Dilation properties for weighted modulation spaces

In this paper we give a sharp estimate on the norm of the scaling operator $U_{\lambda}f(x)=f(\lambda x)$ acting on the weighted modulation spaces $\M{p,q}{s,t}(\R^{d})$. 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 posedeness of the Cauchy problem for these equations. Finally, we provide new embedding results between modulation and Besov spaces.


Introduction
The modulation spaces were introduced by H. Feichtinger [7], by imposing integrability conditions on the short-time Fourier transform (STFT) of tempered distributions. More specifically, for x, ω ∈ R d , we let M ω and T x denote the operators of modulation and translation. Then, the STFT of f with respect to a nonzero window g in the Schwartz class is f (t)g(t − x)e −2πit·ω dt. V g f (x, ω) measures the frequency content of f in a neighborhood of x.
For s 1 , s 2 ∈ R and 1 ≤ p, q ≤ ∞, the weighted modulation space M p,q s 1 ,s 2 (R 2d ) is defined to be the Banach space of all tempered distributions f such that (1.1) f M p,q s 1 ,s 2 = Here and in the sequel, we use the notation v s (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 < ∞, (M p,q s,t ) ′ = M p ′ ,q ′ −s,−t , where p ′ , q ′ denote the dual exponents of p and q, respectively.
When both s = t = 0, we will simply write M p,q = M p,q 0,0 . The weighted L 2 s space is exactly M 2,2 s,0 , while an application of Plancherel's identity shows that the Sobolev space H s 2 coincides with M 2,2 0,s . For further properties and uses of modulation spaces, see Gröchenig's book [9], and we refer to [17] 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 have been investigated and resulted in embedding results of modulation spaces into other function spaces such as the Besov and Sobolev spaces [10,14,15]. Sugimoto and Tomita [14] proved the optimality of certain of the embeddings of modulation spaces into Besov space obtained in [10,15]. These results were obtained as consequence to optimal bounds of U λ M p,q →M p,q [14,Theorem 3.1], where U λ f (·) = f (λ·) for λ > 0.
The operator U λ 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 U λ on the Besov spaces [12,Proposition 3]: The estimate on the norm of U λ on the (unweighted) modulation spaces M p,q (R d ) was first obtained by Sugimoto and Tomita [14]. In this paper, we shall derive optimal lower and upper bounds for the operator U λ on general modulation spaces M p,q t,s (R d ). More specifically, the boundedness of U λ on M p,q t,s is proved in Theorems 3.1, 3.2 and 3.4, and the optimal bounds on U λ M p,q t,s →M p,q t,s 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 U λ , 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 are used, e.,g., 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 [10]. 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 Section 2 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 shall use the notations A B to mean that there exists a constant c > 0 such that A ≤ cB, and A ≍ B means that A B A.
We introduce the indices: if (1/p, 1/q) ∈ I 3 . Next, we prove a lemma that will be used throughout this paper, and which allows us to investigate the action of U λ only on S(R d ). Then Consider now the case p = ∞ or q = ∞. For any given f ∈ M p,q m , consider a sequence f n of Schwartz functions, with f n → f in S ′ (R d ), and (see the proof of Proposition 11.3.4 of [9]). Since f n tends to f in S ′ (R d ), Af n tends to Af in S ′ (R d ), and V ϕ Af n tends to V ϕ Af pointwise. Hence, by Fatou's Lemma, the assumption (2.1) and (2.3), We shall 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 by ϕ(x) = e −π|x| 2 , x ∈ R d . Recall that for 0 < a < 1, the family, Moreover, there exists a dual functionφ ∈ S such that G(φ, a, 1) is also a frame for L 2 and every f ∈ L 2 can be written as It is easy to see from the isometry of the Fourier transform on L 2 and the fact that M ℓ T ak ϕ = T ℓ M −akφ = e 2πiakℓ M −ak T ℓ ϕ, that G(ϕ, 1, a) is a Gabor frame whenever G(ϕ, a, 1) is. The characterization of the modulation spaces by Gabor frame is summarized in the following proposition. We refer to [9, Chapter 9] for a detail treatment of Gabor frames in the context of the modulation spaces. In particular, the next result is proved in [9,Theorem 7.5.3] and describe precisely when the Gaussian function generates a Gabor frame on L 2 . Proposition 2.2. G(ϕ, a, 1) is a Gabor frame for L 2 if and only if 0 < a < 1. In this case, G(ϕ, a, 1) is also a Banach frame for M p,q t,s for all 1 ≤ p, q ≤ ∞, and s, t ∈ R. Moreover, f ∈ M p,q t,s if and only if there exists a sequence {c k,ℓ } k,ℓ∈Z d ∈ ℓ p,q t,s (Z d × Z d ) such that f = k,ℓ∈Z d c k,ℓ ϕ k,ℓ with convergence in the modulation space norm. In addition,

Dilation properties of weighted modulation spaces
We first consider the polynomial weights in the time variables v t (x) = x t = (1 + |x| 2 ) t/2 , t ∈ R.
Theorem 3.1. Let 1 ≤ p, q ≤ ∞, t ∈ R. Then the following are true: (1) There exists a constant C > 0 such that ∀f ∈ M p,q t,0 , λ ≥ 1, We shall 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 We first consider the case λ ≥ 1. Recall the definition of the dilation operator U λ given by U λ f (x) = f (λx). Since the mapping f → · t f is an homeomorphism from M p,q t 0 ,s to M p,q t 0 −t,s , t, t 0 , s ∈ R, see, e.g., [16, Corollary 2.3], we have: Hence, it remains to prove that the pseudodifferential operator with symbol g (t,λ) (x) := x −t λ −1 x t is bounded on M p,q , and that its norm is bounded above by max{1, λ −t }. By [9,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 Indeed, let v (t,λ) (x) = λ −1 x t . Consider the case t ≥ 0. Since λ ≥ 1, we have λ −1 |x| ≤ |x| and v (t,λ) (x) ≤ x t . Analogously, for t < 0, we have v (t,λ) (x) ≤ λ −t x t . Consequently, we get the desired estimates (3.3).
By Leibniz' formula, the estimate |∂ β x t | x t−|β| and (3.3) 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 case 0 < λ ≤ 1. Observe that by [14] we have Moreover, one easily shows that (3.3) still holds using the same arguments along with the fact that v (t,λ) ( Hence, by the proof of (3.1) and [14, Theorem 3.1], we see that This establishes the upper half of (3.2).
We now consider the polynomial weights in the frequency variables v s (ω) = ω s , s ∈ R.
Then the following are true: Here we use the fact that the mapping f → D s f is an homeomorphism from M p,q t,s 0 to M p,q t,s 0 −s , t, s, s 0 ∈ R (see [16,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.
Then the following are true: (1) There exists a constant C > 0 such that ∀f ∈ M p,q vs , λ ≥ 1, We assume s ≥ 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) ∈ I 2 , t ≥ 0).
Moreover, since the result has been proved in [14, 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 mapping f → x, D s f is an homeomorphism from M p,q vs to M p,q , s ∈ R (see [16,Theorem 2.2]). Hence where in the last inequality we used again the dilation properties for unweighted modulation spaces of [14,Theorem 3.1]. Therefore, writing f = x, D −s x, D s f we see that it suffices to prove that the pseudodifferential operator is bounded on M p,q , and its norm is bounded above by max{1, λ −s } max{1, λ s } = max{λ s , λ −s }. To this end, we observe that, if s is an even integer, λ −1 x, λD s is a finite sum of operators of the form λ k x α D β , with |k| ≤ s and |α| + |β| ≤ s. Now, Shubin's pseudo-differential calculus [13] shows that the operators x α D β x, D −s have bounded symbols, together with all their derivatives, so that they are bounded on M p,q . The proof is completed by taking into account the additional factor λ k .
Finally, it is relatively straightforward to give optimal estimates for the dilation operator U λ on the Wiener amalgam spaces W (F L p s , L q t ). These spaces are images of modulation spaces under Fourier transform, that is F M p,q t,s = W (F L p s , L q t ). It is also worth noticing that the indices µ 1 and µ 2 obey the following relations, Using the above relations along with the definition of the Wiener amalgam spaces, as well as the behavior of the Fourier transform under dilation, i.e., f λ = λ −d (f ) 1 λ and Corollary 3.3 we obtain the following result Proposition 3.5. Let 1 ≤ p, q ≤ ∞, t, s ∈ R. Then the following are true:

Sharpness of Theorems 3.1 and 3.2.
In this section we prove the sharpness of Theorems 3.1 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. Recall that ϕ(x) = e −π|x| 2 for x ∈ R d , and that ϕ λ (x) = U λ ϕ(x) = ϕ(λx).
Proof. We shall only prove the first two estimates, as the last two are proved similarly. By some straightforward computations, (see, e.g., [9, Lemma 1.5.2]) we get If 0 < λ ≤ 1, then Then there exists a constant C > 0 such that f M p,q t,0 ≤ C, uniformly with respect to λ. Moreover, Proof. We have The last inequality follows from the fact that the weight · t is · −t -moderate which implies that x + λe 1 t λ t x −t . This proves the first part of the Lemma. Let us now estimate f λ M p,q t,0 from below. We have Hence, by arguing as above and using (4.5), we have which concludes the proof.
Lemma 4.3. Let 1 ≤ p, q ≤ ∞, ǫ > 0, t ∈ R, and λ > 1. Moreover, assume that Then there exists a constant C > 0 such that f M p,q t,0 ≤ C, uniformly with respect to λ. Moreover, Then there exists a constant C > 0 such that f M p,q t,0 ≤ C, uniformly with respect to λ. Moreover, Proof. We only prove part a) as part b) is obtained similarly. We use Proposition 2.2 to prove that f defined in the lemma belongs to M p,q t,0 . Indeed, G(ϕ, 1, λ −1 ) is a Gabor frame, and the coefficients of f in this frame are given by c k,ℓ = δ k,0 |ℓ| −d/p−ǫ if ℓ = 0 and c 0,0 = 0. It is clear that because q/p ≥ 1. Thus, f ∈ M p,q t,0 with uniform norm (with respect to λ). Given λ > 1, we have Using relation (4.5), Therefore, if λ > 1, from which the proof follows. .
Then, f ∈ M p,q t,0 (R d ) and Then f ∈ M p,∞ t,0 and (4.13) Proof. We only prove part a), i.e., the case 1 ≤ q < ∞ as the case q = ∞ is proved in a similar fashion. Let g ∈ S(R d ) satisfy supp g ⊂ [−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 V g f .
Lemma 4.5. Let 1 ≤ p, q ≤ ∞ be such that (1/p, 1/q) ∈ I 3 . Let ǫ > 0, t < 0, and Then there exists a constant C > 0 such that f M p,q t,0 ≤ C, uniformly with respect to λ. Moreover, Then the conclusions of part a) still hold.
Next, notice that G(ϕ, λ 2 , 1) is a Gabor frame. So, to check that f ∈ M p,q t,0 we only need to verify that the sequence c = {c kℓ } = {|k| − ǫ 2 δ ℓ,0 , k = 0} k,ℓ∈Z d ∈ ℓ p,q t,0 . But, the condition t ≤ −d guarantees this, since Next, as in the proof of Lemma 4.3, we have In this case, which completes the proof of part a). b) If p ≥ 1, the assumptions −d < t < 0 and 1 N < p−1 2 − pt 2d are sufficient to prove that f ∈ M p,q t,0 . In addition, the main estimate is that We now state results similar to the above lemmas when the weight is in the frequency variable.  Then there exists a constant C > 0 such that f M p,q 0,s ≤ C, uniformly with respect to λ. Moreover, Proof. We have where we have used again the fact that the weight · s is · −s -moderate. Thus the functions f have norms in M p,q 0,s uniformly bounded with respect to λ. Let us now estimate f λ M p,q 0,s from below. We have f λ (x) = λ s M e 1 ϕ λ (x).
By using (4.5), we obtain as desired.
To prove (4.21) we follow the proof of Lemma 4.3. In particular, we have from which (4.21) 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 ∈ M p,q 0,s , and that f M p,q 0,s ≤ C. Relation (4.21) now follows from The next lemma is proved similarly to Lemma 4.4, so we omit its proof.
Then f ∈ M p,∞ 0,s and (4.24) Lemma 4.9. Let 1 ≤ p, q ≤ ∞ be such that (1/p, 1/q) ∈ I 3 . Let ǫ > 0, s ≥ 0 and 0 < λ < 1. Assume that p > 1, and choose a positive integer N such that 1 Then, there exists a constant C > 0 such that f M p,q 0,s ≤ C, uniformly with respect to λ. Moreover, Proof. In this case, G(ϕ, λ N , 1) is a frame. The condition 1 The rest of the proof is an adaptation of the proof of Lemma 4.5.
Notice that, the previous lemma excludes the case p = 1. We prove this last case by considering the dual case. Observe that the case (1/∞, 1/∞) ∈ I * 1 ∩I * 3 was already considered in dealing with the region I * 1 .
Lemma 4.10. Let 1 ≤ q ≤ ∞ be such that (1/∞, 1/q) ∈ I * 3 . Let ǫ > 0, s ≤ 0 and λ > 1. a) If 1 < q < 2, choose a positive integer N such that 3 N < q − 1. Define Then there exists a constant C > 0 such that f M p,q 0,s ≤ C, uniformly with respect to λ. Moreover, Then the conclusions of part a) still hold. c) If q = 1 and s ≤ −d, define Then there exists a constant C > 0 such that f M ∞,1 d) If q = 1 and −d < s < 0, choose a positive integer N such that 1 N < −s 2d . Define 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 λ The rest of the proof is an adaptation of the proof of Lemma 4.5. b) Assume that 2 ≤ q < ∞. The proof is similar to the above with the following differences: N > q+2 and λ > 1 imply that λ d(1+ 2−N q ) < 1. In addition, the condition s . Therefore, f ∈ M ∞,q 0,s with f M ∞,q 0,s ≤ 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 s . Therefore, f ∈ M ∞,1 0,s with f M ∞,1 0,s ≤ 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 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.
(i) If u is supported in a compact set K ⊂ R d , then, for every t ∈ R, and λ ≥ 1 (ii) Ifû is supported in a compact set K ⊂ R d , then, for every s ∈ R, and λ ≤ 1 Proof. We use the dilation properties for the Sobolev spaces (Bessel potential spaces) H p s (R d ) (see, e.g., [12, Proposition 3]): where C K > 0 depends only on K (see, e.g., [8,11]). Hence, if λ ≥ 1, (ii) Now letû be supported in a compact set K ⊂ R d . We have u ∈ M p,q ⇔ u ∈ L p , and where C K > 0 depends only on K (again, see, e.g., [8]). Arguing as in part (i) above with 0 < λ ≤ 1, and the proof is completed.

Sharpness of Theorems 3.1 and 3.2.
We are now in position to state and prove the sharpness of the results obtained in Section 3. In particular, Theorem 3.1 is optimal in the following sense: (A) If t ≥ 0 then the following statements hold: Assume that there exist constants C > 0, and α, β ∈ R such that (4.33) ∀f ∈ M p,q t,0 and λ ≥ 1, then, α ≥ dµ 1 (p, q), and β ≤ dµ 2 (p, q) − t.
Proof. It will be enough to prove the upper half of each of the estimates, as the lower halves will follow from the fact that f = U λ U 1/λ f . Moreover, the proof relies on analyzing the examples provided by the previous lemmas, and by considering several cases. Case 1: (1/p, 1/q) ∈ I * 2 , t ≥ 0. In this case we have λ ≥ 1 and µ 1 (p, q) = 1/q − 1. Substitute f (x) = ϕ(x) = e −π|x| 2 in the upper half estimates (4.33) and use Lemma 4.1 to obtain , for all λ ≥ 1. This immediately implies that α ≥ −d(1 − 1/q) = dµ 1 (p, q).
sup g M p,q t,0 , where the supremum is taken over all g ∈ S and g M p,q t,0 = 1; hence, Case 3: (1/p, 1/q) ∈ I 3 , t ≥ 0. In this case we have λ ≤ 1 and µ 2 (p, q) = −2/p + 1/q. First assume that 1 ≤ q < ∞ and that the upper-half estimate in (4.35) holds for all f ∈ M p,q t,0 and 0 < λ < 1, but that β > dµ 2 (p, q) − t. Then there is ǫ > 0 such that β > dµ 2 (p, q) − t + ǫ. For this choice of ǫ > 0, we construct a function f as in (4.10) of Lemma 4.4 such that: for some C > 0 and all 0 < λ ≤ 1. This leads to a contradiction on the choice of ǫ. When q = ∞ the function given by (4.12) of Lemma 4.4 gives the optimal bound.
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) ∈ I 1 , s ≥ 0. In this case, 0 < λ ≤ 1 and µ 2 (p, q) = −1/p. Assume there exist constants C > 0 and β ∈ R such that the upper-half estimate (4.38) holds. Taking the Gaussian f = ϕ as in Lemma 4.1 and using (4.3), we have 0,s , for all 0 < λ ≤ 1. This gives β ≤ −d/p.
We recall that H σ i i = 0, 1, are examples of Fourier multipliers which are defined by where σ is called the symbol. The boundedness of H σ i , i = 0, 1 on modulation spaces was proved in [4,3] and in [6]. Moreover, some related local-in-time well posedness results for certain nonlinear PDEs were also obtained in [3,6] for initial data in modulation spaces.