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

Authors: Yevgeniy V. Galperin
Journal: International Journal of Analysis
 (2013)

---

## Abstract

It is shown that, for a certain range of parameters, embeddings of Fourier-LebesgueLap∩ℱLbq  spaces into modulation spaces Mα,βr,s are compact.


---

## Body

## 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.
Let0<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 Lemma6 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 [4, Theorem 5]).
Assume thatg∈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).
Letp≥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 exponentsp/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 that1≤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 [1]).
Letα,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}.
ThenχBσ∈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.


## 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).
Letp≥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 exponentsp/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 that1≤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 [1]).
Letα,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}.
ThenχBσ∈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 Theorem1, we first establish compactness of certain embeddings between modulation spaces.Lemma 6.
Assume that0<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 regionsAσ={(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.
Lemma5 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 weightedLp 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 that1≤p≤2. Then (a)
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,(b)
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 Lemmas6 and 7 leads to Theorem 1.Proof of Theorem1.
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 existsc>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 choosec>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.



---

*Source: 681573-2013-12-18.xml*