EMBEDDINGS OF SOME CLASSICAL BANACH SPACES INTO MODULATION SPACES
KASSO A. OKOUDJOU
Abstract. We give sufficient conditions for a tempered distribution to belong to certain modulation spaces by showing embeddings of some Besov-Triebel- Lizorkin spaces into modulation spaces. As a consequence we have a new proof that the H¨older-Lipschitz space Cs(Rd) embeds into the modulation space M ∞,1(Rd)whens>d. This embedding plays an important role in interpreting recent modulation space approaches to pseudodifferential operators.
1. Introduction An important question in time-frequency analysis is the choice of a convenient representation together with an adequate measure of the joint time-frequency con- centration of a signal or distribution. The Fourier transform provides only non- localized frequency information, whereas time-frequency analysis seeks a represen- tation that simultaneously displays the time-domain and frequency-domain behav- iors of a signal. The short-time Fourier transform (STFT) or windowed Fourier transform is an important joint time-frequency representation. It shares with the Fourier transform many useful mathematical properties, including linearity and continuity as operators on appropriate spaces. Moreover, properties of the STFT can be transferred to other important time-frequency representation such as the Wigner distribution or the ambiguity function [4]. To quantitatively measure the time-frequency content of a distribution, Feichtinger introduced a class of Banach spaces called the modulation spaces, denoted Mp,q or (for the weighted versions) Mp,q p,q ω , by imposing a mixed L norm on the STFT [1], [8]. The modulation spaces are often the “right spaces” in which to formulate results on time-frequency analysis. We refer to [8] and the references therein for more details about these spaces. The apparently simple definition of the modulation spaces hides the practical problem of how to decide whether or not a distribution belongs to a given modu- lation space. In principle one has to estimate the Lp,q norm of the STFT, which can be a non-trivial task. Therefore it is important to understand the relation- ship between time-frequency content and other properties of distributions, e.g., smoothness properties. Such relationships may appear in the form of embeddings of certain spaces that measure smoothness and/or decay into modulation spaces. For example, Gr¨ochenig in [7] and Hogan and Lakey in [11] derived sufficient con- ditions for membership in the modulation space M1,1 (also called the Feichtinger
Date: February 28, 2002. 2000 Mathematics Subject Classification. Primary 46E35; Secondary 42B35. Key words and phrases. Besov space, modulation space, Sobolev space, short-time Fourier transform, Triebel-Lizorkin space, time-frequency analysis. The author was partially supported by NSF Grant DMS-9970524. 1 2 KASSO A. OKOUDJOU algebra) from certain uncertainty principles related to the STFT. Precisely, it was p ∩F q ⊂M1,1 shown that La Lb under appropriate conditions on p, q and the weight p p | | p F q parameters a, b,whereLa is a weighted L space (with weight (1 + x ) ), and Lb q is the image of Lb under the Fourier transform. Somewhat more general embed- dings involving weighted Lp spaces were given by Galperin and Gr¨ochenig in [5]. Another interesting example appears in [10]. There, Heil, Ramanathan and Topi- wala proved that Cs(R2d) ⊂M∞,1(R2d)fors>2d. This embedding is particularly important in relation to pseudodifferential operator theory. It was shown in [9] that a Weyl or Kohn-Nirenberg pseudodifferential operator σ(D, X) is bounded on Mp,p(Rd), including M2,2 = L2 in particular, if the associated symbol σ(ξ,x) lies in M∞,1(R2d). Thus this embedding recovers and extends the classical Calderon– Vaillancourt-type theorem that σ(D, X) is bounded on L2(Rd)ifσ(ξ,x) ∈Cs(R2d) with s>2d, cf. [4, Theorem 2.73]. From these examples, we can see the need for practical and easily checked condi- tions implying that a distribution belongs to a given modulation space. Our goal in this paper is to provide some sufficient conditions by proving some embeddings of classical Banach spaces such as the Besov, Triebel-Lizorkin, or Sobolev spaces into the modulation spaces. As corollaries, we obtain some embeddings which general- ize the embedding from [10] mentioned above, and moreover, we will give an easy sufficient condition for membership of a distribution in M1,1 in the special case of dimension d =1. While writing this paper, H. Feichtinger informed us of the unpublished Ph.D thesis of Gr¨obner [6], which also obtains some embeddings of Besov spaces into the modulation spaces. Gr¨obner’s results and techniques are distinct from those presented here. Gr¨obner constructs a continuous family of Banach spaces having the modulation spaces at one endpoint and the Besov spaces at the other. Additionally, after our paper was completed we learned that J. Toft [14] has simultaneously obtained some embeddings of Besov spaces into the modulation spaces. Toft’s techniques are again different, as he applies Young-type inequalities to derive the embeddings. Additionally, the embeddings obtained are distinct from ours. In particular, Toft does not recover the embedding from [10], as we do.
2. Definitions and Background 2.1. The Short Time Fourier Transform and the Modulation Spaces. All d function spaces will be definedR over R unless otherwise stated. The Fourier trans- ∈ 1 ˆ −2πit·ω form of f L is f(ω)= Rd f(t)e dt. The Fourier transform is an isomor- 0 phism of the Schwartz space S onto itself, and extends to the spaceR S of tempered ˇ 2πit·x distributions by duality. The inverse Fourier transform is f(x)= Rd f(t)e dt. Translation, modulation, and dilation of a function f are 2πit·ω −d/2 Txf(t)=f(t−x),Mωf(t)=e f(t), and Daf(t)=|a| f(t/a). The Short-Time Fourier Transform (STFT) of a function f with respect to a window g is Z −2πiω·t Vgf(x, ω)= hf,MωTxgi = e g(t − x)f(t)dt, Rd whenever the integral makes sense. Analogously to the Fourier transform, the STFT 0 extends in a distributional sense to f, g ∈ S , cf. [4, Prop. 1.42]. EMBEDDINGS OF SOME CLASSICAL BANACH SPACES INTO MODULATION SPACES 3
Definition 2.1. Given 1 ≤ p, q ≤∞, and given a window function g ∈ S, Mp,q is 0 the space of all distributions f ∈ S for which the following norm is finite: Z Z q/p 1/q p (1) kfkMp,q = |Vgf(x, ω)| dx dω = kVgfkLp,q , Rd Rd with the usual modifications if p or q is infinite. This definition is independent of the choice of the window g in the sense of equivalent norms. For background and information on the basic properties of the modulation spaces we refer to [1], [2], [3], [8]. The next proposition, due to Feichtinger [1], on complex interpolation of modu- lation spaces will be used in the proof of our results.
Proposition 2.2. Let 1 ≤ p0 < ∞, 1 ≤ q0 < ∞, 1 ≤ p1 ≤∞,1≤q1 ≤∞,and 1 1−θ θ 1 1−θ θ θ∈(0, 1).Ifs=(1−θ)s0+θs1, = + ,and = + then p p0 p1 q q0 q1 Mp0,q0 Mp1,q1 Mp,q (2) , θ = . 2.2. The Besov and Triebel-Lizorkin Spaces. Let ψ ∈ S be a function such that 0 ≤ ψ(x) ≤ 1, | |≤ ψ(x)=1, if x 1, ψ(x)=0, if |x|≥3/2. Define φ0(x)=ψ(x), x φ1(x)=ψ(2)−ψ(x), −k+1 φk(x)=φ1(2 x),k=2,3, .... { }∞ ⊂{ ∈Rd k−1 ≤ Then φk k=0 is a partition of unity, and satisfies supp(φk) x :2 |x|≤3·2k−1}. 0 Definition 2.3. Let s ∈ R,1≤q≤∞,andf∈S. ≤ ∞ F s (i) For 1 p< the Triebel-Lizorkin space p,q is defined by: Z X∞ p/q 1/p ˇ ∈Fs ⇐⇒ k k s skq | ˆ |q ∞ (3) f p,q f F p,q = 2 (φkf(x)) dx < . Rd k=0 ≤ ≤∞ Bs (ii) For 1 p , the Besov space p,q is defined by: Z X∞ q/p ˇ ∈Bs ⇐⇒ k k s skq | ˆ |p ∞ (4) f p,q f B p,q = 2 (φkf(x)) dx < . Rd k=0 ∞ F s (iii) For p = , the Triebel-Lizorkin space ∞,q is defined by: (5) X∞ X∞ 1/q ∈Fs ⇐⇒ ∃ { } ∞ ˆ ˇ ksq | |q ∞ f ∞,q f k k =0,f= φkfk , sup 2 fk(x) < , Rd k=0 k=0 with norm X∞ 1/q k k s ksq| |q f F∞,q = inf sup 2 fk(x) , Rd k=0 the infimum being taken over all admissible representations. ≤ ≤∞ Hs (iv) For 1 p , the fractional Sobolev space p is defined by: 4 KASSO A. OKOUDJOU Z 1/p 2 2 ˇ ∈Hs ⇐⇒ k k s | | | s/ ˆ |p ∞ (6) f p f H p = ((1 + x ) f(x)) dx < . R d We refer to [15], [16], [12] and [13] for background and information about the Triebel-Lizorkin, Besov, and Sobolev spaces. Because these spaces have been redis- covered (under different names) by various authors, they have a number of equiv- alent definitions. We collect here some of those results that will be needed in F s Bs the sequel: Propositions 2.4 and 2.5 give equivalent definitions of p,q and p,q, respectively, while Proposition 2.6 is a result on interpolation of Besov spaces. The following result is proved in [15, Proposition 1, 2.3.4]. 0 Proposition 2.4. Let s ∈ R, 1
Proposition 2.6. Let s0,s1 ∈R, 1≤p0,q0,p1,q1 ≤∞and 0 <θ<1. 1 1−θ θ 1 1−θ θ If s =(1−θ)s0+θs1, = + ,and = + then p p0 p1 q q0 q1 Bs0 Bs1 Bs (9) p0,q0 , p1,q1 θ = p,q. The next proposition collects some of the computations involved in the proofs of our results. Part (a) computes the STFT of a Gaussian with respect to a dilated Gaussian. The result is essentially the product of two Gaussians (one in time and the other in frequency). Part (b) shows that the inverse Fourier transform of the Bessel potential 2 −s/2 m−s(x)=(1+|x| ) is in M1,1 for s>d. Because M1,1 is invariant under Fourier transforms we then 1,1 conclude that the Bessel potential m−s itself is in M for s>d. EMBEDDINGS OF SOME CLASSICAL BANACH SPACES INTO MODULATION SPACES 5
2 −πx2 −πx Proposition 2.7. Define g(x)=e and ga(x)=e a .Let Z ∞ 2 1 1 −d+s −( πx + t ) dt G(x)= t 2 e t 4π s s/2 (4π) Γ(s/2) 0 t ∈ Rd for s, a > 0 and x . Then the following hold. d/2 2πi x·ω a a+1 a+1 (a) V a g(x, ω)= e g +1(x) g (ω). g a+1 a a d/2 1 (b) V mˇ − =(2π) V (D 1 G ) ∈ L for s>d. g s g 2π s 1,1 (c) m−s ∈M for s>d.
Proof. (a) is an elementary, but tedious, calculation, so we will omit it. d/2 (b) For s>0wehavethatˇm− (x)=(2π) (D 1 G )(x) (cf. [13, Proposition s 2π s 3.1.2]). Therefore: d/2 Vgmˇ −s(x, ω)=(2π) Vg(D 1 Gs)(x, ω) Z 2π d/2 −2πit·ω =(2π) D 1 Gs(t) e g(t − x)dt, Rd 2π so Z ∞ −d+s u 2 −1 d/2 − 4π (10) kVgmˇ −skL1,1 ≤ C u (u +2π) e du, 0 and the last expression is finite if s>d. (c) Follows from (b) and the comments above.
3. Results
3.1. Discussion of the results. There are several embeddings between the Besov or Triebel-Lizorkin and modulation spaces that can easily be derived. These in- clude: Bs ⊂ p ⊂Mp,p0 ≤ ≤ ≤ ≤∞ (a) p,q L for s>0, 1 p 2, and 1 q , Bs ⊂ p ⊂Mp,p ≤ ≤∞ ≤ ≤∞ (b) p,q L for s>0, 2 p ,and1 q [15], [16], [8]. The embeddings we will prove are more difficult, and require an appropriate norm on the Besov or Triebel-Lizorkin space in consideration, along with a correct choice of the STFT. In particular, the following equivalent forms of the STFT will be useful: (11) ˆ −2πixω −2πixω ˇ −2πixω Vgf(x, ω)= f·Txg (ω)=e Vgˆfˆ(ω,−x) = e fˆ·Tωgˆ = e f∗(Mωg)(x) . Our first two main results are as follows. Theorem 3.1. ≤ ≤ ≤ ≤∞ 2 − Let 1 p 2 and 1 q .Ifs>d(p 1) then Bs ⊂Mp0,p (12) p,q .
Theorem 3.2. ≤ ≤∞ d Let 1 p .Ifs> p0 then Bs ⊂Mp,p0 (13) p,p . 6 KASSO A. OKOUDJOU
Corollary 3.3. ≤ ≤∞ − 2 If 2 p and s>d(1 p ) then Bs ⊂Mp,p0 (14) p,p . Theorem 3.2, which recovers and extends the embedding in [10], follows by iden- F s Bs tifying p,p with p,p and by using Proposition 2.4 as the appropriate definition Bs Bs ⊂ 2 M2,2 ≥ of p,p. However, it does not include the fact that 2,2 L = for s 0. This last embedding is obtained in Corollary 3.3 by using complex interpolation methods. Theorem 3.4. ≤ ≤∞ 1 1 1 If 1 p, q, r with p + q =1+r,then Hs ⊂Mr,1 (15) p for s>d. Theorem 3.4 yields an embedding of the fractional Sobolev space (or Bessel potential space) into the modulation space. This can also be seen as an embedding F s F s Hs of the Triebel-Lizorkin space p,2 into the modulation space since p,2 = p for 1 n X2 Z o (16) W2,1 = f ∈ L1 : f 0,f00 ∈ L1, kfk = |f (k)(x)| dx < ∞ ⊂M1,1, R k=0 and n X2 o (17) W2,∞ = f ∈ L∞ : f 0,f00 ∈ L∞, kfk = sup |f (k)(x)| < ∞ ⊂M∞,1. R k=0 Note added. After completing this work, we learned that the embedding (16) had previously been conjectured by H. Feichtinger and a proof different from ours was obtained by K. Gr¨ochenig. 4. Proofs P Proof of Theorem 3.1. ∈Bs 4.1. Let f p,q, and use (8) to write f = bk where P 1 p k ∞ ksq q /q ∈ ˆ ⊂{| |≤ } k kBs ∼ k k p bk L , supp(bk) x 2 ,and f p,q inf k=0 2 bk L ,wherethe infimum is over all possible such representations of f.Giveng∈S,wehaveusing (11): X∞ −2πixω ˇ Vgf(x, ω)= e (ˆbk · Tωgˆ)(x). k=0 Hence by the Hausdorff-Young inequality, ∞ ∞ X ˇ X k · k 0 ≤ k ˆ · k 0 ≤ kˆ · k p Vgf( ,ω) Lp bk Tωgˆ Lp bk Tωgˆ L . k=0 k=0 EMBEDDINGS OF SOME CLASSICAL BANACH SPACES INTO MODULATION SPACES 7 Therefore, by Minkowski’s inequality, Z X∞ 1/p X∞ p k k 0 ≤ kˆ · k k k p kˆ k p (18) Vgf Lp ,p bk Tωgˆ Lp dω = gˆ L bk L . Rd k=0 k=0 k 0 Now, ˆbk has compact support K ⊂{|x|≤2 }. Since 1 ≤ p ≤ 2 ≤ p ,wehave 0 Lp(K)⊂Lp(K), and ( 1 − 1 ) kd p p0 (19) kˆbkkLp ≤ C 2 kbkkLp , where C is the volume of the ball of center 0 and radius 1. Substituting (19) into (18) and applying H¨older’s inequality yields ∞ 1/q ∞ 1/q0 X X 0 1 1 s q kq d( − 0 − ) k k 0 ≤ ksq k k p p d Vgf Lp ,p C 2 bk Lp 2 , k=0 k=0 ∞ 1/q0 X 0 1 1 s kq d( − 0 − ) ≤ k k s p p d (20) C f Bp,q 2 . k=0 1 − 1 The last term in (20) is finite if and only if s>d(p p0). 4.2. Proof of Theorem 3.2. The case p = 1 is trivial, so assume 1 ∞ X ˇ Vgf(x, ω)= fk ∗ φk · Tωgˆ (x), k=0 so by Young’s convolution inequality, ∞ X ˇ kVgf(., ω)kLp ≤ kfkkLp k φk · Tωgˆ kL1 . k=0 Hence, by Minkowski’s inequality and H¨older’s inequality, Z 0 X∞ 1/p ˇ p0 k k 0 ≤ k k p k · k 1 Vgf Lp,p fk L φk Tωgˆ L dω Rd k=0 ∞ 1/p ∞ Z 1/p0 X X 0 ≤ ksp k kp −ksp0 k · ˇkp 2 fk Lp 2 φk Tωgˆ L1 dω , Rd k=0 k=0 and therefore ∞ Z 1/p0 X 0 −ksp0 ˇ p k k p,p0 ≤k kBs k · k 1 (21) f M f p,p 2 φk Tωgˆ L dω . Rd k=0 8 KASSO A. OKOUDJOU Now we will estimate the terms on the right-hand side of (21). Setting gk(x)= g(2−k+1x), we have: Z 0 1/p ˇ p0 k · k k ˇ k 0 φk Tωgˆ L1 dω = Vgφk L1,p Rd −1 − +1 k ˇ k · k · k 0 = Vgk φ1(2 , 2 ) L1,p (− +1) ( −1) 0 d k d k /p k ˇ k 0 =2 2 Vgk φ1 L1,p d/p −kd/p ≤ k k 1,1 k ˇ k 0 (22) C1 2 2 Vgk g L Vgφ1 L1,p , the last inequality following from the independence of the definition of the modula- tion space with respect to the window used to compute the STFT, cf. [8, Proposition 11.3.2]. Using Proposition 2.7 with a =2k−1,wehavethat 2k−2 d/2 k k 1,1 k k 1,1 (23) Vgk g L = Vga g L =(1+2 ) . Combining (21), (22) and (23) yields: 0 X∞ 1/p −kp0(s−d+d/p) k k p,p0 ≤ k k s (24) f M C2 f Bp,p 2 . k=0 − 1 d The last term in the right-hand side of (24) is finite if and only if s>d(1 p )= p0. 4.3. Proof of Corollary 3.3. We will prove this part by interpolating between the cases p =2,s0 ≥0andp=∞,s1 >d.In particular, we trivially have Bs0 F s0 Hs0 ⊂ 2 M2,2 ≥ (25) 2,2 = 2,2 = 2 L = for s0 0, and applying Theorem 3.2 to p = ∞ yields: Bs1 ⊂M∞,1 (26) ∞,∞ for s1 >d. Hence, to complete the proof we apply (2) and (2.6) with s0 ≥ 0, s1 >d,p0 =q0 = 2, and p1 = q1 = ∞. 2 −s/2 −πx2 4.4. Proof of Theorem 3.4. Let m−s(x)=(1+|x|) and g(x)=e . ∈Hs Then for f p we have: −2πixω ˇ −2πixω ˇ ˇ Vg f (x, ω)=e fˆ· Tωgˆ (x)=e fmˆ s ∗ m−s ·Tωgˆ (x) . Hence, by Young’s convolution inequality, ˇ ˇ kVgf(·,ω)kLr ≤k(fmˆ s)kLp k(m−s ·Tωgˆ)kLq, and so Z Z 1/q ˇ q k k r,1 ≤k k s | · | k k s k k q,1 Vgf L f Hp (m−s Tωgˆ)(x) dx dω = f Hp Vgmˇ −s L . Rd Rd Using (10), we have that kVgmˇ −skLq,1 < ∞ for s>d, which concludes the proof. 4.5. Proof of Corollary 3.5. 2 2,1 2 2,∞ Proof. If d =1thenH1 =W and H∞ = W [13] and so the proof follows from Theorem 3.4 with d =1. EMBEDDINGS OF SOME CLASSICAL BANACH SPACES INTO MODULATION SPACES 9 5. Acknowledgment The author would like to thank Prof. Christopher Heil, for invaluable discussions and his support. He would also like to thank Profs. Hans G. Feichtinger, Karlheinz Gr¨ochenig and Joachim Toft for fruitful discussions. References [1] H. G. Feichtinger, Modulation spaces on locally Abelian groups, Technical Report, University of Vienna, 1983. [2] H. G. Feichtinger and K. Gr¨ochenig, Banach spaces related to integrable group representations and their atomic decompositions I, J. Funct. Anal., 86:307–340, 1989. [3] H. G. Feichtinger and K. Gr¨ochenig, Banach spaces related to integrable group representations and their atomic decompositions II, Monatsh. Math., 108:129–148, 1989. [4] G. B. Folland, “Harmonic Analysis in Phase Space,” Ann. of Math. Studies, Princeton Uni- versity Press, Princeton, NJ, 1989. [5] Y. V. Galperin and K. Gr¨ochenig, Uncertainty principles as embeddings of modulation spaces, J. Math. Anal. Appl., 274(1):181–202, 2002. [6] P. 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Sickel, “Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations,” Walter de Gruyter, New York, 1996. [13] E. M. Stein, “Singular Integrals and Differentiability Properties of Functions,” Princeton University Press, Princeton, NJ, 1970. [14] J. Toft, Embeddings for modulation spaces and Besov spaces, Research Report 2001:11, Blekinge Institute of Technology, 2002. [15] H. Triebel, “Theory of Function Spaces,” Birkh¨auser, Boston, 1983. [16] H. Triebel, “Theory of Function Spaces II,” Birkh¨auser, Boston, 1992. School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332- 0160 USA E-mail address: [email protected]