Compactness properties for modulation spaces

We prove that if $\omega _1$ and $\omega _2$ are moderate weights and $\mascB$ is a suitable (quasi-)Banach function space, then a necessary and sufficient condition for the embedding $i\, :\, M (\omega _1,\mascB )\to M (\omega _2,\mascB )$ between two modulation spaces to be compact is that the quotient $\omega _2/\omega _1$ vanishes at infinity. Moreover we show, that the boundedness of $\omega _2/\omega _1$ a necessary and sufficient condition for the previous embedding to be continuous.


Introduction
In the paper we extend well-known compact embedding properties for classical modulation spaces to a broader family of modulation spaces. These investigations 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 since any similar fact does not hold when the Shubin spaces Q s and Q s−t are replaced by the Sobolev spaces H 2 s and H 2 s−t of Hilbert types. Since Q s = M 2,2 (ω) , ω(X) = (1 + |x| + |ξ|) s (0.1) and H 2 s = M 2,2 (ω) , ω(X) = (1 + |ξ|) s , X = (x, ξ) (0.2) the previous compact embedding properties can also be written by means of modulation spaces. In this context, a more general situation were considered by M. Dörfler, H. Feichtinger and K. Gröchenig who proved in [8,Theorem 5] that if p, q ∈ [1, ∞), and ω 1 and ω 2 are certain moderate weights of polynomial types, then is compact if and only if ω 2 /ω 1 tends to zero at infinity. By choosing ω j in similar ways as in (0.1), the latter compactness result confirms the compactness of embedding i : Q s → Q s−t above by Shubin, as well as confirms the lack of compactness of the embedding i : for Sobolev spaces.
In [5], the compact embedding property [8,Theorem 5] by Dörfler, Feichtinger and Gröchenig were extended in such ways that all moderate weights ω j of polynomial type are included. That is, there are no other restrictions on ω j than there should exists constants N j > 0 such that ω j (X + Y ) ω j (X)(1 + |Y |) N j , j = 1, 2.
In Section 2 we extend these results 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 impose boundedness estimate of exponential types ω j (X + Y ) ω j (X)e r j (|Y |) , j = 1, 2, for some constants c 1 , c 2 > 0. We notice that the latter estimate is less restrective than the condition (0.4), which is assumed in [5,8].
Secondly, B can be any general translation invariant Banach function space without restrictions 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. [5,8], 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 (0.3) is compact when ω 2 /ω 1 tends to zero at infinity, and the conditions on ω 1 and ω 2 are relaxed into a suitable "local moderate condition" (cf. Theorem 2.9 (1)). We refer to [41] and to some extent to [43] for a detailed study of modulation spaces with such relaxed conditions assumptions on the involved weight functions.
Finally we remark that compactness properties for (0.3) can also be obtained by Gabor analysis, which transfers (0.3) into i : ℓ p,q (ω 1 ) → ℓ p,q (ω 2 ) , provided ω 1 and ω 2 are moderate weights. Since it is clear that the latter inclusion map is compact, if and only if ω 2 /ω 1 tends to 0 at infinity. Hence the compactness results in [5,32] as well as some of the results in Section 2 can be deduced in such ways. We emphasise however that such technique can not be used in those situations in Section 2 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 the Gabor analysis seems to be insufficient in such situations.

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 [9-11, 14-16, 22, 36-39].
1.1. Weight functions. A weight or weight function ω on R d is a positive function such that ω, 1/ω ∈ L ∞ loc (R d ).
Let ω and v be weights on R d . Then ω is called v-moderate or moderate, if (1.1) 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. If v can be chosen as polynomial, then ω is called a weight of polynomial type.
The function v is called submultiplicative, if it is even and (1.1) holds We let P E (R d ) be the set of all moderate weights on R d , and P(R d ) be the subset of P E (R d ) which consists of all polynomially moderate functions on R d . We also let P E,s (R d ) (P 0 E,s (R d )) be the set of all weights ω in R d such that for some r > 0 (for every r > 0). We have where the last equality follows from the fact that if ω ∈ P E (R d ) (ω ∈ P 0 E (R d )), then hold true for some r > 0 (for every r > 0) (cf. [23]). In some situations we shall consider a more general class of weights compared to P E . (Cf. [41, Definition 1.1].) holds for some positive constants c and r.
Here the supremum is taken over all α, β ∈ N d and x ∈ R d . One immediately gets, that S t s,h is a Banach space which is contained in S . Moreover S t s,h increases with h, s and t and we have the inclusion S t s,h ֒→ S . We use the notation A ֒→ B for topological spaces A and B satisfying A ⊆ B with continuous embeddings. Furthermore for sufficiently large s, t > 1 2 , or s = t = 1 2 and h S s t,h contains all finite linear combinations of the Hermite functions. On account of the density of such linear combinations in S and in S s , for such choices of s and t.
The inductive and projective limits respectively of S s t,h (R d ) are called Gelfand-Shilov spaces of Beurling respectively Roumieu type and are denoted by where the topology for S s t (R d ) is the strongest possible one such that the inclusion map from The Gelfand-Shilov distribution spaces (S s t ) ′ (R d ) and (Σ s t ) ′ (R d ) are the projective and inductive limit respectively of (S s t,h ) ′ (R d ). This implies that [20]. This is also true in topological sense. In case s = t we set For every admissible s, t > 0 and ε > 0 the next embeddings are true: and (1.8) We recall that Fourier transform of f ∈ L 1 (R d ) is defined by where · , · is the usual scalar product on R d . The map F extends uniquely to homeomorphisms on and to a unitary operator on L 2 (R d ). If we replace the Fourier transform by a partial Fourier transform similar results hold true for s = t.
Gelfand-Shilov spaces and their distribution spaces can be characterized in a convenient way by means of estimates of the short-time Fourier transforms, see e. g. [25,41,43]. Before stating this result, we recall the definition of the short-time Fourier transform. For where (UF )(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. In case f, φ ∈ S s (R d ) the short-time Fourier transform of f can be written as The characterisation of Gelfand-Shilov functions and their distributions are formulated in the next two propositions. The proof of the characterizations can be found in e. g. [25,43] (cf. [25,Theorem 2.7]) and in [41,43]: Then the following is true: ( holds for some r > 0; , if and only if (1.9) holds for every r > 0. Proposition 1.3. Let s, t, s 0 , t 0 > 0 be such that s 0 + t 0 ≥ 1, s 0 ≤ s and t 0 ≤ t. Also let φ ∈ S s t (R d ) \ 0 and f ∈ (S s 0 t 0 ) ′ (R d ). Then the following is true: ( holds for every r > 0; , if and only if (1.10) holds for some r > 0. Remark 1.4. For the short-time Fourier transform the following continuity results hold: For every s > 0, the mapping The same is true if we replace each S s by S or by Σ s (cf. e. g. [35,43]).

Modulation spaces.
In the whole subsection we fix some φ ∈ We summarize some well-known facts of Modulation spaces in the next Proposition. See [11,19,22,42] for the proof. The conjugate exponent of p is given by Then the following is true: (ω) is a quasi-Banach space under the quasi-norm in (1.11) and even a Banach space if p, q ≥ 1. Different choices of φ give rise to equivalent (quasi-)norms; (2) if p 1 ≤ p 2 , q 1 ≤ q 2 and ω 2 ≤ Cω 1 for some constant C, then Because of Proposition 1.5 (1) we are allowed to be rather imprecise concerning the choice of φ ∈ M r (v) \0 in (1.11). For instance let C > 0 be a constant and Ω be a subset of Σ ′ 1 . If we then write, that a M p,q (ω) ≤ C for every a ∈ Ω, we mean that the inequality holds for some choice of φ ∈ M r (v) \ 0 and every a ∈ Ω. Additionally a similar inequality is true for any other choice of φ ∈ M r (v) \ 0, although we may have to replace C by another constant.
1.4. 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 transforms, cf. [11][12][13][14][15].
We recall that a quasi-norm · B of order r ∈ (0, 1] on the vectorspace B is a nonnegative functional on B which satisfies (1.12) 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 (1.12) then on account of [2,29] there is an equivalent quasinorm to · B which additionally satisfies (1.13) From now on we always assume that the quasi-norm of the quasi-Banach space B is chosen in such way that both (1.12) and (1.13) hold.
Definition 1.6. Let B ⊆ L r loc (R d ) be a quasi-Banach space of order r ∈ (0, 1] which contains Σ 1 (R d ) with continuous embedding, and let v 0 ∈ P E (R d ). Then B is called a translation invariant Quasi-Banach Function space on R d (with respect to v), or invariant QBF space on R d , if there is a constant C such that the following conditions are fulfilled: ( , then we call B of Definition 1.6 an invariant QBF-space of Roumieu type (Beurling type) of order s.
By means of (2) in Definition 1.6 we know that f · h ∈ B if f ∈ B and h ∈ L ∞ and additionally (1.15) For r = 1, the invariant QBF space B of Definition 1.6 becomes a Banach space and is called an invariant BF-space (with respect to v).
Because of condition (2) a translation invariant BF-space is a solid BF-space in the sense of (A.3) in [12]. For each invariant BF-space The density of Σ 1 in L 1 (v) provides that the definition of f * ϕ extends uniquely to any f ∈ B and ϕ ∈ L 1 (v) (R d ). Hence (1.16) is also true for such f and ϕ.
The following result shows that v 0 in Definition 1.6 can be replaced by a submultiplicative weight v such that (1.14) is true with v in place of v 0 and the constant C = 1, and such that and such that (1.14) holds with v in place of v 0 , and C = 1. Then The result now follows by letting From now on it is assumed that v and v j are submutliplicative 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.
For translation invariant BF-spaces we make the following observation. Proposition 1.9. Assume that v ∈ P E (R d ), and that B is an invariant BF-space with respect to v such that (1.16) holds true. Then the convolution mapping (v) . Next we define the extended class of modulation spaces, which are of interest for us: , see e.g. (1.8). 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. It can be proved by similar arguments as Proposition 11.3.2 in [22]. Hence we omit the proof. In applications, the quasi-Banach space B is mostly a mixed quasinormed Lebesgue space, which is defined next. Let E = {e 1 , . . . , e d } be an orderd basis of R d and let E ′ = {e ′ 1 , . . . , e ′ d } be such that e j , e ′ k = 2πδ jk , j, k = 1, . . . , d. Then E ′ is called the dual basis of E. The corresponding lattice and dual lattice are There is a matrix T E such that e 1 , . . . , e d and e ′ 1 , . . . , e ′ d are the images of the standard basis under T E and T E ′ = 2π(T −1 E ) t , respectively. We also let κ(E) be the parallelepiped spanned by the basis E.
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 1.12. Then the dis- is finite. Here χ j is the characteristic function of j + κ(E). We also set L p E = L p E,(ω) and ℓ p E = ℓ p E,(ω) when ω = 1.
Definition 1.13. Let E be an ordered basis of the phase space R 2d . Then E is called 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 }.
We also can write h α via for some polynomial p α on R d . p α are called the Hermite polynomial of order α. It is well-known that {h α } α∈N d provides an orthonormal basis for L 2 (R d ).
We now can characterize the Gelfand-Shilov spaces , if and only if the coefficients c α (a) in its Hermite series expansion for some r > 0 (for every r > 0). Various kinds of Fourier-invariant functions and distribution spaces can be obtained by applying suitable topologies on formal power series expansions, cf. e. g. [17,43]. To mention one of them, which is of peculiar interest: for some r > 0, respectively If φ, f ∈ L 2 (R d ), the pairing (f, φ) L 2 (R d ) agree's with the L 2 (R d ) scalar product of those two functions. We remark that H ′ ♭ (R d ) is larger than any Fourier-invariant Gelfand-Shilov distribution space, and H ♭ (R d ) is smaller than any Fourierinvariant Gelfand-Shilov space. We already know, that any f ∈ S s (R d ) (f ∈ Σ s (R d )) with s ≥ 1 2 (s > 1 2 ) can be expressed in a unique way by an expansion (1.18) for every r > 0 (for some r > 0). One reason, why the Pilipović flat space H ♭ (R d ) and its dual H ′ ♭ (R d ) are of particular interest, are is their images under the Bargmann transform. The kernel of the Bargmann transform is given by which is analytic in z. Seen as a function of y, A d belongs to H ♭ (R d ).
We define the Bargmann transform where f, φ = (f, φ) L 2 (R d ) . Due to [43] we know 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 ) and Later on we will need, that the Bargmann and the short-time Fourier transform are linked by the formula This can be shown by straight-forward computations. By means of the operator when F is a function or a suitable element of F ∈ H ′ ♭ (R d ) we can write the Bargmann transform as Definition 1.14. Let φ be as in (1.19), ω be a weight on R 2d , B be an invariant QBF-space with respect to v ∈ P E (R 2d ) on R 2d ≃ C d of order r ∈ (0, 1]. (1) B(ω, B) consists of all F ∈ L r loc (R 2d ) = L r loc (C d ) such that We observe the small restrictions on ω compared to what is the main stream, e. g. that ω should belong to P E (R 2d ) or be moderated by functions which are bounded by polynomials. We still call the space M(ω, B) as the modulation space with respect to ω and B. 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 the weight function is given by (1.19) for such ω.
We have the following.
Proposition 1.15. Let φ be as in (1.19), 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-norm space of Lebesgue types, then M(ω, B) and A(ω, B) are quasi-Banach spaces, which are Banach spaces in the case B is a Banach space.
Proof. From (1.19), (1.20) and Definition 1.14 it follows that V d is an isometric injection from M(ω, B) to A(ω, B). Since any element in A(C d ), and thereby any element in A(ω, B) is a Bargmann transform of an element in H ′ ♭ (R d ), it follows that the image of M(ω, B) under V d 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 [43]. The details are left for the reader.

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 Λ is a lattice, then the discrete Lebesgue spaces ℓ ∞ 0 (Λ) and ℓ ∞ 0,(ω) (Λ) are defined analogously. Lemma 2.1. Let E be an ordered basis of R d and let ω ∈ P E (R d ). Then the following is true: (1) P E (R d ) is a convex cone which is closed under multiplication, division and under compositions with power functions; Similar properties has already been shown in [5, Lemma 2.1] for the smaller weight space P.
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 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 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 ℓ p E increases with p and that if in addition p ∈ (0, ∞) d , then ℓ p E ⊆ ℓ ∞ 0 . We also have the following result, which is an immediate consequence of [45,Theorem 2.5].
be v-moderate, and let B be an invariant BF-space with respect to v 0 . Then M(ω, B) is a Banach space, and . We refer to [42] for the proof.
For the proof the twisted convolution * of two functions F, G ∈ L 1 (R 2d ) defined by is needed. The twisted convolution is continuous as a map between several function spaces, see e. g. [21] or Lemma 3 in [7]. For instance the map (F, G) → F * G is continuous from L 1 (R 2d ) × L 1 (R 2d ) to L 1 (R 2d ), and can be restricted to a continuous map from Σ 1 (R 2d ) × Σ 1 (R 2d ) to Σ 1 (R 2d ). The latter map can be continuously extended to a continuous map from

On account of the Fourier's inversion formula we obtain for all
on Σ ′ 1 (R 2d ). The operator P φ has the following properties: Lemma 2.4. Let φ ∈ Σ 1 (R d ). Then the following is true: (2) P φ in (2.5) restricts to a continuous projection from Σ 1 (R 2d ) to Related results can essentially be found in e. g. [18,21]. In order to be self-contained, we here give a short proof.

Proof. By (2.4) it is clear that
By a straight-forward application of Fourier's inversion formula it follows that , 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 where Φ = |V φ φ| belongs to L 1 (v) (R 2d ) for every choice of v ∈ P E (R 2d ). Hence, a combination of (2) in Definition 1.6 and (1.16) gives P φ F ∈ B, and , and the continuity of P φ on B follows. This gives (3).
Lemma 2.6. Let B be an invariant BF-space on R d and ω ∈ P E (R d ). Then Proof. Let v be as in . By Lemma 2.5, L ∞ (ω·v) ֒→ B (ω) . Since it follows that Σ 1 (R d ) is continuously embedded in B (ω) . By straight-forward computations it follows that both (1) and (2) in Definition 1.6 are fulfilled with B (ω) in place of B provided v has been modified in suitable ways.
Since ω is a moderate function, it follows by the previous lemma Lemma 2.4 (3). Since P φ is continuous on B (ω) and satisfies the mapping properties given in Lemma 2.4, we get Before studying compactness of embeddings between modulation spaces, we first consider the related continuity questions.
Theorem 2.7. 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 or a mixed quasi-normed space of Lebesgue type, and let i be the injection (
The next lemma is related to Remark 2.3 and is needed verify the previous theorem.
Lemma 2.8. Let v be submultiplicative and bounded on R 2d , B be an invariant BF-space with respect v which is continuously embedded in Σ ′ 1 (R 2d ), and let ω ∈ P E (R 2d ). Then M(ω, B) ֒→ M ∞ (ω) (R d ). Proof. Let B ′ be the L 2 -dual of B. Then it follows by straight-forward computation that both B and B ′ are translation invariant Banach spaces of order 1 which contain Σ 1 (R 2d ). Let φ ∈ Σ 1 (R d ) be such that φ L 2 = 1, and let Here we have used the fact that  (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 (2.6) is continuous and all assumptions of the second claim hold. Claim (2) follows if we have proved the boundedness of ω 2 /ω 1 . We prove this boundedness by contradiction. We consider, that there is a sequence ( Let φ be as in (1.19) and set In order to show that the sequence f k is bounded in M(ω 1 , B), we choose a submultiplicative weight v 0 ∈ P E (R 2d ) such that ω 1 is v 0moderate and that v 0 ≥ 1. By (see e. g. [22]), we get This gives where C is independent of k ∈ N. Then the hypothesis provides the boundedness of the sequence {f k } in M(ω 2 , B).
(2.7) for some C > 0. In particular inequality (2.7) yields if we take z = z k which proves the result.
We have now the following extension of [5, Theorem 1.2], which is our main result. Theorem 2.9. 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, and let i be the injection (2.9) Then the following is true: , then the map (2.9) is compact; (2) if in addition ω 1 , ω 2 ∈ P E (R 2d ) and v is bounded, then the map (2.9) is compact, if and only if ω 2 /ω 1 ∈ L ∞ 0 (R 2d ). We need the following lemma for the proof.
is locally uniformly convergent. Proof. By the link between the Bargmann transform and Gaussian windowed short-time Fourier transforms, 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 (1.19) and an application of Cantor's diagonalization principle the result follows if we prove that for each R > 0, there is a subsequence j=1 are locally uniformly bounded on R 2d . In particular, are finite for every weight ω 0 on C d ≃ R 2d . By Cauchy's and Taylor's formulae we have where In particular, if {β l } ∞ l=1 be an enumeration of N d , then for each l ≥ 1, {a j (β l )} ∞ j=1 is a bounded set in C. Hence, for a subsequence I 1 = {k 1,1 , k 1,2 , . . . } of Z + = {1, 2, . . . }, the limit lim m→∞ a k 1,m (β 1 ) exists. By induction it follows that for some family of subsequences which decreases with N, the limit lim m→∞ a k N,m (β n ) exists for every n ≤ N.
This in turn gives sup j≥1 a j (α)z α L ∞ (D R ) ≤ C R 2 −|α| and b(α)z α L ∞ (D R ) ≤ C R 2 −|α| (2.13) Hence, (2.11) and the Taylor series are uniformly convergent on D R , and by using (2.13), it follows by straight-forward computations that F j k tends to F uniformly on D R when k tends to infinity.
Proof of Theorem 2.9. In order to verify (1) we need to show, that a bounded sequence {f j } in M(ω 1 , 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 ω 2 (x, ξ) ω 1 (x, ξ) ≤ 1 k , when (x, ξ) ∈ R 2d \ B k . (2.14) By Lemma 2.10 it follows that if φ(x) = π − d 4 e − 1 2 ·|x| 2 , x ∈ R d , then there is a subsequence {h j } ∞ j=1 of {f j } ∞ j=1 such that {V φ h j } ∞ j=1 converges uniformly on any B k , and converges on the whole R 2d .
We have to prove that 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 R in (2.10) is bounded we have (2.15) where C = C R is the constant in (2.10).
In order to make the right-hand side arbitrarily small, k is first chosen large enough. Then V φ h 1 , V φ h 2 , . . . 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 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 (2.9) 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.
By the proof of Theorem 2.7, it follows that the sequence {f k } ∞ k=1 is bounded in M(ω 1 , B), and by the assumptions {f k } ∞ k=1 is precompact in M(ω 2 , B).
As an immediate consequence of Lemma 2.1 and Theorem 2.9 we get: