We give a sharp estimate on the norm of the scaling operator 𝑈𝜆𝑓(𝑥)=𝑓(𝜆𝑥) acting on the weighted modulation spaces 𝑀𝑝,𝑞𝑠,𝑡(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-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 𝑥,𝜔𝑑, 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 is𝑉𝑔𝑓(𝑥,𝜔)=𝑓,𝑀𝜔𝑇𝑥𝑔=2𝑑𝑓(𝑡)𝑔(𝑡𝑥)𝑒2𝜋𝑖𝑡𝜔𝑑𝑡.(1.1)𝑉𝑔𝑓(𝑥,𝜔) measures the frequency content of 𝑓 in a neighborhood of 𝑥.

For 𝑠1,𝑠2, and 1𝑝,𝑞, the weighted modulation space 𝑠𝑝,𝑞1,𝑠2(2𝑑) is defined to be the Banach space of all tempered distributions 𝑓 such that𝑓𝑠12𝑝,𝑞,𝑠=22||𝑉𝑔||𝑓(𝑥,𝜔)𝑝𝑣𝑠1(𝑥)𝑝𝑑𝑥𝑞/𝑝𝑣𝑠2(𝜔)𝑞𝑑𝜔1/𝑞<.(1.2) Here and in the sequel, we use the notation𝑣𝑠(𝑥)=𝑥𝑠=1+|𝑥|2𝑠/2.(1.3) 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<, (𝑝,𝑞𝑠,𝑡)=𝑝,𝑞𝑠,𝑡, where 𝑝,𝑞 denote the dual exponents of 𝑝 and 𝑞, respectively.

When both 𝑠=𝑡=0, we will simply write 𝑝,𝑞=𝑝,𝑞0,0. The weighted 𝐿2𝑠 space is exactly 2,2𝑠,0, while an application of Plancherel’s identity shows that the Sobolev space 𝑠2 coincides with 2,20,𝑠. 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 [46]. 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 𝑈𝜆𝑝,𝑞𝑝,𝑞 [5, Theorem  3.1], where 𝑈𝜆𝑓()=𝑓(𝜆) for 𝜆>0. In the sequel, we adopt the following notation:𝑓𝜆=𝑈𝜆𝑓.(1.4)

The operator 𝑈𝜆 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. For 𝜆(0,), 𝑠, 𝐶1𝜆𝑑/𝑝min{1,𝜆𝑠}𝑓𝐵𝑠𝑝,𝑞𝑓𝜆𝐵𝑠𝑝,𝑞𝐶𝜆𝑑/𝑝max{1,𝜆𝑠}𝑓𝐵𝑠𝑝,𝑞.(1.5)

On the modulation spaces, the boundedness of 𝑈𝜆 was first investigated by Feichtinger [8, Remark  13] for the Feichtinger algebra, that is, 1,1. In fact, this result is a special case of the boundedness of a general class of automorphisms on 1,1. Recently, the general estimate on the norm of 𝑈𝜆 on the (unweighted) modulation spaces 𝑝,𝑞(𝑑) 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 𝑝,𝑞𝑡,𝑠(𝑑). More specifically, the boundedness of 𝑈𝜆 on 𝑝,𝑞𝑡,𝑠 is proved in Theorems 3.1, 3.2, and 3.4, and the optimal bounds on 𝑈𝜆𝑝,𝑞𝑡,𝑠𝑝,𝑞𝑡,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 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 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 𝐴𝐵 to mean that there exists a constant 𝑐>0 such that 𝐴𝑐𝐵, and 𝐴𝐵 means that 𝐴𝐵𝐴.

2. Preliminary

We will use the set and index terminology of the paper [5]. Namely, for 1𝑝, 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 subsets𝐼11=max𝑝,1𝑝1𝑞,𝐼11=min𝑝,1𝑝1𝑞,𝐼21=max𝑞,121𝑝,𝐼21=min𝑞,121𝑝,𝐼31=max𝑞,121𝑝,𝐼31=min𝑞,121𝑝.(2.1) These sets are displayed in Figure 1.

We introduce the indices:𝜇11(𝑝,𝑞)=𝑝1if𝑝,1𝑞𝐼1,1𝑞11if𝑝,1𝑞𝐼2,2𝑝+1𝑞1if𝑝,1𝑞𝐼3,𝜇21(𝑝,𝑞)=𝑝1if𝑝,1𝑞𝐼1,1𝑞11if𝑝,1𝑞𝐼2,2𝑝+1𝑞1if𝑝,1𝑞𝐼3.(2.2) 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(𝑑).

Lemma 2.1. Let 𝑚 be a polynomial growing weight function and 𝐴 a linear continuous operator from 𝒮(𝑑) to 𝒮(𝑑). Assume that 𝐴𝑓𝑚𝑝,𝑞𝐶𝑓𝑚𝑝,𝑞,𝑓𝒮𝑑.(2.3) Then, 𝐴𝑓𝑚𝑝,𝑞𝐶𝑓𝑚𝑝,𝑞,𝑓𝑚𝑝,𝑞𝑑.(2.4)

Proof. The conclusion is clear if 𝑝,𝑞<, because in that case 𝒮(𝑑) is dense in 𝑚𝑝,𝑞(𝑑).
Consider now the case 𝑝= or 𝑞=. For any given 𝑓𝑚𝑝,𝑞, consider a sequence 𝑓𝑛 of Schwartz functions, with 𝑓𝑛𝑓 in 𝒮(𝑑), and 𝑓𝑛𝑚𝑝,𝑞𝑓𝑚𝑝,𝑞(2.5) (see the proof of Proposition  11.3.4 of [2]). Since 𝑓𝑛 tends to 𝑓 in 𝒮(𝑑), 𝐴𝑓𝑛 tends to 𝐴𝑓 in 𝒮(𝑑), and 𝑉𝜑𝐴𝑓𝑛 tends to 𝑉𝜑𝐴𝑓 pointwise. Hence, by Fatou’s Lemma, the assumptions (2.3) and (2.5) hold,𝐴𝑓𝑚𝑝,𝑞liminf𝑛𝐴𝑓𝑛𝑚𝑝,𝑞liminf𝑛𝑓𝑛𝑚𝑝,𝑞𝑓𝑚𝑝,𝑞.(2.6)

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 𝜑(𝑥)=𝑒𝜋|𝑥|2, 𝑥𝑑. Recall that for 0<𝑎<1, the family𝜑𝒢(𝜑,𝑎,1)=𝑘,()=𝑀𝑇𝑎𝑘𝜑=𝑒2𝜋𝑖𝜑(𝑎𝑘),𝑘,𝑑,(2.7) is a Gabor frame for 𝐿2(𝑑) if and only if there exist 0<𝐴𝐵< such that for all 𝑓𝐿2 we have𝐴𝑓2𝐿2𝑘,𝑑||𝑓,𝜑𝑘,||2𝐵𝑓2𝐿2.(2.8) Moreover, there exists a dual function 𝜑𝒮 such that 𝒢(𝜑,𝑎,1) is also a frame for 𝐿2 and every 𝑓𝐿2 can be written as𝑓=𝑘,𝑑𝑓,𝜑𝑘,𝜑𝑘,=𝑘,𝑑𝑓,𝜑𝑘,𝜑𝑘,.(2.9)

It is easy to see from the isometry of the Fourier transform on 𝐿2 and the fact that 𝑀𝑇𝑎𝑘𝜑=𝑇𝑀𝑎𝑘𝜑=𝑒2𝜋𝑖𝑎𝑘𝑀𝑎𝑘𝑇𝜑, 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 𝑝,𝑞𝑡,𝑠 for all 1𝑝,𝑞, and 𝑠,𝑡. Moreover, 𝑓𝑝,𝑞𝑡,𝑠 if and only if there exists a sequence {𝑐𝑘,}𝑘,𝑑𝑝,𝑞𝑡,𝑠(𝑑×𝑑) such that 𝑓=𝑘,𝑑𝑐𝑘,𝜑𝑘, with convergence in the modulation space norm. In addition, 𝑓𝑝,𝑞𝑡,𝑠𝑐𝑝,𝑞𝑡,𝑠=𝑑𝑘𝑑||𝑐𝑘,||𝑝𝑣𝑡(𝑘)𝑝𝑞/𝑝𝑣𝑠()𝑞1/𝑞.(2.10)

3. Dilation Properties of Weighted Modulation Spaces

We first consider the polynomial weights in the time variables 𝑣𝑡(𝑥)=𝑥𝑡=(1+|𝑥|2)𝑡/2, 𝑡.

Theorem 3.1. Let 1𝑝,𝑞, 𝑡. Then the following are true.(1) There exists a constant 𝐶>0 such that forall𝑓𝑝,𝑞𝑡,0, 𝜆1, 𝐶1𝜆𝑑𝜇2(𝑝,𝑞)min1,𝜆𝑡𝑓𝑝,𝑞𝑡,0𝑓𝜆𝑝,𝑞𝑡,0𝐶𝜆𝑑𝜇1(𝑝,𝑞)max1,𝜆𝑡𝑓𝑝,𝑞𝑡,0.(3.1)(2) There exists a constant 𝐶>0 such that forall𝑓𝑝,𝑞𝑡,0, 0<𝜆1, 𝐶1𝜆𝑑𝜇1(𝑝,𝑞)min1,𝜆𝑡𝑓𝑝,𝑞𝑡,0𝑓𝜆𝑝,𝑞𝑡,0𝐶𝜆𝑑𝜇2(𝑝,𝑞)max1,𝜆𝑡𝑓𝑝,𝑞𝑡,0.(3.2)

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<𝜆1 if and only if 1/𝜆1 and 𝑓=𝑈𝜆𝑈1/𝜆𝑓=𝑈1/𝜆𝑈𝜆𝑓.
We first consider the case 𝜆1. Recall the definition of the dilation operator 𝑈𝜆 given by 𝑈𝜆𝑓(𝑥)=𝑓(𝜆𝑥). Since the mapping 𝑓𝑡𝑓 is an homeomorphism from 𝑡𝑝,𝑞0,𝑠 to 𝑡𝑝,𝑞0𝑡,𝑠, 𝑡,𝑡0,𝑠, see, for example, [9, Corollary  2.3], we have 𝑈𝜆𝑓𝑝,𝑞𝑡,0𝑡𝑈𝜆𝑓𝑝,𝑞.(3.3) Using 𝑡𝑈𝜆𝑓=𝑈𝜆(𝜆1𝑡)𝑓 and the dilation properties for unweighted modulation spaces in [5, Theorem  3.1], we obtain 𝑈𝜆𝜆1𝑡𝑓𝑝,𝑞𝐶𝜆𝑑𝜇1(𝑝,𝑞)𝜆1𝑡𝑓𝑝,𝑞𝜆𝑑𝜇1(𝑝,𝑞)𝑡𝜆1𝑡𝑡𝑓𝑝,𝑞.(3.4)
Hence, it remains to prove that the pseudodifferential operator with symbol 𝑔(𝑡,𝜆)(𝑥)=𝑥𝑡𝜆1𝑥𝑡 is bounded on 𝑝,𝑞 and that its norm is bounded above by max{1,𝜆𝑡}.
By [2, Theorem  14.5.2], this will follow once we prove that 𝑔(𝑡,𝜆)(𝑥),1max{1,𝜆𝑡}. To see this, observe first that ||𝑔(𝑡,𝜆)||(𝑥)max1,𝜆𝑡,𝑥𝑑.(3.5) Indeed, let 𝑣(𝑡,𝜆)(𝑥)=𝜆1𝑥𝑡. Consider the case 𝑡0. Since 𝜆1, we have 𝜆1|𝑥||𝑥| and 𝑣(𝑡,𝜆)(𝑥)𝑥𝑡.
Analogously, for 𝑡<0, we have 𝑣(𝑡,𝜆)(𝑥)𝜆𝑡𝑥𝑡. Consequently, we get the desired estimates (3.5).
Using the inclusion 𝒞𝑑+1(𝑑),1(𝑑), we have 𝑔(𝑡,𝜆)(𝑥),1sup|𝛼|𝑑+1sup𝑥𝑑||𝜕𝛼𝑔(𝑡,𝜆)||.(𝑥)(3.6) By Leibniz’ formula, the estimate |𝜕𝛽𝑥𝑡|𝑥𝑡|𝛽|, and (3.5), we see that this last expression is estimated by max{1,𝜆𝑡}.
This concludes the proof of the upper half of (3.1).
We now consider the case 0<𝜆1. Notice that 𝜆1 if and only if 1/𝜆1, and using the upper half of (3.1) we can write 𝑓𝜆𝑝,𝑞𝑡,0||=sup𝑓𝜆||,𝑔=𝜆𝑑||sup𝑓,𝑔1/𝜆||𝜆𝑑𝑓𝑝,𝑞𝑡,0𝑔sup1/𝜆𝑝,𝑞𝑡,0𝐶max1,𝜆𝑡𝜆𝑑𝑑𝜇2(𝑝,𝑞)𝑓𝑝,𝑞𝑡,0sup𝑔𝑝,𝑞𝑡,0,(3.7) where the supremum is taken over all 𝑔𝒮 and 𝑔𝑝,𝑞𝑡,0=1; hence, 𝑓𝜆𝑝,𝑞𝑡,0𝐶max1,𝜆𝑡𝜆(𝑑+𝜇2(𝑝,𝑞))𝑓𝑝,𝑞𝑡,0=𝐶max1,𝜆𝑡𝜆𝑑𝜇1(𝑝,𝑞)𝑓𝑝,𝑞𝑡,0.(3.8) This establishes the upper half of (3.2).

We now consider the polynomial weights in the frequency variables 𝑣𝑠(𝜔)=𝜔𝑠, 𝑠.

Theorem 3.2. Let 1𝑝,𝑞, 𝑠. Then the following are true.(1) There exists a constant 𝐶>0 such that forall𝑓𝑝,𝑞0,𝑠, 𝜆1, 𝐶1𝜆𝑑𝜇2(𝑝,𝑞)min{1,𝜆𝑠}𝑓𝑝,𝑞0,𝑠𝑓𝜆𝑝,𝑞0,𝑠𝐶𝜆𝑑𝜇1(𝑝,𝑞)max{1,𝜆𝑠}𝑓𝑝,𝑞0,𝑠.(3.9)(2) There exists a constant 𝐶>0 such that forall𝑓𝑝,𝑞0,𝑠, 0<𝜆1, 𝐶1𝜆𝑑𝜇1(𝑝,𝑞)min{1,𝜆𝑠}𝑓𝑝,𝑞0,𝑠𝑓𝜆𝑝,𝑞0,𝑠𝐶𝜆𝑑𝜇2(𝑝,𝑞)max{1,𝜆𝑠}𝑓𝑝,𝑞0,𝑠.(3.10)

Proof. Here we use the fact that the mapping 𝑓𝐷𝑠𝑓 is an homeomorphism from 𝑝,𝑞𝑡,𝑠0 to 𝑝,𝑞𝑡,𝑠0𝑠, 𝑡,𝑠,𝑠0 (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 1𝑝,𝑞, 𝑡,𝑠. Then the following are true.(1) There exists a constant 𝐶>0 such that forall𝑓𝑝,𝑞𝑡,𝑠, 𝜆1, 𝐶1𝜆𝑑𝜇2(𝑝,𝑞)min1,𝜆𝑡min{1,𝜆𝑠}𝑓𝑝,𝑞𝑡,𝑠𝑓𝜆𝑝,𝑞𝑡,𝑠𝐶𝜆𝑑𝜇1(𝑝,𝑞)max1,𝜆𝑡max{1,𝜆𝑠}𝑓𝑝,𝑞𝑡,𝑠.(3.11)(2) There exists a constant 𝐶>0 such that forall𝑓𝑝,𝑞𝑡,𝑠, 0<𝜆1, 𝐶1𝜆𝑑𝜇1(𝑝,𝑞)min1,𝜆𝑡min{1,𝜆𝑠}𝑓𝑝,𝑞𝑡,𝑠𝑓𝜆𝑝,𝑞𝑡,𝑠𝐶𝜆𝑑𝜇2(𝑝,𝑞)max1,𝜆𝑡max{1,𝜆𝑠}𝑓𝑝,𝑞𝑡,𝑠.(3.12)

The following result is an analogue of Corollary 3.3 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. Let 1𝑝,𝑞, 𝑠. Then the following are true.(1) There exists a constant 𝐶>0 such that for all 𝑓𝑣𝑝,𝑞𝑠, 𝜆1, 𝐶1𝜆𝑑𝜇2(𝑝,𝑞)min{𝜆𝑠,𝜆𝑠}𝑓𝑣𝑠𝑝,𝑞𝑓𝜆𝑣𝑠𝑝,𝑞𝐶𝜆𝑑𝜇1(𝑝,𝑞)max{𝜆𝑠,𝜆𝑠}𝑓𝑣𝑠𝑝,𝑞.(3.13)(2) There exists a constant 𝐶>0 such that for all 𝑓𝑣𝑝,𝑞𝑠, 0<𝜆1, 𝐶1𝜆𝑑𝜇1(𝑝,𝑞)min{𝜆𝑠,𝜆𝑠}𝑓𝑣𝑠𝑝,𝑞𝑓𝜆𝑣𝑠𝑝,𝑞𝐶𝜆𝑑𝜇2(𝑝,𝑞)max{𝜆𝑠,𝜆𝑠}𝑓𝑣𝑠𝑝,𝑞.(3.14)

Proof. We assume 𝑠0. 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, 𝑡0).
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 mapping 𝑓𝑥,𝐷𝑠𝑓 is an homeomorphism from 𝑣𝑝,𝑞𝑠 to 𝑝,𝑞, 𝑠 (see [9, Theorem  2.2]). Hence 𝑓𝜆𝑣𝑠𝑝,𝑞𝑥,𝐷𝑠𝑓𝜆𝑝,𝑞=𝑈𝜆𝜆1𝑥,𝜆𝐷𝑠𝑓𝑝,𝑞𝜆𝐶𝑑𝜇1(𝑝,𝑞)𝜆1𝑥,𝜆𝐷𝑠𝑓𝑝,𝑞𝜆,𝜆1𝑑𝜇2(𝑝,𝑞)𝜆1𝑥,𝜆𝐷𝑠𝑓𝑝,𝑞,0<𝜆1,(3.15) where in the last inequality we used again the dilation properties for unweighted modulation spaces of [5, Theorem  3.1]. Therefore, writing 𝑓=𝑥,𝐷𝑠𝑥,𝐷𝑠𝑓, we see that it suffices to prove that the pseudodifferential operator 𝜆1𝑥,𝜆𝐷𝑠𝑥,𝐷𝑠(3.16) is bounded on 𝑝,𝑞, and its norm is bounded above by max{1,𝜆𝑠}max{1,𝜆𝑠}=max{𝜆𝑠,𝜆𝑠}. To this end, we observe that, if 𝑠 is an even integer, 𝜆1𝑥,𝜆𝐷𝑠 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 𝑥𝛼𝐷𝛽𝑥,𝐷𝑠 have bounded symbols, together with all their derivatives, so that they are bounded on 𝑝,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 operator 𝑈𝜆 on the Wiener amalgam spaces 𝑊(𝐿𝑝𝑠,𝐿𝑞𝑡). These spaces are images of modulation spaces under Fourier transform, that is, 𝑝,𝑞𝑡,𝑠=𝑊(𝐿𝑝𝑠,𝐿𝑞𝑡). It is also worth noticing that the indices 𝜇1 and 𝜇2 obey the following relations:𝜇1𝑝,𝑞=1𝜇2(𝑝,𝑞),𝜇2𝑝,𝑞=1𝜇11(𝑝,𝑞)whenever𝑝+1𝑝=1𝑞+1𝑞=1.(3.17) 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, 𝑓𝜆=𝜆𝑑(𝑓)1/𝜆, and Corollary 3.3, we obtain the following result.

Proposition 3.5. Let 1𝑝,𝑞, 𝑡,𝑠. Then the following are true.(1) There exists a constant 𝐶>0 such that for all 𝑓𝑊(𝐿𝑝𝑠,𝐿𝑞𝑡), 𝜆1, 𝐶1𝜆𝑑𝜇2(𝑝,𝑞)min1,𝜆𝑡min{1,𝜆𝑠}𝑓𝑊(𝐿𝑝𝑠,𝐿𝑞𝑡)𝑓𝜆𝑊(𝐿𝑝𝑠,𝐿𝑞𝑡)𝐶𝜆𝑑𝜇1(𝑝,𝑞)max1,𝜆𝑡max{1,𝜆𝑠}𝑓𝑊(𝐿𝑝𝑠,𝐿𝑞𝑡).(3.18)(2) There exists a constant 𝐶>0 such that for all 𝑓𝑊(𝐿𝑝𝑠,𝐿𝑞𝑡), 𝜆1, 𝐶1𝜆𝑑𝜇1(𝑝,𝑞)min1,𝜆𝑡min{1,𝜆𝑠}𝑓𝑊(𝐿𝑝𝑠,𝐿𝑞𝑡)𝑓𝜆𝑊(𝐿𝑝𝑠,𝐿𝑞𝑡)𝐶𝜆𝑑𝜇2(𝑝,𝑞)max1,𝜆𝑡max{1,𝜆𝑠}𝑓𝑊(𝐿𝑝𝑠,𝐿𝑞𝑡).(3.19)

Remark 3.6. For 𝐿𝐺𝐿(𝑑,), the operator 𝑈𝐿 defined by 𝑈𝐿𝑓()=𝑓(𝐿) is clearly bounded on 𝑝,𝑞𝑡,𝑠(𝑑). It is interesting to ask whether versions of Theorems 3.1 and 3.2 can be established for 𝑈𝐿.

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 4.34.5, 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 4.64.10. Finally, in Lemma 4.11, we investigated the property of the dilation operator on compactly supported functions.

Recall that 𝜑(𝑥)=𝑒𝜋|𝑥|2 for 𝑥𝑑, and that 𝜑𝜆(𝑥)=𝑈𝜆𝜑(𝑥)=𝜑(𝜆𝑥).

Lemma 4.1. For 𝑡,𝑠0, one has 𝜑𝜆𝑀𝑝,𝑞𝑡,0𝜆𝑑/𝑝𝑡𝜑,0<𝜆1,(4.1)𝜆𝑀𝑝,𝑞𝑡,0𝜆𝑑(11/𝑞)𝜑,𝜆1,(4.2)𝜆𝑝,𝑞0,𝑠𝜆𝑑/𝑝𝜑,0<𝜆1,(4.3)𝜆𝑝,𝑞0,𝑠𝜆𝑑(11/𝑞)+𝑠,𝜆1.(4.4)

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 ||𝑉𝜑𝜑𝜆(||=𝜆𝑥,𝜔)2+1𝑑/2𝑒𝜋(𝜆2/(𝜆2+1))|𝑥|2𝑒𝜋(1/(𝜆2+1))|𝜔|2.(4.5) Hence, 𝜑𝜆𝑝,𝑞𝑡,0𝑉𝜑𝜑𝜆𝑝,𝑞𝑡,0=𝑝2𝑝𝑞2𝑞𝜆𝑑/𝑝𝜆2+1(𝑑/2)((1/𝑝)+(1/𝑞)1)𝑑𝑒𝜋|𝑥|2𝜆2+1𝜆𝑝𝑥𝑝𝑡𝑑𝑥1/𝑝.(4.6) If 0<𝜆1, then 𝜆𝑡|𝑥|𝑡/2𝜆2+1𝜆2|𝑥|2𝑡/2𝜆1+2+1𝜆2|𝑥|2𝑡/22𝜆𝑡1+|𝑥|2𝑡/2.(4.7) Thus, we have 𝜆𝑡𝑑𝑒𝜋|𝑥|2𝜆2+1𝜆𝑝𝑥𝑝𝑡𝑑𝑥1/𝑝𝜆𝑡,0<𝜆1,(4.8) and the estimate (4.1) follows.
Now, observe that, if 𝜆1, then (𝜆2+1/𝜆𝑝)𝑥𝑥, and (4.2) follows.

Lemma 4.2. For 𝑡0, 𝜆1, consider the family of functions 𝑓(𝑥)=𝜆𝑡𝜑𝑥𝜆𝑒1,𝑒1=(1,0,0,,0).(4.9) Then, there exists a constant 𝐶>0 such that 𝑓𝑝,𝑞𝑡,0𝐶, uniformly with respect to 𝜆. Moreover, 𝑓𝜆𝑝,𝑞𝑡,0𝜆𝑡+𝑑((1/𝑞)1),𝜆1.(4.10)

Proof. We have 𝑓𝑝,𝑞𝑡,0𝑉𝜑𝑓(𝑥,𝜔)𝑥𝑡𝐿𝑝,𝑞=𝜆𝑡𝑉𝜑𝜑𝑥𝜆𝑒1,𝜔𝑥𝑡𝐿𝑝,𝑞=𝜆𝑡𝑉𝜑𝜑(𝑥,𝜔)𝑥+𝜆𝑒1𝑡𝐿𝑝,𝑞𝜆𝑡𝜆𝑡𝑉𝜑𝜑𝑥𝑡𝐿𝑝,𝑞1.(4.11) The last inequality follows from the fact that the weight 𝑡 is 𝑡-moderate which implies that 𝑥+𝜆𝑒1𝑡𝜆𝑡𝑥𝑡. This proves the first part of the lemma. Let us now estimate 𝑓𝜆𝑝,𝑞𝑡,0 from below. We have 𝑓𝜆(𝑥)=𝜆𝑡𝜑𝜆𝑥𝑒1.(4.12) Hence, by arguing as above and using (4.5), we have 𝑓𝜆𝑝,𝑞𝑡,0𝜆𝑡𝑉𝜑𝜑𝜆(𝑥,𝜔)𝑥+𝑒1𝑡𝐿𝑝,𝑞𝜆𝑡𝜆𝑑((1/𝑞)1)𝑒𝜋𝑝|𝑥|2𝑥+𝑒1𝑝𝑡𝑑𝑥1/𝑝𝜆𝑡+𝑑((1/𝑞)1),(4.13) which concludes the proof.

Lemma 4.3. Let 1𝑝,𝑞, 𝜖>0, 𝑡, and 𝜆>1. Moreover, assume that (1/𝑝,1/𝑞)𝐼1.(a) If 𝑡0, define 𝑓(𝑥)=0||||𝑑/𝑝𝜖𝑒2𝜋𝑖𝜆1𝑥𝜑(𝑥)=0||||𝑑/𝑝𝜖𝑀𝜆1𝜑(𝑥),in𝒮𝑑.(4.14) Then, there exists a constant 𝐶>0 such that 𝑓𝑝,𝑞𝑡,0𝐶, uniformly with respect to 𝜆. Moreover, 𝑓𝜆𝑝,𝑞𝑡,0𝜆𝑑/𝑝𝜖,𝜆>1.(4.15)(b) If 𝑡0 define 𝑓(𝑥)=𝑘0||𝑘||𝑑/𝑝𝜖𝑡𝜑(𝑥𝑘)=𝑘0||𝑘||𝑑/𝑝𝜖𝑡𝑇𝑘𝜑(𝑥),in𝒮𝑑.(4.16)Then, there exists a constant 𝐶>0 such that 𝑓𝑝,𝑞𝑡,0𝐶, uniformly with respect to 𝜆. Moreover, 𝑓𝜆𝑝,𝑞𝑡,0𝜆𝑑/𝑝𝜖𝑡,𝜆>1.(4.17)

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 𝑝,𝑞𝑡,0. Indeed, 𝒢(𝜑,1,𝜆1) is a Gabor frame, and the coefficients of 𝑓 in this frame are given by 𝑐𝑘,=𝛿𝑘,0||𝑑/𝑝𝜖 if 0 and 𝑐0,0=0. It is clear that 𝑐𝑘,𝑝,𝑞𝑡,0=𝑑𝑘𝑑||𝑐𝑘,||𝑝𝑘𝑝𝑡𝑞/𝑝1/𝑞=0||||𝑞(𝑑/𝑝𝜖)1/𝑞<,(4.18) because 𝑞/𝑝1. Thus, 𝑓𝑝,𝑞𝑡,0 with uniform norm (with respect to 𝜆).
Given 𝜆>1, we have 𝑓𝜆𝑝,𝑞𝑡,0=sup𝑔𝑝,𝑞𝑡,0=1||𝑓𝜆||,𝑔𝜑2𝑝,𝑞𝑡,0||𝑓𝜆||.,𝜑(4.19) Using relation (4.5), 𝑓𝜆,𝜑=0||||𝑑/𝑝𝜖𝑉𝜑𝜑𝜆(0,)=0||||𝑑/𝑝𝜖1+𝜆2𝑑/2𝑒𝜋||2/(𝜆2+1).(4.20) Therefore, if 𝜆>1, 𝑓𝜆𝑝,𝑞𝑡,0𝐶0||||𝑑/𝑝𝜖1+𝜆2𝑑/2𝑒𝜋||2/(𝜆2+1)𝐶𝜆𝑑0||||𝑑/𝑝𝜖𝑒𝜋||2/(𝜆2+1)𝐶𝜆𝑑||||0<<𝜆||||𝑑/𝑝𝜖𝑒𝜋||2/(𝜆2+1)𝐶𝜆𝑑𝜆𝑑/𝑝𝜖||||0<<𝜆𝑒𝜋𝐶𝜆𝑑𝜆𝑑/𝑝𝜖𝑒𝜋𝜆𝑑=𝐶𝜆𝑑/𝑝+𝜖,(4.21) from which the proof follows.

The next results extend [5, Lemmas  3.9 and 3.10].

Lemma 4.4. Let 1𝑝,𝑞, 𝑡0, 𝜖>0. Suppose that 𝜓𝒮(𝑑) satisfy supp𝜓[1/2,1/2]𝑑 and 𝜓=1 on [1/4,1/4]𝑑.(a) If 1𝑞<, define 𝑓(𝑦)=𝑘𝑑{0}||𝑘||(𝑑/𝑞)𝜖𝑡𝑀𝑘𝑇𝑘𝜓(𝑦),in𝒮𝑑.(4.22) Then, 𝑓𝑝,𝑞𝑡,0(𝑑) and 𝑓𝜆𝑝,𝑞𝑡,0𝜆𝑑((2/𝑝)(1/𝑞))+𝜖𝑡,0<𝜆1.(4.23)(b) If 𝑞=, let 𝑓(𝑦)=𝑘0||𝑘||𝑡𝑀𝑘𝑇𝑘𝜓(𝑦),in𝒮𝑑.(4.24) Then, 𝑓𝑝,𝑡,0 and 𝑓𝜆𝑝,𝑡,0𝜆(2𝑑/𝑝)𝑡,0<𝜆1.(4.25)

Proof. We only prove part (a), that is, the case 1𝑞< as the case 𝑞= is proved in a similar fashion.
Let 𝑔𝒮(𝑑) satisfy supp𝑔[1/8,1/8]𝑑, and |̂𝑔|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 that 𝑓𝑝,𝑞𝑡,0(𝑑). We have ||||𝑑𝑒2𝜋𝑖(𝜔𝑘)𝑦𝜓||||=||||(𝑦𝑘)𝑔(𝑦𝑥)𝑑𝑦𝑑||||𝜓(𝑦𝑘)𝑔(𝑦𝑥)1+𝜔𝑘2𝑑𝐼Δ𝑦𝑑𝑒2𝜋𝑖(𝜔𝑘)𝑦||||=1𝑑𝑦||||1+𝜔𝑘2𝑑|||||||𝛽1+𝛽2||2𝑑𝐶𝛽1,𝛽2𝑑𝜕𝛽1𝑇𝑘𝜓𝜕(𝑦)𝛽2𝑔(𝑥𝑦)𝑒2𝜋𝑖(𝜔𝑘)𝑦|||||𝐶𝑑𝑦||||1+𝜔𝑘2𝑑||𝛽1+𝛽2||2𝑑||𝑇𝑘𝜕𝛽1𝜓||||𝜕𝛽2𝑔||(𝑥).(4.26)
Hence, 𝑓𝑝,𝑞𝑡,0𝑉𝑔𝑓𝐿𝑝,𝑞𝑡,0=𝑑𝑑|||||𝑘0||𝑘||𝑑/𝑞𝜖𝑡𝑑𝑒2𝜋𝑖(𝜔𝑘)𝑦|||||𝜓(𝑦𝑘)𝑔(𝑦𝑥)𝑑𝑦𝑝𝑥𝑡𝑝𝑑𝑥𝑞/𝑝𝑑𝜔1/𝑞𝐶𝑑𝑘0||𝑘||𝑑/𝑞𝜖𝑡1||||1+𝜔𝑘2𝑑||𝛽1+𝛽2||2𝑑||𝑇𝑘𝜕𝛽1𝜓||||𝜕𝛽2𝑔||𝐿𝑝𝑡𝑞𝑑𝜔1/𝑞.(4.27)
Using Young’s inequality: |𝑇𝑘(𝜕𝛽1𝜓)||𝜕𝛽2𝑔|𝐿𝑝𝑡𝑇𝑘𝜕𝛽1𝜓𝐿1𝑡𝜕𝛽2𝑔𝐿𝑝𝑡, and the estimate 𝑇𝑘𝜕𝛽1𝜓𝐿1𝑡𝑘𝑡𝜕𝛽1𝜓𝐿1𝑡, we can control (4.27) by 𝐶𝑑𝑘0||𝑘||𝑑/𝑞𝜖1||||1+𝜔𝑘2𝑑𝑞𝑑𝜔1/𝑞𝐶𝑑+[1/2,1/2]𝑑𝑘0||𝑘||𝑑/𝑞𝜖1||||1+𝜔𝑘2𝑑𝑞𝑑𝜔1/𝑞𝐶𝑑𝑘0||𝑘||𝑑/𝑞𝜖1||||1+𝑘2𝑑𝑞1/𝑞=𝐶||𝑘||𝑑/𝑞𝜖1||𝑘||1+2𝑑𝑞<,(4.28) since {|𝑘|𝑑/𝑞𝜖}𝑘0𝑞.
Next, we prove (4.23). Since 𝑉𝑔𝑓𝜆(𝑥,𝜔)=𝜆𝑑𝑉𝑔𝜆1𝑓(𝜆𝑥,𝜆1𝜔), we obtain 𝑉𝑔𝑓𝜆𝐿𝑝,𝑞𝑡,0=𝜆𝑑(1+1/𝑝1/𝑞)𝑑𝑑|||𝑉𝑔𝜆1|||𝑓(𝑥,𝜔)𝑝𝜆1𝑥𝑝𝑡𝑑𝑥𝑞/𝑝𝑑𝜔1/𝑞.(4.29) Observe that 𝜆1(𝑥)𝜆1||||||||if|𝑥|(4.30) and supp𝑔((𝑥)/𝜆)+[1/4,1/4]𝑑, for all 0<𝜆1, 𝑥+[1/8,1/8]𝑑. Since supp𝜓(𝑘)𝑘+[1/2,1/2]𝑑 and 𝜓(𝑡𝑘)=1 if 𝑡𝑘+[1/4,1/4]𝑑, the inner integral can be estimated as follows: 𝑑|||𝑉𝑔𝜆1|||𝑓(𝑥,𝜔)𝑝𝜆1𝑥𝑝𝑡𝑑𝑥1/𝑝0+[1/8,1/8]𝑑|||||𝑘0||𝑘||𝑑/𝑞𝜖𝑡𝑑𝑒2𝜋𝑖(𝜔𝑘)𝑦𝜓(𝑦𝑘)𝑔𝑦𝑥𝜆|||||𝑑𝑦𝑝×𝜆1𝑥𝑝𝑡𝑑𝑥1/𝑝0||||𝑑/𝑞𝜖𝑡𝜆𝑑||||𝜆̂𝑔(𝜆(𝜔))𝑡||||𝑡𝑝1/𝑝0||||𝑑/𝑞𝜖𝜆𝑑𝑡||||̂𝑔(𝜆(𝜔))𝑝1/𝑝.(4.31) Consequently, 𝑉𝑔𝑓𝜆𝐿𝑝,𝑞𝑡,0=𝜆𝑑(1+1/𝑝1/𝑞)𝑑𝑑|||𝑉𝑔𝜆1|||𝑓(𝑥,𝜔)𝑝𝜆1𝑥𝑝𝑡𝑑𝑥𝑞/𝑝𝑑𝜔1/𝑞𝜆𝑑(1+1/𝑝1/𝑞)𝑑0||||𝑑/𝑞𝜖𝜆𝑑𝑡||||̂𝑔(𝜆(𝜔))𝑝𝑞/𝑝𝑑𝜔1/𝑞=𝜆𝑑𝑡𝑑/𝑞𝜆𝑑(1+1/𝑝1/𝑞)𝑑0||||𝑑/𝑞𝜖||||̂𝑔(𝜔+𝜆)𝑝𝑞/𝑝𝑑𝜔1/𝑞𝜆𝑡𝑑/𝑝|𝜔|1||||1/𝜆||||𝑑/𝑞𝜖||||̂𝑔(𝜔+𝜆)𝑝𝑞/𝑝𝑑𝜔1/𝑞𝜆𝑡𝑑/𝑝|𝜔|1||||1/𝜆||||𝑑/𝑞𝜖𝑝𝑞/𝑝𝑑𝜔1/𝑞=𝜆𝑡𝑑/𝑝||||1/𝜆||||𝑑/𝑞𝜖𝑝1/𝑝𝜆𝑡𝑑/𝑝𝜆𝑑/𝑞+𝜖||||1/𝜆1/𝑝𝜆𝑡2(𝑑/𝑝)+𝑑/𝑞+𝜖,(4.32) which completes the proof.

Lemma 4.5. Let 1𝑝,𝑞 be such that (1/𝑝,1/𝑞)𝐼3. Let 𝜖>0, 𝑡<0, and 0<𝜆<1.(a) If 𝑡𝑑 define 𝑓(𝑥)=𝜆𝑑/𝑞2𝑑/𝑝+2𝑑𝑘0||𝑘||𝜖/2𝑇𝜆2𝑘𝜑(𝑥),in𝒮𝑑.(4.33) Then, there exists a constant 𝐶>0 such that 𝑓𝑝,𝑞𝑡,0𝐶, uniformly with respect to 𝜆. Moreover, 𝑓𝜆𝑝,𝑞𝑡,0𝜆𝑑𝜇2(𝑝,𝑞)+𝜖,0<𝜆<1.(4.34)(b) If 𝑑<𝑡<0, choose a positive integer 𝑁 large enough such that 1/𝑁<(𝑝1)/2𝑝𝑡/2𝑑. Define 𝑓(𝑥)=𝜆𝑑/𝑞𝑘0||𝑘||𝑑(2/𝑁𝑝1)𝜖/𝑁𝑇𝜆𝑁𝑘𝜑(𝑥),in𝒮𝑑.(4.35)Then, the conclusions of part (a) still hold.

Proof. (a) For the range of 𝑝,𝑞 being considered, 𝑑/𝑞+2𝑑2𝑑/𝑝=𝑑𝜇2(𝑝,𝑞)+2𝑑0, and so if 𝜆<1, then 𝜆𝑑/𝑞+2𝑑2𝑑/𝑝<1.
Next, notice that 𝒢(𝜑,𝜆2,1) is a Gabor frame. So, to check that 𝑓𝑝,𝑞𝑡,0, we only need to verify that the sequence 𝑐={𝑐𝑘}={|𝑘|𝜖/2𝛿,0,𝑘0}𝑘,𝑑𝑝,𝑞𝑡,0. But, the condition 𝑡𝑑 guarantees this, since 𝑐𝑝,𝑞𝑡,0=𝜆𝑑/𝑞+2𝑑2𝑑/𝑝𝑘0||𝑘||𝑝𝜖/2||𝑘||1+2𝑝𝑡/21/𝑝𝐶.(4.36)
Next, as in the proof of Lemma 4.3, we have 𝑓𝜆𝑝,𝑞𝑡,0=sup𝑔𝑝,𝑞𝑡,0=1||𝑓𝜆||,𝑔𝜑2𝑝,𝑞𝑡,0||𝑓𝜆||.,𝜑(4.37) In this case, 𝑓𝜆,𝜑=𝜆2𝑑+𝑑𝜇2(𝑝,𝑞)𝑘0||𝑘||𝜖/2𝑉𝜑𝜑𝜆(𝜆𝑘,0)=𝜆2𝑑+𝑑𝜇2(𝑝,𝑞)𝑘0||𝑘||𝜖/21+𝜆2𝑑/2𝑒𝜋𝜆4|𝑘|2/(𝜆2+1).(4.38) Therefore, if 𝜆<1, 𝑓𝜆𝑝,𝑞𝑡,0𝐶𝜆2𝑑+𝑑𝜇2(𝑝,𝑞)𝑘0||𝑘||𝜖/21+𝜆2𝑑/2𝑒𝜋𝜆4|𝑘|2/(𝜆2+1)𝐶𝜆2𝑑+𝑑𝜇2(𝑝,𝑞)𝑘0||𝑘||𝜖/2𝐶𝜆2𝑑+𝑑𝜇2(𝑝,𝑞)||𝑘||0<<1/𝜆2||𝑘||𝜖/2𝐶𝜆2𝑑+𝑑𝜇2(𝑝,𝑞)𝜆𝜖𝜆2𝑑=𝐶𝜆𝑑𝜇2(𝑝,𝑞)+𝜖(4.39) which completes the proof of part (a).
(b) If 𝑝1, the assumptions 𝑑<𝑡<0 and 1/𝑁<(𝑝1)/2𝑝𝑡/2𝑑 are sufficient to prove that 𝑓𝑝,𝑞𝑡,0. In addition, the main estimate is that 𝑓𝜆𝑝,𝑞𝑡,0𝐶𝜆𝑑/𝑞𝑘0||𝑘||𝑑(2/𝑁𝑝1)𝜖/𝑁1+𝜆2𝑑/2𝑒𝜋𝜆2𝑁|𝑘|2/(𝜆2+1)𝐶𝜆𝑑/𝑞||𝑘||0<<1/𝜆𝑁||𝑘||𝑑(2/𝑁𝑝1)𝜖/𝑁𝐶𝜆𝑑𝜇2(𝑝,𝑞)+𝜖.(4.40)

We now state results similar to the above lemmas when the weight is in the frequency variable.

Lemma 4.6. For 𝑠0, 0<𝜆1, consider the family of functions 𝑓(𝑥)=𝜆𝑠𝑀𝜆1𝑒1𝜑(𝑥),𝑒1=(1,0,0,,0).(4.41) Then, there exists a constant 𝐶>0 such that 𝑓𝑝,𝑞0,𝑠𝐶, uniformly with respect to 𝜆. Moreover, 𝑓𝜆𝑝,𝑞0,𝑠𝜆𝑠𝑑/𝑝,0<𝜆1.(4.42)

Proof. We have 𝑓𝑝,𝑞0,𝑠𝑉𝜑𝑓(𝑥,𝜔)𝜔𝑠𝐿𝑝,𝑞=𝜆𝑠𝑉𝜑𝜑𝑥,𝜔𝜆1𝑒1𝜔𝑠𝐿𝑝,𝑞=𝜆𝑠𝑉𝜑𝜑(𝑥,𝜔)𝜔+𝜆1𝑒1𝑠𝐿𝑝,𝑞𝜆𝑠𝜆𝑠𝑉𝜑𝜑𝜔𝑠𝐿𝑝,𝑞1,(4.43) where we have used again the fact that the weight 𝑠 is 𝑠-moderate. Thus, the functions 𝑓 have norms in 𝑝,𝑞0,𝑠 uniformly bounded with respect to 𝜆. Let us now estimate 𝑓𝜆𝑝,𝑞0,𝑠 from below. We have 𝑓𝜆(𝑥)=𝜆𝑠𝑀𝑒1𝜑𝜆(𝑥).(4.44) By using (4.5), we obtain 𝑓𝜆𝑝,𝑞0,𝑠=𝜆𝑠𝑉𝜑𝜑𝜆𝑥,𝜔𝑒1𝜔𝑠𝐿𝑝,𝑞𝜆𝑠𝑑/𝑝𝑒𝜋𝑞|𝜔|2𝜔+𝑒1𝑞𝑠𝑑𝜔1/𝑞𝜆𝑠𝑑/𝑝,(4.45) as desired.

Lemma 4.7. Let 1𝑝,𝑞 be such that (1/𝑝,1/𝑞)𝐼2. Assume that 𝑠0, 𝜖>0 and 𝜆>1.(a) If 𝑞2 and 𝑠0, or 1𝑞2 and 𝑠𝑑, define 𝑓(𝑥)=0||||𝑑(1/𝑞1)𝜖𝑒2𝜋𝑖𝜆1𝑥𝜑(𝑥)=0||||𝑑(1/𝑞1)𝜖𝑀𝜆1𝜑(𝑥),in𝒮𝑑.(4.46)Then, there exists a constant 𝐶>0 such that 𝑓𝑝,𝑞0,𝑠𝐶, uniformly with respect to 𝜆. Moreover, 𝑓𝜆𝑝,𝑞0,𝑠𝜆𝑑(1/𝑞1)𝜖,𝜆>1.(4.47)(b) If 1𝑞2 and 𝑑<𝑠<0, choose a positive integer 𝑁 such that 1/𝑁<𝑠𝑞/𝑑, and define 𝑓(𝑥)=0||||𝑑(1/𝑁𝑞1)𝜖/𝑁𝑒2𝜋𝑖𝜆𝑁𝑥𝜑(𝑥)=0||||𝑑(1/𝑁𝑞1)𝜖/𝑁𝑀𝜆𝑁𝜑(𝑥),in𝒮𝑑.(4.48)Then, the conclusions of part (a) still hold.

Proof. (a) First of all, notice that 𝒢(𝜑,1,𝜆1) is a frame. In addition, 𝑞2 is equivalent to 1/𝑞11/𝑞. Thus, for all 𝑠0, {||𝑑(1/𝑞1)𝜖,0}𝑞𝑠, which ensures that the function 𝑓 defined above belongs to 𝑝,𝑞0,𝑠. This is also true when 1𝑞2 and 𝑠𝑑.
To prove (4.47), we follow the proof of Lemma 4.3. In particular, we have 𝑓𝜆𝑝,𝑞0,𝑠𝐶0||||𝑑(1/𝑞1)𝜖1+𝜆2𝑑/2𝑒𝜋||2/(𝜆2+1)𝐶𝜆𝑑||||0<𝜆||||𝑑(1/𝑞1)𝜖𝑒𝜋||2/(𝜆2+1),(4.49) from which (4.47) follows.
(b) In this case, 𝒢(𝜑,1,𝜆𝑁) is a frame. Moreover, the choice of 𝑁 insures that 𝑑(1/(𝑁𝑞)1)+𝑠<𝑑 which is enough to prove that 𝑓𝑝,𝑞0,𝑠 and that 𝑓𝑝,𝑞0,𝑠𝐶. Relation (4.47) now follows from 𝑓𝜆𝑝,𝑞0,𝑠𝐶0||||𝑑(1/𝑁𝑞1)𝜖/𝑁1+𝜆2𝑑/2𝑒𝜋𝜆2𝑁+2||2/(𝜆2+1)𝐶𝜆𝑑||||0<𝜆𝑁||||𝑑(1/𝑁𝑞1)𝜖/𝑁𝑒𝜋𝜆2𝑁+2||2/(𝜆2+1)𝐶𝜆𝑑(1/𝑞1)𝜖.(4.50)

The next lemma is proved similarly to Lemma 4.4, so we omit its proof.

Lemma 4.8. Let 1𝑝,𝑞, 𝑠0, 𝜖>0. Suppose that 𝜓𝒮(𝑑) satisfies supp𝜓[1/2,1/2]𝑑 and 𝜓=1 on [1/4,1/4]𝑑.
(a) If 1𝑞<, define 𝑓(𝑦)=𝑘𝑑{0}|𝑘|𝑑/𝑞𝜖𝑠𝑀𝑘𝑇𝑘𝜓(𝑦),in𝒮(𝑑). Then, 𝑓𝑝,𝑞0,𝑠(𝑑) and𝑓𝜆𝑝,𝑞0,𝑠𝜆𝑑(2/𝑝1/𝑞)+𝜖+𝑠,0<𝜆1.(4.51)(b) If 𝑞=, let 𝑓(𝑦)=𝑘0||𝑘||𝑠𝑒2𝜋𝑖𝑘𝑦𝑇𝑘𝜓(𝑦),in𝒮𝑑.(4.52) Then, 𝑓𝑝,0,𝑠 and𝑓𝜆𝑝,0,𝑠𝜆2𝑑/𝑝+𝑠,0<𝜆1.(4.53)

Lemma 4.9. Let 1𝑝,𝑞 be such that (1/𝑝,1/𝑞)𝐼3. Let 𝜖>0, 𝑠0 and 0<𝜆<1. Assume that 𝑝>1, and choose a positive integer 𝑁 such that 1/𝑁<(𝑝1)/2. Define 𝑓(𝑥)=𝜆𝑑/𝑞𝑘0||𝑘||𝑑(2/𝑁𝑝1)𝜖/𝑁𝑇𝜆𝑁𝑘𝜑(𝑥),in𝒮𝑑.(4.54) Then, there exists a constant 𝐶>0 such that 𝑓𝑝,𝑞0,𝑠𝐶, uniformly with respect to 𝜆. Moreover, 𝑓𝜆𝑝,𝑞0,𝑠𝜆𝑑𝜇2(𝑝,𝑞)+𝜖.(4.55)

Proof. In this case, 𝒢(𝜑,𝜆𝑁,1) is a frame. The condition 1/𝑁<(𝑝1)/2 is equivalent to 2/𝑁𝑝1<1/𝑝 which is enough to show that {|𝑘|𝑑(2/𝑁𝑝1)𝜖/𝑁}𝑘0𝑝. Therefore, 𝑓𝑝,𝑞0,𝑠 with 𝑓𝑝,𝑞0,𝑠𝐶, 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 case 𝑝=1. We prove this last case by considering the dual case. Observe that the case (1/,1/)𝐼1𝐼3 was already considered in dealing with the region 𝐼1.

Lemma 4.10. Let 1𝑞 be such that (1/,1/𝑞)𝐼3. Let 𝜖>0, 𝑠0 and 𝜆>1.(a) If 1<𝑞<2, choose a positive integer 𝑁 such that 3/𝑁<𝑞1. Define 𝑓(𝑥)=𝜆𝑑(12/𝑞)0||||𝑑(3/𝑁𝑞1)𝜖/𝑁𝑀𝜆𝑁𝜑(𝑥),in𝒮𝑑.(4.56)Then, there exists a constant 𝐶>0 such that 𝑓𝑝,𝑞0,𝑠𝐶, uniformly with respect to 𝜆. Moreover, 𝑓𝜆,𝑞0,𝑠𝜆𝑑/𝑞𝜖.(4.57)(b) If 2𝑞<, choose a positive integer 𝑁 such that 𝑁>2+𝑞. Define 𝑓(𝑥)=𝜆𝑑+𝑑(2𝑁)/𝑞0||||𝑑((𝑁1)/𝑁𝑞1)𝜖/𝑁𝑀𝜆𝑁𝜑(𝑥),in𝒮𝑑.(4.58)Then, the conclusions of part (a) still hold.(c) If 𝑞=1 and 𝑠𝑑, define