On Compactness of Embeddings of Fourier-Lebesgue Spaces into Modulation Spaces

Abstract

It is shown that, for a certain range of parameters, embeddings of Fourier-Lebesgue 𝐿^𝑝_𝑎∩ℱ𝐿^𝑞_𝑏 spaces into modulation spaces 𝑀^𝑟,𝑠_𝛼,𝛽 are compact.

1. Introduction

In [1], Galperin and Gröchenig studied the question of how changing the requirements on smoothness and decay of 𝑓 and 𝑓 affects the lower bound in the uncertainty principle. They derived a class of uncertainty principles in the form 𝑉_𝑔𝑓_{𝐿𝑟,𝑠𝛼,𝛽}≤𝐶𝑓_𝐿^𝑝_𝑎+𝑓_𝐿𝑞𝑏 (1) and partially characterized the range of the parameters 𝑝, 𝑞, 𝑟, 𝑠, 𝑎, 𝑏, 𝛼, and 𝛽 for which (1) holds. Here 𝑓 is the Fourier transform of 𝑓 normalized as ℱ𝑓(𝜔)=𝑓(𝜔)=∫_ℝ𝑑𝑓(𝑥)𝑒^{−2𝜋𝑖𝑥𝜔}𝑑𝑥 and the the quantities ‖𝑓‖_𝐿^𝑝_𝑎=‖(1+|𝑥|)^𝑎𝑓‖_𝐿𝑝 and ‖𝑓‖_𝐿^𝑞_𝑏=‖(1+|𝜔|)^𝑏𝑓‖_𝐿𝑞 are used as measures of the concentration of 𝑓 in time and frequency, respectively. For a fixed 𝑔∈𝒮(ℝ^𝑑), a so-called window function, the STFT of a tempered distribution 𝑓∈𝒮^′(ℝ^𝑑) with respect to 𝑔 is defined by 𝑉_𝑔𝑓(𝑥,𝜔)=_ℝ𝑑𝑓(𝑡)𝑔(𝑡−𝑥)𝑒^{−2𝜋𝑖𝜔⋅𝑡}𝑑𝑡=𝑓,𝑀_𝜔𝑇_𝑥, (2) where the translation and modulation operators are defined by 𝑇_𝑥𝑔(𝑡)=𝑔(𝑡−𝑥) and 𝑀_𝜔𝑔(𝑡)=𝑒^{2𝜋𝑖𝜔𝑡}𝑔(𝑡). To measure the joint time-frequency concentration of a function 𝑓, the mixed weighted Lebesgue norm 𝑉_𝑔𝑓_{𝐿𝑟,𝑠𝛼,𝛽}=_{ℝ𝑑ℝ𝑑}𝑉_𝑔𝑓(𝑥,𝜔)^𝑟(1+|𝑥|)^𝑎𝑟𝑑𝑥^𝑠/𝑟×(1+|𝜔|)^𝛽𝑠𝑑𝜔^1/𝑠 (3) is imposed on the short-time Fourier transform (STFT) of 𝑓. The theory of mixed-norm Lebesgue spaces is developed in [2].

The uncertainty principles of form (1) are equivalent to embeddings of Fourier-Lebesgue spaces into modulation spaces. For a fixed 𝑔∈𝒮(ℝ^𝑑), define 𝑀^𝑟,𝑠_𝛼,𝛽 by the (quasi-)norm ‖𝑓‖_𝑀^𝑟,𝑠_𝛼,𝛽=‖𝑉_𝑔𝑓‖_{𝐿𝑟,𝑠𝛼,𝛽}, and then the uncertainty principle (1) is equivalent to the embedding 𝐿^𝑝_𝑎∩ℱ𝐿^𝑞_𝑏↪𝑀^𝑟,𝑠_𝛼,𝛽, (4) where ℱ𝐿^𝑞_𝑏 denotes the space of the tempered distributions 𝑓 whose Fourier transform 𝑓 is in 𝐿^𝑞_𝑏.

In this paper, we show that embeddings (4) are compact. We prove the following theorem.

Theorem 1. Let 0<𝑟,𝑠≤2, 𝛼,𝛽≥0, and 1≤𝑝,𝑞≤∞. Suppose that 0<𝑟≤𝑝≤∞ and 0<𝑠≤𝑞≤∞.
If 𝑎−𝛼𝑑+1𝑝−1𝑟𝑏−𝛽𝑑+1𝑞−1𝑠1>max𝑟−1𝑞^′+𝛼𝑑,1𝑟−12+𝛼𝑑1×max𝑠−1𝑝^′+𝛽𝑑,1𝑠−12+𝛽𝑑, (5) with all factors being nonnegative, then 𝐿^𝑝_𝑎∩ℱ𝐿^𝑞_𝑏 is compactly embedded in 𝑀^𝑟,𝑠_𝛼,𝛽.

The factors on the left side of (5) are quite natural. When they are both positive, 𝐿^𝑝_𝑎↪𝐿^𝑟_𝛼 and ℱ𝐿^𝑞_𝑏↪ℱ𝐿^𝑠_𝛽. Since this is not enough to guarantee that 𝐿^𝑝_𝑎∩ℱ𝐿^𝑞_𝑏↪𝑀^𝑟,𝑠_𝛼,𝛽 (because the modulation space norm measures the decay of 𝑓 in time and in frequency simultaneously, whereas the Lebesgue space norms of 𝑓 and 𝑓 treat time and frequency as separate inputs), the term on the right side of (5) indicates the exact measure of additional decay that has to be imposed on 𝑓 and 𝑓. However, the strict inequality in (5) implies some excessive decay, which results in tightness of the STFT on sets bounded in 𝐿^𝑝_𝑎∩ℱ𝐿^𝑞_𝑏.

It is interesting to compare Lemma 6 and Theorem 1 with the results obtained in [3, Theorem 3.2], which is concerned with compactness of embeddings into modulation space 𝑀_𝑠^𝑝,𝑞. Whereas the weights 𝑚_𝑠(𝑥,𝜔)=(1+|𝑥|+|𝜔|)^𝑠 used in [3] assume the same rate of decay of the short-time Fourier transform in the time and frequency variables, the weights used in this paper differentiate between these two rates. Thus, the result proved in [3] is not directly applicable to our case.

Our result relies on the following criterion of compactness in modulation spaces in terms of tightness of the STFT.

Theorem 2 (see [4, Theorem 5]). Assume that 𝑔∈𝒮(ℝ^𝑑), 0<𝑟,𝑠<∞, and 𝑆 is a closed and bounded subset of 𝑀^𝑟,𝑠_𝛼,𝛽. Then 𝑆 is compact in 𝑀^𝑟,𝑠_𝛼,𝛽 if and only if, for all 𝜖>0, there exists a compact set 𝑈⊂ℝ^2𝑑, such that sup_𝑓∈𝑆𝜒_𝑈𝑐⋅𝑉_𝑔𝑓_{𝐿𝑟,𝑠𝛼,𝛽}<𝜖. (6)

Remark.  (1) In [4], Theorem 2 was proved in the context of the co-orbit spaces (with more general weight functions) for the case 1≤𝑟,𝑠<∞. However, the same argument works for 𝑀^𝑟,𝑠_𝛼,𝛽, 0<𝑟,𝑠<∞.

(2) For the theory of modulation spaces we refer to [5, Chapter 11-12], [6] and to the original literature [7–9].

(3) It is shown in [1] that condition (5) is optimal. If the inequality is reversed, 𝐿^𝑝_𝑎∩ℱ𝐿^𝑞_𝑏 is not embedded in 𝑀^𝑟,𝑠_𝛼,𝛽.

2. Definitions and Preliminary Results

We first provide the necessary definitions and tools. Our notation and definitions are consistent with those in [5].

2.1. Weights and Mixed Norm Spaces

To alleviate notation, we write ⟨𝑥⟩=1+|𝑥|. We need the following lemma for weighted mixed norm spaces.

Lemma 3 (Hölder’s inequality). Let 𝑝≥𝑟 and 𝑞≥𝑠. Write 𝑡=𝑟(𝑝/𝑟)^′=𝑟𝑝/(𝑝−𝑟) and 𝑢=𝑠(𝑞/𝑠)^′=𝑠𝑞/(𝑞−𝑠). Then ‖𝐹⋅𝐻‖_𝐿^𝑟,𝑠_𝛼,𝛽≤‖𝐹‖_𝐿^𝑝,𝑞_𝑎,𝑏⋅‖𝐻‖_𝐿^𝑡,𝑢_{𝛼−𝑎,𝛽−𝑏}, (7) whenever the right-hand side is finite.

Proof. We write the left-hand side as ‖𝐹⋅𝐻‖_𝐿^𝑟,𝑠_𝛼,𝛽=_{ℝ𝑑ℝ𝑑}|𝐹(𝑥,𝜔)|^𝑟⟨𝑥⟩^𝑎𝑟×|𝐻(𝑥,𝜔)|^𝑟⟨𝑥⟩^{(𝛼−𝑎)𝑟}𝑑𝑥^𝑠/𝑟×⟨𝜔⟩^𝑏𝑠⟨𝜔⟩^{(𝛽−𝑏)𝑠}𝑑𝜔^1/𝑠. (8)
Next apply Hölder’s inequality with exponents 𝑝/𝑟 and (𝑝/𝑟)^′ to the integral in 𝑑𝑥 and with exponents 𝑞/𝑠 and (𝑞/𝑠)^′ to the integral in 𝑑𝜔. This yields ‖𝐹⋅𝐻‖_𝐿^𝑟,𝑠_𝛼,𝛽≤_ℝ𝑑‖𝐹(⋅,𝜔)‖^𝑠_𝐿^𝑝_𝑎⟨𝜔⟩^𝑏𝑠⋅‖𝐻(⋅,𝜔)‖^𝑠_{𝐿𝑡𝛼−𝑎}⟨𝜔⟩^{(𝛽−𝑏)𝑠}𝑑𝜔^1/𝑠≤_ℝ𝑑‖𝐹(⋅,𝜔)‖^𝑞_𝐿𝑝𝑎⟨𝜔⟩^𝑏𝑞𝑑𝜔^1/𝑞⋅_ℝ𝑑‖𝐻(⋅,𝜔)‖^𝑢_{𝐿𝑡𝛼−𝑎}⟨𝜔⟩^{(𝛽−𝑏)𝑢}𝑑𝜔^1/𝑢=‖𝐹‖_𝐿^𝑝,𝑞_𝑎,𝑏⋅‖𝐻‖_𝐿^𝑡,𝑢_{𝛼−𝑎,𝛽−𝑏}, (9) as desired.

We will also use the following elementary embedding.

Lemma 4. Suppose that 1≤𝑟<𝑝≤∞. Then 𝐿^𝑝_𝑎↪𝐿^𝑟_𝛼 if and only if (𝑎−𝛼)/𝑑+1/𝑝−1/𝑟>0.

The following technical lemma about weighted mixed norms of certain characteristic functions is instrumental for the main embedding result.

Lemma 5 (see [1]). Let 𝛼,𝛽∈ℝ, 0<𝑟,𝑠≤∞, and 𝜎>0. Define 𝐴_𝜎={(𝑥,𝜔)∈ℝ^2𝑑∶|𝑥|≥|𝜔|^1/𝜎} and 𝐵_𝜎=ℝ^2𝑑⧵𝐴_𝜎={(𝑥,𝜔)∈ℝ^2𝑑∶|𝑥|<|𝜔|^1/𝜎}.
Then 𝜒_𝐵𝜎∈𝐿^𝑟,𝑠_𝛼,𝛽 provided that 1𝜎𝛼max𝑑+1𝑟+𝛽,0𝑑+1𝑠<0. (10)
Furthermore, 𝜒_𝐴𝜎∈𝐿^𝑟,𝑠_𝛼,𝛽 provided that 𝛼𝑑+1𝑟1<0,𝜎𝛼𝑑+1𝑟+𝛽𝑑+1𝑠<0. (11)

3. Proof of the Main Result

In order to prove Theorem 1, we first establish compactness of certain embeddings between modulation spaces.

Lemma 6. Assume that 0<𝑟,𝑠<∞, 𝛼,𝛽≥0, and 𝑝,𝑞^′∈[𝑟,𝑠^′]. If 𝑎−𝛼𝑑+1𝑝−1𝑟𝑏−𝛽𝑑+1𝑞−1𝑠>𝛼𝑑−1𝑞^′+1𝑟𝛽𝑑−1𝑝^′+1𝑠 (12) with all factors nonnegative, then 𝑀^𝑝,𝑝′_𝑎,0∩𝑀^𝑞′,𝑞_0,𝑏 is compactly embedded in 𝑀^𝑟,𝑠_𝛼,𝛽.

Proof. We split the time-frequency plane into the two regions 𝐴_𝜎={(𝑥,𝜔)∈ℝ^2𝑑∶|𝑥|≥|𝜔|^1/𝜎} and 𝐵_𝜎=ℝ^2𝑑⧵𝐴_𝜎 for some 𝜎>0 to be determined later, and we estimate the modulation space (quasi-)norm of 𝑓 accordingly by 𝑓_𝑀^𝑟,𝑠_𝛼,𝛽=𝑉_𝑔𝑓_{𝐿𝑟,𝑠𝛼,𝛽}≤𝐶𝑉_𝑔𝑓⋅𝜒_{𝐴𝜎𝐿𝑟,𝑠𝛼,𝛽}+𝑉_𝑔𝑓⋅𝜒_{𝐵𝜎𝐿𝑟,𝑠𝛼,𝛽}. (13) We then apply Hölder’s inequality (Lemma 3) to each term and use Lemma 5. Writing 𝑡=𝑟(𝑝/𝑟)^′=𝑟𝑝/(𝑝−𝑟) and 𝑢=𝑠(𝑝^′/𝑠)^′=𝑠𝑝^′/(𝑝^′−𝑠), we obtain that 𝑉_𝑔𝑓⋅𝜒_{𝐴𝜎𝐿𝑟,𝑠𝛼,𝛽}≤𝑉_𝑔𝑓_{𝐿′𝑝,𝑝𝑎,0}⋅𝜒_{𝐴𝜎𝐿𝑡,𝑢𝛼−𝑎,𝛽}=𝑓_{𝑀′𝑝,𝑝𝑎,0}⋅𝜒_{𝐴𝜎𝐿𝑡,𝑢𝛼−𝑎,𝛽}. (14)
Lemma 5 implies that 𝜒_𝐴𝜎∈𝐿^𝑡,𝑢_{𝛼−𝑎,𝛽}, whenever 𝛼−𝑎𝑑+1𝑡=𝛼−𝑎𝑑+1𝑟−1𝑝1<0,𝜎𝛼−𝑎𝑑+1𝑡+𝛽𝑑+1𝑢<0. (15)
Equivalently, 𝑎−𝛼𝑑+1𝑝−1𝑟𝛽>𝜎𝑑−1𝑝^′+1𝑠≥0. (16)
Similarly, we obtain for the second term that 𝑉_𝑔𝑓⋅𝜒_{𝐵𝜎𝐿𝑟,𝑠𝛼,𝛽}≤𝑉_𝑔𝑓_{𝐿𝑞′,𝑞0,𝑏}⋅𝜒_𝐵𝜎𝐿𝑡_,̃𝑢_{𝛼,𝛽−𝑏}, (17) where 𝑡̃=𝑟(𝑞^′/𝑟)^′=𝑟𝑞^′/(𝑞^′−𝑟) and 𝑢=𝑠(𝑞/𝑠)^′=𝑠𝑞/(𝑞−𝑠). By Lemma 5 we have 𝜒_𝐵𝜎𝑡̃∈𝐿^,𝑢_{𝛼,𝛽−𝑏} provided that 1𝜎𝛼max𝑑+1𝑡̃,0+𝛽−𝑏𝑑+1𝑢<0, (18) or equivalently, 𝑏−𝛽𝑑+1𝑞−1𝑠>1𝜎𝛼max𝑑−1𝑞^′+1𝑟=1,0𝜎𝛼𝑑−1𝑞^′+1𝑟≥0. (19) Finally, if (12) holds, then there exists 𝜎>0 so that both (16) and (19) and all factors are positive. Hence, 𝜒_𝐴𝜎∈𝐿^𝑡,𝑢_{𝛼−𝑎,𝛽} and 𝜒_𝐵𝜎𝑡̃∈𝐿^,𝑢_{𝛼,𝛽−𝑏}. It follows that, for a given 𝜖>0, there exist compact sets 𝑈₁,𝑈₂⊂ℝ^2𝑑, such that 𝜒_𝐴𝜎⋅𝜒_{𝑈𝑐1𝐿𝑡,𝑢𝛼−𝑎,𝛽}<𝜖,𝜒2𝐶^𝐵𝜎⋅𝜒_{𝑈𝑐2𝐿}𝑡_,̃𝑢_{𝛼,𝛽−𝑏}<𝜖.2𝐶 (20) Define 𝑈=𝑈₁∪𝑈₂. The combination of (14), (17), and (20) yields that sup_{𝑓∈Ball(𝑀′𝑝,𝑝𝑎,0∩𝑀𝑞′,𝑞0,𝑏)}𝜒_𝑈𝑐⋅𝑉_𝑔𝑓_{𝐿𝑟,𝑠𝛼,𝛽}<𝜖. (21) Therefore, by Theorem 2, the embedding 𝑀^𝑝,𝑝′_𝑎,0∩𝑀^𝑞′,𝑞_0,𝑏↪𝑀^𝑟,𝑠_𝛼,𝛽 is compact.

The next lemma relates weighted 𝐿^𝑝 spaces to modulation spaces and can be considered a version of the Hausdorff-Young inequality for the STFT. For more general inequalities see [10, Section 4.2.1].

Lemma 7 (see [1, 11]). Suppose that 1≤𝑝≤2. Then (a)𝐿^𝑝_𝑎↪𝑀^𝑝,𝑝′_𝑎,0 and 𝑉_𝑔𝑓_{𝐿′𝑝,𝑝𝑎,0}≤𝑓_𝐿^𝑝_𝑎𝑔_𝐿^𝑝_|𝑎|, (22) where 𝐶=𝐶(𝑝,𝑎) is independent of 𝑓 and 𝑔,(b)ℱ𝐿^𝑝_𝑎↪𝑀^𝑝′,𝑝_0,𝑎 and 𝑉_𝑔𝑓_{𝐿𝑝′,𝑝0,𝑎}≤𝑓_𝐿𝑝𝑎𝑔_𝐿^𝑝_|𝑎|, (23) where 𝐶=𝐶(𝑝,𝑎) is independent of 𝑓 and 𝑔.

The combination of Lemmas 6 and 7 leads to Theorem 1.

Proof of Theorem 1. Following the proof of the main result in [1], we distinguish several cases.
Case  1. If 𝑟≤𝑝≤2 and 𝑠≤𝑞≤2, then 𝐿^𝑝_𝑎↪𝑀^𝑝,𝑝′_𝑎,0 and ℱ𝐿^𝑞_𝑏↪𝑀^𝑞′,𝑞_0,𝑏 by Lemma 7. As a consequence of (5), Lemma 6 is applicable and thus 𝑀^𝑝,𝑝′_𝑎,0∩𝑀^𝑞′,𝑞_0,𝑏 is compactly embedded in 𝑀^𝑟,𝑠_𝛼,𝛽. Combining the above, we obtain that 𝐿^𝑝_𝑎∩ℱ𝐿^𝑞_𝑏 is compactly embedded in 𝐿^𝑟,𝑠_𝛼,𝛽.
Case  2.  𝑝>2, 𝑠≤𝑞≤2.
By continuity there exists 𝑐>0 such that 𝑎−𝛼𝑑+1𝑝−1𝑟𝑏−𝛽𝑑+1𝑞−1𝑠>𝑐−𝛼𝑑+12−1𝑟𝑏−𝛽𝑑+1𝑞−1𝑠>1𝑟−1𝑞^′+𝛼𝑑1𝑠−12+𝛽𝑑. (24)
The first inequality in (24) implies that (𝑎−𝑐)/𝑑+1/𝑝−1/2>0 and thus 𝐿^𝑝_𝑎↪𝐿²_𝑐 by Lemma 4. In view of (5) Case  1 is now applicable with 𝐿²_𝑐 instead of 𝐿^𝑝_𝑎, and we obtain the compact embedding 𝐿^𝑝_𝑎∩ℱ𝐿^𝑞_𝑏↪𝐿²_𝑐∩ℱ𝐿^𝑞_𝑏↪𝑀^𝑟,𝑠_𝛼,𝛽. (25)
Case  3.  𝑞>2, 𝑟≤𝑝≤2 is similar.
Case  4.  𝑝,𝑞>2.
By continuity we may choose 𝑐>0 and 𝛾>0, so that the inequalities (𝑎−𝑐)/𝑑+1/𝑝−1/2>0,(𝑏−𝛾)/𝑑+1/𝑞−1/2>0, and 𝑎−𝛼𝑑+1𝑝−1𝑟𝑏−𝛽𝑑+1𝑞−1𝑠>𝑐−𝛼𝑑+12−1𝑟𝛾−𝛽𝑑+12−1𝑠>1𝑟−12+𝛼𝑑1𝑠−12+𝛽𝑑 (26) hold simultaneously. It follows that 𝐿^𝑝_𝑎↪𝐿²_𝑐 and ℱ𝐿^𝑞_𝑏↪ℱ𝐿²_𝛾 by Lemma 4 and that 𝐿²_𝑐∩ℱ𝐿²_𝛾 is compactly embedded in 𝑀^𝑟,𝑠_𝛼,𝛽 by Lemmas 6 and 7. Therefore 𝐿^𝑝_𝑎∩ℱ𝐿^𝑞_𝑏 is compactly embedded in 𝑀^𝑟,𝑠_𝛼,𝛽, as desired. The theorem is proved completely.

Acknowledgments

The author would like to thank K. Gröchenig and H. G. Feichtinger for inspiring discussions and important suggestions.