A sharp stability estimate for tensor tomography in non-positive curvature

We consider the geodesic X-ray transform acting on solenoidal tensor fields on a compact simply connected manifold with strictly convex boundary and non-positive curvature. We establish a stability estimate of the form $L^2\mapsto H^{1/2}_{T}$, where the $H^{1/2}_{T}$-space is defined using the natural parametrization of geodesics as initial boundary points and incoming directions (fan-beam geometry); only tangential derivatives at the boundary are used. The proof is based on the Pestov identity with boundary term localized in frequency.


Introduction
To motivate our results, let us begin with the simplest case of the Radon transform in R 2 in parallel beam geometry (see [Na01] for more details).
The main question we address in the present paper is the existence of a stability estimate analogous to (1.1) but in a geometric setting, namely, when R 2 and the lines in the plane are replaced by a Riemannian manifold and its geodesics. There are two features we wish to preserve from (1.1): one is its L 2 → H 1/2 nature and the other is that the H 1/2 T only incorporates "half of the derivatives" of the target space (space of geodesics).
Let us first be more precise about the geometric setting. The geodesic X-ray transform acts on functions defined on the unit sphere bundle of a compact oriented ddimensional Riemannian manifold (M, g) with smooth boundary ∂M (d ≥ 2). Let SM denote the unit sphere bundle on M, i.e. We define the volume form on SM by dΣ 2d−1 (x, v) = dV d (x) ∧ dS x (v), where dV d is the volume form on M and dS x is the volume form on the fibre S x M. The boundary of SM is ∂SM := {(x, v) ∈ SM : x ∈ ∂M}. On ∂SM the natural volume form is dΣ 2d−2 (x, v) = dV d−1 (x) ∧ dS x (v), where dV d−1 is the volume form on ∂M. We distinguish two subsets of ∂SM (incoming and outgoing directions) where ν(x) is the outward unit normal vector on ∂M at x. It is easy to see that Given (x, v) ∈ SM, we denote by γ x,v the unique geodesic with γ x,v (0) = x anḋ γ x,v (0) = v and let τ (x, v) be the first time when the geodesic γ x,v exits M.
We say that (M, g) is non-trapping if τ (x, v) < ∞ for all (x, v) ∈ SM. In this case the space of geodesics is naturally parametrized by ∂ + SM (fan-beam geometry).
Definition 1.1. The geodesic X-ray transform of a function F ∈ C ∞ (SM) is the function If the manifold (M, g) is non-trapping and has strictly convex boundary, then I : C ∞ (SM) → C ∞ (∂ + SM), and extends as a bounded operator I : H k (SM) → H k (∂ + SM) for all k ≥ 0 [Sh94, Theorem 4.2.1], where the Sobolev spaces are defined using the L 2 -inner products arising from the volume forms introduced above.
We shall consider I acting on special functions F ∈ C ∞ (SM) induced by symmetric tensor fields. We denote by C ∞ (S m (T * M)) the space of smooth covariant symmetric tensor fields of rank m on M with L 2 inner product: where w i 1 ···im = g i 1 j 1 · · · g imjm w j 1 ···jm . There is a natural map ℓ m : C ∞ (S m (T * M)) → C ∞ (SM) given by ℓ m (f )(x, v) := f x (v, . . . , v). We can now define the geodesic ray transform acting on symmetric m-tensors simply by setting I m := I • ℓ m . Let d s = σ∇ be the symmetric inner differentiation, where ∇ is the Levi-Civita connection associated with g, and σ denotes symmetrization. It is easy to check that if f = d s p for some p ∈ C ∞ (S m−1 (T * M)) with p| ∂M = 0, then I m f = 0. The tensor tomography problem asks the following question: are such tensors the only obstructions for I m to be injective? If this is the case, then we say I m is solenoidal injective or s-injective for short. This terminology is explained by the following well known decomposition (cf. [Sh94]). Given f ∈ H k (S m (T * M)), k ≥ 0, there exist uniquely determined f s ∈ H k (S m (T * M)) and p ∈ H k+1 (S m−1 (T * M)), such that where δ s is the divergence. We call f s and d s p the solenoidal part and potential part of f respectively.
There is one important instance in which the tensor tomography problem is solved for tensors of any order m and in any dimension d. This is when we assume in addition that the sectional curvature of M is non-positive. Moreover, in this case a stability estimate is available as follows: . (We note that a manifold as in the theorem is necessarily non-trapping.) There are two notorious differences between the stability estimate above and that in (1.1). Firstly, the stability estimate in Theorem 1.2 has in the right hand side the term f H 1 I m f L 2 when m = 0. Secondly, it is not sharp in the sense that it is L 2 → H 1 . In [BS18] Boman and Sharafutdinov resolved these issues for strictly convex domains in Euclidean space and asked whether the same was true for the more general setting of non-positively curved Riemannian manifolds. This paper provides a positive answer to these questions. Moreover, the 1/2-Sobolev space on the target space of I m is naturally suggested by the geometry and the most relevant L 2 -energy identity for the problem: the Pestov identity. The Pestov identity with boundary term in the way that we shall use it here was derived for instance in [IP18,Lemma 8]. It contains a boundary term given by ∇ is the vertical gradient, and T is a tangential operator defined by where X is the geodesic vector field and h ∇ the full horizontal gradient (we refer to Sections 2 and 3 for the precise definitions). The operator T acts on ∂SM and it only involves horizontal derivatives. This suggests that only horizontal derivatives of I m f on ∂SM should appear in the stability estimate.
We can define the tangential (or horizontal ) H 1 (∂SM)-norm by setting where h ∇ u contains the tangential derivatives in h ∇u along ∂M. For example, if M is a ball in R n with Euclidean metric, then ∂SM = ∂M × S d−1 and where ∇ x is the gradient on ∂M. The space H 1 T (∂ + SM) is defined by restriction, and H 1/2 T (∂ + SM) is defined by complex interpolation between L 2 (∂ + SM) and With this definition we may now state our main result: Theorem 1.3. Let (M, g) be a simply connected compact manifold with strictly convex boundary and non-positive sectional curvature.
. The constant C can be estimated in terms of m and (M, g). In fact, for the related stability result for the transport equation in Theorem 5.1, one can take C = 1.
Most of work in the proof of Theorem 1.3 lies in the upgrade from the H 1 (∂ + SM)norm in Theorem 1.2 to the H 1/2 T (∂ + SM)-norm. The upgrade is possible thanks to the localization in frequency of the Pestov identity first noted in full generality in [PS18] (in two dimensions this was proved in [PSU15]). However, in [PS18] we did not consider the boundary term. It turns out, quite remarkably, that the boundary term (1.2) also localizes in frequency. This allows us to change the norm for I m f from H 1 to H 1/2 T , thus producing the upgrade. We also mention that for dim (M) = 2 the proof would simplify substantially because spherical harmonics decompositions and the T operator are simpler; the two-dimensional proof will be given in [PSU19].
1.1. Related results and alternative approaches. There are many earlier results on stability estimates for I m , using different techniques. One approach is to consider the normal operator I * m I m where the adjoint I * m is computed using a natural L 2 µ -inner product on ∂ + SM suggested by the Santaló formula. When M is free of conjugate points, it turns out that I * m I m is an (elliptic) ΨDO of order −1 on a slightly larger open manifold engulfing M. This approach has produced stability estimates for the normal operator, cf. [SU04], and has proved to be of fundamental importance in the solution of several geometric inverse problems. One drawback is that one needs to work on the slightly extended manifold, unless one is willing to incorporate modified transmission conditions to account for boundary effects [MNP19]. Another drawback is that the approach does not give estimates for the constants due to a compactness argument. Still, quite recently, a sharp stability estimate has been obtained in [AS19], by defining a suitable H 1/2 -norm based on this extension or equivalently on a different parametrization of the space of geodesics. Our approach in Theorem 1.3 deals directly with the boundary and with the space of geodesics in "fan-beam" geometry as given by ∂ + SM. In this sense our theorem addresses the open problem stated at the end of the introduction in [AS19]. Also our tangential derivatives are naturally suggested by the geometry of the problem.
The microlocal approach can actually be pushed further, using scattering calculus and a combination of a local theorem with a global strict convexity assumption as in [UV16,SUV18]. This is also very powerful, and allows even to consider situations with conjugate points as long as d ≥ 3. However, the stability estimates thus produced are L 2 → H 1 .
One drawback of Theorem 1.3 is the curvature assumption. In [AS19] the estimates hold for compact simple manifolds for m = 0, 1 and for m = 2 when I m is known to be injective, e.g. when d = 2 [Sh07,PSU13]. Another possible improvement would be to replace the assumption f ∈ H 1 by f ∈ L 2 and to prove the two-sided inequality For this, one would like to prove that I m is bounded from T . This is true if f vanishes near ∂M since I m is a Fourier integral operator, but it is not clear how to prove this with uniform bounds when the support of f extends up to ∂M.
Finally, we mention that quite recently, Monard [M19] has studied very detailed mapping properties of I 0 for 2D discs of constant curvature at all Sobolev scales; for these cases, he also obtains a stability estimate with a suitable H 1/2 -norm. Further references to stability estimates for I m may be found in [AS19].
Acknowledgements. GPP was supported by EPSRC grant EP/R001898/1 and the Leverhulme trust. MS was supported by the Academy of Finland (Finnish Centre of Excellence in Inverse Modelling and Imaging, grant numbers 312121 and 309963) and by the European Research Council under Horizon 2020 (ERC CoG 770924). This material is based upon work supported by the National Science Foundation under Grant No. 1440140, while the authors were in residence at MSRI in Berkeley, California, during the semester on Microlocal Analysis in 2019.

Geometric preliminaries
In this section we collect some geometric preliminaries for subsequent use.
Unit sphere bundle. We start by recalling some standard notions related to the geometry of the unit sphere bundle. We follow the setup and notation of [PSU15]; for other approaches and background information see [GK80,Sh94,Pa99,Kn02,DS11].
Let (M, g) be a d-dimensional compact Riemannian manifold with or without boundary, having unit sphere bundle π : SM → M, and let X be the geodesic vector field. We equip SM with the Sasaki metric. If V denotes the vertical subbundle given by V = Ker dπ, then there is an orthogonal splitting with respect to the Sasaki metric: The subbundle H is called the horizontal subbundle. Elements in H(x, v) and V(x, v) are canonically identified with elements in the codimension one subspace {v} ⊥ ⊂ T x M by the isomorphisms here K x,v is the connection map coming from Levi-Civita connection. We will use these identifications freely below. We shall denote by Z the set of smooth functions Z : Alternatively we may describe the elements of Z is a follows. Consider the pull-back bundle π * T M over SM and let N denote the subbundle of π * T M whose fiber over (x, v) is given by N (x,v) = {v} ⊥ . Then Z coincides with the smooth sections of the bundle N. Note that N carries a natural scalar product and thus an L 2 -inner product (using the Liouville measure on SM for integration).
Given a smooth function u ∈ C ∞ (SM) we can consider its gradient ∇u with respect to the Sasaki metric. Using the splitting above we may write uniquely in the decomposition (2.1) The geodesic vector X acts on Z as follows: where D/dt is the covariant derivative with respect to Levi-Civita connection and ϕ t is the geodesic flow. With respect to the L 2 -product on N, the formal adjoints h div respectively. Note that since X leaves invariant the volume form of the Sasaki metric we have X * = −X for both actions of X on C ∞ (SM) and Z. Let so that any f ∈ L 2 (SM) has the orthogonal decomposition Decomposition of X. The geodesic vector field has a special behaviour with respect to the decomposition into fibrewise spherical harmonics: it maps Ω m into Ω m−1 ⊕Ω m+1 [GK80, Proposition 3.2]. Hence on Ω m we can write [GK80,Proposition 3.7] the operator X + is overdetermined elliptic (i.e. it has injective principal symbol). We can explain the decomposition X = X − + X + as follows. Fix x ∈ M and consider local coordinates which are geodesic at x (so ∂ x j g kl (x) = 0 for all j, k, l). Then Xu(x, v) = v i ∂u ∂x i . We now use the following basic fact about spherical harmonics: the product of a spherical harmonic of degree m with a spherical harmonic of degree one decomposes as the sum of spherical harmonics of degree m − 1 and m + 1.

Pestov identity with boundary term
We recall the following commutator formulas from [PSU15]: Taking adjoints gives the following commutator formulas on Z: Using these relations one can establish a Pestov identity with boundary term. Let µ(x, v) := v, ν(x) . We let · and (·, ·) denote the L 2 -norm and L 2 -inner product respectively determined by the volume form dΣ 2d−1 on SM; we let (·, ·) ∂SM stand for the L 2 -inner product on ∂SM determined by dΣ 2d−2 .
where P is the quadratic form defined by We can express T using the full horizontal derivative It turns out that T can also be rewritten in such a way that it acts on functions u ∈ C ∞ (∂SM). To see this, consider the operators at the boundary. We also define the horizontal part of ∇ as The following simple lemma is proved in [IP18, Lemma 14]: Lemma 3.2. We have From this form we can clearly see that T : Remark 3.3. In 2D, T u = (Tu)iv, where T is the tangential horizontal vector field (iν, 0) and i is the complex structure of the surface. The vector field T and the vertical vector field V form a commuting frame for ∂SM.
We next rewrite the Pestov identity in terms of X + and X − as in [PS18]. To do this, we need some notation: for a polynomially bounded sequence α = (α l ) ∞ l=0 of real numbers, we define a corresponding "inner product" We also write u 2 α = ∞ l=0 α l u l 2 . (If each α l is positive one gets an actual inner product and norm, but it is notationally convenient to allow zero or negative α l .) The Pestov identity can then be written in the following form. Define The next result extends [PS18,Proposition 4.4] to the case with boundary terms.
Proposition 3.4 (Pestov identity in terms of X ± with boundary term). Let (M, g) be a compact manifold with smooth boundary. If u ∈ C ∞ (SM), then This gives On the other hand, one has Somewhat miraculously, we observe that This means that the two sums above involving [(X + u l−1 , X − u l+1 ) + (X − u l+1 , X + u l−1 )] terms are equal. The Pestov identity from Proposition 3.1 now yields where α l , β l are as in (3.6)-(3.7). The result follows.
Later on we shall need the following useful property.
Lemma 3.5 (Adjoint of T ). The formal adjoint of T : Proof. We use the Pestov identity with boundary term, to claim first that the operator v divT is self adjoint. Proposition 3.4 and the polarization identity imply that and since R is symmetric, it follows that P (u,

Frequency localization
Recall from Section 2 that any u ∈ C ∞ (SM) admits an L 2 -orthogonal decomposition where Ω l corresponds to the set of vertical spherical harmonics of degree l. Since X ± maps Ω l to Ω l±1 , it is immediate that the Pestov identity with boundary term (Proposition 3.4) reduces to the following identity when applied to functions in Ω l (i.e. frequency localized Pestov identity).
Proposition 4.1 (Pestov identity on Ω l with boundary term). Let (M, g) be a compact manifold with smooth boundary, and let l ≥ 0. One has It was proved in [PS18] (and in [PSU15, Appendix B] when dim (M) = 2) that the frequency localized Pestov identity for all l is equivalent with the standard Pestov identity. The same is true in the boundary case: Lemma 4.2. The Pestov identity with boundary term on Ω l is equivalent with the Pestov identity with boundary term in the following sense: for any u ∈ C ∞ (SM), one has The result will follow if we can show that the curvature, Z and P terms localise. Thus Lemma 4.2 is a corollary of the next result.
Proof. The localization of the curvature term was proved [PS18, Lemma 5.4]. We shall prove here that the Z-term localizes. That is enough to obtain also the conclusion for P since Proposition 3.4 and the polarization identity imply that Hence the statements for the curvature and Z-term imply that P (u, w) = 0 when m = l. The last claim follows since P (u, w) = −( v divT u, w). The claim for Z(u) for d = 2 follows from [PS18, Remark 6.5] using the explicit representation for Z(u). To prove the claim when d ≥ 3, recall that Z(u) is the Hence if we think of each v j as 1-form it is enough to analyze the vertical Fourier decomposition of A v ∇w, where A is a scalar 1-form and w = ∂ x j u ∈ Ω m . This is precisely the content of Lemma A.1, and combining the statement of that lemma with (3.9) we see that Since B localizes in frequency, the lemma is proved.

Stability for the transport equation
In this section we will prove the main stability estimate for solutions of the transport equation Xu = f in SM when f has finite degree. In the next section we will give the more standard form where the solenoidal part of f is estimated in terms of I m f . whereas if m ≥ 1, then f − X(u 0 + . . . + u m−1 ) L 2 (SM ) ≤ u H 1/2 T (∂SM ) . 5.1. Shifted Pestov identity with boundary terms. To prove Theorem 5.1 we first assume that m ≥ 1, and discuss the case m = 0 later. We will try to estimate f in terms of u| ∂SM in suitable norms. The starting point is the identity from Proposition 4.1 with l ≥ 1: Since we are assuming non-positive sectional curvature, we have We divide this estimate by α l−1 (always different from zero since l ≥ 1), which corresponds to shifting the estimate down by one half vertical derivatives since α l−1 ∼ l. It follows that The constant Note that D d (l) ≤ 1 for d ≥ 4 and in the remaining cases it is sufficiently close to one for all practical purposes (when reading the proof it may be helpful to think that D d (l) ≡ 1).
Thus we have the following inequality: For l ≥ m we have X − u l+2 + X + u l = 0 and using (5.1) we may write Starting at l = m and iterating this inequality N times leads to Write γ d,m,j = j−1 k=0 D d (m + 2k) 2 and γ d,m,0 = 1. In the notation of [PSU15, Theorem 1.1] one has γ d,m,j = j−1 k=0 C d (m − 1 + 2k) 2 , and thus γ d,m,j ≤ c d where Since X − u l 2 → 0 as l → ∞, we may take the limit as N → ∞ to obtain The argument above gives a completely analogous inequality for X − u m+1 2 , and adding these two inequalities leads to ,j α m+2j , k = 2j + 1. Define q := u 0 + u 1 + · · · + u m−1 . Then the transport equation Xu = f also gives If we assume m = 0, then the equation Xu = f implies X − u 1 = f , and (5.3) gives Thus to prove Theorem 5.1 for m ≥ 0, it remains to bound the right hand side of (5.6) for all m ≥ 1.

5.2.
The right hand side of (5.6): the space H 1/2 T (∂SM). Let m ≥ 1. Motivated by (5.6) and the fact that P is defined in terms of the (horizontal) tangential operator T , we define a natural H 1/2 -space as follows. Define the H 1 T (∂SM)-norm by setting The space H

Now we use the key property of localization given by Lemma 4.3 to observe that
We define an operator B = B m : C ∞ (∂SM) → C ∞ (∂SM) by setting Since m ≥ 1, the constant b m,l is well defined also when l = 0. Now (5.6) becomes Here is the main claim: Proof. We may write By the definitions, it suffices to show that By interpolation, this follows from the next two inequalities To prove these estimates we first establish the property Indeed using the definition of B, To prove (5.12) we will show that λ m+l b 2 m,l ≤ 1 for m ≥ 1 and l ≥ 0. If m = 1 and l = 0, then λ 1 b 2 1,0 = λ 1 α 2 0 = 1 d−1 ≤ 1, so we may assume m, l ≥ 1. One has . Observe that simplifying (3.6) gives α l = (2l+d−2)(l+d−1) l+d−2 for all l ≥ 1. We thus have, writing k = m + l ≥ 2, 2 ) 2 ≥ 1 and hence λ k using k ≥ 2. Combining these estimates with (5.2), we have The estimate (5.12) follows. Since − v ∇B is the adjoint of B v div, using (5.12) and (5.8) yields thus proving (5.10).
Finally to prove (5.11), we note that B v divT = v divT B by Lemma 4.3. Using Lemma 3.5 we may write where in the penultimate line we used (5.12) and (5.8).

Stability for the solenoidal part
We now rewrite Theorem 5.1 in terms of the solenoidal part of f and extend the result to H 1 regularity. Recall that the map To simplify the notation we shall the drop the tildes, identify f withf , q withq and use that the L 2 -norms are equivalent.
We first collect regularity properties of solutions of transport equations involving H 1 tensor fields.
where ϕ t is the geodesic flow on SM. One has u f ∈ C ∞ (SM \ S(∂M)) ∩ C(SM) since the same is true for τ , and (6.1) holds for u f . By Then the properties (6.1) remain valid for f ∈ H 1 (the boundary value of u f is in H 1/2 (∂SM) by the trace theorem). Since If vanishes on the boundary of ∂ + SM when f ∈ C ∞ , one has If ∈ H 1 0 (∂ + SM) first for f ∈ C ∞ and then for f ∈ H 1 by density. Since u f | ∂SM = E 0 (If ) where E 0 denotes extension by zero from ∂ + SM to ∂SM, we have u f | ∂SM ∈ H 1 (∂SM) when f ∈ H 1 .
Next we give a version of Theorem 5.1 for H 1 tensor fields.
Theorem 6.2. Let (M, g) be a simply connected compact manifold with strictly convex boundary and sectional curvature ≤ 0, and let f ∈ H 1 (S m (T * M)). If m = 0, then whereas if m ≥ 1, then . Here C only depends on d and m.
Proof. Let m ≥ 1 (the case m = 0 is analogous). Going back to (5.3) and using Lemma 5.3, one has the inequality Since functions in H 1 (SM) have traces in H 1/2 (∂SM), and hence also in H 1/2 T (∂SM), the above inequality holds for u ∈ H 1 (SM) by density. Then it is enough to take u = u f , where u f ∈ H 1 (SM) by Lemma 6.1, and to note that by (5.5) and by equivalence of the L 2 norms . The estimate (6.2) for m = 0 is already in the form that we want, so we will focus on the case m ≥ 1. Using the potential and solenoidal decomposition, we may write f = f s + d s p where δ s f s = 0 and p is an (m − 1)-tensor such that p| ∂M = 0. Let w = p − q. Then integrating by parts Next we observe that for any ε > 0 (6.5) 2|(ι ν f s , q) ∂M | ≤ 1 ε q 2 H 1/2 (∂M ) + ε ι ν f s 2 H −1/2 (∂M ) . We now claim: Proof. This is a duality argument, but it is important that δ s f s = 0. Combining (6.3)-(6.5) with Lemma 6.3 and choosing ε small enough, it follows that . The next two lemmas will be useful when estimating the last term on the right.
Lemma 6.4. Given m ≥ 0, there is a constant C > 0 such that for any tensor q of order m q H 1/2 (∂M ) ≤ C ℓ m q H 1/2 T (∂SM ) . Proof. Recall that we identify symmetric m-tensors with functions in k=0 Ω m−2k via ℓ m as explained at the beginning of this section. By interpolation, it is enough k=0 Ω m−2k . The first inequality follows from the equivalence of the L 2 -norms. For the second inequality, observe that locally a symmetric m-tensor field can be written as q = q i 1 ...im dx i 1 ⊗ · · · ⊗ dx im . The H 1 -norm of q in ∂M consists of the L 2 -norm of q plus the L 2 norm of the components q i 1 ...im (x) tangentially to M. Locally ℓ m q has the form q i 1 ...im v i 1 . . . v im . When we apply h ∇ to ℓ m q all the tangential derivatives in the direction of ∂M will appear. There will also be some vertical derivatives (involving the Christoffel symbols), but since ℓ m q is a polynomial of degree m in v, these terms can all be controlled by the L 2 -norm of ℓ m q. Thus q H 1 ℓ m q H 1 T follows, and this may be rewritten as ℓ −1 m h H 1 h H 1 T . Lemma 6.5 (The H 1 T (∂SM) norm localizes in frequency). One has for all u ∈ H 1 T (∂SM). In particular, m l=0 u l H 1 T (∂SM ) ≤ u H 1 T (∂SM ) when m ≥ 0. Proof. The proof is somewhat indirect and is based on the following observations.
(1) Let W be a vector field on M and let W = (W, 0) be its horizontal lift to SM. Then ∆W = W∆ where ∆ is the vertical Laplacian. This can be seen by taking a geodesic coordinate neighbourhood around a point x, so that ∂ j g kl (x) = 0 for all j, k.l. In that case if we write W = w i (x)∂ x i , then (Wu)(x, v) = w i (x) ∂u ∂x i and thus W : Ω m → Ω m . (Another way to prove this is to check that [δ j , ∆] = 0, using the notation and commutator formulas in This allows us to naturally extend to U ε the exterior unit normal ν to a vector field, still denoted by ν. (3) A smooth function u ∈ C ∞ (∂SM) can be extended to a smooth function u ν ∈ C ∞ (SU ε ) simply by making it constant on the orbits of the flow of ν, the horizontal lift of ν. By item (1) we have (6.7) (u m ) ν = (u ν ) m (4) Let f t be the flow of ν in SM, and let V ε be the neighbourhood of ∂SM in Let u ∈ C ∞ (∂SM). The fact that ν(u ν ) = 0 implies that Integrating over ∂SM and using that V ε = SU ε by (4), this gives . We now recall that it is possible to write for any w ∈ C ∞ (SU ε ) (all norms in L 2 (SU ε )), cf. [LRS18, proof of Lemma 5.1]: where A(d, l) = 2 + d−2 2 and B(d, l) = 2 + 1 d+l−2 . By Lemma 4.3 one may write Z(w) 2 = Z(w m ) 2 , and thus using (6.9) for w = u ν and w = (u ν ) m for each m we deduce Dividing this by ε and taking the limit as ε → 0, the identities (6.7) and (6.8) yield This implies the desired claim for u ∈ C ∞ (∂SM), and the result follows since C ∞ (∂SM) is dense in H 1 T (∂SM). (The density claim can be proved by using a partition of unity, convolution approximation in coordinate neighborhoods, and the Friedrichs lemma [Hö85, Lemma 17.1.5].) We now put the arguments above together to derive: Theorem 6.6. Let (M, g) be a simply connected compact manifold with strictly convex boundary and non-positive sectional curvature. Given m ≥ 0 there is a constant C > 0 such that for any f ∈ H 1 (S m (T * M)), one has Proof. For m = 0 this is just (6.2), so we assume m ≥ 1. Combining (6.6) and Lemma 6.4 we derive Recall that ℓ m−1 q = l≤m−1 u f l . Interpolating the bound l≤m−1 u l L 2 (∂SM ) ≤ u L 2 (∂SM ) for u ∈ L 2 (∂SM) with the bound in Lemma 6.5 gives l≤m−1 u l H 1/2 and the result follows by taking u = u f .
We can refine this further and prove Theorem 1.3 in the introduction. Define the space H 1/2 T (∂ + SM) as the interpolation space between H 1 T (∂ + SM) and L 2 (∂ + SM). Proof of Theorem 1.3. Theorem 6.6 gives (6.10) where E 0 is extension by zero from ∂ + SM to ∂SM. We define H 1 T,0 (∂ + SM) as the closure of C ∞ c ((∂ + SM) int ) with respect to the H 1 T -norm, and H 1/2 T,0 (∂ + SM) as the interpolation space between L 2 and H 1 T,0 (∂ + SM). Since E 0 is bounded H 1 T,0 (∂ + SM) → H 1 T (∂SM) and L 2 (∂ + SM) → L 2 (∂SM), it is also bounded (6.11) Also, our approach with the Pestov identity as it stands is unable to produce stability estimates for higher order Sobolev norms.

Appendix A. Vector spherical harmonics
In Lemma 4.3, we want to prove the frequency localization statement where u m ∈ Ω m and w l ∈ Ω l . As discussed in the proof of Lemma 4.3, this follows from the next localization statement.
for any u m ∈ Ω m and w l ∈ Ω l .
The proof is based on understanding vector spherical harmonics expansions on SM. This will be done next, and the proof of Lemma A.1 will be given in the end of this appendix.
If u m ∈ Ω m , then v ∇u m is an element of the space Thus each Z(x, · ) is a vector field on S x M, and using the Sasaki metric it can be identified with a 1-form on S x M. The Hodge Laplacian dδ + δd acting on 1-forms on each S x M induces a vertical Laplacian on Z, Spherical harmonics expansions on Z are eigenfunction expansions of ∆ 1 . However, since each S x M is isometric to the standard round sphere S d−1 , it is really enough to understand spherical harmonics expansions of 1-forms on S d−1 as studied e.g. in [Bo09]. The case d = dim (M) = 2 is easy and reduces to Fourier expansions on the circle. The cases d = 3 and d ≥ 4 will have different features.
The case d = 2. Using the formulas in [PSU15,Appendix B] where λ m = m(m + d − 2) = m 2 , and one has the orthogonal decomposition In particular, any Z ∈ Z has the L 2 -orthogonal expansion The case d = 3. It turns out that when d = 3, the study of vector spherical harmonics still reduces to Fourier expansions on functions but there is extra structure.
Thus let d = dim (M) = 3, and fix x ∈ M. Recall that Z(x, · ) is a vector field on S x M. Since S x M is two-dimensional (and isometric to S 2 ), there is a well-defined rotation by 90 • on each T v (S x M) which induces a map * : where * SxM is the Hodge star operator on S x M and the musical isomorphisms act with respect to the metric g x on S x M.
Spherical harmonics expansions on Z are described as follows: , and one has the orthogonal decomposition N), one has the L 2 -orthogonal decomposition In particular, any Z ∈ Z has the L 2 -orthogonal expansion Proof. It is enough to work on S x M for fixed x, and since S x M is isometric to S 2 it is enough to determine the eigenvalues and eigenspaces of the Hodge Laplacian ∆ = dδ + δd in S 2 when acting on 1-forms. Recall that δ = − * d * and * * = (−1) k on k-forms.
If α is a 1-form on S 2 , one has the Hodge decomposition α = du + δβ where u is a 0-form and β is a 2-form (there are no nontrivial harmonic 1-forms on S 2 ). Moreover, β = − * w for some 0-form w and one has α = du + * dw.
Since u and w are smooth functions on S 2 , they have spherical harmonics expansions The Hodge Laplacian satisfies ∆d = d∆ and ∆ * d = * d∆. It follows that In the L 2 inner product on S 2 one has (du m , dw l ) = (∆u m , w l ) = λ m (u m , w l ) if u m ∈ Ω m , w l ∈ Ω l . Thus it follows that Next we will see how multiplication by A(x, v) ∈ Ω 1 affects Fourier decompositions on Z. Recall that on C ∞ (SM), multiplication by A splits as A = A + + A − where A ± : Ω m → Ω m±1 . In the case of Z, there is a new operator A 0 which allows us to describe how multiplication by A affects Fourier expansions.
Proof. Again it is enough to fix x and to work with differential forms on S 2 . So we are reduced to the following: if A(v) = a j v j is a scalar function in S 2 with a j ∈ C, and if d, δ and ∆ = dδ + δd are the corresponding operators on S 2 , we need to show that the operator A 0 (u) = dA, * du maps Ω m (S 2 ) to itself, and for any u ∈ C ∞ (S 2 ) we have The last statement will then follow since A commutes with * . Define vector fields on S 2 , ∂ j u = ∂ y j (u(y/|y|))| S 2 (see [PSU15, Appendix A] for more details). Then du = ∂ j u dv j , and * du is the cross product of v and du, which can be written as * du = (V j u) dv j with V 1 = v 2 ∂ 3 − v 3 ∂ 2 etc. In the following we will raise and lower indices with respect to the Euclidean metric in R 3 (i.e. write v j for v j etc). Since It follows that each V j maps Ω m (S 2 ) to itself, and the same is then true for A 0 . It remains to show (A.2) for u ∈ C ∞ (S 2 ). We begin with the Hodge decomposition where α, β ∈ C ∞ (S 2 ) are determined from ∆α = δ(Adu), To solve for the Fourier components of α and β, we expand the right hand sides of the above equations in Fourier series. One has Expanding u in spherical harmonics and using that ∆A = (d − 1)A we get (Note that (δ(Adu)) 0 = 1 2 (A − ∆u 1 − (d − 1)A − u 1 ) = 0.) It follows that for l ≥ 1, For the second equation, we have − * d(Adu) = − * (dA ∧ du) = * (dA ∧ * ( * du)) = dA, * du = A 0 u and therefore for l ≥ 1 one has β l = 1 λ l A 0 u l .
The case d ≥ 4. If d = dim (M) ≥ 4, the Laplacian ∆ 1 acting on Z will have two sets of eigenvalues corresponding to the Hodge demposition of 1-forms. If x is fixed, then S x M is isometric to S d−1 and any 1-form α on S x M has a Hodge decomposition where u is a 0-form and β is an exact 2-form on S x M. Correspondingly, any Z ∈ Z has the Hodge decomposition for some u ∈ C ∞ (SM) and β ∈ Z 2 , where Z 2 is the set of smooth 2-forms β x on each S x M varying smoothly with respect to x, and v δβ is the vertical codifferential of β identified with an element of Z. In fact one can take β ∈ Z 2,d , where Z 2,d is the set of 2-forms β ∈ Z 2 so that β x is exact for each x.
The following result is a consequence of [Bo09]. where Ω 2,d m = {β ∈ Z 2,d ; ∆ 2 β = µ m β} with ∆ 2 corresponding to the Hodge Laplacian on 2-forms on each S x M. In particular, any Z ∈ Z has the L 2 -orthogonal expansion where u m ∈ Ω m and β m ∈ Ω 2,d m . Next we see how multiplication by A ∈ Ω 1 affects Fourier decompositions. If Z, W ∈ Z we write Z ∧ W for the element of Z 2 obtained as the wedge product of the 1-forms corresponding to Z and W .
The proof uses the following lemma.
Lemma A.6. If u and v are 1-forms on a Riemannian manifold (M, g) and if ∆ = dδ + δd is the Hodge Laplacian, then where {E 1 , . . . , E n } is any local orthonormal frame, ∇ is the Levi-Civita connection and R is the Riemann curvature tensor.
Proof. Follows either by relating ∆ to the connection Laplacian via a Weitzenbock identity, or by computations in normal coordinates.
Proof of Lemma A.5. It is enough to fix x and to work with differential forms on S d−1 . Thus we are reduced to the following: if A(v) = a j v j is a scalar function in S d−1 with a j ∈ C, and if d, δ and ∆ = dδ + δd are the corresponding operators on S d−1 , we need to show that the operator maps Ω m (S d−1 ) to Ω 2,d m (S d−1 ), and for any u ∈ C ∞ (S d−1 ) we have Since A 0 u = d(A du), A 0 maps into the set of exact 2-forms. Let u ∈ Ω m . By Lemma A.6, for any local orthonormal frame {E j } of T (S d−1 ) we have using that u ∈ Ω m and A ∈ Ω 1 . Now if v ∈ S d−1 and w ∈ T v S d−1 with |w| = 1, and if γ(t) is the geodesic on S d−1 withγ(0) = w, one has ∇dA| v (w, w) = d 2 dt 2 A(γ(t)) t=0 = d 2 dt 2 (a j γ j (t)) t=0 = −A(v) using that geodesics are great circles. Thus ∇ E j dA| v (w) = −A(v) E j , w , which gives that ∇ E j dA| v = −A(v)E ♭ j and Also, on the sphere we have R(u ♯ , v ♯ , · , · ) = −u ∧ v. These facts imply that Thus A 0 maps Ω m to Ω 2,d m . We next show (A.3) for u ∈ C ∞ (S d−1 ). We begin with the Hodge decomposition A du = dα + δβ where the function α ∈ C ∞ (S d−1 ) and the exact 2-form β ∈ C ∞ (S d−1 , Λ 2 ) are determined from ∆α = δ(Adu), ∆β = d(Adu).