# Remarks on the Unimodular Fourier Multipliers onα-Modulation Spaces

Authors: Guoping Zhao; Jiecheng Chen; Weichao Guo
Journal: Journal of Function Spaces
 (2014)

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## Abstract

We study the boundedness properties of the Fourier multiplier operatoreiμ(D) on α-modulation spaces Mp,qs,α(0≤α<1) and Besov spaces Bp,qs(Mp,qs,1). We improve the conditions for the boundedness of Fourier multipliers with compact supports and for the boundedness of eiμ(D) on Mp,qs,α. If μ is a radial function ϕ(|ξ|)  and ϕ satisfies some size condition, we obtain the sufficient and necessary conditions for the boundedness of eiϕ(|D|) between Mp1,q1s1,α and Mp2,q2s2,α.


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## Body

## 1. Introduction

LetF and F-1 denote the Fourier transform and the inverse Fourier transform, respectively. For a bounded function m, the Fourier multiplier operator associated with m is defined by
(1)Tm(f)(x)=m(D)f=F-1(mFf)
on all Schwartz functions f∈S(Rn), where m is called the symbol or multiplier of Tm. Fourier multipliers arise naturally from the formal solution of linear partial differential equations and from the summabilities of Fourier series. The boundedness properties of a Fourier multiplier in various function or distribution spaces contribute an important research topic in harmonic analysis, as well as many significant applications in partial differential equations.LetX and Y be two function/distribution spaces with norms (or quasinorm) ∥·∥X and ∥·∥Y, respectively. A bounded function m is called a Fourier multiplier from X to Y, if there exists a constant C>0 such that
(2)∥Tm(f)∥Y≤C∥f∥X,
for all f∈S(Rn). We use the above definition to avoid the situation where S(Rn) is not dense in Mp,qs,α when p=∞ or q=∞.In this paper, we will study the unimodular Fourier multipliers on theα-modulation space Mp,qs,α(α∈[0,1]) (see Section 2 for the definition of Mp,qs,α). Particularly, we will focus on the unimodular Fourier multipliers with symbol eiμ(ξ) for real-valued functions μ. These multipliers arise when one solves the Cauchy problem for some dispersive equations. For example, for the Cauchy problem of (linear) Klein-Gordon equations
(3)utt+(I-Δ)u=0,u(0,x)=u0,ut(0,x)=u1,(t,x)∈R×Rn, the formal solution is given by
(4)u(t,x)=K′(t)u0+K(t)u1,
where
(5)K(t)=G(t)-G(-t)2i(1+|D|2)1/2
and the Klein-Gordon semigroups are defined by
(6)G(t)=eit(1+|D|2)1/2.The modulation spaces were introduced by Feichtinger [1] in 1983 by the short-time Fourier transform. Now, people have recognized that the modulation spaces are very important function spaces, since they play more and more significant roles not only in harmonic analysis, but also in the study of partial differential equations. On the other hand, Besov space Bp,qs is also a popular working frame in harmonic analysis and partial differential equations. In 1992, Gröbner introduced the α-modulation space Mp,qs,α [2], that is an intermediate space between these two types of spaces with respect to the parameters α∈[0,1]. Modulation spaces are special α-modulation spaces in the case α=0, and the (inhomogeneous) Besov space Bp,qs can be regarded as the limit case of Mp,qs,α as α→1 (see [2]). So, for the sake of convenience, we can view the Besov spaces as special α-modulation spaces and use Mp,qs,1 to denote the inhomogeneous Besov space Bp,qs.It is known thatei|D|β is not bounded on any Lebesgue space Lp and Besov spaces Bp,qs, except for p=2 or β=1 and n=1, (see [3, 4]). However, ei|D|β is bounded on the modulation space Mp,qs=Mp,qs,0 for all 1≤p,q≤∞, s∈R (see Bényi et al. [5]). Hence, the modulation spaces play an alternative role in the study of unimodular Fourier multipliers. In [5], the authors proved that if 0≤β≤2, ei|D|β is bounded on Mp,qs for all 1≤p,q≤∞, s∈R. Furthermore, in the case β>2, Miyachi et al. [6] showed that, for 1≤p,q≤∞ and s1,s2∈R, ei|D|β is bounded from Mp,qs1 to Mp,qs2 if and only if s1-s2≥(β-2)n|1/p-1/2|. The reader also can see [7–11] for more results in this topic.Since thea-modulation space Mp,qs,α is an extension of the classical modulation space and it is a natural bridge connecting the modulation spaces and the Besov spaces (see [12, 13]), in a recent paper [14], we study the boundedness of eiμ(D) on function spaces Mp,qs,α and establish a sufficient and necessary boundedness theorem by assuming that μ is a homogenous function. Thus, it will be interesting to study eiμ(D) when μ is not a homogenous function. This motivates us to seek some sharp condition to ensure the boundedness on Mp,qs,α for the unimodular multiplier eiμ(D) when μ is not homogenous. In this note, we will focus on the case that μ is a radial but not homogeneous function. We remark that, for a radial function μ, the operator eiμ(D) not only is a generation for the Schrödinger semigroup ei|D|β, but also works for the Klein-Gordon semigroup with symbol eiμ, where μ(ξ)=(1+|ξ|2)1/2 is not homogeneous.We now present our main results.Theorem 1.
Letδ>0, L∈N, L≥[n/2]+1, β>0, and
(7)Sp=Sp(β)=(1p-12)max⁡{(β-2)n+2αn,0}.
Assume that μ is a real-valued function of class CN(N≥L,[n/2]+3) on Rn∖{0} which satisfies
(8)|∂γμ(ξ)|≤Cγ|ξ|δ-|γ|,0<|ξ|≤1,|γ|=L,|∂γμ(ξ)|≤Cγ|ξ|β-|γ|,|ξ|>1,2≤|γ|≤[n2]+3.
Suppose also that 1≤p,q≤∞, si∈R, α∈[0,1], for i=1,2, and satisfies s1-s2≥|Sp|. Then one has
(9)∥eiμ(D)f∥Mp,qs2,α≤C∥f∥Mp,qs1,α,
where the constant C is independent of f.Corollary 2.
Letδ>0, L∈N, L≥[n/2]+1, θ>0, λ>0, and
(10)Sp(β)=(1p-12)max⁡{(β-2)n+2αn,0}.
Assume that ϕ is a real-valued CN(R∖{0})(N≥L,[n/2]+3) function satisfying
(11)|∂kϕ(r)|≤Ck|r|δ-k,0<|r|≤1,k=L,|∂kϕ(r)|≤Ck|r|θ-|k|,|r|>1,1≤|k|≤[n2]+3.
Let h:Rn∖{0}→R∖{0} be a smooth positive homogeneous function with degree λ. Suppose 1≤p,q≤∞, si∈R, α∈[0,1] for i=1,2, and satisfies s1-s2≥|Sp(λθ)|. Then one has
(12)∥eiϕ(h(D))f∥Mp,qs2,α≤C∥f∥Mp,qs1,α,
where the constant C is independent of f.Theorem 3.
Letδ>0, L∈N, L≥[n/2]+1, β>0, β≠1, λ>0, and
(13)Sp=Sp(β)=(1p-12)max⁡{(β-2)n+2αn,0}.
Assume that ϕ is a real-valued CN(R+)(N≥L,[n/2]+3) function which satisfies
(14)|∂kϕ(r)|≤Ck|r|δ-k,0<|r|≤1,k=L,(15)|∂kϕ(r)|≤Ck|r|β-|k|,|r|>1,1≤|k|≤[n2]+3,(16)|∂kϕ(r)|~|r|β-k,|r|>1,k=1,2.
Let 1≤pi,qi≤∞, si∈R, α∈[0,1] for i=1,2. Then
(17)∥eiϕ(|D|)f∥Mp2,q2s2,α≲∥f∥Mp1,q1s1,α
holds for all f if and only if
(18)1p2≤1p1,s2-nαp2+max⁡{Sp2,Sp1′}=s1-nαp1,1q2≤1q1
or
(19)1p2≤1p1,s2-nαp2+max⁡{Sp2,Sp1′}<s1-nαp1,s2-nαp2+max⁡⁡{Sp2,Sp1′}+n(1-α)q2<s1-nαp1+n(1-α)q1.We list two examples to illustrate the assumptions in our theorems. First, the functionμ(ξ)=(1+∑j=1n‍λj|ξ|jλ)θ(λj≥0,λ>0,θ>0) satisfies the assumptions in Theorem 1 and Corollary 2 for β=λθ>0, while μ(ξ) is not radial and not homogeneous. Another function is μ(ξ)=μ(|ξ|) with μ(r)=(1+rλ)θ(λ,θ>0). This function satisfies the assumptions in Theorem 3 for β=λθ, λθ≠1. One may also observe that if β=1, there exists no C2(R+) function ϕ(r), which satisfies the size condition (16). If the reader checks the main theorems in [5, 6], it is not difficult to see that our theorems are a substantial improvement and extension of the known results, even in the case α=0.The paper is organized as follows. In Section2, we recall some definitions and basic properties. In Section 3, we obtain an improvement of results in [5, 6] by studying more general Fourier multipliers eiμ(D), in which we do not need to assume lower order derivatives of μ(ξ) near 0. This new results will be used to achieve a more general result for the boundedness of eiμ(D) on spaces Mp,qs,α. In Section 4, by assuming radial condition on μ(ξ), we deduce a dual estimate of eiϕ(|D|), and then we use the method in [14] to give a sharp result for the boundedness of eiϕ(|D|) between Mp1,q1s1,α and Mp2,q2s2,α.


## 2. Preliminaries

We start this section by recalling some notations. LetC be a positive constant that may depend on the indices n,pi,qi,si,α,β. The notation X≲Y denotes the statement that X≤CY, the notation X~Y means the statement X≲Y≲X, and the notation X⋍Y denotes the statement X=CY. For a multi-index k=(k1,k2,…,kn)∈Zn, we denote |k|∞≔sup⁡i=1,2,…,n|ki| and 〈k〉≔(1+|k|2)1/2.LetS≔S(Rn) be the Schwartz space and S′≔S′(Rn) the space of all tempered distributions. We define the Fourier transform Ff and the inverse Fourier F-1f of f∈S(Rn) by
(20)Ff(ξ)=f^(ξ)=∫Rn‍f(x)e-2πix·ξdx,F-1f(x)=f^(-x)=∫Rn‍f(ξ)e2πix·ξdξ.To describe the function spaces discussed in this note, we first give the partition of unity on frequency space forα∈[0,1). We suppose c>0 and C>0 are two appropriate constants and choose a Schwartz function sequence {ηkα}k∈Zn satisfying
(21)|ηkα(ξ)|≥0,if|ξ-k〈k〉α/(1-α)|<c〈k〉α/(1-α);supp⁡ηkα⊂{ξ:|ξ-k〈k〉α/(1-α)|<C〈k〉α/(1-α)};∑k∈Zn‍ηkα(ξ)≡1,∀ξ∈Rn;|∂γηkα(ξ)|≤C|α|〈k〉-α|γ|/(1-α),n∀ξ∈Rn,γ∈(Z+∪{0})n.
Then {ηkα(ξ)}k∈Zn constitutes a smooth decomposition of unity. The frequency decomposition operators associated with above function sequence can be defined by
(22)□kα≔F-1ηkαF
for k∈Zn. Let 1≤p,q≤∞, s∈R, and α∈[0,1); the α-modulation space associated with above decomposition is defined by
(23)Mp,qs,α(Rn)={(∑k∈Zn‍〈k〉sq/(1-α)∥Wkαf∥pq)1/qf∈S′(Rn):∥f∥Mp,qs,α(Rn)nnn=(∑k∈Zn‍〈k〉sq/(1-α)∥□kαf∥pq)1/q<∞}
with the usual modifications when q=∞. For the sake of simplicity, in this note, we always denote Mp,qs=Mp,qs,0 and ηk(ξ)=ηk0(ξ).We introduce the dyadic decomposition ofRn in order to define the Besov space. Let φ(ξ) be a smooth bump function supported in the ball {ξ:|ξ|<3/2} and be identically equal to 1 on the ball {ξ:|ξ|≤4/3}. We denote
(24)ψ(ξ)=φ(ξ)-φ(2ξ),
and a function sequence
(25)ψj(ξ)=ψ(2-jξ),j∈Nψ0(ξ)=1-∑j∈N‍ψj(ξ)=φ(ξ).
For all integers j∈N, we define the Littlewood-Paley operators
(26)Δjf^=ψj(ξ)f^(ξ),Δ0f^=φ(ξ)f(ξ).
Let 1≤p,q≤∞, and s∈R. For f∈S′ we set the the (inhomogeneous) Besov space space norm by
(27)∥f∥Bp,qs=(∑j=0∞‍2jsq∥Δjf∥Lpq)1/q.
The (inhomogeneous) Besov space is the space of all tempered distributions f for which the quantity ∥f∥Bp,qs is finite. We recall the following embedding results.Lemma 4 (embedding [12, 13]).
Let1≤pi,qi≤∞, si∈R, for i=1,2, α∈[0,1]. If
(28)1p2≤1p1,s2-nαp2=s1-nαp1,1q2≤1q1
or
(29)1p2≤1p1,s2-nαp2<s1-nαp1,s2-nαp2+n(1-α)q2<s1-nαp1+n(1-α)q1,
one has
(30)Mp1,q1s1,α⊆Mp2,q2s2,α.We also recall some results obtained in [6, 14], respectively.Lemma 5 (see Lemma 3.2 in [6]).
Letϵ>0 and let μ be a C[n/2]+1 function on Rn∖{0} satisfying
(31)|∂γμ(ξ)|≤Cγ|ξ|ϵ-γ
for |γ|≤[n/2]+1. Then F-1[ηeiμ(ξ)]∈L1(Rn) for each η∈Cc∞(Rn).Lemma 6 (see Theorem  1.1 in [14]).
Letβ>0, ϵ>0, and
(32)Sp=Sp(β)=(1p-12)max⁡{(β-2)n+2αn,0}.
Assume that μ is a real-valued function of class C[n/2]+3 on Rn∖{0} which satisfies
(33)|∂γμ(ξ)|≤Cγ|ξ|ϵ-|γ|,0<|ξ|≤1,|γ|≤[n2]+1,|∂γμ(ξ)|≤Cγ|ξ|β-|γ|,|ξ|>1,2≤|γ|≤[n2]+3.
Suppose also that 1≤p,q≤∞, si∈R, α∈[0,1], for i=1,2, and they satisfy s1-s2≥|Sp|. Then we have
(34)∥eiμ(D)f∥Mp,qs2,α≤C∥f∥Mp,qs1,α,
where the constant C is independent of f.


## 3. Sufficient Condition of the Boundedness ofeiμ(D)

The goal of this section is to prove Theorem1 and Corollary 2. We will start with the following derivative lemma for showing that the lower order derivative near 0 does not interrupt the boundedness of eiμ(D) on α-modulation spaces.Lemma 7 (derivative lemma).
LetL∈Z, L≥0, δ>0. Suppose that m∈CL(Rn∖{0}) satisfying
(35)|∂γm(ξ)|≤Cγ|ξ|δ-|γ|,0<|ξ|≤1,|γ|=L.
Then the limit
(36)a=lim⁡ξ→0m(ξ)=m(0)
exists, and for any ϵ∈(0,1)∩(0,δ], we have
(37)|∂γ(m-a)(ξ)|≤Cγ|ξ|ϵ-|γ|,0<|ξ|≤1,|γ|≤L.Proof.
We will state the proof for the casesL=0,1,2; the other cases can be deduced by a similar argument and an easy induction.
ForL=0, one can observe directly that a=0=lim⁡ξ→0m(ξ).
ForL=1, fix ϵ∈(0,1)∩(0,δ). For any ξ1,ξ2∈B(0,1), |ξ1|≤|ξ2|, we can find a simple piecewise smooth curve
(38)Γ:[0,l]⟶Rn,
which is jointed by two curves Γ1and Γ2 with length l=|Γ|, where Γ1 is the straight line connecting points ξ1 and (|ξ2|/|ξ1|)ξ1 and Γ2 is shortest curve on the great circle connecting points (|ξ2|/|ξ1|)·ξ1 and ξ2, such that
(39)Γ(0)=ξ1,Γ(|ξ2|-|ξ1|)=|ξ2||ξ1|ξ1,Γ(l)=ξ2,|Γ′(t)|=1
for all t∈[0,l]. We have
(40)|m(ξ2)-m(ξ1)|=|m(Γ(l))-m(Γ(0))|≤|m(Γ(l))-m(Γ(|ξ2|-|ξ1|))|+|m(Γ(|ξ2|-|ξ1|))-m(Γ(0))|,|m(Γ(|ξ2|-|ξ1|))-m(Γ(0))|=|∫0|ξ2|-|ξ1|‍(∇m)(Γ(t))·Γ′(t)dt|≲∫0|ξ2|-|ξ1|‍|(∇m)(Γ(t))|dt≲∫0|ξ2|-|ξ1|‍(|ξ1|+t)ϵ-1dt=|ξ2|ϵ-|ξ1|ϵϵ,|m(Γ(l))-m(Γ(|ξ2|-|ξ1|))|=|∫|ξ2|-|ξ1|l‍(∇m)(Γ(t))·Γ′(t)dt|≲∫|ξ2|-|ξ1|l‍|(∇m)(Γ(t))|dt≲∫|ξ2|-|ξ1|l‍|ξ2|ϵ-1dt≲|ξ2|ϵ.
Hence,
(41)|m(ξ2)-m(ξ1)|≲|ξ2|ϵ-|ξ1|ϵϵ+|ξ2|ϵ⟶0
as ξ1,ξ2→0. So the limit
(42)a=lim⁡ξ→0m(ξ)=m(0)
exists and
(43)|m(ξ)-a|=lim⁡η→0|m(ξ)-m(η)|≤|ξ|ϵϵ+|ξ|ϵ≲|ξ|ϵ.
For L=2, fix ϵ∈(0,1)∩(0,δ). For any ξ∈B(0,1), let Γ be the straight line connecting ξ and ξ/|ξ|, such that
(44)Γ(0)=ξ,Γ(1-|ξ|)=ξ|ξ|,|Γ′(t)|=1.
For any j=1,2,…,n, we have
(45)|∂jm(ξ)-∂jm(ξ|ξ|)|=|(∂jm)(Γ(1-|ξ|))-(∂jm)(Γ(0))|=|∫01-|ξ|‍(∇∂jm)(Γ(t))·Γ′(t)dt|≲∫01-|ξ|‍|(∇∂jm)(Γ(t))|dt≲∫01-|ξ|‍(|ξ|+t)ϵ-2dt=1-|ξ|ϵ-1ϵ-1.
It follows that
(46)|∂jm(ξ)|≤|∂jm(ξ)-∂jm(ξ|ξ|)|+|∂jm(ξ|ξ|)|≲1-|ξ|ϵ-1ϵ-1+1≲|ξ|ϵ-1.
Now we finish the proof by repeating the L=1 case.We are in a position to give the proof of Theorem1.Proof of Theorem1.
In virtue of the above lemma, since(47)|∂γμ(ξ)|≤Cγ|ξ|δ-|γ|,0<|ξ|≤1,|γ|=L,
and L≥[n/2]+1, for fixed ϵ∈(0,1)∩(0,δ], we have
(48)|∂γ(μ-μ(0))(ξ)|≤Cγ|ξ|ϵ-|γ|,0<|ξ|≤1
for any |γ|≤[n/2]+1.
Using Lemma5, we know that
(49)F-1[ηeiμ(ξ)]=F-1[ηei(μ-μ(0))(ξ)]∈L1.
Finally, we use Lemma6 to complete the proof.Proof of Corollary2.
By the assumptions of Corollary2, we can use Lemma 7 to deduce that
(50)|∂k(ϕ-ϕ(0))(r)|≤Ck|r|ϵ-k,0<|r|≤1
for fixed ϵ∈(0,1)∩(0,δ] and any k≤[n/2]+1. The assumption of h(ξ) implies that
(51)|h(ξ)|~|ξ|λ.
It follows then
(52)|∂γ(ϕ∘h)(ξ)|≲|ξ|λϵ-|γ|,0<|ξ|≤1,
for any |γ|=[n/2]+1.
On the other hand, one can deduce that(53)|∂γ(ϕ∘h)(ξ)|≲|ξ|λθ-|γ|,|ξ|≥1,
for 2≤|γ|≤[n/2]+3.
Finally, the conclusion is deduced by Theorem1.


## 4. Sharpness of the Conditions for the Boundenness ofeiϕ(|ξ|)

In this section, we give the proof of Theorem3. The key point is that we can obtain a dual estimate for eiϕ(|ξ|) under some size condition on ϕ. By combining the dual estimate with Theorem 1, we get the simultaneous asymptotic estimates of ∥f∥Mp,qs,α and ∥eiμ(D)∥Mp,qs,α. Then the proof can be finished by the method in [14]. We first start with the dual estimate on Besov spaces.Lemma 8 (dual estimate forBp,qs).
Supposeβ>0, β≠1. Assume that ϕ is a real-valued C[n/2]+3 function which satisfies the assumptions of Theorem 3. Then one has
(54)∥F-1[ψj(ξ)eiϕ(|ξ|)]∥L∞≤C2j(β-2)(-n/2)
for all j∈N.Proof.
Using the change of variables, we have(55)∥F-1[ψj(ξ)eiϕ(|ξ|)]∥L∞=2jn∥F-1[ψ(ξ)eiϕ(2j|ξ|)]∥L∞.
Use the polar coordinates,
(56)F-1[ψ(ξ)eiϕ(2j|ξ|)](x)=∫Rn‍ψ(ξ)eiϕ(2j|ξ|)e2πix·ξdξ=∫0∞‍ψ(r)eiϕ(2jr)rn-1∫Sn-1‍e2πirx·ξ′dσ(ξ′)dr=∫0∞‍ψ(r)eiϕ(2jr)rn-1dσ^(-rx)dr.
Recall that the Fourier transform of the area measure satisfies
(57)dσ^(-rx)=Ce2πir|x||rx|(n-1)/2+Ce-2πir|x||rx|(n-1)/2+O(|rx|-n/2).
The support of ψ yields 1/|rx|(n-1)/2≲1/|x|(n-1)/2 and O(|rx|-n/2)≲|x|-n/2.
For the case|x|≳2jβ, we only need to show that
(58)|∫0∞‍ψ(r)ei[ϕ(2jr)±2πir|x|]rn-1dr|≲2-jβ(n/2),
which is a direct conclusion by the fact that
(59)d2(ϕ(2jr)±2πir|x|)dr2≳2jβ
and the Van der Corput lemma.
For the case that|x|≲2jβ, we define
(60)L≔∑j=1n‍ξj∂ji[2j|ξ|ϕ′(2j|ξ|)+∑j=1n‍2πxjξj]
and notice that
(61)L(ei[ϕ(2jr)±2πir|x|])=ei[ϕ(2jr)±2πir|x|].
Then the inequality
(62)|∫Rn‍ψ(ξ)eiϕ(2j|ξ|)e2πix·ξdξ|≲(2jβ)-n/2
follows by an integration by parts.
Hence, for allj∈N,
(63)∥F-1[ψj(ξ)eiϕ(|ξ|)]∥L∞≲2j(B-2)(-n/2).Lemma 9 (dual estimate forMp,qs,α, α∈[0,1)).
Supposeβ>0, β≠1. Assume that ϕ is a real-valued C[n/2]+3 function satisfying the assumptions of Theorem 3. Then there exists a sufficiently large constant R such that
(64)∥F-1[ηkα(ξ)eiϕ(|ξ|)]∥L∞≤C〈k〉((β-2)/(1-α))(-n/2),
for all k∈Zn with 〈k〉α/(1-α)k∈Rn∖B(0,R), where the constant C is independent of k.Proof.
For sufficiently largek, we use estimate on some Δj to estimate □kα. Choose a j satisfying 〈k〉1/(1-α)~2j. An easy computation shows that
(65)∥F-1[ηkα(ξ)eiϕ(|ξ|)]∥L∞≲∥F-1[ψj(ξ)ηkα(ξ)eiϕ(|ξ|)]∥L∞≲∥F-1[ψj(ξ)eiϕ(|ξ|)]∥L∞≲2j(β-2)(-n/2)~〈k〉((β-2)/(1-α))(-n/2).Now, we give the asymptotic estimates of∥eiϕ(|D|)f∥Mp,qs,α and ∥f∥Mp,qs,α. These results can be verified by the same methods in [14].Lemma 10 (asymptotic estimates of∥eiϕ(|D|)f∥Mp,qs,α(α∈[0,1))).
Supposeβ>0, β≠1. Assume that ϕ is a real-valued function which satisfies the assumptions of Theorem 3. we can find function sequences {fλ}, {hk}, and {gN} such that
(66)∥eiϕ(|D|)fλ∥Mp,qs,α~λn(1-(1/p))asλ⟶0,∥eiϕ(|D|)hk∥Mp,qs,α~〈k〉(s+Sp+αn(1-1/p))/(1-α)as|k|⟶∞,∥eiϕ(|D|)gN∥Mp,qs,α~N((s+Sp+αn(1-1/p))/(1-α))+(n/q)asN⟶∞,s-nαp+Sp+n(1-α)q>-nα,∥eiϕ(|D|)gN∥Mp,qs,α~(ln⁡N)1/qasN⟶∞,s-nαp+Sp+n(1-α)q=-nα.
Moreover, we can easily verify that
(67)∥fλ∥Mp,qs,α~λn(1-(1/p))asλ⟶0,∥hk∥Mp,qs,α~〈k〉(s+αn(1-1/p))/(1-α)as|k|⟶∞,∥gN∥Mp,qs,α~N((s+αn(1-1/p)/(1-α))+(n/q)asN⟶∞,s-nαp+n(1-α)q>-nα,∥gN∥Mp,qs,α~(ln⁡N)1/qasN⟶∞,s-nαp+n(1-α)q=-nα.Proof.
In the case that0<β≤Λα is trivial, we suppose that β≥Λα in this proof. We only show the proof of (66).
Denote byh^(ξ) a Cc∞(Rn) function with sufficiently small support near zero. Let
(68)f^λ=h^(ξλ),λ≥1.
Using Theorem 1, we deduce that
(69)∥fλ∥Mp,qs-|Sp|,α≲∥eiϕ(|D|)fλ∥Mp,qs,α≲∥fλ∥Mp,qs+|Sp|,α.
Notice that
(70)∥fλ∥Mp,qs-|Sp|,α=∥fλ∥Mp,qs+|Sp|,α=∥□0αfλ∥Lp=λn∥h(λx)∥Lp=λn(1-(1/p))∥h∥Lp;
so we have
(71)∥eiϕ(|D|)fλ∥Mp,qs,α~λn(1-(1/p))asλ⟶0.
For the second equation, we denote
(72)h^k(ξ)=h^(ξ-〈k〉α/(1-α)k〈k〉α/(1-α))
and let f^k(ξ)=h^k(ξ), where k∈Zn with 〈k〉α/(1-α)k∈Γ∖B(0,R).
If1≤p≤2, we use Theorem 1 and Lemma 9 to deduce that
(73)∥fk∥Mp′,qs+(1/p-1/2)(β-2)n,α≲∥eiϕ(|D|)fk∥Mp,qs,α≲∥fk∥Mp,qs+Sp,α.
If 2≤p≤∞, we deduce that
(74)∥fk∥Mp,qs+Sp,α≲∥eiϕ(|D|)fk∥Mp,qs,α≲∥fk∥Mp′,qs+(1/p-1/2)(β-2)n,α.
By a direct calculation, we obtain
(75)∥fk∥Mp,qs+Sp,α=〈k〉(s+Sp)/(1-α)∥hk∥Lp000000000=〈k〉(s+Sp+αn(1-1/p))/(1-α)∥h∥Lp,∥fk∥Mp′,qs+(1/p-1/2)(β-2)n,α000000000=〈k〉(s+(1/p-1/2)(β-2)n)/(1-α)∥hk∥Lp′000000000=〈k〉(s+Sp+αn(1-1/p))/(1-α)∥h∥Lp′.
Then the asymptotic formula follows.
For the third and fourth asymptotic formulas, let(76)f^N(ξ)=∑k∈AN‍h^k(ξ),
where AN={k∈Zn:|k|≤N,〈k〉α/(1-α)k∈Rn∖B(0,R)}. By a direct calculation, we obtain
(77)∥fN∥Mp,qs+Sp,α≃(∑k∈AN‍〈k〉((s+Sp+αn(1-1/p))/(1-α))q)1/q≃∥fN∥Mp′,qs+(1/p-1/2)(β-2)n,α.
As above, Theorem 1 and Lemma 9 yield
(78)∥eiϕ(|D|)fN∥Mp,qs,α~(∑k∈AN‍〈k〉((s+Sp+αn(1-1/p))/(1-α))q)1/q.
Finally, the claim follows by
(79)(∑k∈AN‍〈k〉((s+Sp+αn(1-1/p))/(1-α))q)1/q~N((s+Sp+αn(1-1/p))/(1-α))+(n/q),
for s-(nα/p)+Sp+(n(1-α)/q)>-nα, and
(80)(∑k∈AN‍〈k〉((s+Sp+αn(1-1/p))/(1-α))q)1/q~(ln⁡N)1/q
for s-(nα/p)+Sp+(n(1-α)/q)=-nα.For the caseα=1, we have the following lemma; since its proof is similar to that of Lemma 10, we leave the detail to the reader.Lemma 11 (asymptotic estimates of∥eiϕ(|D|)f∥Bp,qs).
Supposeβ>0, β≠1. Assume that ϕ is a real-valued function which satisfies the assumptions of Theorem 3. We can find function sequences {fλ}, {hk}, and {gN} such that
(81)∥eiϕ(|D|)fλ∥Bp,qs~λn(1-(1/p))asλ⟶0,∥eiϕ(|D|)hj∥Bp,qs~2j[s+Sp+n(1-1/p)]as|j|⟶∞,∥eiϕ(|D|)gN∥Bp,qs~2N[s+Sp+n(1-1/p)]asN⟶∞,s-np+Sp>-n,∥eiϕ(|D|)gN∥Bp,qs~N1/qasN⟶∞,s-np+Sp=-n.
Moreover, we can easily verify that
(82)∥fλ∥Bp,qs~λn(1-(1/p))asλ⟶0,∥hj∥Bp,qs~2j[s+n(1-1/p)]as|j|⟶∞,∥gN∥Bp,qs~2N[s+n(1-1/p)]asN⟶∞,s-np>-n,∥gN∥Bp,qs~N1/qasN⟶∞,s-np=-n.Now, we give the sketch of proof for Theorem3.Proof of Theorem3.
As in [14], if
(83)∥eiϕ(|D|)f∥Mp2,q2s2,α≲∥f∥Mp1,q1s1,α
holds for all f, we can use Lemmas 10 and 11 to obtain (18) or (19).
On the other hand, if (18) or (19) holds, we can use the embedding lemma (Lemma 4), the dual estimates (Lemma 8 and Lemma 9), and Theorem 1 to obtain the boundedness of eiϕ(|D|) from Mp1,q1s1,α1 to Mp2,q2s2,α2.



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*Source: 106267-2014-07-17.xml*