‖2^−𝑗𝑠𝑎_𝑗^𝑁𝑗=0‖‖^{𝑝′(𝑥)−𝑞′}_ℓ𝑞′d𝑥_ℝ𝑛𝑁_𝑘=0≤𝑐‖𝑔‖𝑎_𝑘𝑏_{𝑘(𝐹𝑠𝑝(⋅),𝑞)′}d𝑥=⟨𝑔̃,𝑓⟩‖‖2^𝑘𝑠𝑏_𝑘∗𝜑_𝑘^𝑁0‖‖_{𝐿𝑝(⋅)(ℓ𝑞)}≤𝑐‖𝑔‖_{(𝐹𝑠𝑝(⋅),𝑞)}^′‖‖2^𝑘𝑠𝑏_𝑘^𝑁𝑘=0‖‖_{𝐿𝑝(⋅)(ℓ𝑞)}.(5.11) We set 𝛽=⎧⎪⎨⎪⎩esssup^{𝑥∈ℝ𝑛}essinf_{𝑥∈ℝ𝑛}𝑝^(𝑥),if‖‖2^−𝑘𝑠𝑎_𝑘^𝑁𝑘=0‖‖_{𝐿𝑝′(⋅)(ℓ𝑞′)}≤1,𝑝^(𝑥),if‖‖{2^−𝑘𝑠𝑎_𝑘}^𝑁𝑘=0‖‖_{𝐿𝑝′(⋅)(ℓ𝑞′)}>1.(5.12) Then, it follows from Theorem 3.4, (5.11), and (5.10) that ‖‖2^−𝑘𝑠𝑎_𝑘^𝑁𝑘=0‖‖^{𝐿𝛽−𝛼}_{𝑝′(⋅)(ℓ𝑞′)}≤𝑐‖𝑔‖_{(𝐹𝑠𝑝(⋅),𝑞)}^′.(5.13) If 𝑝^(𝑥) is not a constant function, then 𝛽−𝛼≠1. So, we normalize the 𝑎_𝑘. We put 𝑎^𝑘=𝑎_𝑘/𝜆 and 𝑏^𝑘=sgn𝑎^𝑘⋅||𝑎^𝑘||^𝑞′−12^{−𝑠𝑘′𝑞′}‖‖2^−𝑗𝑠𝑎^𝑗^𝑁𝑗=0‖‖^{𝑝′(⋅)−𝑞′}_ℓ𝑞′.(5.14) Then, we have ‖‖2^𝑘𝑠𝑏^𝑘^𝑁𝑘=0‖‖2‖‖_{𝐿𝑝(⋅)(ℓ𝑞)}≤1,_−𝑘𝑠𝑎^𝑘^𝑁𝑘=0‖‖^{𝐿𝛽−𝛼}_{𝑝′(⋅)(ℓ𝑞′)}≤𝑐‖‖‖𝑔𝜆‖‖‖_(𝐹^{𝑠𝑝(⋅),𝑞}₎^′.(5.15) Hence we have ‖‖2^−𝑘𝑠𝑎_𝑘^𝑁𝑘=0‖‖_{𝐿𝑝′(⋅)(ℓ𝑞′)}≤𝑐‖𝑔‖_{(𝐹𝑠𝑝(⋅),𝑞)}^′.(5.16) The last estimate, (4.32), (4.37), and Corollary 4.3 lead us to 𝑔̃∈𝐹^𝑝−𝑠_{′(⋅),𝑞′},‖𝑔̃‖_𝐹^{𝑝′′−𝑠(⋅),𝑞}≤𝑐‖𝑔‖_{(𝐹𝑠𝑝(⋅),𝑞)}^′.(5.17) We specialize the function 𝜌(𝑥) in Definition 4.7 by setting (ℱ𝜌)(𝜉)=1 for 1√2+2𝛿≤||𝜉||≤√2−2𝛿,(5.18) where 𝑓 is a sufficiently small positive number. Then, we find a function 𝜌̃(𝑥)∈𝒮(ℝ^𝑛) such that 0≤ℱ𝜌̃∈𝐶^∞0𝜉∈ℝ^𝑛∶√2−2𝛿<||𝜉||<√2+5𝛿,(5.19)(ℱ𝜌+ℱ𝜌̃+ℱ𝜌₁)=1 for 1/√2+2𝛿<|𝜉|≤2(√2−2𝛿). As in (4.28), we construct 𝜌̃_𝑘(𝑥), and, as in (4.32), we put 𝑔̃̃=^∞_𝑘=0ℱ⁻¹ℱ𝜌̃_𝑘ℱ𝑔(5.20) in 𝒮^(ℝ^𝑛). 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∈𝐹^𝑝−𝑠_{′(⋅),𝑞′}(ℝ^𝑛)∶‖𝑔‖_𝐹^{−𝑠𝑝′(⋅),𝑞′}≤𝑐‖𝑔‖_{(𝐹𝑠𝑝(⋅),𝑞)}^′.(5.21) This completes Theorem 5.1 for the 𝐹^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛). Step 3. Finally, we will prove the 𝐵^{𝑠𝑝(⋅),𝑞} case. Let 𝑓∈𝒮(ℝ^𝑛) and 𝑔∈𝐵^{−𝑠𝑝(⋅),𝑞}(ℝ^𝑛). Then, we have 𝑔∈(𝐵^{𝑠𝑝(⋅),𝑞}(ℝ^𝑛))′ and see that there exists a positive number 𝑐 such that ‖𝑔‖_{(𝐵𝑠𝑝(⋅),𝑞)}^′≤𝑐‖𝑔‖_𝐵^{𝑝′′−𝑠(⋅),𝑞},(5.22) by the same manner as in the Step 1. Step 4. Let 𝑔∈(𝐵^{𝑠𝑝(⋅),𝑞})^ and 𝑔̃ as in (4.32). Then, we have 𝑔̃∈(𝐵^{𝑠𝑝(⋅),𝑞})^ and ‖𝑔̃‖_{(𝐵𝑠𝑝(⋅),𝑞)}^′≤𝑐‖𝑔‖_{(𝐵𝑠𝑝(⋅),𝑞)}^′(5.23) by the same argument in the proof of Lemma 4.9. Furthermore, Lemma 4.10 also holds for 𝑔∈(𝐵^{𝑠𝑝(⋅),𝑞})^ by the same argument in the proof of Lemma 4.10. Let 𝑎_𝑘 as in (4.37). Then, 𝑎_𝑘∈𝐿^𝑝′(⋅)(ℝ^𝑛). We set 𝑏_𝑘(𝑥)=sgn𝑎_𝑘⋅||𝑎_𝑘(𝑥)||^{𝑝′(𝑥)−1}2^{−𝑠𝑘𝑞′}‖‖𝑎_𝑘‖‖^{𝑞′−𝑝′𝑝(𝑥)}_′(⋅),(5.24) for 𝑘=0,1,2,…. Then, we have ‖𝑏_𝑘‖_𝐿^𝑝(⋅)≤2^{−𝑠𝑘𝑞′}‖𝑎_𝑘‖^{𝑞′𝐿/𝑞}_𝑝′(⋅) by the following calculating: _ℝ𝑛⎛⎜⎜⎝||𝑏_𝑘(𝑥)||2^{−𝑘𝑠𝑞′}‖‖𝑎_𝑘‖‖^{𝑞′𝐿/𝑞}_𝑝′(⋅)⎞⎟⎟⎠^𝑝(𝑥)d𝑥=_ℝ𝑛⎛⎜⎜⎝||𝑎_𝑘(𝑥)||^{𝑝′(𝑥)−1}‖‖𝑎_𝑘‖‖^{𝑞′−𝑝′𝐿(𝑥)}_𝑝′(⋅)‖‖𝑎_𝑘‖‖^{𝐿𝑞/𝑞}_𝑝′(⋅)⎞⎟⎟⎠^𝑝(𝑥)=d𝑥_ℝ𝑛||𝑎_𝑘(𝑥)||^{𝑝′(𝑥)}‖‖𝑎_𝑘‖‖^{(𝑞′/𝑞)𝑝(𝑥)−𝑝′𝐿(𝑥)}_𝑝′(⋅)‖‖𝑎_𝑘‖‖^{(𝑞′𝐿/𝑞)𝑝(𝑥)}_𝑝′(⋅)=d𝑥_ℝ𝑛||𝑎^𝑘(𝑥)||‖‖𝑎_𝑘‖‖_{𝐿𝑝′(⋅)}^{𝑝′(𝑥)}d𝑥=1.(5.25) Hence, we have ‖‖2^𝑘𝑠𝑏_𝑘^𝑁0‖‖_{ℓ𝑞(𝐿𝑝(⋅))}≤^𝑁_𝑘=02^{𝑠𝑘𝑞(1−𝑞′)}‖‖𝑎_𝑘‖‖^𝑞′_{𝐿𝑝′(⋅)}^1/𝑞=^𝑁_𝑘=02^{−𝑠𝑘𝑞′}‖‖𝑎_𝑘‖‖^𝑞′_{𝐿𝑝′(⋅)}^{𝑞′/(𝑞′⋅𝑞)}=‖‖2^−𝑘𝑠𝑎_𝑘^𝑁0‖‖^{𝑞′ℓ/𝑞}_{𝑞′(𝐿𝑝′(⋅))}.(5.26) We have _ℝ𝑛^𝑁_𝑘=0𝑎_𝑘𝑏_𝑘d𝑥=^𝑁_𝑘=02^{−𝑠𝑘𝑞′}_ℝ𝑛=||𝑎_𝑘(𝑥)||^{𝑝′(𝑥)}‖‖𝑎_𝑘‖‖^{𝑞′−𝑝′𝐿(𝑥)}_𝑝′(⋅)d𝑥_𝑁_𝑘=02^{−𝑠𝑘𝑞′}‖‖𝑎_𝑘‖‖^𝑞′_{𝐿𝑝′(⋅)}=‖‖2^−𝑘𝑠𝑎_𝑘^𝑁0‖‖^𝑞′_{ℓ𝑞′(𝐿𝑝′(⋅))}.(5.27) Hence, we have ‖‖2^−𝑘𝑠𝑎_𝑘^𝑁0‖‖^𝑞′_{ℓ𝑞′𝐿𝑝′(⋅)}≤‖𝑔‖_{(𝐵𝑠𝑝(⋅),𝑞)}^′‖‖2^−𝑘𝑠𝑎_𝑘^𝑁0‖‖^{𝑞′ℓ/𝑞}_{𝑞′𝐿𝑝′(⋅)}(5.28) by the same manner as in Step 2. It follows from this, as in Step 2, that 𝑔∈𝐵^𝑝−𝑠_{′(⋅),𝑞′}(ℝ^𝑛),‖𝑔‖_𝐵^{𝑝′′−𝑠(⋅),𝑞}≤𝑐‖𝑔‖_{(𝐵𝑠𝑝(⋅),𝑞)}^′.(5.29) This complete the proof.

The next corollary is an immediate consequence of Theorems 4.6 and 5.1.

Corollary 5.2. Let 𝑠≥0 and p(⋅)∈ℬ(ℝ^𝑛). Then, 𝐿^{𝑠,𝑝(⋅)}(ℝ^𝑛)^=𝐹^𝑝−𝑠_′(⋅),2(ℝ^𝑛).(5.30) In addition, for 𝑘∈ℕ₀, one has 𝑊^{𝑘,𝑝(⋅)}(ℝ^𝑛)^=𝐹^𝑝−𝑘_′(⋅),2(ℝ^𝑛).(5.31) Hence, 𝐿^{𝑠,𝑝(⋅)}(ℝ^𝑛) and 𝑊^{𝑘,𝑝(⋅)}(ℝ^𝑛) are reflexive.

Remark 5.3. Let Ω be ℝ^𝑛 or any measurable subset of ℝ^𝑛. As we mentioned in Section 1, Kováčik and Rákosník [9] showed that the dual of 𝐿^𝑝(⋅)(Ω) is 𝐿^𝑝(⋅)(Ω) if 𝑝(⋅) is a measurable function on Ω with a range in [1,∞] and 𝑝(⋅)∈𝐿^∞(Ω). Furthermore, 𝐿^𝑝(⋅)(Ω) is reflexive if 1<𝑝₋≤𝑝₊<∞. There is no assumption of boundedness of Hardy-Littlewood maximal operator ℳ on 𝐿^𝑝(⋅)(Ω).

Acknowledgments

The authors would like to express his gratitude to Professor Y. Kobayashi for the great support in any manner and for his valuable suggestions in many discussions. He is also thankful to the anonymous referees for their reading carefully and their advices in an appropriate manner.