Dilation Properties for Weighted Modulation Spaces
We give a sharp estimate on the norm of the scaling operator acting on the weighted modulation spaces . 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.
The modulation spaces were introduced by Feichtinger , by imposing integrability conditions on the short-time Fourier transform (STFT) of tempered distributions. More specifically, for , we let and denote the operators of modulation and translation. Then, the STFT of with respect to a nonzero window in the Schwartz class is measures the frequency content of in a neighborhood of .
For , and , the weighted modulation space is defined to be the Banach space of all tempered distributions such that Here and in the sequel, we use the notation The definition of modulation space is independent of the choice of the window , 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 , , , where denote the dual exponents of and , respectively.
When both , we will simply write . The weighted space is exactly , while an application of Plancherel’s identity shows that the Sobolev space coincides with . For further properties and uses of modulation spaces, see Gröchenig’s book , and we refer to  for equivalent definitions of the modulation spaces for all .
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  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 [5, Theorem 3.1], where for . In the sequel, we adopt the following notation:
The operator 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 on the Besov spaces [7, Proposition 3].
Theorem 1.1. For , ,
On the modulation spaces, the boundedness of was first investigated by Feichtinger [8, Remark 13] for the Feichtinger algebra, that is, . In fact, this result is a special case of the boundedness of a general class of automorphisms on . Recently, the general estimate on the norm of on the (unweighted) modulation spaces was obtained by Sugimoto and Tomita . In this paper, we will derive optimal lower and upper bounds for the operator on general modulation spaces . More specifically, the boundedness of on is proved in Theorems 3.1, 3.2, and 3.4, and the optimal bounds on 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 , 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 . 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 will use the notations to mean that there exists a constant such that , and means that .
We introduce the indices: Next, we prove a lemma that will be used throughout this paper, and which allows us to investigate the action of only on .
Lemma 2.1. Let be a polynomial growing weight function and a linear continuous operator from to . Assume that Then,
Proof. The conclusion is clear if , because in that case is dense in .
Consider now the case or . For any given , consider a sequence of Schwartz functions, with in , and (see the proof of Proposition 11.3.4 of ). Since tends to in , tends to in , and tends to pointwise. Hence, by Fatou’s Lemma, the assumptions (2.3) and (2.5) hold,
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 by , . Recall that for , the family is a Gabor frame for if and only if there exist such that for all we have Moreover, there exists a dual function such that is also a frame for and every can be written as
It is easy to see from the isometry of the Fourier transform on and the fact that , that is a Gabor frame whenever 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 .
Proposition 2.2. is a Gabor frame for if and only if . In this case, is also a Banach frame for for all , and . Moreover, if and only if there exists a sequence such that with convergence in the modulation space norm. In addition,
3. Dilation Properties of Weighted Modulation Spaces
We first consider the polynomial weights in the time variables , .
Theorem 3.1. Let , . Then the following are true.(1) There exists a constant such that , , (2) There exists a constant such that , ,
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 if and only if and .
We first consider the case . Recall the definition of the dilation operator given by . Since the mapping is an homeomorphism from to , , see, for example, [9, Corollary 2.3], we have Using and the dilation properties for unweighted modulation spaces in [5, Theorem 3.1], we obtain
Hence, it remains to prove that the pseudodifferential operator with symbol is bounded on and that its norm is bounded above by .
By [2, Theorem 14.5.2], this will follow once we prove that . To see this, observe first that Indeed, let . Consider the case . Since , we have and .
Analogously, for , we have . Consequently, we get the desired estimates (3.5).
Using the inclusion , we have By Leibniz’ formula, the estimate , and (3.5), we see that this last expression is estimated by .
This concludes the proof of the upper half of (3.1).
We now consider the case . Notice that if and only if , and using the upper half of (3.1) we can write where the supremum is taken over all and ; hence, This establishes the upper half of (3.2).
We now consider the polynomial weights in the frequency variables , .
Theorem 3.2. Let , . Then the following are true.(1) There exists a constant such that , , (2) There exists a constant such that , ,
The next result follows immediately by combining the last two theorems.
Corollary 3.3. Let , . Then the following are true.(1) There exists a constant such that , , (2) There exists a constant such that , ,
The following result is an analogue of Corollary 3.3 for modulation spaces defined by nonseparable polynomial growing weight function such as .
Theorem 3.4. Let , . Then the following are true.(1) There exists a constant such that for all , , (2) There exists a constant such that for all , ,
Proof. We assume . A duality argument can be used to complete the proof when (notice, this duality argument will be given explicitly below in the proof of the sharpness of Theorem 3.1 in the case , ).
Moreover, since the result has been proved in [5, Theorem 3.1] for , one can use interpolation arguments along with Lemma 2.1 to reduce the proof when is an even integer.
The mapping is an homeomorphism from to , (see [9, Theorem 2.2]). Hence where in the last inequality we used again the dilation properties for unweighted modulation spaces of [5, Theorem 3.1]. Therefore, writing , we see that it suffices to prove that the pseudodifferential operator is bounded on , and its norm is bounded above by . To this end, we observe that, if is an even integer, is a finite sum of operators of the form , with and . Now, Shubin’s pseudodifferential calculus  shows that the operators have bounded symbols, together with all their derivatives, so that they are bounded on . The proof is completed by taking into account the additional factor .
Finally, it is relatively straightforward to give optimal estimates for the dilation operator on the Wiener amalgam spaces . These spaces are images of modulation spaces under Fourier transform, that is, . It is also worth noticing that the indices and 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, that is, , and Corollary 3.3, we obtain the following result.
Proposition 3.5. Let , . Then the following are true.(1) There exists a constant such that for all , , (2) There exists a constant such that for all , ,
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.
4.1. Preliminary Estimates
The next two lemmas involve estimates for the modulation space norms of various modifications of the Gaussian. Together with Lemmas 4.3–4.5, 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 that for , and that .
Lemma 4.1. For , one has
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
If , then
Thus, we have
and the estimate (4.1) follows.
Now, observe that, if , then , and (4.2) follows.
Lemma 4.2. For , , consider the family of functions Then, there exists a constant such that , uniformly with respect to . Moreover,
Proof. We have The last inequality follows from the fact that the weight is -moderate which implies that . This proves the first part of the lemma. Let us now estimate from below. We have Hence, by arguing as above and using (4.5), we have which concludes the proof.
Lemma 4.3. Let , , , and . Moreover, assume that .(a) If , define Then, there exists a constant such that , uniformly with respect to . Moreover, (b) If define Then, there exists a constant such that , uniformly with respect to . Moreover,
Proof. We only prove part (a) as part (b) is obtained similarly. We use Proposition 2.2 to prove that defined in the lemma belongs to . Indeed, is a Gabor frame, and the coefficients of in this frame are given by if and . It is clear that
because . Thus, with uniform norm (with respect to ).
Given , we have Using relation (4.5), Therefore, if , from which the proof follows.
The next results extend [5, Lemmas 3.9 and 3.10].
Lemma 4.4. Let , , . Suppose that satisfy and on .(a) If , define Then, and (b) If , let Then, and
Proof. We only prove part (a), that is, the case as the case is proved in a similar fashion.
Let satisfy , and on . The proof of each part of the Lemma is based on the appropriate estimate for .
Let us first show that . We have
Using Young’s inequality: , and the estimate , we can control (4.27) by since .
Next, we prove (4.23). Since , we obtain Observe that and , for all , . Since and if , the inner integral can be estimated as follows: Consequently, which completes the proof.
Lemma 4.5. Let be such that . Let , , and .(a) If define Then, there exists a constant such that , uniformly with respect to . Moreover, (b) If , choose a positive integer large enough such that . Define Then, the conclusions of part (a) still hold.
Proof. (a) For the range of being considered, , and so if , then .
Next, notice that is a Gabor frame. So, to check that , we only need to verify that the sequence . But, the condition guarantees this, since
Next, as in the proof of Lemma 4.3, we have In this case, Therefore, if , which completes the proof of part (a).
(b) If , the assumptions and are sufficient to prove that . In addition, the main estimate is that
We now state results similar to the above lemmas when the weight is in the frequency variable.
Lemma 4.6. For , , consider the family of functions Then, there exists a constant such that , uniformly with respect to . Moreover,
Proof. We have where we have used again the fact that the weight is -moderate. Thus, the functions have norms in uniformly bounded with respect to . Let us now estimate from below. We have By using (4.5), we obtain as desired.
Lemma 4.7. Let be such that . Assume that , and .(a) If and , or and , define Then, there exists a constant such that , uniformly with respect to . Moreover, (b) If and , choose a positive integer such that , and define Then, the conclusions of part (a) still hold.
Proof. (a) First of all, notice that is a frame. In addition, is equivalent to . Thus, for all , , which ensures that the function defined above belongs to . This is also true when and .
To prove (4.47), we follow the proof of Lemma 4.3. In particular, we have from which (4.47) follows.
(b) In this case, is a frame. Moreover, the choice of insures that which is enough to prove that and that . Relation (4.47) now follows from
The next lemma is proved similarly to Lemma 4.4, so we omit its proof.
Lemma 4.8. Let , , . Suppose that satisfies and on .
(a) If , define . Then, and(b) If , let Then, and
Lemma 4.9. Let be such that . Let , and . Assume that , and choose a positive integer such that . Define Then, there exists a constant such that , uniformly with respect to . Moreover,
Proof. In this case, is a frame. The condition is equivalent to which is enough to show that . Therefore, with , where 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 case . We prove this last case by considering the dual case. Observe that the case was already considered in dealing with the region .
Lemma 4.10. Let be such that . Let , and .(a) If , choose a positive integer such that . Define Then, there exists a constant such that , uniformly with respect to . Moreover, (b) If , choose a positive integer such that . Define Then, the conclusions of part (a) still hold.(c) If and , define