Invariant distributions, Beurling transforms and tensor tomography in higher dimensions

In the recent articles \cite{PSU1,PSU3}, a number of tensor tomography results were proved on two-dimensional manifolds. The purpose of this paper is to extend some of these methods to manifolds of any dimension. A central concept is the surjectivity of the adjoint of the geodesic ray transform, or equivalently the existence of certain distributions that are invariant under geodesic flow. We prove that on any Anosov manifold, one can find invariant distributions with controlled first Fourier coefficients. The proof is based on subelliptic type estimates and a Pestov identity. We present an alternative construction valid on manifolds with nonpositive curvature, based on the fact that a natural Beurling transform on such manifolds turns out to be essentially a contraction. Finally, we obtain uniqueness results in tensor tomography both on simple and Anosov manifolds that improve earlier results by assuming a condition on the terminator value for a modified Jacobi equation.


Introduction
The unifying theme for most results in this paper is that of invariant distributions (distributions invariant under the geodesic flow). We also discuss related harmonic analysis and dynamical concepts, and the applications of these ideas in geometric inverse problems. The paper employs results from several different areas. For the benefit of the reader, we begin with a brief overview of the topics that will appear: • Geodesic flows. The basic setting is a closed oriented Riemannian manifold (M, g).
If SM is the unit tangent bundle, the geodesic flow φ t is a dynamical system on SM generated by the geodesic vector field X on SM. We consider situations where the geodesic flow displays chaotic behavior (ergodicity etc.), with (M, g) satisfying conditions such as: negative or nonpositive sectional curvature, Anosov geodesic flow, no conjugate points, or rank one conditions.
• Invariant distributions. The first item of interest are distributions on SM that are invariant under geodesic flow, i.e. distributional solutions of X w = 0 in SM. If the geodesic flow is ergodic, there are no nonconstant solutions in L 2 (SM), but suitable distributional solutions turn out to be useful. We give two constructions of invariant distributions with controlled first Fourier coefficients: one on nonpositively curved manifolds (via a Beurling transform), and one on manifolds with Anosov geodesic flow (via subelliptic L 2 estimates).
• Harmonic analysis on S M. The constructions of invariant distributions are based on harmonic analysis on SM. We will consider spherical harmonics (or Fourier) expansions of functions in L 2 (SM) with respect to vertical variables. The geodesic vector field X has a splitting X = X + + X − which respects these expansions. The • Dynamical aspects. For L 2 estimates, we need certain notions that interpolate between the "no conjugate points" and "nonpositive curvature" conditions: αcontrolled manifolds, β-conjugate points, and terminator value β T er . These conditions are studied from the dynamical systems point of view, leading to a new characterization of manifolds with Anosov geodesic flow.
• Geometric inverse problems. Our main motivation for studying the above questions comes from several geometric inverse problems related to the ray transform I m , which encodes the integrals of a symmetric m-tensor field over all closed geodesics. Basic questions include the injectivity of I m (up to natural obstruction) and, dually, the surjectivity of I * m . Our results on invariant distributions lead to surjectivity of I * m , and consequently to injectivity of I m and spectral rigidity results, under various conditions. Analogous results are discussed also on compact manifolds with boundary.

Invariant distributions.
Let (M, g) be a compact oriented Riemannian manifold with or without boundary, and let X be the geodesic vector field regarded as a first order differential operator X : C ∞ (SM) → C ∞ (SM) acting on functions on the unit tangent bundle SM. This paper is concerned with weak solutions u to the transport equation Xu = f under various conditions on the metric g. A good understanding of the transport equation is essential in many geometric inverse problems such as tensor tomography, boundary rigidity and spectral rigidity. It also plays a fundamental role in the study of the dynamics of the geodesic flow. For example, ergodicity of the geodesic flow on a closed manifold is just the absence of non-constant L 2 functions w with X w = 0. Similarly, certain solvability results for the equation Xu = f on a manifold with boundary are equivalent with nontrapping conditions for the geodesic flow [15].
For the most part, we will consider closed manifolds (M, g). In this paper we are going to exhibit many invariant distributions (distributional solutions to X w = 0) having prescribed initial Fourier components and Sobolev regularity H −1 . This will be possible when (M, g) is either Anosov or a rank one manifold of non-positive sectional curvature. Both cases contain in particular manifolds of negative sectional curvature. Invariant distributions will be constructed by two mechanisms: one emanates from a fundamental L 2 -identity, called the Pestov identity, and subelliptic type estimates for certain operators associated with the transport equation. The other mechanism consists in the introduction of a Beurling transform acting on trace-free symmetric tensor fields. The key property of this transform, also proved via the Pestov identity, is that it is essentially a contraction when the sectional curvature is non-positive.
Let us recall that the rank, rank(v), of a unit vector v ∈ T x M is the dimension of the space of parallel Jacobi fields along the geodesic γ determined by (x, v) ∈ SM. (A parallel Jacobi field is a parallel vector field that also satisfies the Jacobi equation.) Sinceγ is trivially a parallel Jacobi field, rank(v) ≥ 1 and rank(v) ≥ 2 if and only if there is a non-zero parallel Jacobi field J along γ which is orthogonal toγ . Given such a field, the sectional curvature of the two-plane spanned by J (t) andγ (t) is zero for all t. The rank of (M, g) is defined as the minimum of rank(v) over all v.
Start with an element w k ∈ k with X − w k = 0 (i.e. a solenoidal trace-free symmetric k-tensor). Surjectivity of X − means that there is a unique w k+2 ∈ k+2 such that w k+2 is orthogonal to Ker X − (equivalently, w k+2 has minimal L 2 norm) and Continuing in this fashion we construct a formal solution w = w k + w k+2 + · · · to the transport equation X w = 0. This formal solution can be conveniently expressed using what we call the Beurling transform.
Definition Let (M, g) be a closed manifold without nontrivial conformal Killing tensors. Given k ≥ 0 and f k ∈ k , there is a unique function f k+2 ∈ k+2 orthogonal to Ker X − such that X − f k+2 = −X + f k . We define the Beurling transform as The terminology comes from the fact that if M is two-dimensional, then bundles of symmetric trace-free tensors can be expressed in terms of holomorphic line bundles and X ± correspond to ∂ and ∂ operators. If X ± are split in terms of the η ± operators of Guillemin-Kazhdan [21], then −B corresponds exactly to ∂ −1 ∂ and ∂ −1 ∂ operators as in the classical Beurling transform in the complex plane [2]. See Appendix B for more details. Related transforms have been studied on differential forms in R n [27], on Riemannian manifolds [34], and in hyperbolic space [25]. The Beurling transform considered in this paper (which is essentially the first ladder operator in [22] when n = 2) seems to be different from these. Let us return to invariant distributions: Definition Let (M, g) be a closed manifold without conformal Killing tensors, and let f ∈ k 0 satisfy X − f = 0. The formal invariant distribution starting at f is the formal sum At this point we do not know if the sum converges in any reasonable sense. However, if the manifold has nonpositive sectional curvature it converges nicely. This follows from the fact that the Beurling transform is essentially a contraction on such manifolds and this is the content of our first theorem.
where A n (m 0 ) = ∞ j=0 C n (m 0 + 2 j) is a finite constant satisfying In the theorem we use the mixed norm spaces where as usual m = (1 + m 2 ) 1/2 . Note that the norm of the Beurling transform is always ≤ 1 in dimensions n ≥ 4, is ≤ 1 in two dimensions unless m = 0, and is sufficiently close to 1 in three dimensions so that formal invariant distributions exist in nonpositive curvature. We remark that the same construction gives a more general family of distributions, where finitely many Fourier coefficients are obtained by taking any solutions of X − f k+2 = −X + f k (not necessarily orthogonal to Ker X − ) and the remaining coefficients are obtained from the Beurling transform.
We shall see below that a rank one manifold of non-positive sectional curvature must be free of conformal Killing tensors and thus by Theorem 1.1 given any with X w = 0 and w k 0 = f . We state this explicitly in the following corollary.
Our next task is to discuss distributional solutions to X w = 0 replacing the hypothesis of sectional curvature K ≤ 0 by the Anosov condition. This makes the problem harder but nevertheless we can show by using the Pestov identity and a duality argument: Theorem 1.3 Let (M, g) be an Anosov manifold. Given any f ∈ k for k = 0, 1 with Let us compare the invariant distributions in Theorems 1.1 and 1.3 for the case k = 0, solving the equation X w = 0 with w 0 = f 0 for given f 0 ∈ 0 . The formal invariant distributions in Theorem 1.1 exist on manifolds with K ≤ 0, they lie in , and they are the unique invariant distributions whose Fourier coefficients w k are minimal energy solutions of X − w k+2 = −X + w k . On the other hand, by the results in Sect. 8 the invariant distributions in Theorem 1.3 exist on any Anosov manifold, they lie in L 2 x H −1 v , and they are the unique invariant distributions for which the quantity ∞ m=1 1 m(m+n−2) w m 2 L 2 is minimal. It is an interesting question whether there is any relation between these two classes of invariant distributions.
Very recently, Theorem 1.3 has been improved in [19]. The paper [19] introduces a new set of tools coming from microlocal analysis of Anosov flows and it also gives information on the wave front set of the invariant distributions.
We remark that in general, an arbitrary transitive Anosov flow has a plethora of invariant measures and distributions (e.g. equilibrium states, cf. [31]), but the ones in Theorems 1.1-1.3 are geometric since they really depend on the geometry of the spherical fibration π : SM → M. In the case of surfaces of constant negative curvature these distributions and their regularity are discussed in [1,Sect. 2].
Interlude. One of our main motivations for considering invariant distributions as above has been the fundamental interplay that has been emerging in recent years between injectivity properties of the geodesic ray transform and the existence of special solutions to the transport equation Xu = 0. This interplay has lead to the solution of several long-standing geometric inverse problems, especially in two dimensions [19,20,37,38,41,44,51].
We will now try to explain this interplay in the easier setting of simply connected compact manifolds with strictly convex boundary and without conjugate points. Such manifolds are called simple and an obvious example is the region enclosed by a simple, closed and strictly convex curve in the plane (less obvious examples are small perturbations of this flat region). It is well known that under these assumptions M is topologically a ball and geodesics hit the boundary in finite time, i.e. (M, g) is nontrapping. The notion of simple manifold appears naturally in the context of the boundary rigidity problem [35] and it has been at the center of recent activity on geometric inverse problems. Geodesics where τ (x, v) is the exit time of γ (t, x, v). Even though the definition is given for smooth functions, I acting on L 2 -functions is well-defined too. Integrating or averag-ing along the orbits of a group action is the obvious way to produce invariant objects and this case is no exception: I f naturally gives rise to a first integral of the geodesic flow since it is a function defined on ∂ + (SM) which parametrizes the orbits of the geodesic flow. In general, if h ∈ C(∂ + (SM)), ) is a first integral. There are natural L 2 -inner products so that we can consider the adjoint I * : L 2 (∂ + (SM)) → L 2 (SM) which turns out to be I * h = h .
The situation gets interesting once we start restricting I to relevant subspaces of L 2 (SM), a typical example being L 2 (M), and in this case the resulting geodesic ray transform is denoted by I 0 and I * 0 h is easily seen to be the Fourier coefficient of zero degree of h . Hence surjectivity of I * 0 can be interpreted as the existence of a solution to Xu = 0 with prescribed u 0 . In the context of simple manifolds a fundamental property is that I * 0 I 0 is an elliptic pseudo-differential operator of order −1 in the interior of M [44] and this combined with injectivity results for I 0 gives rise to the desired surjectivity properties for I * 0 . Symmetric tensors can also be seen as interesting subspaces of L 2 (SM) and the same ideas apply. Sometimes it is convenient to use this interplay backwards: if one can construct invariant distributions with certain prescribed Fourier components one might be able to prove injectivity of the relevant geodesic ray transform. In the case of Anosov manifolds the situation is technically more challenging, but the guiding principles remain and this was exploited and explained at great length in [41]. In fact Theorem 1.3 extends [41, Theorem 1.4 and Theorem 1.5] to any dimension, and the result should be regarded as the analogue of the surjectivity result for the adjoint of the geodesic ray transform on simple manifolds acting on functions and one-forms [12,44].
It seems convenient at this point to conclude the interlude and give details about how to define the geodesic ray transform in the context of closed manifolds.
Inverse problems on closed manifolds. Let (M, g) be a closed oriented manifold, and let G be the set of periodic geodesics parametrized by arc length. The ray transform of a symmetric m-tensor field f on M is defined by It is easy to check that I m (dh)(γ ) = 0 for all γ ∈ G if h is a symmetric (m − 1)-tensor and d denotes the symmetrized Levi-Civita covariant derivative acting on symmetric tensors. The tensor tomography problem asks whether these are the only tensors in the kernel of I m . When this occurs I m is said to be solenoidal injective or s-injective. Of course, one would expect a positive answer only on manifolds (M, g) with sufficiently many periodic geodesics. The Anosov manifolds are one reasonable class where this question has been studied.
Our main result in this direction is: Note that the hypotheses imply by Corollary 6.13 that (M, g) is an Anosov manifold. In [10] it was shown that I 0 and I 1 are s-injective on any Anosov manifold. Theorem 1.4 was proved earlier for negative curvature in [21] for n = 2 and in [6] for arbitrary n. For an arbitrary Anosov surface, s-injectivity of I 2 was established in [41], and earlier the same result was proved in [50] if additionally the surface has no focal points. Solenoidal injectivity of I m for an Anosov surface and m ≥ 3 is finally settled in [19]. It is also known that for any n and m the kernel of I m is finite dimensional on any Anosov manifold [10]. Note that for m = 2 the condition in Theorem 1.4 becomes β T er > 2(n+1) 2+n and hence for any Anosov manifold with β T er ∈ [2, ∞], I 2 is s-injective.
There are numerous motivations for considering the tensor tomography problem for an Anosov manifold but perhaps the most notorious one is that of spectral rigidity which involves I 2 . In Guillemin and Kazhdan [21] proved that if (M, g) is an Anosov manifold such that I 2 is s-injective then (M, g) is spectrally rigid. This means that if (g s ) is a smooth family of Riemannian metrics on M for s ∈ (−ε, ε) such that g 0 = g and the spectra of − g s coincide up to multiplicity, Inverse problems on manifolds with boundary. The results up to this point have been about closed manifolds, but our techniques also apply to the case of simple manifolds with boundary. In fact in [41] we advocated a strong analogy between simple and Anosov manifolds and this has proved quite fruitful as we hinted in the interlude. The tensor tomography problem on simple manifolds is well known and we refer to [40,47] for extensive discussions. The geodesic ray transform of a symmetric tensor f is defined by where we abuse notation and we denote by f also the function ( If h is a symmetric (m−1)-tensor field with h| ∂ M = 0, then I m (dh) = 0. The transform I m is said to be s-injective if these are the only elements in the kernel. The terminology arises from the fact that any tensor field f may be written uniquely as f = f s + dh, where f s is a symmetric m-tensor with zero divergence and h is an (m −1)-tensor with h| ∂ M = 0 (cf. [47]). The tensor fields f s and dh are called respectively the solenoidal and potential parts of f . Saying that I m is s-injective is saying precisely that I m is injective on the set of solenoidal tensors.
We can now state our last theorem. This theorem improves the results in [7,42] in which s-injectivity of I m is proved under the weaker condition β T er ≥ m(m+n) 2m+n−1 . It is also known that I m is always sinjective on simple surfaces [38], and I 2 is s-injective for a generic class of simple metrics including real-analytic ones [52], but it remains an open question whether I 2 is s-injective on arbitrary simple manifolds of dimension ≥ 3 (see however the recent papers [20,53] for results under different conditions).
Open questions, structure of the paper. Here we list some open questions related to the topics of this article: • Is I 2 s-injective on simple manifolds with dim M ≥ 3?
• Is I 2 s-injective on Anosov manifolds with dim M ≥ 3?
• Do Anosov manifolds with dim M ≥ 3 have no nontrivial conformal Killing tensors? Can one find more general conditions to ensure this? • Is there any relation between the formal and minimal invariant distributions in Theorems 1.1 and 1.3? • Can one say more about the norm of the Beurling transform?
Much of this paper deals with multidimensional generalizations of the results in [38,41]. The paper is organized as follows. Section 1 is the introduction and states the main results. In Sect. 2 we prove the Pestov identity following the approach of [38]. The result is well known, but we give a simple proof based on three first order operators on SM and their commutator formulas which are multidimensional analogues of the structure equations on the circle bundle of a Riemannian surface. Section 3 follows [11,23] and discusses spherical harmonic expansions in the vertical variable, which generalize Fourier expansions in the angular variable for dim M = 2. Section 4 considers the notion of α-controlled manifolds (introduced in [41] for surfaces), and contains certain useful estimates.
In Sect. 5 we consider the Beurling transform and show that it is essentially a contraction on nonpositively curved manifolds. Section 6 studies β-conjugate points and terminator values, and Sect. 7 shows that β T er ≥ β implies (β − 1)/β-controlled. Sections 8 and 9 are concerned with subelliptic estimates coming from the Pestov identity and the existence of invariant distributions related to surjectivity of I * m , and in Sect. 10 we discuss solenoidal injectivity for I m . All the work up to this point has been on closed manifolds; Sect. 11 considers analogous results on compact manifolds with boundary. There are two appendices. The first appendix contains local coordinate formulas for operators arising in this paper and proves the basic commutator formulas. The second appendix considers the results of this paper specialized to the two-dimensional case, mentions a relation between Pestov and Guillemin-Kazhdan energy identities, and connects the present treatment to the works [38,41].

Commutator formulas and Pestov identity
In this section we introduce and prove the fundamental energy identity which is the basis of a considerable part of our work. This identity is a special case of a more general identity already in the literature. However we take a slightly different approach to its derivation which emphasizes the role of the unit sphere bundle very much in the spirit of [38]. For other presentations see [33,47], [8,Theorem 4.8].
Let (M, g) be a closed Riemannian manifold with unit sphere bundle π : SM → M and as always let X be the geodesic vector field. It is well known that SM carries a canonical metric called the Sasaki metric. If we let V denote 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. We shall use this identification freely below (see for example [33,36] for details on these facts).
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 The derivatives h ∇u and v ∇u are called horizontal and vertical derivatives respectively. Note that this differs slightly from the definitions in [33,47] since here we are considering all objects defined on SM as opposed to T M. One advantage of this is to make more transparent the connection with our approach in [38] for the two-dimensional case.
We shall denote by Z the set of smooth functions Z : Observe that X acts on Z as follows: where φ t is the geodesic flow. Note that Z (t) := Z (φ t (x, v)) is a vector field along the geodesic γ determined by (x, v), so it makes sense to take its covariant derivative with respect to the Levi-Civita connection of M. Since Z ,γ = 0 it follows that D Z dt ,γ = 0 and hence X Z ∈ Z.
Another way to describe the elements of Z is a follows. Consider the pull-back bundle π * T M → SM. 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 . Observe that N carries a natural L 2 -inner product and with respect to this product the formal adjoints of v ∇ : C ∞ (SM) → Z and h ∇ : C ∞ (SM) → Z are denoted by − v div and − 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.
The next lemma contains the basic commutator formulas. The first three of these formulas are the analogues of the structure equations used in [38] in two dimensions.

Lemma 2.1
The following commutator formulas hold on C ∞ (SM): Taking adjoints, we also have the following commutator formulas on Z: These commutator formulas can be extracted from the calculus of semibasic tensor fields [11,47]. For completeness we also prove them in Appendix A, which contains local coordinate expressions for many operators arising in this paper.
We next prove the Pestov identity, following the approach of [38]. We briefly recall the motivation for this identity which comes from showing that the ray transform I 0 on Anosov manifolds is injective. If f ∈ C ∞ (M) satisfies I 0 f = 0, meaning that the integrals of f over periodic geodesics vanish, then the Livsic theorem [13] implies The Pestov identity is the following energy estimate involving the norm v ∇ Xu 2 . It implies that a smooth solution of v ∇ Xu = 0 on an Anosov manifold must be constant, and thus by the above argument any smooth function f on M with I 0 f = 0 must be zero. All norms and inner products will be L 2 .

Proposition 2.2 Let (M, g) be a closed Riemannian manifold. Then
This is the required estimate.

Remark 2.3
The same identity holds, with the same proof, for (M, g) a compact Riemannian manifold with boundary provided that u| ∂(SM) = 0.

Spherical harmonics expansions
Vertical Laplacian. In this section we consider the vertical Laplacian If we fix a point x ∈ M and consider S x M with the inner product determined by Let us recall that for S n−1 endowed with the canonical metric, the Laplacian S n−1 has eigenvalues λ m = m(n + m − 2) for m = 0, 1, 2, . . .. The eigenspace H m of λ m consists of the spherical harmonics of degree m which are in turn the restriction to S n−1 of homogeneous harmonic polynomials of degree m in R n . Hence we have an orthogonal decomposition Using this we can perform a similar orthogonal decomposition for the vertical Laplacian on SM, which on each fibre over M is just the decomposition in S n−1 . We set m := H m (SM) ∩ C ∞ (SM). Then a function u is in m if and only if u = m(m + n − 2)u (for details on this see [11,23]). If u ∈ L 2 (SM), this decomposition will be written as Note that by orthogonality we have the identities (recall that all norms are L 2 ) Decomposition of X . The geodesic vector field behaves nicely with respect to the decomposition into fibrewise spherical harmonics: it maps m into m−1 ⊕ m+1 [23,Proposition 3.2]. Hence on m we can write By [23,Proposition 3.7] the operator X + is overdetermined elliptic (i.e. it has injective principal symbol). One can gain insight into why the decomposition X = X − + X + holds as follows. Fix x ∈ M and consider local coordinates which are geodesic at x (i.e. all Christoffel symbols vanish at x).
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 a spherical harmonics of degree m − 1 and m + 1. Since the v i have degree one, this explains why X maps m to m−1 ⊕ m+1 .
Next we give some basic properties of X ± .
Proof Since X preserves the volume of SM, X * = −X and hence for f ∈ m and h ∈ m+1 we have The self-adjoint operator X − X + is elliptic and its kernel coincides with the kernel of X + . (This is also considered in [11].)

Lemma 3.2 We have an orthogonal decomposition
Clearly X − f is orthogonal to the kernel of X − X + since the kernel of X − X + coincides with the kernel of X + . Hence by ellipticity there is a smooth solution h ∈ m−1 such that

Lemma 3.3 Given u ∈ m we have
Proof This is obvious once we know that X ± : m → m±1 .
Identification with trace free symmetric tensors. There is an identification of m with the smooth trace free symmetric tensor fields of degree m on M which we denote by m [11,23]. More precisely, as in [11] let λ : C ∞ (S * τ ) → C ∞ (SM) be the map which takes a symmetric m-tensor f and maps it into the function The map λ turns out to be an isomorphism between m and m . In fact up to a factor which depends on m and n only it is a linear isometry when the spaces are endowed with the obvious L 2 -inner products, cf. [ for u ∈ m . The expression for X + in terms of tensors is as follows. If d denotes symmetrized covariant derivative (formal adjoint of −δ) and p denotes orthogonal projection onto m+1 then In other words, up to λ, X + is pd and X − is m n+2m−2 δ. The operator X + , at least for m = 1, has many names and is known as the conformal Killing operator, trace-free deformation tensor, or Ahlfors operator.
Under this identification Ker (X + : m → m+1 ) consists of the conformal Killing symmetric tensor fields of rank m, a finite dimensional space. The dimension of this space depends only on the conformal class of the metric.
Pestov identity on m . We will see in Appendix B that in two dimensions, the Pestov identity specialized to functions in m is just the Guillemin-Kazhdan energy identity [21] involving η + and η − . We record here a multidimensional version of this fact.

Proposition 3.4 Let (M, g) be a closed Riemannian manifold. If the Pestov identity is applied to functions in m , one obtains the identity
For the proof, we need a commutator formula for the geodesic vector field and the vertical Laplacian.

Lemma 3.5 The following commutator formula holds:
Proof Using Lemma 2.1 repeatedly we have The result follows directly from Proposition 2.2 and the calculation above.
The Pestov identity on m immediately implies a vanishing theorem for conformal Killing tensors. The following result is proved also in [11, Theorem 1.6] (except for the observation about rank one manifolds). In particular Xu = X − u + X + u = 0. If the geodesic flow is transitive, this implies u = 0.
In [16,Theorem 3.11], P. Eberlein proved that the geodesic flow of a closed rank one manifold of non-positive sectional curvature is transitive, hence these manifolds do not have CKTs.

α-Controlled manifolds
From the Pestov identity in Proposition 2.2, one would like to obtain good lower bounds for v ∇ Xu 2 . On the other side of the identity, the term (n − 1) Xu 2 is always nonnegative. The next definition is concerned with the remaining terms (this notion was introduced in [41] for two-dimensional manifolds).

Definition 4.1
Let α be a real number. We say that a closed Riemannian manifold is the same as the sign of the sectional curvature of the two-plane spanned by v and Z (x, v).
We record the following properties.

Lemma 4.2 Let (M, g) be a closed Riemannian manifold. Then
which is equivalent with nonpositive sectional curvature. The fact that manifolds without conjugate points are 0-controlled will be proved in Proposition 7.1. Similarly, the fact that Anosov manifolds are α-controlled for some α > 0 will be proved in Theorem 7.2.
We conclude this section with a lemma that will allow to get lower bounds by using the α-controlled assumption.
If u ∈ m , we have Observe that degree zero is irrelevant since v ∇u 0 = 0. The proof relies on another lemma: As a consequence, for any u ∈ C ∞ (SM) we have the decomposition Proof By Lemmas 3.3 and 3.5, which proves the first claim. For the second one, we note that Proof of Lemma 4.3 Let u = ∞ l=m u l with m ≥ 2. First note that We use the decomposition in Lemma 4.4, which implies that where w l ∈ l for l ≥ m + 1 are given by and where Z ∈ Z satisfies v div Z = 0. Taking the L 2 norm squared, and noting that The claims for m = 1 or for u ∈ m are essentially the same.

Beurling transform
In this section we will prove Theorem 1.1. The main step is the following inequality, where the point is that the constant in the norm estimate is always ≤ 1 in dimensions n ≥ 4, is ≤ 1 in two dimensions unless m = 1, and is sufficiently close to 1 in three dimensions for all practical purposes.
Lemma 5.1 Let (M, g) be a closed Riemannian manifold having nonpositive sectional curvature. One has for any m ≥ 1 Recall that the Beurling transform can be defined on any manifold (M, g) that is free of nontrivial conformal Killing tensors. We know that this holds on any surface of genus ≥ 2, and more generally on any manifold whose conformal class contains a negatively curved metric [11] or a rank one metric of non-positive curvature (cf. Corollary 3.6).
If (M, g) has no nontrivial conformal Killing tensors, the operator X − : k → k−1 is surjective for all k ≥ 2. Hence given k ≥ 0 and f k ∈ k there is a unique function f k+2 ∈ k+2 orthogonal to Ker(X − ) such that X − f k+2 = −X + f k . We defined the Beurling transform to be the map The next lemma shows that a norm estimate relating X − and X + implies a bound for the Beurling transform whenever it is defined.

Lemma 5.2
Let (M, g) be a closed Riemannian manifold, let m ≥ 0, and assume that for some A > 0 one has If additionally (M, g) has no conformal Killing (m + 1)-tensors, then the Beurling transform is well defined m → m+2 and Proof Let f ∈ m and u = B f , so that X − u = −X + f and u ⊥ Ker(X − ). Then by Lemma 3.2, u = X + v for some v ∈ m+1 . We have By Cauchy-Schwarz and by the norm estimate in the statement, we have This shows that u ≤ A f as required.

1)
where C α is the positive constant satisfying In particular, if (M, g) has nonpositive sectional curvature and if n = 3, then one has For n = 3 and (M, g) of nonpositive sectional curvature one has for u ∈ m and m ≥ 1.
Proof Let u ∈ m with m ≥ 2. The Pestov identity in Proposition 2.2 and the α- Since Xu = X + u + X − u and u ∈ m , orthogonality implies that Collecting these facts gives The constant in brackets on the right is always positive since α ≤ 1, and the constant in brackets on the left is positive if α satisfies the condition in the statement. This proves (5.1). When m = 1 the Pestov identity and Lemma 4.3 yield, by the argument above, which implies (5.2). If (M, g) has nonpositive sectional curvature, then one can take α = 1. Computing C α in this case gives, for u ∈ m with m ≥ 2, Simplifying the constant further implies that The constant is always = 1 if n = 2 or n = 4, is < 1 if n ≥ 5, but is > 1 when n = 3. Proof By Lemma 5.3, the conditions imply that any solution u ∈ m of X + u = 0 also satisfies X − u = 0. Thus Xu = 0, and transitivity of the geodesic flow implies that u is a constant. Since u ∈ m , we get u = 0.
It is now easy to give the proofs of Lemma 5.
where C n (k) := D n (k + 1) and coincides with the definition in the statement of Theorem 1.1. Let w be as in Theorem 1.1. Since w m 0 +2k = B k f , the estimate on Fourier coefficients follows immediately from the fact that the Beurling transform satisfies the inequality above. If ε > 0 we have where A is any constant such that A n (k) ≤ A for all n and k. This shows that w ∈

Symplectic cocycles, Green solutions and terminator values
In this section we wish to give a characterization of the Anosov condition that involves a very simple one parameter family of symplectic cocycles over the geodesic flow. The characterization will not require a perturbation of the underlying metric and it will be in terms of a critical value related to conjugate points of the one-parameter family of cocycles. The results here are generalizations to arbitrary dimensions of the the results in [41,Sect. 7].
Let (M, g) be a closed Riemannian manifold of dimension n and let φ t : SM → SM denote the geodesic flow acting on the unit sphere bundle SM. For details of this we refer to [36]. The linear maps dφ t : (x, v)) define a symplectic cocycle over the geodesic flow with respect to the symplectic form ω := −dα| W . We can now embed the derivative cocycle into a 1-parameter family by considering for each β ∈ R, the β-Jacobi equation: This also defines a symplectic cocycle (on the symplectic vector bundle (W, ω) over SM) where J β ξ is the unique solution to (6.1) with initial conditions ξ = J β ξ (0),J β ξ (0) . Clearly 1 t = dφ t . The cocycle β t is generated by the following one-parameter family of infinitesimal generators: and since R(x, v) is a symmetric linear map, it is immediate that β t is symplectic (see [30] for information on cocycles over dynamical systems). In this section we shall study this family of cocycles putting emphasis on two properties: absence of conjugate points and hyperbolicity. For completeness we first give the following two definitions.

Definition 6.2 The cocycle
β t is said to be hyperbolic if there is a continuous invariant splitting W = E u β ⊕ E s β , and constants C > 0 and 0 < ρ < 1 < η such that for all t > 0 we have In order to simplify the notation we will often drop the subscript β in E s,u β hoping that this will not cause confusion.

Remark 6.3
It is well known that the bundles E s and E u are (n − 1)-dimensional and Lagrangian. For the purposes of Definition 6.2 one could use any norm on W since they are all uniformly equivalent due to the compactness of SM (the constants C, η and ρ would be different though). There is however an obvious choice of inner product on W = H ⊕ V: on each H and V we have the inner product induced by g on {v} ⊥ ; this is the same as the restriction of the Sasaki metric on Of course, saying that 1 t is hyperbolic is the same as saying that (M, g) is an Anosov manifold. The two properties are related by the following: Proof For β = 1 this is exactly the content of Klingenberg's theorem mentioned in the introduction [32]. For arbitrary β this can proved, for example, using the results in [4] as we now explain. Let (SM) denote the bundle of Lagrangian subspaces in W . The subbundles V, E u,s are all Lagrangian sections of this bundle, but with E s,u only continuous. The key property of β t is that it is optical or positively twisted with respect to V. This means that for any Lagrangian subspace λ in W (x, v) intersecting V(x, v) non-trivially, the form Here the term D dt indicates that along a curve ξ(t) = (ξ h (t), ξ v (t)) we just take covariant derivatives of each one of the components ξ h (t), ξ v (t). It is very well-known that the geodesic flow is optical with respect to the vertical distribution, and we can now quickly check that the same is true for β t . Indeed, take η ∈ V (x, v), then using the β-Jacobi equation so we have the desired positive twisting with respect to the vertical distribution. We are now in good shape to apply Theorem 4.8 in [4] to either E s or E u to conclude that there are no β-conjugate points and that both E s and E u are transversal to V. Strictly speaking Theorem 4.8 in [4] is stated for the derivative cocycle of a Hamiltonian flow, but it is plain the proof works just the same for symplectic cocycles as in our context.
Let us describe now the Green limit solutions when β t is free of conjugate points [18]. For two dimensions these constructions are due to E. Hopf [24]. We shall follow the elegant and short exposition in [28] to construct our solutions.
Set T (x, v)). Since dπ : E T (x, v) → {v} ⊥ is an isomorphism, there exists a linear map S T (x, v) : {v} ⊥ → {v} ⊥ such that E T is the graph of S T ; in other words, given w ∈ {v} ⊥ , there exists a unique w ∈ {v} ⊥ such that (w, w ) ∈ E T and we set S T w := w . Since the cocycle is symplectic E T is Lagrangian. This is equivalent to S T being symmetric. The claim is that S T has a limit as T → ∞.
Recall that it is possible to give a partial order to the set of symmetric linear maps by declaring that A B if A − B is positive definite. Then we need to show that S T is monotone and bounded. For t < s, the linear map S s − S t is obviously symmetric and we observe that its signature does not change in the region 0 < t < s since β t has no conjugate points. An elementary estimate shows that S s = − 1 s Id + O(s 2 ) and hence S s − S t is positive definite for 0 < t < s. Consequently S T is monotone increasing as T goes to ∞. Similarly the signature of S s − S t does not change in the region t < 0 < s and by the same estimate, S s − S t is negative definite in this region. Hence S T is bounded by S t for t < 0. We let The graph of U − determines an invariant Lagrangian subbundle E − (stable bundle) which is in general only measurable. Moreover U − is measurable, bounded and satisfies the Riccati equationU These claims are all proved as in the case of β = 1 (geodesic flows). Similarly, by considering β T ((φ −T (x, v)) we obtain an invariant subbundle E + (unstable bundle) and a symmetric map U + also solving the Riccati equation above along geodesics. Let us agree that given two symmetric linear maps A and B, A B means that A − B is a non-negative operator. By construction U + U − since S t S s for t < 0 < s. We summarize these properties in the following lemma.

Moreover, U + − U − is a non-negative operator.
We call the symmetric maps U ± the Green solutions and often we shall use a subscript β to indicate that they are associated with the cocycle β t .

Theorem 6.6 Assume that
Proof For β = 1 this was proved by Eberlein in [17]. To prove the theorem for arbitrary β we shall make use of Theorem 0.2 in [5]. When applied to our situation, it says that  β is E u and E + = E u . Since E s and E u are transversal it follows that E + and E − are also transversal.
Suppose now E + and E − are transversal everywhere. By the argument above, any non-trivial solution J of the β-Jacobi equation must be unbounded. Since where J is the unique solution to the β-Jacobi equation with (J (0),J (0)) = ξ , it follows that (6.3) holds and hence β t is hyperbolic. Below we will find convenient to use the following well-known comparison lemma (cf. [45, p. 340]): Lemma 6.8 Let U i (t), i = 0, 1 be solutions of the matrix initial value problemṡ Proof Let U ± β 0 be the Green solutions associated with This already implies that the cocycle aβ 0 t is free of conjugate points. Indeed, let and q ± aβ 0 R. Lemma 6.8 implies that the cocycle aβ 0 t is free of conjugate points. Moreover, it also implies that for all t > −r . By letting r → ∞ we derive and similarly Putting everything together we have Suppose now that This theorem motivates the following definition.

Definition 6.10 Let (M, g) be a closed Riemannian manifold. Let β T er ∈ [0, ∞]
denote the supremum of the values of β ≥ 0 for which β t is free of conjugate points. We call β T er the terminator value of the manifold.
Observe that β T er could be zero. For instance, this would be the case if (M, g) has positive sectional curvature. We complement this definition with the following lemma: Proof The first claim follows from the following general observation: if β 0 t has conjugate points, then there is ε > 0 such that for any β ∈ (β 0 − ε, β 0 + ε) the cocycle β t has conjugate points. This is proved in the same way (in fact it is easier) as the proof that the existence of conjugate points is an open condition on the metric g in the C k -topology for any k ≥ 2; see for example [46,Corollary 1.2].
For the second claim note that obviously sectional curvature K ≤ 0 implies that β T er = ∞. For the converse, suppose that there is a point x ∈ M and a 2-plane σ ⊂ T x M such that the sectional curvature K x (σ ) > 0. Let {v, w} denote an orthonormal basis of σ . Let γ denote the geodesic defined by (x, v) and let e(t) be the parallel transport of w along γ . Consider the 2-plane σ t spanned byγ (t) and e(t).
Since Since there are no β-conjugate points for all β ≥ 0 we must have (see for example [42, p. 46] for all β ≥ 0 which is clearly absurd. We now have the following purely geometric characterization of hyperbolicity (the parameter β is always ≥ 0 in what follows). Recall an orthogonal parallel Jacobi field is a parallel field along a geodesic γ , orthogonal toγ and such that it satisfies the Jacobi equation. Proof We know that if β t is hyperbolic then β ≤ β T er . Since hyperbolicity is an open condition we must have β < β T er . Finally if there is a geodesic with a nonzero orthogonal parallel Jacobi field J , then β t cannot be hyperbolic as such a field is bounded and solves the β-Jacobi equation for any β.
Consider β ∈ (0, β T er ) and assume that β t is not hyperbolic. By Theorem 6.6 there is a geodesic γ along which E + ∩ E − = {0}. Equivalently there is a non-zero β-Jacobi field J such thatJ = U − J = U + J along γ . Let a := β/β T er . Using (6.4) for β 0 = β T er we deduce that along γ we must havė Differentiating the equality U + β J = aU + β T er J with respect to t and using thatJ = U + β J we deriveU Using the Riccati equation we arrive at U + Since U + β T er is symmetric this implies that U + β T er J = 0 and hencė J = 0 which contradicts our hypotheses.
We can certainly rephrase this theorem using the definition of the rank of a unit vector given in the introduction. The condition of having no geodesic with a nonzero orthogonal parallel Jacobi field is equivalent to saying that every unit vector has rank one. As an immediate consequence we obtain the following geometric characterization of Anosov manifolds.

Corollary 6.13 A closed Riemannian manifold (M, g) is Anosov if and only if every unit vector has rank one and β T er > 1.
It is interesting to note that the main example in [3] has the property of being a non-Anosov closed surface with no conjugate points such that every vector has rank one. Hence for this example β T er = 1.

Anosov manifolds are α-controlled
Consider β > 0 such that there are no β-conjugate points and let U be one of the Green solutions satisfying the Riccati equatioṅ Recall that U is measurable, bounded and differentiable along the geodesic flow, see Lemma 6.5. The next identity is very similar to [9, Theorem 3.1], but here we give a short self-contained proof. (M, g) be a closed manifold without β-conjugate points for some β > 0. Then for any Z ∈ Z

Proposition 7.1 Let
Proof Let us write All we need to show is that For this note that X * = −X and that U is a symmetric operator, hence using the Riccati equation we derive for all Z ∈ Z.
Proof Since the geodesic flow is Anosov we have continuous stable and unstable bundles. It is well known that these are invariant, Lagrangian and contained in the kernel of the contact form. Moreover, by Theorem 6.4 the subbundles are transversal to the vertical subbundle. Thus we have two continuous symmetric maps U + and U − satisfying the Riccati equation with the property that the linear map Let A := X Z − U − Z and B := X Z − U + Z . Using Eq. (7.1) we see that A = B . Solving for Z we obtain Since U ± are continuous, there is a constant a > 0 independent of Z such that Z ≤ a A .
From these inequalities and (7.1) the existence of α easily follows.

0
In this section we will show that the Anosov condition allows to upgrade the Pestov identity in Proposition 2.2 to a subelliptic (H 1 to L 2 ) estimate for the operator v ∇ X . From this point on, we will use the notation Note that P maps C ∞ (SM) to Z = C ∞ (SM, N ) (see Sect. 2 for the definitions). Also recall from the discussion before Proposition 2.2 that uniqueness results for P imply injectivity for the geodesic ray transform I 0 . The subelliptic estimate may then be viewed as a quantitative injectivity result for I 0 .
If E is a subspace of D (SM), we write E for the subspace of those v ∈ E with (v, 1) = 0.

Theorem 8.1 Let (M, g) be an Anosov manifold. Then
More generally, for u ∈ H 1 (SM) with Pu ∈ L 2 (SM) one has Vol(SM) SM u is the average of u.
Proof Let u ∈ C ∞ (SM). We combine the Pestov identity in Proposition 2.2 with for some α > 0 by the Anosov property. Thus we obtain The subelliptic estimate also requires a Pu . Hence By the Poincaré inequality for closed Riemannian manifolds, one also has Hence Using the Poincaré inequality in the form the above argument also implies the inequality Since P has smooth coefficients, this last inequality holds also for any w ∈ H 1 (SM) with Pw ∈ L 2 (SM) by convolution approximation and the Friedrichs lemma [26, Lemma 17.1.5] (one covers SM by coordinate neighborhoods, uses a subordinate partition of unity, and performs the convolution approximations in the coordinate charts).
We now use the subelliptic estimate to obtain a solvability result for P * = X v div. In [41] this was done by a Hahn-Banach argument in the two-dimensional case. Here we give an alternative Hilbert space argument based on the Riesz representation theorem, and we also give a notion of uniqueness for solutions.

Lemma 8.2 Let (M, g) be an Anosov manifold. For any f
There is a unique solution satisfying one of the following equivalent conditions: Proof Define the space This is an inner product space, since (u, u) A = 0 for u ∈ A implies u = 0 by Theorem 8.1. It is also a Hilbert space: if (u j ) is a Cauchy sequence in A then it is a Cauchy sequence in H 1 by Theorem 8.1, hence (u j ) converges in H 1 to some u ∈ H 1 and also (Pu j ) converges to some w in L 2 . Then Pu j → Pu in H −1 and also Pu j → w in H −1 , showing that Pu = w ∈ L 2 and u j → u in A.
(The expression on the right is the distributional pairing.) This functional satisfies by Theorem 8.1 Thus l is continuous on A, and by the Riesz representation theorem there is u ∈ A satisfying For any w ∈ C ∞ (SM) we have Therefore h = Pu solves P * h = f in SM in the sense of distributions (since both f and P * h are orthogonal to constants) and h L 2 f H −1 . We next observe that h is the unique solution of P * h = f which is in the range of P acting on A. Indeed, if both h = Pu andh = Pũ solve the equation, then P * P(u −ũ) = 0 so (P(u −ũ), Pw) = 0 for smooth w. We can use the Friedrichs lemma to approximate u −ũ by smooth functions in the A norm as in the proof of Theorem 8.1. This shows P(u−ũ) 2 = 0, so u =ũ by Theorem 8.1. The equivalence of (a), (b), (c) follows since anyh ∈ L 2 can be expressed as the orthogonal sum h = Pu +w where u ∈ A and w ∈ L 2 with P * w = 0 (this follows since P : A → L 2 is bounded with closed range and the orthocomplement of Ran(P) is Ker(P * ), one can again approximate u by smooth functions in the A norm).
Let ω n−1 denote the volume of the (n − 1)-dimensional unit sphere. Given w ∈ D (SM), let w 0 ∈ D (M) be defined by We can now prove surjectivity of I * 0 . See [41,Sect. 1.2] for an explanation why the next result is indeed equivalent to surjectivity of I * 0 , and observe that There is a unique such w for which the quantity ∞ m=1 1 m(m+n−2) w m 2 is minimal, and this w satisfies w L 2 ∇a for some a ∈ L 2 (SM) and m(m + n − 1) ,

The decomposition
Using these facts in the Pestov identity above shows that Applying the α-controlled assumption and Lemma 4.3 yields for m ≥ 2 If α > α m,n , the constants in brackets are positive and it follows that If m = 1 we obtain (The last two identities hold for u ∈ C ∞ (SM).) The subelliptic estimate for Q m now follows from the above discussion and the basic subelliptic estimate for P given in Theorem 8.1.
The subelliptic estimate implies a solvability result for Q * m = X T ≥m+1 v div. The proofs are similar to those in Sect. 8 (the next results could also be made slightly more precise as in Sect. 8; we omit the details). But now for any u ∈ C ∞ (SM) we have This leads to invariant distributions starting at any a m with X − a m = 0:

Injectivity of I m
In this section we would like to prove the following injectivity result. This has two immediate corollaries stated in terms of the terminator value of (M, g). (M, g) be a closed Riemannian manifold such that every unit vector has rank one, and let m ≥ 2. Suppose in addition that β T er ≥ m(m+n−1) 2m+n−2 and that there are no non-trivial conformal Killing tensors of order m + 1 (this is always true if β T er > m(m+n−1) 2m+n−2 ). Then I m is s-injective.
The assumption on α implies that the first constant in brackets is positive and the second is nonnegative. If we use that Q m u = 0 we derive: Using the commutator formula in Lemma 3.5 we may write The geodesic flow is Anosov and therefore transitive. This implies From this we easily derive that u k = 0 for all k = m + 1. Thus Xu = Xu m+1 = X − u m+1 which implies that X + u m+1 = 0. It follows that u m+1 is a conformal Killing tensor of order m + 1. By hypothesis u m+1 = 0 and hence u = 0. This implies that a has degree m − 1 and f = Xa. Rewriting this in terms of symmetric tensors via the map λ in Sect. 3, we see that f = da is a potential tensor.

Manifolds with boundary
As mentioned in the introduction, the methods in this paper also apply to manifolds with boundary. The main changes when going from closed manifolds to the boundary case include the assumption that test functions need to vanish on the boundary whenever appropriate, and that in the boundary case there are no nontrivial conformal Killing tensors vanishing on the boundary [11].
We now indicate what kinds of results can be achieved in the boundary case. In this section we will assume that (M, g) is a compact oriented Riemannian manifold with smooth boundary, and n = dim M ≥ 2. The results in Sect. 2 are valid in the boundary case, and the Pestov identity now holds for any u ∈ C ∞ (SM) with u| ∂(SM) = 0. The spherical harmonics expansions u = ∞ m=0 u m and the decomposition X = X + + X − in Sect. 3 remain unchanged. One has the adjoint identity when u ∈ m , w ∈ m+1 , and one of u, w vanishes on ∂(SM). Notice that in general one only needs boundary conditions in the x variable, since one can integrate by parts freely in the v variable (the fibres are compact with no boundary).
Proof By the results of [11], the problem is an elliptic boundary value problem on sections of trace-free symmetric m-tensors (hence on m ), with trivial kernel when m ≥ 1. This shows (a) and also (b) since As in Sect. 4, the manifold with boundary (M, g) is said to be α-controlled if Then Y x,v is a vector field along the geodesic γ x,v : [0, τ (x, v)] → M starting from (x, v), it is orthogonal toγ and vanishes at the endpoints, and we have x, v)).

It follows from the Santaló formula that
where I γ is the index form on a geodesic γ : [0, T ] → M, for all normal vector fields Y along γ x,v vanishing at the endpoints. Thus for some α > 0 by choosing δ small enough. Finally, nonpositive sectional curvature is equivalent with having (R Z, Z ) ≤ 0 for all Z ∈ Z with Z | ∂(SM) = 0, which is equivalent with 1-controlled.

Remark 11.3
The same proof as above shows that if there are no β-conjugate points The key ingredient is the positivity of the index form with parameter β which was established for example in [42, p. 46] or [7,Lemma 5.3].
If (M, g) is compact with boundary, by Lemma 11.1 we can define the Beurling transform for any m ≥ 0 by where f m+2 ∈ m+2 is the unique solution of the equation that is orthogonal to Ker(X − ) (equivalently, the unique solution with minimal L 2 norm, or the unique solution of the form X + v m+1 where v m+1 | ∂(SM) = 0). Lemma 5.3 remains true if one considers u ∈ m with u| ∂(SM) = 0. The arguments in Sect. 5 now show that in nonpositive curvature, the Beurling transform is essentially a contraction and formal invariant distributions starting at any f exist. Theorem 11.4 Let (M, g) be a compact manifold with boundary, and assume that the sectional curvatures are nonpositive. The Beurling transform satisfies (SM) for any ε > 0, and the Fourier coefficients of w satisfy We have seen in Lemma 11.2 that simple manifolds are always α-controlled for some positive α. For these manifolds, the methods in Sects. 8 and 9 yield the following subelliptic estimates and existence results for invariant distributions. Then we have If a m ∈ m with X − a m = 0, there is w ∈ H −1 (SM) with X w = 0 and w m = a m . Remark 11.7 In the case of simple manifolds, one actually expects to find invariant distributions that are C ∞ functions. (Invariant distributions are solutions of X w = 0 in SM, and on simple manifolds with boundary there are many solutions in C ∞ (SM).
In contrast, on closed manifolds with ergodic geodesic flow any solution w ∈ L 2 (SM) is constant and thus nontrivial solutions must be distributions.) One has the following results in the direction of Theorems 11.4-11.6: [39].
These results rely on the ellipticity of the normal operator I * m I m and a solenoidal extension argument. If (M, g) is simple, the ellipticity of I * m I m acting on solenoidal tensor fields is known for any m [49], but it is not clear how to perform the solenoidal extension as in [12]. Thus, at present, we are not able to produce C ∞ invariant distributions in the setting of Theorem 11.6 if dim M ≥ 3 (although a weaker result similar to [43, Theorem 4.2] seems possible).
Finally, modifying the arguments in Sect. 10 appropriately yields the following solenoidal injectivity result. Then I m is s-injective. Remark 11.9 The same argument proving Theorem 11.8 also gives the following result: let (M, g) be a compact manifold with non-empty boundary, nonpositive sectional curvature and with property that any smooth q with Xq = 0 and q| ∂(SM) = 0 must vanish. If u ∈ C ∞ (SM) solves Xu = f in SM with u| ∂(SM) = 0 where f ∈ C ∞ (SM) has degree m, then u has degree m − 1.
Theorem 1.6 in the Introduction follows directly from Theorem 11.8 and Remark 11.3.
To be precise, we identify vector fields V on SM with the corresponding fields i * V on T M, where i : SM → T M is the natural inclusion. Equip SM with the restriction of the Sasaki metric from T M. The identity d SM i * U = i * d T M U for functions U on T M implies that the gradient on SM is given by whereũ ∈ C ∞ (T M) is any function withũ| SM = u and where the vector field V on SM is expressed as above in the form V = X j δ x j + Y k ∂ y k .
We define vector fields on SM that act on u ∈ C ∞ (SM) by where p : T M\{0} → SM is the projection p(x, y) = (x, y/|y| g(x) ). We see that ∇u has the following form in local coordinates: Commutator formulas. Direct computations in local coordinates give the following formulas for vector fields on T M: We wish to consider corresponding formulas for the vector fields δ j and ∂ j on SM. If u ∈ C ∞ (SM), writeũ(x, y) = (u • p)(x, y) = u(x, y/|y| g ). Homogeneity implies that Since also δ x j (|y| g ) = 0 and ∂ y j (|y| g ) = y j /|y| g , we obtain We also note that v j ∂ j = 0.
Using the identity ∂ x j g ab + g am b jm + g bm a jm = 0 we also obtain We can now compute and (by the formula (11.2) below) Moving on to [X, and (again by (11.2)) In the second term we have −(Xu)(X v l ) − l jk v j (Xu)v k = 0, and when taking the commutator the terms containing X 2 u cancel. It follows that The part in brackets is zero, which yields [X, Adjoints. To prove the last basic commutator formula, it is useful to have local coordinate expressions for the adjoints of X, h ∇, v ∇ on the space Z. The first step is to compute the adjoints of the local vector fields δ j and ∂ j : if u, w ∈ C ∞ (SM) and w vanishes when x is outside a coordinate patch (and additionally w vanishes on ∂(SM) if M has a boundary), we claim that in the L 2 (SM) inner product (δ j u, w) = −(u, (δ j + j )w), (∂ j u, w) = −(u, (∂ j − (n − 1)v j )w). (11.1) Here j = k jk . Assuming these identities, one can check that the adjoint of X on C ∞ (SM) is −X . Moreover, if Z ∈ Z is written as Z (x, v) = Z j (x, v)∂ x j , the vector field X Z is the covariant derivative (with respect to the Levi-Civita connection in (M, g)) Then the adjoint of X on Z is also −X . The adjoints of h ∇ and v ∇ are given in local Given these expressions, we get the final commutator formula: It remains to check (11.1). To do this it is enough to prove that (The proof of (11.1) also uses the identity |g| −1/2 ∂ x j (|g| 1/2 ) = j .) To show the first formula, let u ∈ C ∞ (SM) vanish when x is outside a coordinate patch, and define Writeũ(x, y) = u(x, y/|y| g ), and choose ϕ ∈ C ∞ c ((0, ∞)) so that ∞ 0 ϕ(r )r n−1 dr = 1. Write g(x) for the matrix of g in the x coordinates. We write ω for points in S n−1 , and note that ω → g(x) −1/2 ω is an isometry from S n−1 (with the metric induced by the Euclidean metric e in R n ) onto S x M (with the metric induced by Sasaki metric on T x M, having volume form dT x = |g(x)| 1/2 dx). Therefore Now δ x j (|y| g ) = 0, and it follows by undoing the changes of variables above that as required. The second formula follows from a similar computation as above: now we define and compute Write h(y) = |y| g ϕ(|y| g ). Then The expression in brackets is v j ∞ 0 (ϕ(r ) + r ϕ (r ))r n−1 dr = −(n − 1)v j , and the result follows.

Appendix B: The two-dimensional case
In this section we reconsider the arguments in this paper in the special case of twodimensional manifolds. This discussion allows to connect the present treatment with earlier work in two dimensions, in particular [38,41].
Vector fields. Let (M, g) be a compact oriented Riemann surface with no boundary (the boundary case is analogous, if we additionally assume that test functions vanish on ∂(SM) if appropriate). We have n = dim(M) = 2. For any vector v ∈ S x M, there is a unique vector iv ∈ S x M such that {v, iv} is a positive orthonormal basis of T x M. Let X be the geodesic vector field on SM as before. We can define vector fields V and X ⊥ on SM, acting on u ∈ C ∞ (SM) by Note also that any Z ∈ Z is of the form Z (x, v) = z(x, v)iv for some z ∈ C ∞ (SM). If γ (t) is a unit speed geodesic we have D t [iγ (t)],γ (t) = ∂ t iγ (t),γ (t) = 0, Thus iv is parallel along geodesics, and for Z = z(x, v)iv we have X Z = (X z)iv. The Guillemin-Kazhdan operators [21] are defined as the vector fields η ± := 1 2 (X ± i X ⊥ ).
All these vector fields have simple expressions in isothermal coordinates. Since (M, g) is two-dimensional, near any point there are positively oriented isothermal coordinates (x 1 , x 2 ) so that the metric can be written as ds 2 = e 2λ (dx 2 1 +dx 2 2 ) where λ is a smooth real-valued function of x = (x 1 , x 2 ). This gives coordinates (x 1 , x 2 , θ) on SM where θ is the angle between a unit vector v and ∂/∂ x 1  where ∂/∂z = 1 2 (∂ x 1 − i∂ x 2 ) and ∂/∂z = 1 2 (∂ x 1 + i∂ x 2 ). Commutator formulas. The above discussion shows that the commutator formula [X, Thus the adjoint of V is −V , and the first commutator formula implies that the adjoint of X ⊥ is −X ⊥ . Consequently Spherical harmonics expansions. It is easy to express the operators X ± in terms of η ± . The vertical Laplacian on SM is given by The operator −i V on L 2 (SM) has eigenvalues k ∈ Z with corresponding eigenspaces E k . We write Locally in the (x, θ) coordinates, elements of E k are of the formw(x)e ikθ . Writing If m ≥ 1, the action of X ± on m is given by X ± (e m + e −m ) = η ± e m + η ∓ e −m , e j ∈ j , and for m = 0 we have X + | 0 = η + + η − , X − | 0 = 0. In the two-dimensional case it will be convenient to work with the k spaces (the corresponding results in terms of the m spaces will follow easily).

Beurling transform and invariant distributions.
Recall that w ∈ D (SM) is called invariant if X w = 0. If the geodesic flow is ergodic, these are genuinely distributions since any w ∈ L 1 (SM) that satisfies X w = 0 must be constant. For Riemann surfaces, one can look at distributions with one-sided Fourier series; let us consider the case where w k = 0 for k < k 0 , for some integer k 0 ≥ 0. For such a distribution, the equation X w = 0 reduces to countably many equations for the Fourier coefficients (by parity it is enough to look at w k 0 +2 j for j ≥ 0): On a Riemann surface with genus ≥ 2, the operator η + : k−1 → k is injective and its adjoint η − : k → k−1 is surjective for k ≥ 2 by conformal invariance (there is a constant negative curvature metric in the conformal class, and these have no conformal Killing tensors). Also for k ≥ 2 we have the L 2 -orthogonal splitting k = Ker(η − | k ) ⊕ η + k−1 .
If k ≥ 0, we define the Beurling transform where f k+2 is the unique function in k+2 orthogonal to Ker(η − | k+2 ) (equivalently, the L 2 -minimal solution) satisfying η − f k+2 = −η + f k . Note that in R 2 , one thinks of η − as ∂ and of η + as ∂, so B + is formally the operator −∂ −1 ∂ which is the usual Beurling transform up to minus sign. Note also that B + is the first ladder operator from [22]. If k ≥ 0 one has the analogous operator where f −k−2 is the L 2 -minimal solution of η + f −k−2 = −η − f −k . The relation to the Beurling transform in Sect. 5 is On a closed surface of genus ≥ 2, one can always formally solve the countably many equations for w k . If we take the minimal energy solution for each equation, we arrive at the formal invariant distributions. We restrict our attention to B + (the case of B − is analogous). Definition 11.10 Let (M, g) be a closed oriented surface with genus ≥ 2, let k 0 ≥ 0, and let f ∈ k 0 satisfy η − f = 0. The formal invariant distribution starting at f is the formal sum w = ∞ j=0 (B + ) j f.
As before, it is not clear if the sum converges in any reasonable sense. However, if the surface has nonpositive curvature it does converge nicely. This follows from the fact that the Beurling transform B + is a contraction on such surfaces, and to prove this we use the Guillemin-Kazhdan energy identity [21]: Lemma 11.11 Let (M, g) be a closed Riemann surface. Then Proof In [21] one has the commutator formula This implies that, for u ∈ C ∞ (SM), Lemma 11.12 Let (M, g) be a closed surface, and assume that K ≤ 0. Then for any k ≥ 0 we have Since K ≤ 0 we get X − u 2 ≤ 2 X + u 2 . If K = 0 we have equality if and only if η − f 1 = η + f −1 . Identifying 1 with 1-forms on M, this means that the 1-form f 1 − f −1 is divergence free and thus f 1 − f −1 = * da 0 + h for some a 0 ∈ C ∞ (M) and some harmonic 1-form h. Consequently, if K = 0 then equality holds exactly when u = da 0 + h for some a 0 ∈ C ∞ (M) and some harmonic 1-form h. (Note that h = h 1 + h −1 is a harmonic 1-form if and only if η − h 1 = η + h −1 = 0.) The inequalities for B follow as in Sect. 5.
Remark 11.14 We note that the inequality X − u ≤ √ 2 X + u for u ∈ 1 differs by the factor √ 2 with inequality (5.2) in [23, p. 173] for n = 2 and p = 1 which gives X − u ≤ X + u . Since our inequality in Lemma 11.13 is shown to be sharp in the flat case this indicates an algebraic mistake in the calculation of the constants in [23].
Pestov and Guillemin-Kazhdan energy identities. We conclude this section by discussing the relation between two basic energy identities. The Pestov identity from Proposition 2.2 takes the following form in two dimensions: The Guillemin-Kazhdan energy identity in Lemma 11.11 looks as follows: As discussed in [38], the Pestov identity is essentially the commutator formula [X V, V X] = −X 2 + V K V , whereas the Guillemin-Kazhdan identity follows from the commutator formula [η + , η − ] = i 2 K V . We now show that the Pestov identity applied to u ∈ k is just the Guillemin-Kazhdan identity for u ∈ k . Indeed, we compute V Xu 2 = V η + u 2 + V η − u 2 = (k + 1) 2 η + u 2 + (k − 1) 2 η − u 2 and X V u 2 −(K V u, V u) + Xu 2 = k 2 ( η + u 2 + η − u 2 ) + ik(K V u, u) The Pestov identity and simple algebra show that This is the Guillemin-Kazhdan identity if k = 0. In the converse direction, assume that we know the Guillemin-Kazhdan identity for each k , Multiplying by 2k and summing gives 2k( η + u k 2 − η − u k 2 ) = ik(K V u k , u k ).
On the other hand, the Pestov identity for u = ∞ k=−∞ u k reads