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Localization Operators and Exponential Weights for Modulation Spaces

Elena Cordero Stevan Pilipovi´ c Luigi Rodino Nenad Teofanov

Vienna, Preprint ESI 1704 (2005)

Supported by the Austrian Federal Ministry of Education, Science and Culture Available via http://www.esi.ac.at

October 13, 2005

Localization operators and exponential weights for modulation spaces Elena Cordero, Stevan Pilipovi´c, Luigi Rodino and Nenad Teofanov Abstract. We study localization operators within the framework of ultradistributions. More precisely, given a symbol a and two windows ϕ1 , ϕ2 , we ′ investigate the multilinear mapping from (a, ϕ1 , ϕ2 ) ∈ S (1) (R2d ) × S (1)(Rd ) × 1 ,ϕ2 S (1) (Rd ) to the localization operator Aϕ . Results are formulated in terms a of modulation spaces with weights which may have exponential growth. We 1 ,ϕ2 give sufficient and necessary conditions for Aϕ to be bounded or to bea long 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 Aaϕ1 ,ϕ2 , where a is the symbol of the operator and ϕ1 , ϕ2 are the analysis and synthesis windows, 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 2 time-frequency plane. We recall that the case ϕ1 , ϕ2 = e−πkxk comes back to 1 ,ϕ2 comes as the result of a Berezin [1]; in the latter framework the operator Aϕ a quantization procedure and is used in the PDE context, see Shubin [24]. From this point of view, it is natural to compare localization operators with classical pseudodifferential operators, as defined originally by H¨ ormander, Kohn and Nirenberg, and generalized and used by many authors, see H¨ ormander [18]. If we consider in particular Lσ , the Weyl pseudodifferential operator with symbol σ, a precise 1 ,ϕ2 = Lσ if connection is obtained by observing that Aϕ a (1.1)

σ = a ∗ W (ϕ2 , ϕ1 ) ,

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. ˇ of Serbia. This research is supported by FIRB (MIUR, Italy) and MNZZS

2

Elena Cordero, Stevan Pilipovi´c, Luigi Rodino and Nenad Teofanov

where W (ϕ2 , ϕ1 ) is the cross-Wigner distribution defined below (see (3.2)). It follows from (1.1) that, if ϕ1 , ϕ2 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 L2 -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 windows p,q ϕ1 , ϕ2 was studied in detail, by using modulation spaces Mm of Feichtinger [9]. As a striking example, it was observed in [6] that any Schwartz distribution with 1 ,ϕ2 , if ϕ1 , ϕ2 ∈ S. compact support a ∈ E ′ gives a trace class operator Aϕ a 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. [17, 21, 26]; namely, we introduce Mw1 s , with ws 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 L2 -boundedness. In this way we extend [6], obtaining in particular that for windows ϕ1 , ϕ2 ∈ S {1} , every ultra-distribution with compact support a ∈ Et′ , t > 1 (see [22, 23]), defines 1 ,ϕ2 a trace class operator Aϕ . For example, we may consider an infinite sum of a derivatives of the δ distribution at the origin: a=

X

µ∈N2d 0

1 µ ∂ δ, (µ!)r

which belongs to Et′ , 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, kf(x)eh·|x| kL∞ < ∞

and kfˆ(ω)ek·|ω| kL∞ < ∞. ′

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 (1.2)

Tx f(t) = f(t − x)

and Mω f(t) = e2πiωt f(t) .

For a fixed non-zero g ∈ S (1) (Rd ) the short-time Fourier transform (STFT) of f ∈ S (1)(Rd ) with respect to the window g is given by Z (1.3) Vg f(x, ω) = hf, Mω Tx gi = f(t) g(t − x) e−2πiωt dt , Rd

Localization operators and exponential weights for modulation spaces

3

′

which can be extended to f ∈ S (1) (Rd ) by duality. For a fixed g ∈ S (1) (Rd ) the following characterization of S (1) holds: f ∈ S (1)(Rd ) ⇔ Vg f ∈ S (1) (R2d ).

For z = (x, ω) ∈ R2d , and s > 0, we consider the exponential weights ws (z) := ′ e . Note that ws belongs to S (1) but it is not a tempered distribution. For ′ f ∈ S (1) (Rd ), we define the modulation space Mw1 s (Rd ) by the norm skzk

kfkMw1 s := k(Vg f)ws kL1 (R2d ) < ∞

∞ for some (hence all) non-zero g ∈ S (1)(Rd ). Its dual space M1/w (Rd ) possesses s the norm ∞ := k(Vg f)1/ws kL∞ (R2d ) < ∞ . kfkM1/w s

(1)

It can be shown that S can be represented as projective limit of weighted ′ modulation spaces, see Proposition 2.3. This implies that S (1) can be viewed as the union of the corresponding duals. Namely, we have \ [ ′ ∞ S (1) = Mw1 s and S (1) = . M1/w s s≥0

s≥0

1 ,ϕ2 Aϕ a

with symbol a and windows The time-frequency localization operator ϕ1 , ϕ2 is defined to be Z 1 ,ϕ2 (1.4) Aϕ f(t) = a(x, ω)Vϕ1 f(x, ω)Mω Tx ϕ2 (t) dxdω a R2d

or, in the weak sense, (1.5)

1 ,ϕ2 hAϕ f, gi = haVϕ1 f, Vϕ2 gi = ha, Vϕ1 f Vϕ2 gi, a ′

f, g ∈ S (1) (Rd ) .

By (1.5) if a ∈ S (1) (R2d ) and ϕ1 , ϕ2 ∈ S (1)(Rd ), then (1.4) is a well-defined ′ continuous operator from S (1) (Rd ) to S (1) (Rd ). Our results for the boundedness of a localization operator can be simplified as follows. ∞ 1 ,ϕ2 is bounded (R2d ), and ϕ1 , ϕ2 ∈ Mw1 s (Rd ), then Aϕ Theorem 1.1. If a ∈ M1/w a s 2 d on L (R ) (and on each modulation space as well), with operator norm at most 1 ,ϕ2 ∞ kϕ1 kMw1 s kϕ2 kMw1 s . kop ≤ CkakM1/w kAϕ a s

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 [21, 25, 26] for the sub-exponential case. In Section 3 we prove Theorem 1.1 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 2 case, i.e. localization operators with Gaussian windows: ϕ1 (t) = ϕ2 (t) = e−πktk . Actually, this choice of the windows suggests that we could take as symbols even

4

Elena Cordero, Stevan Pilipovi´c, 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 r should allow a growth at infinity of the type eckzk , 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 Rd . Given a vector x = ∂ . (x1 , . . . , xd ) ∈ Rd , the partial derivative with respect to xj is denoted by ∂j = ∂x j d p Given a multi-index p = (p1 , . . . , pd ) ≥ 0, i.e., p ∈ N0 and pj ≥ 0, we write ∂ = Qd ∂1p1 · · · ∂dpd ; moreover, we write xp = (x1 , . . . , xd )(p1 ,...,pd ) = i=1 xpi i . We shall P denote the Euclidean norm by kxk, the ℓ1 -norm by kxk1 = di=1 |xi | and the ℓ∞ norm by kxk∞ = max{|x1|, . . . , |xd|}; the absolute value of a vector x means the Pd vector of absolute values |x| = (|x1 |, . . . , |xd|). We write h·|x|1/α = i=1 hi |xi |1/αi . Moreover, for p ∈ Nd0 and α ∈ Rd+ , we set (p!)α = (p1 !)α1 . . . (pd !)αd , while as standard p! = p1 ! . . . pd !. In the sequel, a real number r ∈ R+ may play the role of the vector with constant components rj = r, so for α ∈ Rd+ , by writing α > r we mean αj > r for all j = 1, . . . , d. The Schwartz class is denoted by S(Rd ), the space of tempered distributi′ ons by S ′ (Rd ). We use the brackets hf, gi to denote the extension to S (1) (Rd ) × R S (1) (Rd ) of the inner product hf, gi = f(t)g(t)dt on L2 (Rd ). The space of smooth functions with compact support on Rd is denoted by D(Rd ). The Fourier transform R is normalized to be fˆ(ω) = Ff(ω) = f(t)e−2πitω dt. We denote by h·is the polynomial weights h(x, ω)is = hzis = (1 + x2 + ω2 )s/2 ,

z = (x, ω) ∈ R2d ,

s∈R

and the product weights h·is ⊗ h·is signifying hxis hωis . The singular values {sk (L)}∞ L ∈ B(L2 (Rd )) are k=1 of a compact operator √ the eigenvalues of the positive self-adjoint operator L∗ L. For 1 ≤ p < ∞, the Schatten class Sp is the space of all compact operators whose singular values lie in lp . For consistency, we define S∞ := B(L2 (Rd )) to be the space of bounded operators on L2 (Rd ). In particular, S2 is the space of Hilbert-Schmidt operators, and S1 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 B1 ֒→ B2 denotes the continuous and dense embedding of the topological vector space B1 into B2 .

2. Function spaces 2.1. 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 [13, 17, 20, 25].

Localization operators and exponential weights for modulation spaces

5

Definition 2.1. Let α, β ∈ Rd+ , and assume A1 , . . . , Ad , B1 , . . . , Bd > 0. Then, the α,B α,B Gelfand-Shilov type space Sβ,A = Sβ,A (Rd ) is defined by α,B Sβ,A = {f ∈ C ∞ (Rd ) | (∃C > 0) kxp∂ q fkL∞ ≤ CAp (p!)α B q (q!)β , ∀p, q ∈ Nd0 }.

Their projective and inductive limits are denoted by Σα β = proj

lim

A>0,B>0

α,B Sβ,A ; Sβα := ind

lim

A>0,B>0

α,B Sβ,A .

The above spaces are contained in S, and moreover for any α, β ≥ 0 the Fourier transform is a topological isomorphism between Sαβ and Sβα (F(Sαβ ) = Sβα ) and extends to a continuous linear transform from (Sαβ )′ onto (Sβα )′ . In particular, if α = β and α ≥ 1/2 then F(Sαα ) = Sαα . Similar assertions hold for Σα β . Due to this fact, corresponding dual spaces are referred to as tempered ultra-distributions (of Beurling or Roumieu type). Note that Sβα is nontrivial if and only if α + β > 1, or α + β = 1 and {α} α αβ > 0. We will use the notation S (α) = Σα = Sαα . We have Σα α, S β ֒→ Sβ ֒→ S. The following result is a characterization of Gelfand-Shilov type spaces, see [3, 13, 17, 19]. Theorem 2.2. Let α, β ∈ Rd+ , α, β ≥ 1/2, i.e., αi , βi ≥ 1/2, for every i = 1, . . . , d. The following conditions are equivalent: a) f ∈ Sβα (resp. f ∈ Σα β ). b) There exist (resp. for every) constants A1 , . . . , Ad , B1 , . . . , Bd > 0, kxp fkL∞ ≤ CAp (p!)β

and

kωq fˆkL∞ ≤ CB q (q!)α ,

∀p, q ∈ Nd0 ,

for some C > 0. c) There exist (resp. for every) constants A1 , . . . , Ad , B1 , . . . , Bd > 0, kxpfkL∞ ≤ CAp (p!)β

and

k∂ q fkL∞ ≤ CB q (q!)α ,

∀p, q ∈ Nd0 ,

for some C > 0. d) There exist (resp. for every) constants h1 , . . . , hd, k1 , . . . , kd > 0, (2.1)

kfeh·|x|

1/β

kL∞ < ∞

and

kfˆek·|ω|

1/α

kL∞ < ∞.

e) There exist (resp. for every) constants h1 , . . . , hd, B1 , . . . , Bd > 0, k(∂ q f)eh·|x|

(2.2)

1/β

kL∞ < CB q (q!)α ,

for some C > 0. Gelfand-Shilov type spaces enjoy the embedding property below. Lemma 2.1. For every α1 < α2 and β1 < β2 we have (2.3)

2 Sβα11 ֒→ Σα β2 .

6

Elena Cordero, Stevan Pilipovi´c, Luigi Rodino and Nenad Teofanov 1/β1

Proof. If f ∈ Sβα11 , there exists λ ∈ Rd , λ > 0, such that kfeλ·|x| kL∞ < ∞. Since β1 < β2 , for every h1 , · · · , hd > 0 we have h·|x|1/β2 < λ·|x|1/β1 for |x| large enough, 1/β2 1/β1 and hence there exists Ch > 0 such that eh·|x| ≤ Ch eλ·|x| . Therefore, the 1/α2 1/β2 kL∞ < ∞. kL∞ is finite. The same arguments prove kfˆek·|ω| norm kfeh·|x| α2 By (2.1) we conclude f ∈ Σβ2 . Lemma 2.2. Let g ∈ S (1)(Rd ) \ {0}. Then for f ∈ (S (1) )′ (Rd ), the following characterization holds: f ∈ S (1)(Rd ) ⇔ Vg f ∈ S (1) (R2d ).

(2.4)

For the proof we refer to [25, Theorems 3.8-9]. 2.2. Modulation Spaces The modulation space norms traditionally measure the joint time-frequency distribution of f ∈ S ′ , we refer, for instance, to [9], [14, Ch. 11-13] and the original literature quoted there for various properties and applications. It is usually 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. Note that the general approach introduced already in [9] includes the weights of sub-exponential growth (see (2.6)). We refer to [10, 11] for related but even more general constructions, based on the general theory of coorbit spaces. Weight Functions. In the sequel v will always be a continuous, positive, even, submultiplicative function (submultiplicative weight), i.e., v(0) = 1, v(z) = v(−z), and v(z1 + z2 ) ≤ v(z1 )v(z2 ), for all z, z1 , z2 ∈ R2d. 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) ∈ R2d. Submultiplicativity implies that v(z) is dominated by an exponential function, i.e. (2.5)

∃ C, k > 0 such that v(z) ≤ Cekkzk,

z ∈ R2d .

For example, every weight of the form (2.6)

b

v(z) = eskzk (1 + kzk)a logr (e + kzk)

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 Mv . A positive, even weight function m on R2d belongs to Mv if it satisfies the condition m(z1 + z2 ) ≤ Cv(z1 )m(z2 ) We note that this definition implies that 1/m ∈ Mv .

1 v

∀z1 , z2 ∈ R2d .

. m . v, m 6= 0 everywhere, and that

Localization operators and exponential weights for modulation spaces

7

For our investigation of localization operators we will mostly use the exponential weights defined by (2.7)

ws (z)

=

ws (x, ω) = esk(x,ω)k,

(2.8)

τs (z)

=

τs (x, ω) = eskωk .

z = (x, ω) ∈ R2d ,

Notice that arguing on R4d we may read (2.9)

τs (z, ζ) = ws (ζ)

which will be used in the sequel.

z, ζ ∈ R2d ,

Definition 2.1. Given a non-zero window g ∈ S (1) (Rd ), a weight function m ∈ Mv p,q defined on R2d , and 1 ≤ p, q ≤ ∞, the modulation space Mm (Rd ) consists of all ′ 2d tempered ultra-distributions f ∈ S (1) (Rd ) such that Vg f ∈ Lp,q m (R ) (weighted p,q mixed-norm spaces). The norm on Mm is q/p !1/q Z Z p p p,q = kV fk p,q = kfkMm |Vg f(x, ω)| m(x, ω) dx dω , g Lm Rd

Rd

(obvious changes if p = ∞ or q = ∞).

Note that, for f, g ∈ S (1) (Rd ) the above integral is convergent thanks to (2.4). Namely, in view of (2.5) for a given m ∈ Mv there exist l > 0 such that m(x, ω) ≤ Celk(x,ω)k and therefore Z Z q/p dω |Vg f(x, ω)|p m(x, ω)p dx Rd d R Z Z q/p p lpk(x,ω)k dω < ∞ |Vg f(x, ω)| e dx ≤C Rd Rd

since by (2.4) and Theorem 2.2, d) we have |Vg f(x, ω)| < Ce−sk(x,ω)k for every p,q s > 0. This implies S (1) ⊂ Mm . p p,p If p = q, we write Mm instead of Mm , and if m(z) ≡ 1 on R2d, then we p,q p p,q p,p write M and M for Mm and Mm . p,q In the next proposition we show that Mm (Rd ) are Banach spaces whose definition is independent of the choice of the window g ∈ Mv1 \ {0}. In order to do so, we need the adjoint of the short-time Fourier transform. 2d ∗ For given window g ∈ S (1) and a function F (x, ξ) ∈ Lp,q m (R ) we define Vg F by hVg∗ F, fi := hF, Vg fi, whenever the duality is well defined. In our context, [14, Proposition 11.3.2.] can be rewritten as follows. Proposition 2.2. Fix m ∈ Mv and g, ψ ∈ S (1), with hg, ψi = 6 0. Then d 2d p,q (R ), and (R ) → M a) Vg∗ : Lp,q m m (2.10)

p,q ≤ CkV gk 1 kF k p,q . kVg∗ F kMm ψ Lv Lm

8

Elena Cordero, Stevan Pilipovi´c, Luigi Rodino and Nenad Teofanov p,q = hg, ψi−1 V ∗ V , where I p,q stands b) The inversion formula holds: IMm Mm g ψ for the identity operator. p,q c) Mm (Rd ) are Banach spaces whose definition is independent on the choice of g ∈ S (1) \ {0}. d) The space of admissible windows can be extended from S (1) to Mv1 .

Proof. a) Clearly, |hVg∗ F, fi| = |hF, Vg fi| ≤ kF kLp,q kVg fkLp′ ,q ′ , m 1/m

′

where p =

p−1 p

′

and q =

q−1 . q

Moreover, there exists C > 0 such that 1/m ≤ Cv

which together with (2.5) implies that there exist l > 0 such that ke−lkzk kLp′ ,q ′ < 1/m

∞. Now, f ∈ S (1) implies that Vg f ∈ S (1) (R2d ), and we obtain (2.11)

|hVg∗ F, fi| ≤ kF kLp,q kelkzk Vg fk∞ ke−lkzk kLp′ ,q ′ < ∞. m 1/m

Vg∗ F

Therefore, is a well defined (tempered) ultra-distribution. Proceeding then as in the proof of [14, Proposition 11.3.2 (a)] we come to the desired result. b) It follows by the same arguments of [14, Proposition 11.3.2 (b)] by replacing ′ S ′ with S (1) . c) Now that a) and b) are settled c) follows almost directly from the corresponding result [14, Proposition 11.3.2 (c), Theorem 11.3.5 (a)]. p,q d) For this part we need the density of S (1) in Mm . This fact is not obvious and will be proven in a more general setting in [4]. Then d) follows by using standard arguments of [14, Theorem 11.3.7]. 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 el0 kzk for certain l0 > 0. This would not be sufficient to ensure kelkzk Vg fk∞ < ∞ in (2.11). However, the space S {1} will play an important role in the last section, see Corollary 4.3. b) Proposition 2.2 d) actually implies that Definition 2.1 coincides with the usual definition of modulation spaces with weights of polinomial and sub-exponential growth (see, for example [5, 9, 14, 21]). To benefit of non-expert readers, we recall that the class of modulation spaces contains the following well-known function spaces: 2 d 2 d s 2 d Weighted L2 -spaces: Mhxi s (R ) = Ls (R ) = {f : f(x)hxi ∈ L (R )}, s ∈ R. Sobolev spaces: M 2 s (Rd ) = H s (Rd ) = {f : fˆ(ω)hωis ∈ L2 (Rd )}, s ∈ R. hωi

2 d 2 d s d d Shubin-Sobolev spaces [24, 2]: Mh(x,ω)i s (R ) = Ls (R ) ∩ H (R ) = Qs (R ). Feichtinger’s algebra: M 1 (Rd ) = S0 (Rd ). T 1 d The characterization of the Schwartz class given in [16]: S(Rd ) = s≥0 Mh·i s (R ) ′ d and the corresponding description of the space of tempered distributions S (R ) =

Localization operators and exponential weights for modulation spaces S

9

∞ d s≥0 M1/h·is (R )

can now be reformulated for Gelfand-Shilov spaces and tempered ultra-distributions. Proposition\ 2.3. Let 1 ≤ p, q ≤[ ∞, and let ws be given by (2.7). We have p,q (1)′ a) S (1) = , S = Mwp,q M1/w . s s s≥0

b) S

{1}

=

[

s>0

s≥0

, Mwp,q s

S

{1}′

=

\

p,q M1/w . s

s>0

The proof is just a rearrangement of [25, Theorem 4.3], written for the case 1/t of sub-exponential weights esk(x,ω)k , s ≥ 0 and t > 1. Here we observe the case t = 1. We finally recall a convolution relation for modulation spaces [6, Proposition 2.4]. Proposition 2.4. Let m ∈ Mv defined on R2d and let m1 (x) = m(x, 0) and m2 (ω) = m(0, ω), the restrictions to Rd × {0} and {0} × Rd , and likewise for v. Let ν(ω) > 0 be an arbitrary weight function on Rd and 1 ≤ p, q, r, s, t ≤ ∞. If 1 1 1 + −1 = , p q r

and

1 1 + ′ = 1, t t

then (2.12)

′

p,st d r,s d Mm (Rd ) ∗ Mvq,st −1 (R ) ֒→ Mm (R ) 1 ⊗ν 1 ⊗v2 ν

r,s . kfk ′ with norm inequality kf ∗ hkMm M p,st khkM q,st m1 ⊗ν

.

v1 ⊗v2 ν −1

3. Sufficient and necessary conditions ′

If we consider a ultra-distributional symbol a ∈ S (1) , provided that the windows 1 ,ϕ2 is a wellϕ1 , ϕ2 belong to the smooth space S (1) , the localization operators Aϕ a ′ ϕ1 ,ϕ2 (1) (1) defined and bounded mapping Aa : S −→ S . This is easily seen by the characterization (2.4), the weak definition of a localization operator (1.5) and the fact that S (1) is closed under multiplication. 1 ,ϕ2 Next, we study the regularity properties of the operator Aϕ on L2 , within a 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 Lσ , with symbol σ ∈ S (1) (R2d ), is defined by (3.1)

hLσ f, gi = hσ, W (g, f)i,

f, g ∈ S (1) (Rd ),

where the cross-Wigner distribution W (f, g) is defined to be Z t t (3.2) W (f, g)(x, ω) = f(x + )g(x − )e−2πiωt dt. 2 2

10

Elena Cordero, Stevan Pilipovi´c, Luigi Rodino and Nenad Teofanov

1 ,ϕ2 Every localization operator Aϕ can be represented as a Weyl operator as a ϕ1 ,ϕ2 1 ,ϕ2 is = La∗W (ϕ2 ,ϕ1 ) , that is the Weyl symbol of Aϕ follows [2, 7, 12, 24]: Aa a given by (1.1). We will use the following boundedness and Schatten class Sp results for the Weyl calculus in terms of modulation spaces. For the proof we refer to [6, 15].

Theorem 3.1. a) If σ ∈ M ∞,1 (R2d ), then Lσ is bounded on M p,q (Rd ), 1 ≤ p, q ≤ ∞, with a uniform estimate kLσ kop . kσkM ∞,1 for the operator norm. In particular, Lσ is bounded on L2 (Rd ). b) If σ ∈ M 1 (R2d ), then Lσ ∈ S1 and kLσ kS 1 . kσkM 1 . c) If 1 ≤ p ≤ 2 and σ ∈ M p(R2d ), then Lσ ∈ Sp and kLσ kSp . kσkM p . ′ d) If 2 ≤ p ≤ ∞ and σ ∈ M p,p (R2d ), then Lσ ∈ Sp and kLσ kSp . kσkM p,p′ . We present a reformulation of [6, Proposition 2.5] in terms of modulation spaces with exponential weights. The proof is omitted since the density of S (1) in Mw1 s allows to follow the proof given there. We then state and prove Theorem 1.1 in a more general form. Lemma 3.1. Let ws , τs be the weights defined in (2.7) and (2.8). If 1 ≤ p ≤ ∞, (R2d ), and s ≥ 0, ϕ1 ∈ Mw1 s (Rd ) and ϕ2 ∈ Mwp s (Rd ), then W (ϕ2 , ϕ1 ) ∈ Mτ1,p s kW (ϕ2 , ϕ1 )kMτ1,p . kϕ1 kMw1 s kϕ2 kMwp s .

(3.3)

s

∞ Theorem 3.2. a) Let s ≥ 0, a ∈ M1/τ (R2d ), ϕ1 , ϕ2 ∈ Mw1 s (Rd ). Then Aaϕ1 ,ϕ2 is s p,q d bounded on M (R ) for all 1 ≤ p, q ≤ ∞, and the operator norm satisfies the uniform estimate 1 ,ϕ2 ∞ kAϕ kϕ1 kMw1 s kϕ2 kMw1 s . kop . kakM1/τ a s

p,∞ 1 ,ϕ2 is bounded from M1/τ b) If 1 ≤ p ≤ 2, then the mapping (a, ϕ1 , ϕ2 ) 7→ Aϕ (R2d )× a s Mw1 s (Rd ) × Mwp s (Rd ) into Sp , in other words, 1 ,ϕ2 p,∞ kϕ k 1 kϕ2 kM p . kAϕ kSp . kakM1/τ 1 Mw a ws s s

p,∞ 1 ,ϕ2 is bounded from M1/τ c) If 2 ≤ p ≤ ∞, then the mapping (a, ϕ1 , ϕ2 ) 7→ Aϕ × a s ′

Mw1 s × Mwp s into Sp , and

1 ,ϕ2 p,∞ kϕ k ′ . 1 kϕ2 k kSp . kakM1/τ kAϕ 1 Mw a Mp s s

ws

Proof. a) We use the convolution relation (2.12) to show that the Weyl symbol 1 ,ϕ2 a ∗ W (ϕ2 , ϕ1 ) of Aϕ is in M ∞,1 . If ϕ1 , ϕ2 ∈ Mw1 s (Rd ), then by (3.3), we have a 2d ∞ 1 W (ϕ2 , ϕ1 ) ∈ Mτs (R ). Applying Proposition 2.4 in the form M1/τ ∗Mτ1s ⊆ M ∞,1 , s we obtain σ = a ∗ W (ϕ2 , ϕ1 ) ∈ M ∞,1 . The result now follows from Theorem 3.1 a). Similarly, the proof of b) and c) is based on results of Proposition 2.4 and Theorem 3.1, items b) - d).

Localization operators and exponential weights for modulation spaces

11

For the sake of completeness, we state the necessary boundedness result, which follows by straightforward modifications of [6, Theorem 4.3]. ′

Theorem 3.3. Let a ∈ S (1) (R2d ) and s ≥ 0. If there exists a constant C = C(a) > 0 depending only on a such that (3.4)

1 ,ϕ2 kAϕ kS∞ ≤ C kϕ1 kMw1 s kϕ2 kMw1 s a

∞ . for all ϕ1 , ϕ2 ∈ S (1) (Rd ), then a ∈ M1/τ s

4. Ultra-distributions with Compact Support as Symbols As an application we treat ultra-distributions with compact support, denoted here by Et′ , t > 1. The Gelfand-Shilov space S {t} is embedded in the corresponding ′ ′ Gevrey function space and therefore we have Et′ ⊂ S {t} ⊂ S (t) . Hence, an element ′ from Et′ , t > 1 can be viewed as an element of S (1) . We skip the precise definition of Et′ , see e.g. [22, Definition 1.5.5, Example 1.8.4 (b)]. In fact, the following structure theorem, obtained by a slight generalization of [22, Theorem 1.5.6] (see also [23]) to the anisotropic case, will be sufficient for our purposes. Theorem 4.1. Let t ∈ Rd , t > 1. Every u ∈ Et′ (Rd ) can be represented as X (4.1) u= ∂ α µα , α∈Nd 0

where µα is a measure satisfying, for every ǫ = (ǫ1 , . . . , ǫd ) > 0, Z |dµα | ≤ Cǫ ǫα (α!)−t , (4.2) K

for a suitable constant Cǫ > 0 and a suitable compact set K ⊂ Rd , independent of α. Proposition 4.2. Let t ∈ Rd , t > 1, and a ∈ Et′ (Rd ). Then its STFT with respect to any window g ∈ S (1) satisfies the estimate |Vg a(x, ω)| . e−h·|x| et·|2πω|

1/t

,

for every h > 0. Proof. Note that, for every g ∈ S (1) , we have X α α (2πiω)β Mω Tx ∂yα−β g(y) . ∂ (Mω Tx g)(y) = β β≤α

12

Elena Cordero, Stevan Pilipovi´c, Luigi Rodino and Nenad Teofanov

Then, using the representation (4.1), the STFT of a ∈ Et′ can be estimated as follows: |Vg a(x, ω)| X X ≤ |h∂ α µα , Mω Tx gi| = |hµα , ∂ α (Mω Tx g)i| α∈Nd 0

≤

α∈Nd 0

Z X X α β (Mω Tx ∂yα−β g)(y)| |dµα (y)|. |(2πiω) | β Rd d

α∈N0 β≤α

Now we use the estimate (2.2), where we may assume Bj = 1 for all j = 1, . . . , d, to get: |Mω Tx ∂yα−β g(y)| = |Tx ∂yα−β g(y)| ≤ C(α − β)!e−h·|y−x| , X α = 2kαk1 , so that we have for every h > 0. Furthermore, recall β β≤α

|Vg a(x, ω)| Z X X |2πω|β t t kαk1 (β!) [(α − β)!] e−h·|y−x| |dµα(y)|. . 2 (β!)t Rd d α∈N0

β≤α

¿From the representation theorem (4.2) we get, for every ǫ > 0, Z Z e−h·|y−x| |dµα (y)| ≤ e−h·|x| eh·|y| |dµα (y)| ≤ Ch,K,ǫ e−h·|x| ǫα (α!)−t . K

Since β!(α − β)! ≤ α!, we obtain −h·|x|

|Vg a(x, ω)| . e

X  |2πω|β/t t (2ǫ) , β! d

X

α∈N0

α

β≤α

for every ǫ > 0. We then choose ǫ < 1/2 and the following estimate assures the desired result: t  X  |2πω|β/t t X  |2πω|β/t t X |2πω|β/t 1/t  = et·|2πω| . ≤ ≤ β! β! β! d d β≤α

β∈N0

β∈N0

Corollary 4.3. Let t ∈ Rd , t > 1 and a ∈ Et′ (R2d ). If the window functions ϕ1 , ϕ2 ∈ 1 ,ϕ2 is a trace class operator. S {1} (Rd ), then Aϕ a

Proof. If ϕ1 , ϕ2 ∈ S {1} (Rd ), Proposition 2.3 b), with p = q = 1, assures the existence of an ǫ > 0 such that ϕ1 , ϕ2 ∈ Mw1 ǫ (Rd ). On the other hand, for kωk > Cǫ (where Cǫ is a suitable positive constant depending on ǫ) we can write t · |2πω|1/t =

d X i=1

ti |2πωi |1/ti ≤ ǫkωk ;

Localization operators and exponential weights for modulation spaces

13

1,∞ then, the estimate of Proposition 4.2 gives a ∈ M1/τ (R2d ). Finally, since ϕ1 , ϕ2 ∈ ǫ

1,∞ Mw1 ǫ (Rd ) and a ∈ M1/τ (R2d ), Theorem 3.2 b), written for the case p = 1, implies ǫ ϕ1 ,ϕ2 that the operator Aa is a trace class operator.

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Elena Cordero, Stevan Pilipovi´c, Luigi Rodino and Nenad Teofanov

[18] L. H¨ ormander. The Analysis of Linear Partial Differential Operators I,II,III,IV. Springer-Verlag, Berlin, 1983-1985. [19] A. Kami´ nski, D. Periˇsi´c, S. Pilipovi´c, On various integral transformations of tempered ultradistributions Demonstratio Math. 33(3): 641-655, 2000. [20] S. Pilipovi´c, Tempered ultradistributions Boll. Un. Mat. Ital. 7(2-B): 235–251, 1988. [21] S. Pilipovi´c and N. Teofanov, Pseudodifferential operators on ultra-modulation spaces J. Funct. Anal. 208:194–228, 2004. [22] L. Rodino, Linear Partial Differential Operators in Gevrey Spaces, World Scientific, Singapore, 1993. ´ [23] C. Roumieu, Sur quelques extensions de la notion de distribution Ann. Sci. Ecole Norm, Sup. 3(77): 41-121, 1960. [24] M. A. Shubin. Pseudodifferential Operators and Spectral Theory. Springer-Verlag, Berlin, second edition, 2001. [25] 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, M.W. Wong, editors, Birkh¨ auser, 164:173–191, 2006. [26] 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 Schr¨ odinger Institute (ESI), Wien. On the base of their Junior Research Fellowship 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 E-mail address: [email protected], [email protected] E-mail address: [email protected], [email protected]