# Dilation Properties for Weighted Modulation Spaces

Authors: Elena Cordero; Kasso A. Okoudjou
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

We give a sharp estimate on the norm of the scaling
operatorUλf(x)=f(λx) acting on the weighted modulation spaces Ms,tp,q(ℝd). In particular, we recover and extend recent results by Sugimoto and Tomita in the
unweighted case. As an application of our results, we estimate the growth in
time of solutions of the wave and vibrating plate equations, which is of interest
when considering the well-posedness of the Cauchy problem for these equations.
Finally, we provide new embedding results between modulation and Besov spaces.


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## Body

## 1. Introduction

The modulation spaces were introduced by Feichtinger [1], by imposing integrability conditions on the short-time Fourier transform (STFT) of tempered distributions. More specifically, for x,ω∈ℝd, we let Mω and Tx denote the operators of modulation and translation. Then, the STFT of f with respect to a nonzero window g in the Schwartz class is(1.1)Vgf(x,ω)=〈f,MωTxg〉=∫R2df(t)g(t-x)¯e-2πit⋅ωdt.Vgf(x,ω) measures the frequency content of f in a neighborhood of x.Fors1,s2∈ℝ, and 1≤p,q≤∞, the weighted modulation space ℳs1,s2p,q(ℝ2d) is defined to be the Banach space of all tempered distributions f such that(1.2)‖f‖Ms1,s2p,q=(∫R2(∫R2|Vgf(x,ω)|pvs1(x)pdx)q/pvs2(ω)qdω)1/q<∞.
Here and in the sequel, we use the notation(1.3)vs(x)=〈x〉s=(1+|x|2)s/2.
The definition of modulation space is independent of the choice of the window g, in the sense that different window functions yield equivalent modulation-space norms. Furthermore, the dual of a modulation space is also a modulation space, if p<∞, q<∞, (ℳs,tp,q)′=ℳ-s,-tp′,q′, where p′,q′ denote the dual exponents of p and q, respectively.When boths=t=0, we will simply write ℳp,q=ℳ0,0p,q. The weighted Ls2 space is exactly ℳs,02,2, while an application of Plancherel’s identity shows that the Sobolev space ℋ2s coincides with ℳ0,s2,2. For further properties and uses of modulation spaces, see Gröchenig’s book [2], and we refer to [3] for equivalent definitions of the modulation spaces for all 0<p,q≤∞.The modulation spaces appeared in recent years in various areas of mathematics and engineering. Their relationship with other function spaces has been investigated and resulted in embedding results of modulation spaces into other function spaces such as the Besov and Sobolev spaces [4–6]. Sugimoto and Tomita [5] proved the optimality of certain of the embeddings of modulation spaces into Besov space obtained in [4, 6]. These results were obtained as consequence to optimal bounds of ∥Uλ∥ℳp,q→ℳp,q [5, Theorem  3.1], where Uλf(·)=f(λ·) for λ>0. In the sequel, we adopt the following notation:(1.4)fλ:=Uλf.The operatorUλ has been investigated on many other function spaces including the Besov spaces. For purpose of comparison with our results, we include the following results summarizing the behavior of Uλ on the Besov spaces [7, Proposition  3].Theorem 1.1.
Forλ∈(0,∞), s∈ℝ,
(1.5)C-1λ-d/pmin⁡{1,λs}‖f‖Bsp,q≤‖fλ‖Bsp,q≤Cλ-d/pmax⁡{1,λs}‖f‖Bsp,q.On the modulation spaces, the boundedness ofUλ was first investigated by Feichtinger [8, Remark  13] for the Feichtinger algebra, that is, ℳ1,1. In fact, this result is a special case of the boundedness of a general class of automorphisms on ℳ1,1. Recently, the general estimate on the norm of Uλ on the (unweighted) modulation spaces ℳp,q(ℝd) was obtained by Sugimoto and Tomita [5]. In this paper, we will derive optimal lower and upper bounds for the operator Uλ on general modulation spaces ℳt,sp,q(ℝd). More specifically, the boundedness of Uλ on ℳt,sp,q is proved in Theorems 3.1, 3.2, and 3.4, and the optimal bounds on ∥Uλ∥ℳt,sp,q→ℳt,sp,q are established by Theorems 4.12 and 4.13. We wish to point out that it is not trivial to prove sharp bounds on the norm of the operator Uλ, as one has to construct examples of functions in the modulation spaces that achieve the desired optimal estimates. We construct such examples by exploiting the properties of Gabor frames generated by the Gaussian window. It is likely that the functions that we construct can play some role in other areas of analysis where the modulation is used, for example, time-frequency analysis of pseudodifferential operators and PDEs.Interesting applications concern Strichartz estimates for dispersive equations such as the wave equation and the vibrating plate equation on Wiener amalgam and modulation spaces, where the time parameter of the Fourier multiplier symbol is considered as scaling factor. We plan to investigate such applications in a subsequent paper.Finally, we prove new embeddings between modulation spaces and Besov spaces, generalizing some of the results of [4]. Although strictly speaking this is not an application of the above dilation results, it is clearly in the spirit of the main topic of the present paper, so that we devote a short subsection to the problem.Our paper is organized as follows. In Section2, we set up the notation and prove some preliminary results needed to establish our theorems. In Section 3, we prove the complete scaling of weighted modulation spaces. In Section 4, the sharpness of our results are proved, and in Section 5 we point out the applications of our main results.Finally, we will use the notationsA≲B to mean that there exists a constant c>0 such that A≤cB, and A≍B means that A≲B≲A.


## 2. Preliminary

We will use the set and index terminology of the paper [5]. Namely, for 1≤p≤∞, let p′ be the conjugate exponent of p (1/p+1/p′=1). For (1/p,1/q)∈[0,1]×[0,1], we define the subsets(2.1)I1=max⁡(1p,1p′)≤1q,I1*=min⁡(1p,1p′)≥1q,I2=max⁡(1q,12)≤1p′,I2*=min⁡(1q,12)≥1p′,I3=max⁡(1q,12)≤1p,I3*=min⁡(1q,12)≥1p.
These sets are displayed in Figure 1.The index sets.
(a)(b)We introduce the indices:(2.2)μ1(p,q)={-1pif(1p,1q)∈I1*,1q-1if(1p,1q)∈I2*,-2p+1qif(1p,1q)∈I3*,μ2(p,q)={-1pif(1p,1q)∈I1,1q-1if(1p,1q)∈I2,-2p+1qif(1p,1q)∈I3.
Next, we prove a lemma that will be used throughout this paper, and which allows us to investigate the action of Uλ only on 𝒮(ℝd).Lemma 2.1.
Letm be a polynomial growing weight function and A a linear continuous operator from 𝒮′(ℝd) to 𝒮′(ℝd). Assume that
(2.3)‖Af‖Mmp,q≤C‖f‖Mmp,q,∀f∈S(Rd).
Then,
(2.4)‖Af‖Mmp,q≤C‖f‖Mmp,q,∀f∈Mmp,q(Rd).Proof.
The conclusion is clear ifp,q<∞, because in that case 𝒮(ℝd) is dense in ℳmp,q(ℝd).
Consider now the casep=∞ or q=∞. For any given f∈ℳmp,q, consider a sequence fn of Schwartz functions, with fn→f in 𝒮′(ℝd), and
(2.5)‖fn‖Mmp,q≲‖f‖Mmp,q
(see the proof of Proposition  11.3.4 of [2]). Since fn tends to f in 𝒮′(ℝd), Afn tends to Af in 𝒮′(ℝd), and VφAfn tends to VφAf pointwise. Hence, by Fatou’s Lemma, the assumptions (2.3) and (2.5) hold,(2.6)‖Af‖Mmp,q≤liminf⁡n→∞‖Afn‖Mmp,q≲liminf⁡n→∞‖fn‖Mmp,q≲‖f‖Mmp,q.We will also make use of the following characterization of the modulation spaces by Gabor frames generated by the Gaussian function, which will be denoted through the paper byφ(x)=e-π|x|2, x∈ℝd. Recall that for 0<a<1, the family(2.7)G(φ,a,1)={φk,l(⋅)=MlTakφ=e2πil⋅φ(⋅-ak),k,l∈Zd},
is a Gabor frame for L2(ℝd) if and only if there exist 0<A≤B<∞ such that for all f∈L2 we have(2.8)A‖f‖L22≤∑k,l∈Zd|〈f,φk,l〉|2≤B‖f‖L22.
Moreover, there exists a dual function φ̃∈𝒮 such that 𝒢(φ̃,a,1) is also a frame for L2 and every f∈L2 can be written as(2.9)f=∑k,l∈Zd〈f,φ̃k,l〉φk,l=∑k,l∈Zd〈f,φk,l〉φ̃k,l.It is easy to see from the isometry of the Fourier transform onL2 and the fact that MℓTakφ̂=TℓM-akφ̂=e2πiakℓM-akTℓφ, that 𝒢(φ,1,a) is a Gabor frame whenever 𝒢(φ,a,1)is one. The characterization of the modulation spaces by Gabor frame is summarized in the following proposition. We refer to [2, Chapter  9] for a detail treatment of Gabor frames in the context of the modulation spaces. In particular, the next result is proved in [2, Theorem  7.5.3] and describes precisely when the Gaussian function generates a Gabor frame on L2.Proposition 2.2.
𝒢(φ,a,1) is a Gabor frame for L2 if and only if  0<a<1. In this case, 𝒢(φ,a,1) is also a Banach frame for ℳt,sp,q for all  1≤p,q≤∞, and s,t∈ℝ. Moreover, f∈ℳt,sp,q if and only if there exists a sequence {ck,ℓ}k,ℓ∈ℤd∈ℓt,sp,q(ℤd×ℤd) such that f=∑k,ℓ∈ℤdck,ℓφk,ℓ with convergence in the modulation space norm. In addition,
(2.10)‖f‖Mt,sp,q≍‖c‖lt,sp,q:=(∑l∈Zd(∑k∈Zd|ck,l|pvt(k)p)q/pvs(l)q)1/q.


## 3. Dilation Properties of Weighted Modulation Spaces

We first consider the polynomial weights in the time variablesvt(x)=〈x〉t=(1+|x|2)t/2, t∈ℝ.Theorem 3.1.
Let1≤p,q≤∞, t∈ℝ. Then the following are true.(1)
There exists a constantC>0 such that  forallf∈ℳt,0p,q, λ≥1,
(3.1)C-1λdμ2(p,q)min⁡{1,λ-t}‖f‖Mt,0p,q≤‖fλ‖Mt,0p,q≤Cλdμ1(p,q)max⁡{1,λ-t}‖f‖Mt,0p,q.(2)
There exists a constantC>0 such that forallf∈ℳt,0p,q, 0<λ≤1,
(3.2)C-1λdμ1(p,q)min⁡{1,λ-t}‖f‖Mt,0p,q≤‖fλ‖Mt,0p,q≤Cλdμ2(p,q)max⁡{1,λ-t}‖f‖Mt,0p,q.Proof.
We will only prove the upper halves of each of the estimates (3.1) and (3.2). The lower halves will follow from the fact that 0<λ≤1 if and only if 1/λ≥1 and f=UλU1/λf=U1/λUλf.
We first consider the caseλ≥1. Recall the definition of the dilation operator Uλ given by Uλf(x)=f(λx). Since the mapping f↦〈·〉tf is an homeomorphism from ℳt0,sp,q to ℳt0-t,sp,q, t,t0,s∈ℝ, see, for example, [9, Corollary  2.3], we have
(3.3)‖Uλf‖Mt,0p,q≍‖〈⋅〉tUλf‖Mp,q.
Using 〈·〉tUλf=Uλ(〈λ-1·〉t)f and the dilation properties for unweighted modulation spaces in [5, Theorem  3.1], we obtain
(3.4)‖Uλ(〈λ-1⋅〉tf)‖Mp,q≤Cλdμ1(p,q)‖〈λ-1⋅〉tf‖Mp,q≍λdμ1(p,q)‖〈⋅〉-t〈λ-1⋅〉t(〈⋅〉tf)‖Mp,q.
Hence, it remains to prove that the pseudodifferential operator with symbolg(t,λ)(x):=〈x〉-t〈λ-1x〉t is bounded on ℳp,q and that its norm is bounded above by max{1,λ-t}.
By [2, Theorem  14.5.2], this will follow once we prove that ∥g(t,λ)(x)∥ℳ∞,1≲max{1,λ-t}. To see this, observe first that
(3.5)|g(t,λ)(x)|≲max⁡{1,λ-t},∀x∈Rd.
Indeed, let v(t,λ)(x)=〈λ-1x〉t. Consider the case t≥0. Since λ≥1, we have λ-1|x|≤|x| and v(t,λ)(x)≤〈x〉t.
Analogously, fort<0, we have v(t,λ)(x)≤λ-t〈x〉t. Consequently, we get the desired estimates (3.5).
Using the inclusion𝒞d+1(ℝd)↪ℳ∞,1(ℝd), we have
(3.6)‖g(t,λ)(x)‖M∞,1≲sup⁡|α|≤d+1sup⁡x∈Rd|∂αg(t,λ)(x)|.
By Leibniz’ formula, the estimate |∂β〈x〉t|≲〈x〉t-|β|, and (3.5), we see that this last expression is estimated by max{1,λ-t}.
This concludes the proof of the upper half of (3.1).
We now consider the case0<λ≤1. Notice that λ≤1 if and only if 1/λ≥1, and using the upper half of (3.1) we can write
(3.7)‖fλ‖Mt,0p,q=sup⁡|〈fλ,g〉|=λ-dsup⁡|〈f,g1/λ〉|≤λ-d‖f‖Mt,0p,qsup⁡‖g1/λ‖M-t,0p′,q′≤Cmax⁡{1,λ-t}λ-d-dμ2(p′,q′)‖f‖Mt,0p,qsup⁡‖g‖M-t,0p′,q′,
where the supremum is taken over all g∈𝒮 and ∥g∥ℳ-t,0p′,q′=1; hence,
(3.8)‖fλ‖Mt,0p,q≤Cmax⁡{1,λ-t}λ-(d+μ2(p′,q′))‖f‖Mt,0p,q=Cmax⁡{1,λ-t}λdμ1(p,q)‖f‖Mt,0p,q.
This establishes the upper half of (3.2).We now consider the polynomial weights in the frequency variablesvs(ω)=〈ω〉s, s∈ℝ.Theorem 3.2.
Let1≤p,q≤∞, s∈ℝ. Then the following are true.(1)
There exists a constantC>0 such that forallf∈ℳ0,sp,q, λ≥1,
(3.9)C-1λdμ2(p,q)min⁡{1,λs}‖f‖M0,sp,q≤‖fλ‖M0,sp,q≤Cλdμ1(p,q)max⁡{1,λs}‖f‖M0,sp,q.(2)
There exists a constantC>0 such that forallf∈ℳ0,sp,q, 0<λ≤1,
(3.10)C-1λdμ1(p,q)min⁡{1,λs}‖f‖M0,sp,q≤‖fλ‖M0,sp,q≤Cλdμ2(p,q)max⁡{1,λs}‖f‖M0,sp,q.Proof.
Here we use the fact that the mappingf↦〈D〉sf is an homeomorphism from ℳt,s0p,q to ℳt,s0-sp,q, t,s,s0∈ℝ (see [9, Corollary  2.3]). The rest of the proof uses similar arguments as those in Theorem 3.1.The next result follows immediately by combining the last two theorems.Corollary 3.3.
Let1≤p,q≤∞, t,s∈ℝ. Then the following are true.(1)
There exists a constantC>0 such that forallf∈ℳt,sp,q, λ≥1,
(3.11)C-1λdμ2(p,q)min⁡{1,λ-t}min⁡{1,λs}‖f‖Mt,sp,q≤‖fλ‖Mt,sp,q≤Cλdμ1(p,q)max⁡{1,λ-t}max⁡{1,λs}‖f‖Mt,sp,q.(2)
There exists a constantC>0 such that forallf∈ℳt,sp,q, 0<λ≤1,
(3.12)C-1λdμ1(p,q)min⁡{1,λ-t}min⁡{1,λs}‖f‖Mt,sp,q≤‖fλ‖Mt,sp,q≤Cλdμ2(p,q)max⁡{1,λ-t}max⁡{1,λs}‖f‖Mt,sp,q.The following result is an analogue of Corollary3.3 for modulation spaces defined by nonseparable polynomial growing weight function such as vs(x,ω):=〈(x,ω)〉s=(1+|x|2+|ω|2)s/2.Theorem 3.4.
Let1≤p,q≤∞, s∈ℝ. Then the following are true.(1)
There exists a constantC>0 such that for all f∈ℳvsp,q, λ≥1,
(3.13)C-1λdμ2(p,q)min⁡{λ-s,λs}‖f‖Mvsp,q≤‖fλ‖Mvsp,q≤Cλdμ1(p,q)max⁡{λ-s,λs}‖f‖Mvsp,q.(2)
There exists a constantC>0 such that for all f∈ℳvsp,q, 0<λ≤1,
(3.14)C-1λdμ1(p,q)min⁡{λ-s,λs}‖f‖Mvsp,q≤‖fλ‖Mvsp,q≤Cλdμ2(p,q)max⁡{λ-s,λs}‖f‖Mvsp,q.Proof.
We assumes≥0. A duality argument can be used to complete the proof when s<0 (notice, this duality argument will be given explicitly below in the proof of the sharpness of Theorem 3.1 in the case (1/p,1/q)∈I2, t≥0).
Moreover, since the result has been proved in [5, Theorem  3.1] for s=0, one can use interpolation arguments along with Lemma 2.1 to reduce the proof when s is an even integer.
The mappingf↦〈x,D〉sf is an homeomorphism from ℳvsp,q to ℳp,q, s∈ℝ (see [9, Theorem  2.2]). Hence
(3.15)‖fλ‖Mvsp,q≍‖〈x,D〉sfλ‖Mp,q=‖Uλ(〈λ-1x,λD〉sf)‖Mp,q≤C{λdμ1(p,q)‖〈λ-1x,λD〉sf‖Mp,q,λ≥1λdμ2(p,q)‖〈λ-1x,λD〉sf‖Mp,q,0<λ≤1,
where in the last inequality we used again the dilation properties for unweighted modulation spaces of [5, Theorem  3.1]. Therefore, writing f=〈x,D〉-s〈x,D〉sf, we see that it suffices to prove that the pseudodifferential operator
(3.16)〈λ-1x,λD〉s〈x,D〉-s
is bounded on ℳp,q, and its norm is bounded above by max{1,λ-s}max{1,λs}=max{λs,λ-s}. To this end, we observe that, if s is an even integer, 〈λ-1x,λD〉s is a finite sum of operators of the form λkxαDβ, with |k|≤s and |α|+|β|≤s. Now, Shubin’s pseudodifferential calculus [10] shows that the operators xαDβ〈x,D〉-s have bounded symbols, together with all their derivatives, so that they are bounded on ℳp,q. The proof is completed by taking into account the additional factor λk.Finally, it is relatively straightforward to give optimal estimates for the dilation operatorUλ on the Wiener amalgam spaces W(ℱLsp,Ltq). These spaces are images of modulation spaces under Fourier transform, that is, ℱℳt,sp,q=W(ℱLsp,Ltq). It is also worth noticing that the indices μ1 and μ2 obey the following relations:(3.17)μ1(p′,q′)=-1-μ2(p,q),μ2(p′,q′)=-1-μ1(p,q)whenever1p+1p′=1q+1q′=1.
Using the above relations along with the definition of the Wiener amalgam spaces, as well as the behavior of the Fourier transform under dilation, that is, fλ̂=λ-d(f̂)1/λ, and Corollary 3.3, we obtain the following result.Proposition 3.5.
Let1≤p,q≤∞, t,s∈ℝ. Then the following are true.(1)
There exists a constantC>0 such that for all f∈W(ℱLsp,Ltq), λ≥1,
(3.18)C-1λdμ2(p′,q′)min⁡{1,λt}min⁡{1,λ-s}‖f‖W(FLsp,Ltq)≤‖fλ‖W(FLsp,Ltq)≤Cλdμ1(p′,q′)max⁡{1,λt}max⁡{1,λ-s}‖f‖W(FLsp,Ltq).(2)
There exists a constantC>0 such that for all f∈W(ℱLsp,Ltq), λ≤1,
(3.19)C-1λdμ1(p′,q′)min⁡{1,λt}min⁡{1,λ-s}‖f‖W(FLsp,Ltq)≤‖fλ‖W(FLsp,Ltq)≤Cλdμ2(p′,q′)max⁡{1,λt}max⁡{1,λ-s}‖f‖W(FLsp,Ltq).Remark 3.6.
ForL∈GL(d,ℝ), the operator UL defined by ULf(·)=f(L·) is clearly bounded on ℳt,sp,q(ℝd). It is interesting to ask whether versions of Theorems 3.1 and 3.2 can be established for UL.


## 4. Sharpness of Theorems3.1 and 3.2

In this section, we prove the sharpness of Theorems3.1 and 3.2. The sharpness of Theorem 3.4 is proved by modifying the examples constructed in the next subsection. Therefore, we omit it. But we first prove some preliminary lemmas in which we construct functions that achieve the optimal bound.


### 4.1. Preliminary Estimates

The next two lemmas involve estimates for the modulation space norms of various modifications of the Gaussian. Together with Lemmas4.3–4.5, they provide examples of functions that achieve the optimal bound under the dilation operator on weighted modulation spaces with weight on the space parameter. Similar constructions for weighted modulation spaces with weight on the frequency parameter are contained in Lemmas 4.6–4.10. Finally, in Lemma 4.11, we investigated the property of the dilation operator on compactly supported functions.Recall thatφ(x)=e-π|x|2 for x∈ℝd, and that φλ(x)=Uλφ(x)=φ(λx).Lemma 4.1.
Fort,s≥0, one has
(4.1)‖φλ‖Mt,0p,q≍λ-d/p-t,0<λ≤1,(4.2)‖φλ‖Mt,0p,q≍λ-d(1-1/q),λ≥1,(4.3)‖φλ‖M0,sp,q≍λ-d/p,0<λ≤1,(4.4)‖φλ‖M0,sp,q≍λ-d(1-1/q)+s,λ≥1.Proof.
We will only prove the first two estimates, as the last two, are proved similarly. By some straightforward computations (see, e.g., [2, Lemma  1.5.2]), we get
(4.5)|Vφφλ(x,ω)|=(λ2+1)-d/2e-π(λ2/(λ2+1))|x|2e-π(1/(λ2+1))|ω|2.
Hence,
(4.6)‖φλ‖Mt,0p,q≍‖Vφφλ‖Mt,0p,q=p-2pq-2qλ-d/p(λ2+1)(d/2)((1/p)+(1/q)-1)(∫Rde-π|x|2〈λ2+1λpx〉ptdx)1/p.
If 0<λ≤1, then
(4.7)λ-t|x|t/2≤(λ2+1λ2|x|2)t/2≤(1+λ2+1λ2|x|2)t/2≤2λ-t(1+|x|2)t/2.
Thus, we have
(4.8)λ-t≲(∫Rde-π|x|2〈λ2+1λpx〉ptdx)1/p≲λ-t,0<λ≤1,
and the estimate (4.1) follows.
Now, observe that, ifλ≥1, then 〈(λ2+1/λp)x〉≍〈x〉, and (4.2) follows.Lemma 4.2.
Fort≤0, λ≥1, consider the family of functions
(4.9)f(x)=λ-tφ(x-λe1),e1=(1,0,0,…,0).
Then, there exists a constant C>0 such that ∥f∥ℳt,0p,q≤C, uniformly with respect to λ. Moreover,
(4.10)‖fλ‖Mt,0p,q≳λ-t+d((1/q)-1),∀λ≥1.Proof.
We have(4.11)‖f‖Mt,0p,q≍‖Vφf(x,ω)〈x〉t‖Lp,q=λ-t‖Vφφ(x-λe1,ω)〈x〉t‖Lp,q=λ-t‖Vφφ(x,ω)〈x+λe1〉t‖Lp,q≲λ-tλt‖Vφφ〈x〉-t‖Lp,q≲1.
The last inequality follows from the fact that the weight 〈·〉t is 〈·〉-t-moderate which implies that 〈x+λe1〉t≲λt〈x〉-t. This proves the first part of the lemma. Let us now estimate ∥fλ∥ℳt,0p,q from below. We have
(4.12)fλ(x)=λ-tφλ(x-e1).
Hence, by arguing as above and using (4.5), we have
(4.13)‖fλ‖Mt,0p,q≍λ-t‖Vφφλ(x,ω)〈x+e1〉t‖Lp,q≳λ-tλd((1/q)-1)(∫e-πp|x|2〈x+e1〉ptdx)1/p≳λ-t+d((1/q)-1),
which concludes the proof.Lemma 4.3.
Let1≤p,q≤∞, ϵ>0, t∈ℝ, and λ>1. Moreover, assume that (1/p,1/q)∈I1*.(a)
Ift≥0, define
(4.14)f(x)=∑l≠0|l|-d/p-ϵe2πiλ-1l⋅xφ(x)=∑l≠0|l|-d/p-ϵMλ-1lφ(x),inS′(Rd).
Then, there exists a constant C>0 such that ∥f∥ℳt,0p,q≤C, uniformly with respect to λ. Moreover,
(4.15)‖fλ‖Mt,0p,q≳λ-d/p-ϵ,∀λ>1.(b)
Ift≤0 define
(4.16)f(x)=∑k≠0|k|-d/p-ϵ-tφ(x-k)=∑k≠0|k|-d/p-ϵ-tTkφ(x),inS′(Rd).Then, there exists a constant C>0 such that ∥f∥ℳt,0p,q≤C, uniformly with respect to λ. Moreover,
(4.17)‖fλ‖Mt,0p,q≳λ-d/p-ϵ-t,∀λ>1.Proof.
We only prove part (a) as part (b) is obtained similarly. We use Proposition2.2 to prove that f defined in the lemma belongs to ℳt,0p,q. Indeed, 𝒢(φ,1,λ-1) is a Gabor frame, and the coefficients of f in this frame are given by ck,ℓ=δk,0|ℓ|-d/p-ϵ if ℓ≠0 and c0,0=0. It is clear that
(4.18)‖ck,l‖lt,0p,q=(∑l∈Zd(∑k∈Zd|ck,l|p〈k〉pt)q/p)1/q=(∑l≠0|l|q(-d/p-ϵ))1/q<∞,
because q/p≥1. Thus, f∈ℳt,0p,q with uniform norm (with respect to λ).
Givenλ>1, we have
(4.19)‖fλ‖Mt,0p,q=sup⁡‖g‖M-t,0p′,q′=1|〈fλ,g〉|≥‖φ‖M-t,0p′,q′-2|〈fλ,φ〉|.
Using relation (4.5),
(4.20)〈fλ,φ〉=∑l≠0|l|-d/p-ϵVφφλ(0,l)=∑l≠0|l|-d/p-ϵ(1+λ2)-d/2e-π|l|2/(λ2+1).
Therefore, if λ>1,
(4.21)‖fλ‖Mt,0p,q≥C∑l≠0|l|-d/p-ϵ(1+λ2)-d/2e-π|l|2/(λ2+1)≥Cλ-d∑l≠0|l|-d/p-ϵe-π|l|2/(λ2+1)≥Cλ-d∑0<|l|<λ|l|-d/p-ϵe-π|l|2/(λ2+1)≥Cλ-dλ-d/p-ϵ∑0<|l|<λe-π≥Cλ-dλ-d/p-ϵe-πλd=Cλ-d/p+ϵ,
from which the proof follows.The next results extend [5, Lemmas  3.9 and 3.10].Lemma 4.4.
Let1≤p,q≤∞, t≥0, ϵ>0. Suppose that ψ∈𝒮(ℝd) satisfy suppψ⊂[-1/2,1/2]d and ψ=1 on [-1/4,1/4]d.(a)
If1≤q<∞, define
(4.22)f(y)=∑k∈Zd∖{0}|k|-(d/q)-ϵ-tMkTkψ(y),inS′(Rd).
Then, f∈ℳt,0p,q(ℝd) and
(4.23)‖fλ‖Mt,0p,q≳λ-d((2/p)-(1/q))+ϵ-t,∀0<λ≤1.(b)
Ifq=∞, let
(4.24)f(y)=∑k≠0|k|-tMkTkψ(y),inS′(Rd).
Then, f∈ℳt,0p,∞ and
(4.25)‖fλ‖Mt,0p,∞≳λ-(2d/p)-t,∀0<λ≤1.Proof.
We only prove part (a), that is, the case1≤q<∞ as the case q=∞ is proved in a similar fashion.
Letg∈𝒮(ℝd) satisfy suppg⊂[-1/8,1/8]d, and |ĝ|≥1 on [-2,2]d. The proof of each part of the Lemma is based on the appropriate estimate for Vgf.
Let us first show thatf∈ℳt,0p,q(ℝd). We have
(4.26)|∫Rde-2πi(ω-k)yψ(y-k)g(y-x)dy|=|∫Rdψ(y-k)g(y-x){(1+|ω-k|2)-d(I-Δy)de-2πi(ω-k)y}dy|=1(1+|ω-k|2)d|∑|β1+β2|≤2dCβ1,β2∫Rd∂β1(Tkψ)(y)(∂β2g)(x-y)e-2πi(ω-k)ydy|≤C(1+|ω-k|2)d∑|β1+β2|≤2d(|Tk(∂β1ψ)|*|∂β2g|)(x).
Hence,(4.27)‖f‖Mt,0p,q≍‖Vgf‖Lt,0p,q=(∫Rd(∫Rd|∑k≠0|k|-d/q-ϵ-t∫Rde-2πi(ω-k)yψ(y-k)g(y-x)dy|p〈x〉tpdx)q/pdω)1/q≤C(∫Rd(∑k≠0|k|-d/q-ϵ-t1(1+|ω-k|2)d∑|β1+β2|≤2d‖|Tk(∂β1ψ)|*|∂β2g|‖Ltp)qdω)1/q.
Using Young’s inequality:∥|Tk(∂β1ψ)|*|∂β2g|∥Ltp≲∥Tk∂β1ψ∥Lt1∥∂β2g∥Ltp, and the estimate ∥Tk∂β1ψ∥Lt1≤〈k〉t∥∂β1ψ∥Lt1, we can control (4.27) by
(4.28)C(∫Rd(∑k≠0|k|-d/q-ϵ1(1+|ω-k|2)d)qdω)1/q≤C(∑l∈Zd∫l+[-1/2,1/2]d(∑k≠0|k|-d/q-ϵ1(1+|ω-k|2)d)qdω)1/q≤C̃(∑l∈Zd(∑k≠0|k|-d/q-ϵ1(1+|l-k|2)d)q)1/q=C̃‖|k|-d/q-ϵ*1(1+|k|2)d‖lq<∞,
since {|k|-d/q-ϵ}k≠0∈ℓq.
Next, we prove (4.23). Since Vgfλ(x,ω)=λ-dVgλ-1f(λx,λ-1ω), we obtain
(4.29)‖Vgfλ‖Lt,0p,q=λ-d(1+1/p-1/q)(∫Rd(∫Rd|Vgλ-1f(x,ω)|p〈λ-1x〉ptdx)q/pdω)1/q.
Observe that
(4.30)〈λ-1(x)〉≳λ-1|l|if|x|≍|l|
and suppg((·-x)/λ)⊂ℓ+[-1/4,1/4]d, for all 0<λ≤1, x∈ℓ+[-1/8,1/8]d. Since suppψ(·-k)⊂k+[-1/2,1/2]d and ψ(t-k)=1 if t∈k+[-1/4,1/4]d, the inner integral can be estimated as follows:
(4.31)(∫Rd|Vgλ-1f(x,ω)|p〈λ-1x〉ptdx)1/p≥(∑l≠0∫l+[-1/8,1/8]d|∑k≠0|k|-d/q-ϵ-t∫Rde-2πi(ω-k)yψ(y-k)g(y-xλ)¯dy|p×〈λ-1x〉ptdx∑l≠0∫l+[-1/8,1/8]d|∑k≠0|k|-d/q-ϵ-t∫Rde-2πi(ω-k)yψ(y-k)g(y-xλ)¯dy|p)1/p≳(∑l≠0(|l|-d/q-ϵ-tλd|ĝ(-λ(ω-l))|λ-t|l|t)p)1/p≳(∑l≠0(|l|-d/q-ϵλd-t|ĝ(-λ(ω-l))|)p)1/p.
Consequently,
(4.32)‖Vgfλ‖Lt,0p,q=λ-d(1+1/p-1/q)(∫Rd(∫Rd|Vgλ-1f(x,ω)|p〈λ-1x〉ptdx)q/pdω)1/q≳λ-d(1+1/p-1/q)(∫Rd(∑l≠0(|l|-d/q-ϵλd-t|ĝ(-λ(ω-l))|)p)q/pdω)1/q=λd-t-d/qλ-d(1+1/p-1/q)(∫Rd(∑l≠0(|l|-d/q-ϵ|ĝ(ω+λl)|)p)q/pdω)1/q≳λ-t-d/p(∫|ω|≤1(∑|l|≤1/λ(|l|-d/q-ϵ|ĝ(ω+λl)|)p)q/pdω)1/q≳λ-t-d/p(∫|ω|≤1(∑|l|≤1/λ(|l|-d/q-ϵ)p)q/pdω)1/q=λ-t-d/p(∑|l|≤1/λ(|l|-d/q-ϵ)p)1/p≳λ-t-d/pλd/q+ϵ(∑|l|≤1/λ)1/p≳λ-t-2(d/p)+d/q+ϵ,
which completes the proof.Lemma 4.5.
Let1≤p,q≤∞ be such that (1/p,1/q)∈I3. Let ϵ>0, t<0, and 0<λ<1.(a)
Ift≤-d define
(4.33)f(x)=λd/q-2d/p+2d∑k≠0|k|-ϵ/2Tλ2kφ(x),inS′(Rd).
Then, there exists a constant C>0 such that ∥f∥ℳt,0p,q≤C, uniformly with respect to λ. Moreover,
(4.34)‖fλ‖Mt,0p,q≳λdμ2(p,q)+ϵ,∀0<λ<1.(b)
If-d<t<0, choose a positive integer N large enough such that 1/N<(p-1)/2-pt/2d. Define
(4.35)f(x)=λd/q∑k≠0|k|d(2/Np-1)-ϵ/NTλNkφ(x),inS′(Rd).Then, the conclusions of part (a) still hold.Proof.
(a) For the range ofp,q being considered, d/q+2d-2d/p=dμ2(p,q)+2d≥0, and so if λ<1, then λd/q+2d-2d/p<1.
Next, notice that𝒢(φ,λ2,1) is a Gabor frame. So, to check that f∈ℳt,0p,q, we only need to verify that the sequence c={ckℓ}={|k|-ϵ/2δℓ,0,k≠0}k,ℓ∈ℤd∈ℓt,0p,q. But, the condition t≤-d guarantees this, since
(4.36)‖c‖lt,0p,q=λd/q+2d-2d/p(∑k≠0|k|-pϵ/2(1+|k|2)pt/2)1/p≤C.
Next, as in the proof of Lemma4.3, we have
(4.37)‖fλ‖Mt,0p,q=sup⁡‖g‖M-t,0p′,q′=1|〈fλ,g〉|≥‖φ‖M-t,0p′,q′-2|〈fλ,φ〉|.
In this case,
(4.38)〈fλ,φ〉=λ2d+dμ2(p,q)∑k≠0|k|-ϵ/2Vφφλ(λk,0)=λ2d+dμ2(p,q)∑k≠0|k|-ϵ/2(1+λ2)-d/2e-πλ4|k|2/(λ2+1).
Therefore, if λ<1,
(4.39)‖fλ‖Mt,0p,q≥Cλ2d+dμ2(p,q)∑k≠0|k|-ϵ/2(1+λ2)-d/2e-πλ4|k|2/(λ2+1)≥Cλ2d+dμ2(p,q)∑k≠0|k|-ϵ/2≥Cλ2d+dμ2(p,q)∑0<|k|<1/λ2|k|-ϵ/2≥Cλ2d+dμ2(p,q)λϵλ-2d=Cλdμ2(p,q)+ϵ
which completes the proof of part (a).
(b) Ifp≥1, the assumptions -d<t<0 and 1/N<(p-1)/2-pt/2d are sufficient to prove that f∈ℳt,0p,q. In addition, the main estimate is that
(4.40)‖fλ‖Mt,0p,q≥Cλd/q∑k≠0|k|d(2/Np-1)-ϵ/N(1+λ2)-d/2e-πλ2N|k|2/(λ2+1)≥Cλd/q∑0<|k|<1/λN|k|d(2/Np-1)-ϵ/N≥Cλdμ2(p,q)+ϵ.We now state results similar to the above lemmas when the weight is in the frequency variable.Lemma 4.6.
Fors≤0, 0<λ≤1, consider the family of functions
(4.41)f(x)=λsMλ-1e1φ(x),e1=(1,0,0,…,0).
Then, there exists a constant C>0 such that ∥f∥ℳ0,sp,q≤C, uniformly with respect to λ. Moreover,
(4.42)‖fλ‖M0,sp,q≳λs-d/p,∀0<λ≤1.Proof.
We have(4.43)‖f‖M0,sp,q≍‖Vφf(x,ω)〈ω〉s‖Lp,q=λs‖Vφφ(x,ω-λ-1e1)〈ω〉s‖Lp,q=λs‖Vφφ(x,ω)〈ω+λ-1e1〉s‖Lp,q≲λsλ-s‖Vφφ〈ω〉-s‖Lp,q≲1,
where we have used again the fact that the weight 〈·〉s is 〈·〉-s-moderate. Thus, the functions f have norms in ℳ0,sp,q uniformly bounded with respect to λ. Let us now estimate ∥fλ∥ℳ0,sp,q from below. We have
(4.44)fλ(x)=λsMe1φλ(x).
By using (4.5), we obtain
(4.45)‖fλ‖M0,sp,q=λs‖Vφφλ(x,ω-e1)〈ω〉s‖Lp,q≳λs-d/p(∫e-πq|ω|2〈ω+e1〉qsdω)1/q≳λs-d/p,
as desired.Lemma 4.7.
Let1≤p,q≤∞ be such that (1/p,1/q)∈I2*. Assume that s≤0, ϵ>0 and λ>1.(a)
Ifq≥2 and s≤0, or 1≤q≤2 and s≤-d, define
(4.46)f(x)=∑l≠0|l|d(1/q-1)-ϵe2πiλ-1l⋅xφ(x)=∑l≠0|l|d(1/q-1)-ϵMλ-1lφ(x),inS′(Rd).Then, there exists a constant C>0 such that ∥f∥ℳ0,sp,q≤C, uniformly with respect to λ. Moreover,
(4.47)‖fλ‖M0,sp,q≳λd(1/q-1)-ϵ,∀λ>1.(b)
If1≤q≤2 and -d<s<0, choose a positive integer N such that 1/N<-sq/d, and define
(4.48)f(x)=∑l≠0|l|d(1/Nq-1)-ϵ/Ne2πiλ-Nl⋅xφ(x)=∑l≠0|l|d(1/Nq-1)-ϵ/NMλ-Nlφ(x),inS′(Rd).Then, the conclusions of part (a) still hold.Proof.
(a) First of all, notice that𝒢(φ,1,λ-1) is a frame. In addition, q≥2 is equivalent to 1/q-1≤-1/q. Thus, for all s≤0, {|ℓ|d(1/q-1)-ϵ,ℓ≠0}∈ℓsq, which ensures that the function f defined above belongs to ℳ0,sp,q. This is also true when 1≤q≤2 and s≤-d.
To prove (4.47), we follow the proof of Lemma 4.3. In particular, we have
(4.49)‖fλ‖M0,sp,q≥C∑l≠0|l|d(1/q-1)-ϵ(1+λ2)-d/2e-π|l|2/(λ2+1)≥Cλ-d∑0<|l|≤λ|l|d(1/q-1)-ϵe-π|l|2/(λ2+1),
from which (4.47) follows.
(b) In this case,𝒢(φ,1,λ-N) is a frame. Moreover, the choice of N insures that d(1/(Nq)-1)+s<-d which is enough to prove that f∈ℳ0,sp,q and that ∥f∥ℳ0,sp,q≤C. Relation (4.47) now follows from
(4.50)‖fλ‖M0,sp,q≥C∑l≠0|l|d(1/Nq-1)-ϵ/N(1+λ2)-d/2e-πλ-2N+2|l|2/(λ2+1)≥Cλ-d∑0<|l|≤λN|l|d(1/Nq-1)-ϵ/Ne-πλ-2N+2|l|2/(λ2+1)≥Cλd(1/q-1)-ϵ.The next lemma is proved similarly to Lemma4.4, so we omit its proof.Lemma 4.8.
Let1≤p,q≤∞, s≤0, ϵ>0. Suppose that ψ∈𝒮(ℝd) satisfies suppψ⊂[-1/2,1/2]d and ψ=1 on [-1/4,1/4]d.
(a)
If1≤q<∞, define f(y)=∑k∈ℤd∖{0}|k|-d/q-ϵ-sMkTkψ(y),in𝒮′(ℝd). Then, f∈ℳ0,sp,q(ℝd) and(4.51)‖fλ‖M0,sp,q≳λ-d(2/p-1/q)+ϵ+s,∀0<λ≤1.(b)
Ifq=∞, let
(4.52)f(y)=∑k≠0|k|-se2πikyTkψ(y),inS′(Rd).
Then, f∈ℳ0,sp,∞ and(4.53)‖fλ‖M0,sp,∞≳λ-2d/p+s,∀0<λ≤1.Lemma 4.9.
Let1≤p,q≤∞ be such that (1/p,1/q)∈I3. Let ϵ>0, s≥0 and 0<λ<1. Assume that p>1, and choose a positive integer N such that 1/N<(p-1)/2. Define
(4.54)f(x)=λd/q∑k≠0|k|d(2/Np-1)-ϵ/NTλNkφ(x),inS′(Rd).
Then, there exists a constant C>0 such that ∥f∥ℳ0,sp,q≤C, uniformly with respect to λ. Moreover,
(4.55)‖fλ‖M0,sp,q≳λdμ2(p,q)+ϵ.Proof.
In this case,𝒢(φ,λN,1) is a frame. The condition 1/N<(p-1)/2 is equivalent to 2/Np-1<-1/p which is enough to show that {|k|d(2/Np-1)-ϵ/N}k≠0∈ℓp. Therefore, f∈ℳ0,sp,q with ∥f∥ℳ0,sp,q≤C, where C is a universal constant. The rest of the proof is an adaptation of the proof of Lemma 4.5.Notice that the previous lemma excludes the casep=1. We prove this last case by considering the dual case. Observe that the case (1/∞,1/∞)∈I1*∩I3*  was already considered in dealing with the region I1*.Lemma 4.10.
Let1≤q≤∞ be such that (1/∞,1/q)∈I3*. Let ϵ>0, s≤0 and λ>1.(a)
If1<q<2, choose a positive integer N such that 3/N<q-1. Define
(4.56)f(x)=λd(1-2/q)∑l≠0|l|d(3/Nq-1)-ϵ/NMλ-Nlφ(x),inS′(Rd).Then, there exists a constant C>0 such that ∥f∥ℳ0,sp,q≤C, uniformly with respect to λ. Moreover,
(4.57)‖fλ‖M0,s∞,q≳λd/q-ϵ.(b)
If2≤q<∞, choose a positive integer N such that N>2+q. Define
(4.58)f(x)=λd+d(2-N)/q∑l≠0|l|d((N-1)/Nq-1)-ϵ/NMλ-Nlφ(x),inS′(Rd).Then, the conclusions of part (a) still hold.(c)
Ifq=1 and s≤-d, define(4.59)f(x)=∑l≠0|l|-ϵ/2Mλ-2lφ(x),inS′(Rd).
Then, there exists a constant C>0 such that ∥f∥ℳ0,s∞,1≤C, uniformly with respect to λ. Moreover,
(4.60)‖fλ‖M0,s∞,1≳λd-ϵ.(d)
Ifq=1 and -d<s<0, choose a positive integer N such that 1/N<-s/2d. Define(4.61)f(x)=∑l≠0|l|d(2/N-1)-ϵ/NMλ-Nlφ(x),inS'(Rd).
Then, the conclusions of part (c) still hold.Proof.
(a) In this case,𝒢(φ,1,λ-N) is a frame. The hypotheses 1<q<2 and λ>1 imply that λd(1-2/q)<1. In addition, the condition 3/N<q-1 is equivalent to 3/Nq-1<-1/q which is enough to show that {|ℓ|d(3/Nq-1)-ϵ/N}ℓ≠0∈ℓsq. Therefore, f∈ℳ0,s∞,q with ∥f∥ℳ0,s∞,q≤C, where C is a universal constant. The rest of the proof is an adaptation of the proof of Lemma 4.5.
(b) Assume that2≤q<∞. The proof is similar to the above with the following differences: N>q+2 and λ>1 imply that λd(1+(2-N)/q)<1. In addition, the condition q≥2 implies that (N-1)/Nq-1<-1/q. This is enough to show that {|ℓ|d((N-1)/Nq-1)-ϵ/N}ℓ≠0∈ℓsq. Therefore, f∈ℳ0,s∞,q with ∥f∥ℳ0,s∞,q≤C, where C is a universal constant.
(c) In this case,𝒢(φ,1,λ-2) is a frame. The fact that s≤-d implies that {|ℓ|-ϵ/2}ℓ≠0∈ℓs1. Therefore, f∈ℳ0,s∞,1 with ∥f∥ℳ0,s∞,1≤C, where C is a universal constant. The rest of the proof is an adaptation of the proof of Lemma 4.5.
(d) In this case,𝒢(φ,1,λ-N) is a frame. The fact that -d<s<0 and the choice of N imply that d(2/N-1)+s<-d. Therefore, {|ℓ|d(2/N-1)-ϵ/2}ℓ≠0∈ℓs1. Therefore, f∈ℳ0,s∞,1 with ∥f∥ℳ0,s∞,1≤C, where C is a universal constant. The rest of the proof is an adaptation of the proof of Lemma 4.5.We finish this subsection by proving lower bound estimates for the dilation of functions that are compactly supported either in the time or in the frequency variables.Lemma 4.11.
Letu∈𝒮(ℝd), λ∈(0,∞), and 1≤p,q≤∞.(i)
Ifu is supported in a compact set K⊂ℝd, then, for every t∈ℝ and λ≥1,(4.62)‖uλ‖Mt,0p,q≳λ-d(1-1/q)min⁡{1,λ-t}.(ii)
Ifû is supported in a compact set K⊂ℝd, then, for every s∈ℝ and λ≤1,(4.63)‖uλ‖M0,sp,q≳Cλ-d/pmin⁡{1,λs}.Proof.
We use the dilation properties for the Sobolev spaces (Bessel potential spaces)Hsp(ℝd) (see, e.g., [7, Proposition  3]):
(4.64)C-1λ-d/pmin⁡{1,λs}‖u‖Hsp≤‖uλ‖Hsp≤Cλ-d/pmax⁡{1,λs}‖u‖Hsp,1≤p≤∞,s>0.(i)
Letu be supported in a compact set K⊂ℝd, we have u∈ℳp,q⇔u∈ℱLq, and
(4.65)CK-1‖u‖Mp,q≤‖u‖FLq≤CK‖u‖Mp,q,
where CK>0 depends only on K (see, e.g., [11, 12]). Hence, if λ≥1,
(4.66)‖uλ‖Mt,0p,q≍‖〈⋅〉tuλ‖Mp,q≍‖〈⋅〉tuλ‖FLq≍‖F-1(uλ)‖Htq=λ-d‖(F-1u)λ-1‖Htq≥λ-d(λ-1)-d/qmin⁡{1,λ-t}.(ii)
Now letû be supported in a compact set K⊂ℝd. We have u∈ℳp,q⇔u∈Lp, and
(4.67)CK-1‖u‖Mp,q≤‖u‖Lp≤CK‖u‖Mp,q,
where CK>0 depends only on K (again, see, e.g., [11]). Arguing as in part (i) above with 0<λ≤1,(4.68)‖uλ‖M0,sp,q≍‖〈D〉suλ‖Mp,q≍‖〈D〉suλ‖Lp≍‖uλ‖Hsp≥Cλ-d/pmin⁡{1,λs}‖u‖Hsp
and the proof is completed.


### 4.2. Sharpness of Theorems3.1 and 3.2

We are now in position to state and prove the sharpness of the results obtained in Section3. In particular, Theorem 3.1 is optimal in the following sense.Theorem 4.12.
Let1≤p,q≤∞.
(A)
Ift≥0 then the following statements hold.
Assume that there exist constantsC>0 and α,β∈ℝ such that(4.69)C-1λβ‖f‖Mt,0p,q≤‖Uλf‖Mt,0p,q≤Cλα‖f‖Mt,0p,q,∀f∈Mt,0p,q,λ≥1,
then, α≥dμ1(p,q), and β≤dμ2(p,q)-t.
Assume that there exist constantsC>0 and α,β∈ℝ such that(4.70)C-1λα‖f‖Mt,0p,q≤‖Uλf‖Mt,0p,q≤Cλβ‖f‖Mt,0p,q,∀f∈Mt,0p,q,0<λ≤1,
then, α≥dμ1(p,q), and β≤dμ2(p,q)-t.(B)
Ift≤0 then the following statements hold.
Assume that there exist constantsC>0 and α,β∈ℝ such that(4.71)C-1λβ‖f‖Mt,0p,q≤‖Uλf‖Mt,0p,q≤Cλα‖f‖Mt,0p,q,∀f∈Mt,0p,q,λ≥1,
then, α≥dμ1(p,q)-t, and β≤dμ2(p,q).
Assume that there exist constantsC>0 and α,β∈ℝ such that(4.72)C-1λα‖f‖Mt,0p,q≤‖Uλf‖Mt,0p,q≤Cλβ‖f‖Mt,0p,q,∀f∈Mt,0p,q,0<λ≤1,
then, α≥dμ1(p,q)-t, and β≤dμ2(p,q).Proof.
It will be enough to prove the upper half of each of the estimates, as the lower halves will follow from the fact thatf=UλU1/λf. Moreover, the proof relies on analyzing the examples provided by the previous lemmas and by considering several cases.
Case 1 ((1/p,1/q)∈I2*, t≥0).
In this case, we haveλ≥1 and μ1(p,q)=1/q-1. Substitute f(x)=φ(x)=e-π|x|2 in the upper half estimates (4.69) and use Lemma 4.1 to obtain
(4.73)λ-d(1-1/q)≲‖φλ‖Mt,0p,q≤Cλα‖φ‖Mt,0p,q,
for all λ≥1. This immediately implies that α≥-d(1-1/q)=dμ1(p,q).Case 2 ((1/p,1/q)∈I2, t≤0).
This is the dual case to the previous case and can be handled as follows. In this case, we haveλ≤1 and μ2(p,q)=1/q-1. Assume that the upper-half estimate in (4.72) holds. Notice that (1/p,1/q)∈I2 if and only if (1/p′,1/q′)∈I2* and that λ≤1 if and only if 1/λ≥1.
(4.74)‖f1/λ‖M-t,0p′,q′=sup⁡|〈f1/λ,g〉|=λdsup⁡|〈f,gλ〉|≤λd‖f‖M-t,0p′,q′sup⁡‖gλ‖Mt,0p,q≤λd+β‖f‖M-t,0p′,q′sup⁡‖g‖Mt,0p,q,
where the supremum is taken over all g∈𝒮 and ∥g∥ℳt,0p,q=1; hence,
(4.75)‖f1/λ‖M-t,0p′,q′≤λd+β‖f‖M-t,0p′,q′.
Thus, from Case 1 above, -β-d≥dμ1(p′,q′)=d/q′-d. Hence, β≤dμ2(p,q).Case 3 ((1/p,1/q)∈I3, t≥0).
In this case, we haveλ≤1 and μ2(p,q)=-2/p+1/q. First assume that 1≤q<∞ and that the upper-half estimate in (4.70) holds for all f∈ℳt,0p,q and 0<λ<1 but that β>dμ2(p,q)-t. Then there is ϵ>0 such that β>dμ2(p,q)-t+ϵ. For this choice of ϵ>0, we construct a function f as in (4.22) of Lemma 4.4 such that
(4.76)λdμ2(p,q)-t+ϵ≲‖fλ‖Mt,0p,q≤Cλβ‖f‖Mt,0p,q
for some C>0 and all 0<λ≤1. This leads to a contradiction on the choice of ϵ.
Whenq=∞, the function given by (4.24) of Lemma 4.4 gives the optimal bound.Case 4 ((1/p,1/q)∈I3*, t≤0).
In this case,λ≥1, and μ1(p,q)=-2/p+1/q. This is the dual of Case 3, and a duality argument similar to the used in Case 2 above gives the result.Case 5 ((1/p,1/q)∈I1*,t≤0).
In this case,λ≥1, and μ1(p,q)=-1/p. Assume that the upper-half estimate in (4.71) holds and that α<dμ1(p,q)-t. Then, choose ϵ>0 and construct a function f as in part (b) of Lemma 4.3. A contradiction immediately follows.Case 6 ((1/p,1/q)∈I1, t≥0).
In this case,λ≤1, and μ2(p,q)=-1/p. This is the dual of Case 5.Case 7 ((1/p,1/q)∈I1*, t≥0).
In this case,λ≥1, and μ1(p,q)=-1/p. Assume that the upper-half estimate in (4.69) holds for all f∈ℳt,0p,q and λ>1, but that α<dμ1(p,q). Then, there is ϵ>0 such that α<dμ1(p,q)-ϵ. For this choice of ϵ>0, we can now construct a function f as in Lemma 4.3, part (a), such that
(4.77)λdμ1(p,q)-ϵ≲‖fλ‖Mt,0p,q≤Cλα‖f‖Mt,0p,q
for some C>0 and all λ≥1. This leads to a contradiction on the choice of ϵ.Case 8 ((1/p,1/q)∈I1, t≤0).
In this case,λ≤1, and μ2(p,q)=-1/p. This is the dual of Case 7.Case 9 ((1/p,1/q)∈I2*, t≤0).
In this case,λ≥1, and μ1(p,q)=1/q-1. The function constructed in Lemma 4.2 leads to the result.Case 10 ((1/p,1/q)∈I2, t≥0).
In this case,λ≤1, and μ2(p,q)=1/q-1. This is the dual of Case 9.Case 11 ((1/p,1/q)∈I3, t≤0).
In this case,λ≤1, and μ1(p,q)=-2/p+1/q, and Lemma 4.5 can be used to conclude.Case 12 ((1/p,1/q)∈I3*,t≥0).
In this case,λ≥1, and μ2(p,q)=-2/p+1/q. This is the dual of Case 11.We next consider the sharpness of Theorem3.2.Theorem 4.13.
Let1≤p,q≤∞.(A)
Ifs≥0 then the following statements hold.Assume that there exist constants C>0 and α,β∈ℝ such that
(4.78)C-1λβ‖f‖M0,sp,q≤‖Uλf‖M0,sp,q≤Cλα‖f‖M0,sp,q,∀f∈M0,sp,q,λ≥1,
then, α≥dμ1(p,q)+s, and β≤dμ2(p,q).Assume that there exist constants C>0 and α,β∈ℝ such that
(4.79)C-1λα‖f‖M0,sp,q≤‖Uλf‖M0,sp,q≤Cλβ‖f‖M0,sp,q,∀f∈M0,sp,q,0<λ≤1,
then, α≥dμ1(p,q)+s, and β≤dμ2(p,q).(B)
Ifs≤0 then the following statements hold.Assume that there exist constants C>0 and α,β∈ℝ such that
(4.80)C-1λβ‖f‖M0,sp,q≤‖Uλf‖M0,sp,q≤Cλα‖f‖M0,sp,q,∀f∈M0,sp,q,λ≥1,
then, α≥dμ1(p,q), and β≤dμ2(p,q)+s.Assume that there exist constants C>0 and α,β∈ℝ such that
(4.81)C-1λα‖f‖M0,sp,q≤‖Uλf‖M0,sp,q≤Cλβ‖f‖M0,sp,q,∀f∈M0,sp,q,0<λ≤1,
then, α≥dμ1(p,q), and β≤dμ2(p,q)+s.Proof.
As for the time weights, it is enough to prove the upper half of each estimates. Moreover, in what follows we consider only 6 of the 12 cases to be proved, since the others are obtained by the same duality argument used in the previous theorem.
Case 1 ((1/p,1/q)∈I1, s≥0).
In this case,0<λ≤1 and μ2(p,q)=-1/p. Assume there exist constants C>0 and β∈ℝ such that the upper-half estimate (4.79) holds. Taking the Gaussian f=φ as in Lemma 4.1 and using (4.3), we have
(4.82)λ-d/p≲‖φλ‖M0,sp,q≲λβ‖φ‖M0,sp,q,
for all 0<λ≤1. This gives β≤-d/p.Case 2 ((1/p,1/q)∈I1, s≤0).
Here,λ≤1, and we test the upper-half estimate (4.81) on the family of functions (4.41). Using (4.42), we obtain β≤s-d/p.Case 3 ((1/p,1/q)∈I2*, s≥0).
Hereλ≥1, μ1(p,q)=1/q-1. We assume the upper-half estimate (4.78) and test it on the dilated Gaussian function in (4.4), obtaining α≥d(1/q-1)+s.Case 4 ((1/p,1/q)∈I2*, s≤0).
Here,λ≥1, μ1(p,q)=1/q-1. We use a contradiction argument based on Lemma 4.7.Case 5 ((1/p,1/q)∈I3,s≥0).
Here,λ≤1, μ2(p,q)=-2/p+1/q. The sharpness is obtained by testing the upper-half estimate (4.79) on the family of functions fλ, defined in Lemma 4.9 when p>1.
Ifp=1, we consider the dual case, that is (1/∞,1/q)∈I3*, s≤0. Here λ≥1, μ1(∞,q)=1/q. We use a contradiction argument based on Lemma 4.10.Case 6 ((1/p,1/q)∈I3, s≤0).
Here,λ≤1, μ2(p,q)=-2/p+1/q. The sharpness is obtained by testing the upper-half estimate (4.81) on the family of functions fλ, defined in Lemma 4.8.


## 4.1. Preliminary Estimates

The next two lemmas involve estimates for the modulation space norms of various modifications of the Gaussian. Together with Lemmas4.3–4.5, they provide examples of functions that achieve the optimal bound under the dilation operator on weighted modulation spaces with weight on the space parameter. Similar constructions for weighted modulation spaces with weight on the frequency parameter are contained in Lemmas 4.6–4.10. Finally, in Lemma 4.11, we investigated the property of the dilation operator on compactly supported functions.Recall thatφ(x)=e-π|x|2 for x∈ℝd, and that φλ(x)=Uλφ(x)=φ(λx).Lemma 4.1.
Fort,s≥0, one has
(4.1)‖φλ‖Mt,0p,q≍λ-d/p-t,0<λ≤1,(4.2)‖φλ‖Mt,0p,q≍λ-d(1-1/q),λ≥1,(4.3)‖φλ‖M0,sp,q≍λ-d/p,0<λ≤1,(4.4)‖φλ‖M0,sp,q≍λ-d(1-1/q)+s,λ≥1.Proof.
We will only prove the first two estimates, as the last two, are proved similarly. By some straightforward computations (see, e.g., [2, Lemma  1.5.2]), we get
(4.5)|Vφφλ(x,ω)|=(λ2+1)-d/2e-π(λ2/(λ2+1))|x|2e-π(1/(λ2+1))|ω|2.
Hence,
(4.6)‖φλ‖Mt,0p,q≍‖Vφφλ‖Mt,0p,q=p-2pq-2qλ-d/p(λ2+1)(d/2)((1/p)+(1/q)-1)(∫Rde-π|x|2〈λ2+1λpx〉ptdx)1/p.
If 0<λ≤1, then
(4.7)λ-t|x|t/2≤(λ2+1λ2|x|2)t/2≤(1+λ2+1λ2|x|2)t/2≤2λ-t(1+|x|2)t/2.
Thus, we have
(4.8)λ-t≲(∫Rde-π|x|2〈λ2+1λpx〉ptdx)1/p≲λ-t,0<λ≤1,
and the estimate (4.1) follows.
Now, observe that, ifλ≥1, then 〈(λ2+1/λp)x〉≍〈x〉, and (4.2) follows.Lemma 4.2.
Fort≤0, λ≥1, consider the family of functions
(4.9)f(x)=λ-tφ(x-λe1),e1=(1,0,0,…,0).
Then, there exists a constant C>0 such that ∥f∥ℳt,0p,q≤C, uniformly with respect to λ. Moreover,
(4.10)‖fλ‖Mt,0p,q≳λ-t+d((1/q)-1),∀λ≥1.Proof.
We have(4.11)‖f‖Mt,0p,q≍‖Vφf(x,ω)〈x〉t‖Lp,q=λ-t‖Vφφ(x-λe1,ω)〈x〉t‖Lp,q=λ-t‖Vφφ(x,ω)〈x+λe1〉t‖Lp,q≲λ-tλt‖Vφφ〈x〉-t‖Lp,q≲1.
The last inequality follows from the fact that the weight 〈·〉t is 〈·〉-t-moderate which implies that 〈x+λe1〉t≲λt〈x〉-t. This proves the first part of the lemma. Let us now estimate ∥fλ∥ℳt,0p,q from below. We have
(4.12)fλ(x)=λ-tφλ(x-e1).
Hence, by arguing as above and using (4.5), we have
(4.13)‖fλ‖Mt,0p,q≍λ-t‖Vφφλ(x,ω)〈x+e1〉t‖Lp,q≳λ-tλd((1/q)-1)(∫e-πp|x|2〈x+e1〉ptdx)1/p≳λ-t+d((1/q)-1),
which concludes the proof.Lemma 4.3.
Let1≤p,q≤∞, ϵ>0, t∈ℝ, and λ>1. Moreover, assume that (1/p,1/q)∈I1*.(a)
Ift≥0, define
(4.14)f(x)=∑l≠0|l|-d/p-ϵe2πiλ-1l⋅xφ(x)=∑l≠0|l|-d/p-ϵMλ-1lφ(x),inS′(Rd).
Then, there exists a constant C>0 such that ∥f∥ℳt,0p,q≤C, uniformly with respect to λ. Moreover,
(4.15)‖fλ‖Mt,0p,q≳λ-d/p-ϵ,∀λ>1.(b)
Ift≤0 define
(4.16)f(x)=∑k≠0|k|-d/p-ϵ-tφ(x-k)=∑k≠0|k|-d/p-ϵ-tTkφ(x),inS′(Rd).Then, there exists a constant C>0 such that ∥f∥ℳt,0p,q≤C, uniformly with respect to λ. Moreover,
(4.17)‖fλ‖Mt,0p,q≳λ-d/p-ϵ-t,∀λ>1.Proof.
We only prove part (a) as part (b) is obtained similarly. We use Proposition2.2 to prove that f defined in the lemma belongs to ℳt,0p,q. Indeed, 𝒢(φ,1,λ-1) is a Gabor frame, and the coefficients of f in this frame are given by ck,ℓ=δk,0|ℓ|-d/p-ϵ if ℓ≠0 and c0,0=0. It is clear that
(4.18)‖ck,l‖lt,0p,q=(∑l∈Zd(∑k∈Zd|ck,l|p〈k〉pt)q/p)1/q=(∑l≠0|l|q(-d/p-ϵ))1/q<∞,
because q/p≥1. Thus, f∈ℳt,0p,q with uniform norm (with respect to λ).
Givenλ>1, we have
(4.19)‖fλ‖Mt,0p,q=sup⁡‖g‖M-t,0p′,q′=1|〈fλ,g〉|≥‖φ‖M-t,0p′,q′-2|〈fλ,φ〉|.
Using relation (4.5),
(4.20)〈fλ,φ〉=∑l≠0|l|-d/p-ϵVφφλ(0,l)=∑l≠0|l|-d/p-ϵ(1+λ2)-d/2e-π|l|2/(λ2+1).
Therefore, if λ>1,
(4.21)‖fλ‖Mt,0p,q≥C∑l≠0|l|-d/p-ϵ(1+λ2)-d/2e-π|l|2/(λ2+1)≥Cλ-d∑l≠0|l|-d/p-ϵe-π|l|2/(λ2+1)≥Cλ-d∑0<|l|<λ|l|-d/p-ϵe-π|l|2/(λ2+1)≥Cλ-dλ-d/p-ϵ∑0<|l|<λe-π≥Cλ-dλ-d/p-ϵe-πλd=Cλ-d/p+ϵ,
from which the proof follows.The next results extend [5, Lemmas  3.9 and 3.10].Lemma 4.4.
Let1≤p,q≤∞, t≥0, ϵ>0. Suppose that ψ∈𝒮(ℝd) satisfy suppψ⊂[-1/2,1/2]d and ψ=1 on [-1/4,1/4]d.(a)
If1≤q<∞, define
(4.22)f(y)=∑k∈Zd∖{0}|k|-(d/q)-ϵ-tMkTkψ(y),inS′(Rd).
Then, f∈ℳt,0p,q(ℝd) and
(4.23)‖fλ‖Mt,0p,q≳λ-d((2/p)-(1/q))+ϵ-t,∀0<λ≤1.(b)
Ifq=∞, let
(4.24)f(y)=∑k≠0|k|-tMkTkψ(y),inS′(Rd).
Then, f∈ℳt,0p,∞ and
(4.25)‖fλ‖Mt,0p,∞≳λ-(2d/p)-t,∀0<λ≤1.Proof.
We only prove part (a), that is, the case1≤q<∞ as the case q=∞ is proved in a similar fashion.
Letg∈𝒮(ℝd) satisfy suppg⊂[-1/8,1/8]d, and |ĝ|≥1 on [-2,2]d. The proof of each part of the Lemma is based on the appropriate estimate for Vgf.
Let us first show thatf∈ℳt,0p,q(ℝd). We have
(4.26)|∫Rde-2πi(ω-k)yψ(y-k)g(y-x)dy|=|∫Rdψ(y-k)g(y-x){(1+|ω-k|2)-d(I-Δy)de-2πi(ω-k)y}dy|=1(1+|ω-k|2)d|∑|β1+β2|≤2dCβ1,β2∫Rd∂β1(Tkψ)(y)(∂β2g)(x-y)e-2πi(ω-k)ydy|≤C(1+|ω-k|2)d∑|β1+β2|≤2d(|Tk(∂β1ψ)|*|∂β2g|)(x).
Hence,(4.27)‖f‖Mt,0p,q≍‖Vgf‖Lt,0p,q=(∫Rd(∫Rd|∑k≠0|k|-d/q-ϵ-t∫Rde-2πi(ω-k)yψ(y-k)g(y-x)dy|p〈x〉tpdx)q/pdω)1/q≤C(∫Rd(∑k≠0|k|-d/q-ϵ-t1(1+|ω-k|2)d∑|β1+β2|≤2d‖|Tk(∂β1ψ)|*|∂β2g|‖Ltp)qdω)1/q.
Using Young’s inequality:∥|Tk(∂β1ψ)|*|∂β2g|∥Ltp≲∥Tk∂β1ψ∥Lt1∥∂β2g∥Ltp, and the estimate ∥Tk∂β1ψ∥Lt1≤〈k〉t∥∂β1ψ∥Lt1, we can control (4.27) by
(4.28)C(∫Rd(∑k≠0|k|-d/q-ϵ1(1+|ω-k|2)d)qdω)1/q≤C(∑l∈Zd∫l+[-1/2,1/2]d(∑k≠0|k|-d/q-ϵ1(1+|ω-k|2)d)qdω)1/q≤C̃(∑l∈Zd(∑k≠0|k|-d/q-ϵ1(1+|l-k|2)d)q)1/q=C̃‖|k|-d/q-ϵ*1(1+|k|2)d‖lq<∞,
since {|k|-d/q-ϵ}k≠0∈ℓq.
Next, we prove (4.23). Since Vgfλ(x,ω)=λ-dVgλ-1f(λx,λ-1ω), we obtain
(4.29)‖Vgfλ‖Lt,0p,q=λ-d(1+1/p-1/q)(∫Rd(∫Rd|Vgλ-1f(x,ω)|p〈λ-1x〉ptdx)q/pdω)1/q.
Observe that
(4.30)〈λ-1(x)〉≳λ-1|l|if|x|≍|l|
and suppg((·-x)/λ)⊂ℓ+[-1/4,1/4]d, for all 0<λ≤1, x∈ℓ+[-1/8,1/8]d. Since suppψ(·-k)⊂k+[-1/2,1/2]d and ψ(t-k)=1 if t∈k+[-1/4,1/4]d, the inner integral can be estimated as follows:
(4.31)(∫Rd|Vgλ-1f(x,ω)|p〈λ-1x〉ptdx)1/p≥(∑l≠0∫l+[-1/8,1/8]d|∑k≠0|k|-d/q-ϵ-t∫Rde-2πi(ω-k)yψ(y-k)g(y-xλ)¯dy|p×〈λ-1x〉ptdx∑l≠0∫l+[-1/8,1/8]d|∑k≠0|k|-d/q-ϵ-t∫Rde-2πi(ω-k)yψ(y-k)g(y-xλ)¯dy|p)1/p≳(∑l≠0(|l|-d/q-ϵ-tλd|ĝ(-λ(ω-l))|λ-t|l|t)p)1/p≳(∑l≠0(|l|-d/q-ϵλd-t|ĝ(-λ(ω-l))|)p)1/p.
Consequently,
(4.32)‖Vgfλ‖Lt,0p,q=λ-d(1+1/p-1/q)(∫Rd(∫Rd|Vgλ-1f(x,ω)|p〈λ-1x〉ptdx)q/pdω)1/q≳λ-d(1+1/p-1/q)(∫Rd(∑l≠0(|l|-d/q-ϵλd-t|ĝ(-λ(ω-l))|)p)q/pdω)1/q=λd-t-d/qλ-d(1+1/p-1/q)(∫Rd(∑l≠0(|l|-d/q-ϵ|ĝ(ω+λl)|)p)q/pdω)1/q≳λ-t-d/p(∫|ω|≤1(∑|l|≤1/λ(|l|-d/q-ϵ|ĝ(ω+λl)|)p)q/pdω)1/q≳λ-t-d/p(∫|ω|≤1(∑|l|≤1/λ(|l|-d/q-ϵ)p)q/pdω)1/q=λ-t-d/p(∑|l|≤1/λ(|l|-d/q-ϵ)p)1/p≳λ-t-d/pλd/q+ϵ(∑|l|≤1/λ)1/p≳λ-t-2(d/p)+d/q+ϵ,
which completes the proof.Lemma 4.5.
Let1≤p,q≤∞ be such that (1/p,1/q)∈I3. Let ϵ>0, t<0, and 0<λ<1.(a)
Ift≤-d define
(4.33)f(x)=λd/q-2d/p+2d∑k≠0|k|-ϵ/2Tλ2kφ(x),inS′(Rd).
Then, there exists a constant C>0 such that ∥f∥ℳt,0p,q≤C, uniformly with respect to λ. Moreover,
(4.34)‖fλ‖Mt,0p,q≳λdμ2(p,q)+ϵ,∀0<λ<1.(b)
If-d<t<0, choose a positive integer N large enough such that 1/N<(p-1)/2-pt/2d. Define
(4.35)f(x)=λd/q∑k≠0|k|d(2/Np-1)-ϵ/NTλNkφ(x),inS′(Rd).Then, the conclusions of part (a) still hold.Proof.
(a) For the range ofp,q being considered, d/q+2d-2d/p=dμ2(p,q)+2d≥0, and so if λ<1, then λd/q+2d-2d/p<1.
Next, notice that𝒢(φ,λ2,1) is a Gabor frame. So, to check that f∈ℳt,0p,q, we only need to verify that the sequence c={ckℓ}={|k|-ϵ/2δℓ,0,k≠0}k,ℓ∈ℤd∈ℓt,0p,q. But, the condition t≤-d guarantees this, since
(4.36)‖c‖lt,0p,q=λd/q+2d-2d/p(∑k≠0|k|-pϵ/2(1+|k|2)pt/2)1/p≤C.
Next, as in the proof of Lemma4.3, we have
(4.37)‖fλ‖Mt,0p,q=sup⁡‖g‖M-t,0p′,q′=1|〈fλ,g〉|≥‖φ‖M-t,0p′,q′-2|〈fλ,φ〉|.
In this case,
(4.38)〈fλ,φ〉=λ2d+dμ2(p,q)∑k≠0|k|-ϵ/2Vφφλ(λk,0)=λ2d+dμ2(p,q)∑k≠0|k|-ϵ/2(1+λ2)-d/2e-πλ4|k|2/(λ2+1).
Therefore, if λ<1,
(4.39)‖fλ‖Mt,0p,q≥Cλ2d+dμ2(p,q)∑k≠0|k|-ϵ/2(1+λ2)-d/2e-πλ4|k|2/(λ2+1)≥Cλ2d+dμ2(p,q)∑k≠0|k|-ϵ/2≥Cλ2d+dμ2(p,q)∑0<|k|<1/λ2|k|-ϵ/2≥Cλ2d+dμ2(p,q)λϵλ-2d=Cλdμ2(p,q)+ϵ
which completes the proof of part (a).
(b) Ifp≥1, the assumptions -d<t<0 and 1/N<(p-1)/2-pt/2d are sufficient to prove that f∈ℳt,0p,q. In addition, the main estimate is that
(4.40)‖fλ‖Mt,0p,q≥Cλd/q∑k≠0|k|d(2/Np-1)-ϵ/N(1+λ2)-d/2e-πλ2N|k|2/(λ2+1)≥Cλd/q∑0<|k|<1/λN|k|d(2/Np-1)-ϵ/N≥Cλdμ2(p,q)+ϵ.We now state results similar to the above lemmas when the weight is in the frequency variable.Lemma 4.6.
Fors≤0, 0<λ≤1, consider the family of functions
(4.41)f(x)=λsMλ-1e1φ(x),e1=(1,0,0,…,0).
Then, there exists a constant C>0 such that ∥f∥ℳ0,sp,q≤C, uniformly with respect to λ. Moreover,
(4.42)‖fλ‖M0,sp,q≳λs-d/p,∀0<λ≤1.Proof.
We have(4.43)‖f‖M0,sp,q≍‖Vφf(x,ω)〈ω〉s‖Lp,q=λs‖Vφφ(x,ω-λ-1e1)〈ω〉s‖Lp,q=λs‖Vφφ(x,ω)〈ω+λ-1e1〉s‖Lp,q≲λsλ-s‖Vφφ〈ω〉-s‖Lp,q≲1,
where we have used again the fact that the weight 〈·〉s is 〈·〉-s-moderate. Thus, the functions f have norms in ℳ0,sp,q uniformly bounded with respect to λ. Let us now estimate ∥fλ∥ℳ0,sp,q from below. We have
(4.44)fλ(x)=λsMe1φλ(x).
By using (4.5), we obtain
(4.45)‖fλ‖M0,sp,q=λs‖Vφφλ(x,ω-e1)〈ω〉s‖Lp,q≳λs-d/p(∫e-πq|ω|2〈ω+e1〉qsdω)1/q≳λs-d/p,
as desired.Lemma 4.7.
Let1≤p,q≤∞ be such that (1/p,1/q)∈I2*. Assume that s≤0, ϵ>0 and λ>1.(a)
Ifq≥2 and s≤0, or 1≤q≤2 and s≤-d, define
(4.46)f(x)=∑l≠0|l|d(1/q-1)-ϵe2πiλ-1l⋅xφ(x)=∑l≠0|l|d(1/q-1)-ϵMλ-1lφ(x),inS′(Rd).Then, there exists a constant C>0 such that ∥f∥ℳ0,sp,q≤C, uniformly with respect to λ. Moreover,
(4.47)‖fλ‖M0,sp,q≳λd(1/q-1)-ϵ,∀λ>1.(b)
If1≤q≤2 and -d<s<0, choose a positive integer N such that 1/N<-sq/d, and define
(4.48)f(x)=∑l≠0|l|d(1/Nq-1)-ϵ/Ne2πiλ-Nl⋅xφ(x)=∑l≠0|l|d(1/Nq-1)-ϵ/NMλ-Nlφ(x),inS′(Rd).Then, the conclusions of part (a) still hold.Proof.
(a) First of all, notice that𝒢(φ,1,λ-1) is a frame. In addition, q≥2 is equivalent to 1/q-1≤-1/q. Thus, for all s≤0, {|ℓ|d(1/q-1)-ϵ,ℓ≠0}∈ℓsq, which ensures that the function f defined above belongs to ℳ0,sp,q. This is also true when 1≤q≤2 and s≤-d.
To prove (4.47), we follow the proof of Lemma 4.3. In particular, we have
(4.49)‖fλ‖M0,sp,q≥C∑l≠0|l|d(1/q-1)-ϵ(1+λ2)-d/2e-π|l|2/(λ2+1)≥Cλ-d∑0<|l|≤λ|l|d(1/q-1)-ϵe-π|l|2/(λ2+1),
from which (4.47) follows.
(b) In this case,𝒢(φ,1,λ-N) is a frame. Moreover, the choice of N insures that d(1/(Nq)-1)+s<-d which is enough to prove that f∈ℳ0,sp,q and that ∥f∥ℳ0,sp,q≤C. Relation (4.47) now follows from
(4.50)‖fλ‖M0,sp,q≥C∑l≠0|l|d(1/Nq-1)-ϵ/N(1+λ2)-d/2e-πλ-2N+2|l|2/(λ2+1)≥Cλ-d∑0<|l|≤λN|l|d(1/Nq-1)-ϵ/Ne-πλ-2N+2|l|2/(λ2+1)≥Cλd(1/q-1)-ϵ.The next lemma is proved similarly to Lemma4.4, so we omit its proof.Lemma 4.8.
Let1≤p,q≤∞, s≤0, ϵ>0. Suppose that ψ∈𝒮(ℝd) satisfies suppψ⊂[-1/2,1/2]d and ψ=1 on [-1/4,1/4]d.
(a)
If1≤q<∞, define f(y)=∑k∈ℤd∖{0}|k|-d/q-ϵ-sMkTkψ(y),in𝒮′(ℝd). Then, f∈ℳ0,sp,q(ℝd) and(4.51)‖fλ‖M0,sp,q≳λ-d(2/p-1/q)+ϵ+s,∀0<λ≤1.(b)
Ifq=∞, let
(4.52)f(y)=∑k≠0|k|-se2πikyTkψ(y),inS′(Rd).
Then, f∈ℳ0,sp,∞ and(4.53)‖fλ‖M0,sp,∞≳λ-2d/p+s,∀0<λ≤1.Lemma 4.9.
Let1≤p,q≤∞ be such that (1/p,1/q)∈I3. Let ϵ>0, s≥0 and 0<λ<1. Assume that p>1, and choose a positive integer N such that 1/N<(p-1)/2. Define
(4.54)f(x)=λd/q∑k≠0|k|d(2/Np-1)-ϵ/NTλNkφ(x),inS′(Rd).
Then, there exists a constant C>0 such that ∥f∥ℳ0,sp,q≤C, uniformly with respect to λ. Moreover,
(4.55)‖fλ‖M0,sp,q≳λdμ2(p,q)+ϵ.Proof.
In this case,𝒢(φ,λN,1) is a frame. The condition 1/N<(p-1)/2 is equivalent to 2/Np-1<-1/p which is enough to show that {|k|d(2/Np-1)-ϵ/N}k≠0∈ℓp. Therefore, f∈ℳ0,sp,q with ∥f∥ℳ0,sp,q≤C, where C is a universal constant. The rest of the proof is an adaptation of the proof of Lemma 4.5.Notice that the previous lemma excludes the casep=1. We prove this last case by considering the dual case. Observe that the case (1/∞,1/∞)∈I1*∩I3*  was already considered in dealing with the region I1*.Lemma 4.10.
Let1≤q≤∞ be such that (1/∞,1/q)∈I3*. Let ϵ>0, s≤0 and λ>1.(a)
If1<q<2, choose a positive integer N such that 3/N<q-1. Define
(4.56)f(x)=λd(1-2/q)∑l≠0|l|d(3/Nq-1)-ϵ/NMλ-Nlφ(x),inS′(Rd).Then, there exists a constant C>0 such that ∥f∥ℳ0,sp,q≤C, uniformly with respect to λ. Moreover,
(4.57)‖fλ‖M0,s∞,q≳λd/q-ϵ.(b)
If2≤q<∞, choose a positive integer N such that N>2+q. Define
(4.58)f(x)=λd+d(2-N)/q∑l≠0|l|d((N-1)/Nq-1)-ϵ/NMλ-Nlφ(x),inS′(Rd).Then, the conclusions of part (a) still hold.(c)
Ifq=1 and s≤-d, define(4.59)f(x)=∑l≠0|l|-ϵ/2Mλ-2lφ(x),inS′(Rd).
Then, there exists a constant C>0 such that ∥f∥ℳ0,s∞,1≤C, uniformly with respect to λ. Moreover,
(4.60)‖fλ‖M0,s∞,1≳λd-ϵ.(d)
Ifq=1 and -d<s<0, choose a positive integer N such that 1/N<-s/2d. Define(4.61)f(x)=∑l≠0|l|d(2/N-1)-ϵ/NMλ-Nlφ(x),inS'(Rd).
Then, the conclusions of part (c) still hold.Proof.
(a) In this case,𝒢(φ,1,λ-N) is a frame. The hypotheses 1<q<2 and λ>1 imply that λd(1-2/q)<1. In addition, the condition 3/N<q-1 is equivalent to 3/Nq-1<-1/q which is enough to show that {|ℓ|d(3/Nq-1)-ϵ/N}ℓ≠0∈ℓsq. Therefore, f∈ℳ0,s∞,q with ∥f∥ℳ0,s∞,q≤C, where C is a universal constant. The rest of the proof is an adaptation of the proof of Lemma 4.5.
(b) Assume that2≤q<∞. The proof is similar to the above with the following differences: N>q+2 and λ>1 imply that λd(1+(2-N)/q)<1. In addition, the condition q≥2 implies that (N-1)/Nq-1<-1/q. This is enough to show that {|ℓ|d((N-1)/Nq-1)-ϵ/N}ℓ≠0∈ℓsq. Therefore, f∈ℳ0,s∞,q with ∥f∥ℳ0,s∞,q≤C, where C is a universal constant.
(c) In this case,𝒢(φ,1,λ-2) is a frame. The fact that s≤-d implies that {|ℓ|-ϵ/2}ℓ≠0∈ℓs1. Therefore, f∈ℳ0,s∞,1 with ∥f∥ℳ0,s∞,1≤C, where C is a universal constant. The rest of the proof is an adaptation of the proof of Lemma 4.5.
(d) In this case,𝒢(φ,1,λ-N) is a frame. The fact that -d<s<0 and the choice of N imply that d(2/N-1)+s<-d. Therefore, {|ℓ|d(2/N-1)-ϵ/2}ℓ≠0∈ℓs1. Therefore, f∈ℳ0,s∞,1 with ∥f∥ℳ0,s∞,1≤C, where C is a universal constant. The rest of the proof is an adaptation of the proof of Lemma 4.5.We finish this subsection by proving lower bound estimates for the dilation of functions that are compactly supported either in the time or in the frequency variables.Lemma 4.11.
Letu∈𝒮(ℝd), λ∈(0,∞), and 1≤p,q≤∞.(i)
Ifu is supported in a compact set K⊂ℝd, then, for every t∈ℝ and λ≥1,(4.62)‖uλ‖Mt,0p,q≳λ-d(1-1/q)min⁡{1,λ-t}.(ii)
Ifû is supported in a compact set K⊂ℝd, then, for every s∈ℝ and λ≤1,(4.63)‖uλ‖M0,sp,q≳Cλ-d/pmin⁡{1,λs}.Proof.
We use the dilation properties for the Sobolev spaces (Bessel potential spaces)Hsp(ℝd) (see, e.g., [7, Proposition  3]):
(4.64)C-1λ-d/pmin⁡{1,λs}‖u‖Hsp≤‖uλ‖Hsp≤Cλ-d/pmax⁡{1,λs}‖u‖Hsp,1≤p≤∞,s>0.(i)
Letu be supported in a compact set K⊂ℝd, we have u∈ℳp,q⇔u∈ℱLq, and
(4.65)CK-1‖u‖Mp,q≤‖u‖FLq≤CK‖u‖Mp,q,
where CK>0 depends only on K (see, e.g., [11, 12]). Hence, if λ≥1,
(4.66)‖uλ‖Mt,0p,q≍‖〈⋅〉tuλ‖Mp,q≍‖〈⋅〉tuλ‖FLq≍‖F-1(uλ)‖Htq=λ-d‖(F-1u)λ-1‖Htq≥λ-d(λ-1)-d/qmin⁡{1,λ-t}.(ii)
Now letû be supported in a compact set K⊂ℝd. We have u∈ℳp,q⇔u∈Lp, and
(4.67)CK-1‖u‖Mp,q≤‖u‖Lp≤CK‖u‖Mp,q,
where CK>0 depends only on K (again, see, e.g., [11]). Arguing as in part (i) above with 0<λ≤1,(4.68)‖uλ‖M0,sp,q≍‖〈D〉suλ‖Mp,q≍‖〈D〉suλ‖Lp≍‖uλ‖Hsp≥Cλ-d/pmin⁡{1,λs}‖u‖Hsp
and the proof is completed.


## 4.2. Sharpness of Theorems3.1 and 3.2

We are now in position to state and prove the sharpness of the results obtained in Section3. In particular, Theorem 3.1 is optimal in the following sense.Theorem 4.12.
Let1≤p,q≤∞.
(A)
Ift≥0 then the following statements hold.
Assume that there exist constantsC>0 and α,β∈ℝ such that(4.69)C-1λβ‖f‖Mt,0p,q≤‖Uλf‖Mt,0p,q≤Cλα‖f‖Mt,0p,q,∀f∈Mt,0p,q,λ≥1,
then, α≥dμ1(p,q), and β≤dμ2(p,q)-t.
Assume that there exist constantsC>0 and α,β∈ℝ such that(4.70)C-1λα‖f‖Mt,0p,q≤‖Uλf‖Mt,0p,q≤Cλβ‖f‖Mt,0p,q,∀f∈Mt,0p,q,0<λ≤1,
then, α≥dμ1(p,q), and β≤dμ2(p,q)-t.(B)
Ift≤0 then the following statements hold.
Assume that there exist constantsC>0 and α,β∈ℝ such that(4.71)C-1λβ‖f‖Mt,0p,q≤‖Uλf‖Mt,0p,q≤Cλα‖f‖Mt,0p,q,∀f∈Mt,0p,q,λ≥1,
then, α≥dμ1(p,q)-t, and β≤dμ2(p,q).
Assume that there exist constantsC>0 and α,β∈ℝ such that(4.72)C-1λα‖f‖Mt,0p,q≤‖Uλf‖Mt,0p,q≤Cλβ‖f‖Mt,0p,q,∀f∈Mt,0p,q,0<λ≤1,
then, α≥dμ1(p,q)-t, and β≤dμ2(p,q).Proof.
It will be enough to prove the upper half of each of the estimates, as the lower halves will follow from the fact thatf=UλU1/λf. Moreover, the proof relies on analyzing the examples provided by the previous lemmas and by considering several cases.
Case 1 ((1/p,1/q)∈I2*, t≥0).
In this case, we haveλ≥1 and μ1(p,q)=1/q-1. Substitute f(x)=φ(x)=e-π|x|2 in the upper half estimates (4.69) and use Lemma 4.1 to obtain
(4.73)λ-d(1-1/q)≲‖φλ‖Mt,0p,q≤Cλα‖φ‖Mt,0p,q,
for all λ≥1. This immediately implies that α≥-d(1-1/q)=dμ1(p,q).Case 2 ((1/p,1/q)∈I2, t≤0).
This is the dual case to the previous case and can be handled as follows. In this case, we haveλ≤1 and μ2(p,q)=1/q-1. Assume that the upper-half estimate in (4.72) holds. Notice that (1/p,1/q)∈I2 if and only if (1/p′,1/q′)∈I2* and that λ≤1 if and only if 1/λ≥1.
(4.74)‖f1/λ‖M-t,0p′,q′=sup⁡|〈f1/λ,g〉|=λdsup⁡|〈f,gλ〉|≤λd‖f‖M-t,0p′,q′sup⁡‖gλ‖Mt,0p,q≤λd+β‖f‖M-t,0p′,q′sup⁡‖g‖Mt,0p,q,
where the supremum is taken over all g∈𝒮 and ∥g∥ℳt,0p,q=1; hence,
(4.75)‖f1/λ‖M-t,0p′,q′≤λd+β‖f‖M-t,0p′,q′.
Thus, from Case 1 above, -β-d≥dμ1(p′,q′)=d/q′-d. Hence, β≤dμ2(p,q).Case 3 ((1/p,1/q)∈I3, t≥0).
In this case, we haveλ≤1 and μ2(p,q)=-2/p+1/q. First assume that 1≤q<∞ and that the upper-half estimate in (4.70) holds for all f∈ℳt,0p,q and 0<λ<1 but that β>dμ2(p,q)-t. Then there is ϵ>0 such that β>dμ2(p,q)-t+ϵ. For this choice of ϵ>0, we construct a function f as in (4.22) of Lemma 4.4 such that
(4.76)λdμ2(p,q)-t+ϵ≲‖fλ‖Mt,0p,q≤Cλβ‖f‖Mt,0p,q
for some C>0 and all 0<λ≤1. This leads to a contradiction on the choice of ϵ.
Whenq=∞, the function given by (4.24) of Lemma 4.4 gives the optimal bound.Case 4 ((1/p,1/q)∈I3*, t≤0).
In this case,λ≥1, and μ1(p,q)=-2/p+1/q. This is the dual of Case 3, and a duality argument similar to the used in Case 2 above gives the result.Case 5 ((1/p,1/q)∈I1*,t≤0).
In this case,λ≥1, and μ1(p,q)=-1/p. Assume that the upper-half estimate in (4.71) holds and that α<dμ1(p,q)-t. Then, choose ϵ>0 and construct a function f as in part (b) of Lemma 4.3. A contradiction immediately follows.Case 6 ((1/p,1/q)∈I1, t≥0).
In this case,λ≤1, and μ2(p,q)=-1/p. This is the dual of Case 5.Case 7 ((1/p,1/q)∈I1*, t≥0).
In this case,λ≥1, and μ1(p,q)=-1/p. Assume that the upper-half estimate in (4.69) holds for all f∈ℳt,0p,q and λ>1, but that α<dμ1(p,q). Then, there is ϵ>0 such that α<dμ1(p,q)-ϵ. For this choice of ϵ>0, we can now construct a function f as in Lemma 4.3, part (a), such that
(4.77)λdμ1(p,q)-ϵ≲‖fλ‖Mt,0p,q≤Cλα‖f‖Mt,0p,q
for some C>0 and all λ≥1. This leads to a contradiction on the choice of ϵ.Case 8 ((1/p,1/q)∈I1, t≤0).
In this case,λ≤1, and μ2(p,q)=-1/p. This is the dual of Case 7.Case 9 ((1/p,1/q)∈I2*, t≤0).
In this case,λ≥1, and μ1(p,q)=1/q-1. The function constructed in Lemma 4.2 leads to the result.Case 10 ((1/p,1/q)∈I2, t≥0).
In this case,λ≤1, and μ2(p,q)=1/q-1. This is the dual of Case 9.Case 11 ((1/p,1/q)∈I3, t≤0).
In this case,λ≤1, and μ1(p,q)=-2/p+1/q, and Lemma 4.5 can be used to conclude.Case 12 ((1/p,1/q)∈I3*,t≥0).
In this case,λ≥1, and μ2(p,q)=-2/p+1/q. This is the dual of Case 11.We next consider the sharpness of Theorem3.2.Theorem 4.13.
Let1≤p,q≤∞.(A)
Ifs≥0 then the following statements hold.Assume that there exist constants C>0 and α,β∈ℝ such that
(4.78)C-1λβ‖f‖M0,sp,q≤‖Uλf‖M0,sp,q≤Cλα‖f‖M0,sp,q,∀f∈M0,sp,q,λ≥1,
then, α≥dμ1(p,q)+s, and β≤dμ2(p,q).Assume that there exist constants C>0 and α,β∈ℝ such that
(4.79)C-1λα‖f‖M0,sp,q≤‖Uλf‖M0,sp,q≤Cλβ‖f‖M0,sp,q,∀f∈M0,sp,q,0<λ≤1,
then, α≥dμ1(p,q)+s, and β≤dμ2(p,q).(B)
Ifs≤0 then the following statements hold.Assume that there exist constants C>0 and α,β∈ℝ such that
(4.80)C-1λβ‖f‖M0,sp,q≤‖Uλf‖M0,sp,q≤Cλα‖f‖M0,sp,q,∀f∈M0,sp,q,λ≥1,
then, α≥dμ1(p,q), and β≤dμ2(p,q)+s.Assume that there exist constants C>0 and α,β∈ℝ such that
(4.81)C-1λα‖f‖M0,sp,q≤‖Uλf‖M0,sp,q≤Cλβ‖f‖M0,sp,q,∀f∈M0,sp,q,0<λ≤1,
then, α≥dμ1(p,q), and β≤dμ2(p,q)+s.Proof.
As for the time weights, it is enough to prove the upper half of each estimates. Moreover, in what follows we consider only 6 of the 12 cases to be proved, since the others are obtained by the same duality argument used in the previous theorem.
Case 1 ((1/p,1/q)∈I1, s≥0).
In this case,0<λ≤1 and μ2(p,q)=-1/p. Assume there exist constants C>0 and β∈ℝ such that the upper-half estimate (4.79) holds. Taking the Gaussian f=φ as in Lemma 4.1 and using (4.3), we have
(4.82)λ-d/p≲‖φλ‖M0,sp,q≲λβ‖φ‖M0,sp,q,
for all 0<λ≤1. This gives β≤-d/p.Case 2 ((1/p,1/q)∈I1, s≤0).
Here,λ≤1, and we test the upper-half estimate (4.81) on the family of functions (4.41). Using (4.42), we obtain β≤s-d/p.Case 3 ((1/p,1/q)∈I2*, s≥0).
Hereλ≥1, μ1(p,q)=1/q-1. We assume the upper-half estimate (4.78) and test it on the dilated Gaussian function in (4.4), obtaining α≥d(1/q-1)+s.Case 4 ((1/p,1/q)∈I2*, s≤0).
Here,λ≥1, μ1(p,q)=1/q-1. We use a contradiction argument based on Lemma 4.7.Case 5 ((1/p,1/q)∈I3,s≥0).
Here,λ≤1, μ2(p,q)=-2/p+1/q. The sharpness is obtained by testing the upper-half estimate (4.79) on the family of functions fλ, defined in Lemma 4.9 when p>1.
Ifp=1, we consider the dual case, that is (1/∞,1/q)∈I3*, s≤0. Here λ≥1, μ1(∞,q)=1/q. We use a contradiction argument based on Lemma 4.10.Case 6 ((1/p,1/q)∈I3, s≤0).
Here,λ≤1, μ2(p,q)=-2/p+1/q. The sharpness is obtained by testing the upper-half estimate (4.81) on the family of functions fλ, defined in Lemma 4.8.


## 5. Applications

### 5.1. Applications to Dispersive Equations

#### 5.1.1. Wave Equation

Let us first recall the Cauchy problem for the wave equation:(5.1)∂t2u-Δxu=0u(0,x)=u0(x),∂tu(0,x)=u1(x),
with t≥0, x∈ℝd, d≥1, Δx=∂x12+⋯∂xd2. The formal solution u(t,x) is given by (5.2)u(t,x)=∫Rde2πixξcos⁡(2πt|ξ|)u0̂(ξ)dξ+∫Rde2πixξsin⁡(2πt|ξ|)2π|ξ|u1̂(ξ)dξ,=Hσ0u0(x)+Hσ1(x)
with, σ0(ξ)=cos(2πt|ξ|) and σ1(ξ)=sin(2πt|ξ|)/2π|ξ|.We recall thatHσii=0,1, are examples of Fourier multipliers which are defined by(5.3)Hσf(x)=∫Rde2πixξσ(ξ)f̂(ξ)dξ,
where σ is called the symbol.The boundedness ofHσi,i=0,1 on modulation spaces was proved in [13, 14] and in [15]. Moreover, some related local-in-time well-posedness results for certain nonlinear PDEs were also obtained in [14, 15] for initial data in modulation spaces.Proposition 5.1.
Lets∈ℝ, and 1≤p,q≤∞. Then, the solution u(t,x) of (5.1) with initial data (u0,u1)∈ℳ0,sp,q×ℳ0,s-1p,q satisfies
(5.4)‖u(t,⋅)‖M0,sp,q≤C0(1+t)d+1‖u0‖M0,sp,q+C1t(1+t)d+1‖u1‖M0,s-1p,q,
where C0 and C1 are only functions of the dimension d.Proof.
It was proved in [13] that σ0(ξ)∈W(ℱL1,L∞) and in [15] that σ1(ξ)∈W(ℱL1,L1∞). In addition, it was shown in [15] that the solution satisfies
(5.5)‖u(t,⋅)‖M0,sp,q≤‖Hσ0u0‖M0,sp,q+‖Hσ1u1‖M0,sp,q≤‖σ0‖W(FL1,L∞)‖u0‖M0,sp,q+‖σ1‖W(FL1,L1∞)‖u1‖M0,s-1p,q≤C0(t)‖u0‖M0,sp,q+C1(t)‖u1‖M0,s-1p,q.
We can now use the results proved in Section 3 to estimate C0(t) and C1(t). More specifically, setting σ0̃(ξ)=cos|ξ|, for t>0, we can write σ0(ξ)=(σ0̃)2πt. Using (3.10) with μ1(∞,1)=1, μ2(∞,1)=0, we have, for every R>0,
(5.6)‖(σ0̃)2πt‖W(FL1,L∞)≤{C0,R‖σ0̃‖W(FL1,L∞),t≤RC0,R′td+1‖σ0̃‖W(FL1,L∞),t≥R.
Hence,
(5.7)C0(t)≤{C0,R,0≤t≤RC0,R′td+1,t≥R.
Setting σ1̃(ξ)=sin|ξ|/|ξ|, for t>0, we can write σ1(ξ)=t(σ1̃)2πt and, for every R>0,
(5.8)‖(σ1̃)2πt‖W(FL1,L1∞)≤{C1,R‖σ1̃‖W(FL1,L1∞),t≤RC1,R′td+1‖σ1̃‖W(FL1,L1∞),t≥R.
Hence,
(5.9)C1(t)≤{C1,Rt,0≤t≤RC1,R′td+2,t≥R,
and the estimate (5.4) becomes
(5.10)‖u(t,⋅)‖M0,sp,q≤C0(1+t)d+1‖u0‖M0,sp,q+C1t(1+t)d+1‖u1‖M0,s-1p,q,t>0.


#### 5.1.2. Vibrating Plate Equation

Consider now the following Cauchy problem for the vibrating plate equation(5.11)∂t2u+Δx2u=0,u(0,x)=u0(x),∂tu(0,x)=u1(x),
with t≥0, x∈ℝd, d≥1. The formal solution u(t,x) is given by (5.12)u(t,x)=∫Rde2πixξcos⁡(4π2t|ξ|2)u0̂(ξ)dξ+∫Rde2πixξsin⁡(4π2t|ξ|2)4π2|ξ|2u1̂(ξ)dξ,
and satisfies the following estimate.Proposition 5.2.
Lets∈ℝ, and 1≤p,q≤∞. Then, the solution u(t,x) of (5.11) with initial data (u0,u1)∈ℳ0,sp,q×ℳ0,s-2p,q satisfies
(5.13)‖u(t,⋅)‖M0,sp,q≤C0(1+t)d/2‖u0‖M0,sp,q+C1t(1+t)d/2+1‖u1‖M0,s-2p,q,
where C0 and C1 are only functions of the dimension d.Proof.
Here the solution is the sum of two Fourier multipliersu=H0u0+H1u1 having symbols σ0(ξ)=cos(4π2t|ξ|2)∈W(ℱL1,L∞) (see [13]) and σ1(ξ)=sin(4π2t|ξ|2)/4π2|ξ|2∈W(ℱL1,L2∞) (see [16]).
Sinceσ0(ξ)=cos(|ξ|2)2πt and σ1(ξ)=t(sin(|ξ|2)/|ξ|2)2πt, using the same arguments as for the wave equation, we obtain
(5.14)‖u(t,⋅)‖M0,sp,q≤C0(1+t)d/2‖u0‖M0,sp,q+C1t(1+t)d/2+1‖u1‖M0,s-2p,q,t>0.


### 5.2. Embedding of Besov Spaces into Modulation Spaces

We generalize some results of [4]. But first, we recall the inclusion relations between Besov spaces and modulation spaces (see [5, 17]). Consider the following indices, where μi, i=1,2 were defined in Section 2: (5.15)ν1(p,q)=μ1(p,q)+1p,ν2(p,q)=μ2(p,q)+1p.The following result was proved in [6, Theorem  3.1] and in [17, Theorem  1.1].Theorem 5.3.
Let1≤p,q≤∞ and s∈ℝ.(i)
Ifs≥dν1(p,q), then Bsp,q(ℝd)↪ℳp,q(ℝd).(ii)
Ifs≤dν2(p,q), then ℳp,q(ℝd)↪Bsp,q(ℝd).The next results improve those in [4, Theorem  3.1].Theorem 5.4.
Let1≤p≤2.(i)
Ifs≥d(1/p-1/p′) and 1≤q≤p, then Bsp,q↪ℳp.(ii)
Ifs>d(1/p-1/p′) and 1≤q≤∞, then Bsp,q↪ℳp.Proof.
(i) Fors≥d(1/p-1/p′)=dν1(p,p), Theorem 5.3 says that Bsp,p↪ℳp,p. However, the inclusion relations for Besov spaces give Bsp,q↪Bsp,p, for q≤p. Hence the result follows.
(ii) Ifs>d(1/p-1/p′)≥0, and q≤p, then this is exactly as (i) above. If p≤q, then there exists an ϵ>0 such that s=d(1/p-1/p′)+ϵ and
(5.16)Bsp,q=Bd(1/p-1/p′)+ϵp,q↪Bd(1/p-1/p′)p,p↪Mp,
where the last inclusion follows from (i).The next results improve those in [4, Theorem  3.2].Theorem 5.5.
(i) Let1≤p≤2, s>0. Then, Bsp,q↪ℳp,p′, for all 1≤q≤∞.
(ii) If2≤p≤∞, s>d(1/p′-1/p), then Bsp,q↪ℳp,p′, for all 1≤q≤∞.Proof.
(i) For1≤p≤2, ν1(p,p′)=0 and using Theorem 5.3, we obtain Bp,p′↪ℳp,p′. Since Bsp,q↪Bp,p′, for all 1≤q≤∞, s>0, the result follows.
(ii) If2≤p≤∞,
(5.17)ν1(p,p′)=1p′-1p≤1p′.
Hence, if s≥d(1/p′-1/p), Theorem 5.3 gives Bsp,p′↪ℳp,p′. If s>d(1/p′-1/p), the inclusion relations for Besov spaces give Bsp,q↪Bd(1/p′-1/p)p,p′. This is easy to see if q≤p′. On the other hand, if q>p′, it follows by an application of Hölder's inequality for ℓp spaces. In any case, this concludes the proof.


## 5.1. Applications to Dispersive Equations

### 5.1.1. Wave Equation

Let us first recall the Cauchy problem for the wave equation:(5.1)∂t2u-Δxu=0u(0,x)=u0(x),∂tu(0,x)=u1(x),
with t≥0, x∈ℝd, d≥1, Δx=∂x12+⋯∂xd2. The formal solution u(t,x) is given by (5.2)u(t,x)=∫Rde2πixξcos⁡(2πt|ξ|)u0̂(ξ)dξ+∫Rde2πixξsin⁡(2πt|ξ|)2π|ξ|u1̂(ξ)dξ,=Hσ0u0(x)+Hσ1(x)
with, σ0(ξ)=cos(2πt|ξ|) and σ1(ξ)=sin(2πt|ξ|)/2π|ξ|.We recall thatHσii=0,1, are examples of Fourier multipliers which are defined by(5.3)Hσf(x)=∫Rde2πixξσ(ξ)f̂(ξ)dξ,
where σ is called the symbol.The boundedness ofHσi,i=0,1 on modulation spaces was proved in [13, 14] and in [15]. Moreover, some related local-in-time well-posedness results for certain nonlinear PDEs were also obtained in [14, 15] for initial data in modulation spaces.Proposition 5.1.
Lets∈ℝ, and 1≤p,q≤∞. Then, the solution u(t,x) of (5.1) with initial data (u0,u1)∈ℳ0,sp,q×ℳ0,s-1p,q satisfies
(5.4)‖u(t,⋅)‖M0,sp,q≤C0(1+t)d+1‖u0‖M0,sp,q+C1t(1+t)d+1‖u1‖M0,s-1p,q,
where C0 and C1 are only functions of the dimension d.Proof.
It was proved in [13] that σ0(ξ)∈W(ℱL1,L∞) and in [15] that σ1(ξ)∈W(ℱL1,L1∞). In addition, it was shown in [15] that the solution satisfies
(5.5)‖u(t,⋅)‖M0,sp,q≤‖Hσ0u0‖M0,sp,q+‖Hσ1u1‖M0,sp,q≤‖σ0‖W(FL1,L∞)‖u0‖M0,sp,q+‖σ1‖W(FL1,L1∞)‖u1‖M0,s-1p,q≤C0(t)‖u0‖M0,sp,q+C1(t)‖u1‖M0,s-1p,q.
We can now use the results proved in Section 3 to estimate C0(t) and C1(t). More specifically, setting σ0̃(ξ)=cos|ξ|, for t>0, we can write σ0(ξ)=(σ0̃)2πt. Using (3.10) with μ1(∞,1)=1, μ2(∞,1)=0, we have, for every R>0,
(5.6)‖(σ0̃)2πt‖W(FL1,L∞)≤{C0,R‖σ0̃‖W(FL1,L∞),t≤RC0,R′td+1‖σ0̃‖W(FL1,L∞),t≥R.
Hence,
(5.7)C0(t)≤{C0,R,0≤t≤RC0,R′td+1,t≥R.
Setting σ1̃(ξ)=sin|ξ|/|ξ|, for t>0, we can write σ1(ξ)=t(σ1̃)2πt and, for every R>0,
(5.8)‖(σ1̃)2πt‖W(FL1,L1∞)≤{C1,R‖σ1̃‖W(FL1,L1∞),t≤RC1,R′td+1‖σ1̃‖W(FL1,L1∞),t≥R.
Hence,
(5.9)C1(t)≤{C1,Rt,0≤t≤RC1,R′td+2,t≥R,
and the estimate (5.4) becomes
(5.10)‖u(t,⋅)‖M0,sp,q≤C0(1+t)d+1‖u0‖M0,sp,q+C1t(1+t)d+1‖u1‖M0,s-1p,q,t>0.


### 5.1.2. Vibrating Plate Equation

Consider now the following Cauchy problem for the vibrating plate equation(5.11)∂t2u+Δx2u=0,u(0,x)=u0(x),∂tu(0,x)=u1(x),
with t≥0, x∈ℝd, d≥1. The formal solution u(t,x) is given by (5.12)u(t,x)=∫Rde2πixξcos⁡(4π2t|ξ|2)u0̂(ξ)dξ+∫Rde2πixξsin⁡(4π2t|ξ|2)4π2|ξ|2u1̂(ξ)dξ,
and satisfies the following estimate.Proposition 5.2.
Lets∈ℝ, and 1≤p,q≤∞. Then, the solution u(t,x) of (5.11) with initial data (u0,u1)∈ℳ0,sp,q×ℳ0,s-2p,q satisfies
(5.13)‖u(t,⋅)‖M0,sp,q≤C0(1+t)d/2‖u0‖M0,sp,q+C1t(1+t)d/2+1‖u1‖M0,s-2p,q,
where C0 and C1 are only functions of the dimension d.Proof.
Here the solution is the sum of two Fourier multipliersu=H0u0+H1u1 having symbols σ0(ξ)=cos(4π2t|ξ|2)∈W(ℱL1,L∞) (see [13]) and σ1(ξ)=sin(4π2t|ξ|2)/4π2|ξ|2∈W(ℱL1,L2∞) (see [16]).
Sinceσ0(ξ)=cos(|ξ|2)2πt and σ1(ξ)=t(sin(|ξ|2)/|ξ|2)2πt, using the same arguments as for the wave equation, we obtain
(5.14)‖u(t,⋅)‖M0,sp,q≤C0(1+t)d/2‖u0‖M0,sp,q+C1t(1+t)d/2+1‖u1‖M0,s-2p,q,t>0.


## 5.1.1. Wave Equation

Let us first recall the Cauchy problem for the wave equation:(5.1)∂t2u-Δxu=0u(0,x)=u0(x),∂tu(0,x)=u1(x),
with t≥0, x∈ℝd, d≥1, Δx=∂x12+⋯∂xd2. The formal solution u(t,x) is given by (5.2)u(t,x)=∫Rde2πixξcos⁡(2πt|ξ|)u0̂(ξ)dξ+∫Rde2πixξsin⁡(2πt|ξ|)2π|ξ|u1̂(ξ)dξ,=Hσ0u0(x)+Hσ1(x)
with, σ0(ξ)=cos(2πt|ξ|) and σ1(ξ)=sin(2πt|ξ|)/2π|ξ|.We recall thatHσii=0,1, are examples of Fourier multipliers which are defined by(5.3)Hσf(x)=∫Rde2πixξσ(ξ)f̂(ξ)dξ,
where σ is called the symbol.The boundedness ofHσi,i=0,1 on modulation spaces was proved in [13, 14] and in [15]. Moreover, some related local-in-time well-posedness results for certain nonlinear PDEs were also obtained in [14, 15] for initial data in modulation spaces.Proposition 5.1.
Lets∈ℝ, and 1≤p,q≤∞. Then, the solution u(t,x) of (5.1) with initial data (u0,u1)∈ℳ0,sp,q×ℳ0,s-1p,q satisfies
(5.4)‖u(t,⋅)‖M0,sp,q≤C0(1+t)d+1‖u0‖M0,sp,q+C1t(1+t)d+1‖u1‖M0,s-1p,q,
where C0 and C1 are only functions of the dimension d.Proof.
It was proved in [13] that σ0(ξ)∈W(ℱL1,L∞) and in [15] that σ1(ξ)∈W(ℱL1,L1∞). In addition, it was shown in [15] that the solution satisfies
(5.5)‖u(t,⋅)‖M0,sp,q≤‖Hσ0u0‖M0,sp,q+‖Hσ1u1‖M0,sp,q≤‖σ0‖W(FL1,L∞)‖u0‖M0,sp,q+‖σ1‖W(FL1,L1∞)‖u1‖M0,s-1p,q≤C0(t)‖u0‖M0,sp,q+C1(t)‖u1‖M0,s-1p,q.
We can now use the results proved in Section 3 to estimate C0(t) and C1(t). More specifically, setting σ0̃(ξ)=cos|ξ|, for t>0, we can write σ0(ξ)=(σ0̃)2πt. Using (3.10) with μ1(∞,1)=1, μ2(∞,1)=0, we have, for every R>0,
(5.6)‖(σ0̃)2πt‖W(FL1,L∞)≤{C0,R‖σ0̃‖W(FL1,L∞),t≤RC0,R′td+1‖σ0̃‖W(FL1,L∞),t≥R.
Hence,
(5.7)C0(t)≤{C0,R,0≤t≤RC0,R′td+1,t≥R.
Setting σ1̃(ξ)=sin|ξ|/|ξ|, for t>0, we can write σ1(ξ)=t(σ1̃)2πt and, for every R>0,
(5.8)‖(σ1̃)2πt‖W(FL1,L1∞)≤{C1,R‖σ1̃‖W(FL1,L1∞),t≤RC1,R′td+1‖σ1̃‖W(FL1,L1∞),t≥R.
Hence,
(5.9)C1(t)≤{C1,Rt,0≤t≤RC1,R′td+2,t≥R,
and the estimate (5.4) becomes
(5.10)‖u(t,⋅)‖M0,sp,q≤C0(1+t)d+1‖u0‖M0,sp,q+C1t(1+t)d+1‖u1‖M0,s-1p,q,t>0.


## 5.1.2. Vibrating Plate Equation

Consider now the following Cauchy problem for the vibrating plate equation(5.11)∂t2u+Δx2u=0,u(0,x)=u0(x),∂tu(0,x)=u1(x),
with t≥0, x∈ℝd, d≥1. The formal solution u(t,x) is given by (5.12)u(t,x)=∫Rde2πixξcos⁡(4π2t|ξ|2)u0̂(ξ)dξ+∫Rde2πixξsin⁡(4π2t|ξ|2)4π2|ξ|2u1̂(ξ)dξ,
and satisfies the following estimate.Proposition 5.2.
Lets∈ℝ, and 1≤p,q≤∞. Then, the solution u(t,x) of (5.11) with initial data (u0,u1)∈ℳ0,sp,q×ℳ0,s-2p,q satisfies
(5.13)‖u(t,⋅)‖M0,sp,q≤C0(1+t)d/2‖u0‖M0,sp,q+C1t(1+t)d/2+1‖u1‖M0,s-2p,q,
where C0 and C1 are only functions of the dimension d.Proof.
Here the solution is the sum of two Fourier multipliersu=H0u0+H1u1 having symbols σ0(ξ)=cos(4π2t|ξ|2)∈W(ℱL1,L∞) (see [13]) and σ1(ξ)=sin(4π2t|ξ|2)/4π2|ξ|2∈W(ℱL1,L2∞) (see [16]).
Sinceσ0(ξ)=cos(|ξ|2)2πt and σ1(ξ)=t(sin(|ξ|2)/|ξ|2)2πt, using the same arguments as for the wave equation, we obtain
(5.14)‖u(t,⋅)‖M0,sp,q≤C0(1+t)d/2‖u0‖M0,sp,q+C1t(1+t)d/2+1‖u1‖M0,s-2p,q,t>0.


## 5.2. Embedding of Besov Spaces into Modulation Spaces

We generalize some results of [4]. But first, we recall the inclusion relations between Besov spaces and modulation spaces (see [5, 17]). Consider the following indices, where μi, i=1,2 were defined in Section 2: (5.15)ν1(p,q)=μ1(p,q)+1p,ν2(p,q)=μ2(p,q)+1p.The following result was proved in [6, Theorem  3.1] and in [17, Theorem  1.1].Theorem 5.3.
Let1≤p,q≤∞ and s∈ℝ.(i)
Ifs≥dν1(p,q), then Bsp,q(ℝd)↪ℳp,q(ℝd).(ii)
Ifs≤dν2(p,q), then ℳp,q(ℝd)↪Bsp,q(ℝd).The next results improve those in [4, Theorem  3.1].Theorem 5.4.
Let1≤p≤2.(i)
Ifs≥d(1/p-1/p′) and 1≤q≤p, then Bsp,q↪ℳp.(ii)
Ifs>d(1/p-1/p′) and 1≤q≤∞, then Bsp,q↪ℳp.Proof.
(i) Fors≥d(1/p-1/p′)=dν1(p,p), Theorem 5.3 says that Bsp,p↪ℳp,p. However, the inclusion relations for Besov spaces give Bsp,q↪Bsp,p, for q≤p. Hence the result follows.
(ii) Ifs>d(1/p-1/p′)≥0, and q≤p, then this is exactly as (i) above. If p≤q, then there exists an ϵ>0 such that s=d(1/p-1/p′)+ϵ and
(5.16)Bsp,q=Bd(1/p-1/p′)+ϵp,q↪Bd(1/p-1/p′)p,p↪Mp,
where the last inclusion follows from (i).The next results improve those in [4, Theorem  3.2].Theorem 5.5.
(i) Let1≤p≤2, s>0. Then, Bsp,q↪ℳp,p′, for all 1≤q≤∞.
(ii) If2≤p≤∞, s>d(1/p′-1/p), then Bsp,q↪ℳp,p′, for all 1≤q≤∞.Proof.
(i) For1≤p≤2, ν1(p,p′)=0 and using Theorem 5.3, we obtain Bp,p′↪ℳp,p′. Since Bsp,q↪Bp,p′, for all 1≤q≤∞, s>0, the result follows.
(ii) If2≤p≤∞,
(5.17)ν1(p,p′)=1p′-1p≤1p′.
Hence, if s≥d(1/p′-1/p), Theorem 5.3 gives Bsp,p′↪ℳp,p′. If s>d(1/p′-1/p), the inclusion relations for Besov spaces give Bsp,q↪Bd(1/p′-1/p)p,p′. This is easy to see if q≤p′. On the other hand, if q>p′, it follows by an application of Hölder's inequality for ℓp spaces. In any case, this concludes the proof.



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*Source: 145491-2012-01-11.xml*