Calderón-type inequalities for affine frames

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Applied and Computational Harmonic Analysis www.elsevier.com/locate/acha

Calderón-type inequalities for affine frames Davide Barbieri a , Eugenio Hernández a,∗ , Azita Mayeli b a b

Universidad Autónoma de Madrid, 28049 Madrid, Spain City University of New York, Queensborough and the Graduate Center, New York, USA

a r t i c l e

i n f o

Article history: Received 27 November 2017 Received in revised form 26 December 2018 Accepted 29 July 2019 Available online xxxx Communicated by Stephane Jaffard Keywords: Frames in LCA groups Calderón condition for frames Gabor systems

a b s t r a c t We prove sharp upper and lower bounds for generalized Calderón’s sums associated to frames on LCA groups generated by affine actions of cocompact subgroup translations and general measurable families of automorphisms. The proof makes use of techniques of analysis on metric spaces, and relies on a counting estimate of lattice points inside metric balls. We will deduce as special cases Calderón-type inequalities for families of expanding automorphisms as well as for LCA-Gabor systems. © 2019 Elsevier Inc. All rights reserved.

1. Introduction If ψ ∈ L2 (R) and a ∈ R+ \ {1}, several well-known conditions for discrete wavelets systems are given in terms of the Calderón’s sum Cψ (ξ) =



 j ξ)|2 . |ψ(a

j∈Z j

A necessary condition for an affine system Ψ = {a 2 ψ(aj · −k)}j,k∈Z to be an orthonormal basis of L2 (R) (see [8], [26]) is Cψ (ξ) = 1 a.e. ξ ∈ R which is also sufficient to prove that the system is complete once it is known to be orthonormal (see [23], [1]). On the other hand, a necessary condition for Ψ to be a frame with constants A and B is given by (see [5]) * Corresponding author. E-mail addresses: [email protected] (D. Barbieri), [email protected] (E. Hernández), [email protected] (A. Mayeli). https://doi.org/10.1016/j.acha.2019.07.004 1063-5203/© 2019 Elsevier Inc. All rights reserved.

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A ≤ Cψ (ξ) ≤ B

a.e. ξ ∈ R. j

(1)

A consequence of (1) is that the so-called coaffine systems {a 2 ψ(aj (· − k))}j,k∈Z can not form a frame of L2 (R) (see [11]). Let us also recall that the Calderón’s sum is actually related to the well-known Calderón’s admissibility condition characterizing continuous wavelets (see e.g. [28]). The results on orthonormal wavelets on R have been extended to several degrees of generality, including characterizations of tight frames for generalized shift-invariant systems on Rn [14], and on LCA groups [20], and more recently these results were obtained under weaker hypotheses in a setting that unifies discrete and continuous systems [18]. The setting of general frames is much less studied, but we note that very recently, a closely related result to the one discussed in the present paper has been obtained in [7, Theorem 6.2] with completely different techniques. In this paper we consider frames in subspaces of L2 (G), where G is an LCA group, generated by translations by a cocompact subgroup Γ and by a family of automorphisms, endowed with a Borel measure, that formally replaces dilations. The generality of the setting also allows us to consider modulations, hence including Gabor-type systems, after a proper choice of automorphisms and subspaces of L2(G), see Section 5.  has an invariant metric that satisfies the Our main requirements on these objects are that the dual group G Lebesgue’s differentiation theorem, and that the family of automorphisms be bi-Lipschitz, with measurable upper and lower constants, with respect to such an invariant distance. The main result of this paper is Theorem 17, which proves that affine frames in this setting satisfy inequalities that generalize (1). The proof relies on a variant of a classical strategy which consists of testing the frame condition on a family of functions whose Fourier transforms define an approximation of the identity, and then obtain the inequalities in the limit by Lebesgue’s differentiation theorem. This strategy can be performed whenever two crucial hypotheses are met. The first hypothesis is a counting estimate, that requires the number of points of the discrete annihilator  balls to grow at most as much as the jacobian of the dilation. This Γ⊥ inside the dilation of small G condition, that we call Property X, see Definition 13, is similar to the lattice counting estimate studied recently in [2] for automorphisms given by powers of a matrix. The lattice counting estimate appeared in previous works, notably [14], [9], [20], and in [2] the authors prove that the estimate is equivalent to the automorphisms to be expanding on a subspace (and not contracting on the complementary). In the present paper, in Definition 18, we introduce a general notion of expansiveness, which generalizes that of expanding on a subspace to all bi-Lipschitz families of automorphisms, see Proposition 32. We then prove in Theorem 20 that this notion of expansiveness is actually sufficient to obtain Property X. The proof relies on a technical counting estimate, given by Lemma 2, which holds for any discrete subgroup of an LCA group and that is actually optimal, as shown in Lemma 31. However, we can prove Property X also in nonexpanding settings. In Section 4.3 we indeed obtain Property X for shearlets, which are systems of wavelets with composite dilations that are not expanding, and in Lemma 29 we can obtain it for Gabor systems, after an appropriate adjustment of the involved Hilbert space. The second hypothesis which is required for the proof of Theorem 17 is a local integrability condition for the mother wavelet. This requirement appears naturally when applying Lebesgue’s differentiation, since it provides the local integrability of one of the terms of the estimate, actually a remainder which does vanish. Essentially the same hypothesis but in a slightly different setting, that of generalized shift-invariant systems on R, was thoroughly discussed in the recent paper [4], and it was related to the so-called LIC and α-LIC conditions introduced respectively in [14] and in [18]. Here, in Theorem 22, we provide sufficient conditions on the family of automorphisms, in terms of a slightly stronger notion of expansiveness, that makes this local integrability condition to hold automatically for all mother wavelets. This condition is not always necessary, since for example in the nonexpanding case of Gabor systems we get local integrability for free. However, it may be interesting to observe that essentially the same idea of the proof was used in [4] to obtain this local

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integrability for all wavelets on R (which satisfy a special case of the mentioned stronger expansiveness), and in our opinion the core of this argument can be found also in [5]. Once Theorem 17 is established, we can then obtain Calderón’s inequalities for expanding automorphisms in Corollary 21, as well as for Gabor systems in Theorem 30, by showing that Property X holds. We now compare our results with similar ones that have appeared in the literature on the subject. In [18] the authors work in a different setting and prove the upper bound of the Calderón sum for Bessel sequences, or treat the case of dual frames. We treat the case of general frames to find the lower bound, and allow for continuous families of automorphism. In [17] the authors consider Gabor systems, and in this setting their assumptions are more general than ours, while our results apply to other systems as well. They prove the upper and lower bound for the Calderón sum for Gabor systems. Our Theorem 30 proves the same result in a setting that is less general with respect to Gabor systems, but using completely different techniques that apply to more general systems. In [2] the authors work in Rn , where they obtain the Calderón condition for Parseval frames and dual frames. They study several issues concerning a counting estimate similar to Property X, and they prove that it is equivalent to the expanding on a subspace condition. We treat general second countable LCA groups and general families of automorphisms, with a definition of expanding that implies Property X, but is not equivalent although it includes expanding on a subspace matrices. In [14] the authors work in Rn , they prove the upper bound for the Calderón sum for Bessel sequences, and they obtain the Calderón condition for Parseval frames or dual frames. Acknowledgments This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 777822. D. Barbieri and E. Hernández were supported by Grants MTM2016-76566-P (MINECO, Spain). A. Mayeli was supported by PSC-CUNY grant 60623-00 48. The authors would like to thank Jakob Lemvig for interesting and helpful comments on a previous version of this paper. 2. Preliminaries Let G be a locally compact and second countable abelian group, and let us denote with + its composition law. Let  f(ξ) = f (x)ξ, xdμG (x) G

 be the G-Fourier transform, where we denote by ξ, x ∈ T the pairing with a character ξ ∈ G. Let Γ be a cocompact closed subgroup of G. Let us fix a Haar measure μG on G and, since G/Γ is compact, let us fix the normalized measure κ on G/Γ, i.e. such that κ(G/Γ) = 1. Let us then set a Haar measure μΓ on Γ in such a way that the corresponding Weil’s formula holds with constant 1:      f (x)dμG (x) = f (y + γ)dκ(y) dμΓ (γ). G

Γ

G/Γ

 : λ, γ = 1 ∀ γ ∈ Γ} denote the annihilator of Γ, which is a closed subgroup of G.  Let Γ⊥ = {λ ∈ G ⊥  is discrete, so it is countable because G  is second Observe that, since G/Γ is compact, then Γ ≈ (G/Γ) countable.

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We will make use of the following standard result relating the multiplicative constants of Haar measures with Plancherel theorems on subgroups. It was originally given by Weil [27], a proof can be found also in [16, (31.46), (c)], and the argument to obtain it is reported in Appendix B.  that makes the Lemma 1. Let (G, μG ) and (Γ, μΓ ) be as above, and let νG be the Haar measure on G 2 2  G-Fourier transform a unitary operator from L (G) to L (G), i.e. 

 |f (x)|2 dμG (x) =

|f(ξ)|2 dνG (ξ)

∀ f ∈ L2 (G).

 G

G

 be any ν  -measurable section of G/Γ  ⊥ , i.e. a fundamental set. Let Ω ⊂ G G Then νG (Ω) > 0 and the following identities hold  i.

φ(ξ)dνG (ξ) =  G



 

Ω

 ∀ φ ∈ L1 (G).

λ∈Γ⊥ Ω



|φ(ξ)|2 dνG (ξ) =

ii.

φ(ξ + λ)dνG (ξ)



| Γ

φ(ξ)γ, ξdνG (ξ)|2 dμΓ (γ)

∀ φ ∈ L2 (Ω).

Ω

2.1. Counting estimates on metric abelian groups  is locally compact and second countable, then, it is metrizable, its metric can be chosen to be Since G invariant under the group action, and it is complete1 (see [15, Theorem (8.3)], [24]). Let us denote with ×G  → R+ one of these metrics and with dG : G  : d  (ξ, ξ0 ) < r} B(ξ0 , r) = {ξ ∈ G G  of radius r > 0 and center ξ0 . Note that, denoting by e the identity element of G,  by a dG -metric ball in G the invariance of the metric we have B(ξ0 , r) = B(e, r) + ξ0 . In the following sections we will need a counting estimate for the number of the Γ⊥ lattice points that lie  We will deduce it in some relevant settings inside metric balls deformed by an automorphism α  ∈ Aut(G). from a basic counting lemma that we now present. In order to obtain it, let us introduce the following notation: given r > 0, denote by (Γ⊥ )rα =

 

 α B(e, r) + λ .

λ∈Γ⊥

 let us define Also, for Ω a fundamental domain for Γ⊥ in G, Ωrα = Ω ∩ (Γ⊥ )rα . Note that Ωrα has a finite measure since, item i., Lemma 1, we get 1

Actually it is proper: closed balls are compact.

(2)

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 νG (Ωrα )

dνG (ξ) ≤

= Ω∩(Γ⊥ )rα 

=

 

λ∈Γ⊥Ω+λ

5





dνG (ξ)

  λ∈Γ⊥ Ω∩ αB(e,r)+λ 



χαB(e,r) (ξ)dνG (ξ) = 

χαB(e,r) (ξ)dνG (ξ) = νG ( αB(e, r)).   G

We can now state our counting lemma in general terms as follows.  be a second countable LCA group with Haar measure ν  , let Γ⊥ be a discrete subgroup of Lemma 2. Let G G  Then   ∈ Aut(G). G, and let α 

Γ⊥ ∩ α B(e, r) ≤

1 ν  ( αB(e, 2r)) νG (Ωrα ) G

∀r>0

 a ν  -measurable section of G/Γ  ⊥. where Ωrα is as in (2) for any Ω ⊂ G G  and r > 0, let us denote by Proof. For all ξ ∈ G   Sαr (ξ) = Γ⊥ ∩ ( αB(e, r) − ξ) = λ∈Γ⊥

χαB(e,r)−ξ (λ) = 

 λ∈Γ⊥

χαB(e,r)+λ (ξ) 

and observe that supp Sαr ⊂ (Γ⊥ )rα . By i., Lemma 1 we then have 

χα(B(e,r)) (ξ)dνG (ξ) = 

νG ( α(B(e, r))) =  G





λ∈Γ⊥ Ω

r

Sαr (ξ)dνG (ξ).

(3)

r

Ωrα 

Ω

χα(B(e,r)) (ξ + λ)dνG (ξ) 



Sαr (ξ)dνG (ξ) ≥

Sαr (ξ)dνG (ξ) =

=

 

Ωα2

r

Now, by the definition of (Γ⊥ )α2 , for ξ ∈ Ωα2 there exists at least one λξ ∈ Γ⊥ such that r r ξ ∈ α B(e, ) + λξ = α B( α −1 (λξ ), ). 2 2

(4)

r

We claim that, when ξ ∈ Ωα2 , and for λξ as above, it holds r α B(e, ) − λξ ⊂ α B(e, r) − ξ. 2 r

Claim (5) implies that, when ξ ∈ Ωα2 , we can estimate Sαrˆ (ξ) from below as follows:   r Sαr (ξ) = Γ⊥ ∩ ( αB(e, r) − ξ) ≥ Γ⊥ ∩ ( αB(e, ) − λξ ) 2  r r ⊥ 2 = Γ ∩α B(e, ) = Sα (e). 2 By inserting this into (3) we then get the desired estimate r

r

νG ( αB(e, r)) ≥ νG (Ωα2 )Sα2 (e). In order to prove (5), let ζ = α (η) ∈ α B(e, 2r ) − λξ . Then

(5)

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dG (η, − α −1 (ξ)) ≤ dG (η, − α −1 (λξ )) + dG (− α −1 (λξ ), − α −1 (ξ)) r r r α −1 (λξ ), α  −1 (ξ)) < + < + dG ( 2 2 2 where the last inequality is due to (4). This implies that η ∈ B(− α −1 (ξ), r), so ζ ∈ α B(e, r) − ξ.  2.2. Measurable families of bi-Lipschitz automorphisms Let G be a locally compact and second countable abelian group. Let (H, ς) be a measure space, and let α : H → Aut(G) be a measurable family of G-automorphisms, i.e. assume that (x, h) → αh (x) is measurable in (G, μG ) × (H, ς). Let δ : H → R+ be the ς-measurable map satisfying (see [15, (15.26)]) 

 f (αh (x))dμG (x) = δ(h) G

f (x)dμG (x)

∀ f ∈ L1 (G).

G

 by2 Let us define the dual action α  : H → Aut(G)  x ∈ G. ξ ∈ G,

 αh (ξ), x = ξ, αh−1 (x),

 read as follows. It is well-known that α -changes of variables on G  it holds Lemma 3. For all φ ∈ L1 (G) 

−1



φ( αh (ξ))dνG (ξ) = δ(h)  G

φ(ξ)dνG (ξ).  G

Sketch of the proof. Since μG and νG are such that the G-Fourier transform is unitary, for all f ∈ L2 (G) we have    2 2 −1  |f (ξ)| dνG (ξ) = |f (x)| dμG (x) = δ(h) |f (αh (x))|2 dμG (x)  G

= δ(h)−1  = δ(h)  G



 |

G

G

f (αh (x))ξ, xdμG (x)|2 dνG (ξ)

 G

G

|

f (x)ξ, αh−1 (x)dμG (x)|2 dνG (ξ) = δ(h)





|f( αh (ξ))|2 dνG (ξ)

 G

G

 with values in R+ .  which proves the desired identity for all φ ∈ L1 (G) 2

When H is a group and α is a homomorphism, this too is a homomorphism, since −1

−1

α  h1 ( α  h2 (ξ)), x = α  h2 (ξ), αh1 (x) = ξ, αh1 h2 (x) = α  h1 h2 (ξ), x. .

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Remark 4. Note that, by Lemma 3, we have  ∀ r > 0, ∀ h ∈ H. νG ( αh B(ξ0 , r)) = δ(h)νG (B(ξ0 , r)) ∀ ξ0 ∈ G,  due to the action of α In order to control the deformation on G  we will use the notion of bi-Lipschitz maps. If (E, d) is a metric space, a bijection f : E → E is called bi-Lipschitz if there exist 0 <  ≤ L < ∞ such that d(x, y) ≤ d(f (x), f (y)) ≤ Ld(x, y) ∀ x, y ∈ E.

(6)

For bi-Lipschitz maps, one has the following known result. Lemma 5. Let f : E → E be a bi-Lipschitz map on the metric space (E, d) and for x ∈ E, r > 0 let us denote by B(x, r) = {y ∈ E : d(y, x) < r}. Then B(f (x), r) ⊂ f (B(x, r)) ⊂ B(f (x), Lr)

∀ x ∈ E , ∀ r > 0.

Proof. To prove the inclusion from above, let y ∈ f (B(x, r)), i.e. y = f (z) for some z ∈ E such that d(z, x) < r. Then, by the right hand side inequality in (6), we have d(y, f (x)) = d(f (z), f (x)) ≤ Ld(z, r) < Lr so y ∈ B(f (x), Lr). To prove the inclusion from below, let y ∈ B(f (x), r), and let z = f −1 (y). Then, by the left hand side inequality in (6), we have d(z, x) ≤ d(f (z), f (x)) = d(y, f (x)) < r so z ∈ B(x, r), i.e. y = f (z) ∈ f (B(x, r)).  The class of families of automorphisms that we will consider is that satisfying the following definition. Definition 6. We say that the map α  is measurably bi-Lipschitz if there exist two ς-measurable functions + +  : H → R and L : H → R satisfying (h)dG (ξ, e) ≤ dG ( αh (ξ), e) ≤ L(h)dG (ξ, e)

 ∀ ξ ∈ G.

(7)

For such families, by Lemma 5, we have B( αh (ξ0 ), (h)r) ⊂ α h B(ξ0 , r) ⊂ B( αh (ξ0 ), L(h)r)

 ∀ h ∈ H. ∀ ξ0 ∈ G,

(8)

2.3. The frame condition Let G be a locally compact and second countable abelian group, and let us denote with + its composition law. Let Γ be a cocompact subgroup of G. Let T : Γ → U(L2 (G)) be the unitary representation of Γtranslations on L2 (G), i.e. T (γ)f (x) = f (x − γ), and let D : H → U(L2 (G)) be the unitary operator-valued map given by D(h)f (x) = δ(h)− 2 f (αh (x)) 1

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 : H → U(L2 (G))  the unitary operator-valued map where H and α are as in §2.2. Let us also denote with D  . It reads explicitly  f = D(h)f obtained by conjugation with the Fourier transform, i.e. such that D(h)  f(ξ) = δ(h) 12 f( D(h) αh (ξ))

(9)

since, by definition of δ and α , we have  (ξ) = D(h)f



D(h)f (x)ξ, xdμG (x) = δ(h)− 2 1

G 1



= δ(h) 2

 f (αh (x))ξ, xdμG (x) G

f (x)ξ, αh−1 (x)dμG (x) = δ(h) 2 f( αh (ξ)). 1

G

The two main notions that we will relate in the next sections are the following generalized definitions of affine frame and of Calderón’s sum. Definition 7. With the above notations, let H be a closed subspace of L2 (G) with the induced norm. For ψ ∈ L2 (G), we say that the affine system

 AΓ,H (ψ) = D(h)T (γ)ψ : γ ∈ Γ , h ∈ H is a ς-frame for H with constants 0 < A ≤ B < ∞ if AΓ,H (ψ) ⊂ H and Af 2L2 (G) ≤

  H

|f, D(h)T (γ)ψL2 (G) |2 dμΓ (γ) dς(h) ≤ Bf 2L2 (G)

(10)

Γ

holds for all f ∈ H . If only the right inequality holds for all f ∈ H , we say that AΓ,H (ψ) is a ς-Bessel system for H . Definition 8. For ψ ∈ L2 (G), we define the ς-Calderón sum for AΓ,H (ψ) by  Cψ (ξ) =

 αh (ξ))|2 dς(h). |ψ(

H

As for standard affine systems, with the next lemma we provide a trivial but key equation which has a fundamental role in the following computations. Lemma 9. If AΓ,H (ψ) is a ς-Bessel system for H ⊂ L2 (G), then for all f ∈ H the central term in (10) reads   IΓ,H (f, ψ) = |f, D(h)T (γ)ψL2 (G) |2 dμΓ (γ) dς(h) H



Γ

δ(h)

= H

−1

 Ω

|



 + λ)|2 dν  (ξ) dς(h) f( αh−1 (ξ + λ))ψ(ξ G

λ∈Γ⊥

 stands for a fundamental domain for G/Γ  ⊥. where Ω ⊂ G

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 we have Proof. By Plancherel theorem on L2 (G) and the unitarity property of D(h) IΓ,H (f, ψ) =

  H

Γ

H

Γ

 

=

2  T |f, D(h) (γ)ψL2 (G) | dμ (γ) dς(h) Γ  2  −1 f, T |D(h) | dμ (γ) dς(h) (γ)ψL2 (G) Γ 

 −1 f(ξ) = δ(h)− 12 f( where, by (9), D(h) αh−1 (ξ)), and T (γ)ψ(ξ) =



 ψ(x − γ)ξ, xdμG (x) = ξ, γψ(ξ).

G

Thus, by i., Lemma 1, we have −  −1 f, T D(h) (γ)ψL2 (G)  = δ(h) 2 1

= δ(h)− 2 1



⎛ ⎝

Ω



 f( αh−1 (ξ))ψ(ξ)ξ, γdνG (ξ)

 G



⎞  + λ)⎠ ξ, γdν  (ξ). f( αh−1 (ξ + λ))ψ(ξ G

λ∈Γ⊥

Since AΓ,H (ψ) is a Bessel system, then the left hand side in the above chain of identities, as a function of γ, belongs to L2 (Γ) for ς-a.e. h ∈ H, so that ξ →



 + λ) f( αh−1 (ξ + λ))ψ(ξ

λ∈Γ⊥

belongs to L2 (Ω), and the conclusion follows by ii., Lemma 1.



3. Boundedness of the Calderón’s sum In this section we will make use of the following standing assumptions:  Γ, Γ⊥ and their Haar measures are as in §2. I) The groups G, G,  has an invariant metric d  such that (G,  d  , ν  ) is weakly doubling,3 i.e. that there exist II) The group G G G G r0 > 0 and C > 0 such that νG (B(e, 2r)) ≤ CνG (B(e, r))

∀ r < r0 .

(11)

 is measurably bi-Lipschitz with respect to d  and, in order III) The automorphisms map α  : H → Aut(G) G to make use of Fubini’s theorem, the Borel measure ς on H is σ-finite. 3.1. The test space  and  > 0 let us consider the unit norm function f ξ0 ∈ L2 (G) Following the original idea of [5], for ξ0 ∈ G  given by 3 Note that this is sufficient to have Lebesgue’s differentiation theorem, see [13, Theorem 3.4.3]. A different condition that is weaker than doubling and allows to obtain a Lebesgue’s differentiation theorem is used in [16, Theorem (44.18)] and [3].

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 fξ0 = 

1 χ . νG (B(ξ0 , )) B(ξ0 ,)

By Lemma 9 and Remark 4, whenever AΓ,H (ψ) is a ς-Bessel system for H ⊂ L2 (G), and if fξ0 belongs to H , we get   IΓ,H (fξ0 , ψ)



|

= H Ω

λ∈Γ⊥

χα

h B(ξ0 ,)

 + λ)|2 dνG (ξ)dς(h) . (ξ + λ)ψ(ξ νG ( αh B(ξ0 , ))

(12)

If AΓ,H (ψ) is a frame with constants A and B, then (12) is bounded from above and below by the same constants. We will show that it is possible to obtain from this representation formula the same bounds for the Calderón’s sum in the limit when  → 0. Since we will be dealing with such a special class of test functions, whose G-Fourier transform is the  we can make precise the requirement fξ0 ∈ H with normalized characteristic function of a metric ball in G, the following definition. ⊂ Definition 10. Let H be a Hilbert subspace of L2 (G) endowed with the induced norm. Denoting by H 2  L (G) the image of H under the G-Fourier transform, for 0 > 0, define  : = {ξ ∈ G GH 0 that is the set characterized by fξ0 ∈ H

χB(ξ,) ∈ H ∀  < 0 }

⇐⇒ ξ0 ∈ GH and  < 0 . 0

In order to avoid trivial pitfalls or too technical assumptions, we will always implicitly assume that such has a positive νG measure. In the next sections we will see that this is actually the case in relevant GH 0  be a ball of radius r > 0 . The Paley-Wiener space situations. As a preliminary toy example, let Br ⊂ G 2 Hr   Hr = {f ∈ L (G) : suppf ⊂ Br } is such that G0 = Br−0 . A crucial observation, already present in [5], is that, by restricting to a subset of H which makes the automorphisms a uniformly bounded Lipschitz family, the quantity IΓ,HM (fξ0 , ψ) actually approximates the Calderón’s sum for small . Lemma 11. Let AΓ,H (ψ) be an affine system as in Definition 7. If AΓ,H (ψ) is a ς-Bessel system for H ⊂ L2 (G) and, for M > 0, HM = {h ∈ H : L(h) < M }, then there exists 0 > 0 such that IΓ,HM (fξ0 , ψ) =

1 νG (B(ξ0 , ))





 αh (ξ))|2 dς(h)dν  (ξ) |ψ( G

(13)

B(ξ0 ,) HM

for all ξ0 ∈ GH and all  < 0 . 0 Proof. Without loss of generality, let us choose a fundamental set Ω that contains a neighborhood of α h (ξ0 ). Since, by (8), we have that α h B(ξ0 , ) ⊂ B( αh (ξ0 ), M )

∀ h ∈ HM ,

then, for  sufficiently small, we get α h B(ξ0 , ) ⊂ Ω. For such an , and for h ∈ HM , the sum in (12) contains only one term. Indeed, if λ = e then ( αh B(ξ0 , ) + λ) ∩ Ω = ∅, so χα h B(ξ0 ,) (ξ + λ) = 0 for all ξ ∈ Ω. Thus,   IΓ,HM (fξ0 , ψ) = HM Ω

 2 dν  (ξ) χα h B(ξ0 ,) (ξ)|ψ(ξ)| G

dς(h) νG ( αh B(ξ0 , ))

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 2 dν  (ξ) |ψ(ξ)| G

= HM α  h B(ξ0 ,)





=

11

dς(h) νG ( αh B(ξ0 , ))

 αh (ξ))|2 dν  (ξ) |ψ( G

HM B(ξ0 ,)

dς(h) . νG (B(ξ0 , ))

Since IΓ,HM (fξ0 , ψ) ≤ IΓ,H (fξ0 , ψ) is bounded by the ς-Bessel hypothesis, the claim follows by Fubini’s theorem.  3.2. Bessel bound The upper bound for the Calderón’s sum can be obtained directly from (13) without any additional assumption, as shown by the following proposition. Proposition 12. Let AΓ,H (ψ) be a ς-Bessel system for H ⊂ L2 (G) with constant B. Then Cψ ≤ B almost everywhere in GH . 0   αh (ξ))|2 dς(h) belongs to L1 (G H ). Thus, by applying Proof. By Lemma 11 the function ξ → HM |ψ( loc 0 Lebesgue’s differentiation theorem at the limit for  → 0 and then taking the limit for M → ∞ in (13), we get lim lim IΓ,HM (fξ0 , ψ) = Cψ (ξ0 ) for νG -a.e. ξ0 ∈ GH . 0

M →∞ →0

The result follows because, for all  > 0 and all M > 0, we have IΓ,HM (fξ0 , ψ) ≤ IΓ,H (fξ0 , ψ) ≤ B.  3.3. Main theorem In order to obtain the bound from below of Calderón’s sum for frames, we will need a technical assumption on the estimate of the number of lattice points inside a metric ball deformed by the family of automorphisms. A similar estimate is the central object of study in the recent paper [2] dealing with affine wavelets in Rn . Its role for the study of affine systems was also noted in [14, Lemma 5.11] and, for LCA groups, in [20, Lemma 4.11], but it could be found in disguise even in [5, page 271]. αh }h∈H Definition 13. Let Γ be a cocompact subgroup of G. We say that a measurably bi-Lipschitz family {  of G-automorphisms has Property X for Γ⊥ if there exist r > 0, M > 0, and a constant C > 0 such that 

Γ⊥ ∩ α h B(e, r) ≤ 1 + Cδ(h) (14) c for all h ∈ HM = {h ∈ H : L(h) > M }.

Remark 14. Observe that the estimate (14) takes into account that at least the lattice point e ∈ Γ⊥ ∩ α h B(e, ) has to be counted. Moreover, if (14) holds for a given constant C, then it holds with the same constant C for all r < r and all M > M . In the following sections we will deduce Property X for relevant systems from our counting estimate of Lemma 2. However, a simple paradigmatic example of a setting where it does not hold may be considered now.

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12

   = R2 , let Γ⊥ = Z2 , let H = Z and let αj = 2 10 j . Then δ(j) = det(αj ) = 1 and Example 15. Let G 0 2 c HM = {|j| > log2 M }. That (14) does not hold can be seen because, for any fixed , for large values of j the deformed balls are allowed to contain an arbitrarily large number of lattice points. The main step in order to obtain the lower bound consists of obtaining an approximation to the full Calderón’s sum, up to a remainder, from the representation formula (12). Proposition 16. Let AΓ,H (ψ) be a ς-frame of H ⊂ L2 (G) with constants 0 < A ≤ B < ∞, and let { αh }h∈H have Property X. Then there exist an 0 > 0 and an M0 > 0 such that 1 A≤ νG (B(ξ0 , ))





 αh (ξ))|2 dς(h)dν  (ξ) + Rψ (, ξ0 ) |ψ( M G

B(ξ0 ,) H

for all ξ0 ∈ GH , all  < 0 , and all M > M0 , with 0 ψ RM (, ξ0 )

C = νG (B(ξ0 , ))



 c HM

 2 dν  (ξ)dς(h)  ψ(ξ)| |D(h) G

(15)

B(ξ0 ,)

 was defined in (9). where C is given by (14) and D c Proof. Since H = HM  HM , then ξ c (f 0 , ψ). IΓ,H (fξ0 , ψ) = IΓ,HM (fξ0 , ψ) + IΓ,HM 

(16)

The first term can be treated by Lemma 11, so for some 0 > 0 we have IΓ,HM (fξ0 , ψ) =

1 νG (B(ξ0 , ))





 αh (ξ))|2 dς(h)dν  (ξ) |ψ( G

B(ξ0 ,) HM

for all  < 0 . Let us then focus on the second term. By Cauchy-Schwarz inequality we can write, for the inner summation in (12), 

|

λ∈Γ⊥

 + λ)|2 ≤ S (h, ξ) χα h B(ξ0 ,) (ξ + λ)ψ(ξ

 λ∈Γ⊥

 + λ)|2 χα h B(ξ0 ,) (ξ + λ)|ψ(ξ

where S (h, ξ) =

 λ∈Γ⊥

χα h B(ξ0 ,) (ξ + λ).

Note that this is actually a finite sum. Moreover, for all λ ∈ Γ⊥ we have S (h, ξ) = S (h, ξ + λ), so by i., Lemma 1 we get   ξ c (f 0 , ψ) ≤ IΓ,HM 

=

   c Ω λ∈Γ⊥ HM

S (h, ξ)

 λ∈Γ⊥

c Ω HM

S (h, ξ + λ)χα

h B(ξ0 ,)

 + λ)|2 χα h B(ξ0 ,) (ξ + λ)|ψ(ξ

dνG (ξ)dς(h) νG ( αh B(ξ0 , ))

 + λ)|2 dνG (ξ)dς(h) (ξ + λ)|ψ(ξ νG ( αh B(ξ0 , ))

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  S (h, ξ)χα

=

h B(ξ0 ,)

c HM  G





= c HM α  h B(ξ0 ,)

 2 (ξ)|ψ(ξ)|

13

dνG (ξ)dς(h) νG ( αh B(ξ0 , ))

 2 dνG (ξ)dς(h) . S (h, ξ)|ψ(ξ)| νG ( αh B(ξ0 , ))

By the group invariance of the metric dG , when ξ ∈ α h B(ξ0 , ) we have

 S (h, ξ) ≤ λ ∈ Γ⊥ : α h B(ξ0 , ) ∩ ( αh B(ξ0 , ) + λ) = ∅

 = λ ∈ Γ⊥ : α h B(ξ0 , ) ∩ α h B(ξ0 + α h−1 (λ), ) = ∅

 = λ ∈ Γ⊥ : B(ξ0 , ) ∩ B(ξ0 + α h−1 (λ), ) = ∅

  ≤ λ ∈ Γ⊥ : dG ( αh−1 (λ), e) < 2 = α h−1 Γ⊥ ∩ B(e, 2)     = α h−1 Γ⊥ ∩ α h B(e, 2) = Γ⊥ ∩ α h B(e, 2) ≤ 1 + Cδ(h)

(17)

where the last inequality is Property X. From (16) and (17) and arguing as in Lemma 11, we have then obtained   1 ξ0  αh (ξ))|2 dς(h)dν  (ξ) + Rψ (, ξ0 ) IΓ,H (f , ψ) ≤ |ψ( M G νG (B(ξ0 , )) B(ξ0 ,) H

where the remainder term reads





ψ RM (, ξ0 ) = C c HM α  h B(ξ0 ,)

 2 |ψ(ξ)|

δ(h) dν  (ξ)dς(h) νG ( αh B(ξ0 , )) G

which coincides with (15) by Remark 4.  We are now ready to prove our main theorem, which provides sharp bounds for the Calderón’s sum as the frame bounds. For this, we require an hypothesis of local integrability for a term which, whenever δ = 1, differs from the Calderón’s sum by such a factor. The role of this hypothesis has been discussed in a recent paper concerning generalized shift-invariant systems on the real line [4], where the authors prove that it is strictly related to the so-called LIC condition introduced in [14] and α-LIC condition introduced in [18]. In the present setting, it is the minimal assumption needed to make use of Fubini’s theorem and prove that the remainder vanishes in the limit. In the next section, we will see that this local integrability condition holds for rather general classes of automorphisms. Theorem 17. Let AΓ,H (ψ) be a ς-frame of H ⊂ L2 (G) with constants 0 < A ≤ B < ∞, and suppose  with upper Lipschitz constants that { αh }h∈H is a measurably bi-Lipschitz family of automorphisms of G, {L(h)}h∈H , satisfying Property X. If there exist M > 0 and 0 > 0 such that   2 dς(h) ∈ L1 (G H  O)  ψ(ξ)| ΨM (ξ) = |D(h) (18) loc 0 c HM

c for some zero-measure set O, where HM = {h ∈ H : L(h) > M } and GH is as in Definition 10, then 0

A ≤ Cψ (ξ) ≤ B

for νG − a.e. ξ ∈ GH . 0

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14

Proof. The upper bound is provided by Proposition 12. For the lower bound, if (18) holds, then by Fubini’s theorem the remainder term (15) reads ψ RM (, ξ0 ) =

C νG (B(ξ0 , ))

 ΨM (ξ)dνG (ξ). B(ξ0 ,)

Since, by Proposition 16, we have 1 A≤ νG (B(ξ0 , ))





 αh (ξ))|2 dς(h)dν  (ξ) + Rψ (, ξ0 ) |ψ( M G

B(ξ0 ,) H

and we can apply Lebesgue’s differentiation theorem to both terms, by taking the limit as  → 0 we get A ≤ Cψ (ξ0 ) + C ΨM (ξ0 )

a.e. ξ0 ∈ GH . 0

Since the only term depending on M is ΨM , the proof is concluded by showing that this quantity vanishes when M tends to infinity. Note that if condition (18) holds for a given M , it holds for all larger values of M. c as the disjoint union In order to see that, let M ∈ N, so that can write HM c HM =

 Denoting by φn (ξ0 ) =

∞

∞

n=M

n=M

 {h ∈ H : n < L(h) ≤ n + 1} =  Σ . n

 0 )|2 dς(h), we then get  ψ(ξ |D(h)

Σn

ΨM (ξ0 ) =

∞ 

φn (ξ0 ).

n=M

By (18), for a.e. ξ0 ∈ GH we have that ΨM (ξ0 ) is finite. So, in particular, each term φn (ξ0 ) is finite, and 0 limn→∞ φn (ξ0 ) = 0 for a.e. ξ0 ∈ GH . Thus 0 lim ΨM (ξ0 ) = 0

M →∞

concluding the proof.  4. Expanding automorphisms  for any 0 > 0. We will make In this section we will consider H to be the whole L2 (G), so that GH =G 0 use of a weak notion of expanding automorphisms, which we introduce in terms of the Lipschitz constants defined in (7).  Definition 18. A measurably bi-Lipschitz family of G-automorphisms { αh }h∈H , with bi-Lipschitz constants {(h)}h∈H and {L(h)}h∈H , is called expanding if there exist M0 , N0 > 0 such that, for all M > M0 L(h) > M ⇒ (h) > N0 . We say that it is uniformly expanding if there exist an M > 0 and a monotone increasing function f : R+ → R+ such that

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15

L(h) > M ⇒ L(h) ≤ f ((h)). In particular, if { αh }h∈H is uniformly expanding, then it is expanding. This notion of expanding imposes that (h) can not be arbitrarily small when L(h) is arbitrarily large. This means that any automorphism of the family can not put near two points in the space while moving away two other points at arbitrary distances. It thus rules out cases such as that of Example 15. This general formulation is compatible with the more classical ones considered in [14], [9] and [2], in the sense that the expanding matrices of these works are expanding for the current definition, see Proposition 32. L(h) to be bounded on H, (h) as this would exclude relevant cases such as anisotropic dilations. This ratio, which defines the quasiconformality (see e.g. [13, §14.2]) of α h , is indeed allowed to be arbitrarily large with h in H. Remark 19. This notion of expanding does not require the deformation ratio

4.1. Property X and Calderón’s bounds We show here that the introduced notion of expanding automorphisms allows us to obtain Property X from the counting Lemma 2.  Theorem 20. Let { αh }h∈H be an expanding family of G-automorphisms. Then, for all Γ cocompact subgroup ⊥ of G, it has Property X for Γ . Proof. Let Γ be a cocompact subgroup of G. We introduce the following shorthand notation derived from  and r > 0, let us call (2): if Ω is a fundamental domain for Γ⊥ in G,   ω(r) = νG (Ωr1 ) = νG Ω ∩ B(λ, r) .

(19)

λ∈Γ⊥

By (8) and Lemma 2, and recalling Remark 4, we have that for all N > 0  ν (B(e, 2r))  δ(h) ∀ r > 0 , ∀ h ∈ KN ,

Γ⊥ ∩ α h B(e, r) ≤ G ω(N r) where KN = {h ∈ H : (h) > N }. Let us observe now that we can choose  and N so that ω(N ) = νG (B(e, N )). Indeed, without loss of generality assume that Ω contains a neighborhood of e ∈ Γ⊥ . Then for  sufficiently small we get B(e, ) ⊂ Ω, and no other ball B(λ, ) in (19), for λ = e, intersects Ω. Therefore, for all h ∈ KN  ν  (B(e, 2)) ν  (B(e, 2)) δ(h) ≤ 1 + G δ(h).

Γ⊥ ∩ α h B(e, ) ≤ G νG (B(e, N )) νG (B(e, N )) Now, by the weak doubling property (11), if we assume without loss of generality that N = 2−q , we get νG (B(e, 2)) ≤ C q+1 . νG (B(e, N )) Thus, we have proved that for any  sufficiently small, and for any N > 0, there exists C such that

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Γ⊥ ∩ α h B(e, ) ≤ 1 + Cδ(h) for all h ∈ KN . The proof now follows as a consequence of the expanding property, which implies that there c exists an M > 0 such that HM = {h ∈ H : L(h) > M } ⊂ KN .  As a consequence of Theorems 17 and 20 we get then the following Calderón’s inequalities. αh }h∈H be an Corollary 21. Let AΓ,H (ψ) be a ς-frame of L2 (G) with constants 0 < A ≤ B < ∞, and let {  with upper Lipschitz constants {L(h)}h∈H . If expanding family of bi-Lipschitz automorphisms of G,  ΨM (ξ) =

 2 dς(h) ∈ L1 (G  ψ(ξ)|   O) |D(h) loc

(20)

c HM

c for some zero-measure set O and some M > 0, where HM = {h ∈ H : L(h) > M }, then

 for νG − a.e. ξ ∈ G.

A ≤ Cψ (ξ) ≤ B 4.2. Sufficient conditions for local integrability

We provide here a sufficient condition for the local integrability (20) in the case of a uniformly expanding family of automorphisms. Theorem 22. Let { αh }h∈H be uniformly expanding. If ∃ M > 0 such that   uc (t) = ς ({h ∈ H : t ≤ L(h) ≤ f (ct)}) ∈ L∞ [M, +∞) for all c > 1, then (20) is satisfied for all ψ ∈ L2 (G), with O = {e}. Proof. We prove local integrability by showing that 



 2 dν  (ξ)dς(h) < ∞ ∀ξ0 = e. |ψ(ξ)| G

EM (, ξ0 ) = c HM α  h B(ξ0 ,)

Let ξ ∈ α h B(ξ0 , ), i.e. dG ( αh−1 (ξ), ξ0 ) < . By (7) and (8) we have dG (ξ, e) ≤ dG ( αh (ξ0 ), e) + dG (ξ, α h (ξ0 )) ≤ L(h)(dG (ξ0 , e) + dG ( αh−1 (ξ), ξ0 )) ≤ L(h)(dG (ξ0 , e) + ) dG (ξ, e) ≥ (h)dG ( αh−1 (ξ), e) ≥ (h)(dG (ξ0 , e) − dG ( αh−1 (ξ), ξ0 )) ≥ (h)(dG (ξ0 , e) − ) ≥ f −1 (L(h))(dG (ξ0 , e) − ). Denoting by C(r, R) = B(e, R)  B(e, r), we can then write 



EM (, ξ0 ) ≤

 2 dν  (ξ)dς(h) |ψ(ξ)| G

c HM C(rh ,Rh )

where rh = f −1 (L(h))(d(ξ0 , e) − ) and Rh = L(h)(d(ξ0 , e) + ).

(21)

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dG (ξ, e)

dG  (ξ,e) Δ+

c HM

HM

≤ L(h) ≤ f

17

 dG (ξ,e)  Δ−

−1 (L(h)) dG  (ξ,e)=(Δ−)f

dG  (ξ,e)=(Δ+)L(h)

B (Δ + )M A

(Δ − )f −1 (M )

A

Δ+ f ( Δ− M)

M

L(h)

Fig. 1. Domain of integration for EM (, ξ0 ), with Δ = dG (ξ0 , e).

The domain of integration can now be split as in Fig. 1, so that   2   2 dν  (ξ)dς(h) = E (1) + E (2) . EM (, ξ0 ) ≤ |ψ(ξ)| dνG (ξ)dς(h) + |ψ(ξ)| G AA

B

 and H as follows: denoting For the first term, using Fubini’s theorem, we exchange the integrations in G c by F (t1 , t2 ) = ς ({h ∈ HM : t1 ≤ L(h) ≤ t2 })  E

(1)

=F



Δ+ M M, f Δ−







  C (Δ−)f −1 (M ),(Δ+)M

= uc (M )   C (Δ−)f −1 (M ),(Δ+)M

 2 dν  (ξ) |ψ(ξ)| G

 2 dν  (ξ) |ψ(ξ)| G

where we have used the shorthand notation Δ = dG (ξ0 , e), and c =

Δ+ . Δ−

For the second term we can proceed analogously, obtaining      2 F dG (ξ, e) , f dG (ξ, e) dνG (ξ) E (2) = |ψ(ξ)| Δ+ Δ−  GB(e,(Δ+)M )



 2 uc |ψ(ξ)|

=  GB(e,(Δ+)M )

again with c =



dG (ξ, e) Δ+

 dνG (ξ)

Δ+ . Assuming (21), we then have Δ− EM (, ξ0 ) ≤







ess sup uc [M,+∞)

 2 dν  (ξ) |ψ(ξ)| G

 GB(e,M (Δ−))

which is finite.  4.3. Examples In order to have a concrete picture of the presented results we briefly discuss here some relatively simple examples of wavelet-type frames.

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4.3.1. Semi-continuous wavelets on L2 (R) Let G = R, consider as discrete subgroup Γ = Z, and as a set of automorphisms the full dilations group H = R+ , with α a ξ = aξ and a measure that is absolutely continuous with respect to the Lebesgue one, i.e. dς(a) = m(a)da for some positive m ∈ L1 (R+ ). In this case, (a) = L(a) = a, and α  is uniformly expanding with f (x) = x. Condition (18) reads +∞ +∞   η √ 1 2 2   dη ΨM (ξ) = | aψ(aξ)| m(a)da = 2 |ψ(η)| ηm ξ ξ M

Mξ a→∞

which is convergent for all ψ ∈ L2 (G) and ξ = 0 only if m(a)  O( a1 ). This is the same result one gets from condition (21), since ct uc (t) =

m(a)da. t

In this case, the Calderón’s integral reads +∞  2  Cψ (ξ) = |ψ(aξ)| m(a)da. M

4.3.2. Discrete wavelets on L2 (Rn ) with anisotropic dilations j ξ = Aj ξ for a matrix Let G = Rn , and let Γ = Zn . As set of automorphisms let us take H = Z, with α A ∈ GLn (R) (see also Appendix A for the case of so-called expanding on a subspace matrices). By [14, Lemma 5.1], there exist 0 < λ < λ and C > 0 such that the bi-Lipschitz constants are 1 j j λ , L(j) = Cλ , C 1 j (j) = λ , L(j) = Cλj , C

(j) =

j≥0 j ≤ 0.

So if either λ > 1 or λ < 1, then α  is uniformly expanding, with  f (x) =

C(Cx)logλ λ C(Cx)logλ λ

x>1 . x≤1

Let λ > 1, and set M > 1. The quantity uc in (21) reads (we let the constant c change freely) j

uc (t) = ς({j ∈ Z : t ≤ Cλ ≤ C(Cct)logλ λ }) = ς({j ∈ Z : logλ

t ≤ j ≤ (logλ λ)(logλ Cct)}) C

that is the measure of a finite set. Thus, if we choose ς = m(j)× counting, condition (21) reads simply lim supj→∞ m(j) < ∞ For comparison, let us try to check directly condition (18). The dilation operator for ψ ∈ L2 (Rn ) reads  = | det A| 2j ψ(Aj ξ)  ψ(ξ) D(j) with δ(j) = | det A|j so, for N (M ) = logλ

M C ,

we have

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ΨM (ξ) =

19

 j ξ)|2 m(j) | det A|j |ψ(A

j=N (M )

and, for any K ⊂ Rn compact, 



+∞ 

ΨM (ξ)dξ =

K j=N (M )

K



+∞ 

 j ξ)|2 m(j)dξ = δ(j)|ψ(A

m(j)

j=N (M )

 2 dξ. |ψ(ξ)|

Aj K

Let d = minξ∈K |ξ| and D = maxξ∈K |ξ|, so that K ⊂ B(0, D)  B(0, d) and Aj K ⊂ Aj B(0, D)  Aj B(0, d). The Aj image of a ball B(0, r) is contained in a (hyper)ellipsoid with small semi-minor axis C1 λj r and semij

j

major axis Cλ r. Thus a sufficient condition for K ∩ Aj K = ∅ is Cλ d > D. If d > 0, that is 0 ∈ / K, then D there are at most logλ ( dC ) terms in the sum that correspond to integrations over domains that superpose. D Let then S = logλ ( Cd ), so that  ΨM (ξ)dξ =

S 



+∞ 

m(l + jS + N (M ))

l=1 j=N (M )

K

 2 dξ. |ψ(ξ)|

Al+jS+N (M ) K

 If m is a constant, since the inner sum runs over disjoint domains, we have K ΨM (ξ)dξ ≤ m S ψ2L2 (Rn ) , while in general we have boundedness if lim supj→∞ m(j) < ∞. We would like to observe explicitly that this last argument actually mimics the proof of Theorem 22, and can be found to be very close to the one used in [4, Lemma 4.1]. 4.3.3. Wavelets with composite dilations For G = Rn and Γ a cocompact subgroup, the present setting allows us to consider {αh }h∈H to be any subset of GLn (R) parametrized by H, with an appropriate measure ς which behaves well with respect to the constants (7). This includes the case of so-called composite dilations [10], with no restriction on the discreteness of H. A notable relevant case is provided by shearlets [22], [21]: let G = R2 , let Γ = Z2 , and let H be any −1 subset of R × R+ . For a ∈ R+ and s ∈ R, the shearlets automorphisms read αa,s (x) = A−1 a Ss (x), with a 0 1 s 3 anisotropic dilations Aa = 0 √a and shears Ss = 0 1 . Thus δ(a, s) = a 2 , and  α a,s (ξ) = Aa Ss (ξ) = t

a 0 √ √ a s a

    aξ1 ξ1 √ = √ . ξ2 s aξ1 + aξ2

The optimal bi-Lipschitz constants with respect to the Euclidean metric are given by the singular values of the matrix Aa t Ss , i.e. by the square root of the eigenvalues of  M (a, s) =

√  a s a a √ √ 0 a s a

 0 √ a

 =

a2 + s2 a sa sa a



so that L(a, s) =

 12   a/2 a + s2 + 1 + (a + s2 + 1)2 − 4a

(a, s) =

 12   a/2 a + s2 + 1 − (a + s2 + 1)2 − 4a .

Since, for any fixed a, as s becomes large we can have arbitrarily large values of L with arbitrarily small values of , this family of automorphisms does not satisfy the expanding condition. Nevertheless, it is easy

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 √  a



a

Fig. 2. Left: the ∞ ball B∞ (0, ) = {ξ ∈ R2 : max{|ξ1 |, |ξ2 |} < }. Right: a set α  a,s B∞ (0, ) and the integer lattice.

to see that it satisfies Property X. Indeed, if for simplicity we consider the invariant metric provided by the ∞ distance, whose balls are squares B∞ (0, ) = {ξ ∈ R2 : max{|ξ1 |, |ξ2 |} < }, we can easily check (see e.g. Fig. 2) that   √

Z2 ∩ α a,s B∞ (0, ) ≤ (2a + 1)(2 a + 1) so, if we consider 0 = 12 , for all  < 0 either we have a ≥ 1 or we have only one point inside the ball. If a ≥ 1 we then have  √  3 1+ a a2

Z ∩α a,s B∞ (0, ) ≤ 1 + 2 2 + a 

where

√ 1+ a a



2

≤ 2, so that we can choose C = 4( + 1).

5. Gabor-type frames In this section we will show how the results of §3 allow us to address the case of Gabor frames, even if modulations can not be directly realized in terms of spatial automorphisms. Nevertheless, by considering an appropriate Hilbert subspace and the set of automorphisms that define the Heisenberg group as a semidirect product, one can recover the discussed setting. We will then show that, even if these automorphisms are not expanding, they possess Property X as a simple consequence of Lemma 2. 5.1. LCA-Gabor systems and automorphisms For simplicity, let us identify T = {z ∈ C : |z| = 1} and use exponential coordinates on it. For any LCA group R, let G = R × T , with translations given by T (q, e2πiϕ )f (x, e2πiθ ) = f (x − q, e2πi(θ−ϕ) ) ,

f ∈ L2 (G)

 and let α : P → Aut(G) where we use the multiplicative notation for the composition on T . Let then P ⊂ R, be given by αp (x, e2πiθ ) = (x, e2πi(θ+p,x) ) where we denote by e2πip,x ∈ T the action of the characters of R. Note that this action is the one that defines the (reduced) Heisenberg group (see e.g. [25, §4.1] or [6, §1.3, §1.11]) as the semidirect product

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  R × T . Note that, for this action, we have δ = 1. Let also ς be a σ-finite Borel measure on P such that R this αp is measurable.  × R → U(L2 (R)) be the projective representation Definition 23. For κ ∈ Z  {0}, let πκ : R πκ (p, q)g(x) = e2πiκp,x g(x − q) and let Λ be a cocompact closed subgroup of R. We say that the Gabor system Gκ (g, Λ, P) = {πκ (p, q)g : q ∈ Λ , p ∈ P} is a ς-frame of L2 (R) with constants 0 < A ≤ B < ∞ if Au2L2 (R) ≤

  P

|u, πκ (p, q)gL2 (R) |2 dμΛ (q) dς(p) ≤ Bu2L2 (R)

Λ

for all u ∈ L2 (R). These systems were thoroughly studied in the recent work [17]. Definition 24. For κ ∈ Z, let Hκ be the Hilbert subspace of L2 (G) given by Hκ = {f ∈ L2 (R × T ) : f (x, e2πiθ ) = u(x)e2πiκθ , u ∈ L2 (R)}

and, for f ∈ Hκ , let us call u its R-representative. With this notation, we can provide the main observation that allows us to relate Gabor frames to the affine frames of Definition 7. Lemma 25. Let Λ be a cocompact closed subgroup of R, and let Γ = Λ × T . Let ψ ∈ Hκ , and let g ∈ L2 (R) be its R-representative. Then the system AΓ,P (ψ) is a ς-frame of Hκ with constants 0 < A ≤ B < ∞ if and only if the Gabor system Gκ (g, Λ, P) is a ς-frame of L2 (R) with the same constants. Proof. Let f, ψ ∈ Hκ , and let u, g ∈ L2 (R) be their R-representatives. Then  f, D(p)T (q, e2πiϕ )ψL2 (G) =

u(x)e2πiκθ g(x − q)e−2πiκ(θ−ϕ+p,x) dμR (x)dθ

T ×R

= e2πiκϕ u, πκ (p, q)gL2 (R) . The proof follows by noting that f L2 (G) = uL2 (R) .  5.2. Property X and Calderón’s bounds  × Z can be computed from the duality =R The dual action α  on G (ξ, k), αp−1 (x, θ) = e2πi(ξ,x+k(θ−p,x)) ,  × Z and (x, θ) ∈ R × T , so that where (ξ, k) ∈ R

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α p (ξ, k) = (ξ − kp, k)

(22)

 composition p + p + · · · + p if k > 0. If k < 0, compose |k| times −p, where by kp we mean the iterated R    k

 while if k = 0 that is the neutral element of R.  Given an invariant distance dR  on R satisfying (11), the distance dG ((ξ, k), (η, l)) = dR  (ξ, η) + |k − l|  let us also denote by  and satisfies (11). For ξ ∈ R, is invariant on G ξ = dR  (ξ, eR  ). Lemma 26. The map α p is bi-Lipschitz with optimal constants L(p) = 1 + p ,

(p) =

1 1 = . 1 + p L(p)

dR  (ξ, kp) + |k| , and ξ + |k| (ξ,k) =0

Proof. By definition, we have L(p) = sup

dR ξ + |k| p + |k| p  (ξ, kp) + |k| ≤ =1+ ≤ 1 + p , ξ + |k| ξ + |k| 1 + ξ/|k| and the sup is a max attained at ξ = eR . dR  (ξ, kp) + |k| we have On the other hand, for (p) = inf ξ + |k| (ξ,k) =0 dR ξ − kp + |k|  (ξ, kp) + |k| ≥ = ξ + |k| ξ − kp + |k| p + |k| 1+

1 |k| ξ−kp+|k|

p

≥

1 , 1 + p

and the inf is a min attained at any ξ = kp.  Thus, in order to comply with assumption III) of §3 and make use of the results of that section, we need to require that the function  ·  be ς-measurable on P. Remark 27. Since in this case  =

1 L

is optimal, then these automorphisms are not expanding. Indeed (p) < M ⇐⇒ L(p) >

1 . M

Although these automorphisms do not satisfy the expanding property, we shall prove that they satisfy Property X, and consequently the results of §3 can be applied. This is proved in the following two lemmata. Lemma 28. For all  < 1, we have

χB((ξ0 ,k0 ),) (ξ, k) = χB (ξ0 ,) (ξ)δk,k0 . R

In particular, we have  , k = κ}  : ξ∈R G1Hκ = {(ξ, k) ∈ G  for any κ ∈ Z. which is isomorphic to R

(23)

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  is defined by Proof. A G-metric ball of radius  and center (ξ0 , k0 ) ∈ G dR  (ξ, ξ0 ) + |k − k0 | <  which reduces, for  < 1, to B((ξ0 , k0 ), ) = {ξ ∈ BR  (ξ0 , ) , k = k0 } hence proving (23). As a consequence, for  < 1, if f = χB((ξ0 ,k0 ),) then f belongs to Hκ if and only if k0 = κ.  Lemma 29. The automorphisms (22) have Property X. Proof. Since for any  < 1 we have α p B((ξ0 , k0 ), ) = B((ξ0 − k0 p, k0 ), ) , then α p B(e, ) = B(e, ). Thus, since e = (eR  , 0), we get

(Γ⊥ ∩ α p B(e, )) = {Γ⊥ ∩ B(e, )}. Since this does not depend on p, and since in this situation δ = 1, we then get the estimate (14) for all p ∈ P, for all radii r < 1.  This in turn provides a proof of the following result for Gabor-type frames, which was recently proved in [17, Corollary 5.6] with completely different techniques, and generalizes several previous statements such as [12, Prop. 4.1.4]. Without loss of generality, we have fixed κ = 1. Theorem 30. For g ∈ L2 (R), let {π(p, q)g : p ∈ P , q ∈ Λ} be a ς-frame of L2 (R) with constants 0 < A ≤ B < ∞. Then   A ≤ | g (ξ − p)|2 dς(p) ≤ B a.e. ξ ∈ R. P

Proof. By Lemma 25, the hypothesis of having a Gabor frame of L2 (R) is equivalent to having an affine frame of H1 . Now, since δ = 1, then the quantity we have called ΨM coincides with the Calderón’s sum, so its local integrability is a direct consequence of the Bessel inequality, and hence it is always verified for frames. Lemma 29 allows us to use Theorem 17, so the conclusion follows by Lemma 28.  Appendix A. More on the expanding property We discuss here some related issues concerning expanding automorphisms and the optimality of Lemma 2. A.1. On the optimality of Lemma 2 Let us observe first that, for any fixed α , we can find a small enough r0 such that νG (Ωrα ) = νG ( αB(e, r)). Thus, by Lemma 2 and Remark 4

(Γ⊥ ∩ α B(e, r)) ≤

νG (B(e, 2r)) νG (B(e, r))

∀r < r0 .

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Note that the ratio on the right hand side is always larger than 1, but, by (11), it will be bounded by the doubling constant. On the other hand, when Γ⊥ is a uniform lattice, Ω can be chosen to be compact and in this case, for any fixed α , we can find a large enough r1 such that Ωrα = Ω for all r > r1 . Thus, by Lemma 2,

(Γ⊥ ∩ α B(e, r)) ≤ cνG ( αB(e, 2r)) ∀r > r1 where c =

1 . νG (Ω)

We now prove that a lower estimate also holds.

 be a second countable LCA group with Haar measure ν  , let Γ⊥ be a discrete subgroup Lemma 31. Let G G  Let also d  : G ×G  → R+ be an invariant metric, and let B(ξ0 , r) = {ξ ∈ G  :  and let α of G,  ∈ Aut(G). G  dG (ξ, ξ0 ) < r} for ξ0 ∈ G, r > 0. Then 

Γ⊥ ∩ α B(e, 2r) ≥

1 ν  ( αB(e, r)) νG (Ωrα ) G

∀r>0

 a ν  -measurable section of G/Γ  ⊥. where Ωrα is as in (2) for any Ω ⊂ G G Proof. By the same argument that leads to (3) we can also get  Sαr (ξ)dνG (ξ) ≤ νG (Ωrα ) sup Sαr (ξ).

νG ( α(B(e, r))) =

ξ∈Ωrα

Ωrα

The proof can then be concluded by showing that Sαr (ξ) ≤ Sα2r  (e)

∀ ξ ∈ Ωrα .

To see this, let λξ be such that ξ ∈ α B(e, r) + λξ . Then     α B(e, r) − ξ ⊂ α B(e, 2r) − λξ   because, for any ζ = α (η) ∈ α B(e, r) − ξ we have dG (η, − α−1 (λξ )) ≤ dG (η, − α−1 (ξ)) + dG ( α−1 (ξ), α −1 (λξ )) < 2r. 2r The proof then follows because for all λ ∈ Γ⊥ we have Sα2r  (λ) = Sα  (e).



A.2. On the definition of expanding on a subspace In [14], with the correction discussed in [9], a definition of a matrix in Rn that is expanding on a subspace is given, that allows to obtain a lattice counting estimate given by [14, Lemma 5.11] and [9, Lemma 3.3], discussed later on. Their notion of expansiveness is as follows: given a non-zero linear subspace F ⊂ Rn , a matrix A ∈ GLn (R) is expanding on F if there exists a linear subspace E ⊂ Rn such that i. ii. iii. iv.

Rn = F + E and F ∩ E = {0} A(F ) = F and A(E) = E ∃ 0 < k ≤ 1 < γ < ∞ such that |Aj x| ≥ kγ j |x| ∀j ≥ 0, ∀x ∈ F ∃ a > 0 such that |Aj x| ≥ a|x| ∀j ≥ 0, ∀x ∈ E.

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The classical definition of expanding matrix can be obtained in the special case of E = {0}, that is F = Rn , and reduces to iii. or, equivalently, to saying that all eigenvalues λ of A are such that |λ| > 1 (see also §4.3). Proposition 32. Let A ∈ GLn (R) be expanding on a subspace. Then it is expanding in the sense of Definition 18. Proof. Since, by i., any x ∈ Rn can be written as x = xF + xE with xF ∈ F and xE ∈ E, points iii. and iv. imply that |Aj x| ≥ kγ j |xF | + a|xE | ≥ min{kγ j , a}|x| ∀j ≥ 0 for all x ∈ Rn . Let us call (j) = min{kγ j , a}. By [14, Lemma 5.1], we also have that there exists β > A such that, calling L(j) = β j , we get (j)|x| ≤ |Aj x| ≤ L(j)|x|

∀x ∈ Rn .

Assume now, by contradiction, that for all M0 , N0 > 0 there exists an M > M0 such that L(j) > M while (j) < N0 . For a large value of M0 , this means that j must be a large positive integer. But (j) can not be smaller than a, so any N0 < a provides a contradiction.  We observe that the converse of Proposition 32 does not hold because, for example, a collection of rotations satisfies Definition 18 but is not expanding on a subspace. We also remark that in the present work we are not restricting ourselves to uniform (full-rank) lattices, but we consider the larger class of annihilators of cocompact subgroups (see also [3] for a thorough discussion of this point). A.3. On other notions of expansiveness on LCA groups A lattice counting estimate similar to Property X is obtained in LCA groups, for uniform lattices and   on an LCA group G expanding automorphisms, in [20, Lemma 4.11]. There, an automorphism A ∈ Aut(G) is called expanding if dG (A(ξ), e) ≥ c dG (ξ, e) ∀ ξ = e , for some c > 1. This generalizes classical expanding matrices, and it immediately implies that the family {Aj }j∈Z is uniformly expanding according to Definition 18. We finally note that a notion of expansiveness also plays an important role in topological dynamical systems  of a countable group H and ergodic theory, see e.g. [19]. In this context, an action α  : H → Aut(G)  is said to be expanding if there exists a neighborhood U of e such that on a locally compact group G   is compact and connected, and it admits an expansive action h (U ) = {e}. By [19, Theorem 7.3], if G h∈H α  for H abelian and finitely generated, then G is abelian. However, in this setting, this notion of expansiveness is different from that of Definition 18, since it includes cases that we exclude such as that of Example 15. Appendix B. Proof of Lemma 1 For the sake of completeness, we provide here a proof of Lemma 1.   Proof. Point i. is a direct consequence of the definition of fundamental set, since νG G



(Ω + λ) = 0.

λ∈Γ⊥

The issue of proving point ii. consists of showing that the measures μΓ and νG make the Γ-Fourier transform

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a unitary map from L2 (Γ) to L2 (Ω). To see this, let us choose a Haar measure θ on Ω such that the Γ-Fourier transform is unitary with μΓ on Γ, and let us choose a Haar measure θ0 on Γ⊥ such that the Γ⊥ -Fourier transform is unitary with κ on G/Γ. By uniqueness of the Haar measure, there exist two positive constants c and c0 such that θ = c · νG ,

θ0 = c0 · counting measure.

We only need to prove that c = 1. By [16, (31.46), (c)], we then have that  φ(ξ)dνG (ξ) = c c0

  λ∈Γ⊥

 G

φ(ξ + λ)dνG (ξ)

 ∀ φ ∈ L1 (G).

Ω

Now, since i. holds, we get c c0 = 1. Moreover, by definition of θ0 we have 



|f (y)|2 dκ(y) = c0

 |

f (y)λ, ydκ|2 .

λ∈Γ⊥ G/Γ

G/Γ

In the above identity we can choose f ≡ 1 because G/Γ is compact, hence obtaining, by [15, Lemma (23.19)], κ(G/Γ) = c0



 |

λ, ydκ|2 = c0 κ(G/Γ)2 .

λ∈Γ⊥ G/Γ

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