Compactness Properties for Modulation Spaces

We prove that if ω1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _1$$\end{document} and ω2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _2$$\end{document} are moderate weights and B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {B}$$\end{document} is a suitable (quasi-)Banach function space, then a necessary and sufficient condition for the embedding i:M(ω1,B)→M(ω2,B)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i\, :\, M (\omega _1,\mathscr {B})\rightarrow M (\omega _2,\mathscr {B})$$\end{document} between two modulation spaces to be compact is that the quotient ω2/ω1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _2/\omega _1$$\end{document} vanishes at infinity. Moreover we show, that the boundedness of ω2/ω1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _2/\omega _1$$\end{document} is a necessary and sufficient condition for the previous embedding to be continuous.


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 configuration space R d , 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. modulation space norm measures in a certain way the phase or time-frequency content. 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 : Q s → Q s−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 is, for t > 0 it is well-known that the embedding i : H 2 s → H 2 s−t is continuous but not compact. Since in [9]. It is well-known that this map is well-defined and continuous when ω 2 ω 1 , see e.g. [25]. In [9,Theorem 5] it was proved that if p, q ∈ [1, ∞), and ω 1 and ω 2 are certain moderate weights of polynomial type, then (1.3) is compact if and only if ω 2 /ω 1 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 : Q s → Q s−t above by Shubin, as well as it confirms the lack of compactness of the embedding i : H 2 s → H 2 s−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 (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 ω j (X + Y ) ω j (X )e r j |Y | , j = 1, 2, for some constants r 1 , r 2 > 0.
Secondly, B can be any general translation invariant Banach function space without the restriction that M(ω, B) should be of the form M p,q (ω) . We may also have M(ω, B) = M p,q (ω) , but in contrast to [3,9], we here allow p and q to be smaller than 1.
Here we notice that if p < 1 or q < 1, then M p,q (ω) fails to be a Banach space because of absence of convex topological structures.
Thirdly, we show that (1.3) is compact when ω 2 /ω 1 tends to zero at infinity, when the assumptions on ω 1 and ω 2 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] for a 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 A (ω 0 ) is continuous and bijective from M r , ρ > 0, and p 1 and p 2 are positive polynomials on R d .
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 : Since it is clear that the latter inclusion map is compact, if and only if ω 2 /ω 1 tends to 0 at infinity, it follows that the compactness results in [3,32] as well as some of 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 continuously embedded in a weighted modulation space of the form 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.

Weight Functions
A weight or weight function ω on R d is a positive function such that ω, for some constant c > 0 which is independent of θ in the domain of f and g. If v 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) We let P E (R d ) be the set of all moderate weights on R d , and P(R d ) be the subset for some r > 0 (for every r > 0), and set P 0 where the last equality follows from the fact that if 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 P E given in [ and (2.5)

Gelfand-Shilov Spaces
Let 0 < h, s ∈ R. Then we denote the set of all functions f ∈ C ∞ (R d ) such that is the dual of s (R d ) as proved in [22]. By the definitions we have We let · , · denote the dual form between a topological vector space and its dual. If z, w ∈ C d , then z, w is defined by The Fourier transform of f ∈ L 1 (R d ) is given by The map F extends uniquely to homeomorphisms on S (R d ), S s (R d ) and on s (R d ). Furthermore, F restricts to homeomorphisms on S (R d ), S s (R d ), s (R d ), and to a unitary operator on L 2 (R d ). Similar results hold true for partial Fourier transforms. For where (U F)(x, y) = F(y, y − x). Here F 2 F denotes the partial Fourier transform of F(x, y) ∈ S s (R 2d ) with respect to the y variable (see (A.1) in [5]).
In the case f ∈ S s (R d ), V φ f can be written as Proposition 2.2 Let s, s 0 ≥ 1 2 be such that s 0 ≤ s. Also let φ ∈ S s 0 (R d )\0 and f ∈ S s 0 (R d ). Then the following is true: holds for some r > 0; (2) if in addition s 0 > 1 2 and φ ∈ s 0 (R d ), then f ∈ s (R d ), if and only if (2.10) holds for every r > 0.

Proposition 2.3
Let s, s 0 ≥ 1 2 be such that s 0 ≤ s. Also let φ ∈ S s 0 (R d )\0 and f ∈ S s 0 (R d ). Then the following is true: if and only if (2.11) holds for some r > 0.

Remark 2.4 For every
The same is true if we replace each S s by S or by s . This is admitted by formula (2.9) (cf. e. g. [35,42]).

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 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 (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.
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 Then the following is true:

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][13][14][15][16].
If the weight v 0 even is an element of P E,s (R d ) (P 0 E,s (R d )), then we call B of Definition 2.6 an invariant QBF-space of Roumieu type (Beurling type) of order r .
Note here that an invariant 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 The following result shows that v 0 in Definition 2.6 can be replaced by a submultiplicative weight v such that (2.15) is true with v in place of v 0 and the constant C = 1, and such that Then The result now follows by letting From now on it is assumed that v and v j are submultiplicative weights if nothing else is stated.
Then L p,q 1 and L p,q 2 are translation invariant BF-spaces with respect to v = 1.
Next we define the extended class of modulation spaces, which are of main interest for us. (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]. We refer to [12,[15][16][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 Then E is called the dual basis of E. The corresponding lattice and dual lattice are We also let κ(E) be the parallelepiped spanned by the basis E. We define for each q = (q 1 , . . . , q d ) ∈ (0, ∞] d , max(q) = max(q 1 , . . . , q d ) and min(q) = min(q 1 , . . . , q d ).
is finite, and 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 (ω) and p E = p E,(ω) when ω = 1. Definition 2.12 Let E be an ordered basis of the phase space R 2d . Then E is called weakly phase split if there is a subset E 0 ⊆ E such that the span of E 0 equals { (x, 0) ∈ R 2d ; x ∈ R d } and the span of E\E 0 equals { (0, ξ) ∈ R 2d ; ξ ∈ R d }.

Pilipović Flat Spaces, Modulation Spaces Outside Time-Frequency Analysis and the Bargmann Transform
Besides the characterization by means of the short-time Fourier transform in Proposition 2.2, Gelfand-Shilov spaces also can be characterized via Hermite function expansions. Recall that the Hermite function of order α ∈ N d is given by , if and only if the coefficients c α ( f ) in its Hermite series expansion In [19,42] various kinds of Fourier-invariant function and distribution spaces are obtained by applying suitable topologies on formal power series expansions. In particular, the Pilipović flat space H (R d ), denoted by H 1 (R d ) in [42], and its dual H (R d ), are defined by all formal expansions (2.19) such that |c α ( f )| r |α| α! − 1 2 for some r > 0, respectively We remark that H (R d ) is larger than any Fourier-invariant Gelfand-Shilov distribution space, and H (R d ) is smaller than any Fourier-invariant Gelfand-Shilov space.
We notice that H (R d ) and H (R d ) possess interesting mapping properties under the Bargmann transform. In fact, the Bargmann kernel is given by Due to [42] we have that V d is bijective between H (R d ) and A(C d ), the set of all entire functions on C d , and restricts to a bijective map from H (R d ) to Later on we need that the Bargmann and the short-time Fourier transform are linked by the formula 20) which can be shown by straight-forward computations. By means of the operator where F is a function or a suitable element of F ∈ H (R d ) we can write the Bargmann transform as

Definition 2.13
Let φ be as in (2.20), ω be a weight on R 2d , B be an invariant QBFspace with respect to v ∈ P E (R 2d ) on R 2d C d of order r ∈ (0, 1], and let U V be given by (2.21). Then We observe the smaller restrictions on ω compared to what is the main stream. For example, in Definition 2.13 it is not assumed that ω should belong to P E (R 2d ) or P(R 2d ). 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 P E . For that reason we always assume that φ is given by (2.20) for such ω.
We have the following. Proposition 2.14 Let φ be as in (2.20), ω be a weight on R 2d and let B be an invariant QBF-space with respect to v ∈ P E (R 2d ). Then the following is true: (

1) the map V d 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 P Q 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.

Compactness Properties for Modulation Spaces
This section is devoted to the questions under which sufficient and necessary conditions the inclusion map 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 for moderate weight functions. In what follows let when ω is a weight on R d . We also set L ∞ 0 = L ∞ 0,(ω) when ω = 1. If is a lattice, then the discrete Lebesgue spaces ∞ 0 ( ) and ∞ 0,(ω) ( ) are defined analogously.

Lemma 3.1 Let E be an ordered basis of R d and let ω ∈ P E (R d ). Then the following is true:
(  (2). Let κ(E) be the (closed) parallelepiped spanned by E and let ϑ ∈ P E (R d ). By using the map ϑ → ϑ · ω, we reduce ourself to the case when ω = 1. The moderateness of ϑ ∈ P E (R 2d ) implies that

1) P E (R d ) is a convex cone which is closed under multiplication, division and under compositions with power functions;
Hence, if χ j is the characteristic function of j + κ(E) and then ϑ ϑ 0 , giving that The assertion now follows from the fact that p E increases with p and that if in addition p ∈ (0, ∞) d , then p E ⊆ ∞ 0 . We also need the following extension of [43, Theorem 2.5].  M(ω, B) is a quasi-Banach space. Moreover M(ω, L p E (R 2d )) is increasing with p. In particular, (3.3) is improved in [41] into

Proposition 3.2 Let
For the proof of Proposition 3.2 we need to consider the twisted convolution, * , defined by 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] or Lemma 3 in [7]. For example the map T is continuous from L 1 (R 2d ) × L 1 (R 2d ) to L 1 (R 2d ). For any s > 0 it restricts to continuous mappings (3.4) and by duality it follows that these mappings extend to continuous mappings In fact, for some map . 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 are algebras under compositions. Here T K 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 , we are also interested of the operator P φ , given by it follows from the continuity properties of the twisted convolution above that P φ in (3.7) is continuous on s (R 2d ) and on s (R 2d ). Similar facts hold true with S s and S s in place of s and s , respectively, at each place. The operator P φ has the following properties.

Lemma 3.4 Let s
. Then the following is true: (2) P φ in (3.7) restricts to a continuous projection from s (R 2d ) to (3) if B is an invariant BF-space on R 2d , then P φ is continuous on B.
Similar facts hold true with S s and S s in place of s and s , respectively, at each place.
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.
By (3.6) it is clear that P φ is the identity map on V φ ( s (R d )) and thereby on By the continuity properties of V φ on s and s , it follows that V * φ is continuous from s (R 2d ) to s (R d ) and restricts to a continuous map from s (R 2d ) to s (R d ). By a straight-forward application of Fourier's inversion formula it follows that and V φ ( s (R d )), respectively. This gives (1) and (2). If B is an invariant BF-space on R 2d and F ∈ B, then it follows from the definitions that Hence, a combination of (2) in Definition 2.6 and (2.17) gives P φ F ∈ B and for some v ∈ P E (R 2d ), and the continuity of P φ on B follows. This gives (3).
In what follows we let where B be an invariant BF-space on R d .

Lemma 3.6 Let B be an invariant BF-space on
Proof It is obvious that B (ω) is complete. Let v be as in Lemma 3.5. By Lemma 3.5, we obtain that 1 (R d ) 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.

Proof of Proposition 3.2 Let
Since ω is a moderate function, it follows by the previous lemma that B (ω) is an invariant BF-space.
If we assume that B is an invariant QBF-space (instead of invariant BF-space) with respect of v 0 , 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.

Theorem 3.7
Let ω 1 and ω 2 be weights on R 2d , B be an invariant BF-space on R 2d with respect to v ∈ P E (R 2d ) or a mixed quasi-normed space of Lebesgue type. Then the following is true: The next lemma is related to Remark 3.3 and is needed to verify the previous theorem. Proof Let B be the L 2 -dual of B and let f ∈ 1 (R d ). Then it follows by straightforward computations that both B and B are translation invariant Banach spaces of order 1 which contain 1 (R 2d ). Let φ ∈ 1 (R d ) be such that φ L 2 = 1 and let Together with Proposition 2.5 we get (1) is an immediate consequence of the boundedness of ω 2 /ω 1 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 ω 2 /ω 1 , which we aim to deduce by contradiction. Therefore suppose that ω 2 /ω 1 is unbounded and M(ω 1 , B) is continuously embedded in M(ω 2 , B). Then there is a sequence (x k , ξ k ) ∈ R 2d with |(x k , ξ k )| → ∞ when k → ∞ and such that

Proof of Theorem 3.7 Claim
≥ k for all k ∈ N. (3.9) Let φ be as in (2.20) and set Also let v 0 ∈ P E (R 2d ) be submultiplicative and such that ω 1 is v 0 -moderate and that v 0 ≥ 1. By which follows by straight-forward computations, see e. g. [25], we get This gives where C is independent of k ∈ N. Then the hypothesis provides the boundedness of for some C > 0. In particular by letting X = X k in (3.11) we obtain which contradicts (3.9) and proves the result.
We have now the following extension of [3,Theorem 1.2], which is our main result.

Theorem 3.9
Let ω 1 , ω 2 ∈ P Q (R 2d ), v ∈ P E (R 2d ) be submultiplicative, B be an invariant BF-space on R 2d with respect to v or a mixed quasi-normed space of Lebesgue type. Then the following is true: We need the following lemma for the proof.

Lemma 3.10 Let B be an invariant BF space on
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 F j = V d f j in place of V φ f j . For any R > 0, let D R be the poly-disc Cantor's diagonalization principle the result follows if we prove that for each By [40,Theorem 3.2], we get the boundedness of and thereby {F j } ∞ j=1 are locally uniformly bounded on R 2d . In particular, (3.13) are finite for every weight ω 0 on C d R 2d . By Cauchy's and Taylor's formulae we have where Hence, (3.14) and the Taylor series are uniformly convergent on D R , and by using (3.16), we obtain by straight-forward computations that F j k tends to F uniformly on D R when k tends to infinity.

Proof of Theorem 3.9
In order to verify (1) we need to show that a bounded sequence B) has a convergent subsequence in M(ω 2 , B). By means of the assumptions there is a sequence of increasing balls B k , k ∈ Z + , centered at the origin with radius tending to +∞ as k → ∞ such that converges uniformly on any B k , and converges on the whole R 2d .
Claim (1) follows if we prove h m 1 − h m 2 M(ω 2 ,B) → 0 as m 1 , m 2 → ∞. Let χ k be the characteristic function of B k , k ≥ 1. From the fact that C = C R in (3.13) is bounded we have In order to make the right-hand side arbitrarily small, k is first chosen large enough.
. is a sequence of bounded continuous functions converging uniformly on the compact set B k . Since ω 2 is a weight and B is an invariant BF-space we obtain that tends to zero as m 1 and m 2 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 ω 2 /ω 1 turns to zero at infinity. We prove this claim by contradiction.
Suppose there is a sequence ( Let ψ ∈ 1 (R d ), φ be as in (2.20) and let { f k } ∞ k=1 be as in (3.10). By the proof of Theorem 3.7, we get the boundedness of { f k } ∞ k=1 in M(ω 1 , B), and by the assumptions, we have |V φ ψ| e −r | · | for every r > 0 by Proposition 2.2(2). From the fact ω 1 e −r 0 | · | for some r 0 > 0 we get as k → ∞, which implies that f k tends to zero in 1 (R d ). Hence the only possible as j → ∞. Taking X = X k j in the previous inequality provides which contradicts (3.19) and proves (2).
As an immediate consequence of Lemma 3.1 and Theorem 3.9 we get the following.

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.
If ω ∈ P 0 E (R 2d ) and p, q ∈ (0, ∞], then it is proved in [44] that the definition of Op A (a) above is uniquely extended in such ways that is continuous. In [1,Section 6] it is also proved that is continuous for some ω 1 such that ω 1 /ω 0 tends to zero at infinity. Finally, in [1,Section 5] it is also proved that is a continuous bijection with continuous inverse. A combination of (4.4), the fact that ω 1 /ω 0 tends to zero at infinity and Theorem 3.9 then shows that 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 Ind(Op A (ω 0 )) = Ind(Tp φ (ω 0 )) = 0.
Here the last equality follows from the fact that (4.5) is a continuous bijection.
If ω 0 is given by (4.1), then it follows by straight-forward computations that Op A (ω 0 ) is injective. Since Ind(Op A (ω 0 )) = 0, it follows that the map (4.3) in this case is bijective.

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 quasi-normed space of Lebesgue type, and ω 1 , ω 2 ∈ P E (R 2d ). An open question here concerns wether such characterizations can be deduced when ω 1 and ω 2 are allowed to belong to a broader weight class than P E (R 2d ).
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.