# Dual of Modulation Spaces with Variable Smoothness and Integrability

Authors: Hua Zhu
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this article, we first give a proof for the denseness of the Schwartz class in the modulation spaces with variable smoothness and integrability. Then, we study the dual spaces of such modulation spaces.


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## Body

## 1. Introduction

The modulation spacesMp,qs were introduced by Feichtinger [1] on a locally compact Abelian group in 1983 through short-time Fourier transform. His original motivation for modulation spaces was to introduce a new theory of function spaces and to offer an alternative to the class of Besov spaces. In recent years, it is gradually recognized that the modulation spaces are very useful for studying time-frequency behavior of functions. Therefore, the modulation spaces, α-modulation spaces, and their applications have received a lot of attention and research, such as [2–10] and the references therein. Particularly, in [11–14], Wang and other authors showed that from PDE point of view, the combination of frequency-uniform decomposition operators and Banach function spaces ℓqXℝn is important in making nonlinear estimates, where X is a Banach function space defined on ℝn.On the other hand, function spaces with variable exponents have received extensive attention recently. Even though the study on variable Lebesgue spaces can be traced back to [15, 16] by Orlicz, the modern development started from the article [17] by Kováčik and Rákosník in 1991. In [18], Fan and Zhao obtained the results in [17] again through the method of Musielak-Orlicz spaces. Thereafter, variable Lebesgue and Sobolev spaces have been widely studied (see, for example, [19–23]). In addition, function spaces with variable exponents have a wealth of applications in many fields, such as in fluid dynamics [24], image processing [25], and partial differential equations [26].The function spaces with variable smoothness and variable integrability were firstly introduced by Diening et al. in [27], where they studied Triebel-Lizorkin spaces with variable exponents Fp·,q·α·ℝn. Then, Almeida and Hästö introduced the Besov space with variable smoothness and integrability Bp·,q·α·ℝn in [28]. Since then, many articles about these function spaces appeared, such as [29, 30]. In the past few years, many function spaces with variable exponents have appeared, such as Besov-type spaces with variable exponent, Bessel potential spaces with variable exponent, and Hardy spaces with variable exponent (see [31–35]). Recently, we studied the modulation spaces with variable smoothness and integrability Mp·,q·s· and gave some properties about these spaces in [36]. Since the modulation spaces and the function spaces with variable exponents have rich applications, we believe that the modulation spaces with variable exponents will also have many application areas, and we will continue to explore these application areas, especially in partial differential equations and time frequency analysis.The dual is an important content when we study function spaces; for example, Triebel [37] has obtained duality of the usual Besov spaces and applied it to real interpolation and Sobolev embedding, Izuki [38] has given the duality of Herz spaces with variable exponent and applied it to characterize the above spaces by wavelet expansions. In [8, 12], the dual of modulation spaces was studied, respectively. In [39], Izuki and Noi were concerned with the dual of Triebel-Lizorkin spaces Fp·,q·s· and Besov spaces Bp·,q·s· with variable exponents. In this paper, we will study the dual of modulation spaces Mp·,q·s· with variable smoothness and integrability.The paper is organized as follows. In Section2, we review some notions and notations about semimodular spaces and function spaces with variable exponents. In the theories of function spaces, the research on denseness of the Schwartz class has always been an important topic, by which we can obtain many conclusions such as duality of function spaces and boundedness of some operators. Therefore, in Section 3, we study the denseness of the Schwartz class in the modulation spaces with variable smoothness and integrability. In Section 4, we give the dual of modulation spaces with variable exponents.


## 2. Preliminaries

In this section, we review some notions and conventions and state some basic results. Throughout this article, we letC denote constants that are independent of the main parameters involved but whose value may differ from line to line. By A~B, we mean that there exists a positive constant C such that 1/C≤A/B≤C. The symbol A≲B means that A≤CB. The symbol s for s∈ℝ denotes the maximal integer not more than s. We also set ℕ≡1,2,⋯ and ℤ+≡ℕ∪0. We write x=1+x21/2 and xo=1+x1+x2+⋯+xn for x∈ℝn. It is easy to see that x~xo. For any multi-index α=α1,α2,⋯,αn, we denote Dα=∂1α1∂2α2⋯∂nαn, and for k=k1,k2,⋯,kn, we denote k∞=maxi=1,⋯,nki. We also denote the sequence Lebesgue space by ℓp and Lebesgue space by Lp≔Lpℝn for which the norm is written by ·p.LetS≔Sℝn be the Schwartz function space and S′≔S′ℝn be its strongly topological dual space which is also known as the space of all tempered distributions. For f∈S, we define the Fourier transform Ff and the inverse Fourier transform F−1f, respectively, by
(1)Ffξ=f^ξ=∫ℝnfxe−2πix·ξdx,F−1fx=∫ℝnfξe2πix·ξdξ.


### 2.1. Modular Spaces

In what follows, letX be a vector space over ℝ or ℂ. The function spaces studied in this paper fit into the framework semimodular spaces, and we refer to monograph [23] for a detailed exposition of these concepts.Definition 1.
A functionϱ:X⟶0,∞ is called a semimodular on X if it satisfies
(2)ϱ0=0.(i)
ϱλf=ϱf for all f∈X, λ∈ℝorℂ with λ=1(ii)
ϱλf=0 for all λ>0 implies f=0(iii)
λ↦ϱλf is left-continuous on 0,∞ for every f∈XA semimodularϱ is called a modular if ϱf=0 implies f=0, and it is called continuous if the mapping λ↦ϱλf is continuous on 0,∞ for every f∈X. A semimodular ρ can also be qualified by the term (quasi)convex; that is, for all f,g∈X and θ∈0,1, there exists A such that
(3)ϱθf+1−θg≤Aθϱf+1−θϱg,where A=1 in the convex case and A∈1,∞ in the quasiconvex case. By semimodular, we can obtain a normed space as follows:Definition 2.
Ifρ is a (semi)modular on X, then Xϱ≔f∈X:∃λ>0,s.t.ϱλf<∞ is called a (semi)modular space.In [11], the authors have proven that theXϱ is a (quasi)normed space with the Luxemburg (quasi)norm fϱ≔infλ>0:ϱf/λ≤1, where the infimum of the empty set is infinity by definition. The following conclusion can be found in [23], and we omit the proof here.Theorem 3 (norm-modular unit ball property).
Letϱ be a semimodular on X and f∈X. Then, fϱ≤1 if and only if ϱf≤1. If ϱ is continuous, then fϱ<1 and ϱf<1 are equivalent, so are fϱ=1 and ϱf=1.


### 2.2. Function Spaces with Variable Exponents

A measurable functionp·:ℝn⟶0,∞ is called a variable exponent function if it is bounded away from zero; namely, the range of the px is c,∞ for some c>0. For a measurable function p· and a measurable set Ω⊂ℝn, let pΩ−≔essinfΩpx and pΩ+≔esssupΩpx. For simplicity, we abbreviate p−=pℝn− and p+=pℝn+.We denote byP0 for the set of all measurable functions p·:ℝn⟶0,∞ such that 0<p−≤p+<∞ and denote by P for the set of all measurable functions p·:ℝn⟶0,∞ such that 1<p−≤p+<∞.In order to make the Hardy-Littlewood maximal function bounded in the variable exponent Lebesgue spaces, one need to add some conditions to the variable exponent function, that is, so-called log-Hölder continuity, which was first introduced in [40].Definition 4.
Letp·:ℝn⟶ℝ.
(i)
If there existsclog>0 such that(4)px−py≤clogloge+1/x−y,for all x,y∈ℝn, then p· is called locally log-Hölder continuous, abbreviated as p∈Cloclog.
(ii)
Ifp· is locally log-Hölder continuous and there exists p∞∈ℝ such that(5)px−p∞≤clogloge+x,for all x∈ℝn, then p· is called globally log-Hölder continuous, abbreviated as p∈Clog.If a variable exponentp∈P satisfies 1/p∈Clog, we say that it belongs to the class Plog. The class P0log is defined similarly.Remark 5.
(i)
One can notice that all functionsp∈Cloclog always belong to L∞(ii)
Letp∈P0, then p∈Clog if and only if 1/p∈Clog. If p satisfies (5), then p∞=limx⟶∞px(iii)
We define the conjugate exponent functionp′· by the formula 1/p·+1/p′·=1. If p· is in Clog, then p′· is also in ClogWe define(6)φpt=tp,ifp∈0,∞,0,ifp=∞andt≤1,∞,ifp=∞andt>1,and we adopt the convention 1∞=0 in order that φp is left-continuous. The variable exponent modular of a measurable function f on ℝn is defined by
(7)ρp·f≔∫ℝnφpxfxdx.According to Definition2, one can define the corresponding semimodular space, namely, the variable exponent Lebesgue space which is denoted by Lp·ℝn, and the Luxemburg (quasi)norm of the Lp·ℝn is defined by
(8)fp·≔infλ>0:ϱp·fλ≤1.Now let us recall the mixed Lebesgue sequence spaceℓq·Lp· which was introduced by Almeida and Hästö in [28].Definition 6.
Letp,q∈P0 and Ω be a measurable subset of ℝn. The mixed Lebesgue sequence space ℓq·Lp·Ω is the collection of all sequences fjj∈ℕ of Lp·Ω-functions such that
(9)fjjℓq·Lp·Ω≔infλ>0:ϱℓq·Lp·fjχΩλj≤1<∞,where
(10)ϱℓq·Lp·fjj≔∑j∈ℕinfμj>0:ϱp·fjμj1/q·≤1,with the convention λ1/∞=1 for all λ>0.Remark 7.
Letp,q∈P0.
(i)
Ifq+<∞, then infλ>0:ϱp·f/λ1/q·≤1=fq·p·/q·, and we use the notation(11)ϱℓq·Lp·fjj=∑jfjq·p·/q·.(ii)
By Proposition 3.3 of [28], if q∈0,∞ is constant, then we have fjjℓqLp·=fjp·ℓq(iii)
In [28], Almeida and Hästö proved that ·ℓq·Lp· is a quasinorm for all p·,q·∈P, and ·ℓq·Lp· is a norm when 1/p·+1/q·≤1 pointwise or q is a constant. In [30], Kempka and Vybíral proved that ·ℓq·Lp· is a norm if p·,q·∈P satisfy either 1≤qx≤px≤∞ for almost every x∈ℝn or px≥1, and q∈1,∞ is a constant almost everywhereTo define modulation space with variable exponents, we need some general definitions from the constant exponent case. Fork∈ℤn, let Qk be the unit cube with the center at k; then, Qkk∈ℤn constitutes a decomposition of ℝn. Let ϕ∈Sℝn and ϕ:ℝn⟶0,1 be a smooth function satisfying ϕξ=1 for ξ∞≤1/2 and ϕξ=0 for ξ∞≥1. Let ϕk be a translation of ϕ: ϕkξ=ϕξ−k,k∈ℤn. Then, we see that ϕkξ=1 in Qk and ∑k∈ℤnϕkξ≥1 for all ξ∈ℝn. If we denote φkξ=ϕkξ∑k∈ℤnϕkξ−1, for k∈ℤn, then we have
(12)φkξ≥c,∀ξ∈Qk,suppφk⊂ξ:ξ−k∞≤1,,∑k∈ℤnφkξ≡1,∀ξ∈ℝn,Dαφkξ≤Cα,∀ξ∈ℝn,α∈ℕ∪0n.We denoteY=φkk∈ℤn:φkk∈ℤnsatisfies3.Y is nonempty, and for every sequence φkk∈ℤn∈Y, one can construct an operator sequence as follows:
(13)□k≔F−1φkF,k∈ℤn.□kk∈ℤn are said to be frequency-uniform decomposition operators. Let s∈ℝ and 0<p,q≤∞; the modulation space can be defined as
(14)Mp,qsℝn=f∈S′ℝn:fMp,qs=∑k∈ℤnksq□kfpq1/q<∞.Further details about the frequency-uniform decomposition techniques and their applications to PDE can be found in the book [13] and articles [11, 12, 14].Definition 8.
Letφkk∈ℤn∈Y and □kk∈ℤn be the corresponding frequency-uniform decomposition operators. For p,q∈P0log and s∈Cloclog, the modulation space with variable smoothness and integrability Mp·,q·s· is defined to be the set of all distributions f∈S′ such that
(15)fMp·,q·s·φ≔ks·□kfkℓq·Lp·<∞.For above modulation space, we can define the following modular:(16)ϱMp·,q·s·φf≔ϱℓq·Lp·ks·□kfk,which can be used to define the norm. In [36], we have shown that the space given by Definition 8 is independent of the choice of φkk∈ℤn∈Y and the corresponding frequency-uniform decomposition operators. Thus, we can choose φkk∈ℤn∈Y according to our requirements, and we will omit φ in the notation of the norm and modular.


## 2.1. Modular Spaces

In what follows, letX be a vector space over ℝ or ℂ. The function spaces studied in this paper fit into the framework semimodular spaces, and we refer to monograph [23] for a detailed exposition of these concepts.Definition 1.
A functionϱ:X⟶0,∞ is called a semimodular on X if it satisfies
(2)ϱ0=0.(i)
ϱλf=ϱf for all f∈X, λ∈ℝorℂ with λ=1(ii)
ϱλf=0 for all λ>0 implies f=0(iii)
λ↦ϱλf is left-continuous on 0,∞ for every f∈XA semimodularϱ is called a modular if ϱf=0 implies f=0, and it is called continuous if the mapping λ↦ϱλf is continuous on 0,∞ for every f∈X. A semimodular ρ can also be qualified by the term (quasi)convex; that is, for all f,g∈X and θ∈0,1, there exists A such that
(3)ϱθf+1−θg≤Aθϱf+1−θϱg,where A=1 in the convex case and A∈1,∞ in the quasiconvex case. By semimodular, we can obtain a normed space as follows:Definition 2.
Ifρ is a (semi)modular on X, then Xϱ≔f∈X:∃λ>0,s.t.ϱλf<∞ is called a (semi)modular space.In [11], the authors have proven that theXϱ is a (quasi)normed space with the Luxemburg (quasi)norm fϱ≔infλ>0:ϱf/λ≤1, where the infimum of the empty set is infinity by definition. The following conclusion can be found in [23], and we omit the proof here.Theorem 3 (norm-modular unit ball property).
Letϱ be a semimodular on X and f∈X. Then, fϱ≤1 if and only if ϱf≤1. If ϱ is continuous, then fϱ<1 and ϱf<1 are equivalent, so are fϱ=1 and ϱf=1.


## 2.2. Function Spaces with Variable Exponents

A measurable functionp·:ℝn⟶0,∞ is called a variable exponent function if it is bounded away from zero; namely, the range of the px is c,∞ for some c>0. For a measurable function p· and a measurable set Ω⊂ℝn, let pΩ−≔essinfΩpx and pΩ+≔esssupΩpx. For simplicity, we abbreviate p−=pℝn− and p+=pℝn+.We denote byP0 for the set of all measurable functions p·:ℝn⟶0,∞ such that 0<p−≤p+<∞ and denote by P for the set of all measurable functions p·:ℝn⟶0,∞ such that 1<p−≤p+<∞.In order to make the Hardy-Littlewood maximal function bounded in the variable exponent Lebesgue spaces, one need to add some conditions to the variable exponent function, that is, so-called log-Hölder continuity, which was first introduced in [40].Definition 4.
Letp·:ℝn⟶ℝ.
(i)
If there existsclog>0 such that(4)px−py≤clogloge+1/x−y,for all x,y∈ℝn, then p· is called locally log-Hölder continuous, abbreviated as p∈Cloclog.
(ii)
Ifp· is locally log-Hölder continuous and there exists p∞∈ℝ such that(5)px−p∞≤clogloge+x,for all x∈ℝn, then p· is called globally log-Hölder continuous, abbreviated as p∈Clog.If a variable exponentp∈P satisfies 1/p∈Clog, we say that it belongs to the class Plog. The class P0log is defined similarly.Remark 5.
(i)
One can notice that all functionsp∈Cloclog always belong to L∞(ii)
Letp∈P0, then p∈Clog if and only if 1/p∈Clog. If p satisfies (5), then p∞=limx⟶∞px(iii)
We define the conjugate exponent functionp′· by the formula 1/p·+1/p′·=1. If p· is in Clog, then p′· is also in ClogWe define(6)φpt=tp,ifp∈0,∞,0,ifp=∞andt≤1,∞,ifp=∞andt>1,and we adopt the convention 1∞=0 in order that φp is left-continuous. The variable exponent modular of a measurable function f on ℝn is defined by
(7)ρp·f≔∫ℝnφpxfxdx.According to Definition2, one can define the corresponding semimodular space, namely, the variable exponent Lebesgue space which is denoted by Lp·ℝn, and the Luxemburg (quasi)norm of the Lp·ℝn is defined by
(8)fp·≔infλ>0:ϱp·fλ≤1.Now let us recall the mixed Lebesgue sequence spaceℓq·Lp· which was introduced by Almeida and Hästö in [28].Definition 6.
Letp,q∈P0 and Ω be a measurable subset of ℝn. The mixed Lebesgue sequence space ℓq·Lp·Ω is the collection of all sequences fjj∈ℕ of Lp·Ω-functions such that
(9)fjjℓq·Lp·Ω≔infλ>0:ϱℓq·Lp·fjχΩλj≤1<∞,where
(10)ϱℓq·Lp·fjj≔∑j∈ℕinfμj>0:ϱp·fjμj1/q·≤1,with the convention λ1/∞=1 for all λ>0.Remark 7.
Letp,q∈P0.
(i)
Ifq+<∞, then infλ>0:ϱp·f/λ1/q·≤1=fq·p·/q·, and we use the notation(11)ϱℓq·Lp·fjj=∑jfjq·p·/q·.(ii)
By Proposition 3.3 of [28], if q∈0,∞ is constant, then we have fjjℓqLp·=fjp·ℓq(iii)
In [28], Almeida and Hästö proved that ·ℓq·Lp· is a quasinorm for all p·,q·∈P, and ·ℓq·Lp· is a norm when 1/p·+1/q·≤1 pointwise or q is a constant. In [30], Kempka and Vybíral proved that ·ℓq·Lp· is a norm if p·,q·∈P satisfy either 1≤qx≤px≤∞ for almost every x∈ℝn or px≥1, and q∈1,∞ is a constant almost everywhereTo define modulation space with variable exponents, we need some general definitions from the constant exponent case. Fork∈ℤn, let Qk be the unit cube with the center at k; then, Qkk∈ℤn constitutes a decomposition of ℝn. Let ϕ∈Sℝn and ϕ:ℝn⟶0,1 be a smooth function satisfying ϕξ=1 for ξ∞≤1/2 and ϕξ=0 for ξ∞≥1. Let ϕk be a translation of ϕ: ϕkξ=ϕξ−k,k∈ℤn. Then, we see that ϕkξ=1 in Qk and ∑k∈ℤnϕkξ≥1 for all ξ∈ℝn. If we denote φkξ=ϕkξ∑k∈ℤnϕkξ−1, for k∈ℤn, then we have
(12)φkξ≥c,∀ξ∈Qk,suppφk⊂ξ:ξ−k∞≤1,,∑k∈ℤnφkξ≡1,∀ξ∈ℝn,Dαφkξ≤Cα,∀ξ∈ℝn,α∈ℕ∪0n.We denoteY=φkk∈ℤn:φkk∈ℤnsatisfies3.Y is nonempty, and for every sequence φkk∈ℤn∈Y, one can construct an operator sequence as follows:
(13)□k≔F−1φkF,k∈ℤn.□kk∈ℤn are said to be frequency-uniform decomposition operators. Let s∈ℝ and 0<p,q≤∞; the modulation space can be defined as
(14)Mp,qsℝn=f∈S′ℝn:fMp,qs=∑k∈ℤnksq□kfpq1/q<∞.Further details about the frequency-uniform decomposition techniques and their applications to PDE can be found in the book [13] and articles [11, 12, 14].Definition 8.
Letφkk∈ℤn∈Y and □kk∈ℤn be the corresponding frequency-uniform decomposition operators. For p,q∈P0log and s∈Cloclog, the modulation space with variable smoothness and integrability Mp·,q·s· is defined to be the set of all distributions f∈S′ such that
(15)fMp·,q·s·φ≔ks·□kfkℓq·Lp·<∞.For above modulation space, we can define the following modular:(16)ϱMp·,q·s·φf≔ϱℓq·Lp·ks·□kfk,which can be used to define the norm. In [36], we have shown that the space given by Definition 8 is independent of the choice of φkk∈ℤn∈Y and the corresponding frequency-uniform decomposition operators. Thus, we can choose φkk∈ℤn∈Y according to our requirements, and we will omit φ in the notation of the norm and modular.


## 3. Density

In [28], the authors showed that the maximal function is not a good tool in the variable exponent space ℓq·Lp·; hence, they used so-called η-functions which were also used in [27]. Similarly, in our article, we define the so-called θ-functions on ℝn by
(17)θk,mx≔rkn1+rkxm,with k∈ℤn, m>0, and rk≔n1+k∞. Note that θk,m∈L1 when m>n and that θk,m1=cm is independent of k. These functions are different from the η-functions since we use the uniform decomposition of ℝn rather than the dyadic decomposition.Now let us review some useful results aboutθ-functions which have been proven in [36].Lemma 9 (see [36]).
Lets·∈Cloclog and k∈ℤn; then, there exists a positive constant C such that
(18)ksxθk,m+Rx−y≤Cksyθk,mx−y,for all x,y∈ℝn and R≥clogs, where clogs is the constant from (4) for s·.Remark 10.
By Lemma9, we can move the term inside the convolution as follows:
(19)ksxθk,m+R∗fx≤Cθk,m∗ks·fx,which helps us to treat the variable smoothness in many cases.Lemma 11 (see [36]).
Letp,q∈Plog, for m>n and every sequence fkk∈ℤn of Lloc1-functions, there exists a constant C>0 such that
(20)θk,2m∗fkkℓq·Lp·≤Cfkkℓq·Lp·.Remark 12.
In some cases, although we need to require thatp−,q−≥1, we can weaken this condition by the following identity:
(21)fkkℓq·Lp·=fkrkℓq·/rLp·/r1/r.In [36], we have proven that Sℝn↪Mp·,q·s·ℝn, and in this section, we will prove that Sℝn is also dense in Mp·,q·s·ℝn. For this purpose, we need the following conclusions as in [41]. The first one is the generalization of Lemma A.6 in [27] and Lemma 3.4 in [36].Lemma 13.
Letr>0, k∈ℤn, and m>n. Then, for all x,z∈ℝn and g∈S′ with suppg^⊂ξ:ξ−k∞≤1, there exists a constant C=Cr,m,n>0 such that
(22)gx−z1+rkzm/r≤Cθk,m∗grx1/r.Proof.
As in the proof of Lemma 3.4 of [36], for rk=n1+k∞, there exists v∈ℕ such that 2v≤rk<2v+1, which implies ξ:ξ−k∞≤1⊂ξ:ξ≤2v+1. Then, for u∈ℤn and a fixed dyadic cube Q=Qv,u≔x∈ℝn:2−vui≤xi<2−vui+1,i=1,2,⋯,n, when x−z∈Q, we have
(23)gx−zr≤supw∈Qgwr≤C2vn∑l∈ℤn1+l−m∫Qv,u+lgyrdy.
In addition, forx−z∈Qv,u and y∈Qv,u+l, we have x−z−y~2−vl when l is large enough, which implies 1+2vx−z−y~1+l. Since
(24)1+2vx−ym≤1+2vx−z−ym1+2vzm,we get
(25)supw∈Qgwr≤C2vn∑l∈ℤn∫Qv,u+l1+2vx−z−y−mgyrdy≤C2vn∑l∈ℤn∫Qv,u+l1+2vx−y−m1+2vzmgyrdy≤C1+rkzm∑l∈ℤn∫Qv,u+lrkn1+rkx−y−mgyrdy≤C1+rkzm∫ℝnrkn1+rkx−y−mgyrdy=C1+rkzmθk,m∗grx.
Hence, forx−z∈Qv,u, we have
(26)gx−z1+rkzm/r≤Cθk,m∗grx1/r,where C=Cr,m,n depends only on r, m, and n. For any x,z∈ℝn, there exists a u′∈ℤn such that x−z∈Qv,u′. Then, we get the desired conclusion.Definition 14.
(i)
LetΩ be a compact subset of ℝn; then, we denote the space of all elements f∈Sℝn with suppFf⊂Ω by SΩℝn(ii)
Letp·∈P0log and Ω=Ωkk=0∞ be a sequence of compact subsets of ℝn; then, we denote by LΩp·ℝn the space of all sequences fkk=0∞ in S′ℝn such that suppFfk⊂Ωk and fkp·<∞ for k=0,1,2,⋯Lemma 15.
Letp,q∈P0log, s∈Cloclog, and Ω=Ωkk=0∞ be a sequence of compact subsets of ℝn such that Ωk⊂ξ∈ℝn:ξ−k∞≤1. If 0<r<minp−,q− and m>2n+2clogsminp−,q−, then for all fkk∈ℤn⊂LΩp·ℝn, there exists a constant C such that
(27)supz∈ℝnks·−zfk·−z1+rkzm/rk∈ℤnℓq·Lp·≤Cks·fkk∈ℤnℓq·Lp·.Proof.
Letfkk∈ℤn∈LΩp·ℝn and R≥clogs; then, for any k∈ℤn by Lemmas 9 and 13, we have
(28)ksx−zfkx−z1+rkzm/r≤max2m/r,1ksx−zfkx−z1+rkzm/r≲max2m/r,1ksxfkx−z1+rkzm−Rr/r≲max2m/r,1ksxθk,m−Rr∗fkrx1/r≲max2m/r,1θk,m−2Rry∗krsyfkyrx1/r.
Therefore, for0<r<minp−,q− and m>2n+2Rr, by Lemma 11, we obtain
(29)supz∈ℝnks·−zfk·−z1+rkzm/rk∈ℤnℓq·Lp·≤Cθk,m−2Rry∗krsyfkyr·k∈ℝnℓq·/rLp·/r1/r≤Ckrs·fk·rk∈ℤnℓq·/rLp·/r1/r=Cks·fkk∈ℤnℓq·Lp·,which completes the proof.For a real numbers, we denote
(30)H2sℝn=f∈S′ℝn:fH2s=1+·2s/2Ff·L2<∞.Proposition 16.
Letp,q∈P0log, s∈Cloclog, and Ω=Ωkk=0∞ be a sequence of compact subsets of ℝn such that Ωk⊂ξ∈ℝn:ξ−k∞≤1. If t>n/2+2n+3clogsminp−,q−/minp−,q−, then for all fkk∈ℤn⊂LΩp·ℝn and Mk·k∈ℤn⊂H2tℝn, there exists a constant C such that
(31)ks·F−1MkFfkk∈ℤnℓq·Lp·≤CsupkMkrk·H2tks·fkk∈ℤnℓq·Lp·.Proof.
According to Lemma15, by the similar argument in the proof of Theorem 4.15 of [41], we have
(32)ksxF−1MkFfkx−z≲∫ℝnksxF−1Mkx−z−y1+rkx−ym+Rr/rfky1+rkx−ym+Rr/rdy≲∫ℝnF−1Mkx−z−y1+rkx−ym/rksyfky1+rkx−ym+Rr/rdy≲supu∈ℝnksufku1+rkx−um/r∫ℝnF−1Mkx−z−y1+rkx−ym+Rr/rdy.
Since(33)1+rkx−ym+Rr/r≲1+rkx−y−zm+Rr/r1+rkzm+Rr/r,then for 0<r<minp−,q− and t>n/2+m+Rr/r, by the same argument in 1.6.3 of [42], we have
(34)∫ℝnF−1Mkx−z−y1+rkx−ym+Rr/rdy≲∫ℝnF−1Mkx−z−y1+rkx−y−zm+Rr/r1+rkzm+Rr/rdy≲1+rkzm+Rr/rMkrk·H2t.
Thus,(35)supz∈ℝnksxF−1MkFfkx−z1+rkzm+Rr/r≲supz∈ℝnksx−zfkx−z1+rkzm/rMkrk·H2t.
In addition, since(36)ksxF−1MkFfkx≤supz∈ℝnksxF−1MkFfkx−z1+rkzm+Rr/r,then together with above inequality, (33) and Lemma 15, we can get the conclusion.Remark 17.
In fact, in the conclusion of the above lemma, the “rk” in Mkrk·H2t can be replaced by Ωk≔supx,y∈Ωkx−y.Theorem 18 (density).
Letp,q∈P0log and s∈Cloclog, then Sℝn is dense in Mp·,q·s·ℝn.Proof.
Letf∈Mp·,q·s·, for φkk∈ℤn∈Y and N∈ℤ+, we put
(37)fN=∑k≤N□kf=∑k≤NF−1φkFf,where k≔k1+k2+⋯+kn for k∈ℤn. Then, we have fN∈Mp·,q·s·. In fact, by Proposition 16, we obtain
(38)fNMp·,q·s·=ls·□lfNl∈ℤnℓq·Lp·≲ls·∑r∞≤1□l□l+rfl≤N+nℓq·Lp·≲ls·□lfl∈ℤnℓq·Lp·<∞.
Consequently,(39)f−fNMp·,q·s·≲ls·∑r∞≤1□l□l+rfl>N−nℓq·Lp·≲ls·□lfl>N−nℓq·Lp·⟶0,when N⟶∞, in which the last limit can be deduced by Lemma 2.2 of [43]. Hence, fN⟶f in Mp·,q·s· when N⟶∞.
Next, we should approximatefN by some functions in Sℝn for N∈ℤ+. Let ψ∈Sℝn satisfy ψ0=1 and suppFψ⊂ξ∈ℝn:ξ≤1. Then, for any N∈ℤ+ and n∈ℕ, we have ψ·/nfN∈Sℝn and
(40)suppFfN−ψ·/nfN⊂ξ∈ℝn:ξ∞≤N+3.
Since1−ψ·/n∞=1−ψ∞<∞, we have
(41)limn⟶∞fN−ψ·/nfNp·=0,by Lemma 3.2.8 of [23]. Now, we prove that ψ·/nfNn∈ℕ⊂Sℝn is an approximation of fN in Mp·,q·s·. By (38), we have
(42)fN−ψ·/nfNMp·,q·s·=ls·□lfN−ψ·/nfNl∈ℤnℓq·Lp·=ls·□lfN−ψ·/nfNl∞≤N+3ℓq·Lp·.
Letl0=N+3,0,0,⋯,0, then N+3≤rl0=nN+4 and ξ∈ℝn:ξ∞≤N+3⊂ξ∈ℝn:ξ≤rl0. For each l∈ℤn with l∞≤N+3, let gl=fN−ψ·/nfN; then, gl∈LΩl0p·, where Ωl0=ξ∈ℝn:ξ≤rl0. By Remark 17 and the embedding properties of Mp·,q·s·, we get
(43)fN−ψ·/nfNMp·,q·s·=ls·□lfN−ψ·/nfNl∞≤N+3ℓq·Lp·≲supl∞≤N+3φlΩl0·H2tls·fN−ψ·/nfNl∞≤N+3ℓq·Lp·≲supl∞≤N+3φlΩl0·H2tls+fN−ψ·/nfNl∞≤N+3ℓq−Lp·≲supl∞≤N+3φlΩl0·H2t∑l∞≤N+3ls+q−1/q−fN−ψ·/nfNp·.
Then, combining (39) and supl∞≤N+3φlΩl0·H2t<∞, we obtain
(44)limn⟶∞fN−ψ·/nfNMp·,q·s·=0.
Therefore,Sℝn is dense in Mp·,q·s·ℝn.


## 4. Dual Spaces ofMp·,q·s·

For a quasi-Banach spaceX, we denote the dual space of X by X∗. In this section, we show that Mp·,q·s·∗=Mp′·,q′·−s· for p,q∈Plog and s∈Cloclog.Lemma 19.
Letp,q∈P, fkk∈ℤn, and gkk∈ℤn be sequences of locally Lebesgue integrable functions satisfying fkk∈ℤnℓq·Lp·<∞ and gkk∈ℤnℓq′·Lp′·<∞. Then, we have
(45)∑k∈ℤn∫ℝnfkxgkxdx≤21+1p−−1p+fkk∈ℤnℓq·Lp·gkk∈ℤnℓq′·Lp′·.The above lemma has been proven in [39]; hence, we omit the proof here.Proposition 20.
Letp,q∈P and s∈Cloclog. We denote by M~p·,q·s· the collection of all f∈S′ℝn satisfying that there exists fkk∈ℤn⊂Lp·ℝn such that f=∑k∈ℤn□kfk and
(46)ks·fkk∈ℤnℓq·Lp·<∞.
If we define(47)fM~p·,q·s·=infks·fkk∈ℤnℓq·Lp·:f∈M~p·,q·s·,then M~p·,q·s·=Mp·,q·s·, and ·M~p·,q·s· is an equivalent norm on Mp·,q·s·.Proof.
Letf∈Mp·,q·s· and write fk=∑l∞≤1□k+lf; then, f=∑k∈ℤn□kfk and
(48)ks·fkk∈ℤnℓq·Lp·≲fMp·,q·s·,which implies Mp·,q·s·⊂M~p·,q·s·. On the other hand, for any f∈M~p·,q·s·, by Proposition 16, we have
(49)ks·□kfk∈ℤnℓq·Lp·≲∑l∞≤1ks·□k+lfk+lk∈ℤnℓq·Lp·≲∑l∞≤1ks·fk+lk∈ℤnℓq·Lp·≲ks·fkk∈ℤnℓq·Lp·,by which we can deduce the conclusion of the proposition.Let us define(50)ℓs·q·ℤn,Lp·=f=fkxk∈ℤn:fℓs·q·Lp·<∞,where
(51)fℓs·q·Lp·=ks·fkk∈ℤnℓq·Lp·.Then, we have the following proposition about dual spaces.Proposition 21.
Letp,q∈P and s∈Cloclog. Then,
(52)ℓs·q·ℤn,Lp·∗=ℓ−s·q′·ℤn,Lp′·.
Moreover,g∈ℓs·q·ℤn,Lp·∗ is equivalent to
(53)g,f=∑k∈ℤn∫ℝngkxfkxdx,for all f=fkk∈ℤn∈ℓs·q·ℤn,Lp·, in which
(54)g=gkk∈ℤn∈ℓ−s·q′·ℤn,Lp′·,gℓs·q·Lp·∗=gkℓ−s·q′·Lp′·.Proof.
Firstly, by Lemma19, we have ℓs·q·ℤn,Lp·∗⊃ℓ−s·q′·ℤn,Lp′· and
(55)g,f=∑k∈ℤn∫ℝngkxfkxdx,for all f=fkk∈ℤn∈ℓs·q·ℤn,Lp·, g=gkk∈ℤn∈ℓ−s·q′·ℤn,Lp′·.
On the other hand, for anyg∈ℓs·q·ℤn,Lp·∗, let us define gk by
(56)gk,fk=g,0,⋯,0,fk,0,0,⋯,where f=fkk∈ℤn∈ℓs·q·ℤn,Lp·. It follows that gk∈Lp·∗=Lp′·, whence
(57)gk,fk=∫ℝngkxfkxdx.
Therefore,(58)g,f=∑k∈ℤn∫ℝngkxfkxdx.
We assume thatgk≠0 for all k∈ℤn. Then, for any N∈ℕ, let us define fN=fkk∈ℤn as follows: when k>N, we put fk=0; when k≤N, we set gk′x=gkx/λ, where λ=k−s·gkk≤Nℓq′·Lp′·, and set
(59)fk=sgngk′xk−sxq′xgk′xq′xp′x−1/q′x×k−sxk−s·q′·gk′·q′·p′·/q′·1−p′x/q′x.
Then, by Proposition 2.21 of [22], we have
(60)∫ℝnksxqxfkxqxk−s·q′·gk′·q′·p′·/q′·px/qxdx=∫ℝnk−sxq′xgk′xq′xp′x/q′xk−s·q′·gk′·q′·p′·/q′·px/qx−p′x/q′xk−s·q′·gk′·q′·p′·/q′·px/qxdx=∫ℝnk−sxq′xgk′xq′xk−s·q′·gk′·q′·p′·/q′·p′x/q′xdx=1,which implies
(61)ks·q·fk·q·p·/q·≤k−s·q′·gk′·q′·p′·/q′·.
Thus, by the definition ofg′k, we have
(62)ρℓq·Lp·ks·fk·k≤N=∑k≤Nks·q·fk·q·p·/q·≤ρℓq′·Lp′·k−s·gk′·k≤N≤1,from which we can get
(63)fNℓs·q·Lp·=ks·fkk≤Nℓq·Lp·≤1.
By Proposition 3.5 of [28], we know that ρℓq′·Lp′·· is continuous. Therefore, by k−s·gk′k≤Nℓq′·Lp′·=1, we have
(64)∑k≤Nk−s·q′·g′k·q′·p′·/q′·=ρℓq′·Lp′·k−s·g′kk≤N=1.
Then, it is easy to see that(65)g,fN=∑k∈ℤn∫ℝngkxfkxdx=λ∑k≤N∫ℝngk′xfkxdx=λ∑k≤Nk−s·q′·g′k·q′·p′·/q′··∫ℝnk−sxq′xgk′xq′xk−s·q′·g′k·q′·p′·/q′·p′x/q′xdx=λ.
Hence, for anyN∈ℕ, by (61), we have
(66)k−s·gkk≤Nℓq′·Lp′·=λ=g,fN≤gℓs·q·Lp·∗fNℓs·q·Lp·≤gℓs·q·Lp·∗,which implies g∈ℓ−s·q′·ℤn,Lp′· and ℓs·q·ℤn,Lp·∗⊂ℓ−s·q′·ℤn,Lp′·.Theorem 22 (dual space).
Letp,q∈Plog and s∈Cloclog; then, we have
(67)Mp·,q·s·∗=Mp′·,q′·−s·.Proof.
Firstly, we prove thatMp′·,q′·−s·⊂Mp·,q·s·∗. For any g∈Mp′·,q′·−s·⊂S′ℝn and φ∈Sℝn, by Lemma 19, we have
(68)g,φ=∑k∈ℤn∑l∞≤1□k+l∗g,□kφ∑k∈ℤn∫ℝn□k∗gx□kφxdx∑k∈ℤn∫ℝnk−sx□k∗gx·ksx□kφxdxk−sx□k∗gxℓq′·Lp′·ksx□kφxℓq·Lp·gMp′·,q′·−s·fMp·,q·s·,where □k∗≔FσkF−1. Since Sℝn is dense in Mp·,q·s·ℝn, we obtain Mp′·,q′·−s·⊂Mp·,q·s·∗.
Now, let us prove thatMp·,q·s·∗⊂Mp′·,q′·−s·. It is easy to see that, for f∈Mp·,q·s·,
(69)f↦□kfk∈ℤn∈ℓs·q·ℤn,Lp·is an isometric mapping from Mp·,q·s· into a subspace X of ℓs·q·ℤn,Lp·. Hence, for any g∈Mp·,q·s·∗, we can regard it as a continuous functional on X, which can be extended onto ℓs·q·ℤn,Lp· with the same norm. Then, by Proposition 21, for any f=fkk∈ℤn∈ℓs·q·ℤn,Lp·, we have
(70)g,f=∑k∈ℤn∫ℝngkxfkxdx,where gkk∈ℤn∈ℓ−s·q′·ℤn,Lp′· and
(71)gMp·,q·s·∗=gkk∈ℤnℓ−s·q′·Lp′·.
Since□k∗φk∈ℤn∈ℓs·q·ℤn,Lp· for any φ∈Sℝn, we have
(72)g,φ=∑k∈ℤn∫ℝngkx□k∗φxdx=∫ℝn∑k∈ℤn□kgkxφxdx,which implies g=∑k∈ℤn□kgkx. Thus, by Proposition 20, we obtain
(73)gMp′·,q′·−s·≲gkk∈ℤnℓ−s·q′·Lp′·=gMp·,q·s·∗,by which we can get Mp·,q·s·∗⊂Mp′·,q′·−s·.



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