Localization operators and exponential weights for modulation spaces

Transcription

1 Localization operators and exponential weights for modulation spaces Elena Cordero, Stevan Pilipović, Luigi Rodino and Nenad Teofanov Abstract We study localization operators within the framework of ultradistributions More precisely, given a symbol a and two windo ϕ 1, ϕ 2, we investigate the multilinear mapping from (a, ϕ 1, ϕ 2) S (1) (R 2d ) S (1) (R d ) S (1) (R d ) to the localization operator A ϕ 1,ϕ 2 a Results are formulated in terms of modulation spaces with weights which may have exponential growth We give sufficient and necessary conditions for A ϕ 1,ϕ 2 a to be bounded or to belong to a Schatten class As an application, we study symbols defined by ultra-distributions with compact support, that give trace class localization operators 1 Introduction Time-frequency localization operators are pseudo-differential operators A ϕ1,ϕ2 a, where a is the symbol of the operator and ϕ 1, ϕ 2 are the analysis and synthesis windo, respectively (see below for an explicit expression) In the literature they are also known as Anti-Wick operators, Gabor multipliers, Toeplitz operators or wave packets The terminology Time-frequency localization operators is due to Daubechies [8], who used them as a mathematical tool to localize a signal in the time-frequency plane We recall that the case ϕ 1, ϕ 2 = e π x 2 comes back to Berezin [1]; in the latter framework the operator A ϕ1,ϕ2 a comes as the result of a quantization procedure and is used in the PDE context, see Shubin [22] From this point of view, it is natural to compare localization operators with classical pseudodifferential operators, as defined originally by Hörmander, Kohn and Nirenberg, and generalized and used by many authors, see Hörmander [16] If we consider in particular L σ, the Weyl pseudodifferential operator with symbol σ, a precise connection is obtained by observing that A ϕ1,ϕ2 a = L σ if (11) σ = a W (ϕ 2, ϕ 1 ), 1991 Mathematics Subject Classification 47G30,35S05,46F05,44A05 Key words and phrases Localization operator, ultra-distributions, Gelfand-Shilov type spaces, modulation space, Wigner distribution, short-time Fourier transform, Schatten class This research is supported by FIRB (MIUR, Italy) and MNZŽS of Serbia

2 2 Elena Cordero, Stevan Pilipović, Luigi Rodino and Nenad Teofanov where W (ϕ 2, ϕ 1 ) is the cross-wigner distribution defined below (see (32)) It follo from (11) that, if ϕ 1, ϕ 2 are very smooth, for instance if they belong to the Schwartz space S, then even very rough symbols a for the localization operator give rise to smooth Weyl symbols σ So, with respect to the classical pseudodifferential calculus, we may allow singular symbols for localization operators and nevertheless obtain good properties, in particular L 2 -boundedness This was the basic idea in [6], where continuity and Schatten properties were investigated in the frame of S, the space of tempered distributions In particular, in [6] the interplay between the roughness of the symbol a and the time-frequency concentration of the windo ϕ 1, ϕ 2 was studied in detail, by using modulation spaces Mm p,q of Feichtinger [9] As a striking example, it was observed in [6] that any Schwartz distribution with compact support a E gives a trace class operator Aa ϕ1,ϕ2, if ϕ 1, ϕ 2 S In the present paper we carry the results of [6] to, say, their extreme consequences Window functions are assumed to belong to ultra-modulation spaces, cf [15, 19, 24]; namely, we introduce Mw 1 s, with w s exponential weight depending on s, 0 s < (the limit case s = 0 gives the unweighted space), having as projective and inductive limit the Gelfand-Shilov space S (1), S {1} respectively, subspaces of the Schwartz set S Then, we may allow symbols a which are ultra-distributions in S (1), and find sufficient and necessary conditions for the L 2 -boundedness In this way we extend [6], obtaining in particular that for windo ϕ 1, ϕ 2 S {1}, every ultra-distribution with compact support a E t, t > 1 (see [20, 21]), defines a trace class operator A ϕ1,ϕ2 a For example, we may consider an infinite sum of derivatives of the δ distribution at the origin: a = µ N 2d 0 1 (µ!) r µ δ, which belongs to E t, if r > t To state our main result more precisely, let us recall some definitions First, recall that Gelfand-Shilov type space S (1) can be defined as a class of analytic functions f such that for every h, k > 0, f(x)e h x L < and ˆf(ω)e k ω L < Its strong dual is a space of tempered ultra-distributions of Beurling type, S (1) The introduction of modulation spaces requires some time-frequency tools Translation and modulation operators are given by (12) T x f(t) = f(t x) and M ω f(t) = e 2πiωt f(t) For a fixed non-zero g S (1) (R d ) the short-time Fourier transform (STFT) of f S (1) (R d ) with respect to the window g is given by (13) V g f(x, ω) = f, M ω T x g = f(t) g(t x) e 2πiωt dt, R d

3 Localization operators and exponential weights for modulation spaces 3 which can be extended to f S (1) (R d ) by duality For a fixed g S (1) (R d ) the following characterization of S (1) holds: f S (1) (R d ) V g f S (1) (R 2d ) For z = (x, ω) R 2d, and s > 0, we consider the exponential weights w s (z) := e s z Note that w s belongs to S (1) but it is not a tempered distribution For f S (1) (R d ), we define the modulation space Mw 1 s (R d ) by the norm f M 1 := (V g f)w s L 1 (R 2d ) < for some (hence all) non-zero g S (1) (R d ) Its dual space M1/w s (R d ) possesses the norm f M := (V g f)1/w s 1/ L (R 2d ) < It can be shown that S (1) can be represented as projective limit of weighted modulation spaces, see Proposition 23 This implies that S (1) can be viewed as the union of the corresponding duals Namely, we have S (1) = s 0 M 1 w s and S (1) = s 0 M 1/w s The time-frequency localization operator A ϕ1,ϕ2 a with symbol a and windo ϕ 1, ϕ 2 is defined to be (14) A ϕ1,ϕ2 a f(t) = a(x, ω)v ϕ1 f(x, ω)m ω T x ϕ 2 (t) dxdω R 2d or, in the weak sense, (15) A ϕ1,ϕ2 a f, g = av ϕ1 f, V ϕ2 g = a, V ϕ1 f V ϕ2 g, f, g S (1) (R d ) By (15) if a S (1) (R 2d ) and ϕ 1, ϕ 2 S (1) (R d ), then (14) is a well-defined continuous operator from S (1) (R d ) to S (1) (R d ) Our results for the boundedness of a localization operator can be simplified as follo Theorem 11 If a M1/w s (R 2d ), and ϕ 1, ϕ 2 Mw 1 s (R d ), then Aa ϕ1,ϕ2 is bounded on L 2 (R d ) (and on each modulation space as well), with operator norm at most A ϕ1,ϕ2 a op C a M ϕ 1 1/ M 1 ϕ 2 M 1 Section 2 is devoted to a more detailed study of the involved function spaces Novelty here is the extension to exponential weights of the results of [19, 23, 24] for the sub-exponential case In Section 3 we prove Theorem 11 and give more general results Parts of the proofs are only sketched, because they are easy modifications of [6] In Section 4 we prove that ultra-distributions with compact support as symbols give trace class operators Let us finally observe that our results apply in particular to the anti-wick case, ie localization operators with Gaussian windo: ϕ 1 (t) = ϕ 2 (t) = e π t 2 Actually, this choice of the windo suggests that we could take as symbols even

4 4 Elena Cordero, Stevan Pilipović, Luigi Rodino and Nenad Teofanov more general ultra-distributions, in the duals of the Gelfand-Shilov class S (α), with 1/2 α < 1 Nevertheless, the weights of the corresponding modulation spaces should allow a growth at infinity of the type e c z r, c > 0 and 1 < r 2, loosing the sub-multiplicative property, which is essential in our arguments To be definite, we set the notation we shall use (and have already used) in this paper Notation We define xy = x y, the scalar product on R d Given a vector x = (x 1,, x d ) R d, the partial derivative with respect to x j is denoted by j = x j Given a multi-index p = (p 1,, p d ) 0, ie, p N d 0 and p j 0, we write p = p1 1 p d d ; moreover, we write xp = (x 1,, x d ) (p1,,pd) = d i=1 xpi i We shall denote the Euclidean norm by x, the l 1 -norm by x 1 = d i=1 x i and the l - norm by x = max{ x 1,, x d }; the absolute value of a vector x means the vector of absolute values x = ( x 1,, x d ) We write h x 1/α = d i=1 h i x i 1/αi Moreover, for p N d 0 and α R d +, we set (p!) α = (p 1!) α1 (p d!) α d, while as standard p! = p 1! p d! In the sequel, a real number r R + may play the role of the vector with constant components r j = r, so for α R d +, by writing α > r we mean α j > r for all j = 1,, d The Schwartz class is denoted by S(R d ), the space of tempered distributions by S (R d ) We use the brackets f, g to denote the extension to S (1) (R d ) S (1) (R d ) of the inner product f, g = f(t)g(t)dt on L 2 (R d ) The space of smooth functions with compact support on R d is denoted by D(R d ) The Fourier transform is normalized to be ˆf(ω) = Ff(ω) = f(t)e 2πitω dt We denote by s the polynomial weights (x, ω) s = z s = (1 + x 2 + ω 2 ) s/2, z = (x, ω) R 2d, s R and the product weights s s signifying x s ω s The singular values {s k (L)} k=1 of a compact operator L B(L2 (R d )) are the eigenvalues of the positive self-adjoint operator L L For 1 p <, the Schatten class S p is the space of all compact operators whose singular values lie in l p For consistency, we define S := B(L 2 (R d )) to be the space of bounded operators on L 2 (R d ) In particular, S 2 is the space of Hilbert-Schmidt operators, and S 1 is the space of trace class operators Throughout the paper, we shall use the notation A B to indicate A cb for a suitable constant c > 0, whereas A B means that c 1 A B ca for some c 1 The symbol B 1 B 2 denotes the continuous and dense embedding of the topological vector space B 1 into B 2 2 Function spaces 21 Gelfand-Shilov Type Spaces We introduce the definition and properties of Gelfand-Shilov spaces which will be used in the sequel For more details we refer the reader to [11, 15, 18, 23]

5 Localization operators and exponential weights for modulation spaces 5 Definition 21 Let α, β R d +, and assume A 1,, A d, B 1,, B d > 0 Then, the Gelfand-Shilov type space S α,b β,a = Sα,B β,a (Rd ) is defined by S α,b β,a = {f C (R d ) ( C > 0) x p q f L CA p (p!) α B q (q!) β, p, q N d 0} Their projective and inductive limits are denoted by Σ α β = proj lim A>0,B>0 Sα,B β,a ; Sα β := ind lim A>0,B>0 Sα,B β,a The above spaces are contained in S, and moreover for any α, β 0 the Fourier transform is a topological isomorphism between Sα β and Sβ α (F(Sβ α) = Sβ α) and extends to a continuous linear transform from (Sα) β onto (Sβ α) In particular, if α = β and α 1/2 then F(Sα) α = Sα α Similar assertions hold for Σ α β Due to this fact, corresponding dual spaces are referred to as tempered ultra-distributions (of Beurling or Roumieu type) Note that Sβ α is nontrivial if and only if α + β > 1, or α + β = 1 and αβ > 0 We will use the notation S (α) = Σ α α, S {α} = Sα α We have Σ α β Sα β S The following result is a characterization of Gelfand-Shilov type spaces, see [3, 11, 15, 17] Theorem 22 Let α, β R d +, α, β 1/2, ie, α i, β i 1/2, for every i = 1,, d The following conditions are equivalent: a) f S α β (resp f Σα β ) b) There exist (resp for every) constants A 1,, A d, B 1,, B d > 0, x p f L CA p (p!) β and ω q ˆf L CB q (q!) α, p, q N d 0, for some C > 0 c) There exist (resp for every) constants A 1,, A d, B 1,, B d > 0, x p f L CA p (p!) β and q f L CB q (q!) α, p, q N d 0, for some C > 0 d) There exist (resp for every) constants h 1,, h d, k 1,, k d > 0, (21) fe h x 1/β L < and ˆfe k ω 1/α L < e) There exist (resp for every) constants h 1,, h d, B 1,, B d > 0, (22) ( q f)e h x 1/β L < CB q (q!) α, for some C > 0 Gelfand-Shilov type spaces enjoy the embedding property below Lemma 21 For every α 1 < α 2 and β 1 < β 2 we have (23) S α1 β 1 Σ α2 β 2

6 6 Elena Cordero, Stevan Pilipović, Luigi Rodino and Nenad Teofanov Proof If f S α1 β 1, there exists λ R d, λ > 0, such that fe λ x 1/β 1 L < Since β 1 < β 2, for every h 1,, h d > 0 we have h x 1/β2 < λ x 1/β1 for x large enough, and hence there exists C h > 0 such that e h x 1/β 2 C h e λ x 1/β 1 Therefore, the norm fe h x 1/β 2 L is finite The same arguments prove ˆfe k ω 1/α 2 L < By (21) we conclude f Σ α2 β 2 Lemma 22 Let g S (1) (R d ) \ {0} Then for f (S (1) ) (R d ), the following characterization holds: (24) f S (1) (R d ) V g f S (1) (R 2d ) For the proof we refer to [23, Theorems 38-9] 22 Modulation Spaces The modulation space norms traditionally measure the joint time-frequency distribution of f S, we refer, for instance, to [9], [12, Ch 11-13] and the original literature quoted there for various properties and applications In that setting it is sufficient to observe modulation spaces with weights which admit at most polynomial growth at infinity However the study of ultra-distributions requires a more general approach that includes the weights of exponential growth Weight Functions In the sequel v will always be a continuous, positive, even, submultiplicative function (submultiplicative weight), ie, v(0) = 1, v(z) = v( z), and v(z 1 + z 2 ) v(z 1 )v(z 2 ), for all z, z 1, z 2 R 2d Moreover, v is assumed to be even in each group of coordinates, that is, v(x, ω) = v( ω, x) = v( x, ω), for any (x, ω) R 2d Submultiplicativity implies that v(z) is dominated by an exponential function, ie (25) C, k > 0 such that v(z) Ce k z, z R 2d For example, every weight of the form v(z) = e s z b (1 + z ) a log r (e + z ) for parameters a, r, s 0, 0 b 1 satisfies the above conditions Associated to every submultiplicative weight we consider the class of so-called v-moderate weights M v A positive, even weight function m on R 2d belongs to M v if it satisfies the condition m(z 1 + z 2 ) Cv(z 1 )m(z 2 ) z 1, z 2 R 2d We note that this definition implies that 1 v m v, m 0 everywhere, and that 1/m M v For our investigation of localization operators we will mostly use the exponential weights defined by (26) (27) w s (z) = w s (x, ω) = e s (x,ω), z = (x, ω) R 2d, τ s (z) = τ s (x, ω) = e s ω Notice that arguing on R 4d we may read (28) τ s (z, ζ) = w s (ζ) z, ζ R 2d,

7 Localization operators and exponential weights for modulation spaces 7 which will be used in the sequel Definition 21 Given a non-zero window g S (1) (R d ), a weight function m M v defined on R 2d, and 1 p, q, the modulation space Mm p,q (R d ) consists of all tempered ultra-distributions f S (1) (R d ) such that V g f L p,q m (R 2d ) (weighted mixed-norm spaces) The norm on Mm p,q is ( ( ) ) q/p 1/q f M p,q = V m gf L p,q = V m g f(x, ω) p m(x, ω) p dx dω, R d R d (obvious changes if p = or q = ) Note that, for f, g S (1) (R d ) the above integral is convergent thanks to (24) Namely, in view of (25) for a given m M v there exist l > 0 such that m(x, ω) Ce l (x,ω) and therefore ( ) q/p V g f(x, ω) p m(x, ω) p dx dω R d R d ( ) q/p C V g f(x, ω) p e lp (x,ω) dx dω R d R < d since by (24) and Theorem 22, d) we have V g f(x, ω) < Ce s (x,ω) for every s > 0 This implies S (1) Mm p,q If p = q, we write Mm p instead of Mm p,p, and if m(z) 1 on R 2d, then we write M p,q and M p for Mm p,q and Mm p,p In the next proposition we show that Mm p,q (R d ) are Banach spaces whose definition is independent of the choice of the window g Mv 1 \ {0} In order to do so, we need the adjoint of the short-time Fourier transform For given window g S (1) and a function F (x, ξ) L p,q m (R 2d ) we define Vg F by Vg F, f := F, V g f, whenever the duality is well defined In our context, [12, Proposition 1132] can be rewritten as follo Proposition 22 Fix m M v and g, ψ S (1), with g, ψ 0 Then a) V g : L p,q m (R 2d ) M p,q m (R d ), and (29) Vg F M p,q C V m ψg L 1 v F L p,q m b) The inversion formula holds: I M p,q m for the identity operator = g, ψ 1 V g V ψ, where I M p,q m stands c) The definition of Mm p,q is independent on the choice of g S (1) \ {0} and different windo yield equivalent norms d) The space of admissible windo can be extended from S (1) to Mv 1

8 8 Elena Cordero, Stevan Pilipović, Luigi Rodino and Nenad Teofanov Proof a) Clearly, where p = p 1 p Vg F, f = F, V g f F L p,q V m gf L p,q, and q = q 1 q Moreover, there exists C > 0 such that 1/m Cv < which together with (25) implies that there exist l > 0 such that e l z L p,q 1/m Now, f S (1) implies that V g f S (1) (R 2d ), and we obtain (210) Vg F, f F L p,q el z V m g f e l z L p,q < 1/m Therefore, Vg F is a well defined (tempered) ultra-distribution Proceeding then as in the proof of [12, Proposition 1132 a)] we come to the desired result b) It follo by the same arguments of [12, Proposition 1132 b)] by replacing S with S (1) c) Now that a) and b) are settled c) follo almost directly from the corresponding result [12, Proposition 1132 c)] d) For this part we need the density of S (1) in Mm p,q This fact is not obvious and will be proven in a more general setting in [4] Then d) follo by using standard arguments of [12, Theorem 1137] REMARK: a) The preceding proofs motivate our initial choice of S (1) for the window functions, instead of S {1} More precisely, in the case of the inductive limit of Gelfand-Shilov type spaces, the short-time Fourier transform decays faster than e l0 z for certain l 0 > 0 This would not be sufficient to ensure e l z V g f < in (210) However, the space S {1} will play an important role in the last section, see Corollary 43 b) Proposition 22 d) actually implies that Definition 21 coincides with the usual definition of modulation spaces with weights of polinomial and sub-exponential are Banach spaces This follo from Proposition 22 and arguments used in [12, Theorem 1135] To benefit of non-expert readers, we recall that the class of modulation spaces contains the following well-known function spaces: Weighted L 2 -spaces: M x 2 (R d ) = L 2 s(r d ) = {f : f(x) x s L 2 (R d )}, s R s growth (see, for example [5, 9, 12, 19]) We do not prove that M p,q m Sobolev spaces: M 2 ω s (R d ) = H s (R d ) = {f : ˆf(ω) ω s L 2 (R d )}, s R Shubin-Sobolev spaces [22, 2]: M 2 (x,ω) s (R d ) = L 2 s(r d ) H s (R d ) = Q s (R d ) Feichtinger s algebra: M 1 (R d ) = S 0 (R d ) The characterization of the Schwartz class given in [14]: S(R d ) = s 0 M 1 s (R d ) and the corresponding description of the space of tempered distributions S (R d ) = s 0 M 1/ s (R d ) can now be reformulated for Gelfand-Shilov spaces and tempered ultra-distributions 1/m

9 Localization operators and exponential weights for modulation spaces 9 Proposition 23 Let 1 p, q, and let w s be given by (26) We have a) S (1) = Mw p,q s, S (1) = M p,q 1/w s s 0 s 0 b) S {1} = s>0 Mw p,q s, S {1} = M p,q 1/w s s>0 The proof is just a rearrangement of [23, Theorem 43], written for the case of sub-exponential weights e s (x,ω) 1/t, s 0 and t > 1 Here we observe the case t = 1 We finally recall a convolution relation for modulation spaces [6, Proposition 24] Proposition 24 Let m M v defined on R 2d and let m 1 (x) = m(x, 0) and m 2 (ω) = m(0, ω), the restrictions to R d {0} and {0} R d, and likewise for v Let ν(ω) > 0 be an arbitrary weight function on R d and 1 p, q, r, s, t If then 1 p + 1 q 1 = 1 r, and 1 t + 1 t = 1, (211) M p,st m 1 ν(r d ) M q,st v 1 v 2ν 1 (R d ) M r,s m (R d ) with norm inequality f h M r,s m f M p,st h m 1 ν M q,st v 1 v 2 ν 1 3 Sufficient and necessary conditions If we consider a ultra-distributional symbol a S (1), provided that the windo ϕ 1, ϕ 2 belong to the smooth space S (1), the localization operators A ϕ1,ϕ2 a is a welldefined and bounded mapping A ϕ1,ϕ2 a : S (1) S (1) This is easily seen by the characterization (24), the weak definition of a localization operator (15) and the fact that S (1) is closed under multiplication Next, we study the regularity properties of the operator A ϕ1,ϕ2 a on L 2, within the framework of ultra-modulation spaces This shall be achieved by passing through the connection between localization operators and Weyl operators Namely, let us recall that the Weyl operator L σ, with symbol σ S (1) (R 2d ), is defined by (31) L σ f, g = σ, W (g, f), f, g S (1) (R d ), where the cross-wigner distribution W (f, g) is defined to be (32) W (f, g)(x, ω) = f(x + t 2 )g(x t 2 )e 2πiωt dt Every localization operator A ϕ1,ϕ2 a follo [2, 7, 10, 22]: A ϕ1,ϕ2 a given by (11) can be represented as a Weyl operator as = L a W (ϕ2,ϕ 1), that is the Weyl symbol of A ϕ1,ϕ2 a is

10 10 Elena Cordero, Stevan Pilipović, Luigi Rodino and Nenad Teofanov We will use the following boundedness and Schatten class S p results for the Weyl calculus in terms of modulation spaces For the proof we refer to [6, 13] Theorem 31 a) If σ M,1 (R 2d ), then L σ is bounded on M p,q (R d ), 1 p, q, with a uniform estimate L σ op σ M,1 for the operator norm In particular, L σ is bounded on L 2 (R d ) b) If σ M 1 (R 2d ), then L σ S 1 and L σ S 1 σ M 1 c) If 1 p 2 and σ M p (R 2d ), then L σ S p and L σ Sp σ M p d) If 2 p and σ M p,p (R 2d ), then L σ S p and L σ Sp σ M p,p We present a reformulation of [6, Proposition 25] in terms of modulation spaces with exponential weights The proof is omitted since the density of S (1) in Mw 1 s allo to follow the proof given there We then state and prove Theorem 11 in a more general form Lemma 31 Let w s, τ s be the weights defined in (26) and (27) If 1 p, s 0, ϕ 1 Mw 1 s (R d ) and ϕ 2 Mw p s (R d ), then W (ϕ 2, ϕ 1 ) Mτ 1,p s (R 2d ), and (33) W (ϕ 2, ϕ 1 ) M 1,p τs ϕ 1 M 1 ϕ 2 M p Theorem 32 a) Let s 0, a M1/τ s (R 2d ), ϕ 1, ϕ 2 Mw 1 s (R d ) Then A ϕ1,ϕ2 a is bounded on M p,q (R d ) for all 1 p, q, and the operator norm satisfies the uniform estimate A ϕ1,ϕ2 a op a M ϕ 1 1/τs M 1 ϕ 2 M 1 b) If 1 p 2, then the mapping (a, ϕ 1, ϕ 2 ) A ϕ1,ϕ2 a Mw 1 s (R d ) Mw p s (R d ) into S p, in other words, a Sp a M p, ϕ 1 M 1 ϕ 2 M p A ϕ1,ϕ2 1/τs c) If 2 p, then the mapping (a, ϕ 1, ϕ 2 ) Aa ϕ1,ϕ2 Mw 1 s Mw p s into S p, and A ϕ1,ϕ2 a Sp a M p, 1/τs ϕ 1 M 1 ϕ 2 M p is bounded from M p, 1/τ s (R 2d ) is bounded from M p, 1/τ s Proof a) We use the convolution relation (211) to show that the Weyl symbol a W (ϕ 2, ϕ 1 ) of A ϕ1,ϕ2 a is in M,1 If ϕ 1, ϕ 2 Mw 1 s (R d ), then by (33), we have W (ϕ 2, ϕ 1 ) Mτ 1 s (R 2d ) Applying Proposition 24 in the form M1/τ s Mτ 1 s M,1, we obtain σ = a W (ϕ 2, ϕ 1 ) M,1 The result now follo from Theorem 31 a) Similarly, the proof of b) and c) is based on results of Proposition 24 and Theorem 31, items b) - d) For the sake of completeness, we state the necessary boundedness result, which follo by straightforward modifications of [6, Theorem 43]

11 Localization operators and exponential weights for modulation spaces 11 Theorem 33 Let a S (1) (R 2d ) and s 0 If there exists a constant C = C(a) > 0 depending only on a such that (34) A ϕ1,ϕ2 a S C ϕ 1 M 1 ϕ 2 M 1 for all ϕ 1, ϕ 2 S (1) (R d ), then a M 1/τ s 4 Ultra-distributions with Compact Support as Symbols As an application we treat ultra-distributions with compact support, denoted here by E t, t > 1 The Gelfand-Shilov space S t is embedded in the corresponding Gevrey function space and therefore in view of Lemma 21 we have E t S {t} S (t) Hence, an element from E t, t > 1 can be viewed as an element of S (1) We skip the precise definition of E t,, see eg [20, Definition 155, Example 184 (b)] In fact, the following structure theorem, obtained by a slight generalization of [20, Theorem 156] (see also [21]) to the anisotropic case, will be sufficient for our purposes Theorem 41 Let t R d, t > 1 Every u E t(r d ) can be represented as (41) u = α µ α, α N d 0 where µ α is a measure satisfying, for every ɛ = (ɛ 1,, ɛ d ) > 0, (42) dµ α C ɛ ɛ α (α!) t, K for a suitable constant C ɛ > 0 and a suitable compact set K R d, independent of α Proposition 42 Let t R d, t > 1, and a E t(r d ) Then its STFT with respect to any window g S (1) satisfies the estimate for every h > 0 V g a(x, ω) e h x e t 2πω 1/t, Proof Note that, for every g S (1), we have α (M ω T x g)(y) = ( ) α (2πiω) β M ω T x y α β g(y) β β α

12 12 Elena Cordero, Stevan Pilipović, Luigi Rodino and Nenad Teofanov Then, using the representation (41), the STFT of a E t can be estimated as follo: V g a(x, ω) α µ α, M ω T x g = µ α, α (M ω T x g) α N d 0 α N d β α 0 ( ) α (2πiω) β β α N d 0 (M ω T x y α β R d g)(y) dµ α (y) Now we use the estimate (22), where we may assume B j = 1 for all j = 1,, d, to get: M ω T x y α β g(y) = T x y α β g(y) C(α β)!e h y x, for every h > 0 Furthermore, recall ( ) α = 2 α 1, so that we have β β α V g a(x, ω) 2 α 1 α N d 0 β α 2πω β (β!) t (β!)t [(α β)!] t e h y x dµ α (y) R d From the representation theorem (42) we get, for every ɛ > 0, e h y x dµ α (y) e h x e h y dµ α (y) C h,k,ɛ e h x ɛ α (α!) t Since β!(α β)! α!, we obtain V g a(x, ω) e h x (2ɛ) [ ] α 2πω β/t t β!, β α K α N d 0 for every ɛ > 0 We then choose ɛ < 1/2 and the following estimate assures the desired result: [ ] 2πω β/t t β! [ ] 2πω β/t t β! t 2πω β/t = e t 2πω 1/t β! β α β N d 0 β N d 0 Corollary 43 Let t R d, t > 1 and a E t(r 2d ) If the window functions ϕ 1, ϕ 2 S {1} (R d ), then A ϕ1,ϕ2 a is a trace class operator Proof If ϕ 1, ϕ 2 S {1} (R d ), Proposition 23 b), with p = q = 1, assures the existence of an ɛ > 0 such that ϕ 1, ϕ 2 Mw 1 ɛ (R d ) On the other hand, for ω > C ɛ (where C ɛ is a suitable positive constant depending on ɛ) we can write d t 2πω 1/t = t i 2πω i 1/ti ɛ ω ; i=1

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14 14 Elena Cordero, Stevan Pilipović, Luigi Rodino and Nenad Teofanov [19] S Pilipović and N Teofanov, Pseudodifferential operators on ultra-modulation spaces J Funct Anal 208: , 2004 [20] L Rodino, Linear Partial Differential Operators in Gevrey Spaces, World Scientific, Singapore, 1993 [21] C Roumieu, Sur quelques extensions de la notion de distribution Ann Sci École Norm, Sup 3(77): , 1960 [22] M A Shubin Pseudodifferential Operators and Spectral Theory Springer-Verlag, Berlin, second edition, 2001 [23] N Teofanov, Ultradistributions and time-frequency analysis, in Pseudo-differential Operators and Related Topics, Operator Theory: Advances and Applications, P Boggiatto, L Rodino, J Toft, MW Wong, editors, Birkhäuser, 164: , 2006 [24] N Teofanov, Some remarks on ultra-modulation spaces, ESI preprint series 1685:1 13, 2005 Acknowledgment E Cordero and N Teofanov are grateful to the Erwin Schrödinger Institute (ESI), Wien On the base of their Junior Research Fellohip they visited ESI during the spring of 2005 and started there the work on the paper Department of Mathematics and Informatics, University of Novi Sad, Serbia and Montenegro Department of Mathematics, University of Torino, Italy address: elenacordero@unitoit, pilipovic@imnsacyu address: luigirodino@unitoit, tnenad@imnsacyu