Dilation Properties for Weighted Modulation Spaces

Abstract

We give a sharp estimate on the norm of the scaling operator 𝑈_𝜆𝑓(𝑥)=𝑓(𝜆𝑥) acting on the weighted modulation spaces 𝑀^{𝑝,𝑞𝑠,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-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,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 is𝑉_𝑔𝑓(𝑥,𝜔)=⟨𝑓,𝑀_𝜔𝑇_𝑥𝑔⟩=_ℝ2𝑑𝑓(𝑡)𝑔(𝑡−𝑥)𝑒^{−2𝜋𝑖𝑡⋅𝜔}𝑑𝑡.(1.1)𝑉_𝑔𝑓(𝑥,𝜔) measures the frequency content of 𝑓 in a neighborhood of 𝑥.

For 𝑠₁,𝑠₂∈ℝ, and 1≤𝑝,𝑞≤∞, the weighted modulation space ℳ^{𝑠𝑝,𝑞}_1,𝑠2(ℝ^2𝑑) is defined to be the Banach space of all tempered distributions 𝑓 such that‖𝑓‖_ℳ^{𝑠12𝑝,𝑞,𝑠}=_ℝ2_ℝ2||𝑉_𝑔𝑓(𝑥,𝜔)||^𝑝𝑣_𝑠1(𝑥)^𝑝𝑑𝑥^𝑞/𝑝𝑣_𝑠2(𝜔)^𝑞𝑑𝜔^1/𝑞<∞.(1.2) Here and in the sequel, we use the notation𝑣_𝑠(𝑥)=⟨𝑥⟩^𝑠=1+|𝑥|²^𝑠/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<∞, (ℳ^{𝑝,𝑞𝑠,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,𝑞0,0. The weighted 𝐿^2𝑠 space is exactly ℳ^2,2𝑠,0, while an application of Plancherel’s identity shows that the Sobolev space ℋ^𝑠2 coincides with ℳ^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 [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 ‖𝑈_𝜆‖_ℳ^𝑝,𝑞_→ℳ^𝑝,𝑞 [5, Theorem  3.1], where 𝑈_𝜆𝑓(⋅)=𝑓(𝜆⋅) for 𝜆>0. In the sequel, we adopt the following notation:𝑓_𝜆∶=𝑈_𝜆𝑓.(1.4)

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 [7, Proposition  3].

Theorem 1.1. For 𝜆∈(0,∞), 𝑠∈ℝ, 𝐶⁻¹𝜆^−𝑑/𝑝min{1,𝜆^𝑠}‖𝑓‖_𝐵^{𝑠𝑝,𝑞}≤‖‖𝑓_𝜆‖‖_𝐵^{𝑠𝑝,𝑞}≤𝐶𝜆^−𝑑/𝑝max{1,𝜆^𝑠}‖𝑓‖_𝐵^{𝑠𝑝,𝑞}.(1.5)

On the modulation spaces, the boundedness of 𝑈_𝜆 was first investigated by Feichtinger [8, Remark  13] for the Feichtinger algebra, that is, ℳ^1,1. In fact, this result is a special case of the boundedness of a general class of automorphisms on ℳ^1,1. Recently, the general estimate on the norm of 𝑈_𝜆 on the (unweighted) modulation spaces ℳ^𝑝,𝑞(ℝ^𝑑) was obtained by Sugimoto and Tomita [5]. In this paper, we will derive optimal lower and upper bounds for the operator 𝑈_𝜆 on general modulation spaces ℳ^{𝑝,𝑞𝑡,𝑠}(ℝ^𝑑). More specifically, the boundedness of 𝑈_𝜆 on ℳ^{𝑝,𝑞𝑡,𝑠} is proved in Theorems 3.1, 3.2, and 3.4, and the optimal bounds on ‖𝑈_𝜆‖_ℳ^{𝑝,𝑞𝑡,𝑠}_→ℳ^{𝑝,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 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 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 𝐴≲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 subsets𝐼₁=max1𝑝,𝑝1^≤1𝑞,𝐼^∗1=min1𝐼₂𝑝,𝑝1^≥1𝑞,=max1𝑞,12≤𝑝1^,𝐼^∗2=min1𝑞,12≥𝑝1^𝐼₃,=max1𝑞,12≤1𝑝,𝐼^∗3=min1𝑞,12≥1𝑝.(2.1) These sets are displayed in Figure 1.

(a)

(b)

Figure 1 

The index sets.

We introduce the indices:𝜇₁(𝑝,𝑞)=⎧⎪⎪⎪⎨⎪⎪⎪⎩−1𝑝if1𝑝,1𝑞∈𝐼^∗11,𝑞−1if1𝑝,1𝑞∈𝐼^∗2−2,𝑝+1𝑞if1𝑝,1𝑞∈𝐼^∗3𝜇₂,(𝑝,𝑞)=⎧⎪⎪⎪⎨⎪⎪⎪⎩−1𝑝if1𝑝,1𝑞∈𝐼¹1,𝑞−1if1𝑝,1𝑞∈𝐼²−2,𝑝+1𝑞if1𝑝,1𝑞∈𝐼³.(2.2) 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).

Lemma 2.1. Let 𝑚 be a polynomial growing weight function and 𝐴 a linear continuous operator from 𝒮^(ℝ^𝑑) to 𝒮^(ℝ^𝑑). Assume that ‖𝐴𝑓‖_ℳ^{𝑚𝑝,𝑞}≤𝐶‖𝑓‖_ℳ^{𝑚𝑝,𝑞},∀𝑓∈𝒮ℝ^𝑑.(2.3) Then, ‖𝐴𝑓‖_ℳ^{𝑚𝑝,𝑞}≤𝐶‖𝑓‖_ℳ^{𝑚𝑝,𝑞},∀𝑓∈ℳ^{𝑚𝑝,𝑞}ℝ^𝑑.(2.4)

Proof. The conclusion is clear if 𝑝,𝑞<∞, because in that case 𝒮(ℝ^𝑑) is dense in ℳ^{𝑚𝑝,𝑞}(ℝ^𝑑).
Consider now the case 𝑝=∞ or 𝑞=∞. For any given 𝑓∈ℳ^{𝑚𝑝,𝑞}, consider a sequence 𝑓_𝑛 of Schwartz functions, with 𝑓_𝑛→𝑓 in 𝒮^(ℝ^𝑑), and ‖‖𝑓_𝑛‖‖_ℳ^{𝑚𝑝,𝑞}≲‖𝑓‖_ℳ^{𝑚𝑝,𝑞}(2.5) (see the proof of Proposition  11.3.4 of [2]). Since 𝑓_𝑛 tends to 𝑓 in 𝒮^(ℝ^𝑑), 𝐴𝑓_𝑛 tends to 𝐴𝑓 in 𝒮^(ℝ^𝑑), and 𝑉_𝜑𝐴𝑓_𝑛 tends to 𝑉_𝜑𝐴𝑓 pointwise. Hence, by Fatou’s Lemma, the assumptions (2.3) and (2.5) hold,‖𝐴𝑓‖_ℳ^{𝑚𝑝,𝑞}≤liminf_𝑛→∞‖‖𝐴𝑓_𝑛‖‖_ℳ^{𝑚𝑝,𝑞}≲liminf_𝑛→∞‖‖𝑓_𝑛‖‖_ℳ^{𝑚𝑝,𝑞}≲‖𝑓‖_ℳ^{𝑚𝑝,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)=𝑒^{−𝜋|𝑥|2}, 𝑥∈ℝ^𝑑. Recall that for 0<𝑎<1, the family𝒢(𝜑,𝑎,1)=𝜑_𝑘,ℓ(⋅)=𝑀_ℓ𝑇_𝑎𝑘𝜑=𝑒^{2𝜋𝑖ℓ⋅}𝜑(⋅−𝑎𝑘),𝑘,ℓ∈ℤ^𝑑,(2.7) is a Gabor frame for 𝐿²(ℝ^𝑑) if and only if there exist 0<𝐴≤𝐵<∞ such that for all 𝑓∈𝐿² we have𝐴‖𝑓‖^2𝐿₂≤_{𝑘,ℓ∈ℤ𝑑}||𝑓,𝜑_𝑘,ℓ||²≤𝐵‖𝑓‖^2𝐿₂.(2.8) Moreover, there exists a dual function 𝜑∈𝒮 such that 𝒢(𝜑,𝑎,1) is also a frame for 𝐿² and every 𝑓∈𝐿² can be written as𝑓=_{𝑘,ℓ∈ℤ𝑑}𝑓,𝜑_𝑘,ℓ𝜑_𝑘,ℓ=_{𝑘,ℓ∈ℤ𝑑}𝑓,𝜑_𝑘,ℓ𝜑_𝑘,ℓ.(2.9)

It is easy to see from the isometry of the Fourier transform on 𝐿² and the fact that 𝑀_ℓ𝑇_𝑎𝑘𝜑=𝑇_ℓ𝑀_−𝑎𝑘𝜑=𝑒^{2𝜋𝑖𝑎𝑘ℓ}𝑀_−𝑎𝑘𝑇_ℓ𝜑, 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 ℳ^{𝑝,𝑞𝑡,𝑠} for all 1≤𝑝,𝑞≤∞, and 𝑠,𝑡∈ℝ. Moreover, 𝑓∈ℳ^{𝑝,𝑞𝑡,𝑠} if and only if there exists a sequence {𝑐_𝑘,ℓ}_{𝑘,ℓ∈ℤ}^𝑑∈ℓ^{𝑝,𝑞𝑡,𝑠}(ℤ^𝑑×ℤ^𝑑) such that 𝑓=∑_{𝑘,ℓ∈ℤ𝑑}𝑐_𝑘,ℓ𝜑_𝑘,ℓ with convergence in the modulation space norm. In addition, ‖𝑓‖_ℳ^{𝑝,𝑞𝑡,𝑠}≍‖𝑐‖_ℓ^{𝑝,𝑞𝑡,𝑠}∶=⎛⎜⎜⎝_{ℓ∈ℤ𝑑}_{𝑘∈ℤ𝑑}||𝑐_𝑘,ℓ||^𝑝𝑣_𝑡(𝑘)^𝑝^𝑞/𝑝𝑣_𝑠(ℓ)^𝑞⎞⎟⎟⎠^1/𝑞.(2.10)

3. 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 forall𝑓∈ℳ^{𝑝,𝑞𝑡,0}, 𝜆≥1, 𝐶⁻¹𝜆^{𝑑𝜇2(𝑝,𝑞)}min1,𝜆^−𝑡‖𝑓‖_ℳ^{𝑝,𝑞𝑡,0}≤‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞𝑡,0}≤𝐶𝜆^{𝑑𝜇1(𝑝,𝑞)}max1,𝜆^−𝑡‖𝑓‖_ℳ^{𝑝,𝑞𝑡,0}.(3.1)(2) There exists a constant 𝐶>0 such that forall𝑓∈ℳ^{𝑝,𝑞𝑡,0}, 0<𝜆≤1, 𝐶⁻¹𝜆^{𝑑𝜇1(𝑝,𝑞)}min1,𝜆^−𝑡‖𝑓‖_ℳ^{𝑝,𝑞𝑡,0}≤‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞𝑡,0}≤𝐶𝜆^{𝑑𝜇2(𝑝,𝑞)}max1,𝜆^−𝑡‖𝑓‖_ℳ^{𝑝,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<𝜆≤1 if and only if 1/𝜆≥1 and 𝑓=𝑈_𝜆𝑈_1/𝜆𝑓=𝑈_1/𝜆𝑈_𝜆𝑓.
We first consider the case 𝜆≥1. Recall the definition of the dilation operator 𝑈_𝜆 given by 𝑈_𝜆𝑓(𝑥)=𝑓(𝜆𝑥). Since the mapping 𝑓↦⟨⋅⟩^𝑡𝑓 is an homeomorphism from ℳ^{𝑡𝑝,𝑞}_0,𝑠 to ℳ^{𝑡𝑝,𝑞}_{0−𝑡,𝑠}, 𝑡,𝑡₀,𝑠∈ℝ, see, for example, [9, Corollary  2.3], we have ‖‖𝑈_𝜆𝑓‖‖_ℳ^{𝑝,𝑞𝑡,0}≍‖‖⟨⋅⟩^𝑡𝑈_𝜆𝑓‖‖_ℳ𝑝,𝑞.(3.3) Using ⟨⋅⟩^𝑡𝑈_𝜆𝑓=𝑈_𝜆(⟨𝜆⁻¹⋅⟩^𝑡)𝑓 and the dilation properties for unweighted modulation spaces in [5, Theorem  3.1], we obtain ‖‖𝑈_𝜆𝜆⁻¹⋅^𝑡𝑓‖‖_ℳ𝑝,𝑞≤𝐶𝜆^{𝑑𝜇1(𝑝,𝑞)}‖‖𝜆⁻¹⋅^𝑡𝑓‖‖_ℳ𝑝,𝑞≍𝜆^{𝑑𝜇1(𝑝,𝑞)}‖‖⟨⋅⟩^−𝑡𝜆⁻¹⋅^𝑡⟨⋅⟩^𝑡𝑓‖‖_ℳ𝑝,𝑞.(3.4)
Hence, it remains to prove that the pseudodifferential operator with symbol 𝑔^(𝑡,𝜆)(𝑥)∶=⟨𝑥⟩^−𝑡⟨𝜆⁻¹𝑥⟩^𝑡 is bounded on ℳ^𝑝,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 ||𝑔^(𝑡,𝜆)(𝑥)||≲max1,𝜆^−𝑡,∀𝑥∈ℝ^𝑑.(3.5) 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, for 𝑡<0, we have 𝑣^(𝑡,𝜆)(𝑥)≤𝜆^−𝑡⟨𝑥⟩^𝑡. Consequently, we get the desired estimates (3.5).
Using the inclusion 𝒞^𝑑+1(ℝ^𝑑)↪ℳ^∞,1(ℝ^𝑑), we have ‖‖𝑔^(𝑡,𝜆)(𝑥)‖‖_ℳ∞,1≲sup_{|𝛼|≤𝑑+1}sup_{𝑥∈ℝ𝑑}||𝜕^𝛼𝑔^(𝑡,𝜆)(𝑥)||.(3.6) 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 case 0<𝜆≤1. Notice that 𝜆≤1 if and only if 1/𝜆≥1, and using the upper half of (3.1) we can write ‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞𝑡,0}=sup||⟨𝑓_𝜆,𝑔⟩||=𝜆^−𝑑sup||⟨𝑓,𝑔_1/𝜆⟩||≤𝜆^−𝑑‖𝑓‖_ℳ^{𝑝,𝑞𝑡,0}sup‖‖𝑔_1/𝜆‖‖_{ℳ𝑝′′,𝑞−𝑡,0}≤𝐶max1,𝜆^−𝑡𝜆^{−𝑑−𝑑𝜇2(𝑝′,𝑞′)}‖𝑓‖_ℳ^{𝑝,𝑞𝑡,0}sup‖𝑔‖_{ℳ𝑝′′,𝑞−𝑡,0},(3.7) where the supremum is taken over all 𝑔∈𝒮 and ‖𝑔‖_{ℳ𝑝′′,𝑞−𝑡,0}=1; hence, ‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞𝑡,0}≤𝐶max1,𝜆^−𝑡𝜆^{−(𝑑+𝜇2(𝑝′,𝑞′))}‖𝑓‖_ℳ^{𝑝,𝑞𝑡,0}=𝐶max1,𝜆^−𝑡𝜆^{𝑑𝜇1(𝑝,𝑞)}‖𝑓‖_ℳ^{𝑝,𝑞𝑡,0}.(3.8) 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.

Theorem 3.2. Let 1≤𝑝,𝑞≤∞, 𝑠∈ℝ. Then the following are true.(1) There exists a constant 𝐶>0 such that forall𝑓∈ℳ^{𝑝,𝑞0,𝑠}, 𝜆≥1, 𝐶⁻¹𝜆^{𝑑𝜇2(𝑝,𝑞)}min{1,𝜆^𝑠}‖𝑓‖_ℳ^{𝑝,𝑞0,𝑠}≤‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞0,𝑠}≤𝐶𝜆^{𝑑𝜇1(𝑝,𝑞)}max{1,𝜆^𝑠}‖𝑓‖_ℳ^{𝑝,𝑞0,𝑠}.(3.9)(2) There exists a constant 𝐶>0 such that forall𝑓∈ℳ^{𝑝,𝑞0,𝑠}, 0<𝜆≤1, 𝐶⁻¹𝜆^{𝑑𝜇1(𝑝,𝑞)}min{1,𝜆^𝑠}‖𝑓‖_ℳ^{𝑝,𝑞0,𝑠}≤‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞0,𝑠}≤𝐶𝜆^{𝑑𝜇2(𝑝,𝑞)}max{1,𝜆^𝑠}‖𝑓‖_ℳ^{𝑝,𝑞0,𝑠}.(3.10)

Proof. Here we use the fact that the mapping 𝑓↦⟨𝐷⟩^𝑠𝑓 is an homeomorphism from ℳ^{𝑝,𝑞𝑡,𝑠}₀ to ℳ^{𝑝,𝑞𝑡,𝑠}_0−𝑠, 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. Let 1≤𝑝,𝑞≤∞, 𝑡,𝑠∈ℝ. Then the following are true.(1) There exists a constant 𝐶>0 such that forall𝑓∈ℳ^{𝑝,𝑞𝑡,𝑠}, 𝜆≥1, 𝐶⁻¹𝜆^{𝑑𝜇2(𝑝,𝑞)}min1,𝜆^−𝑡min{1,𝜆^𝑠}‖𝑓‖_ℳ^{𝑝,𝑞𝑡,𝑠}≤‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞𝑡,𝑠}≤𝐶𝜆^{𝑑𝜇1(𝑝,𝑞)}max1,𝜆^−𝑡max{1,𝜆^𝑠}‖𝑓‖_ℳ^{𝑝,𝑞𝑡,𝑠}.(3.11)(2) There exists a constant 𝐶>0 such that forall𝑓∈ℳ^{𝑝,𝑞𝑡,𝑠}, 0<𝜆≤1, 𝐶⁻¹𝜆^{𝑑𝜇1(𝑝,𝑞)}min1,𝜆^−𝑡min{1,𝜆^𝑠}‖𝑓‖_ℳ^{𝑝,𝑞𝑡,𝑠}≤‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞𝑡,𝑠}≤𝐶𝜆^{𝑑𝜇2(𝑝,𝑞)}max1,𝜆^−𝑡max{1,𝜆^𝑠}‖𝑓‖_ℳ^{𝑝,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 𝑣_𝑠(𝑥,𝜔)∶=⟨(𝑥,𝜔)⟩^𝑠=(1+|𝑥|²+|𝜔|²)^𝑠/2.

Theorem 3.4. Let 1≤𝑝,𝑞≤∞, 𝑠∈ℝ. Then the following are true.(1) There exists a constant 𝐶>0 such that for all 𝑓∈ℳ^{𝑣𝑝,𝑞}_𝑠, 𝜆≥1, 𝐶⁻¹𝜆^{𝑑𝜇2(𝑝,𝑞)}min{𝜆^−𝑠,𝜆^𝑠}‖𝑓‖_ℳ^{𝑣𝑠𝑝,𝑞}≤‖‖𝑓_𝜆‖‖_ℳ^{𝑣𝑠𝑝,𝑞}≤𝐶𝜆^{𝑑𝜇1(𝑝,𝑞)}max{𝜆^−𝑠,𝜆^𝑠}‖𝑓‖_ℳ^{𝑣𝑠𝑝,𝑞}.(3.13)(2) There exists a constant 𝐶>0 such that for all 𝑓∈ℳ^{𝑣𝑝,𝑞}_𝑠, 0<𝜆≤1, 𝐶⁻¹𝜆^{𝑑𝜇1(𝑝,𝑞)}min{𝜆^−𝑠,𝜆^𝑠}‖𝑓‖_ℳ^{𝑣𝑠𝑝,𝑞}≤‖‖𝑓_𝜆‖‖_ℳ^{𝑣𝑠𝑝,𝑞}≤𝐶𝜆^{𝑑𝜇2(𝑝,𝑞)}max{𝜆^−𝑠,𝜆^𝑠}‖𝑓‖_ℳ^{𝑣𝑠𝑝,𝑞}.(3.14)

Proof. 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)∈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 mapping 𝑓↦⟨𝑥,𝐷⟩^𝑠𝑓 is an homeomorphism from ℳ^{𝑣𝑝,𝑞}_𝑠 to ℳ^𝑝,𝑞, 𝑠∈ℝ (see [9, Theorem  2.2]). Hence ‖‖𝑓_𝜆‖‖_ℳ^{𝑣𝑠𝑝,𝑞}≍‖‖⟨𝑥,𝐷⟩^𝑠𝑓_𝜆‖‖_ℳ𝑝,𝑞=‖‖𝑈_𝜆𝜆⁻¹𝑥,𝜆𝐷^𝑠𝑓‖‖_ℳ𝑝,𝑞≤𝐶𝜆^{𝑑𝜇1(𝑝,𝑞)}𝜆‖‖𝜆⁻¹𝑥,𝜆𝐷^𝑠𝑓‖‖_ℳ𝑝,𝑞,𝜆≥1_{𝑑𝜇2(𝑝,𝑞)}‖‖𝜆⁻¹𝑥,𝜆𝐷^𝑠𝑓‖‖_ℳ𝑝,𝑞,0<𝜆≤1,(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,𝐷⟩^−𝑠⟨𝑥,𝐷⟩^𝑠𝑓, we see that it suffices to prove that the pseudodifferential operator 𝜆⁻¹𝑥,𝜆𝐷^𝑠⟨𝑥,𝐷⟩^−𝑠(3.16) 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⁻¹𝑥,𝜆𝐷⟩^𝑠 is a finite sum of operators of the form 𝜆^𝑘𝑥^𝛼𝐷^𝛽, with |𝑘|≤𝑠 and |𝛼|+|𝛽|≤𝑠. Now, Shubin’s pseudodifferential calculus [10] shows that the operators 𝑥^𝛼𝐷^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(ℱ𝐿^𝑝𝑠,𝐿^𝑞𝑡). These spaces are images of modulation spaces under Fourier transform, that is, ℱℳ^{𝑝,𝑞𝑡,𝑠}=𝑊(ℱ𝐿^𝑝𝑠,𝐿^𝑞𝑡). It is also worth noticing that the indices 𝜇₁ and 𝜇₂ obey the following relations:𝜇₁𝑝^,𝑞^=−1−𝜇₂(𝑝,𝑞),𝜇₂𝑝^,𝑞^=−1−𝜇₁(𝑝,𝑞)whenever1𝑝+𝑝1^=1𝑞+𝑞1^=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, 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.(1) There exists a constant 𝐶>0 such that for all 𝑓∈𝑊(ℱ𝐿^𝑝𝑠,𝐿^𝑞𝑡), 𝜆≥1, 𝐶⁻¹𝜆^{𝑑𝜇2(𝑝′,𝑞′)}min1,𝜆^𝑡min{1,𝜆^−𝑠}‖𝑓‖_𝑊(ℱ𝐿^𝑝𝑠_,𝐿^𝑞𝑡₎≤‖‖𝑓_𝜆‖‖_𝑊(ℱ𝐿^𝑝𝑠_,𝐿^𝑞𝑡₎≤𝐶𝜆^{𝑑𝜇1(𝑝′,𝑞′)}max1,𝜆^𝑡max{1,𝜆^−𝑠}‖𝑓‖_𝑊(ℱ𝐿^𝑝𝑠_,𝐿^𝑞𝑡₎.(3.18)(2) There exists a constant 𝐶>0 such that for all 𝑓∈𝑊(ℱ𝐿^𝑝𝑠,𝐿^𝑞𝑡), 𝜆≤1, 𝐶⁻¹𝜆^{𝑑𝜇1(𝑝′,𝑞′)}min1,𝜆^𝑡min{1,𝜆^−𝑠}‖𝑓‖_𝑊(ℱ𝐿^𝑝𝑠_,𝐿^𝑞𝑡₎≤‖‖𝑓_𝜆‖‖_𝑊(ℱ𝐿^𝑝𝑠_,𝐿^𝑞𝑡₎≤𝐶𝜆^{𝑑𝜇2(𝑝′,𝑞′)}max1,𝜆^𝑡max{1,𝜆^−𝑠}‖𝑓‖_𝑊(ℱ𝐿^𝑝𝑠_,𝐿^𝑞𝑡₎.(3.19)

Remark 3.6. For 𝐿∈𝐺𝐿(𝑑,ℝ), the operator 𝑈_𝐿 defined by 𝑈_𝐿𝑓(⋅)=𝑓(𝐿⋅) is clearly bounded on ℳ^{𝑝,𝑞𝑡,𝑠}(ℝ^𝑑). It is interesting to ask whether versions of Theorems 3.1 and 3.2 can be established for 𝑈_𝐿.

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 𝜑(𝑥)=𝑒^{−𝜋|𝑥|2} for 𝑥∈ℝ^𝑑, and that 𝜑_𝜆(𝑥)=𝑈_𝜆𝜑(𝑥)=𝜑(𝜆𝑥).

Lemma 4.1. For 𝑡,𝑠≥0, one has ‖‖‖‖𝜑𝜑_𝜆‖‖_𝑀^{𝑝,𝑞𝑡,0}≍𝜆^{−𝑑/𝑝−𝑡},0<𝜆≤1,(4.1)_𝜆‖‖𝜑‖‖_𝑀^{𝑝,𝑞𝑡,0}≍𝜆^{−𝑑(1−1/𝑞)},𝜆≥1,(4.2)_𝜆‖‖𝜑‖‖_ℳ^{𝑝,𝑞0,𝑠}≍𝜆^−𝑑/𝑝,0<𝜆≤1,(4.3)_𝜆‖‖_ℳ^{𝑝,𝑞0,𝑠}≍𝜆^{−𝑑(1−1/𝑞)+𝑠},𝜆≥1.(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 ||𝑉_𝜑𝜑_𝜆(𝑥,𝜔)||=𝜆²+1^−𝑑/2𝑒^{−𝜋(𝜆2/(𝜆2+1))|𝑥|2}𝑒^{−𝜋(1/(𝜆2+1))|𝜔|2}.(4.5) Hence, ‖‖𝜑_𝜆‖‖_ℳ^{𝑝,𝑞𝑡,0}≍‖‖𝑉_𝜑𝜑_𝜆‖‖_{ℳ𝑝,𝑞𝑡,0}=𝑝^−2𝑝𝑞^−2𝑞𝜆^−𝑑/𝑝𝜆²+1^{(𝑑/2)((1/𝑝)+(1/𝑞)−1)}_ℝ^𝑑𝑒^{−𝜋|𝑥|2}√𝜆²+1𝜆√𝑝𝑥^𝑝𝑡𝑑𝑥^1/𝑝.(4.6) If 0<𝜆≤1, then 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.(4.7) Thus, we have 𝜆^−𝑡≲_ℝ𝑑𝑒^{−𝜋|𝑥|2}√𝜆²+1𝜆√𝑝𝑥^𝑝𝑡𝑑𝑥^1/𝑝≲𝜆^−𝑡,0<𝜆≤1,(4.8) and the estimate (4.1) follows.
Now, observe that, if 𝜆≥1, then ⟨(√𝜆²+1/𝜆√𝑝)𝑥⟩≍⟨𝑥⟩, and (4.2) follows.

Lemma 4.2. For 𝑡≤0, 𝜆≥1, consider the family of functions 𝑓(𝑥)=𝜆^−𝑡𝜑𝑥−𝜆𝑒₁,𝑒₁=(1,0,0,…,0).(4.9) Then, there exists a constant 𝐶>0 such that ‖𝑓‖_ℳ^{𝑝,𝑞𝑡,0}≤𝐶, uniformly with respect to 𝜆. Moreover, ‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞𝑡,0}≳𝜆^{−𝑡+𝑑((1/𝑞)−1)},∀𝜆≥1.(4.10)

Proof. We have ‖𝑓‖_ℳ^{𝑝,𝑞𝑡,0}≍‖‖𝑉_𝜑𝑓(𝑥,𝜔)⟨𝑥⟩^𝑡‖‖_{𝐿𝑝,𝑞}=𝜆^−𝑡‖‖𝑉_𝜑𝜑𝑥−𝜆𝑒₁,𝜔⟨𝑥⟩^𝑡‖‖_{𝐿𝑝,𝑞}=𝜆^−𝑡‖‖𝑉_𝜑𝜑(𝑥,𝜔)⟨𝑥+𝜆𝑒₁⟩^𝑡‖‖_{𝐿𝑝,𝑞}≲𝜆^−𝑡𝜆^𝑡‖‖𝑉_𝜑𝜑⟨𝑥⟩^−𝑡‖‖_{𝐿𝑝,𝑞}≲1.(4.11) The last inequality follows from the fact that the weight ⟨⋅⟩^𝑡 is ⟨⋅⟩^−𝑡-moderate which implies that ⟨𝑥+𝜆𝑒₁⟩^𝑡≲𝜆^𝑡⟨𝑥⟩^−𝑡. This proves the first part of the lemma. Let us now estimate ‖𝑓_𝜆‖_ℳ^{𝑝,𝑞𝑡,0} from below. We have 𝑓_𝜆(𝑥)=𝜆^−𝑡𝜑_𝜆𝑥−𝑒₁.(4.12) Hence, by arguing as above and using (4.5), we have ‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞𝑡,0}≍𝜆^−𝑡‖‖𝑉_𝜑𝜑_𝜆(𝑥,𝜔)⟨𝑥+𝑒₁⟩^𝑡‖‖_{𝐿𝑝,𝑞}≳𝜆^−𝑡𝜆^{𝑑((1/𝑞)−1)}𝑒^{−𝜋𝑝|𝑥|2}⟨𝑥+𝑒₁⟩^𝑝𝑡𝑑𝑥^1/𝑝≳𝜆^{−𝑡+𝑑((1/𝑞)−1)},(4.13) which concludes the proof.

Lemma 4.3. Let 1≤𝑝,𝑞≤∞, 𝜖>0, 𝑡∈ℝ, and 𝜆>1. Moreover, assume that (1/𝑝,1/𝑞)∈𝐼^∗1.(a) If 𝑡≥0, define 𝑓(𝑥)=_ℓ≠0||ℓ||^{−𝑑/𝑝−𝜖}𝑒^{2𝜋𝑖𝜆−1ℓ⋅𝑥}𝜑(𝑥)=_ℓ≠0||ℓ||^{−𝑑/𝑝−𝜖}𝑀_𝜆⁻¹_ℓ𝜑(𝑥),in𝒮^ℝ^𝑑.(4.14) Then, there exists a constant 𝐶>0 such that ‖𝑓‖_ℳ^{𝑝,𝑞𝑡,0}≤𝐶, uniformly with respect to 𝜆. Moreover, ‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞𝑡,0}≳𝜆^{−𝑑/𝑝−𝜖},∀𝜆>1.(4.15)(b) If 𝑡≤0 define 𝑓(𝑥)=_𝑘≠0||𝑘||^{−𝑑/𝑝−𝜖−𝑡}𝜑(𝑥−𝑘)=_𝑘≠0||𝑘||^{−𝑑/𝑝−𝜖−𝑡}𝑇_𝑘𝜑(𝑥),in𝒮^ℝ^𝑑.(4.16)Then, there exists a constant 𝐶>0 such that ‖𝑓‖_ℳ^{𝑝,𝑞𝑡,0}≤𝐶, uniformly with respect to 𝜆. Moreover, ‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞𝑡,0}≳𝜆^{−𝑑/𝑝−𝜖−𝑡},∀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 ℳ^{𝑝,𝑞𝑡,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−𝜖} if ℓ≠0 and 𝑐_0,0=0. It is clear that ‖‖𝑐_𝑘,ℓ‖‖_{ℓ𝑝,𝑞𝑡,0}=⎛⎜⎜⎝_{ℓ∈ℤ𝑑}_{𝑘∈ℤ𝑑}||𝑐_𝑘,ℓ||^𝑝⟨𝑘⟩^𝑝𝑡^𝑞/𝑝⎞⎟⎟⎠^1/𝑞=_ℓ≠0||ℓ||^{𝑞(−𝑑/𝑝−𝜖)}^1/𝑞<∞,(4.18) because 𝑞/𝑝≥1. Thus, 𝑓∈ℳ^{𝑝,𝑞𝑡,0} with uniform norm (with respect to 𝜆).
Given 𝜆>1, we have ‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞𝑡,0}=sup_{‖𝑔‖ℳ𝑝′′,𝑞−𝑡,0=1}||⟨𝑓_𝜆,𝑔⟩||≥‖𝜑‖^ℳ−2_{𝑝′,𝑞′−𝑡,0}||⟨𝑓_𝜆,𝜑⟩||.(4.19) Using relation (4.5), ⟨𝑓_𝜆,𝜑⟩=_ℓ≠0||ℓ||^{−𝑑/𝑝−𝜖}𝑉_𝜑𝜑_𝜆(0,ℓ)=_ℓ≠0||ℓ||^{−𝑑/𝑝−𝜖}1+𝜆²^−𝑑/2𝑒^{−𝜋|ℓ|2/(𝜆2+1)}.(4.20) Therefore, if 𝜆>1, ‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞𝑡,0}≥𝐶_ℓ≠0||ℓ||^{−𝑑/𝑝−𝜖}1+𝜆²^−𝑑/2𝑒^{−𝜋|ℓ|2/(𝜆2+1)}≥𝐶𝜆^−𝑑_ℓ_≠0||ℓ||^{−𝑑/𝑝−𝜖}𝑒^{−𝜋|ℓ|2/(𝜆2+1)}≥𝐶𝜆^−𝑑_||_ℓ||0<<𝜆||ℓ||^{−𝑑/𝑝−𝜖}𝑒^{−𝜋|ℓ|2/(𝜆2+1)}≥𝐶𝜆^−𝑑𝜆^{−𝑑/𝑝−𝜖}_||_ℓ||0<<𝜆𝑒^−𝜋≥𝐶𝜆^−𝑑𝜆^{−𝑑/𝑝−𝜖}𝑒^−𝜋𝜆^𝑑=𝐶𝜆^{−𝑑/𝑝+𝜖},(4.21) from which the proof follows.

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

Lemma 4.4. Let 1≤𝑝,𝑞≤∞, 𝑡≥0, 𝜖>0. Suppose that 𝜓∈𝒮(ℝ^𝑑) satisfy supp𝜓⊂[−1/2,1/2]^𝑑 and 𝜓=1 on [−1/4,1/4]^𝑑.(a) If 1≤𝑞<∞, define 𝑓(𝑦)=_{𝑘∈ℤ𝑑⧵{0}}||𝑘||^{−(𝑑/𝑞)−𝜖−𝑡}𝑀_𝑘𝑇_𝑘𝜓(𝑦),in𝒮^ℝ^𝑑.(4.22) Then, 𝑓∈ℳ^{𝑝,𝑞𝑡,0}(ℝ^𝑑) and ‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞𝑡,0}≳𝜆^{−𝑑((2/𝑝)−(1/𝑞))+𝜖−𝑡},∀0<𝜆≤1.(4.23)(b) If 𝑞=∞, let 𝑓(𝑦)=_𝑘≠0||𝑘||^−𝑡𝑀_𝑘𝑇_𝑘𝜓(𝑦),in𝒮^ℝ^𝑑.(4.24) Then, 𝑓∈ℳ^{𝑝,∞𝑡,0} and ‖‖𝑓_𝜆‖‖_ℳ^{𝑝,∞𝑡,0}≳𝜆^{−(2𝑑/𝑝)−𝑡},∀0<𝜆≤1.(4.25)

Proof. We only prove part (a), that is, the case 1≤𝑞<∞ as the case 𝑞=∞ is proved in a similar fashion.
Let 𝑔∈𝒮(ℝ^𝑑) 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 𝑉_𝑔𝑓.
Let us first show that 𝑓∈ℳ^{𝑝,𝑞𝑡,0}(ℝ^𝑑). We have ||||_ℝ𝑑=𝑒||||^{−2𝜋𝑖(𝜔−𝑘)𝑦}𝜓(𝑦−𝑘)𝑔(𝑦−𝑥)𝑑𝑦_ℝ𝑑||||𝜓(𝑦−𝑘)𝑔(𝑦−𝑥)1+||𝜔−𝑘||²^−𝑑𝐼−Δ_𝑦^𝑑𝑒^{−2𝜋𝑖(𝜔−𝑘)𝑦}=1𝑑𝑦||||1+||𝜔−𝑘||²^𝑑|||||_{||𝛽1+𝛽2||≤2𝑑}𝐶_𝛽1,𝛽2_ℝ𝑑≤𝐶𝜕^𝛽1𝑇_𝑘𝜓(𝑦)𝜕^𝛽2𝑔(𝑥−𝑦)𝑒^{−2𝜋𝑖(𝜔−𝑘)𝑦}𝑑𝑦|||||1+||𝜔−𝑘||²^𝑑_||𝛽1+_{𝛽2||≤2𝑑}||𝑇_𝑘𝜕^𝛽1𝜓||∗||𝜕^𝛽2𝑔||(𝑥).(4.26)
Hence, ‖𝑓‖_ℳ^{𝑝,𝑞𝑡,0}≍‖‖𝑉_𝑔𝑓‖‖_{𝐿𝑝,𝑞𝑡,0}=⎛⎜⎜⎝_ℝ𝑑⎛⎜⎜⎝_ℝ𝑑|||||_𝑘≠0||𝑘||^{−𝑑/𝑞−𝜖−𝑡}_ℝ𝑑𝑒^{−2𝜋𝑖(𝜔−𝑘)𝑦}𝜓(𝑦−𝑘)𝑔(𝑦−𝑥)𝑑𝑦|||||^𝑝⟨𝑥⟩^𝑡𝑝𝑑𝑥⎞⎟⎟⎠^𝑞/𝑝𝑑𝜔⎞⎟⎟⎠^1/𝑞≤𝐶⎛⎜⎜⎜⎝_ℝ𝑑⎛⎜⎜⎜⎝_𝑘≠0||𝑘||^{−𝑑/𝑞−𝜖−𝑡}11+||𝜔−𝑘||²^𝑑_||𝛽1+_{𝛽2||≤2𝑑}‖‖||𝑇_𝑘𝜕^𝛽1𝜓||∗||𝜕^𝛽2𝑔||‖‖_𝐿𝑝𝑡⎞⎟⎟⎟⎠^𝑞𝑑𝜔⎞⎟⎟⎟⎠^1/𝑞.(4.27)
Using Young’s inequality: ‖|𝑇_𝑘(𝜕^𝛽1𝜓)|∗|𝜕^𝛽2𝑔|‖_𝐿^𝑝𝑡≲‖𝑇_𝑘𝜕^𝛽1𝜓‖_𝐿^1𝑡‖𝜕^𝛽2𝑔‖_𝐿^𝑝𝑡, and the estimate ‖𝑇_𝑘𝜕^𝛽1𝜓‖_𝐿^1𝑡≤⟨𝑘⟩^𝑡‖𝜕^𝛽1𝜓‖_𝐿^1𝑡, we can control (4.27) by 𝐶⎛⎜⎜⎜⎝_ℝ𝑑⎛⎜⎜⎜⎝_𝑘≠0||𝑘||^{−𝑑/𝑞−𝜖}11+||𝜔−𝑘||²^𝑑⎞⎟⎟⎟⎠^𝑞𝑑𝜔⎞⎟⎟⎟⎠^1/𝑞≤𝐶⎛⎜⎜⎜⎝_{ℓ∈ℤ𝑑}_{ℓ+[−1/2,1/2]𝑑}⎛⎜⎜⎜⎝_𝑘≠0||𝑘||^{−𝑑/𝑞−𝜖}11+||𝜔−𝑘||²^𝑑⎞⎟⎟⎟⎠^𝑞𝑑𝜔⎞⎟⎟⎟⎠^1/𝑞≤𝐶⎛⎜⎜⎜⎝_{ℓ∈ℤ𝑑}⎛⎜⎜⎜⎝_𝑘≠0||𝑘||^{−𝑑/𝑞−𝜖}11+||ℓ−𝑘||²^𝑑⎞⎟⎟⎟⎠^𝑞⎞⎟⎟⎟⎠^1/𝑞=𝐶‖‖‖‖‖||𝑘||^{−𝑑/𝑞−𝜖}∗11+||𝑘||²^𝑑‖‖‖‖‖_ℓ𝑞<∞,(4.28) since {|𝑘|^{−𝑑/𝑞−𝜖}}_𝑘≠0∈ℓ^𝑞.
Next, we prove (4.23). Since 𝑉_𝑔𝑓_𝜆(𝑥,𝜔)=𝜆^−𝑑𝑉_𝑔𝜆−1𝑓(𝜆𝑥,𝜆⁻¹𝜔), we obtain ‖‖𝑉_𝑔𝑓_𝜆‖‖_{𝐿𝑝,𝑞𝑡,0}=𝜆^{−𝑑(1+1/𝑝−1/𝑞)}_ℝ^𝑑_ℝ𝑑|||𝑉_𝑔𝜆−1𝑓(𝑥,𝜔)|||^𝑝𝜆⁻¹𝑥^𝑝𝑡𝑑𝑥^𝑞/𝑝𝑑𝜔^1/𝑞.(4.29) Observe that 𝜆⁻¹(𝑥)≳𝜆⁻¹||ℓ||if|𝑥|≍||ℓ||(4.30) 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: _ℝ𝑑|||𝑉_𝑔𝜆−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)𝑦}𝜓(𝑦−𝑘)𝑔𝑦−𝑥𝜆𝑑𝑦|||||^𝑝×𝜆⁻¹𝑥^𝑝𝑡𝑑𝑥⎞⎟⎟⎠^1/𝑝≳_ℓ≠0||ℓ||^{−𝑑/𝑞−𝜖−𝑡}𝜆^𝑑||𝑔̂(−𝜆(𝜔−ℓ))||𝜆^−𝑡||ℓ||^𝑡^𝑝^1/𝑝≳_ℓ≠0||ℓ||^{−𝑑/𝑞−𝜖}𝜆^𝑑−𝑡||𝑔̂(−𝜆(𝜔−ℓ))||^𝑝^1/𝑝.(4.31) Consequently, ‖‖𝑉_𝑔𝑓_𝜆‖‖_{𝐿𝑝,𝑞𝑡,0}=𝜆^{−𝑑(1+1/𝑝−1/𝑞)}_ℝ^𝑑_ℝ𝑑|||𝑉_𝑔𝜆−1𝑓(𝑥,𝜔)|||^𝑝𝜆⁻¹𝑥^𝑝𝑡𝑑𝑥^𝑞/𝑝𝑑𝜔^1/𝑞≳𝜆^{−𝑑(1+1/𝑝−1/𝑞)}⎛⎜⎜⎝_ℝ𝑑_ℓ≠0||ℓ||^{−𝑑/𝑞−𝜖}𝜆^𝑑−𝑡||𝑔̂(−𝜆(𝜔−ℓ))||^𝑝^𝑞/𝑝𝑑𝜔⎞⎟⎟⎠^1/𝑞=𝜆^{𝑑−𝑡−𝑑/𝑞}𝜆^{−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/𝑝}𝜆^{𝑑/𝑞+𝜖}⎛⎜⎜⎝_{||ℓ||≤1/𝜆}⎞⎟⎟⎠^1/𝑝≳𝜆^{−𝑡−2(𝑑/𝑝)+𝑑/𝑞+𝜖},(4.32) which completes the proof.

Lemma 4.5. Let 1≤𝑝,𝑞≤∞ be such that (1/𝑝,1/𝑞)∈𝐼₃. Let 𝜖>0, 𝑡<0, and 0<𝜆<1.(a) If 𝑡≤−𝑑 define 𝑓(𝑥)=𝜆^{𝑑/𝑞−2𝑑/𝑝+2𝑑}_𝑘_≠0||𝑘||^−𝜖/2𝑇_𝜆²_𝑘𝜑(𝑥),in𝒮^ℝ^𝑑.(4.33) Then, there exists a constant 𝐶>0 such that ‖𝑓‖_ℳ^{𝑝,𝑞𝑡,0}≤𝐶, uniformly with respect to 𝜆. Moreover, ‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞𝑡,0}≳𝜆^{𝑑𝜇2(𝑝,𝑞)+𝜖},∀0<𝜆<1.(4.34)(b) If −𝑑<𝑡<0, choose a positive integer 𝑁 large enough such that 1/𝑁<(𝑝−1)/2−𝑝𝑡/2𝑑. Define 𝑓(𝑥)=𝜆^𝑑/𝑞_𝑘_≠0||𝑘||^{𝑑(2/𝑁𝑝−1)−𝜖/𝑁}𝑇_𝜆^𝑁_𝑘𝜑(𝑥),in𝒮^ℝ^𝑑.(4.35)Then, the conclusions of part (a) still hold.

Proof. (a) For the range of 𝑝,𝑞 being considered, 𝑑/𝑞+2𝑑−2𝑑/𝑝=𝑑𝜇₂(𝑝,𝑞)+2𝑑≥0, and so if 𝜆<1, then 𝜆^{𝑑/𝑞+2𝑑−2𝑑/𝑝}<1.
Next, notice that 𝒢(𝜑,𝜆²,1) is a Gabor frame. So, to check that 𝑓∈ℳ^{𝑝,𝑞𝑡,0}, we only need to verify that the sequence 𝑐={𝑐_𝑘ℓ}={|𝑘|^−𝜖/2𝛿_ℓ,0,𝑘≠0}_{𝑘,ℓ∈ℤ}^𝑑∈ℓ^{𝑝,𝑞𝑡,0}. But, the condition 𝑡≤−𝑑 guarantees this, since ‖𝑐‖_ℓ^{𝑝,𝑞𝑡,0}=𝜆^{𝑑/𝑞+2𝑑−2𝑑/𝑝}_𝑘≠0||𝑘||^{−𝑝𝜖/2}1+||𝑘||²^𝑝𝑡/2^1/𝑝≤𝐶.(4.36)
Next, as in the proof of Lemma 4.3, we have ‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞𝑡,0}=sup_{‖𝑔‖ℳ𝑝′′,𝑞−𝑡,0=1}||⟨𝑓_𝜆,𝑔⟩||≥‖𝜑‖^ℳ−2_{𝑝′,𝑞′−𝑡,0}||⟨𝑓_𝜆,𝜑⟩||.(4.37) In this case, ⟨𝑓_𝜆,𝜑⟩=𝜆^{2𝑑+𝑑𝜇2(𝑝,𝑞)}_𝑘_≠0=𝜆_{2𝑑+𝑑𝜇2(𝑝,𝑞)}||𝑘||^−𝜖/2𝑉_𝜑𝜑_𝜆(𝜆𝑘,0)_𝑘_≠0||𝑘||^−𝜖/21+𝜆²^−𝑑/2𝑒^{−𝜋𝜆4|𝑘|2/(𝜆2+1)}.(4.38) Therefore, if 𝜆<1, ‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞𝑡,0}≥𝐶𝜆^{2𝑑+𝑑𝜇2(𝑝,𝑞)}_𝑘_≠0||𝑘||^−𝜖/21+𝜆²^−𝑑/2𝑒^{−𝜋𝜆4|𝑘|2/(𝜆2+1)}≥𝐶𝜆^{2𝑑+𝑑𝜇2(𝑝,𝑞)}_𝑘_≠0||𝑘||^−𝜖/2≥𝐶𝜆^{2𝑑+𝑑𝜇2(𝑝,𝑞)}_||𝑘|_|0<<1/𝜆2||𝑘||^−𝜖/2≥𝐶𝜆^{2𝑑+𝑑𝜇2(𝑝,𝑞)}𝜆^𝜖𝜆^−2𝑑=𝐶𝜆^{𝑑𝜇2(𝑝,𝑞)+𝜖}(4.39) 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,𝑞𝑡,0}. In addition, the main estimate is that ‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞𝑡,0}≥𝐶𝜆^𝑑/𝑞_𝑘_≠0||𝑘||^{𝑑(2/𝑁𝑝−1)−𝜖/𝑁}1+𝜆²^−𝑑/2𝑒^{−𝜋𝜆2𝑁|𝑘|2/(𝜆2+1)}≥𝐶𝜆^𝑑/𝑞_||𝑘||_{0<<1/𝜆𝑁}||𝑘||^{𝑑(2/𝑁𝑝−1)−𝜖/𝑁}≥𝐶𝜆^{𝑑𝜇2(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 𝑠≤0, 0<𝜆≤1, consider the family of functions 𝑓(𝑥)=𝜆^𝑠𝑀_𝜆⁻¹_𝑒1𝜑(𝑥),𝑒₁=(1,0,0,…,0).(4.41) Then, there exists a constant 𝐶>0 such that ‖𝑓‖_ℳ^{𝑝,𝑞0,𝑠}≤𝐶, uniformly with respect to 𝜆. Moreover, ‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞0,𝑠}≳𝜆^{𝑠−𝑑/𝑝},∀0<𝜆≤1.(4.42)

Proof. We have ‖𝑓‖_ℳ^{𝑝,𝑞0,𝑠}≍‖‖𝑉_𝜑𝑓(𝑥,𝜔)⟨𝜔⟩^𝑠‖‖_{𝐿𝑝,𝑞}=𝜆^𝑠‖‖𝑉_𝜑𝜑𝑥,𝜔−𝜆⁻¹𝑒₁⟨𝜔⟩^𝑠‖‖_{𝐿𝑝,𝑞}=𝜆^𝑠‖‖𝑉_𝜑𝜑(𝑥,𝜔)𝜔+𝜆⁻¹𝑒₁^𝑠‖‖_{𝐿𝑝,𝑞}≲𝜆^𝑠𝜆^−𝑠‖‖𝑉_𝜑𝜑⟨𝜔⟩^−𝑠‖‖_{𝐿𝑝,𝑞}≲1,(4.43) where we have used again the fact that the weight ⟨⋅⟩^𝑠 is ⟨⋅⟩^−𝑠-moderate. Thus, the functions 𝑓 have norms in ℳ^{𝑝,𝑞0,𝑠} uniformly bounded with respect to 𝜆. Let us now estimate ‖𝑓_𝜆‖_ℳ^{𝑝,𝑞0,𝑠} from below. We have 𝑓_𝜆(𝑥)=𝜆^𝑠𝑀_𝑒1𝜑_𝜆(𝑥).(4.44) By using (4.5), we obtain ‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞0,𝑠}=𝜆^𝑠‖‖𝑉_𝜑𝜑_𝜆𝑥,𝜔−𝑒₁⟨𝜔⟩^𝑠‖‖_{𝐿𝑝,𝑞}≳𝜆^{𝑠−𝑑/𝑝}𝑒^{−𝜋𝑞|𝜔|2}⟨𝜔+𝑒₁⟩^𝑞𝑠𝑑𝜔^1/𝑞≳𝜆^{𝑠−𝑑/𝑝},(4.45) as desired.

Lemma 4.7. Let 1≤𝑝,𝑞≤∞ be such that (1/𝑝,1/𝑞)∈𝐼^∗2. Assume that 𝑠≤0, 𝜖>0 and 𝜆>1.(a) If 𝑞≥2 and 𝑠≤0, or 1≤𝑞≤2 and 𝑠≤−𝑑, define 𝑓(𝑥)=_ℓ≠0||ℓ||^{𝑑(1/𝑞−1)−𝜖}𝑒^{2𝜋𝑖𝜆−1ℓ⋅𝑥}𝜑(𝑥)=_ℓ≠0||ℓ||^{𝑑(1/𝑞−1)−𝜖}𝑀_𝜆⁻¹_ℓ𝜑(𝑥),in𝒮^ℝ^𝑑.(4.46)Then, there exists a constant 𝐶>0 such that ‖𝑓‖_ℳ^{𝑝,𝑞0,𝑠}≤𝐶, uniformly with respect to 𝜆. Moreover, ‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞0,𝑠}≳𝜆^{𝑑(1/𝑞−1)−𝜖},∀𝜆>1.(4.47)(b) If 1≤𝑞≤2 and −𝑑<𝑠<0, choose a positive integer 𝑁 such that 1/𝑁<−𝑠𝑞/𝑑, and define 𝑓(𝑥)=_ℓ≠0||ℓ||^{𝑑(1/𝑁𝑞−1)−𝜖/𝑁}𝑒^{2𝜋𝑖𝜆−𝑁ℓ⋅𝑥}𝜑(𝑥)=_ℓ≠0||ℓ||^{𝑑(1/𝑁𝑞−1)−𝜖/𝑁}𝑀_𝜆^−𝑁_ℓ𝜑(𝑥),in𝒮^ℝ^𝑑.(4.48)Then, the conclusions of part (a) still hold.

Proof. (a) First of all, notice that 𝒢(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}∈ℓ^𝑞𝑠, which ensures that the function 𝑓 defined above belongs to ℳ^{𝑝,𝑞0,𝑠}. 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 ‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞0,𝑠}≥𝐶_ℓ≠0||ℓ||^{𝑑(1/𝑞−1)−𝜖}1+𝜆²^−𝑑/2𝑒^{−𝜋|ℓ|2/(𝜆2+1)}≥𝐶𝜆^−𝑑_||_{ℓ||0<≤𝜆}||ℓ||^{𝑑(1/𝑞−1)−𝜖}𝑒^{−𝜋|ℓ|2/(𝜆2+1)},(4.49) 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∈ℳ^{𝑝,𝑞0,𝑠} and that ‖𝑓‖_ℳ^{𝑝,𝑞0,𝑠}≤𝐶. Relation (4.47) now follows from ‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞0,𝑠}≥𝐶_ℓ≠0||ℓ||^{𝑑(1/𝑁𝑞−1)−𝜖/𝑁}1+𝜆²^−𝑑/2𝑒^{−𝜋𝜆−2𝑁+2|ℓ|2/(𝜆2+1)}≥𝐶𝜆^−𝑑_||_{ℓ||0<≤𝜆𝑁}||ℓ||^{𝑑(1/𝑁𝑞−1)−𝜖/𝑁}𝑒^{−𝜋𝜆−2𝑁+2|ℓ|2/(𝜆2+1)}≥𝐶𝜆^{𝑑(1/𝑞−1)−𝜖}.(4.50)

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

Lemma 4.8. Let 1≤𝑝,𝑞≤∞, 𝑠≤0, 𝜖>0. Suppose that 𝜓∈𝒮(ℝ^𝑑) satisfies supp𝜓⊂[−1/2,1/2]^𝑑 and 𝜓=1 on [−1/4,1/4]^𝑑.
(a) If 1≤𝑞<∞, define 𝑓(𝑦)=∑_{𝑘∈ℤ𝑑⧵{0}}|𝑘|^{−𝑑/𝑞−𝜖−𝑠}𝑀_𝑘𝑇_𝑘𝜓(𝑦),in𝒮^(ℝ^𝑑). Then, 𝑓∈ℳ^{𝑝,𝑞0,𝑠}(ℝ^𝑑) and‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞0,𝑠}≳𝜆^{−𝑑(2/𝑝−1/𝑞)+𝜖+𝑠},∀0<𝜆≤1.(4.51)(b) If 𝑞=∞, let 𝑓(𝑦)=_𝑘≠0||𝑘||^−𝑠𝑒^{2𝜋𝑖𝑘𝑦}𝑇_𝑘𝜓(𝑦),in𝒮^ℝ^𝑑.(4.52) Then, 𝑓∈ℳ^{𝑝,∞0,𝑠} and‖‖𝑓_𝜆‖‖_ℳ^{𝑝,∞0,𝑠}≳𝜆^{−2𝑑/𝑝+𝑠},∀0<𝜆≤1.(4.53)

Lemma 4.9. Let 1≤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 𝑓(𝑥)=𝜆^𝑑/𝑞_𝑘_≠0||𝑘||^{𝑑(2/𝑁𝑝−1)−𝜖/𝑁}𝑇_𝜆^𝑁_𝑘𝜑(𝑥),in𝒮^ℝ^𝑑.(4.54) Then, there exists a constant 𝐶>0 such that ‖𝑓‖_ℳ^{𝑝,𝑞0,𝑠}≤𝐶, uniformly with respect to 𝜆. Moreover, ‖‖𝑓_𝜆‖‖_ℳ^{𝑝,𝑞0,𝑠}≳𝜆^{𝑑𝜇2(𝑝,𝑞)+𝜖}.(4.55)

Proof. In this case, 𝒢(𝜑,𝜆^𝑁,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∈ℓ^𝑝. Therefore, 𝑓∈ℳ^{𝑝,𝑞0,𝑠} with ‖𝑓‖_ℳ^{𝑝,𝑞0,𝑠}≤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 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 𝜖>0, 𝑠≤0 and 𝜆>1.(a) If 1<𝑞<2, choose a positive integer 𝑁 such that 3/𝑁<𝑞−1. Define 𝑓(𝑥)=𝜆^{𝑑(1−2/𝑞)}_ℓ_≠0||ℓ||^{𝑑(3/𝑁𝑞−1)−𝜖/𝑁}𝑀_𝜆^−𝑁_ℓ𝜑(𝑥),in𝒮^ℝ^𝑑.(4.56)Then, there exists a constant 𝐶>0 such that ‖𝑓‖_ℳ^{𝑝,𝑞0,𝑠}≤𝐶, uniformly with respect to 𝜆. Moreover, ‖‖𝑓_𝜆‖‖_ℳ^{∞,𝑞0,𝑠}≳𝜆^{𝑑/𝑞−𝜖}.(4.57)(b) If 2≤𝑞<∞, choose a positive integer 𝑁 such that 𝑁>2+𝑞. Define 𝑓(𝑥)=𝜆^{𝑑+𝑑(2−𝑁)/𝑞}_ℓ_≠0||ℓ||^{𝑑((𝑁−1)/𝑁𝑞−1)−𝜖/𝑁}𝑀_𝜆^−𝑁_ℓ𝜑(𝑥),in𝒮^ℝ^𝑑.(4.58)Then, the conclusions of part (a) still hold.(c) If 𝑞=1 and 𝑠≤−𝑑, define