On Compactness of Embeddings of Fourier-Lebesgue Spaces into Modulation Spaces
It is shown that, for a certain range of parameters, embeddings of Fourier-Lebesgue spaces into modulation spaces are compact.
In , Galperin and Gröchenig studied the question of how changing the requirements on smoothness and decay of and affects the lower bound in the uncertainty principle. They derived a class of uncertainty principles in the form and partially characterized the range of the parameters , , , , , , , and for which (1) holds. Here is the Fourier transform of normalized as and the the quantities and are used as measures of the concentration of in time and frequency, respectively. For a fixed , a so-called window function, the STFT of a tempered distribution with respect to is defined by where the translation and modulation operators are defined by and . To measure the joint time-frequency concentration of a function , the mixed weighted Lebesgue norm is imposed on the short-time Fourier transform (STFT) of . The theory of mixed-norm Lebesgue spaces is developed in .
The uncertainty principles of form (1) are equivalent to embeddings of Fourier-Lebesgue spaces into modulation spaces. For a fixed , define by the (quasi-)norm , and then the uncertainty principle (1) is equivalent to the embedding where denotes the space of the tempered distributions whose Fourier transform is in .
In this paper, we show that embeddings (4) are compact. We prove the following theorem.
Theorem 1. Let , , and . Suppose that and .
If with all factors being nonnegative, then is compactly embedded in .
The factors on the left side of (5) are quite natural. When they are both positive, and . Since this is not enough to guarantee that (because the modulation space norm measures the decay of in time and in frequency simultaneously, whereas the Lebesgue space norms of and 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 and . However, the strict inequality in (5) implies some excessive decay, which results in tightness of the STFT on sets bounded in .
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 . Whereas the weights used in  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  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 , , and is a closed and bounded subset of . Then is compact in if and only if, for all , there exists a compact set , such that
2. Definitions and Preliminary Results
We first provide the necessary definitions and tools. Our notation and definitions are consistent with those in .
2.1. Weights and Mixed Norm Spaces
To alleviate notation, we write . We need the following lemma for weighted mixed norm spaces.
Lemma 3 (Hölder’s inequality). Let and . Write and . Then whenever the right-hand side is finite.
Proof. We write the left-hand side as
Next apply Hölder’s inequality with exponents and to the integral in and with exponents and to the integral in . This yields as desired.
We will also use the following elementary embedding.
Lemma 4. Suppose that . Then if and only if .
The following technical lemma about weighted mixed norms of certain characteristic functions is instrumental for the main embedding result.
Lemma 5 (see ). Let , , and . Define and .
Then provided that
Furthermore, provided that
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 , , and . If with all factors nonnegative, then is compactly embedded in .
Proof. We split the time-frequency plane into the two regions and for some to be determined later, and we estimate the modulation space (quasi-)norm of accordingly by
We then apply Hölder’s inequality (Lemma 3) to each term and use Lemma 5. Writing and , we obtain that
Lemma 5 implies that , whenever
Similarly, we obtain for the second term that where and . By Lemma 5 we have provided that or equivalently, Finally, if (12) holds, then there exists so that both (16) and (19) and all factors are positive. Hence, and . It follows that, for a given , there exist compact sets , such that Define . The combination of (14), (17), and (20) yields that Therefore, by Theorem 2, the embedding is compact.
The next lemma relates weighted 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 ].
Proof of Theorem 1. Following the proof of the main result in , we distinguish several cases.
Case 1. If and , then and by Lemma 7. As a consequence of (5), Lemma 6 is applicable and thus is compactly embedded in . Combining the above, we obtain that is compactly embedded in .
Case 2. , .
By continuity there exists such that
The first inequality in (24) implies that and thus by Lemma 4. In view of (5) Case 1 is now applicable with instead of , and we obtain the compact embedding
Case 3. , is similar.
Case 4. .
By continuity we may choose and , so that the inequalities , and hold simultaneously. It follows that and by Lemma 4 and that is compactly embedded in by Lemmas 6 and 7. Therefore is compactly embedded in , as desired. The theorem is proved completely.
The author would like to thank K. Gröchenig and H. G. Feichtinger for inspiring discussions and important suggestions.
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