It is shown that, for a certain range of parameters, embeddings of Fourier-Lebesgue Lap∩ℱLbq spaces into modulation spaces Mα,βr,s 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
(1)∥Vgf∥Lα,βr,s≤C(∥f∥Lap+∥f^∥Lbq)
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-2πixωdx and the the quantities ∥f∥Lap=∥(1+|x|)af∥Lp and ∥f^∥Lbq=∥(1+|ω|)bf^∥Lq are used as measures of the concentration of f in time and frequency, respectively. For a fixed g∈𝒮(ℝd), a so-called window function, the STFT of a tempered distribution f∈𝒮′(ℝd) with respect to g is defined by
(2)Vgf(x,ω)=∫ℝdf(t)g(t-x)¯e-2πiω·tdt=〈f,MωTx〉,
where the translation and modulation operators are defined by Txg(t)=g(t-x) and Mωg(t)=e2πiωtg(t). To measure the joint time-frequency concentration of a function f, the mixed weighted Lebesgue norm
(3)∥Vgf∥Lα,βr,s=(∫ℝd(∫ℝd|Vgf(x,ω)|r(1+|x|)ardx)s/r∥Vgf∥Lα,βr,s=11×(1+|ω|)βsdω(∫ℝd(∫ℝd|Vgf(x,ω)|r(1+|x|)ardx)s/r)1/s
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(ℝd), define Mα,βr,s by the (quasi-)norm ∥f∥Mα,βr,s=∥Vgf∥Lα,βr,s, and then the uncertainty principle (1) is equivalent to the embedding
(4)Lap∩ℱLbq↪Mα,βr,s,
where ℱLbq denotes the space of the tempered distributions f whose Fourier transform f^ is in Lbq.

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
(5)(a-αd+1p-1r)(b-βd+1q-1s)>max{(1r-1q′+αd),(1r-12+αd)}×max{(1s-1p′+βd),(1s-12+βd)},
with all factors being nonnegative, then Lap∩ℱLbq is compactly embedded in Mα,βr,s.

The factors on the left side of (5) are quite natural. When they are both positive, Lap↪Lαr and ℱLbq↪ℱLβs. Since this is not enough to guarantee that Lap∩ℱLbq↪Mα,βr,s (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 Lap∩ℱLbq.

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 Msp,q. Whereas the weights ms(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 [<xref ref-type="bibr" rid="B2">4</xref>, Theorem 5]).

Assume that g∈𝒮(ℝd), 0<r,s<∞, and S is a closed and bounded subset of Mα,βr,s. Then S is compact in Mα,βr,s if and only if, for all ϵ>0, there exists a compact set U⊂ℝ2d, such that
(6)supf∈S∥χUc·Vgf∥Lα,βr,s<ϵ.

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 Mα,βr,s, 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, Lap∩ℱLbq is not embedded in Mα,βr,s.

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
(7)∥F·H∥Lα,βr,s≤∥F∥La,bp,q·∥H∥Lα-a,β-bt,u,
whenever the right-hand side is finite.

Proof.

We write the left-hand side as
(8)∥F·H∥Lα,βr,s=(∑xx∫ℝd(∑xx∫ℝd|F(x,ω)|r〈x〉ar111111111111111111111×|H(x,ω)|r〈x〉(α-a)rdx∑xx)s/r∥F·H∥Lα,βr,s=11111×〈ω〉bs〈ω〉(β-b)sdω(∑xx))1/s.

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
(9)∥F·H∥Lα,βr,s≤(∫ℝd∥F(·,ω)∥Laps〈ω〉bs1111111111111111·∥H(·,ω)∥Lα-ats〈ω〉(β-b)sdω∫ℝd)1/s∥F·H∥Lα,βr,s≤(∫ℝd∥F(·,ω)∥Lapq〈ω〉bqdω)1/q∥F·H∥Lα,βr,s=·(∫ℝd∥H(·,ω)∥Lα-atu〈ω〉(β-b)udω)1/u∥F·H∥Lα,βr,s=∥F∥La,bp,q·∥H∥Lα-a,β-bt,u,
as desired.

We will also use the following elementary embedding.

Lemma 4.

Suppose that 1≤r<p≤∞. Then Lap↪Lαr 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 [<xref ref-type="bibr" rid="B3">1</xref>]).

Let α,β∈ℝ, 0<r,s≤∞, and σ>0. Define Aσ={(x,ω)∈ℝ2d:|x|≥|ω|1/σ} and Bσ=ℝ2d∖Aσ={(x,ω)∈ℝ2d:|x|<|ω|1/σ}.

Then χBσ∈Lα,βr,s provided that
(10)1σmax{αd+1r,0}+βd+1s<0.

Furthermore, χAσ∈Lα,βr,s provided that
(11)αd+1r<0,1σ(αd+1r)+βd+1s<0.

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
(12)(a-αd+1p-1r)(b-βd+1q-1s)>(αd-1q′+1r)(βd-1p′+1s)
with all factors nonnegative, then Ma,0p,p′∩M0,bq′,q is compactly embedded in Mα,βr,s.

Proof.

We split the time-frequency plane into the two regions Aσ={(x,ω)∈ℝ2d:|x|≥|ω|1/σ} and Bσ=ℝ2d∖Aσ for some σ>0 to be determined later, and we estimate the modulation space (quasi-)norm of f accordingly by
(13)∥f∥Mα,βr,s=∥Vgf∥Lα,βr,s∥f∥Mα,βr,s≤C(∥Vgf·χAσ∥Lα,βr,s+∥Vgf·χBσ∥Lα,βr,s).
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
(14)∥Vgf·χAσ∥Lα,βr,s≤∥Vgf∥La,0p,p′·∥χAσ∥Lα-a,βt,u∥Vgf·χAσ∥Lα,βr,s=∥f∥Ma,0p,p′·∥χAσ∥Lα-a,βt,u.

Lemma 5 implies that χAσ∈Lα-a,βt,u, whenever
(15)α-ad+1t=α-ad+1r-1p<0,1σ(α-ad+1t)+βd+1u<0.

Equivalently,
(16)a-αd+1p-1r>σ(βd-1p′+1s)≥0.

Similarly, we obtain for the second term that
(17)∥Vgf·χBσ∥Lα,βr,s≤∥Vgf∥L0,bq′,q·∥χBσ∥Lα,β-bt~,u~,
where t~=r(q′/r)′=rq′/(q′-r) and u~=s(q/s)′=sq/(q-s). By Lemma 5 we have χBσ∈Lα,β-bt~,u~ provided that
(18)1σmax{αd+1t~,0}+β-bd+1u~<0,
or equivalently,
(19)b-βd+1q-1s>1σmax{αd-1q′+1r,0}b-βd+1q-1s=1σ(αd-1q′+1r)≥0.
Finally, if (12) holds, then there exists σ>0 so that both (16) and (19) and all factors are positive. Hence, χAσ∈Lα-a,βt,u and χBσ∈Lα,β-bt~,u~. It follows that, for a given ϵ>0, there exist compact sets U1,U2⊂ℝ2d, such that
(20)∥χAσ·χU1c∥Lα-a,βt,u<ϵ2C,∥χBσ·χU2c∥Lα,β-bt~,u~<ϵ2C.
Define U=U1∪U2. The combination of (14), (17), and (20) yields that
(21)supf∈Ball(Ma,0p,p′∩M0,bq′,q)∥χUc·Vgf∥Lα,βr,s<ϵ.
Therefore, by Theorem 2, the embedding Ma,0p,p′∩M0,bq′,q↪Mα,βr,s is compact.

The next lemma relates weighted Lp 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 [<xref ref-type="bibr" rid="B3">1</xref>, <xref ref-type="bibr" rid="B5">11</xref>]).

Suppose that 1≤p≤2. Then

Lap↪Ma,0p,p′ and
(22)∥Vgf∥La,0p,p′≤∥f∥Lap∥g∥L|a|p,
where C=C(p,a) is independent of f and g,

ℱLap↪M0,ap′,p and
(23)∥Vgf∥L0,ap′,p≤∥f^∥Lap∥g^∥L|a|p,
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 <xref ref-type="statement" rid="thm1.1">1</xref>.

Following the proof of the main result in [1], we distinguish several cases.

Case 1. If r≤p≤2 and s≤q≤2, then Lap↪Ma,0p,p′ and ℱLbq↪M0,bq′,q by Lemma 7. As a consequence of (5), Lemma 6 is applicable and thus Ma,0p,p′∩M0,bq′,q is compactly embedded in Mα,βr,s. Combining the above, we obtain that Lap∩ℱLbq is compactly embedded in Lα,βr,s.

Case 2.p>2, s≤q≤2.

By continuity there exists c>0 such that
(24)(a-αd+1p-1r)(b-βd+1q-1s)>(c-αd+12-1r)(b-βd+1q-1s)>(1r-1q′+αd)(1s-12+βd).

The first inequality in (24) implies that (a-c)/d+1/p-1/2>0 and thus Lap↪Lc2 by Lemma 4. In view of (5) Case 1 is now applicable with Lc2 instead of Lap, and we obtain the compact embedding
(25)Lap∩ℱLbq↪Lc2∩ℱLbq↪Mα,βr,s.

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
(26)(a-αd+1p-1r)(b-βd+1q-1s)>(c-αd+12-1r)(γ-βd+12-1s)>(1r-12+αd)(1s-12+βd)
hold simultaneously. It follows that Lap↪Lc2 and ℱLbq↪ℱLγ2 by Lemma 4 and that Lc2∩ℱLγ2 is compactly embedded in Mα,βr,s by Lemmas 6 and 7. Therefore Lap∩ℱLbq is compactly embedded in Mα,βr,s, 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.

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