Journal of Function Spaces and ApplicationsVolume 2012 (2012), Article ID 145491, 29 pagesdoi:10.1155/2012/145491

Research Article

Dilation Properties for Weighted Modulation Spaces

Elena Cordero¹ and Kasso A. Okoudjou²

¹Department of Mathematics, University of Torino, Via Carlo Alberto 10, 10123 Torino, Italy²Department of Mathematics, University of Maryland, College Park, MD 20742, USA

Received 31 January 2011; Accepted 7 March 2011

Academic Editor: Hans G. Feichtinger

Copyright © 2012 Elena Cordero and Kasso A. Okoudjou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

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 , 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 [2], and we refer to [3] 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 [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 [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 [5]. 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 [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 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 .

2. Preliminary

We will use the set and index terminology of the paper [5]. Namely, for , let be the conjugate exponent of (). For , we define the subsets These sets are displayed in Figure 1.

(a)

(b)

Figure 1: The index sets.

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 [2]). 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 , ,

Proof. Here we use the fact that the mapping is an homeomorphism from to , (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. 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 [10] 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 , ,

Remark 3.6. For , the operator defined by is clearly bounded on . It is interesting to ask whether versions of Theorems 3.1 and 3.2 can be established for .

4. 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.

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 Hence, 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 Hence, 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 Then, there exists a constant such that , uniformly with respect to . Moreover, (d) If and , choose a positive integer such that . Define Then, the conclusions of part (c) still hold.

Proof. (a) In this case, is a frame. The hypotheses and imply that . In addition, 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.(b) Assume that . The proof is similar to the above with the following differences: and imply that . In addition, the condition implies that . This is enough to show that . Therefore, with , where is a universal constant. (c) In this case, is a frame. The fact that implies that . Therefore, with , where is a universal constant. The rest of the proof is an adaptation of the proof of Lemma 4.5.(d) In this case, is a frame. The fact that and the choice of imply that . Therefore, . Therefore, with , where 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. Let , , and . (i) If is supported in a compact set , then, for every and , (ii) If is supported in a compact set , then, for every and ,

Proof. We use the dilation properties for the Sobolev spaces (Bessel potential spaces) (see, e.g., [7, Proposition  3]): (i) Let be supported in a compact set , we have , and where depends only on (see, e.g., [11, 12]). Hence, if , (ii) Now let be supported in a compact set . We have , and where depends only on (again, see, e.g., [11]). Arguing as in part (i) above with , and the proof is completed.

4.2. 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.

Theorem 4.12. Let . (A) If then the following statements hold. Assume that there exist constants and such that then, , and . Assume that there exist constants and such that then, , and . (B) If then the following statements hold. Assume that there exist constants and such that then, , and . Assume that there exist constants and such that then, , and .

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 . Moreover, the proof relies on analyzing the examples provided by the previous lemmas and by considering several cases.Case 1 (, ). In this case, we have and . Substitute in the upper half estimates (4.69) and use Lemma 4.1 to obtain for all . This immediately implies that .Case 2 (, ). This is the dual case to the previous case and can be handled as follows. In this case, we have and . Assume that the upper-half estimate in (4.72) holds. Notice that if and only if and that if and only if . where the supremum is taken over all and ; hence, Thus, from Case 1 above, . Hence, .Case 3 (, ). In this case, we have and . First assume that and that the upper-half estimate in (4.70) holds for all and but that . Then there is such that . For this choice of , we construct a function as in (4.22) of Lemma 4.4 such that for some and all . This leads to a contradiction on the choice of .When , the function given by (4.24) of Lemma 4.4 gives the optimal bound.Case 4 (, ). In this case, , and . This is the dual of Case 3, and a duality argument similar to the used in Case 2 above gives the result.Case 5 (). In this case, , and . Assume that the upper-half estimate in (4.71) holds and that . Then, choose and construct a function as in part (b) of Lemma 4.3. A contradiction immediately follows.Case 6 (, ). In this case, , and . This is the dual of Case 5.Case 7 (, ). In this case, , and . Assume that the upper-half estimate in (4.69) holds for all and , but that . Then, there is such that . For this choice of , we can now construct a function as in Lemma 4.3, part (a), such that for some and all . This leads to a contradiction on the choice of .Case 8 (, ). In this case, , and . This is the dual of Case 7.Case 9 (, ). In this case, , and . The function constructed in Lemma 4.2 leads to the result.Case 10 (, ). In this case, , and . This is the dual of Case 9.Case 11 (, ). In this case, , and , and Lemma 4.5 can be used to conclude.Case 12 (). In this case, , and . This is the dual of Case 11.

We next consider the sharpness of Theorem 3.2.

Theorem 4.13. Let . (A) If then the following statements hold.Assume that there exist constants and such that then, , and .Assume that there exist constants and such that then, , and .(B) If then the following statements hold.Assume that there exist constants and such that then, , and .Assume that there exist constants and such that then, , and .

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 (, ). In this case, and . Assume there exist constants and such that the upper-half estimate (4.79) holds. Taking the Gaussian as in Lemma 4.1 and using (4.3), we have for all . This gives .Case 2 (, ). Here, , and we test the upper-half estimate (4.81) on the family of functions (4.41). Using (4.42), we obtain .Case 3 (, ). Here , . We assume the upper-half estimate (4.78) and test it on the dilated Gaussian function in (4.4), obtaining .Case 4 (, ). Here, , . We use a contradiction argument based on Lemma 4.7.Case 5 (). Here, , . The sharpness is obtained by testing the upper-half estimate (4.79) on the family of functions , defined in Lemma 4.9 when .If , we consider the dual case, that is , . Here , . We use a contradiction argument based on Lemma 4.10.Case 6 (, ). Here, , . The sharpness is obtained by testing the upper-half estimate (4.81) on the family of functions , 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: with , , , . The formal solution is given by with, and .

We recall that , are examples of Fourier multipliers which are defined by where is called the symbol.

The boundedness of 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. Let , and . Then, the solution of (5.1) with initial data satisfies where and are only functions of the dimension .

Proof. It was proved in [13] that and in [15] that . In addition, it was shown in [15] that the solution satisfies We can now use the results proved in Section 3 to estimate and . More specifically, setting , for , we can write . Using (3.10) with , , we have, for every , Hence, Setting , for , we can write and, for every , Hence, and the estimate (5.4) becomes

5.1.2. Vibrating Plate Equation

Consider now the following Cauchy problem for the vibrating plate equation with , , . The formal solution is given by and satisfies the following estimate.

Proposition 5.2. Let , and . Then, the solution of (5.11) with initial data satisfies where and are only functions of the dimension .

Proof. Here the solution is the sum of two Fourier multipliers having symbols (see [13]) and (see [16]).Since and , using the same arguments as for the wave equation, we obtain

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 , were defined in Section 2:

The following result was proved in [6, Theorem  3.1] and in [17, Theorem  1.1].

Theorem 5.3. Let and . (i) If , then .(ii) If , then .

The next results improve those in [4, Theorem  3.1].

Theorem 5.4. Let . (i) If and , then .(ii) If and , then .

Proof. (i) For , Theorem 5.3 says that . However, the inclusion relations for Besov spaces give , for . Hence the result follows.(ii) If , and , then this is exactly as (i) above. If , then there exists an such that and where the last inclusion follows from (i).

The next results improve those in [4, Theorem  3.2].

Theorem 5.5. (i) Let , . Then, , for all . (ii) If , , then , for all .

Proof. (i) For , and using Theorem 5.3, we obtain . Since , for all , , the result follows.(ii) If , Hence, if , Theorem 5.3 gives . If , the inclusion relations for Besov spaces give . This is easy to see if . On the other hand, if , it follows by an application of Hölder's inequality for spaces. In any case, this concludes the proof.

Acknowledgments

The authors would like to thank Fabio Nicola for helpful discussions. They are grateful to the anonymous referees for their valuable comments. K. A. Okoudjou would also like to acknowledge the partial support of the Alexander von Humboldt foundation. K. A. Okoudjou partially is supported by ONR Grant N000140910324 and by RASA from the Graduate School of UMCP.

References

H. G. Feichtinger, “Modulation spaces on locally Abelian groups. Technical Report (University of Vienna, 1983),” in Proceedings of the International Conference on Wavelets and Applications, pp. 99–140, Chennai, India, 2003.

K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser Boston., Boston, Mass, USA, 2001.

H. Triebel, “Modulation spaces on the Euclidean n-space,” Zeitschrift für Analysis und Ihre Anwendungen, vol. 2, no. 5, pp. 443–457, 1983.

K. A. Okoudjou, “Embedding of some classical Banach spaces into modulation spaces,” Proceedings of the American Mathematical Society, vol. 132, no. 6, pp. 1639–1647, 2004.

M. Sugimoto and N. Tomita, “The dilation property of modulation spaces and their inclusion relation with Besov spaces,” Journal of Functional Analysis, vol. 248, no. 1, pp. 79–106, 2007.

J. Toft, “Continuity properties for modulation spaces, with applications to pseudo-differential calculus—I,” Journal of Functional Analysis, vol. 207, no. 2, pp. 399–429, 2004.

T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, vol. 3, Walter de Gruyter & Co., Berlin, Germany, 1996.

H. G. Feichtinger, “On a new Segal algebra,” Monatshefte für Mathematik, vol. 92, no. 4, pp. 269–289, 1981.

J. Toft, “Continuity properties for modulation spaces, with applications to pseudo-differential calculus—II,” Annals of Global Analysis and Geometry, vol. 26, no. 1, pp. 73–106, 2004.

M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer, Berlin, Germany, 1987.

H. G. Feichtinger, “Atomic characterizations of modulation spaces through Gabor-type representations,” The Rocky Mountain Journal of Mathematics, vol. 19, no. 1, pp. 113–125, 1989, Proceedings of the International Conference on Constructive Function Theory

K. A. Okoudjou, “A Beurling-Helson type theorem for modulation spaces,” Journal of Function Spaces and Applications, vol. 7, no. 1, pp. 33–41, 2009.

Á. Bényi, K. Gröchenig, K. A. Okoudjou, and L. G. Rogers, “Unimodular Fourier multipliers for modulation spaces,” Journal of Functional Analysis, vol. 246, no. 2, pp. 366–384, 2007.

Á. Bényi and K. A. Okoudjou, “Local well-posedness of nonlinear dispersive equations on modulation spaces,” Bulletin of the London Mathematical Society, vol. 41, no. 3, pp. 549–558, 2009.

E. Cordero and F. Nicola, “Remarks on Fourier multipliers and applications to the wave equation,” Journal of Mathematical Analysis and Applications, vol. 353, no. 2, pp. 583–591, 2009.

E. Cordero and D. Zucco, “The Cauchy problem for the vibrating plate equation in modulation spaces,” Journal of Pseudo-Differential Operators Applications, vol. 2, no. 3, pp. 343–354, 2011.

W. Baoxiang and H. Chunyan, “Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations,” Journal of Differential Equations, vol. 239, no. 1, pp. 213–250, 2007.