Ruelle-Pollicott Resonances for Manifolds with Hyperbolic Cusps

We present new methods to construct a Ruelle-Pollicott spectrum for the geodesic flow on manifolds with strictly negative curvature and a finite number of hyperbolic cusps.

as well as its Laplace transform̂ , ( ) which is holomorphic for Re( ) > 0. Pollicott [Pol85] and Ruelle [Rue87] proved that for Axiom A flows, this Laplace transform̂ , extends meromorphically to a small strip Re( ) > − for a certain class of measures. Its poles are called Ruelle-Pollicott resonances of with respect to . In the following decade, several works [Hay90,Fri95,Rug96,Kit99] were dedicated to obtaining sharp bounds on the maximal strip on which the continuation is possible in terms of the regularity of the flow. More recently it has been understood that these resonances can be seen as the discrete spectrum in the usual sense of the generator of the flow on some carefully chosen Banach spaces. They appear as the poles of the meromorphic continuation of the resolvent kernel. See [Liv04,BL07,FS11,GLP13a,DZ16,DG16], and also [BKL02,GL06,BT07,BT08,FRS08,Bal18] for the related case of hyperbolic diffeomorphisms. A wide generality of dynamical systems is considered in these articles, however all these results have in common that they assume the system has a compact trapped set: Y. G. Bonthonneau Proof. Theorem 1 applies to the smooth manifold Γ =Γ∖ (2, ℝ). The corollary then follows by definition of smooth functions and distributions on orbifolds and because the resolvent commutes with isometries.
We now want to mention some results related to Theorem 1: In order to study eigenvalues of the Laplacian on moduli spaces Avila and Gouëzel [AG13] develop a functional analytic framework for the Teichmüller flows which are also a class of dynamical systems with noncompact finite volume trapped set. They obtain a meromorphic continuation of the resolvent to a neighbourhood of zero (cf. [AG13,Prop 3.3]). It would probably be possible to adapt their method to geodesic flows on cusp manifolds in order to obtain a continuation to a small strip along the imaginary axis (instead of ℂ in our case). However their functional analytic tools are quite different from ours.
Another series of related results have been obtained for the special case of surfaces of constant negative curvature with cusps. It has been shown in a series of articles, by Mayer, Morita and Pohl [May91,Mor97,Poh15,Poh16] that one can associate the geodesic flow with one dimensional expanding maps, using a carefully chosen discretization. Out of this discretization one can build transfer operators with discrete spectrum and these spectra have interesting relations to number theory and the theory of Maass cusp forms [LZ01,MP13,BLZ15]. One should be able to recover these spectra as a subset 4 of the resonances defined from Theorem 1. It will be subject to further research to establish this connection precisely.
As Ruelle-Pollicott resonances are an important tool to study decay of correlations, let us shortly mention that the question of mixing is not yet satisfactorily answered for our class of cusp manifolds: For constant curvature manifolds with cusps, exponential decay of correlations for the Liouville measure was proved in [Moo87], while for variable curvature only its mixing property is known [DP98]. Two other recent results on the mixing of Weil-Petersson geodesic flows on manifolds with cusp-like singularities 5 have been obtained in [BMMW17a,BMMW17b]. We hope that the analytic tools that we develop in this article will prove to be helpful in the future for studying mixing properties of geodesic flows on manifolds with hyperbolic cusps.
The meromorphic continuation of dynamical zeta functions is another important field where meromorphically continued resolvents of flow vector fields have successfully been applied. If  denotes the set of primitive periodic orbits of a hyperbolic flow and ( ) their lengths then the Ruelle zeta function is defined for Re( ) ≫ 0 by Smale [Sma67] raised the question if for Axiom A flows, the Ruelle zeta function 6 has a meromorphic continuation to ℂ? This question has recently been affirmatively answered 4 More precisely the connection should be to the so called first band of Ruelle-Pollicott resonances cf. [DFG15,GHW18a,GHW18b] 5 Note that their notion of cusp sigularities differs from ours: They consider singularities where the distance to the cusp is bounded, but the curvature is divergent 6 Actually Smale considered a different version of a dynamical zeta function which is rather an analog of Selberg's zeta function. The question of meromorphic continuation is however trivially related to by Dyatlov and Guillarmou [DG18] following a long series of precedent works, that prove meromorphic continuation under additional assumptions [Rue76,PP83,Fri95,GLP13b,DZ16,DG16] (We refer to [Bal18,Zwo18] for a recent overview of the literature). In all the recent accounts [GLP13b,DZ16,DG16,FT17,DG18] of these meromorphically continued zeta functions, a meromorphically continued resolvent was the central ingredient. Consequently, Theorem 1 indicates 7 that the Smale conjecture could hold for geodesic flows on cusp manifolds, i.e. beyond the class of Axiom A flows. So far such a result is only known in the particular case of constant negative curvature where it is a rather direct consequence of Selberg's trace formula.
Contrary to the "classical" Ruelle-Pollicott resonances, the definition of "quantum" resonances of the Laplace Beltrami operator Δ on a cusp manifold has been established for a long time starting with works of Maaß [Maa49] and Selberg [Sel69]. See the introduction of [LP76] for the constant curvature case, and [CdV81,Mül83] for the variable curvature case. In fact the proof of our main result borrows ideas from the definition of quantum resonances (such as the compact Sobolev embedding, Lemma 4.12).
Let us shortly sketch the further ingredients for proving the meromorphic continuation of the resolvent to the whole complex plane (Theorem 1): As a first step we construct a family of anisotropic spaces that are adapted to the hyperbolic structure of the flow. These are Hilbert spaces of distributions on , and ∞ ( ) is dense in each . They are an adaptation of the spaces defined by Faure-Sjöstrand [FS11] and Dyatlov-Zworski [DZ16]. Using a mix of their techniques we obtain much in the same way a first parametrix, which inverts − up to a smoothing remainder. However, this parametrix is -contrary to the compact case -not sufficient for a meromorphic resolvent. Therefore, it was necessary to introduce another technique. We chose to use ideas from Melrose's b-calculus to deal with the explicit form of the generator in the cusp. From the very nature of these techniques, they work independently of the dimension of . It is not entirely clear whether the technique could be applied or not to the case that the curvature is not exactly equal to −1 in the cusps, only close to −1. However, by analogy to the resonances of the Laplace operators we conjecture that the meromorphic continuation to the full complex plane will not hold true when assuming only pinched negative curvature in the cusps.
Let us present the structure of the paper: In Section 1 we introduce the precise settings in which we are working and collect several properties of the geodesic flow on cusp manifolds, that will be crucial in the sequel. To prove our theorem, we then build a first parametrix in Section 2.2 following the arguments of [FS11,DZ16]. The geometric construction of the escape function is presented; however, the technical microlocal lemmas are proved in Appendix A. Section 3 is devoted to introducing techniques adapted from b-calculus and proving the meromorphic continuation of the resolvent of a certain class of translation invariant operators. These operators show up precisely when restricting the geodesic flow to the zeroth Fourier mode in the cusp. In Section 4 such a resolvent is used for the construction of a parametrix (up to compact remainder) of the geodesic flow vector field. Then, using analytic Fredholm theory we conclude on the meromorphic continuation announced in Theorem 1. In Section 3 and 4 we work in a more general setting under a list of assumptions. This should allow for an easy generalization to more general settings (such as fibred or complex hyperbolic cusps) in the future. In Section 5 we finally compute explicitly the indicial roots for the b-operators associated to the geodesic flow on our class of cusp manifolds and we check that all the necessary assumptions in Section 3 and 4 are fulfilled.
Note that in fact we prove more general and more precise versions of Theorem 1. For example we continue the resolvent for a certain class of derivations on vector bundles (cf. Definition 1.4) including the geodesic vector field with smooth potential, Lie derivatives on perpendicular -forms and general associated vector bundles over constant curvature manifolds (cf. Examples 1.5-1.7). Furthermore we give a precise description of the wavefrontset of the resolvent. For a full statement we refer the reader to Theorem 3 and 4.
Acknowledgement We thank Frédéric Faure, Sébastien Gouëzel, Colin Guillarmou, Luis-Miguel Rodrigues, Gabriel Rivière and Viet Dang and the referee for helpful remarks and discussions. We acknowledge the hospitality of the Hausdorff Institute of Mathematics in Bonn where part of this work was done. TW acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) through the Emmy Noether group "Microlocal Methods for Hyperbolic Dynamics"(Grant No. WE 6173/1-1) 1 Geometric preliminaries

The geodesic flow on cusp manifolds
Let us give a precise definition of the manifolds on which we are working. Definition 1.1. A manifold will be called an admissible cusp manifold if the following assumptions hold. First, ( , ) is a ( + 1)-dimensional Riemannian manifold, connected and complete when endowed with the corresponding Riemannian distance. Second, it decomposes as the union 0 ∪ 1 ∪ ⋯ ∪ . 0 is a compact manifold whose boundary 0 is a finite disjoint union of -dimensional torii. At each component = 1 … of 0 is glued the hyperbolic cusp , which takes the form (1.1) (Here, Λ is a lattice in ℝ , and we can impose the normalization condition that it is unimodular). We require that the metric has strictly negative curvature in the whole of , and additionally, we fix for each = 1 … | = 2 + 2 2 . (1.2) Then the sectional curvature is −1 in each cusp, and the volume of is finite. Since the sectional curvature of is pinched, we deduce that its geodesic flow is a uniformly hyperbolic flow 8 on its cosphere bundle. More precisely we have: Proof. Let̃ be the universal cover of . It is a simply connected, complete Riemannian manifold with pinched negative sectional curvature (− 2 < < − 2 < 0), because the noncompact ends are endowed with a constant negative curvature metric. For the same reason all derivatives of the sectional curvature are bounded. Thus Theorem 7.3 and Lemma 7.4 in [PPS15] apply to this situation and they provide the splitting into invariant bundles over * ̃ with the above properties. As the invariant bundles are invariant under isometries, taking the quotient we obtain the desired result.
For the proof of Theorem 1 it will be crucial to have a precise understanding of the geometry and the dynamics on the noncompact ends of * . We therefore start by introducing explicit coordinates on * . In order to simplify the notation we will drop the indices = 1 … that number the cusps.
Recall that a cusp is = [ , ∞[×ℝ ∕Λ and since we have assumed that Λ is unimodular, we have canonical coordinates ∈ [ , ∞[, ∈ ℝ ∕Λ. In many cases it will be convenient to perform the change of variables = log ∈ [log , ∞[ and the metric becomes = 2 + −2 2 . (1.4) A single cusp has the local isometry pseudo-group given by ℝ × ℝ which is realized by linear scaling and translations in the , variables , 0 ( , ) ∶= ( , Using the , variables we can write ∈ * , as = + for ∈ ℝ and ∈ ℝ and the Riemannian norm of such a cotangent vector is given by (1.7) Elements ∈ * , of a cosphere fibre are thus in bijection with ∶= ( , ) ∈ ⊂ ℝ +1 . In particular the cosphere bundle over the cusp is trivializable * ≅ ( , ) × . The usual metric on * , the Sasaki metric (see e.g. [GK02] for an easily accessible introduction), is not a product metric. However, one can check (see the expression of the Sasaki metric in [Bon16, Section C.2]) that it is equivalent to the product metric ⊗ where is the usual metric on the sphere. We will use the product metric in the sequel. For the study of the geodesic flow some more precise variables on the spheres are useful. We choose a orthonormal base of coordinates 1 , … , in ℝ . We fix ( = 1, = 0) ≃ −1 to be zenith, and −1 1 the azimuthal reference. With these conventions, a point ∈ * , is non-ambiguously determined by its inclination -the angle it makes with the zenith -and its azimuthal position, ∈ −1 which is determined by the choice of base in ℝ . As a point in ℝ +1 , = (cos , sin ) ≃ −1 cos + −1 sin ⋅ . We single out two important points, the North pole  ∈ with = 0 that corresponds to the cotangent element −1 = ∈ * pointing into the direction of the cusp and the South pole  corresponding to − −1 = − pointing perpendicularly to the bottom of the cusp.
The dynamics of the geodesic flow vectorfield is illustrated in Figure 2. Let us emphasize two important properties of the geodesic flow dynamics on * : (A) The Hamiltonian is independent of the variable, which implies that the corresponding momentum variable is a constant of motion under the geodesic flow.
(B) The dynamics of the variable ∈ ≅ * , is decoupled from the dynamics on . By property (A) this dynamics is even rotationally invariant around the axis through  and  and it is precisely the gradient flow on the sphere with the obvious height function. This has the following consequence for the dynamics of the geodesic flow on the cusp. Assume that trajectories stop when they reach the lower boundary = . Then the only wandering trajectories are those with =  or = . They correspond to the geodesics that leave or enter the cusp, parallel to the -axis. All other trajectories only rise up to a finite height into the cusp and are thus "trapped". However, by chosing arbitrary close to  this height can be made arbitrary large and the trapped set is noncompact. Since the non-compactness of the trapped set is the central problem in extending the techniques of [FS11,DZ16], these regions around  and  will become crucial in the analysis.
Finally let us add a third remark that is not directly related to the dynamics of the geodesic flow, but rather to its action as a differential operator.
(C) As the geodesic vector field commutes with local isometries, it commutes in particular with the ℝ -action by translation and thus preserves the Fourier modes in the variable. If is an element in the dual lattice Λ * ⊂ ℝ then, restricting the geodesic flow vector field to the Fourier modes yields a differential operator = cos( ) + sin( ) + sin( ) ⋅ . (1.8)

Admissible vector bundles
As mentioned in the introduction we want to prove the meromorphy of the resolvent not only for the geodesic vector field acting on functions but also for a larger class of admissible vector bundles. In order to precisely define these admissible vector bundles let us first recall how to write the noncompact ends * as locally homogeneous spaces: Given a cusp = [ , ∞[×ℝ ∕Λ we will consider the associated full cusp to be the space , = (ℝ + ) × (ℝ ∕Λ ) with the metric defined in Equation (1.2) extended to , in the obvious way. Let = ( + 1, 1), then using the Iwasawa decomposition we can write = ℕ , where = (ℝ + , ⋅), ℕ = (ℝ , +) are abelian groups and = ( + 1) is the maximal compact subgroup in . Then a full cusp is simply the double quotient This sub-bundle of Λ( * ) is invariant under  . Also,  preserves the Liouville one form , which is a contact one-form. In particular, we can identify the action of  on Λ ⟂ ( * ) with the action of  on Λ((ker ) * ). This is also an admissible lift of .

Anisotropic space and first parametrix
The main idea that was presented in [FS11] was to resort to usual semi-classical techniques to prove the meromorphic continuation of the resolvent of the flow generator for Anosov flows on compact manifolds. This is not the only method available for compact manifolds -see [BL07] -but it is the one we will extend to our case. Another paper [DZ16] used propagation of singularities to obtain the wavefront set of the resolvent, in order to simplify the proof of meromorphic continuation of the zeta functions. We will use a mixture of both, since we use the approach of [FS11] to continue the resolvent, and ideas from [DZ16] to obtain the wavefront set of the resolvent. We consider → = * an admissible bundle and  an admissible lift of the geodesic flow on . Since we will use semi-classical techniques, we introduce a small parameter 0 < ℎ ≤ ℎ 0 , and we let ∶= ℎ. We refer to Appendix A where we collect the definition of the notions of microlocal and semiclassical analysis (pseudodifferential operators, symbol classes, . . . ) which we will use in the sequel.
The first result in this section is: Proposition 2.1. For each > 0, and ℎ > 0, we can build a space of -valued distributions on that contains ∞ ( , ), and a pseudo-differential operator microsupported in an arbitrarily small neighbourhood of the zero section in the fibers of * , so that there exists ℎ 0 > 0, so that for 0 < ℎ ≤ ℎ 0 , and for |Im | < ℎ −1∕2 and Re( ) > − , As ℎ varies, the spaces remain the same as vector spaces, with equivalent norms.
The space will take the form (see definition 2.7) In this formula, denotes a so-called escape function, and Op a semi-classical quantization that we define in Appendix A (see equation (A.4)). The construction of will be done first for acting on functions. Then the general case is obtained by tensorizing Op( − ) with the identity ∈ End( ).

Remark 2.2.
As should be clear after reading the proof, the construction of the escape function is local in the sense that it can be done in the universal cover. In particular, Proposition 2.1 should hold in any geometrically finite negatively curved manifold whose universal cover has bounded geometry. We do not prove this general result because that would require the construction of an explicit quantization with uniform bounds on these noncompact spaces. This seemed too much a detour considering that our aim is to study cusp manifolds and that a suitable quantization in this setting has already been developed by the first author in [Bon16].

Building the escape function
In this subsection we want to construct an escape function in complete analogy to [FS11, Lemma 1.2]. As we deal with a noncompact situation we however have to take care that the required uniform bounds hold. The escape function will be a function on the cotangent bundle * and we introduce the decomposition * = * 0 ⊕ * ⊕ * , (2.1) so that * 0 = ℝ where is the Liouville one-form − ⋅ . Furthermore * = ( ⊕ 0 ) ⟂ and * = ( ⊕ 0 ) ⟂ .
We have to introduce some notations regarding the dynamics. We lift the geodesic flow symplectically to the flow It is the Hamiltonian flow associated to the Hamiltonian which is the symbol of − and we denote by Φ its hamiltonian vectorfield. The decompositions (2.1) is preserved by the flow, and (iv) ∈ log ( ): it is an anisotropic symbol of order ( , ), with ∈ 0 ( ) being a 0-homogeneous classical symbol (see Definition A.6 for a definition of classical symbol classes) with In order to prove Lemma 2.3 it will be helpful to restrict Φ to the unit sphere bundle * . In order to do this, let us interpret * ≅ ( * ⧵ {0})∕ℝ where ℝ acts on each fibre by linear multiplication. Then, by homogeneity, Φ factors to a flowΦ ∶ * → * with vector field Φ . By an abuse of notations, we can see * 0 , * and * as subsets of * . From the uniform estimates in Proposition 1.2 we obtain: Finally for any > 0 there is ′ > 0 such that ′ ⊂Φ ( ) and ′ 0, ⊂Φ − ( 0, ). The same statement holds for * 0 ⊕ * and * . proof of lemma 2.3. Let us first construct the weight function . This decomposes into two symmetrical steps.
The actual weight function will be multiplied by a constant, that we will determine at the end. Now comes the second part of the proof: building the symbol . Choose (resp. , 0 ) to be the cone in * generated by ⊂ * (resp. , 0 ). We want to choose a symbol ∈ 1 ( ) to be a positive elliptic symbol, so that outside of | | < , on 0 it equals | |. We also want that on (resp. ) it satisfies Φ log ≥ ∕2 (resp ≤ − ∕2). We would like to set to be just the norm | | in a neighbourhood of * ⊕ * , but this is not suitable because the constant in the estimate (2.3) is not necessarily 1. However, we find that for ( , ) ∈ * , This suggests to pick ′ > 2 log( )∕ and define for ( , ) in a fixed conical neighbour- This is not a norm anymore, but is still 1-homogeneous and smooth -except at 0. On * , log ≤ −3 ∕4, so that if > 0 was chosen small enough, log ≤ − ∕2 in . We also have the corresponding estimates in . We can piece together and | | around 0 to obtain a globally defined elliptic 1-homogeneous symbol. Let be its infimum on We have all the pieces to define ′ > 0 is a constant fixed later and is a ∞ (] − 1, 1[) function, that equals 1 in [−1∕2, 1∕2] and takes values between 0 and 1. It is there to ensure that is smooth at = 0. We can check that ≥ 0. By the properties of from above, we directly deduce that Lemma 2.3(iii) holds. Now, we can compute Φ = − ′ ( Φ ) log 2 Let us discuss the different terms: • If ( , ) ∉ ( 0 ∪ ∪ ): Note that | | and | Φ | are globally bounded by a constant 0 . By property (c) above Φ > 1. By the fact that is elliptic, there is a constant > 0 such that when {| | > }, log(2 ∕ ) > 1 + 2 0 . Then for | | > , we also have log(2 ∕ ) > 1 + 2 0 , thus Φ > ′ for | | > • If ( , ) ∈ : Now we only know that Φ ≥ 0, so we need a uniform lower bound for the second term. But from the choice of , it is precisely there that Φ ∕ > ∕2. Together with the property (e) of above, we deduce Φ > ′ ∕2 for • If ( , ) ∈ : As in the previous case, we obtain Φ > ′ ∕2 for | | > .
• If ( , ) ∈ 0 : As is a function of on 0 and Φ is the Hamiltonian flow of , Let ′ (resp. ′ ) be the conical neighbourhood corresponding to ′ (resp ′ ). We have ′ ⊂ and ′ ⊂ . On ′ (resp. ′ ), = 2 ′ log | | + (1) (resp. −2 ′ log | | + (1)). So we choose ′ ≥ max( 2 , 1). This gives = 2 ′ , and = ′ . At last we have to verify that and are symbols in the right class in the sense of Definition A.6. The weight was constructed as a ∞ function on * , and that is the definition of being in 0 ( ). For to be elliptic in log ( ), it suffices then to check that (1 − (| |)) itself is elliptic in 1 ( ). By definition, this means that ∕| | is a ∞ function on * . That is also a direct consequence of the construction.
Actually, in our case, we can say something a little better, that will simplify the rest of the proof. Proof. Recall from the discussion in Section 1.2 that each cusp can been seen as a subset of the full cusp , = Λ ∖ ∕ . The geodesic flow on the hyperbolic space ∖ or rather on its sphere bundle ( ∕ ) = ∕ is known to be uniformly hyperbolic with analytic stable and unstable bundles̃ ∕ which are invariant under all isometries of the hyperbolic space ∕ i.e. under the left action. Consequently, these bundles descend to the full cusp , and can thus be restricted to the cusps. We call them the stable and unstable bundles corresponding to constant curvature and denote them by ∕ . By the invariance under isometries of hyperbolic space, the bundles ∕ are invariant under the local isometries , 0 defined in equation (1.5).
Let us now explain that ∕ and ∕ become (1∕ ) close high in the cusp. Let us do this for the example of : First note that ⊕ = ⊕ (this is because the contact form of both flows coincides. Now for trajectories whose past is included in the cusp, and have to coincide, so at the bottom of the cusp ( = ) directions that are close to the South pole (i.e incoming trajectories), and are transverse (by continuity of the bundles). Now high in the cusp ( ≫ ) in an arbitrary direction (except in  ), its trajectory, when it exits the cusp has to be almost vertical, i.e. in the neighbourhood of the south poles considered above; Now the uniform hyperbolicity of both splittings implies that and are (1∕ )-close as → +∞. As a consequence to the fact that the bundles ∕ and ∕ become close, when building the functions 0, and 0, , we can actually choose them to be invariant by , 0 high in the cusp -say > 0 .
Since it takes at least a time ∼ log to go from height in the cusp to the compact part The last thing to check is the invariance of . In the cusp, the vector field is also invariant under local isometries of the hyperbolic space, so that also can be chosen to be , 0 invariant for > 0 ′ .
Remark 2.6. We can chose so that it coincides with the in Definition 4.3. It will be smaller than the of point (7) of Proposition A.8.

A first parametrix
Now that we have built our escape function, we focus on building an approximate inverse for − ℎ . Recall that we use semiclassical analysis: We had defined = ℎ and we will work with the semi-classical quantization Op ℎ, acting on sections of , see Appendix A eq. (A.4). For a simpler notation we simply write Op in the sequel. A priori for Op( ) to make sense, we need that is valued in End( ). If is just a function, we can consider Op( ⊗ ). This operator will be denoted by abuse of notations just as Op( ).
Definition 2.7. Let > 0 and the corresponding escape function given by Lemma 2.3. Let > 0. We denote by the set of distributions It is endowed with the norm The space actually does not depend on ℎ or on , but the norm does. As a convention, we denote 0 = 2 ( , ).
We will drop the indices in the notations, to lighten a bit the presentation, and just write (= ). Only at the end of section 4 in the proof of Theorem 3 will we let go 0. From the properties of , we directly obtain the following regularity properties, which show that is an anisotropic space 9 .
Lemma 2.8. For any > 0, > 0 we have the continuous inclusions + ⊂ ⊂ − . Furthermore near * , is microlocally equivalent to and near * , is microlocally equivalent to − . In particular, for ∈ 0 ( , ) (2.5) The differential operator , which is a priori defined on ∞ ( , ) has a unique closed extension[FS11, Lemma A.1] to the domain ∶= { ∈ ∶ ∈ }. The domain 9 The spaces that show up here are distributions that are regular in the * direction and irregular in the * direction. This is no contradiction to precedent works like e.g. [DG16] where the authors continue the resolvent of the operator − and thus obtain the converse regularity properties.
We deduce that ‖( − ℎ ) ‖ ≥ ℎ‖ ‖ . As a consequence, the image of ( − ℎ ) is closed. We deduce that it is the orthogonal of the kernel of the adjoint. We also get that the kernel of ( −ℎ ) is empty. Additionally, we observe that the adjoint of −ℎ satisfies the same sharp Gårding estimate, so that it also is injective, and thus ( − ℎ ) is surjective. We conclude that it is invertible.
For each > 0, we pick (= ), a self-adjoint semi-classical pseudo-differential operator, of the form Op( ), with ∈ 0 equal to 1 in {| | ≤ 2 }, everywhere positive, and supported in {| | < 3 } -the constant was given in Lemma 2.3. This is an absorbing potential. Let us denote by Then we have the key estimate: Proof. We fix a tempered family of functions ∈ ∞ ( , ). We consider the regions and If > 0 is chosen small enough they overlap, so we can build a partition of unity 1 = ell + Gårding , with ell (resp. Gårding ) microsupported in Ω ell (resp. Ω Gårding ), and both 's in Ψ 0 .
In the region Ω ell , the principal symbol of ( ) is elliptic, so we deduce 10 from Proposition A.14.
Now, we can concentrate on the region of interest Ω Gårding . By definition, the action of ( ) on is conjugated by Op( − ) to the action on 2 of We denote bỹ Gårding the operator obtained after conjugation by where 2 is a microlocal cutoff in a slightly bigger neighbourhood of WF ℎ ( Gårding ) and Here, Re is the real part of acting on 2 , and it is an (ℎ) order 0 operator). By Lemma 2.3(i) (recall that { , } = Φ ) we conclude that 2 ∈ Ψ 0+ has non-negative principal symbol and by the sharp Gårding inequality A.10, we deduce that The constant depends on , which depends itself on . Using Cauchy-Schwarz and our assumption Re( ) > − + 1 we get

Now let us consider
10 Note that Proposition A.14(2) is stated in terms of ordinary Sobolev spaces and not in terms of anisotropic Sobolev spaces. The statement on anisotropic spaces can however be deduced by applying Proposition A.14(2) to the conjugated operators Op( − ) −1 ell Op( − ) and Op( − ) −1 ( − − ℎ ) Op( − ) respectively. Note therefore that the conjugation does not affect the ellipticity. This estimate implies that for sufficiently small ℎ, the operator ( ) is injective and has closed range. Performing exactly the same estimates for the adjoint operator, we deduce that ( ) is surjective.
In the case of compact manifolds, the end of the proof of the equivalent of Theorem 1 is based on the fact that by writing we have that − ℎ is invertible modulo a smoothing operator, and smoothing operators are compact on compact manifolds, so − ℎ is invertible modulo compact operator. Hence it is Fredholm, of index 0, and its inverse is a meromorphic family of operators in the parameter.
However, in our case, smoothing operators are not compact. We will present a special ingredient in the next section to overcome this problem. Before that, we consider wavefront sets.
Let us prove that ℛ ( ) ′ =  −∞ → ∞ (ℎ ∞ ). Let be a tempered family of distributions. Let > 0, and , 1 elliptic on respectively Φ ( ) and ∪ 0≤ ≤ Φ ( ). Observe that ℛ ( ) ′ is in all Sobolev spaces because , ′ are compactly microsupported. Then we get by Propagation of Singularities A.16 that for ∈ ℝ By the assumption on the microsupport of and ′ , by taking the microsupport of 1 small enough, we can ensure that 1 Now, we just have to consider what happens when the time becomes larger. For ( , ) ∈ { = 0} ⊂ * there are only two possibilities: Either there is converges to * ∩ * (see Definition A.6 for the radial compactification).
In the first case take sufficiently small such that Φ ( ) ⊂ ell 1 ( ). Thus we can assume that in the propagation estimate (2.7) is microsupported in ell 1 ( ). Taking ′ ∈ Ψ 0 elliptic on the microsupport of , the elliptic estimate (Proposition A.14) gives, Since we can choose ′ such that WF( ′ ) ∩ WF( ′ ) = ∅, the RHS is (ℎ ∞ ). Now, we turn to second case which we will treat using the high regularity radial estimate (Proposition A.18): Note that * ∩ * is a sink in the sense of Definition A.17. Next let us choose ∈ Ψ 0 such that * ∩ * ⊂ ell 1 ( ) and such that WF ℎ ( ) ∩ WF ℎ ( ′ ) = ∅. Then Proposition A.18 provides us with an order 0 operator 1 which is elliptic in a neighbourhood of * ∩ * . Furthermore we can assume WF ℎ ( 1 ) ⊂ .
Since 1 ℛ ′ ∈ and is microsupported in , by Lemma 2.8, we know that 1 ℛ ( ) ′ ∈ and taking > 0 we have the necessary regularity for the sink estimate. We get for any > 0 Finally for sufficiently small and by propagation of singularity for a long enough but finite time we can assume that Φ ( ) ⊂ ell 1 ( ). Combining (2.7) and (2.8) we obtain as desired ‖ ℛ ( ) ′ ‖ = (ℎ ∞ ). We have a final remark for this section Definition-Proposition 2.12. If ≥ 0 and ∈ ℝ, we say that = + is a weight. Such a weight is said to be large if is large, and ∕ is small. We define We get that the conclusion of Proposition 2.10 holds on the space when |Im | < ℎ −1∕2 , Re ≥ − + + 1 for some constant independent of , , and for ℎ > 0 small enough.
The proof is completely analogous to the proof of proposition 2.10.

Continuation of the resolvent for translation invariant operators
In this section, we will be considering a vector bundle over some compact Riemannian manifold → , endowed with a bundle metric and a compatible connection. We will always see the space ℝ × as a fiber bundle over (ℝ) × ( ) , endowed with the product structure. We will also use the natural measure , and 2 (ℝ × ) will be understood as the space of square-integrable sections with respect to this measure.
Let us first see how bundles of this type can be naturally obtained from admissible vector bundles in the sense of Definition 1.4.
In Section 4 we will study sections of these bundles that are independent on the variable ∈ (ℝ ∕Λ ) and these sections are naturally identified with sections of ℝ × ( × ). This shows that studying -independent sections of admissible vector bundles Remark 3.2. For the proof of Theorem 3 on vector bundles one could restrict the discussion of the whole section to the special case in the example above. As all arguments, however, hold without any further complications in the general case of vector bundles → over general compact manifolds we announce and prove all results in this section in this setting. Additionally, we expect that this wider class is likely to show up when studying uniformly hyperbolic flows on fibred cusps.

b-Operators
We will consider a particular class of operators on ℝ × → ℝ × : Recall that by the Schwartz kernel theorem any continuous linear Operator ∶ ∞ (ℝ × ) →  ′ (ℝ × ) is represented by its kernel ∈  ′ (ℝ×ℝ× ⊠ ). We call such an operator a convolution Definition 3.3. The set of semiclassical pseudo-differential operators acting on sections of ℝ × that are convolution operators in the variable will be denoted by Ψ (ℝ × ). It is the set of b-operators.
Such operators that additionally are supported in {| − ′ | ≤ log } will be denoted Ψ , (ℝ × ). We say that they are b-operators with precision . When = 1, the kernels are supported on { = ′ }.
Remark 3.4. Our notion of b-operators is, as its name suggests, strongly inspired by Melrose's b-calculus (see e.g. [Mel93]). However in this article we use a much more restrictive class of operators. Let us shortly explain the relation of our b-operators to the usual class of b-differential operators in the sense of Melrose. Let [0, ∞[ ×ℝ be the simplest model of a manifold with boundary. Then the b-differential operators are those in the algebra of operators generated by b-vectorfields that take the form ( , ) . Using a Taylor expansion, the leading order near the boundary of these operators takes the form 0 ( ) + 0 ( ) . After a variable transformation = log( ) these are in the form 0 ( ) + 0 ( ) . Such operators are then translation invariant in the variable, i.e. are convolution operators. Their kernels take the form In some sense our class of b-operators contains just those which are equal to their leading part in the asymptotic expansion near the boundary of the usual class of b-(pseudo)differential operators. For our purpose this is sufficient and the restriction to this class allows us to concentrate on the difficulties that arise from the fact that we have to construct a parametrix for an operator that is not elliptic (even in a b-calculus sense).
Example 3.5. The generator of the geodesic flow acting on functions supported in a cusp and not depending on is a differential operator acting on the trivial bundle, i.e. on 2 (ℝ× , − ) given by (cf. equation (1.8)) 0 = cos + sin .
In order to make it a b-operator acting on 2 (ℝ × , ) we conjugate it with − ∕2 and get: In order to work in the semiclassical calculus we write ∶= ℎ .
The aim of Section 3 is to show that the resolvent of can be continued meromorphically from Re( ) > 0 to ℂ. In fact, for the reasons discussed in Remark 3.2, we will treat a more general class of operators ∈ Ψ ,0 (ℝ × ) whose precise assumptions will be formulated in Section 3.2 Next, let us introduce symbols and quantizations that lead to b-operators Definition 3.6. Denote by the metric on and by ( * ) = ⊕ the splitting into vertical and horizontal directions w.r.t. the Levi-Civita connection. We endow * (ℝ × ) with the metric described in Definition A.1. Consider its restriction to ( * 0 ℝ) × ( * ) ( , ) . It can be expressed as By Lemma A.2, has bounded geometry. of and a trivialisation ∶ pr −1 → ( ) → × ℝ dim( ) as well as a quadratic partition of unity ∑ 2 = 1, ∈ ∞ ( , ℝ ≥0 ) such a quantization can be written for ∈ ( ) by where Op ℎ,ℝ dim is the usual Weyl quantization on ℝ dim . Now we can use Op ℎ, to define a quantization of b-symbols ∈ (ℝ × ) on ℝ × by which yields an element of Ψ (ℝ × ). If additionally, we choose a smooth cutoff supported in ] − log , log [, equal to 1 in ] − log 1∕2 , log 1∕2 [, we can multiply the kernel of Op ( ) by ( − ′ ), and obtain an operator From this expression we see that all usual properties of quantizations, such as composition formulas, 2 estimates, sharp Gårding inequalities etc that hold for the quantization on ℝ dim +1 (see e.g. [DZ19, Appendix E]) directly transfer to Op ( ). The same holds for Op ( ) because the cutoff away from the diagonal modifies the operator only be an element of ℎ ∞ Ψ −∞ .
Remark 3.8. We will see in Proposition 4.11 that there will be a method to construct bsymbols from any symbol ∈ ( → * ) which is invariant by the local isometries of the cusp , . (Recall that e.g. the escape function had this property).
• is elliptic in ,log for some ∈ 0 .
• = ℎ where is a differential operator independent of ℎ.
Example 3.10. We will see in Section 4 that , and defined in the previous section 2.2 will give rise to a an admissible triple after restricting to -invariant sections. The constant ′ is just , when > 0 is small enough.
Definition 3.11. As in Def-Proposition 2.12, we say that ∈ 0 is a weight if it is of the form + . When we say that a weight is large, it means that > 0 is large, and that ∕ is arbitrarily small. Given a weight and ∈ ℝ, we will work with the space of -valued distributions on The main result in this subsection is the following: Proof. We can apply the same arguments as in the proof of Proposition 2.10. Note that in the positive commutator part, which uses the sharp Gårding inequality, it is important that the real part of (3.5) Here it is crucial that the absolute value of the second line in (3.5) is bounded by (| | + | |) -it would not be the case a priori replacing log⟨ ⟩ by ′ log⟨ ⟩ where ′ ∈ 0 .

The indicial family
For the following constructions it is useful to bear in mind the elementary method of invertion of convolution operator on ℝ. Consider some ∈  ′ (ℝ) compactly supported and the operator ∶ ↦ * . Obviously, the Fourier transform of is just̂ ̂ . To invert , is suffices then to invert̂ ↦̂ ̂ . Our aim is to invert the b-operators introduced in section section 3.1 which motivates us to introduce an analogon to the above appearing Fourier transform: Let ∈ Ψ , (ℝ × ) and ∈ ∞ ( ). For ∈ ℂ, we consider ∕ℎ ( ) ∈ ∞ (ℝ × ). By the support properties of the kernel, is a properly supported pseudodifferential operator and thus defines a continuous operator on ∞ (ℝ × ). Moreover, by the fact that is a convolution operator, ( , ) ↦ − ∕ℎ ( •∕ℎ (•))( , ) is independent of , thus ↦ − 0 ∕ℎ ( •∕ℎ (•))( 0 , ) is independent of 0 and is a well defined smooth section of .
Example 3.14. If is obtained from the geodesic flow on a cusp, i.e. is the operator in equation (3.1), the corresponding indicial family is ( , ) = cos + ℎ 2 cos + sin .
Note that an equivalent description of the indicial family is the following: Fix ∈ ∞ (ℝ) with ∫ ( ) = 1, then the indicial family is the family of operators ( , ) ∶ ∞ ( ) →  ′ ( ) such that for any 1 , 2 ∈ ∞ ( ): This expression is helpful for two purposes: First, by taking complex derivatives of the right hand side with respect to we conclude: Secondly it allows to extend the definition of the indicial families to convolution operators on ℝ × that fail to be in Ψ , . Note that we will work with nonelliptic problems, thus the appearing inverse operators (like for example ( − − ℎ ) −1 from Lemma 3.12) will not be pseudodifferential operators, so it will be crucial to have the following extended definition.
Now, we can define spaces on → : Definition 3.18. Let = + be a weight. We denote by + the space Let us discuss the subscript in the notation of the spaces + . It may seem that these spaces depend on the parameter , and since we want to consider analytic families of operators depending on the parameter , this may be problematic -recall that for the theory in Kato [Kat95] to apply, we need that operators are of type (A), which basically means that they all act on the same domain. To address this problem, we start with It suffices to check that the operators , ′ ) are bounded on 2 for , ′ ∈ ℂ (and depend continuously on , ′ ). Since they are pseudodifferential, and their symbols have the same asymptotics for large (see (3.10)), this is a consequence of usual pseudo-differential arguments.
Then, for ∈] 0 , 1 [, there is a continuous operator ( , . The resulting operator does not depend on , so we denote it by ( ). In the case that ( ) = ( , ) for ∈ Ψ , or some as in Lemma 3.16, we get that ( ) = .
Proof. Let us first check that the given kernel defines a well defined continuous operator ( , ) ∶ ∞ (ℝ × ) →  ′ (ℝ × ): The expression of the kernel means that for 1 , 2 ∈ ∞ ( ), 1 , 2 ∈ ∞ (ℝ) one has As the Fourier transform of compactly supported functions extend holomorphically to ℂ the independence from follows from Cauchy's theorem. The fact that ( , ) can be extended continuously to arbitrary (nonproduct) elements of ∞ (ℝ × ) can be seen by letting any of the 1 , 2 ∈ ∞ ( ) or 1 , 2 ∈ ∞ (ℝ) to zero in the corresponding topologies. Then the temperedness assumption of ( ) implies that (3.13) goes to zero. As to why ( ( )) = , this follows by Fourier inversion after plugging in the definitions (3.6) and (3.13) and a few lines of straightforward calculations. Also the multiplicativity ( 2 1 ) = ( 2 ) ( 1 ) follows from a straightforward calculation which is completely analogous to the calculations needed to show that the Fourier transform of a product is the convolution of Fourier transforms.
After this additional reduction, we are left to prove that This is just an avatar of the Plancherel formula: By the definition of ( ) (see (3.13)) one has for 1 , 2 ∈ ∞ ( ), 1 , 2 ∈ ∞ (ℝ): and from this formula (3.14) can be read off directly.
Finally, we get As a consequence, the family ( , ) is a type (A) family, so that we can apply the results from [Kat95].

Effective continuation
In this last subsection of Section 3, we want to establish a meromorphic continuation of ( − ℎ ) −1 . Before going on with the proof, let us come back to the convolution operator on the real line ∶ ↦ * , with ∈  ′ (ℝ) compactly supported. In the language above ( , ℎ ) =̂ (− ). Since is compactly supported,̂ is an entire function, and acts by multiplication on the whole of ℂ. Given ∈ ℂ, the function (̂ − ) −1 is a meromorphic function. So one can define for ∈ ∞ (ℝ) and 0 ∈ ℝ, Note that in the general notation from Definition-Proposition 3.21 we can identify after setting ℎ = 1, ( 0 , ) = (( ( , ) − ) −1 , ). One finds that ( − ) (0, ) = when is not in the closure of̂ (ℝ). By Cauchy's theorem, ( 0 , ) = ( 1 , ) when̂ (− ) does not take the value in the region Re ∈ [ 0 , 1 ]. Now, consider 1 ∈ ℂ such that (− 1 ) = ,̂ ′ (− 1 ) ≠ 0, and̂ (− ⋅) does not take the value another time in a region Re ∈]Re 1 − , Re 1 + [ for some > 0. Another application of Cauchy's theorem gives Using this argument, one can hope to obtain a meromorphic continuation of the resolvent of a translation invariant operator from the meromorphicity of the resolvent of its indicial family ( , ). This is done by replacing "multiplication" by "action in the variable". This heuristics is at the core of Melrose's b-calculus and will be pursued here. As one can expect, just as it is crucial to follow the solutions of̂ ( ) = for the convolution by , we have to follow the ( , )'s such that ( − ℎ , ℎ ) is not invertible. 1. For fixed ∈ ℂ, the set of ∈ ℂ such that ( − ℎ , ℎ ) −1 is singular is the set of ( -)indicial roots of . It will be denoted by Spec b ( ) and by construction it is independent of ℎ.
3. For affine roots say that a root ( ) = + is positive if > 0 (resp. negative if < 0) and we denote the set of positive/negative roots by Spec b ± ( ).
The remainder of this section is devoted to the proof of Theorem 2. We start with some observations. As a direct consequence of Lemma 3.24, we get: The second statement follows directly from considering (3.15) in the case = 0.
We call ∈ ℝ s-regular if Spec b ( ) ∩ ( + ℝ) = ∅. The above bounds on and imply that for any ∈ ℂ, the set of -regular ∈ ℝ is open and dense. Furthermore for a -regular , Lemma 3.24 implies that which is again bounded with norm (1∕ℎ). Indeed one directly checks that However, ( ) depends strongly on the choice of due to the fact that ( − ℎ , ) −1 has singularities, i.e. that there exist indicial roots. In order to understand the meromorphic continuation one has examine what happens if indicial roots cross the integration contours.
In order to shorten the notation in the sequel it is convenient to define seen as a meromorphic function on ℂ 2 taking values in convolution operators on ℝ× . The ℎ factor is actually chosen such that it becomes ℎ-independent and using the Definition of ( ) (Definition-Proposition3.21) we write (3.20) When freezing the variable, the poles of ( , ⋅) are precisely Spec b ( ). We can integrate over a small closed curve around a pole 0 , enclosing only 0 , and obtain its residue Res( ( , ⋅), 0 ) in the variable. With this notation, we can state an equivalent to equation (3.17): Lemma 3.30. Let −∞ < < ′ < ∞ be -regular for some ∈ ℂ. Then we have Spec b ( , , ′ ) is finite and Proof. That the sets are finite follows from the uniform estimates on , . The identity is a consequence of Cauchy's theorem and (3.20).
The following lemma is crucial in the proof: Proof. Since we already know that we can parametrize the roots without algebraic singularities -in the words of Kato, there is no branching point -this is a direct consequence of Theorem 1.8 in [Kat95,p.70] Now we can come back to the proof of our theorem.
Proof of Theorem 2. First, we focus on the meromorphic continuation of the Schwartz kernel of the resolvent. Recall from Lemma 3.24 that there is such that ( − ℎ , ) is invertible for Re( ) > and in this half plane, we define If 1 < 0 < 2 are such that {Re ∈ [ 1 , 2 ]} does not intersect Spec b ( ) then we deduce that ( ) is bounded on all spaces  , for ∈ [ 1 , 2 ], given that the weight is large enough.
We want to construct a meromorphic continuation of ( ) to all ℂ and therefore we have to take care of the indicial roots that cross the contour at Re( ) = 0. We define the set of positive (resp. negative) visible roots at as Spec b + ( , −∞, 0) and Spec b − ( , 0, ∞), respectively (see Figure 3 for the case of the geodesic flow for cusps).
By the uniform bounds on , , we deduce that for any ∈ ℂ, there are finitely many visible roots.
Let be the set of ∈ ℂ such that 0 is -regular, i.e Spec b ( ) ∩ ℝ = ∅. For ∈ , we set As 0 ( ) is holomorphic on any connected component of and as ( , ( )) are meromorphic by Lemma 3.31 this defines a meromorphic family on . It remains to prove that we can patch the different connected components of (which are vertical strips because the roots are affine) together: Therefore take 0 such that 0 is not 0 -regular, we consider < 0 < ′ small enough such that Spec b ( 0 , , ′ ) ⊂ ℝ. Then, for in a small vertical strip around 0 , and ′ are still -regular. For ∈ we define The equality between the two expressions follows from Lemma 3.30. By construction and by Lemma 3.31 ( ) defines a meromorphic operator on the strip . It only remains to check that on ∩ both definitions of ( ) and ( ) coincide. But this is again a direct consequence of Lemma 3.30. We can thus patch the definitions to a globally meromorphic operator which we denote by ( ). Now we will determine on which functional spaces this meromorphic continuation acts. Let us focus on the structure of the residues of . If we assume that 0 is anindicial root, and that for > 0, there are no other indicial roots in { , | − 0 | ≤ }. In that case, We will need the lemma: Lemma 3.32. For > 0 and ∈ ℝ, we have the equality of spaces The corresponding norms are equivalent with (1) constants as ℎ → 0.
Proof. It suffices to prove that both are bounded on 2 (ℝ × ). However since the quantization is properly supported, these operators are pseudo-differential with symbols in 1 + (ℎ −1 + ). Hence they give rise to bounded operators on 2 (ℝ × ).
With 0 , as above we deduce If is the multiplication by −2 ⟨ ⟩ , the operator in the norm is the composition , so that is a convolution operator whose kernel takes the form Recall that Re > 1 + + (| max ( )| + | |) − and |Im | ≤ ℎ −1∕2 , so we can apply Lemma 3.24. In particular, the indicial operator in the last line is bounded on 2 with norm ( ). Since the kernel of decomposes as a product, we see directly that it is bounded from Re( − 0 ) − ⟨ ⟩ 2 to Re( − 0 ) + ⟨ ⟩ 2 . But since | − 0 | = , it is thus bounded from −2 ⟨ ⟩ 2 to 2 ⟨ ⟩ 2 uniformly in . Finally, since maps 2 to −2 ⟨ ⟩ 2 and 2 ⟨ ⟩ 2 to 2 , we obtain the desired result Else, if ∈ ℂ with Re( ) > such that there are visible roots, let us choose > 0 such that max Note that this is possible because we are in the case ( ) > 0 and thus, as was discussed after (3.18), is strictly monotonous. Now, combining Equations (3.21), (3.22) and (3.23), we deduce that To obtain the boundedness for ∈ ℂ ⧵ , one can use similar arguments. Consider a pole of ( ) corresponding to an indicial root crossing 0 . From the considerations above, it follows that the Laurent expansion has its image contained in the direct sum of (3.25) where 0 , … , are finite dimensional subspaces of 0 ( ), related to the images of the Laurent expansion of ( −ℎ , ) −1 around 0 . In particular, this is finite dimensional.
Note that in the case of a geodesic flow we will see in Section 5 that the resonant states of coming from the indicial resolvent can be explicitely expressed by dirac distributions and homogeneous distributions on the North and South pole of = .

Black box formalism and main theorem
In this section, we introduce a black box formalism in the spirit of [SZ91]. For the same reason as in Section 3 we work in a geometric setting that is more general then the admissible bundles → * from Definition 1.4. Again, this bigger generality comes without any additional effort in the proofs. Let us define the geometric setting of this section.   Let → be a general admissible bundle, with cusps 1 , … , . Take > , and let We have the orthogonal decomposition In Section 3, we used the measure instead of − . In particular, In Equation (4.2), the first term will be regarded as a black box and the second one as the free space. In the black box, we will use the variable (more appropriate for geometric purposes), and in the free space the variable (more appropriate for analysis). In the case of elliptic operators, one can really isolate the black box, because it can be embedded in another space where the relevant operator -mostly the Laplacian -has compact resolvent. However in our case, since being uniformly hyperbolic is a global property, such surgery cannot be performed a priori. It is the fact that the flow is exactly translation invariant that will save us.
We can define extension and restriction operators. Let ∈ ∞ ( , ). We let be the function in ∞ ([log , +∞[ × , ) obtained by restriction to the cusp and averaging in the variable. Conversely, let ∈ ∞ (] log , +∞[ × , ). We consider it as a function ℰ supported in cusp , not depending on . We have ℰ = . We extend these definitions to distributions by duality: for distributions ∈  ′ ( , ) and Note that after this extension we can apply ℰ equally to ∞ ([log , ∞[) and the compostion ℰ is well defined. Given a function ∈ ∞ ([log , +∞[) that is constant near , we can define the associated black box multiplication operator as the operator In this section, we will define a class of operators that preserve this structure, and review some of their properties. Then, we will conclude on the meromorphic extension of the resolvent of admissible such operators.

The class of cusp-b-pseudors
Now that we have added some structure to our space 2 ( , ), we need to determine a reasonnable class of operators that will preserve the structure. First consider a differential operator that commutes with and in each cusp, for > . It thus acts on the space of smooth functions supported in a cusp that do not depend . We denote by 0 , that restriction for each cusp . Then we find that for = 1, … , , acting on  ′ ( , ), We also have the dual statement, acting on ∞ (] , +∞[× , ): Since we want to use anistropic spaces that can only be defined using pseudo-differential operators, we have to accept slightly different relations. Indeed, pseudo-differential operators cannot be exactly supported on the diagonal. If additionally for each = 1, … , , ℰ acts on ∞ (] log( ), +∞[× , ) as the restriction of a translation invariant operator 0 , on sections of ℝ × , that is supported for | − ′ | ≤ log , we say that is a cusp-b-operator. We define which is again translation invariant. In this way, while 0 , acts naturally on 2 (ℝ × , − ), , acts on 2 (ℝ × , ).

Finally, if
∈ Ψ( , ) is also a pseudo-differential operator, we say that is a cusp-b-pseudor, and write ∈ Ψ , ( , ). In what follows the constant will be fixed a priori, it is a geometric data of the problem, and we will mostly not mention it. Let us give a word of explanation. Condition (4.4) implies that if has zero mean value in the variable in each cusp for > , then the mean value of in the variable vanishes when > . The condition (4.5) is the dual version of the assumption: it means that if was supported only in cusps for > , and did not depend on , then would be supported in > , and also not depend on . (4.6) Assume that , does not depend on for = 1 … . Then we say that is a b-symbol of order 0 and write ∈ 0 ( , ). Given a cusp-b-symbol of order 0, we correspondingly define ( , ) the set of cusp-b-symbols of order .
By a direct computation, one gets: We also get Proof. From the considerations in Section A.1.1, we deduce that , satisfies usual symbol estimates on ℝ × . The -invariance follows from the definition.
Let us consider ∈ ( , ) and the corresponding operator Op( ). According to Proposition A.8, by adjusting the parameter ≥ , we get that Op( ) satisfies Equation (4.5). It is not difficult to check that it also satisfies Equation (4.4) for similar reasons. We now consider its restriction to functions supported in the cusp and not depending on . Actually, we want to compute directly {Op( )} , instead of {Op( )} 0 , . Thus we take a function of the form ∕2 ( , ), so that the action of Op( ) on 2 ( ) -with the measure − -will correspond to the action on 2 (ℝ × , ). By definition of the quantization -see equations (A.2) and (A.3) -and already replacing (2 ℎ) − ∫ ⟨ − ′ , ⟩∕ℎ ′ by =0 , we get for − ∕2 Op( ) ∕2 : (recall is the dimension of ). We take the coordinate change = log and = ( + ′ ) ∕2. The volume form becomes wherẽ = . Since , does not depend on , we deduce that Provided that the support of the cutoff chosen after Equation (3.3) is slightly larger than the support of Op , we can find a symbol̃ , ∈ (ℝ × ) such that We have proved

Meromorphic continuation of resolvents of admissible b-operators
In this section, we will need the crucial compactness lemma: Lemma 4.12. Let → be a general admissible bundle as in Definition 4.3. Let > and This is a closed subspace of 1 ( , ) and the injection 1 ( , ) ↪ 2 ( , ) is compact.
To show that it is a norm limit we have to show that for ∈ 1 ( , ), with a constant → 0 as → +∞. We use the Poincaré inequality: consider a unimodular lattice Λ ⊂ ℝ and Λ = ℝ ∕Λ. For̃ ∈ 1 ( Λ ) with ∫̃ = 0, we have Now, with the number of cusps, We have a statement for general weights.
Definition-Proposition 4.13. We pick some smooth function ′ ( ) equal to log for ≤ log , and equal to when > log 2 . Then, given , , ∈ ℝ and the corresponding black box multiplication operator ( ′ ) from (4.3), we define Proof. From the choice of ′ , and Pseudodifferential operator symbol calculus we can reduce directly to the case of = 0 and = 0. Applying ( ( − ′ ) ′ ), we can also reduce to the case < 0 = ′ . Then we can adapt the argument from before, adding a contribution from the 0-th Fourier mode that decays as log ‖ ‖ 2 . Definition 4.14. Let → be a general admissible bundle. Let  be a derivation on sections of extending a vector field on . Also assume that ∶= ℎ ∈ Ψ ( , ). Assume that the flow generated by is uniformly hyperbolic, and that we can construct escape functions ∈ ( , ) for any > 0 satisfying the conclusions (i)-(iv) of Lemma 2.3 as well as the invariance properties from Lemma 2.5. Then we say that is a general admissible operator. We denote by , and * , the corresponding stable and unstable bundles.
Given a general admissible operator the proof of Propositions 2.10 and 2.11 apply, so we get a first parametrix ℛ ( ) ∶= ( − − ℎ ) −1 with norm (ℎ −1 ). Furthermore from Definition 4.5 and Proposition4.11 we deduce straightforwardly: Consequently from Lemma 3.12 and Theorem 2 we deduce that , ( ) ∶= ( , − , − ℎ ) −1 and that , ( ) are analytic, respectively meromorphic families of operators on the appropriate anisotropic spaces. We now choose ∈ ∞ (ℝ) such that ( ) = 0 for < log( ), and ( ) = 1 for > log( 2 ) and define Next, let us define for ∈ ℝ max ( ) ∶= sup max, ( ). (4.7) Recall that max, was defined in Equation (3.18). Also keep in mind that weights are functions of the form = + , and they are large when is large and so is ∕| |. As a consequence, we get the main theorem of this article: It suffices now to show that ( ) is compact on the appropriate space. We will use the fact that if is large, so it ± 1. The first observation is that from a standard resolvent identity, we have for = 1 … , (4.10) This is a bounded operator from − max, ( )⟨ ⟩  −1 to max, ( )⟨ ⟩  +1 (it is smoothing). Now, we compute ( − ℎ )ℛ ′ ( ), and we find that the operator ( ) = 1 ( ) + 2 ( ) writes as the sum of two terms. The first one is This operator is compact on − max ( ) since it maps it continuously to < +1 -here we are applying Theorem 2 crucially.
The other term in ( ) is (4.12) Applying ( − − ℎ ) on the right, we obtain Since ( ) = 1 when > 2 , we get that Using that is smoothing together with Lemma 4.12, we deduce that 3 ( − − ℎ ) −1 is a compact operator on − ′ max ( ) . According to Lemma 3.12, provided is large enough, , ( ) is bounded on spaces  , with < − max ( ). Recall that was chosen to be constant outside a compact set so [ , , − , ] ∶  , 1 →  , 2 is bounded for arbitrary

Explicit computations for the geodesic flow
In this section, we come back to the case of admissible bundles over * with an admissible cusp manifold. Let us denote by max the maximum of Re( ) when ranges in the eigenvalues of the endomorphisms . Then we define max, ( ) = max(0, max − − ∕2) (Note that for functions i.e. being the trivial bundle, we have max = 0. We prove  1.4).
More precisely for any < 0, ∈ ℝ there is a suficiently large such that on Re( ) > , |Im( )| ≤ ℎ 1∕2 the resolvent is a meromorphic family of bounded operators Finally, the wavefront set of ℛ( ) satisfies estimate (4.8) and its polar part satisfies (4.9) as in Theorem 3.
According to the proof of Theorem 3 it suffices to show that the roots are affine in the sense of Definition 3.26. This will be shown in Lemma 5.11.
We will explicitly calculate the indicial roots for an admissible lift of the geodesic flow in the sense of Definition 1.4. We do this in three steps: First we compute the family of indicial operators for admissible lifts. Then we determine the indicial roots for the scalar case, and finally deduce the precise formula for the indicial roots of an admissible vector bundle. , thus we can identify with a functioñ ∶ Λ ∖ → , that is right -equivariant, i.e.̃ (Λ ) = ( −1 )̃ (Λ ). Note that the geodesic flow on * , ≅ Λ ∖ ∕ is given by the right -action and we can write 11

The Indicial operator for admissible lifts
for a suitably normalized ∈ = Lie( ). Let us check that preserves sections that are independent of the variable. Note that w.r.t. the ℕ decomposition, this means that̃ (Λ ) =̃ (Λ ) (cf. Section 1.2). That such functions are preserved under is obvious by (5.2). Consequently is a black-box operator according to Definition 4.5.
Let us thus remove the dependencies in ∈ Λ ∖ ℕ and consider the operator 0 , acting on sections ∈ ∞ (ℝ × , ℝ × ). Further identify these sections with rightinvariant functions̃ ∶ × → . By the ℕ -Iwasawa decomposition we can write any ∈ in a unique way as = ( ) ( ) ( ). With this notation we can write 0 , where we used the identities ( ) = ⋅ ( ) and ( ) = ( ). This formula shows directly that 0 , commutes with translations in the direction and we have thus shown that is a cusp-b-operator according to Definition 4.5. It finally remains to express 0 , in the coordinates , , of ℝ × ≅ × ∕ as introduced above. In particular we have to identify the differential operators In order to pass from 0 , to one simply has to conjugate the differential operator by − ∕2 which creates the additional ∕2 cos term in (5.1) Now from Equation (5.1) and the definition 3.13 of the indicial family we directly obtain:

Finding the indicial roots for functions
In this section, we focus on the action on functions. In that case,  = and = ℎ . Since the flow is the same for each cusp, we can safely drop the dependence in the index . We compute the indicial roots of ( , ) − ℎ . As this operator will frequently show up in the sequel we introduce the shorter notation Le us introduce some notations which we will need to formulate the spectral properties of . Recall that we have introduced the coordinates ( , ) ∈ [0, ]× −1 on . Consider the projection of to the equatorial plane. It is a smooth chart on both strict hemispheres. We denote these smooth restrictions by Note that ( , ) ∶= (sin , ) ∈ [0, 1] × −1 are exactly the radial coordinates in both charts.
For further reference we recall that the Taylor expansion in radial coordinates at 0 for ∈ (ℝ ) can be written in the following fashion Here ∈ ℕ is a multindex, Υ ∈ ∞ ( −1 ) is the monomial , ∈ ℝ , of degree | | restricted to the unit sphere −1 ⊂ ℝ . Let us come back to . According to Lemma 3.20, to determine the indicial roots, it suffices to consider the action of on 0 ( ), that we denote just ( ). Inspecting the Formula (5.4), we see that is a gradient vector field plus a complex potential. Pollicott-Ruelle resonances for Morse-Smale gradient flows were studied in detail by Dang and Rivière [DR16]. In particular, the spaces they defined are quite similar to ( ). Recall that was defined in Lemma 2.3 Lemma 5.3. There is an > 0 such that the following holds.
• Let ∈  ′ ( ) be supported in the -neighbourhood of the North Pole. Then ∈ ( ) if and only if ∈ − ( ).

• Let ∈  ′ ( ) be supported in the -neighbourhood of the South Pole. Then ∈ ( ) if and only if ∈ ( ).
Proof. Let us prove the first assertion: By Definition 3.18 of and standard Pseudodifferential operator arguments it is enough to prove that ,0 ∈ 0 ( ) is constant equal to − in a neighbourhood of  modulo some lower order terms −1+ ( ). By Lemma 3.17 and Proposition 4.11 the leading term is given by Putting everything together we know that at leading order ,0 is constant equal to − around  which implies the first assertion.
The second statement follows from similar arguments.
The following lemma shows that in the charts  , , the operator takes a particularly simple form -recall that here = sin On the Southern Hemisphere, the function 2 tan( ∕2) sin is an analytic, nonzero function and expressed in the ( , )-charts, defined above we have (2 tan( ∕2) sin ) − ( − ℎ ) (2 tan( ∕2) sin ) Proof. The results follows from a straightforward calculation using standard trigonometric identities.
Let ∈ ℕ be a multindex, then the we define the standard dirac distributions on ℝ by Proof. Assume that ∈  ′ ( ) is a distribution that fulfills ( − ℎ ) = 0. Then we can distinguish two cases: Either | ⧵{ ,} = 0 or not. In the first case, must be a linear combination of ( )  ∕ and from (5.9) we deduce that the possible solutions are either = ℎ( + + ∕2) and is a linear combination of ( )  , with | | = or = −ℎ( + + ∕2) and is a linear combination of ( ) , , again with | | = .
Whether these distributional solutions belong to ( ), or not, depends on . Suppose that > + ∕2, ∈ ℕ, then locally around the South Pole, according to As in a neighbourhood of the South Pole, (2 tan( ∕2) sin ) − is a smooth nonvanishing ∞ ( ) function we have to extend − ∕2− ∕ℎ− ⊗ to a -function on ℝ . According to (5.7) this is possible if either Re(− ∕2 − ∕ℎ − ) ≥ or if − ∕2 − ∕ℎ − = for some ∈ ℕ and is a linear combination of Υ with | | = (or in other words, is a homogeneous polynomial of degree ). Note that the first case is ruled out since we assumed that −Re( + ∕ℎ) < − ∕2 + so that we would have < − , and > ∕2, contrary to our assumption. Now, to complete the proof of Proposition 5.5, we have to check that for = ±ℎ( + ∕2 + ), the kernel is not empty. We have already done this in the proof for = ℎ( + ∕2 + ) for ∈ ℕ, so we concentrate on the case that = −ℎ( + ∕2 + ).
Since 2 tan( ∕2)∕ sin is a nonvanishing ∞ ( ) function near  , we conclude that Integrating by parts in the variable, times when Re < −ℎ ∕2, we obtain that The expression in the RHS is obviously meromorphic for Re < ℎ( − )∕2. The poles are situated at = ℎ( − )∕2, with = 0, … , − 1, and they are of order 1. At such a point, we find = −( + + )∕2, and = ℎ( + ∕2 + ). In other words, the poles of ,Υ, correspond to root crossings. This is sufficient to ensure that the indicial roots are exactly the ±ℎ( + ∕2 + ) with ∈ ℕ, and finishes the proof of Proposition 5.5. Now, while not necessary for the proof of the main theorem, we want here to describe the Jordan block structure at the root crossings. Since the residue of ,Υ, at a pole does not depend on the level of regularization (as long as ≥ + 1), we can choose = + 1. Then the residue is given by But as is supported in { < } this is just where the equality can be read off (5.7). Writing = 1∕ ! ∫ −1 Υ( )Υ ( ) , the residue of ,Υ, at = ℎ( − )∕2 is thus given by The finite part of ,Υ, at such a point is a distribution such that coincides with ,Υ,ℎ( − )∕2 in ⧵  . Additionally, since we have for all ( + + ℎ( + ∕2)) ,Υ, = 0, Differentiating in the parameter , we deduce that Consequently the finite part of ,Υ, is an eigendistribution of if all the vanish. If Υ is chosen such that this is not the case, we can however modify in order to get a generalized eigendistribution: Consider the space of distributions supported in { }, of order < . Choosing a basis of such distributions of decreasing order, we find that ℎ( − )∕2 acts on in an upper triangular fashion, and the diagonal coefficients are non singular. We deduce that ℎ( − )∕2 is invertible on . In particular, since ( ℎ( − )∕2 + ℎ + + 2 ) 2 ∈ , we can find ∈ such that ( ℎ( − )∕2 + ℎ + + 2 ) 2 ( + ) = 0. In particular, the kernel is non empty, and there is an order 2 Jordan block.

Indicial roots for fiber bundles
After this study of the action on functions, we come back to the action on admissible vector bundles = × → ∕ ≅ . For the moment let us fix a cusp and drop the index . Note that Definition 1.4 does not assume that is an irreducible representation. However, we can reduce the problem to the irreducible case: Consider the complexified representaion ( , ℂ ) which decomposes into irreducible unitary representations ( , ). We then get Furthermore, using the explicit form of ( , ) from (5.3) and the fact that according to Definition 1.4 the nonscalar zero order term is equivariant, we conclude that ( , ) preserves this splitting. Finally, we have to take into account that we do not want to study the operator acting on 2 but rather on the anisotropic spaces ( , ). Recall that the escape function is a purely scalar symbol. We can pick the quantization so that scalar symbols are mapped to operators that preserve the decomposition (5.12) -that is a lower order term condition. In particular, (Op ( − ), ) acts on 2 ( , × ) as a principally scalar operator. Thus we assume from now on that we have fixed a cusp and that ( , ) is unitary and irreducible. Since it is irreducible, the equivariant term has to be scalar by Schur's Lemma and the indicial operator Similarly we chose a orthonormal parallel frame  ( ) on ⧵{ }. Comparing these two orthonormal frames on the equator = ∕2, ∈ −1 we get a smooth gluing function ℊ ∶ −1 → ( ) such that With this gluing function we can express the transformation under the change of trivialisation for ∈  ′ ( ⧵ { , }, ) as follows Having introduced these orthonormal frames we can prove.
Lemma 5.9. If we fix a cusp and consider = × for an irreducible unitary representation ( , ), then the operator where ∈ End( ) has been identified with a scalar by Schur's lemma and ∈ ℕ.
Proof. Let us reduce the problem to the case of functions, dealt with by Lemma 5.6: Suppose that ∈ ( , ) ⧵ {0} with ( − ℎ , ) = 0. Then one of the following cases holds: First case: supp( )⧵{ , } ≠ ∅. Then we can expand the restriction of to ⧵{ } in the orthonormal trivialisation  and get for scalar distributions  ∈  ′ ( ⧵ { }). From the fact that ∇  = 0 we deduce that ℎ( + ∕2 cos ) + cos − ℎ( − )  = 0 Next, using that ∈ ( , ) and Lemma 5.3 we conclude that  ∈ ( ), in a small neighbourhood around . Furthermore at least one  must be nonvanishing on ⧵ { , }. We are thus precisely in the setting of the second case in Lemma 5.6 and we deduce with the same arguments that such a distribution only exists if − = − ∕2 − ∕ℎ − for some ∈ ℕ and the eigendistributions are precisely given by a linear combination of , ,Υ with | | = .
Second case: supp( ) = {}. Then we use the same trivialisation as above. This would require distributions  ∈ ( ) with supp  = . But as Lemma 5.3 requires these distributions to have positive Sobolev regularity, they have to be zero.
Third case: supp( ) =  . Then using the trivialisation on ⧵ { } we write: We are thus precisely in the setting of the first case in Lemma 5.6 and we deduce that such distributions only exist if ℎ( − ) = − ℎ( + ∕2) for some ∈ ℕ and they are precisely given by linear combinations of ( )  , with | | = . As in the case of functions, we have to care about the extension of those distributions coming from ,Υ , and check which still remain in the kernel of the indicial operator. Therefore the following notation is convenient: Given Υ = (Υ (1) , … , Υ (dim ) ) ∈ (ℝ ( −1 )) dim , define the section ,Υ, ∶=  . In order to understand the extension in the sense of homogenous distributions at the North Pole we use the definition of ,Υ, (Eq. (5.11)) and pass to the trivialisation  : Definition A.4. Let ∈ 0 ( ) be an order zero Kohn Nirenberg symbol which we call an order function. The space of anisotropic symbols log ( , ) consist of those sections ℎ ∶ * ↦ ℒ ( , ) parametrized by a parameter 0 < ℎ ≤ ℎ 0 such that for all ∈ ℕ, there is independent of ℎ such that , Note that the loss of log⟨ ⟩ is necessary for the space of anisotropic symbols to contain sufficiently interesting elements such as for example ⟨ ⟩ ( , ) .
In the sequel we will usually drop the parameters ℎ to simplify the notation unless we want to emphasize dependence on ℎ Consider two symbols , ∈ ( , ). We say that is scalar if it takes the form ′ with ′ ∈ ( , ℝ). In this case, we define the Poisson bracket: where ′ is the Hamiltonian vector field of ′ ∈ ( , ) ⊂ ∞ ( * ). It is important to notice that the set of symbols ( , ℝ), where is an admissible cusp manifold is exactly the same class of symbols as was described in the paper [Bon16]. It is straight forward to show that the proofs therein apply to log (the largest of all classes here).
To close this section, we consider the radial compactification of the cotangent space.
Definition A.6. Let * be the radial compactification of the cotangent space. It has a structure of continuous manifold, but not of ∞ manifold a priori. We consider the map This is a homeomorphism of * to (0, 1) in * and it endows * with a the structure of a smooth manifold with boundary. Let̃ = * and define ̃ norms on * using̃ . Then we define the classical symbols as 0 ( ) ∶= ∞ ( * ) and for ∈ ℤ we set ( ) ∶= ⟨ ⟩ ∞ ( * ).
Note that for the prescribed smooth structure on * , ⟨ ⟩ −1 is a boundary defining function. In particular, because classical symbols are smooth up to the boundary, they have a homogeneous expansion as → ∞.
The remainder worsens to ℎ 2 log(1 + ⟨ ⟩) 2 1 + 2 −2 log in the case of exotic symbols.  The stabilization of Fourier modes is a nice feature from which we profit because we have assumed that the curvature is constant −1 in the cusps. In a more general case of curvature tending to −1, one would have to look for more subtle estimates.

If is hermitian valued,
Remark A.9. The remainder in the product formula can actually be written as a Op ′ ( ), with a (ℎ ∞ −∞ ) symbol, if Op ′ is another quantization built in the same fashion, but where the cutoff away from the diagonal has been changed to another one with sufficiently larger support.
Proof. Proofs for 1.-5. can be found in [Bon16] for non-exotic symbols. The arguments, however all transfer to exotic symbols. Note that the key argument in [Bon16] is that for a symbol on a cusp, in an interval at height 0 , the cusps can be rescaled such that one transfers the problem to an euclidean cylinder. The crucial point is that the symbols transform uniformly in 0 under this rescaling (see [Bon16, Section 1.3]). Furthermore, since we introduced a cutoff away from the diagonal in our quantization (A.3), we can rescale the whole operator from a neighbourhood of = 0 to a fixed Euclidean cylinder with uniform estimates. Much as in [Bon16], the proof of boundedness and other estimates follow from the estimates holding on ℝ . It is also the case for the sharp Gårding estimate.
Let us say a word on the property 6. in Proposition A.8. Inspecting the formula (A.2), we observe that in the variable, the kernel is just a Fourier transform of in the variable. Such an operator commutes with and thus preserves Fourier modes. To be able to use this formula, we just need that the support of does not intersect the support of the cutoff function corresponding to compact charts, hence the condition that is supported in { ≥ }.
Following Lemma 1.8 in [Bon16], we can prove that our operators actually act as pseudo-differential operators, and that our quantization is a quantization in the usual sense: Definition-Proposition A.11. Take ∈ 0 ( , ℝ) scalar. We let Ψ ( , ) be the algebra of operators generated by operators of the form Op( ) with ∈ log ( , ). We call them the algebra of semiclassical pseudodifferential operators (or also just pseudodifferential operators in short).
On Ψ , we have a principal symbol map 0 which is defined independently of the choice of quantization Op as a map 0 ∶ Ψ → log ∕ℎ −1 log , with 0 (Op( )) = [ ]. Once we have fixed a a quantization, we obtain by iterations a full symbol map ∶ Ψ → ∕ℎ ∞ −∞ .
3. As a consequence, WF ℎ ( ) ∩ ell( ) ⊂ WF ℎ ( ) Proof. 1. follows from a standard inductive parametrix construction (see e.g. [DZ19, Proposition E.32]). The notion of -elliptic set has been introduced precisely to assure that the construction yields symbol in the uniform symbol classes.

A.4 Propagation of singularities and other estimates
Throughout the paper, to obtain results on the wavefront sets of several operators, we have used lemmas that were almost identical to some lemmas in [DZ16]. In this section we give the versions on admissible cusp manifolds.
For the most part, the proofs given in the appendix of [DZ16] are also valid in our case. As a consequence, this is a cursory review of some special technicalities, destined to the reader already acquainted with the detail of the arguments in [DZ16].
The constants are (1) ( 0 ) , but we will not need this fact. One can mimick the proof in [DZ16] step by step. Be mindful that Re has to be replaced by −Im , and Im by Re .
Proof. In the whole proof, when working on subsets of * , we will be working with the notion of distance obtained on * obtained by pulling back the distance on (0, 1) ⊂ * by the map defined in Definition A.6. Since ∈ 1 , is a smooth flow for this structure. Additionally, we can always assume that the symbols of , and 1 are in 0 , i.e smooth up to the boundary of * . To start with, applying a partition of unity argument, we can assume that is microsupported in a ball with small radius 0 > 0. Then we can also assume that is microsupported in a 3 0 -neighbourhood of the image (WF ℎ ( )) for some ∈ [0, 0 ], and 1 is microsupported in a 3 0 -neighbourhood of the union ∪ ∈[0, ] (WF ℎ ( )). Since the proof in [DZ16] is based on local considerations along the trajectories of the flow in bounded time, and we are not seeking to determine the behaviour of the constants when the time 0 goes to infinity, we already observe that the estimate holds if is supposed to be microsupported in a fixed compact set of , with constants that depend on the compact set. As a consequence, we can restrict our attention to the case when , , 1 are supported in a fibred cusp end , above a set of the form { > 0 } with 0 arbitrary large, and satisfy symbol estimates with constants not depending on 0 .
( designs a generic point in the generic fiber of → ). Let us do some more reduction. The vector field acts in a uniform ∞ fashion on * , and as such |∇ | ∞ < ∞. Additionally, by symbol estimates, we know that = ( −∞ ). As a consequence, for large enough, an escape function for = ∫ is also an escape function for . In other words, we can assume that does not depend on . Then commutes with . Consider that in the cusp, we have an additional fiber structure. Indeed, write = (ℝ ∕Λ ) × ℝ × . Then we can see * as a fiber bundle, by forgetting the variable. Seeing 0 as a vector bundle over ℝ × , we can also extend as a map * → 0 . Since commutes with , it projects to a vector field 0