Compactness Properties for Modulation Spaces

Home , Modulation space

Complex Analysis and Operator Theory (2019) 13:3521–3548 Complex Analysis https://doi.org/10.1007/s11785-019-00903-4 and Operator Theory

Christine Pfeuﬀer1 · Joachim Toft2

Received: 3 October 2018 / Accepted: 12 February 2019 / Published online: 9 March 2019 © The Author(s) 2019

Abstract We prove that if ω1 and ω2 are moderate weights and B is a suitable (quasi-)Banach function space, then a necessary and sufficient condition for the embedding i : M(ω1, B) → M(ω2, B) between two modulation spaces to be compact is that the quotient ω2/ω1 vanishes at infinity. Moreover we show, that the boundedness of ω2/ω1 is a necessary and sufficient condition for the previous embedding to be continuous.

Keywords Gelfand–Shilov spaces · Distributions · Bargmann transform · Quasi-Banach spaces

Mathematics Subject Classiﬁcation 42B35 · 46B50 · 54D30 · 46Fxx

1 Introduction

A fundamental question in analysis, science and engineering concerns compactness. In the paper we investigate continuity and compactness properties for a broad class of modulation spaces, a family of functions and distribution spaces on the configura- tion space Rd , introduced by Feichtinger [12]. The basic theory of these spaces was thereafter established by Feichtinger and Gröchenig (see [15,16,25] and the references therein). A modulation space is defined by imposing a certain norm or quasi-norm estimate on the short-time Fourier transform of the involved functions and distributions. (See [28] or Sect. 2 for definitions and notations.) Roughly speaking, this means that each

Communicated by Palle Jorgensen.

B Joachim Toft [email protected] Christine Pfeuffer [email protected]

1 Department of Mathematics, University of Regensburg, Regensburg, Germany 2 Department of Mathematics, Linnæus University, Växjö, Sweden 3522 C. Pfeuffer, J. Toft modulation space norm measures in a certain way the phase or time-frequency con- tent. That is, it admits to measure the conﬁguration energy and momentum energy simultaneously, in a certain way. By the design of these spaces it turns out that they are useful in several ﬁelds in analysis, physics and engineering (see [6,8,31,34] and the references therein). Compactness investigations of embeddings between modulation spaces go in some sense back to [32], where M. Shubin proved that if t > 0, then the embedding i : Qs → Qs−t is compact. In the community, the previous compactness property was not obvious because similar facts do not hold for Sobolev spaces of Hilbert type. That > : 2 → 2 is, for t 0 it is well-known that the embedding i Hs Hs−t is continuous but not compact. Since

2,2 s Qs = M(ω),ω(X) = (1 +|x|+|ξ|) (1.1) and

2 = 2,2,ω() = ( +|ξ|)s, = ( ,ξ) Hs M(ω) X 1 X x (1.2) the previous compact embedding properties can also be written by means of modulation spaces. A more general situation was considered by M. Dörﬂer, H. Feichtinger and K. Gröchenig who considered the inclusion map

, , i : M p q (Rd ) → M p q (Rd ) (1.3) (ω1) (ω2) in [9]. It is well-known that this map is well-defined and continuous when ω2 ω1, see e.g. [25]. In [9, Theorem 5] it was proved that if p, q ∈[1, ∞), and ω1 and ω2 are certain moderate weights of polynomial type, then (1.3) is compact if and only if ω2/ω1 tends to zero at infinity. By choosing ω j in similar ways as in (1.1), the latter compactness result confirms the compactness of the embedding i : Qs → Qs−t above by Shubin, as well as it confirms the lack of compactness of the embedding : 2 → 2 i Hs Hs−t for Sobolev spaces. In [3], the compact embedding property [9, Theorem 5] was extended in such a way that all moderate weights ω j of polynomial type are included. That is, there are no other restrictions on ω j than there should exist constants N j > 0 such that

N j 2d ω j (X + Y ) ω j (X)(1 +|Y |) , X, Y ∈ R , j = 1, 2. (1.4)

Moreover, in [3], the Lebesgue exponents p and q in the modulation spaces are allowed to attain ∞. In our main results, Theorems 3.7 and 3.9 in Sect. 3, these continuity and compactness results are extended to involve modulation spaces M(ω, B), which are more general in different ways. Firstly, there are no boundedness estimates of polynomial type for the involved weight ω. In most of our considerations, we require that the weights are moderate, which implies that the condition (1.4) is relaxed into

r j |Y | ω j (X + Y ) ω j (X)e , j = 1, 2, Compactness Properties for Modulation Spaces 3523 for some constants r1, r2 > 0. Secondly, B can be any general translation invariant Banach function space with- p,q out the restriction that M(ω, B) should be of the form M(ω) . We may also have p,q M(ω, B) = M(ω) , but in contrast to [3,9], we here allow p and q to be smaller than 1. p,q Here we notice that if p < 1orq < 1, then M(ω) fails to be a Banach space because of absence of convex topological structures. Thirdly, we show that (1.3) is compact when ω2/ω1 tends to zero at inﬁnity, when the assumptions on ω1 and ω2 are relaxed into a suitable local moderate condition (cf. Theorem 3.9(1) in Sect. 3). We refer to [40] and to some extent to [42]fora detailed study of modulation spaces with such relaxed conditions on the involved weight functions. In Sect. 4 we show how Theorem 3.9 has been applied in [1] to deduce index results and lifting properties for certain pseudo-differential operators. Especially it is here indicated in which way our results on compactness are used to show that the , , operator Op (ω ) is continuous and bijective from M p q to M p q when A 0 (ω) (ω/ω0)

r ρ ω0(x,ξ)≡ p1(x) + p2(ξ) ,

d r,ρ >0, and p1 and p2 are positive polynomials on R . In Sect. 4 we also give some links on possible extensions and generalizations. Finally we remark that for moderate weights, the continuity and compactness properties for (1.3) can also be obtained by Gabor analysis, which transfers (1.3)into

, , i :p q → p q , (ω1) (ω2)

Since it is clear that the latter inclusion map is compact, if and only if ω2/ω1 tends to 0 at inﬁnity, it follows that the compactness results in [3,32]aswellassomeof the results in Sect. 3 can be deduced in such ways. We emphasise however that such technique can not be used in those situations in Sect. 3 when modulation spaces are of the form M(ω, B), where either B is a general BF-space, or ω fails to be moderate, since it seems that Gabor analysis is not applicable on such modulation spaces. Parts of the proof of these continuity and compactness results are based on some properties for modulation spaces which might be of independent interests. We prove that if B is a Banach Function space, then M(ω, B) is a Banach space, which is contin- ∞ ( d ) uously embedded in a weighted modulation space of the form M(1/v) R for certain weights v. (See Proposition 3.2.) We also need some properties for the Bargmann transform when acting on modulation spaces, essentially deduced in [40,42], which are presented in Sect. 2.5.

2 Preliminaries

In this section we discuss basic properties for modulation spaces and other related spaces. The proofs are in many cases omitted since they can be found in [10–12,15– 17,25,36–39]. 3524 C. Pfeuffer, J. Toft

2.1 Weight Functions

A weight or weight function ω on Rd is a positive function such that ω,1/ω ∈ ∞ ( d ) ω v d ω v Lloc R .Let and be weights on R . Then is called -moderate or moderate,if d ω(x1 + x2) ω(x1)v(x2), x1, x2 ∈ R . (2.1) Here f (θ) g(θ) means that f (θ) ≤ cg(θ) for some constant c > 0 which is independent of θ in the domain of f and g.Ifv can be chosen as a polynomial, then ω is called a weight of polynomial type. The function v is called submultiplicative, if it is even and (2.1) holds for ω = v. We notice that (2.1) implies that if v is submultiplicative on Rd , then there is a constant c > 0 such that v(x) ≥ c when x ∈ Rd . d d d We let PE (R ) be the set of all moderate weights on R , and P(R ) be the subset d d of PE (R ) which consists of all polynomially moderate functions on R .Fors > 0, P ( d ) P0 ( d ) ω d also let E,s R ( E,s R ) be the set of all weights in R such that

1 r|x2| s d ω(x1 + x2) ω(x1)e , x1, x2 ∈ R . (2.2)

> > P0 = P0 for some r 0 (for every r 0), and set E E,1.Wehave

0 0 P ⊆ P ⊆ P , ⊆ P ⊆ P when s < s E,s1 E s1 E,s2 E 2 1 and

PE,s = PE when s ≤ 1,

ω ∈ P ( d ) ω ∈ P0 ( d ) where the last equality follows from the fact that if E R ( E R ), then

| | 1 − | | | | ω(x + y) ω(x)er y s and e r x ≤ ω(x) er x , x, y ∈ Rd (2.3) hold true for some r > 0 (for every r > 0) in view of [26] when s ≤ 1. In some situations we shall consider a more general class of weights compared to PE given in [40, Deﬁnition 1.1].

d d Deﬁnition 2.1 The set PQ(R ) consists of all weights ω on R such that for some constants R ≥ 2 and c, r > 0 it holds

− | |2 | |2 e r x ω(x) er x , (2.4) and c ω(x)2 ω(x + y)ω(x − y) ω(x)2 when Rc ≤|x|≤ . (2.5) |y| Compactness Properties for Modulation Spaces 3525

2.2 Gelfand–Shilov Spaces

Let 0 < h, s ∈ R. Then we denote the set of all functions f ∈ C∞(Rd ) such that

|xβ ∂α f (x)| f S ≡ sup < ∞ (2.6) s,h h|α+β|(α! β!)s

d d d by Ss,h(R ). Here the supremum is taken over all α, β ∈ N and x ∈ R . It is clear that Ss,h is a Banach space which increases with h and s, and that Ss,h → S holds. Here A → B means that A ⊆ B with continuous embedding. If > 1 s 2 , then Ss,h and S1/2,h h>0 contains all ﬁnite linear combinations of the Hermite functions. By the density of such S S S ( d ) S ( d ) linear combinations in and in s,h, the dual s,h R of s,h R is a Banach space which contains S (Rd ), for such choices of s. d The inductive and projective limits respectively of Ss,h(R ) are called Gelfand– d Shilov spaces of Beurling respectively Roumieu type and are denoted by Ss(R ) and d s(R ). Hence d d d d Ss(R ) = Ss,h(R ) and s(R ) = Ss,h(R ), (2.7) h>0 h>0

d where the topology for Ss(R ) is the strongest possible one such that the inclusion d d map from Ss,h(R ) to Ss(R ) is continuous, for every choice of h > 0. Equipped with · > ( d ) the seminorms Ss,h , h 0 the space s R is a Fréchet space. Additionally, ( d ) ={ } > 1 S ( d ) ={ } ≥ 1 s R 0 , if and only if s 2 , and s R 0 , if and only if s 2 . S ( d ) ( d ) The Gelfand–Shilov distribution spaces s R and s R are the projective and S ( d ) inductive limit respectively of s,h R . This implies that S ( d ) = S ( d ) ( d ) = S ( d ). s R s,h R and s R s,h R (2.7) h>0 h>0

S ( d ) S ( d ) ( d ) ( d ) Note that s R is the dual of s R , and s R is the dual of s R as proved in [22]. By the deﬁnitions we have

S1/2 → s → Ss → s+ε → S 1 → S → +ε → S → → S / , s > ,ε>0. (2.8) s s s 1 2 2 3526 C. Pfeuffer, J. Toft

We let ·, · denote the dual form between a topological vector space and its dual. If z,w ∈ Cd , then z,w is deﬁned by

d d d z,w = z j w j , z = (z1,...,zd ) ∈ C ,w= (w1,...,wd ) ∈ C . j=1

The Fourier transform of f ∈ L1(Rd ) is given by − d −i x,ξ (F f )(ξ) = f (ξ) ≡ (2π) 2 f (x)e dx. Rd

F S ( d ) S ( d ) ( d ) The map extends uniquely to homeomorphisms on R , s R and on s R . d d d Furthermore, F restricts to homeomorphisms on S (R ), Ss(R ), s(R ), and to a unitary operator on L2(Rd ). Similar results hold true for partial Fourier transforms. φ ∈ S ( d ) ∈ S ( d ) For a ﬁxed s R the short-time Fourier transform Vφ f of f s R with respect to the window function φ is the Gelfand–Shilov distribution on R2d , deﬁned by

− d −i ·,ξ Vφ f (x,ξ)= (2π) 2 f , φ(·−x)e

∈ ( d ) ∈ S ( d ) φ ( d ) S ( d ) If instead f s R or f R , then should belong to s R or R , respectively. We have

Vφ f (x,ξ)≡ (F2(U( f ⊗ φ)))(x,ξ)= F ( f φ(·−x))(ξ), (2.9) where (UF)(x, y) = F(y, y − x).HereF2 F denotes the partial Fourier transform ( , ) ∈ S ( 2d ) of F x y s R with respect to the y variable (see (A.1) in [5]). d In the case f ∈ Ss(R ), Vφ f can be written as − d −i y,ξ Vφ f (x,ξ)= (2π) 2 f (y)φ(y − x)e dy.

The next two propositions characterize Gelfand–Shilov functions and their distributions by suitable estimates on the short-time Fourier transforms of the corresponding distributions. The proofs are omitted since the results are special cases of [27, Theorem 2.7]) and [42, Proposition 2.2]. Proposition 2.2 , ≥ 1 ≤ φ ∈ S ( d )\ Let s s0 2 be such that s0 s. Also let s0 R 0 and f ∈ S (Rd ). Then the following is true: s0 d (1) f ∈ Ss(R ), if and only if

1 1 −r(|x| s +|ξ| s ) |Vφ f (x,ξ)| e (2.10)

holds for some r > 0; Compactness Properties for Modulation Spaces 3527

> 1 φ ∈ ( d ) ∈ ( d ) (2) if in addition s0 2 and s0 R , then f s R , if and only if (2.10) holds for every r > 0. Proposition 2.3 , ≥ 1 ≤ φ ∈ S ( d )\ Let s s0 2 be such that s0 s. Also let s0 R 0 and f ∈ S (Rd ). Then the following is true: s0 ∈ S ( d ) (1) f s R , if and only if

1 1 r(|x| s +|ξ| s ) |Vφ f (x,ξ)| e (2.11)

holds for every r > 0; > 1 φ ∈ ( d ) ∈ ( d ) (2) if in addition s0 2 and s0 R , then f s R , if and only if (2.11) holds for some r > 0.

d Remark 2.4 For every s > 0, the mapping ( f ,φ)→ Vφ f is continuous from Ss(R )× S ( d ) S ( 2d ) S ( d ) × s R to s R and extends uniquely to continuous mappings from s R S ( d ) S ( 2d ) S S s R to s R . The same is true if we replace each s by or by s.Thisis admitted by formula (2.9)(cf.e.g.[35,42]).

2.3 Modulation Spaces

We recall that a quasi-norm · B of order r ∈ (0, 1] on the vector-space B is a nonnegative functional on B which satisﬁes

1 −1 f + g B ≤ 2 r ( f B + g B), f , g ∈ B, α · f B =|α|· f B,α∈ C, f ∈ B (2.12) and

f B = 0 ⇔ f = 0.

The vector space B is called a quasi-Banach space if it is a complete quasi-normed space. If B is a quasi-Banach space with quasi-norm satisfying (2.12) then by [2,29] there is an equivalent quasi-norm to · B which additionally satisﬁes

r r r f + g B ≤ f B + g B, f , g ∈ B. (2.13)

From now on we always assume that the quasi-norm of the quasi-Banach space B is chosen in such a way that both (2.12) and (2.13) hold. d 2d Let φ ∈ 1(R )\0, p, q ∈ (0, ∞] and ω ∈ PE (R ). Then the modulation space p,q ( d ) ∈ ( d ) M(ω) R is deﬁned as the set of all f 1 R such that q/p 1/q p f p,q ≡ |Vφ f (x,ξ)ω(x,ξ)| dx dξ < ∞ (2.14) M(ω)

p p,p p,q p,q p p holds. We set M(ω) = M(ω) , and if ω = 1, we set M = M(ω) and M = M(ω). 3528 C. Pfeuffer, J. Toft

We summarize some well-known facts about Modulation spaces in the next proposition. See [12,21,25,41] for the proof. Here the conjugate exponent of p ∈ (0, ∞] is given by ⎧ ⎪∞ ∈ ( , ], ⎨⎪ when p 0 1 p = when p ∈ (1, ∞), p ⎪ − ⎩⎪ p 1 1 when p =∞.

Proposition 2.5 Let p, q, p j , q j , r ∈ (0, ∞] be such that r ≤ min(1, p, q),j= 1, 2, 2d r d let ω,ω1,ω2,v ∈ PE (R ) be such that ω is v-moderate, φ ∈ M(v)(R )\0 and let ∈ ( d ) f 1 R . Then the following is true: p,q d p,q (1) f ∈ M(ω) (R ) if and only if (2.14) holds. Moreover, M(ω) is a quasi-Banach space under the quasi-norm in (2.14) and a Banach space if p, q ≥ 1. Different choices of φ give rise to equivalent (quasi-)norms; (2) if p1 ≤ p2,q1 ≤ q2 and ω2 ω1, then

, , (Rd ) ⊆ M p1 q1 (Rd ) ⊆ M p2 q2 (Rd ) ⊆ (Rd ); 1 (ω1) (ω2) 1

2 d (3) if in addition p, q ≥ 1, then the L -form on 1(R ) extends uniquely to a contin-

p,q ( d )× p ,q ( d ) , < ∞ uous sesqui-linear form on M(ω) R M(1/ω) R . Furthermore, if p q ,

p,q ( d ) p ,q ( d ) then the dual of M(ω) R can be identiﬁed by M(1/ω) R through this form.

2.4 A Broader Family of Modulation Spaces

In this subsection we introduce a broader class of modulation spaces by imposing certain types of translation invariant solid BF-space norms on the short-time Fourier transform, cf. [12–16].

Deﬁnition 2.6 B ⊆ r ( d ) ∈ ( , ] Let Lloc R be a quasi-Banach space of order r 0 1 which d d contains 1(R ) with continuous embedding, and let v0 ∈ PE (R ). Then B is called d a translation invariant Quasi-Banach Function space on R (with respect to v0), or invariant QBF space on Rd of order r, if there is a constant C such that: (1) if x ∈ Rd and f ∈ B, then f ( ·−x) ∈ B, and

f ( ·−x) B ≤ Cv0(x) f B; (2.15)

, ∈ r ( d ) ∈ B | |≤| | ∈ B (2) if f g Lloc R satisfy g and f g , then f and

f B ≤ C g B.

v P ( d ) P0 ( d ) B If the weight 0 even is an element of E,s R ( E,s R ), then we call of Deﬁnition 2.6 an invariant QBF-space of Roumieu type (Beurling type) of order r. Compactness Properties for Modulation Spaces 3529

By Deﬁnition 2.6(2) we have f · h ∈ B and

f · h B ≤ C f B h L∞ , (2.16) when f ∈ B and h ∈ L∞. The QBF space B is called is called an invariant BF-space (with respect to v0), if ( d ) ( ,ϕ)→ ∗ ϕ it is of order 1, is continuously embedded in 1 R and the map f f is well-deﬁned and continuous from B × L1 (Rd ) to B. Note here that an invariant (v0) QBF space of order r = 1 is a Banach space. Because of condition (2) an invariant BF-space is a solid BF-space in the sense of (A.3) in [13]. For the invariant BF-space B ⊆ 1 ( d ) v Lloc R with respect to 0 we have Minkowski’s inequality, i. e.

1 d f ∗ ϕ B ≤ C f B ϕ 1 , f ∈ B,ϕ∈ L (R ) (2.17) L(v) (v0) for some C > 0 which is independent of f ∈ B and ϕ ∈ L1 (Rd ). (v0) The following result shows that v0 in Deﬁnition 2.6 can be replaced by a submultiplicative weight v such that (2.15) is true with v in place of v0 and the constant C = 1, and such that v(x + y) ≤ v(x)v(y), x, y ∈ Rd . (2.18)

d d Proposition 2.7 Let B be an invariant BF-space on R with respect to v0 ∈ PE (R ). d Then there is a submultiplicative v ∈ PE (R ) which satisﬁes (2.15) and (2.18) with v in place of v0, and C = 1.

Proof Let f ( ·−x) B v (x) ≡ . 1 sup f ∈B f B

Then f ( ·−x − y) B f ( ·−y) B v1(x + y) = sup · ∈B f ( ·−y) B f B f f ( ·−x) B f ( ·−y) B ≤ · = v (x)v (y). sup sup 1 1 f ∈B f B f ∈B f B

The result now follows by letting

v(x) = max(v1(x), v1(−x)).

From now on it is assumed that v and v j are submultiplicative weights if nothing else is stated. 3530 C. Pfeuffer, J. Toft

Example 2.8 , ∈[, ∞] p,q ( 2d ) ∈ 1 ( 2d ) For p q 1 the space L1 R consists of all f Lloc R such that q/p 1/q p f p,q ≡ | f (x,ξ)| dx dξ < ∞. L1

p,q ( 2d ) ∈ 1 ( 2d ) Additionally L2 R is the set of all f Lloc R such that p/q 1/p q f p,q ≡ | f (x,ξ)| dξ dx < ∞. L2

p,q p,q v = Then L1 and L2 are translation invariant BF-spaces with respect to 1. Next we deﬁne the extended class of modulation spaces, which are of main interest for us.

2d 2d Deﬁnition 2.9 Let B be a translation invariant QBF-space on R , ω ∈ PE (R ), φ ∈ ( d )\ (ω, B) ∈ ( d ) and 1 R 0. Then the set M consists of all f 1 R such that

f M(ω,B) ≡ Vφ f ω B < ∞.

p,q ( d ) = (ω, B) B = p,q ( 2d ) Obviously, we have M(ω) R M if L1 R , see e.g. (2.14). We remark that many properties of the classical modulation spaces are also true for M(ω, B). For instance, the deﬁnition of M(ω, B) is independent of the choice of φ when B is a Banach space. This statement is formulated in the next proposition. We omit the proof since it can be proved by similar arguments as in Proposition 11.3.2 in [25].

2d Proposition 2.10 Let B be an invariant BF-space with respect to v0 ∈ PE (R ) and 2d let ω,v ∈ PE (R ) be such that ω is v-moderate. Also let M(ω, B) be the same as in Deﬁnition 2.9, and let φ ∈ M1 (Rd )\0 and f ∈ (Rd ). Then f ∈ M(ω, B) if (v0v) 1 and only if Vφ f · ω ∈ B, and different choices of φ gives rise to equivalent norms of M(ω, B).

We refer to [12,15–17,21,25,30,41] for more facts about modulation spaces. In applications, B is mostly a mixed quasi-normed Lebesgue space, which is deﬁned ={ ,..., } d ={ ,..., } next. Let E e1 ed be an ordered basis of R and let E e1 ed be such that

, = πδ , , = ,..., . e j ek 2 jk j k 1 d

Then E is called the dual basis of E. The corresponding lattice and dual lattice are

d E ={j1e1 +···+ jd ed ; ( j1,..., jd ) ∈ Z } Compactness Properties for Modulation Spaces 3531 and

= ={ι +···+ι ; (ι ,...,ι ) ∈ d }. E E 1e1 d ed 1 d Z

We also let κ(E) be the parallelepiped spanned by the basis E. d We deﬁne for each q = (q1,...,qd ) ∈ (0, ∞] ,

max(q) = max(q1,...,qd ) and min(q) = min(q1,...,qd ).

d d Deﬁnition 2.11 Let E ={e1,...,ed } be an orderd basis of R , ω be a weight on R , = ( ,..., ) ∈ ( , ∞]d = ( , ) ∈ r ( d ) p p1 pd 0 and r min 1 p .If f Lloc R , then

f p ≡ gd−1 L pd (R), L E,(ω)

d−k where gk(zk), zk ∈ R , k = 0,...,d − 1, are inductively deﬁned as

g0(x1,...,xd ) ≡|f (x1e1 +···+xd ed )ω(x1e1 +···+xd ed )| and

gk(zk) ≡ gk−1( · , zk) L pk (R), k = 1,...,d − 1.

p d r d The space L ,(ω)(R ) consists of all f ∈ L (R ) such that f p is ﬁnite, and E loc L E,(ω) is called E-split Lebesgue space (with respect to p and ω). Let E, p and ω be the same as in Deﬁnition 2.11. Then the discrete version p ( ) p ( d ) ={ ( )} E,(ω) E of L E,(ω) R is the set of all sequences a a j j∈E such that the quasi-norm a p ≡ fa p , fa = a( j)χ j , E,(ω) L E,(ω) j∈E

χ + κ( ) p = p is ﬁnite. Here j is the characteristic function of j E .WealsosetL E L E,(ω) p = p ω = and E E,(ω) when 1. Deﬁnition 2.12 Let E be an ordered basis of the phase space R2d . Then E is called weakly phase split if there is a subset E0 ⊆ E such that the span of E0 equals 2d d 2d d { (x, 0) ∈ R ; x ∈ R } and the span of E\E0 equals { (0,ξ)∈ R ; ξ ∈ R }.

2.5 Pilipovi´c Flat Spaces, Modulation Spaces Outside Time-Frequency Analysis and the Bargmann Transform

Besides the characterization by means of the short-time Fourier transform in Propo- sition 2.2, Gelfand–Shilov spaces also can be characterized via Hermite function expansions. Recall that the Hermite function of order α ∈ Nd is given by 3532 C. Pfeuffer, J. Toft

| |2 − d |α| |α| − 1 x α −|x|2 hα(x) = π 4 (−1) (2 α!) 2 e 2 (∂ e ).

{ } 2( d ) It is well-known that hα α∈Nd provides an orthonormal basis for L R and a basis S ( d ) ( d ) > 1 S ( d ) ≥ 1 for R , s R when s 2 , s R when s 2 and their duals. ≥ 1 > 1 ∈ S ( d ) ∈ ( d ) By [23]wehavefors 2 (s 2 ) that f s R ( f s R ), if and only if the coefﬁcients cα( f ) in its Hermite series expansion = ( ) , ( ) = ( , ) f cα f hα cα f f hα L2(Rd ) (2.19) α∈Nd fulﬁll

1 −r|α| 2s |cα( f )| e

d d for some r > 0 (for every r > 0) with convergence in Ss(R ) ( s(R )). Similarly, ∈ S ( d ) ∈ ( d ) f s R ( f s R ), if and only if

1 r|α| 2s |cα( f )| e

> > S ( d ) ( d ) for every r 0 (for some r 0) with convergence in s R ( s R ). In [19,42] various kinds of Fourier-invariant function and distribution spaces are obtained by applying suitable topologies on formal power series expansions. In par- H ( d ) H ( d ) ticular, the Pilipović flat space R , denoted by 1 R in [42], and its dual d H(R ), are defined by all formal expansions (2.19) such that

|α| − 1 |cα( f )| r α! 2 for some r > 0, respectively

|α| 1 |cα( f )| r α! 2

d d for every r > 0. For f ∈ H(R ) and φ ∈ H(R ), we deﬁne ( ,φ) ≡ ( ) (φ). f L2(Rd ) cα f cα α∈Nd

φ, ∈ 2( d ) ( ,φ) 2( d ) If f L R , the pairing f L2(Rd ) agrees with the L R scalar product of those two functions. d We remark that H(R ) is larger than any Fourier-invariant Gelfand–Shilov distribution space, and H(Rd ) is smaller than any Fourier-invariant Gelfand–Shilov space. d d We notice that H(R ) and H(R ) possess interesting mapping properties under the Bargmann transform. In fact, the Bargmann kernel is given by − d 1 2 1 Ad (z, y) = π 4 exp − ( z, z +|y| ) + 2 2 z, y , 2 Compactness Properties for Modulation Spaces 3533

d d which is analytic in z. For ﬁxed z ∈ C , the function y → Ad (z, y) belongs to H(R ). d The Bargmann transform (Vd f )(z) of f ∈ H(R ) is then deﬁned by

(Vd f )(z) = f , Ad (z, · ) .

d d Dueto[42] we have that Vd is bijective between H(R ) and A(C ),thesetofall entire functions on Cd , and restricts to a bijective map from H(Rd ) to

| | { F ∈ A(Cd ) ;|F(z)| eR z , for some R > 0 }.

Later on we need that the Bargmann and the short-time Fourier transform are linked by the formula

d 1 (|x|2+|ξ|2) −i x,ξ 1 1 (Vd f )(x + iξ) = (2π)2 e 2 e (Vφ f )(2 2 x, −2 2 ξ), (2.20) − d − 1 |x|2 d φ(x) = π 4 e 2 , x ∈ R , which can be shown by straight-forward computations. By means of the operator

d 1 (|x|2+|ξ|2) −i x,ξ 1 1 (UV F)(x,ξ)= (2π)2 e 2 e F(2 2 x, −2 2 ξ), (2.21)

d where F is a function or a suitable element of F ∈ H(R ) we can write the Bargmann transform as

(Vd f )(x + iξ) = (UV(Vφ f ))(x,ξ).

Deﬁnition 2.13 Let φ be as in (2.20), ω be a weight on R2d , B be an invariant QBF- 2d 2d d space with respect to v ∈ PE (R ) on R C of order r ∈ (0, 1], and let UV be given by (2.21). Then (ω, B) ∈ r ( 2d ) = r ( d ) (1) B consists of all F Lloc R Lloc C such that

−1 F B(ω,B) ≡ (UV F)ω B < ∞;

(2) A(ω, B) consists of all F ∈ A(Cd ) ∩ B(ω, B) with topology inherited from B(ω, B); d (3) M(ω, B) consists of all f ∈ H(R ) such that

f M(ω,B) ≡ Vφ f · ω B

is ﬁnite. We observe the smaller restrictions on ω compared to what is the main stream. For 2d example, in Deﬁnition 2.13 it is not assumed that ω should belong to PE (R ) or P(R2d ). We still call M(ω, B) a modulation space. In contrast to earlier situations, it seems that M(ω, B) is not invariant under the choice of φ when ω fails to belong to PE . For that reason we always assume that φ is given by (2.20) for such ω. 3534 C. Pfeuffer, J. Toft

We have the following. Proposition 2.14 Let φ be as in (2.20), ω be a weight on R2d and let B be an invariant 2d QBF-space with respect to v ∈ PE (R ). Then the following is true: (1) the map Vd is an isometric bijection from M(ω, B) to A(ω, B); (2) if in addition B is a mixed quasi-normed space of Lebesgue type, then M(ω, B) and A(ω, B) are quasi-Banach spaces, which are Banach spaces when B is a Banach space. For moderate weights, Proposition 2.14 is proved in [18,24] with some completing arguments given in [33]. For the broader weight class PQ in Deﬁnition 2.1, a proof of Proposition 2.14 is given in [40]. We now present a proof which holds for any weight ω.

Proof From (2.20), (2.21) and Deﬁnition 2.13 it follows that Vd is an isometric injection from M(ω, B) to A(ω, B). Since any element in A(Cd ), and thereby any element d in A(ω, B) is a Bargmann transform of an element in H(R ), it follows that the image of M(ω, B) under Vd contains A(ω, B). This gives the stated bijectivity in (1). The completeness of A(ω, B), and thereby of M(ω, B) follows from [42, Theorem 4.8]. This gives (2).

3 Compactness Properties for Modulation Spaces

This section is devoted to the questions under which sufﬁcient and necessary conditions the inclusion map

i : M(ω1, B) → M(ω2, B) is continuous or even compact for suitable invariant QBF-spaces B. As ingredients for the proof of our main results we need to deduce some properties ∞ ( d ) ∈ for moderate weight functions. In what follows let L0,(ω) R be the set of all f ∞ d L(ω)(R ) with the property | ( )ω( )| = , lim ess sup|x|≥R f x x 0 R→∞

ω d ∞ = ∞ ω = when is a weight on R .WealsosetL0 L0,(ω) when 1. If is a lattice, ∞() ∞ () then the discrete Lebesgue spaces 0 and 0,(ω) are deﬁned analogously. d d Lemma 3.1 Let E be an ordered basis of R and let ω ∈ PE (R ). Then the following is true: d (1) PE (R ) is a convex cone which is closed under multiplication, division and under compositions with power functions; P ( d ) ∩ p ( d ) ∈ ( , ∞]d (2) E R L E,(ω) R increases with p 0 , and P ( d ) ∩ p ( d ) ⊆ P ( d ) ∩ ∞ ( d ), ∈ ( , ∞)d . E R L E,(ω) R E R L0,(ω) R p 0 (3.1)

Similar properties have been shown in [3, Lemma 2.1] for the subset P of PE . Compactness Properties for Modulation Spaces 3535

Proof Claim (1) can easily be veriﬁed by means of the deﬁnition of moderate weights. It remains to verify (2). Let κ(E) be the (closed) parallelepiped spanned by E and d let ϑ ∈ PE (R ). By using the map ϑ → ϑ · ω, we reduce ourself to the case when ω = 1. 2d The moderateness of ϑ ∈ PE (R ) implies that

ϑ(x1) ϑ(x1 + x2), when x2 ∈ κ(E). (3.2)

Hence, if χ j is the characteristic function of j + κ(E) and ϑ0(x) = ϑ(j)χ j (x), j∈E then ϑ ϑ0, giving that

ϑ p ϑ0 p ϑ p . L E L E E

p The assertion now follows from the fact that E increases with p and that if in addition ∈ ( , ∞)d p ⊆ ∞ p 0 , then E 0 . We also need the following extension of [43, Theorem 2.5]. 2d 2d Proposition 3.2 Let v,v0 ∈ PE (R ) be submultiplicative, ω ∈ PE (R ) be v- moderate, and let B be an invariant BF-space with respect to v0. Then M(ω, B) is a Banach space, and ∞ M(ω, B) → M (Rd ). (3.3) (1/(v0v)) Remark 3.3 B = p ( 2d ) 2d ∈ If L E R for some weakly phase split basis E of R , p 2d 2d (0, ∞] and ω ∈ PE (R ), then M(ω, B) is a quasi-Banach space. Moreover (ω, p ( 2d )) M L E R is increasing with p. In particular, (3.3) is improved in [41]into

(ω, p ( 2d )) → ∞ ( 2d ). M L E R M(ω) R

For the proof of Proposition 3.2 we need to consider the twisted convolution, ∗, deﬁned by − d −i x−y,η (F ∗ G)(x,ξ)= (2π) 2 F(x − y,ξ − η)G(y,η)e dydη R2d when F, G ∈ L1(R2d ). The twisted convolution map T , which takes suitable pairs of functions and distributions (F, G) into F ∗ G, is continuous between several function and distribution spaces, see e. g. [25]orLemma3in[7]. For example the map T is continuous from L1(R2d ) × L1(R2d ) to L1(R2d ). For any s > 0 it restricts to continuous mappings

2d 2d 2d T : Ss(R ) × Ss(R ) → Ss(R ), (3.4) 2d 2d 2d T : s(R ) × s(R ) → s(R ), 3536 C. Pfeuffer, J. Toft and by duality it follows that these mappings extend to continuous mappings

2d 2d 2d T : S (R ) × Ss(R ) → S (R ), s s (3.5) T : ( 2d ) × ( 2d ) → ( 2d ). s R s R s R

In fact, for some map A∗ we have A∗(F ∗ G) = (A∗ F) ◦ (A∗G). This map A∗ is similar to the map A in Sections 1 and 2 in [4], and by identifying operators with their kernels, A∗ consists of pullbacks of partial Fourier transforms and non-degenerate linear transformations on the phase space. Since such pullbacks are homeomorphic on any Fourier invariant Gelfand–Shilov space, (3.4) follows from the fact that the sets

2d 2d { TK ; K ∈ Ss(R ) } and { TK ; K ∈ s(R ) } are algebras under compositions. Here TK denotes the linear operator with distribution kernel K . The twisted convolution is convenient to use when changing window functions in short-time Fourier transforms. In fact by Fourier’s inversion formula and some straight-forward computations one has

(φ ,φ ) · = ( )∗ ( φ ) 3 1 L2 Vφ2 f Vφ1 f Vφ2 3 (3.6)

∈ ( d ) φ ,φ ,φ ∈ ( d ) φ ∈ ( d )\ for every f s R and 1 2 3 s R . For any s R 0, we are also interested of the operator Pφ, given by

−2 2d Pφ F ≡ φ F ∗ (Vφφ), F ∈ (R ). L2(Rd ) s (3.7)

φ ∈ ( 2d ) φ ,φ ∈ ( d ) Since Vφ2 3 s R when 2 3 s R , it follows from the continuity prop- 2d erties of the twisted convolution above that Pφ in (3.7) is continuous on s(R ) and ( 2d ) S S on s R . Similar facts hold true with s and s in place of s and s, respectively, at each place. The operator Pφ has the following properties. d Lemma 3.4 Let s > 0, φ ∈ s(R )\0. Then the following is true: ( 2d ) (1) Pφ in (3.7) is a continuous projection from s R to ( ( d )) ≡{ ; ∈ ( d ) }⊆ ( 2d ) ∞( 2d ); Vφ s R Vφ f f s R s R C R

2d (2) Pφ in (3.7) restricts to a continuous projection from s(R ) to

d d 2d Vφ( s(R )) ≡{Vφ f ; f ∈ s(R ) }⊆ s(R );

(3) if B is an invariant BF-space on R2d , then Pφ is continuous on B. S S Similar facts hold true with s and s in place of s and s, respectively, at each place. Compactness Properties for Modulation Spaces 3537

Related results can essentially be found in e. g. [20,24]. In order to be self-contained, we here give a short proof. Proof We only prove the result in the Beurling case. The Roumieu case follows by similar arguments and is left for the reader. ( ( d )) By (3.6) it is clear that Pφ is the identity map on Vφ s R and thereby on d Vφ( s(R )). ∗ 2 ∗ Let Vφ be the L -adjoint of Vφ. That is, Vφ F satisﬁes

( ∗ ,ψ) = ( , ψ) , ∈ ( 2d ), ψ ∈ ( d ). Vφ F L2(Rd ) F Vφ L2(R2d ) F s R s R ∗ By the continuity properties of Vφ on s and s, it follows that Vφ is continuous from ( 2d ) ( d ) ( 2d ) ( d ) s R to s R and restricts to a continuous map from s R to s R . By a straight-forward application of Fourier’s inversion formula it follows that

−2 ∗ Pφ F = Vφ f f = φ V F, when L2 φ

( 2d ) ( 2d ) ( ( d )) which shows that the images of s R and s R under Pφ equal Vφ s R d and Vφ( s(R )), respectively. This gives (1) and (2). If B is an invariant BF-space on R2d and F ∈ B, then it follows from the deﬁnitions that

|Pφ F| |F|∗,

1 2d 2d where =|Vφφ| belongs to L(v)(R ) for every choice of v ∈ PE (R ). Hence, a combination of (2) in Deﬁnition 2.6 and (2.17)givesPφ F ∈ B and

Pφ F B F B 1 L(v)

2d for some v ∈ PE (R ), and the continuity of Pφ on B follows. This gives (3). Lemma 3.5 If B is an invariant BF-space of Rd , then

∞ d L(v)(R ) → B

d for some v ∈ PE (R ). d Proof Since 1(R ) is continuously embedded in B we have β |·|/ ( ) · h0 ∞ D f e L f B sup |β| β∈ d β! N h0

−2|·|/h −| · |2/2 for some h0 > 0. Let ω = e 0 , ω0 = ω ∗ e and let v = 1/ω0. Then d d ω ∈ PE (R ), and [1, Proposition 1.6] shows that ω0 ∈ PE (R ) and that

β |β| −2|·|/h0 |D ω0| h β! e 3538 C. Pfeuffer, J. Toft holds for every h > 0. By choosing h < h0 we get β |·|/ |β| |·|/ ( ω ) · h0 ∞ β! ω · h0 ∞ ω D 0 e L h 0 e L 0 B sup |β| sup |β| β∈ d β! β∈ d β! N h0 N h0 −2|·|/h0 |·|/h0 e · e L∞ < ∞.

∞ d Hence, if f ∈ L(v)(R ), then | f | ω0 which implies that f ∈ B in view of Deﬁnition 2.6(2). Furthermore,

ω · v ∞ ∞ f B 0 B f L f L(v) and the result follows.

In what follows we let

B ≡{ ∈ 1 ( d ) ; · ω ∈ B }, (ω) f Lloc R f where B be an invariant BF-space on Rd .

d d Lemma 3.6 Let B be an invariant BF-space on R and ω ∈ PE (R ). Then B(ω) is an invariant BF-space under the norm

→ ≡ · ω . f f B(ω) f B

Proof It is obvious that B(ω) is complete. Let v be as in Lemma 3.5. By Lemma 3.5, ∞ L(ω·v) → B(ω). Since

d ∞ d 1(R ) → L(ω·v)(R ),

d we obtain that 1(R ) is continuously embedded in B(ω). By straight-forward computations it follows that both (1) and (2) in Definition 2.6 are fulfilled with B(ω) in place of B provided v in that definition has been modified in suitable ways.

d Proof of Proposition 3.2 Let φ ∈ 1(R )\0 be ﬁxed. Since ω is a moderate function, it follows by the previous lemma that B(ω) is an invariant BF-space. { }∞ (ω, B) { }∞ Let f j j=1 be a Cauchy sequence in M . Then Vφ f j j=1 is a Cauchy sequence in B(ω). Since B(ω) is a Banach space, there is a unique F ∈ B(ω) such that

− = . lim Vφ f j F B(ω) 0 j→∞

−2 ∗ f = φ V F Vφ f = Pφ F B(ω) Let L2 φ . Then belongs to in view of Lemma 3.4 (3). Since Pφ is continuous on B(ω) and satisﬁes the mapping properties given in Lemma 3.4, we get Compactness Properties for Modulation Spaces 3539

− = ( − ) lim f j f M(ω,B) lim Vφ f j f B(ω) j→∞ j→∞ = ( − ) − = . lim Pφ Vφ f j F B(ω) lim Vφ f j F B(ω) 0 j→∞ j→∞

Hence, f j → f in M(ω, B), and the completeness of M(ω, B) follows. Conse- quently, M(ω, B) is a Banach space. The embedding (3.3) is an immediate consequence of [43, Theorem 2.5] and the fact that M(ω, B) is a Banach space. If we assume that B is an invariant QBF-space (instead of invariant BF-space) with respect of v0, then it seems to be an open question wether (3.3) might be violated or not. Before studying compactness of embeddings between modulation spaces, we ﬁrst consider related continuity questions. 2d 2d Theorem 3.7 Let ω1 and ω2 be weights on R , B be an invariant BF-space on R 2d with respect to v ∈ PE (R ) or a mixed quasi-normed space of Lebesgue type. Then the following is true: ∞ 2d (1) if ω2/ω1 ∈ L (R ), then M(ω1, B) ⊆ M(ω2, B) and the injection

i : M(ω1, B) → M(ω2, B). (3.8)

is continuous; 2d (2) if in addition ω1,ω2 ∈ PE (R ) and v is bounded, then the map (3.8) is a ∞ 2d continuous injection, if and only if ω2/ω1 ∈ L (R ). The next lemma is related to Remark 3.3 and is needed to verify the previous theorem. Lemma 3.8 Let v be submultiplicative and bounded on R2d , B be an invariant BF- 2d ∞ d space and let ω ∈ PE (R ). Then M(ω, B) → M(ω)(R ). Proof B 2 B ∈ ( d ) Let be the L -dual of and let f 1 R . Then it follows by straight- forward computations that both B and B are translation invariant Banach spaces of ( 2d ) φ ∈ ( d ) φ = order 1 which contain 1 R .Let 1 R be such that L2 1 and let d ={g ∈ 1(R ) ; g 1 ≤ 1 }. M(1/ω)

By Feichtinger’s minimality principle (cf. the extension [43, Theorem 2.4] of [24, Theorem 12.1.9]) one gets

Vφ g/ω B g M(1/ω,B ) g 1 < ∞. M(1/ω)

Together with Proposition 2.5 we get

∞ |( , ) d |= |( φ · ω, φ /ω) d | f M(ω) sup f g L2(R ) sup V f V g L2(R2 ) g∈ g∈

≤ sup Vφ f · ω B Vφ g/ω B Vφ f · ω B f M(ω,B). g∈ 3540 C. Pfeuffer, J. Toft

Proof of Theorem 3.7 Claim (1) is an immediate consequence of the boundedness of ω2/ω1 and of B being an invariant BF-space. Assume instead that the embedding i in (3.8) is continuous and the assumptions of the second claim hold. Claim (2) follows if we prove the boundedness of ω2/ω1, which we aim to deduce by contradiction. Therefore suppose that ω2/ω1 is unbounded and M(ω1, B) is continuously embedded in M(ω2, B). Then there is a sequence (xk,ξk) ∈ 2d R with |(xk,ξk)|→∞when k →∞and such that

ω (x ,ξ ) 2 k k ≥ k for all k ∈ N. (3.9) ω1(xk,ξk) Let φ be as in (2.20) and set

1 i ·,ξk fk = e φ(·−xk), Xk = (xk,ξk). (3.10) ω1(Xk)

2d Also let v0 ∈ PE (R ) be submultiplicative and such that ω1 is v0-moderate and that v0 ≥ 1. By

i ·,ξ i x,η−ξ Vφ(e f ( ·−x))(y,η)= e (Vφ f )(y − x,η− ξ), which follows by straight-forward computations, see e. g. [25], we get

i ·,ξ ( ·− ) ≤ ω ( ,ξ) , ∈ (v , B). e f x M(ω1,B) C 1 x f M(v0,B) f M 0

This gives

1 ·,ξ = i k φ(·− ) ≤ φ < ∞, fk M(ω1,B) e xk M(ω1,B) C M(v0,B) ω1(Xk) where C is independent of k ∈ N. Then the hypothesis provides the boundedness of the sequence { fk} in M(ω2, B). Since M(ω , B) → M∞ due to Lemma 3.8 we have 2 (ω2) ω ( )|( )( )|≤ ≤ ∈ N sup 2 X Vφ fk X C fk M(ω2,B) C for all k (3.11) X∈R2d for some C > 0. In particular by letting X = Xk in (3.11) we obtain

We have now the following extension of [3, Theorem 1.2], which is our main result.

2d 2d Theorem 3.9 Let ω1,ω2 ∈ PQ(R ), v ∈ PE (R ) be submultiplicative, B be an invariant BF-space on R2d with respect to v or a mixed quasi-normed space of Lebesgue type. Then the following is true: ω /ω ∈ ∞( 2d ) (1) if 2 1 L0 R , then the injection (3.8) is compact; 2d (2) if in addition ω1,ω2 ∈ PE (R ) and v is bounded, then the injection (3.8) is ω /ω ∈ ∞( 2d ) compact, if and only if 2 1 L0 R . We need the following lemma for the proof.

d − d − 1 ·|x|2 d Lemma 3.10 Let B be an invariant BF space on R2 , φ(x) = π 4 e 2 ,x∈ R , ω ∈ P ( 2d ) { }∞ ⊆ ( d ) (ω, B) Q R and let f j j=1 1 R be a bounded set in M . Then { }∞ { }∞ { }∞ there is a subsequence f jk k=1 of f j j=1 such that Vφ f jk k=1 is locally uniformly convergent.

Proof By the link (2.20) between the Bargmann transform and the Gaussian windowed short-time Fourier transform, the result follows if we prove the assertion with Fj = Vd f j in place of Vφ f j . For any R > 0, let DR be the poly-disc

≡{ ( ,ξ)∈ 2d ; 2 + ξ 2 < 2, = ,..., } DR x R x j j R j 1 d in R2d which we identify with

{ + ξ ∈ d ; 2 + ξ 2 < 2, = ,..., } x i C x j j R j 1 d in Cd . By Cantor’s diagonalization principle the result follows if we prove that for each > { }∞ { }∞ { }∞ R 0, there is a subsequence f jk k=1 of f j j=1 such that Fjk k=1 is uniformly convergent on DR. By [40, Theorem 3.2], we get the boundedness of { f }∞ in M∞ (Rd ) for some j j=1 (ω0) ω ∈ P ( 2d ) { }∞ { }∞ choice of 0 Q R . Hence, Vφ f j j=1 and thereby Fj j=1 are locally uniformly bounded on R2d . In particular,

≡ ∞ ≡ ω ∞ CR sup Fj L (D2R ) and CR,ω0 sup Fj 0 L (D2R ) (3.13) j≥1 j≥1

d 2d are ﬁnite for every weight ω0 on C R . By Cauchy’s and Taylor’s formulae we have α Fj (z) = a j (α)z , z ∈ DR, (3.14) α∈Nd where

−|α| d |a j (α)|≤CR(2R) ,α∈ N . (3.15) 3542 C. Pfeuffer, J. Toft

{β }∞ d ≥ { (β )}∞ In particular, if l l=1 is an enumeration of N , then for each l 1, a j l j=1 is a bounded set in C. Hence, for a subsequence I1 ={k1,1, k1,2,...} of Z+ ={1, 2,...}, the limit

lim ak , (β1) m→∞ 1 m exists. By induction it follows that for some family of subsequences

IN ={kN,1, kN,2,...}⊆Z+, which decreases with N, the limit

lim ak , (βn) m→∞ N m exists for every n ≤ N. { }∞ By Cantor’s diagonal principle, there is a subsequence jk k=1 of Z+ and sequence { (α)} b α∈Nd such that

lim a j (α) = b(α). k→∞ k

By (3.15) we get

−|α| |b(α)|≤CR(2R) .

This in turn gives

α −|α| α −|α| (α) ∞ ≤ (α) ∞ ≤ . sup a j z L (DR ) CR2 and b z L (DR ) CR2 (3.16) j≥1

Hence, (3.14) and the Taylor series F(z) ≡ b(α)zα α∈Nd are uniformly convergent on DR, and by using (3.16), we obtain by straight-forward computations that Fjk tends to F uniformly on DR when k tends to inﬁnity.

Proof of Theorem 3.9 In order to verify (1) we need to show that a bounded sequence { }∞ (ω , B) (ω , B) f j j=1 in M 1 has a convergent subsequence in M 2 . By means of the assumptions there is a sequence of increasing balls Bk, k ∈ Z+, centered at the origin with radius tending to +∞ as k →∞such that

ω ( ,ξ) 2 x 1 2d ≤ , when (x,ξ)∈ R \Bk. (3.17) ω1(x,ξ) k Compactness Properties for Modulation Spaces 3543

− d − 1 ·|x|2 d By Lemma 3.10 it follows that if φ(x) = π 4 e 2 , x ∈ R , then there is a { }∞ { }∞ { }∞ subsequence h j j=1 of f j j=1 such that Vφh j j=1 converges uniformly on any Bk, and converges on the whole R2d . ( ) − → , →∞ χ Claim 1 follows if we prove hm1 hm2 M(ω2,B) 0asm1 m2 .Let k be the characteristic function of Bk, k ≥ 1. From the fact that C = CR in (3.13)is bounded we have

− = − hm hm (ω ,B) Vφhm Vφhm B(ω ) 1 2 M 2 1 2 2 ( − )χ + − / Vφhm Vφhm k B(ω ) Vφhm Vφhm B(ω ) k 1 2 2 1 2 1 ≤ ( − )χ + / . Vφhm Vφhm k B(ω ) 2C k (3.18) 1 2 2

In order to make the right-hand side arbitrarily small, k is first chosen large enough. Note that Vφh1, Vφh2,...is a sequence of bounded continuous functions converging uniformly on the compact set Bk. Since ω2 is a weight and B is an invariant BF-space we obtain that − χ = − ω χ Vφhm Vφhm k B(ω ) Vφhm Vφhm 2 k B 1 2 2 1 2 | ( ,ξ)− ( ,ξ) ω ( ,ξ)| χ sup Vφhm1 x Vφhm2 x 2 x k B (x,ξ)∈Bk | ( ,ξ)− ( ,ξ)| sup Vφhm1 x Vφhm2 x (x,ξ)∈Bk tends to zero as m1 and m2 tend to infinity. This proves (1). In order to verify (2) we suppose that the embedding i in (3.8) is compact and all assumptions of the second claim hold. From the first part of the proof, the result follows if we prove that ω2/ω1 turns to zero at infinity. We prove this claim by contradiction. 2d Suppose there is a sequence (xk,ξk) ∈ R with |(xk,ξk)|→∞if k →∞and a C > 0 fulfilling ω (x ,ξ ) 2 k k ≥ C for all k ∈ N. (3.19) ω1(xk,ξk) ψ ∈ ( d ), φ { }∞ Let 1 R be as in (2.20) and let fk k=1 be as in (3.10). { }∞ (ω , B) By the proof of Theorem 3.7, we get the boundedness of fk k=1 in M 1 , { }∞ (ω , B) and by the assumptions, fk k=1 is precompact in M 2 . d −r|·| Since φ ∈ 1(R ) we have |Vφψ| e for every r > 0 by Proposition 2.2(2). −r |·| From the fact ω1 e 0 for some r0 > 0 we get 1 ψ(x) fk(x) dx = (Vφψ)(xk,ξk) → 0, ω1(xk,ξk)

→∞ ( d ) as k , which implies that fk tends to zero in 1 R . Hence the only possible (ω , B) { }∞ limit point in M 2 of fk k=1 is zero. { }∞ (ω , B) { }∞ As fk k=1 is precompact in M 2 , we can extract a subsequence fk j j=1 which converges to zero in M(ω2, B). 3544 C. Pfeuffer, J. Toft

Since M(ω , B) → M∞ due to Lemma 3.8 we have 2 (ω2) ω ( )|( ( ))( )|≤ → sup 2 X Vφ fk j X C fk j M(ω2,B) 0 (3.20) X∈R2d →∞ = as j . Taking X Xk j in the previous inequality provides ω ( ) ω ( ) 2 Xk j d 2 Xk j = ( π)2 |(Vφφ)( )| ω ( ) 2 ω ( ) 0 1 Xk j 1 Xk j ω ( ) d 2 Xk j i ·,ξ = (2π)2 |(Vφ(e k j φ(·−x ))(X )| ω ( ) k j k j 1 Xk j d = ( π)2 ω ( )|( ( ))( )|→ , 2 2 Xk j Vφ fk j Xk j 0 (3.21) which contradicts (3.19) and proves (2). As an immediate consequence of Lemma 3.1 and Theorem 3.9 we get the following. 2n Corollary 3.11 Assume that ω1,ω2 ∈ PE (R ), and that p, p0, q, q0 ∈ (0, ∞] are p ,q 2d such that p0, q0 < ∞. Also assume that ω2/ω1 ∈ L 0 0 (R ). Then the embedding (3.8) is compact.

4 Applications and Open Questions

Compactness is a fundamental property in analysis, science and engineering, as remarked in the introduction. In this section we make a review on how the compactness results from the previous section are applied in [1] to deduce index and lifting results for pseudo-differential operators and Toeplitz operators. Thereafter we give some links on some open questions and further developments.

4.1 Applications to Index Results for Pseudo-Differential and Toeplitz Operators

d 2d Let A be a real d×d matrix and φ ∈ 1(R )\0 be ﬁxed, and let a ∈ 1(R ). We recall ( ) ( ) that the pseudo-differential operator OpA a and the Toeplitz operator Tpφ a are the ( d ) ( d ) linear and continuous operators from 1 R to 1 R , deﬁned by the formulae ( ( ) )( ) = ( π)−d ( − ( − ), ξ) ( ) i x−y,ξ ξ OpA a f x 2 a x A x y f y e dyd R2d and

( ( ) , ) = ( · , ) , , ∈ ( d ). Tpφ a f g L2(Rd ) a Vφ f Vφ g L2(R2d ) f g 1 R

(Cf. [1,5,7,25,28,32,34–40,43,44].) The deﬁnitions of pseudo-differential and Toeplitz operators extend in different ways. For example, let p1 and p2 be positive polynomials d on R of degrees n1 and n2, and let Compactness Properties for Modulation Spaces 3545

r1 ρ1 ω0(x,ξ)= p1(x) + p2(ξ) (4.1) or r2 ρ2 r ω0(x,ξ)= exp (p1(x) + p2(ξ) ) (4.2) for some r, r j ,ρj > 0for j = 1, 2, which satisfy

r · max(r2n1,ρ2n2)<1.

ω ∈ P0 ( 2d ) , ∈ ( , ∞] If E R and p q 0 , then it is proved in [44] that the deﬁnition of ( ) OpA a above is uniquely extended in such ways that

, , Op (ω ) : M p q (Rd ) → M p q (Rd ) (4.3) A 0 (ω) (ω/ω0) is continuous. In [1, Section 6] it is also proved that

p,q d p,q d Op (ω ) − Tpφ(ω ) : M (R ) → M (R ) (4.4) A 0 0 (ω) (ω/ω1) is continuous for some ω1 such that ω1/ω0 tends to zero at inﬁnity. Finally, in [1, Section 5] it is also proved that

p,q d p,q d Tpφ(ω ) : M (R ) → M (R ) (4.5) 0 (ω) (ω/ω0) is a continuous bijection with continuous inverse. A combination of (4.4), the fact that ω1/ω0 tends to zero at inﬁnity and Theorem 3.9 then shows that

p,q d p,q d Op (ω ) − Tpφ(ω ) : M (R ) → M (R ) (4.6) A 0 0 (ω) (ω/ω0) is compact. In particular, if in addition p, q ≥ 1, then the involved modulation spaces are Banach spaces. Hence, Fredholm’s theorem shows that the indices of the operators in (4.3) and (4.5) satisfy

( (ω )) = ( (ω )) = . Ind OpA 0 Ind Tpφ 0 0

Here the last equality follows from the fact that (4.5) is a continuous bijection. If ω0 is given by (4.1), then it follows by straight-forward computations that (ω ) ( (ω )) = OpA 0 is injective. Since Ind OpA 0 0, it follows that the map (4.3)inthis case is bijective.

4.2 Open Questions and Further Developments

The main objective in the paper is Theorem 3.9 which completely characterizes compactness for the injection map (3.8) when B is either an invariant BF-space or a mixed 3546 C. Pfeuffer, J. Toft

2d quasi-normed space of Lebesgue type, and ω1,ω2 ∈ PE (R ). An open question here concerns wether such characterizations can be deduced when ω1 and ω2 are allowed 2d to belong to a broader weight class than PE (R ). In fact, for general weights, Theorem 3.9 gives some sufﬁcient but no necessary conditions for the map (3.8) to be compact. An other open question concerns wether Theorem 3.9 holds for any QBF-space B with respect to v = 1 and not only when B is either an invariant BF-space or a mixed quasi-normed space of Lebesgue type.

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