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.


Introduction
In [1], Galperin and Gröchenig studied the question of how changing the requirements on smoothness and decay of  and f 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 f is the Fourier transform of  normalized as F() = f() = ∫ R  () −2  and the the quantities ‖‖    = ‖(1 + ||)  ‖   and ‖ f‖    = ‖(1 + ||)  f‖   are used as measures of the concentration of  in time and frequency, respectively.For a fixed  ∈ S(R  ), a so-called window function, the STFT of a tempered distribution  ∈ S  (R  ) with respect to  is defined by where the translation and modulation operators are defined by   () = ( − ) and   () =  2 ().
is imposed on the short-time Fourier transform (STFT) of .
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  ∈ S(R  ), define  , , by the (quasi-)norm ‖‖  , , = ‖  ‖  , , , and then the uncertainty principle (1) is equivalent to the embedding where F   denotes the space of the tempered distributions  whose Fourier transform f is in    .In this paper, we show that embeddings (4) are compact.We prove the following theorem.Theorem 1.Let 0 < ,  ≤ 2, ,  ≥ 0, and 1 ≤ ,  ≤ ∞.Suppose that 0 <  ≤  ≤ ∞ and 0 <  ≤  ≤ ∞. (because the modulation space norm measures the decay of  in time and in frequency simultaneously, whereas the Lebesgue space norms of  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  and f.However, the strict inequality in (5) implies some excessive decay, which results in tightness of the STFT on sets bounded in    ∩ F   .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   (, ) = (1 + || + ||)  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.

International Journal of Analysis
Our result relies on the following criterion of compactness in modulation spaces in terms of tightness of the STFT.
(3) It is shown in [1] that condition (5) is optimal.If the inequality is reversed,    ∩ F   is not embedded in  , , .

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

Weights and Mixed
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.
The following technical lemma about weighted mixed norms of certain characteristic functions is instrumental for the main embedding result.
By Lemma 5 we have    ∈  t,ũ ,− provided that or equivalently, Finally, if (12) holds, then there exists  > 0 so that both ( 16) and ( 19) and all factors are positive.Hence,    ∈ where  = (, ) is independent of  and .
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.