Partial data inverse problems for the Hodge Laplacian

We prove uniqueness results for a Calderon type inverse problem for the Hodge Laplacian acting on graded forms on certain manifolds in three dimensions. In particular, we show that partial measurements of the relative-to-absolute or absolute-to-relative boundary value maps uniquely determine a zeroth order potential. The method is based on Carleman estimates for the Hodge Laplacian with relative or absolute boundary conditions, and on the construction of complex geometric optics solutions which reduce the Calderon type problem to a tensor tomography problem for 2-tensors. The arguments in this paper allow to establish partial data results for elliptic systems that generalize the scalar results due to Kenig-Sjostrand-Uhlmann.

The Calderón problem with partial data corresponds to the case where one can only make measurements on subsets of the boundary. Let Γ D and Γ N be open subsets of ∂Ω, and assume that we measure voltages on Γ D and currents on Γ N . If the potential is grounded on ∂Ω \ Γ D but can be prescribed on Γ D , the partial boundary measurements are given by the partial DN map Λ DN γ f | ΓN for all f ∈ H 1/2 (∂Ω) with supp(f ) ⊂ Γ D . If instead the set ∂Ω \ Γ N is insulated but we can freely prescribe currents on Γ N , then we know the partial ND map The basic uniqueness question is whether a (sufficiently smooth) conductivity is determined by such boundary measurements. We remark that in the partial data case there seems to be no direct way of obtaining the partial DN map from the partial ND map or vice versa, and the two cases need to be considered separately. By now there are many uniqueness results for the Calderón problem with partial data involving varying assumptions on the sets Γ D and Γ N . For further information we refer to the recent survey [KS13] for results in dimensions n ≥ 3 and the surveys [GT13], [IY13b] for the case n = 2. We only list here some of the main results for the partial DN map: • n ≥ 3, Γ D can be possibly very small but Γ N has to be slightly larger than the complement of Γ N [KSU07] • n ≥ 3, Γ D = Γ N = Γ and the complement of Γ has to be part of a hyperplane of a sphere [Is07] • n = 2, Γ D = Γ N = Γ can be an arbitrary open set [IUY10] • n ≥ 3, Γ D = Γ N = Γ and the complement of Γ has to be (conformally) flat in one direction and a certain ray transform needs to be injective [KS12] (a special case of this was proved independently in [IY13a]) The approach of [KSU07] is based on Carleman estimates with boundary terms and the approach of [Is07] is based on reflection arguments. The paper [KS12] combines these two approaches and extends both. There seem to be fewer results for the partial ND map, see [Ch12], [HPS12], [Is07], [IUY12]. In fact, in dimensions n ≥ 3 the Carleman estimate approach for the partial ND map seems to be more involved than for the partial DN map.
The purpose of this paper is to consider analogous partial data results for elliptic systems. In the full data case (Γ D = Γ N = ∂Ω), many uniqueness results are available for linear elliptic systems such as the Maxwell system [OPS93], [KSU11a], [CZ12], Dirac systems [NT00], [ST09], Schrödinger equation with Yang-Mills potentials [Es01], elasticity [NU94], [NU03], [ER02], and equations in fluid flow [HLW07], [LW07]. In contrast, the only partial data results for such systems in dimensions n ≥ 3 that we are aware of are [COS09] for the Maxwell system and [ST10] for the Dirac system. One reason for the lack of partial data results for systems is the fact that Carleman estimates for systems often come with boundary terms that do not seem helpful for partial data inverse problems (see [El08], [ST09] for some such estimates). In two dimensions, a number of partial data results for systems are available [AGTU11], [IY12b].
In this paper we establish partial data results analogous to [KSU07] for systems involving the Hodge Laplacian for graded differential forms, on certain Riemannian manifolds in dimensions n ≥ 3. These are elliptic systems that generalize the scalar Schrödinger equation (−△ + q)u = 0 and are very close to the time-harmonic Maxwell equations when n = 3. The main technical contribution is a Carleman estimate for the Hodge Laplacian, with limiting Carleman weights, that has boundary terms involving the relative and absolute boundary values of graded forms. The boundary terms are of such a form that allows to carry over the Carleman estimate approach of [KSU07] to the Hodge Laplace system.
In a sense, to deal with boundary terms for systems in a flexible way, one first needs a good understanding of the different splittings of Cauchy data in the scalar case. This seems to be related to understanding both the scalar DN and ND maps simultaneously. Therefore the methods developed in [Ch12] for the partial ND map, involving Fourier analysis to treat the boundary terms in Carleman estimates, will be crucial in our approach. We expect that the methods developed in this paper open the way for obtaining partial data results via Carleman estimates for other elliptic systems as well.
The plan of this document is as follows. Section 1 is the introduction, and Section 2 contains precise statements of the main results. Section 3 collects notation and identities used throughout the paper. Sections 4-6 will be devoted to the proofs of the Carleman estimates. In Section 4, we will give the basic integration by parts argument for k-forms and simplify the boundary terms. In Section 5, we prove the Carleman estimates for 0-forms using the arguments from [Ch12], [KS12]. We will conclude the argument in Section 6 by showing that the Carleman estimates for graded forms follow from an induction argument, given the corresponding result for 0-forms. In Section 7 we will construct relevant complex geometrical optics solutions, following the ideas in [DKSaU09]. In Section 8 we will present the Green's theorem argument and give the density result based on injectivity of a tensor tomography problem which finishes the proofs of Theorems 2.1 and 2.2. Section 9 will contain the proof of Theorem 2.3 and make some remarks about the case of dimensions n ≥ 4. Section 10 contains proofs of some technical lemmas.
Acknowledgements. F.C. and M.S. were supported in part by the Academy of Finland (Finnish Centre of Excellence in Inverse Problems Research) and an ERC Starting Grant, and L.T. was partly supported by the Academy of Finland and Vetenskapsrådet. The authors would like to thank the organizers of the Institut Mittag-Leffler program on Inverse Problems in 2013, where part of this work was carried out.

Statement of results
The results in this paper are new even in Euclidean space, but it will be convenient to state them on certain Riemannian manifolds following [DKSaU09], [KS12], [DKLS13]. Suppose that (M 0 , g 0 ) is a compact oriented manifold with smooth boundary, and consider a manifold T = R × M 0 equipped with a Riemannian metric of the form g = c(e ⊕ g 0 ), where c is a smooth conformal factor and (R, e) is the real line with Euclidean metric. A compact manifold (M, g) of dimension n ≥ 3, with boundary ∂M , is said to be CTA (conformally transversally anisotropic) if it can be expressed as a submanifold of such a T . A CTA manifold is called admissible if additionally (M 0 , g 0 ) can be chosen to be simple, meaning that ∂M 0 is strictly convex and for any point x ∈ M 0 , the exponential map exp x is a diffeomorphism from some closed neighbourhood of 0 in T x M 0 onto M 0 (see [Sh94]).
Let Λ k M be the kth exterior power of the cotangent bundle on M , and let ΛM be the corresponding graded algebra. The corresponding spaces of sections (smooth differential forms) are denoted by Ω k M and ΩM . We will define △ to be the Hodge Laplacian on M , acting on graded forms: Here d is the exterior derivative and δ is the codifferential (adjoint of d in the L 2 inner product). Suppose Q is an L ∞ endomorphism of ΛM ; that is, Q associates to almost every point x ∈ M a linear map Q(x) from Λ x M to itself, and the map x → Q(x) is bounded and measurable. We will also consider continuous endomorphisms, meaning that x → Q(x) is continuous in M .
We would like to consider boundary value problems for the operator −△ + Q. In order to do this, we will define the tangential trace t : ΩM → Ω∂M by where i : ∂M → M is the natural inclusion map. Then the first natural boundary value problem to consider for −△ + Q, acting on graded forms u, is the relative boundary problem If Q is such that 0 is not an eigenvalue for this problem, then the uniqueness of solutions to this problem defines a relative-to-absolute map The second natural boundary value problem to consider is the absolute boundary value problem Assuming that 0 is not an eigenvalue, this defines an absolute-to-relative map For a reference on wellposedness of the relative and absolute boundary value problems with the Hodge Laplacian, see [Ta99,Section 5.9].
These maps both give rise to a Calderón type inverse problem, which asks if knowledge of N RA Q or N AR Q suffices to determine Q. If we restrict ourselves to considering the case of zero-forms only and if Q acts on zero-forms by multiplication by a function q ∈ L ∞ (M ), then the relative-to-absolute and absolute-to-relative maps become the DN and ND maps, respectively, for the Schrödinger equation where u is now a function on M and △ is the Laplace-Beltrami operator on functions. Our problem is therefore a generalization of the standard partial data problem for the scalar Schrödinger equation on a compact manifold with boundary.
Let us review some earlier results for the Schrödinger problem in the scalar case. In dimensions n ≥ 3, where M is Euclidean, Sylvester and Uhlmann proved that knowledge of the full DN map uniquely determines the potential q in the paper [SU87]. Versions of this problem on admissible and CTA manifolds as defined above have been considered in [DKSaU09] and [DKLS13]. Partial data results for the DN map have been proven in [BU01], [Is07], and [KSU07] for the Euclidean case, and more recently in [KS12], the last of which contains the previous three results and extends them to the manifold case. Improved results in the linearized case are in [DKSjU09]. Partial data results for the ND map, analogous to the ones in [KSU07], were proven in [Ch12]. Other partial data results for scalar equations with first order potentials as well were obtained in [DKSjU07] and [Ch11], and some of those techniques will be useful to us in this paper as well. In dimension n = 2, the Schrödinger problem with full data was solved in [Bu08] and with local data in [IUY10], and the manifold case also with local data was considered in [GT11].
For the Hodge Laplacian acting on graded forms, we are not aware of previous results dealing with the determination of a potential from the relative-to-absolute or absolute-to-relative maps. However, [KLU11] reconstructs a real analytic metric from these maps in the case of no potential, and [SS12], [Sho13], [BS08], and [JL05] recover various kinds of topological information about the manifold from variants of these maps, again in the case of no potential. We remark that full data problems for the Hodge Laplacian in Euclidean space are very similar to the scalar case (see Section 9), but the partial data problem even in Euclidean space is more involved.
In order to describe the main results precisely, we will define 'front' and 'back' sets of the boundary ∂M as in [KSU07]. If M ⊂ T = R × M 0 is CTA, we can use coordinates (x 1 , x ′ ) where x 1 is the Euclidean variable, and define the function ϕ : T → R by ϕ(x 1 , x ′ ) = x 1 . As discussed in [DKSaU09], ϕ is a natural limiting Carleman weight in M . Now define Then the main results of this paper are the following.
Theorem 2.1. Let M ⊂ R × M 0 be a three dimensional admissible manifold with conformal factor c = 1, and let Q 1 and Q 2 be continuous endomorphisms of ΛM such that N RA Q1 , N RA Q2 are defined. Let Γ + ⊂ ∂M be a neighbourhood of ∂M + , and let Γ − ⊂ ∂M be a neighbourhood of ∂M − . Suppose that Theorem 2.2. Let M be a three dimensional admissible manifold with conformal factor c = 1, and let Q 1 and Q 2 be continuous endomorphisms of ΛM such that N AR Q1 , N AR Q2 are defined. Let Γ + ⊂ ∂M be a neighbourhood of ∂M + , and let Γ − ⊂ ∂M be a neighbourhood of ∂M − . Suppose that In the case that M is a domain in Euclidean space, we can also extend the results to higher dimensions.
Theorem 2.3. Let M be a bounded smooth domain in R n , with n ≥ 3, and let Q 1 and Q 2 be continuous endomorphisms of ΛM such that N RA Q1 , N RA Q2 are defined. Fix a unit vector α, and let ϕ(x) = α · x. Let Γ + ⊂ ∂M be a neighbourhood of ∂M + , and let Γ − ⊂ ∂M be a neighbourhood of The same result holds if we replace the relative-to-absolute map with the absolute-to-relative one.
Theorem 2.1 is a generalization to certain systems of the scalar partial data result of [KSU07] for the DN map, and similarly Theorem 2.2 is an extension to systems of the scalar result of [Ch12] for the ND map. To be precise, the above theorems are stated for the linear Carleman weight and not for the logarithmic weight as in [KSU07], [Ch12]. This restriction comes from the lack of conformal invariance of the full Hodge Laplacian. However, in the scalar case we could use the conformal invariance of the scalar Schrödinger operator together with a reduction from [KS12] to recover the logarithmic weight results of [KSU07], [Ch12] from the above theorems.
The proof of Theorems 2.1 and 2.2 involve three main ingredients -the construction of complex geometric optics (CGO) solutions, a Green's theorem argument, and a density argument relating this inverse problem to a tensor tomography problem. Both the construction of CGO solutions and the Green's theorem argument require appropriate Carleman estimates.
To describe them, we will introduce the following notation. For a CTA manifold M , let N be the inward pointing normal vector field along ∂M . We can extend N to be a vector field in a neighborhood of ∂M by parallel transporting along normal geodesics, and then to a vector field on M by multiplying by a cutoff function. For u ∈ ΩM we will let where N ♭ is the 1-form corresponding to N and i N is the interior product, and Let ∇ denote the Levi-Civita connection on M , and ∇ ′ denote the pullback connection on the boundary. Let Then the Carleman estimates are as follows.
Theorem 2.4. Let M be a CTA manifold, and let Q be a L ∞ endomorphism of ΛM . Define Γ + ⊂ ∂M to be a neighbourhood of ∂M + . Suppose u ∈ H 2 (M, ΛM ) satisfies the boundary conditions u| Γ+ = 0 to first order for some smooth endomorphism σ independent of h. Then there exists h 0 such that if 0 < h < h 0 , Here H 1 signifies the semiclassical H 1 space with semiclassical parameter h, and for instance The constant implied in the sign is meant to be independent of h. Note that the last boundary condition in (2.1) can be rewritten as Theorem 2.5. Let M be a CTA manifold, and let Q be a L ∞ endomorphism of ΛM . Define Γ + ⊂ ∂M to be a neighbourhood of ∂M + . Suppose u ∈ H 2 (M, ΛM ) satisfies the boundary conditions for some smooth endomorphism σ independent of h. Then there exists h 0 such that if 0 < h < h 0 , . Note that Theorem 2.5 is actually Theorem 2.4 with u replaced by * u, and vice versa. Therefore it suffices to prove Theorem 2.5 only. It is also worth noting that the Carleman estimates are proved for CTA manifolds in general, with no restriction on either the dimension, the conformal factor, or the transversal manifold (M 0 , g 0 ). Theorems 2.4 and 2.5 are extensions to the Hodge Laplace system of the Carleman estimates in [KSU07] and [Ch12], respectively.

Notation and identities
Let (M, g) be a smooth (= C ∞ ) n-dimensional Riemannian manifold with or without boundary. All manifolds will be assumed to be oriented. We write v, w for the g-inner product of tangent vectors, and |v| = v, v 1/2 for the g-norm. If x = (x 1 , . . . , x n ) are local coordinates and ∂ j the corresponding vector fields, we write g jk = ∂ j , ∂ k for the metric in these coordinates. The determinant of (g jk ) is denoted by |g|, and (g jk ) is the matrix inverse of (g jk ).
We shall sometimes do computations in normal coordinates. These are coordinates x defined in a neighborhood of a point p ∈ M int such that x(p) = 0 and geodesics through p correspond to rays through the origin in the x coordinates. The metric in these coordinates satisfies g jk (0) = δ jk , ∂ l g jk (0) = 0.
The Einstein convention of summing over repeated upper and lower indices will be used. We convert vector fields to 1-forms and vice versa by the musical isomorphisms, which are given by The set of smooth k-forms on M is denoted by Ω k M , and the graded algebra of differential forms is written as ΩM = ⊕ n k=0 Ω k M. The set of k-forms with L 2 or H s coefficients are denoted by L 2 (M, Λ k M ) and H s (M, Λ k M ), respectively. Here H s for s ∈ R are the usual Sobolev spaces on M . The inner product · , · and norm | · | are extended to forms and more generally tensors on M in the usual way, and we also extend the inner product · , · to complex valued tensors as a complex bilinear form.
Let d : Ω k M → Ω k+1 M be the exterior derivative, and let * : Ω k M → Ω n−k M be the Hodge star operator. We introduce the sesquilinear inner product on Ω k M , Here dV = * 1 = |g| 1/2 dx 1 · · · dx n is the volume form. The codifferential δ : Ω k M → Ω k−1 M is defined as the formal adjoint of d in the inner product on real valued forms, so that (dη|ζ) = (η|δζ), for η ∈ Ω k−1 M, ζ ∈ Ω k M compactly supported and real.
If X is a vector field, the interior product i X : If ξ is a 1-form then the interior product i ξ = i ξ ♯ is the formal adjoint of ξ∧ in the inner product on real valued forms, and on k-forms it has the expression i ξ = (−1) n(k−1) * ξ ∧ * .
The interior and exterior products interact by the formula where α is a k-form and β an m-form. In particular if α and ξ are one-forms then In addition, the differential and codifferential satisfy the following product rules: The Hodge Laplacian on k-forms is defined by It satisfies ∆ * = * ∆. The above quantities may be naturally extended to graded forms. We will also have to deal with forms that are not compactly supported on M . We have already introduced the tangential trace t : ΩM → Ω∂M by so if u is a graded form on M , then tu is a graded form on ∂M . Then (tu|tv) ∂M is interpreted in the same manner as (u|v) M above. If u and v are graded forms on M , we will also define where dS is the volume form on ∂M . Now if η ∈ Ω k−1 M and ζ ∈ Ω k M then d and δ satisfy the following integration by parts formulas.
Here ν denotes both the unit outer normal of ∂M and the corresponding 1-form. Applying these formulas for the Hodge Laplacian gives where u and v are k-forms, or graded forms. We can also redo the integration by parts to write the boundary terms in terms of absolute and relative boundary conditions, so The Levi-Civita connection, defined on tensors in M , is denoted by ∇ and it satisfies ∇ X * = * ∇ X . We will sometimes write ∇f (where f is any function) for the metric gradient of f , defined by If X is a vector field and η, ζ are differential forms we have We can also express d using the ∇ operator, as follows: if ω is a k-form on M , and X 1 , .., X k+1 are vector fields on M , then whereX l means that we omit the X l argument. Moreover if e 1 , . . . , e n are an orthonormal frame of For the statements of the Carleman estimates, we introduced the notation where N is a smooth vector field which coincides with the inward pointing normal vector field at the boundary ∂M , and is extended into M by parallel transport. Note that i N u = 0, N ∧ u ⊥ = 0, and tu ⊥ = 0 at ∂M . In addition, if u and v are graded forms on M , then If X is a vector field, we can break down X into parallel and perpendicular components in the same way by using (X ♭ ) ♯ and (X ♭ ⊥ ) ♯ . The ⊥ and signs are interchanged by the Hodge star operator: * (u ) = ( * u) ⊥ and * (u ⊥ ) = ( * u) .
Note that by its definition in terms of parallel transport, ∇ N N = 0. Thus ∇ N commutes with N ∧ and i N . If we view ∂M as a submanifold embedded into M , then T M splits into T ∂M ⊕ N ∂M , where T ∂M is the tangent bundle of ∂M and N ∂M is the normal bundle. Then the second fundamental form II : T ∂M ⊕ T ∂M → N ∂M of ∂M relative to this embedding is defined by The second fundamental form can also be defined in terms of the shape operator s : T ∂M → T ∂M by s(X) = ∇ X N. Then II(X, Y ) = (s(X)|Y )N. These two operators carry information about the shape of the ∂M in M , and thus show up in our boundary computations. Now we move to some more specific technical formulas used in the paper. For proofs of the following lemmas, see the appendix. We begin with a simple computation.
Lemma 3.1. If ξ and η are real valued 1-forms on M and if u is a k-form, then We also give an expression for the conjugated Laplacian.
Next, an expansion for the expression tδ.
We also need an expansion for tδB, where B is the operator Lemma 3.5. If u ∈ Ω k (M ) is such that tu = 0, then Finally, we will need to do a computation to split the Hodge Laplacian into normal and tangential parts. To do this, we will take advantage of a Weitzenbock identity, which says that where R is a zeroth order linear operator depending only on the curvature of M , △ is the Hodge Laplacian, and△ is the connection Laplacian: ∆u := ∇ * ∇u.
We then have the following result for∆.

Carleman estimates and boundary terms
As noted in the introduction, Theorem 2.4 follows from Theorem 2.5, so it suffice to show that we can prove Theorem 2.5.
In proving the Carleman estimates, it will suffice to work with smooth sections of ΛM and apply a density argument to get the final result. Let Ω k (M ) denote the space of smooth sections of Λ k M , and Ω(M ) denote the space of smooth sections of ΛM .
In this section we give an initial form of the Carleman estimates by using an integration by parts argument as in [KSU07]. To do this, we will first need to understand the relevant boundary terms. We will use the integration by parts formulas for u, v ∈ Ω(M ).
As in [KSU07], we will need to work with the convexified weight By Lemma 3.2 we can write this as A + iB where A and B are self-adjoint operators given by Let · indicate the L 2 norm on M , unless otherwise stated. Then, for u ∈ Ω k (M ), Integrating by parts gives and after a short computation This finishes the basic integration by parts argument and shows the following: Now we invoke the absolute boundary conditions to estimate the non-boundary terms and to simplify the boundary terms in (4.3). It is enough to consider differential forms u ∈ Ω k (M ) for fixed k.
for some smooth bounded endomorphism σ whose bounds are uniform in h.
Then the non-boundary terms in (4.3) satisfy for h ≪ ε ≪ 1. Also, the boundary terms in (4.3) have the form ∂M . for any large enough K independent of h.
Proof of Proposition 4.2. We will prove (4.5) first. The argument follows the one given in [Ch12], for scalar functions.
Note that A and B have the same scalar principal symbols as they do for zero-forms: that is, given a local basis dx 1 , . . . , dx n for the cotangent space with dx where E 1 and E 0 are first and zeroth order operators, respectively, with uniform bounds in h and ε. Therefore locally where R is a first order operator with uniform bounds in h and ε. Choosing a partition of unity χ 1 , . . . , χ m of M such that this operation can be performed near each supp(χ j ), the argument for scalar functions in the proof of Proposition 3.1 from [Ch12] implies that where Q is a second order operator. Recall that so using integration by parts with the above formula, we get By Lemma 3.1 we obtain The absolute boundary condition says that t * u = 0, so u ⊥ = 0 at the boundary. Therefore

The boundary term in the last expression is bounded by
Since t * u = 0, we have i N u, u ⊥ = 0 at the boundary, and thus where in the last step we invoked the absolute boundary condition. Therefore Meanwhile, since t * u = 0 on ∂M we can write Using the absolute boundary conditions again, we have αε with α large and fixed. Putting this together with the inequality for (i[A, B]u|u) and Gaffney's inequality u H 1 ∼ u L 2 + hdu L 2 + hδu L 2 when t * u = 0, we obtain that for h ≪ ε ≪ 1. This proves (4.5).
We will now show the expression (4.6) for the boundary terms in (4.3). Recall that these boundary terms are given by Moreover, if u satisfies the absolute boundary conditions (4.4), then * u satisfies the relative boundary conditions and vice versa. Therefore it suffices to prove that if u satisfies (4.8) then the boundary terms (4.7) become for any large enough K independent of h. So let's return to (4.7), and assume u satisfies (4.8). The condition tu = 0 implies that the first term ih(hdBu|ν ∧ u) ∂M is zero. Therefore we are left with We calculate each of the terms individually. Firstly, Therefore, Moreover, by Lemma 3.3, We can write this as where R 2 satisfies the bound on R in (4.10). Secondly, Expanding B, this becomes Since tu = 0, the last expression is equal to part has the same type of bound as in (4.10), so where R 3 has the same bound as in (4.10). Thirdly, By Lemma 3.5, The terms on the last two lines, when paired with ihti N u, are bounded by (4.10). Moreover, using the boundary conditions in the form of equation (4.12) on the shows that this too is bounded by (4.10). Therefore we need only worry about the first three terms.
For the −ih(hδ ′ tBu|ti N u) term, we can integrate by parts to get , and so this term is bounded by (4.10).
For the 2h 3 (∇ ′ (∇ϕc) t∇ N i N u|ti N u) ∂M term, we can use equation (4.12) to get and then use Cauchy-Schwartz, so this term is bounded by (4.10) too. Therefore where R 1 is bounded by (4.10). Finally, Using the Weitzenbock identity, we can write −2h The second term is bounded by (4.10). For the first term, we can apply Lemma 3.6 to get The last term is bounded again by (4.10) and we can integrate by parts in the△ ′ part to get something bounded by (4.10) as well. Therefore where R 4 is bounded by (4.10). Now putting this together with (4.13), (4.15), and (4.16), we get that the boundary terms in (4.3) have the form We can replace ϕ c by ϕ and incorporate the error into R without affecting the bound on R, to get and the proposition follows.

The 0-form case
We will now prove Theorem 2.5 in the 0-form case. In the case where (M, g) is a domain in Euclidean space, Theorem 2.5 for 0-forms is the Carleman estimate given in [Ch12, Theorem 1.3]. In this section we will deal with the added complication of being on a CTA manifold, rather than in Euclidean space.
If u is a zero form, then i N u = 0, so u ⊥ = 0 and u = u . Theorem 2.5 reduces to the estimate where Q ∈ L ∞ (M ) and 0 < h < h 0 , for functions u ∈ H 2 (M ) with u| Γ+ = 0 to first order and By arguing as in the beginning of Section 6 below, the estimate (5.1) will be a consequence of the following proposition.
Proposition 5.1. Suppose u is a function in H 2 (M ) which satisfies the following boundary conditions: , We will prove this proposition in the case where the metric g has the form g = e ⊕ g 0 . However, if g were of the form g = c(e ⊕ g 0 ), we could write Therefore the proposition remains true even in the case when the conformal factor is not constant. More generally, the proofs of the Carleman estimates work for any smooth conformal factor, and thus as noted earlier, the Carleman estimates hold on CTA manifolds in general.

The operators.
Here we introduce the operators we will use in the proof of Proposition 5.1. Similar operators are found in [Ch11] and [Ch12]. Suppose F (ξ) is a complex valued function on R n−1 , with the properties that |F (ξ)|, ReF (ξ) ≃ 1 + |ξ|. Fix coordinates (x 1 , x ′ ) on R n , and define R n + to be the subset of R n with x 1 > 0. Define S(R n + ) as the set of restrictions to R n + of Schwartz functions on R n . Finally, if u ∈ S(R n + ), then defineû(x 1 , ξ) to be the semiclassical Fourier transform of u in the x ′ variables only. Now for u ∈ S(R n + ), define J by This has adjoint J * defined by These operators have right inverses given by Now we have the following boundedness result, given in [Ch12].
Note that similar mapping properties hold between H 1 (R n + ) and H 2 (R n + ), by the same reasoning. We'll record the other operator fact from [Ch12] here, too.
|α| for all multiindices α and β, and for 0 ≤ j ≤ k. In other words, each ∂ j y a(x, ξ, y) is a symbol on R n−1 of order m, with bounds uniform in y, for 0 ≤ j ≤ k. Then we can define an operator A on Schwartz functions in R n by applying the pseudodifferential operator on R n−1 with symbol a(x, ξ, y), defined by the Kohn-Nirenberg quantization, to f (x, y) for each fixed y.
For this section we'll make two additional assumptions on f and M 0 .
First, we'll assume that g 0 is nearly constant: that there exists a choice of coordinates on the subset P (M ) which consists of the projection of M onto M 0 , such that when represented in these coordinates, where δ is a positive constant to be chosen later. Second, we'll assume that f is such that ∇ g0 f is nearly constant on P (M ): that there exists a constant vector field K on T M 0 such that where δ is the same constant from above. The choice of δ will depend ultimately only on K. In the next subsection we'll see how to remove these two assumptions. Now we can do the change of variables ( . DefineM ′ andΓ ′ + to be the images of M and Γ + respectively, under this map. Note that {x 1 ≥ f (x ′ )} maps to (0, ∞) × M 0 , and Γ c + maps to a subset of 0 × M 0 . Now it suffices to prove the following proposition.
where σ is smooth and bounded onM ′ . Then . Note that onM ′ , α is very close to 1. This proposition implies Proposition 5.1, in the graph case described above.
Proof of Proposition 5.1 in the graph case. Suppose u ∈ H 2 (M ), and u satisfies (5.2). Let w be the function onM defined by w( , and w satisfies (5.4). Therefore by Proposition 5.4, . Now by a change of variables, where E 1 is a first order semiclassical differential operator. Therefore by a change of variables, Putting this all together gives We can do a second change of variables to move to Euclidean space. By our assumption on M 0 , we can choose coordinates on P (M ′ ) = P (M ) such that Now we have a change of variables giving a map from P (M ′ ) to a subset of R n−1 , and hence a map fromM ′ to a subset of R n + , where the image ofΓ ′ + lies in the plane x 1 = 0. LetM andΓ + be the images ofM ′ andΓ ′ + respectively, under this map. We'll use the notation (x 1 , x ′ ) to describe points in R n + , where now x ′ ranges over R n−1 . Now it suffices to prove the following proposition.
Proposition 5.5. Suppose w ∈ H 2 (M ), and w, ∂ ν w = 0 onΓ + h∂ y w|Γc where σ is smooth and bounded onM , and β and γ are a vector valued and scalar valued function, respectively, which coincide with the coordinate representations of ∇ g0 f and |∇ g0 f | g0 . Then whereL ϕ,ε = (1 + |γ| 2 )h 2 ∂ 2 and L is the second order differential operator in the x ′ variables given by Proposition 5.4 can be obtained from Proposition 5.5 in the same manner as before, with errors from the change of variables being absorbed into the appropriate terms. Therefore it suffices to prove Proposition 5.5.
To do this, we'll split w into small and large frequency parts, using a Fourier transform. Recall that we are assuming Translating down toM , and recalling that g 0 is nearly the identity, we get that there is a constant vector fieldK onM such that |β −K| ≤ C δ and |γ − |K|| ≤ C δ where C δ goes to zero as δ goes to zero. Now choose m 2 > m 1 > 0, and µ 1 and µ 2 such that The eventual choice of µ j and m j will depend only onK.
In the definition, we'll choose the branch of the square root which has non-negative real part, so the branch cut occurs on the negative real axis.
Proof. Now consider the behaviour of A ± (|K|,K, ξ) on the support of ρ, or equivalently, on the support ofŵ s . If η > 0, we can choose µ 2 such that on the support ofŵ s , Then on the support ofŵ s , the expression has real part confined to the interval [−K 2 −m 2 2 , η +m 2 2 ], and imaginary part confined to the interval [−2m 2 , 2m 2 ]. Therefore, by correct choice of η and m 2 , we can ensure .
on the support ofŵ s . This allows us to fix the choice of µ 1 , µ 2 , m 1 , and m 2 . Note that the choices depend only onK, as promised.
The bounds on A ± (|K|,K, ξ) allow us to choose F ± so that F ± = A ± (|K|,K, ξ) on the support ofŵ s , and ReF ± , |F ± | ≃ 1 + |ξ| on R n , with constant depending only on K. Therefore F + and F − both satisfy the conditions on F in Section 2. If T ψ represents the operator with Fourier multiplier ψ (in the x ′ variables), then it follows that the operators h∂ y − T F+ and h∂ y − T F− both have the properties of J * in that section.
Up until now, the operatorL ϕ,ε has only been applied to functions supported inM . However, we can extend the coefficients ofL ϕ,ε to R n + while retaining the |β −K| < C δ and |γ − |K|| ≤ C δ conditions. Then for sufficiently small h and some C δ which goes to zero as δ goes to zero. Meanwhile, Since F ± = A ± (K, K, ξ) on the support ofŵ s , this can be written as . Now by the boundedness properties, , so for small enough δ, L ϕ,ε w s L 2 (R n + ) w s H 2 (R n + ) .
Now we have to deal with the large frequency term.
Proposition 5.7. Suppose w is the extension by zero to R n + of a function in C ∞ (M ) which is 0 in a neighbourhood ofΓ + , and satisfies (5.5), and let w ℓ be defined as above. Then if δ is small enough, h so A ± (a, V, ξ) are roots of the polynomial (1 + |a| 2 )X 2 − 2(1 + iV · ξ)X + (1 − |ξ| 2 ).

Now let's define
).) Again we'll use the branch of the square root with non-negative real part. Now set ζ ∈ C ∞ 0 (R n−1 ) to be a smooth cutoff function such that ζ = 1 if |K · ξ| < 1 2 m 1 and |ξ| < 1 2 |K| Consider the singular support of A ε ± (γ, β, ξ). These are smooth as functions of x and ξ except when the argument of the square root falls on the non-positive real axis. This occurs when β · ξ = 0 and Now for δ sufficiently small, depending onK, this does not occur on the support of 1 − ζ. Therefore G ε ± (γ, β, ξ) = (1 − ζ)A ε ± (γ, β, ξ) + ζ are smooth, and one can check that they are symbols of first order on R n .
Then by properties of pseudodifferential operators, . This last line can be written out as by modfiying E 1 as necessary. Now T ζ w ℓ = 0, so
Now we can check that G + (|K|,K, ξ) satisfies the necessary properties of F from Section 5, so L ϕ,ε w ℓ L 2 (R n+1 .
Now combing the results of Propositions 5.6 and 5.7 gives . for w ∈ C ∞ (M ) such that w ≡ 0 in a neighbourhood ofΓ + , and w satisfies (5.5). A density argument now proves Proposition 5.5, and hence Proposition 5.1, at least under the assumptions on g 0 and f made at the beginning of this section. 5.3. Finishing the proof. Now we need to remove the graph conditions on Γ c + , and the conditions on the metric g 0 . Since Γ + is a neighbourhood of ∂M + , in a small enough neighbourhood U around any point p on Γ c + , Γ c + coincides locally with a subset of a graph of the form x 1 = f (x ′ ), with M ∩ U lying in the set x 1 > f (x ′ ). Moreover, for any δ > 0, if ∇ g0 f (p) = K, then in some small neighbourhood of p, |∇ g0 f − K| g0 < δ. Additionally, since we can choose coordinates at p such that g 0 = I in those coordinates, for any δ > 0 we can ensure that there are coordinates such that |g 0 − I| ≤ δ in a small neighbourhood of p. We can choose δ to be small enough for Proposition 5.1 to hold, by the proof in the previous subsection. Now we can let U j be open sets in R n such that {U 1 , . . . U m } is a finite open cover of Γ c + such that each M ∩ U j has smooth boundary, and each Γ c + ∩ U j is represented as a graph of the form with |∇ g0 f j − K j | g0 < δ j , and there is a choice of coordinates on the projection of M ∩ U j in which |g 0 − I| ≤ δ j , where δ j are small enough for (5.6) Now let χ 1 , . . . χ m be a partition of unity subordinate to U 1 , . . . U m , and for w ∈ H 2 (M ) satisfying (5.2), define w j = χ j w. Then if Γ c + ∩ U j = ∅, w j satisfies (5.6) for some σ, and so . Adding together these estimates gives . This finishes the proof of Proposition 5.1.

The k-form case
We will prove Theorem 2.5 for u ∈ Ω k (M ) by induction. If k = 0, then i N u = 0, so u ⊥ = 0 and u = u . Then Theorem 2.5 for k = 0 becomes the Carleman estimate (5.1) that was established in Section 5.
Note that it suffices to prove Theorem 2.5 for u ∈ Ω k (M ), with the appropriate boundary conditions, for each k, and Q = 0. Then the final theorem follows by adding the resulting estimates and noting that the extra h 2 Qu term on the right can be absorbed into the terms on the left, for sufficiently small h.
6.1. Proof of Theorem 2.5 for k ≥ 1. Suppose u ∈ Ω k (M ) with k ≥ 1. First note that if we impose the boundary conditions (2.2) of Theorem 2.5, substituting the result of Proposition 4.2 into (4.3) gives Recall also from Proposition 4.2 that the non-boundary terms Au 2 + Bu 2 + (i[A, B]u|u) satisfy (6.2) for h ≪ ε ≪ 1. We now return to (6.1) and examine the boundary terms. On Γ c + , there exists ε 1 > 0 such that ∂ ν ϕ < −ε 1 . Using this together with (6.1) and (6.2) gives Since e ϕ 2 2ε is smooth and bounded on M , we get . Thus Theorem 2.5 for k ≥ 1 will follow after we have proved Lemma 6.1.
Proof of Lemma 6.1. For the 0-form case, this follows from Theorem 2.5 for 0-forms, which in this section we are assuming has been proved. Therefore we can seek to prove (6.3) for k-forms by induction on k.
Let k > 0, and assume that (6.3) holds for k − 1 forms satisfying (2.2). Now let U 1 , . . . , U m ⊂ T be an open cover of Γ c + such that each U i ∩ Γ c + has a coordinate patch, and let χ 1 , . . . , χ m be a partition of unity with respect to {U i } such that χ i = 1 near Γ c + and ∇ N χ i = 0 for each i. It will suffice to show that ., e n−1 } be an orthonormal frame for the tangent space, and extend these vector fields into M by parallel transport along normal geodesics.
Observe for all ω ∈ Ω k (U j ∩ Γ c + ) one can write Therefore we can write Then it suffices to show that + . Now we want to apply the induction hypothesis to i ej χ i u || , so we have to check that it satisfies the boundary conditions (2.2). In fact we will have to modify i ej χ i u || slightly to achieve this. Let ρ(x) be a function defined in a neighbourhood of the boundary as the distance to the boundary along a normal geodesic, and extend it to the rest of M by multiplication by a cutoff function. Then the claim is that v = i ej χ i (u || + h(1 − e −ρ h )Zu ) satisfies the absolute boundary conditions (2.2), where Z is an endomorphism yet to be chosen.
Since u satisfies (2.2), i ej χ i u || and i ej χ i (h(1 − e −ρ h )Zu )) both vanish to first order on Γ + . Therefore v does as well.
Since t * i ej χ i u || = 0 on Γ c + , the first term vanishes there as well. Therefore on Γ c the other term vanishes since (1 − e −ρ h ) = 0 at the boundary. Thus 3 and by (2.2), Viewing this as an equation for th∇ N i N ( * u) and substituting into (6.6) gives where here we identify S and σ with their extensions by parallel transport to a neighbourhood of the boundary, then we can replace the dϕ in ti dϕ * u with its normal component: Then Since t * i ej χ i u || = 0 on Γ c + , We can replace u on the right side by Zu ) satisfies the boundary conditions (2.2), and so by the induction hypothesis, Keeping in mind that the second term of v is zero at the boundary, and O(h) elsewhere, we get The commutators of ∆ ϕc with i ej χ i and i ej χ i (1 − e −ρ h )Z are O(h) and first order, so Substituting back into (6.7) gives This proves (6.5), which finishes the proof of the lemma.

Complex geometrical optics solutions
We will begin by constructing CGOs for the relative boundary case. To start, we can use the Carleman estimate from Theorem 2.4 to generate solutions via a Hahn-Banach argument. The notations are as in Section 2.
Proposition 7.1. Let Q be an L ∞ endomorphism on ΛM , and let Γ + be a neighbourhood of ∂M + . For all v ∈ L 2 (M, ΛM ), and f, g ∈ L 2 (M, Λ∂M ) with support in Γ c + , there exists u ∈ L 2 (M, ΛM ) such that . Proof. Suppose w ∈ Ω(M ) satisfies the relative boundary conditions (2.1) with σ = 0, and examine the expression

This is bounded above by
. Then by Theorem 2.4, Therefore on the subspace . By Hahn-Banach, this functional extends to the whole space, and thus there exists a u ∈ L 2 (M, ΛM ) such that Integrating by parts and applying the boundary conditions (2.1) gives Then by choosing the correct w, we can see that To match notations with previous papers, we will begin by rewriting this result, along with the Carleman estimate, in τ notation, as follows.
Theorem 2.4 becomes the following.
for some smooth endomorphism σ independent of τ . Then there exists τ 0 > 0 such that if τ > τ 0 , By choice of coordinates, note that the same theorem holds for τ < 0, with Γ + replaced by Γ − . Then Proposition 7.1 becomes the following.
Proposition 7.3. Let Q be an L ∞ endomorphism on ΛM . For all v ∈ L 2 (M, ΛM ), and f, g ∈ L 2 (Γ c + , ΛΓ c + ), there exists u ∈ L 2 (M, ΛM ) such that . Now we turn to the construction of the CGOs themselves. From now on we will invoke the assumption that the conformal factor c in the definition of M as an admissible manifold satisfies c = 1. Below we will consider complex valued 1-forms, and · , · will denote the complex bilinear extension of the Riemannian inner product to complex valued forms.
We assume that where (M 0 , g 0 ) is a compact (n − 1)-dimensional manifold with smooth boundary. We write x = (x 1 , x ′ ) for points in R × M 0 , where x 1 is the Euclidean coordinate and x ′ is a point in M 0 . Let Q be an L ∞ endomorphism of ΛM . We next wish to construct solutions to the equation where Z is a graded differential form in L 2 (M, ΛM ) having the form Z = e −sx1 (A + R).
Here s = τ + iλ is a complex parameter where τ, λ ∈ R and |τ | is large, the graded form A is a smooth amplitude, and R will be a correction term obtained from the Carleman estimate. Inserting the expression for Z in the equation results in The point is to choose A so that F L 2 (M) = O(1) as |τ | → ∞. By Lemma 3.2, we have F = (−∆ − s 2 + 2s∇ ∂1 + Q)A.
We wish to choose A so that ∇ ∂1 A = 0. The following lemma explains this condition. Below, we identify a differential form in M 0 with the corresponding differential form in R × M 0 which is constant in x 1 .
Lemma 7.4. If u is a k-form in R × M 0 with local coordinate expression u = u I dx I , then If ∇ ∂1 u = 0, then there is a unique decomposition For such a k-form u, one has where ∆ and ∆ x ′ are the Hodge Laplacians in R × M 0 and in M 0 , respectively.
The proof of the lemma may be found in the appendix. Returning to the expression for F , the assumption ∇ ∂1 A = 0 gives that Writing Y k for the k-form part of a graded form Y , and decomposing A k = dx 1 ∧ (A k ) ′ + (A k ) ′′ as in Lemma 7.4, we obtain that Thus, in order to have F L 2 (M) = O(1) as |τ | → ∞, it is enough to find for each k a smooth (k − 1)-form (A k ) ′ and a smooth k-form (A k ) ′′ in M 0 such that If (M 0 , g 0 ) is simple, there is a straightforward quasimode construction for achieving this.
Lemma 7.5. Let (M 0 , g 0 ) be a simple m-dimensional manifold, and let 0 ≤ k ≤ m. Suppose that (M 0 , g 0 ) is another simple manifold with (M 0 , g 0 ) ⊂⊂ (M 0 , g 0 ), fix a point ω ∈M int 0 \ M 0 , and let (r, θ) be polar normal coordinates in (M 0 , g 0 ) with center ω. Suppose that η 1 , . . . , η m is a global orthonormal frame of T * M 0 with η 1 = dr and ∇ ∂r η j = 0 for 2 ≤ j ≤ m, and let {η I } be a corresponding orthonormal frame of Λ k M 0 . Then for any λ ∈ R and for any m k complex functions b I ∈ C ∞ (S m−1 ), the smooth k-form u = e isr |g 0 (r, θ)| −1/4 Proof. We first try to find the quasimode in the form u = e isψ a for some smooth real valued phase function ψ and some smooth k-form a. Lemma 3.2 implies that Let (r, θ) be polar normal coordinates as in the statement of the lemma, and note that g 0 (r, θ) = 1 0 0 h(r, θ) globally in M 0 for some (m − 1) × (m − 1) symmetric positive definite matrix h.
Then ψ ∈ C ∞ (M 0 ) and |dψ| 2 = 1, so that the s 2 term will be zero. We next want to choose a so that 2∇ grad(ψ) a + (∆ x ′ ψ)a = 0. Note that Thus, choosing a = |g 0 | −1/4ã for some k-formã, it is enough to arrange that Using the frame {η j } above, with η 1 = dr, we writẽ for some functions α 1,J and β J in M 0 . Now, the form of the metric implies that ∇ ∂r η 1 = 0, and by assumption ∇ ∂r η j = 0 for 2 ≤ j ≤ m. Therefore In the definitions ofã ′ andã ′′ , we may now choose where b I are the given functions in C ∞ (S m−1 ). The resulting k-form u = e isψ |g 0 | −1/4ã satisfies the required conditions.
The next result gives the full construction of the complex geometrical optics solutions.
Lemma 7.6. Let (M, g) ⊂⊂ (R× M 0 , g) where g = e ⊕ g 0 , assume that (M 0 , g 0 ) is simple, and let Q be an L ∞ endomorphism of ΛM . Let (M 0 , g 0 ) be another simple manifold with (M 0 , g 0 ) ⊂⊂ (M 0 , g 0 ), fix a point ω ∈M int 0 \M 0 , and let (r, θ) be polar normal coordinates in (M 0 , g 0 ) with center ω. Suppose that η 1 , . . . , η n is a global orthonormal frame of T * (R × M 0 ) with η 1 = dx 1 , η 2 = dr, and ∇ ∂r η j = 0 for 3 ≤ j ≤ n, and let {η I } be a corresponding orthonormal frame of Λ(R × M 0 ). Let also λ ∈ R. If |τ | is sufficiently large and if s = τ + iλ, then for any 2 n complex functions b I ∈ C ∞ (S n−2 ) there exists a solution Z ∈ L 2 (M, ΛM ) of the equation Decomposing the k-form part of A as Let η 1 , . . . , η n and {η I } be orthonormal frames as in the statement of the result. We can use Lemma 7.5 to find, for any n−1 k−1 functions b ′ J (θ) and for any n−1 Recalling that A k = η 1 ∧(A k ) ′ +(A k ) ′′ and relabeling functions, this shows that for any n k functions b I ∈ C ∞ (S n−2 ) we may find A k of the form Repeating this construction for all k, we obtain the amplitude A = e isr |g 0 (r, θ)| −1/4 with the same norm estimates as those for A k . Then also F L 2 (M) = O(1). Then Proposition 7.3 allows us to find R with the right properties. This finishes the proof.
Note that if Z is a solution to (−∆ + * Q * −1 )Z = 0 in M , and Z has relative boundary values that vanish on Γ c + , then * Z is a solution to (−∆ + Q) * Z = 0 in M , and * Z has absolute boundary values that vanish on Γ c + . Thus this construction also gives us solutions with vanishing absolute boundary values on Γ c + .

The tensor tomography problem
Now we can begin the proof of Theorems 2.1 and 2.2. First we will use the hypotheses of Theorem 2.1 to obtain some vanishing integrals involving (Q 2 − Q 1 ).
Lemma 8.1. Suppose the hypotheses of Theorem 2.1 hold. Using the notation in Lemma 7.6, let Z j ∈ L 2 (M, ΛM ) be solutions of (−∆ + Q 1 )Z 1 = (−∆ +Q 2 )Z 2 = 0 in M of the form Z 1 = e −sx1 e isr |g 0 | −1/4 I c I (θ)η I + R 1 , Z 2 = e sx1 e isr |g 0 | −1/4 I d I (θ)η I + R 2 with vanishing relative boundary conditions on Γ c − and Γ c + respectively. Then Proof. Let Y be a solution of (−∆ + Q 2 )Y = 0 in M with the same relative boundary conditions as Z 1 ; such a solution exists by the assumption on Q 2 . Then consider the integral Recall from the section on notation and identities that Since the relative boundary values of (Z 1 − Y ) vanish, by definition, the integration by parts formula above implies that Meanwhile, by the hypothesis on N RA Q1 and N RA Q2 , we have that N RA . Therefore the fact that Z 2 has relative boundary values that vanish on Γ c + implies that for each such pair of CGO solutions Z 1 and Z 2 .
Working through the same argument with * Z 1 and * Z 2 gives us the following lemma as well.
Lemma 8.2. Suppose the hypotheses of Theorem 2.2 hold. Using the notation in Lemma 7.6, let * Z j ∈ L 2 (M, ΛM ) be solutions of (−∆ + Q 1 ) * Z 1 = (−∆ +Q 2 ) * Z 2 = 0 in M of the form Then Therefore both of the main theorems reduce to using the condition (QZ 1 , Z 2 ) L 2 (M) = 0 for solutions of the type given in Lemma 7.6 to show that Q = 0.
The next result shows that from the condition (QZ 1 , Z 2 ) L 2 (M) = 0 for solutions of the type given in Lemma 7.6, it follows that certain exponentially attenuated integrals over geodesics in (M 0 , g 0 ) of matrix elements of Q, further Fourier transformed in x 1 , must vanish. Proposition 8.3. Assume the hypotheses in Theorem 2.1 or 2.2, with Q = Q 2 − Q 1 extended by zero to R × M 0 . Fix a geodesic γ : [0, L] → M 0 with γ(0), γ(L) ∈ ∂M 0 , let ∂ r be the vector field in M 0 tangent to geodesic rays starting at γ(0), and suppose that {η I } is an orthonormal frame of Λ(R × M int 0 ) with η 1 = dx 1 , η 2 = dr, and ∇ ∂r η j = 0 for 3 ≤ j ≤ n. (Such a frame always exists.) Then for any λ ∈ R and any I, J one has Proof. Using the notation in Lemma 7.6, let Z j ∈ L 2 (M, ΛM ) be solutions of (−∆ + Q 1 )Z 1 = (−∆ +Q 2 )Z 2 = 0 in M of the form where s = τ + iλ, τ > 0 is large, λ ∈ R, and c I , d I ∈ C ∞ (S n−2 ). We can assume that R j L 2 (M) = O(τ −1 ) as τ → ∞, and that the relative (absolute) boundary values of Z 1 are supported inF and the relative (absolute) boundary values of Z 2 are supported inB. By Lemma 8.1 (Lemma 8.2), we have We now extend the M 0 -geodesic γ toM 0 , choose ω = γ(−ε) for small ε > 0, and choose θ 0 so that γ(t) = (t, θ 0 ). The functions c I and d J can be chosen freely, and by varying them we obtain that ∞ 0 e −2λr ∞ −∞ e −2iλx1 Q(x 1 , r, θ 0 )η I , η J dx 1 dr = 0 for each fixed I and J. Since Q is compactly supported in M int 0 , this implies the required result. It remains to show that a frame {η I } with the required properties exists. Let ω = γ(0), and let (M 0 , g 0 ) be a simple manifold with (M 0 , g 0 ) ⊂⊂ (M 0 , g 0 ) such that theM 0 -geodesic starting at ω in direction ν(ω) never meets M 0 . (It is enough to embed (M 0 , g 0 ) in some closed manifold and to takeM 0 strictly convex and slightly larger than M 0 .) Let (r, θ) be polar normal coordinates inM 0 with center ω = γ(0), fix r 0 > 0 so that the geodesic ball B(ω, r 0 ) is contained inM int 0 , and let θ ∈ S n−2 be the direction of ν(ω). Choose some orthonormal frame η 3 , . . . , η n of the cotangent space of ∂B(ω, r 0 ) \ {(r 0 ,θ)}, and extend these as 1-forms in M int 0 by parallel transporting along integral curves of ∂ r . We thus obtain a global orthonormal frame η 2 , . . . , η n of T * M int 0 with η 2 = dr and ∇ ∂r η j = 0 for 3 ≤ j ≤ n. Moreover, η 1 , . . . , η n will be a global orthonormal frame of T * (R × M int 0 ) inducing an orthonormal frame {η I } of Λ(R × M int 0 ). We will now show how the coefficients are uniquely determined by the integrals in Proposition 8.3. This follows by inverting attenuated ray transforms, a topic of considerable independent interest (see [ABK98], [No02], [BS04] and the survey [Fi03] for results in the Euclidean case, and the survey [PSU13b] and references below for the manifold case). The transform in Proposition 8.3 is not exactly the same kind of attenuated ray transform/Fourier transform as in the scalar case for instance in [DKSaU09], since the matrix element of Q that appears in the integral may actually depend on the geodesic γ (note that the 1-forms η depend on γ). To clarify this point, we fix some global orthonormal frame {ε 1 , . . . , ε n } of T * (R × M 0 ) with ε 1 = dx 1 , and let {ε I } be the corresponding orthonormal frame of Λ(R × M 0 ). Define the matrix elements q I,J = Qε I , ε J .
We will now make use of the methods of [PSU13a] in this tensor tomography problem. We only give the details in the case where Q (and hence f 2,2 ) is C ∞ . The result also holds for continuous Q by using an elliptic regularity result for the normal operator, but in the present weighted case for 2-tensors the required result may not be in the literature. We only say that such a result can be proved by adapting the methods of [HS10] to the 2-tensor case (in particular one needs a solenoidal decomposition f = f s + dβ of a 2-tensor f and a further solenoidal decomposition β = β s + dφ of the 1-form β, and one then shows that the normal operator acting on "solenoidal triples" (f s , β s , φ) is elliptic because the weight comes from a nonvanishing attenuation).
Since f 2,2 is C ∞ , the injectivity result for the attenuated ray transform on symmetric 2-tensors (see [As12], following [PSU13a]) says that where X is the geodesic vector field on (M 0 , g 0 ), and u is a smooth function on the unit circle bundle SM 0 that corresponds to the sum of a 1-tensor and scalar function, with u| ∂M0 = 0.
The same argument now works for the remaining entries of q, and this finishes the proof.

Higher dimensions
In higher dimensions, n > 3, as noted above, everything up to and including the proof of Proposition 8.3 still holds. However, this does not reduce easily into a tensor tomography problem, as in the three-dimensional case, because we cannot choose the basis {η i } so that η 3 , . . . , η 4 to depend on η 2 = dr in a tensorial manner.
More precisely, in general we lack tensors T i for which η i = T i (η 2 , . . . , η 2 ) for i ≥ 3, as is the case in three dimensions. Moreover, even if the results of Proposition 8.3 can be reduced to a tensor tomography problem, there is no guarantee that it will be one for which there are useful injectivity results, since there are very few such results for k-tensors with k > 2.
However, in the Euclidean case we can do better, since we have the extra freedom to vary the Carleman weight ϕ. In particular, we can construct CGOs to reduce the problem in Lemmas 8.1 and 8.2 to a Fourier transform, as has been done for inverse problems for scalar functions, e.g. in [BU01]. Therefore we can conclude this paper by a proof for higher dimensions, in the Euclidean case.
Proof of Theorem 2.3. Fix coordinates x 1 , . . . , x n on R n . The corresponding basis for the cotangent space is dx 1 , . . . , dx n , and this gives a corresponding basis {dx I } for ΛM .
Now note that if f is a scalar function, △(f dx I ) = (△f )dx I . Therefore if α and β are unit vectors such that α · β = 0, then (dx I + r) has relative boundary conditions which vanish on Γ c + . Now if k and ℓ are mutually orthogonal unit vectors which are both orthogonal to α, then we can set β 1 = ℓ + hk and β 2 = ℓ − hk, and create Z 1 = e (−α+iβ 1 )·x h (dx I + r 1 ) and Z 2 = e (α+iβ 2 )·x h (dx I + r 2 ) so that (−△ + Q 1 )Z 1 = (−△ + Q 2 )Z 2 = 0, and Z 1 and Z 2 have relative boundary conditions that vanish on Γ c − and Γ c + respectively. Then Lemma 8.1, together with the hypotheses of Theorem 2.3 implies that This can be done for any k orthogonal to α. Since α can be varied slightly without preventing the relative boundary conditions of the solutions from vanishing on the correct set, this is in fact true for k in an open set, from which we can conclude that Q 1 = Q 2 on M .
The absolute boundary value version works similarly, with the appropriate change in the CGOs.
Proof of Lemma 3.1. We will prove that The lemma will follow immediately from this by polarization. It is enough to prove the identity at a fixed point. Choose a positive orthonormal basis ε 1 , . . . , ε n of covectors such that ξ = |ξ|ε 1 , and note that if I = {i 1 , . . . , i k } with i 1 < . . . < i k , we have ε 1 ∧ i ε 1 ε I = ε I , 1 ∈ I, 0, otherwise and i ε 1 (ε 1 ∧ ε I ) = ε I , 1 / ∈ I, 0, otherwise. This proves the result. It follows that e sρ (−∆)(e −sρ u) = e sρ (dδ + δd)(e −sρ u) Writing ρ = φ + iψ where φ and ψ are real valued, Lemma 3.1 implies that It remains to show that To prove this, we compute in Riemannian normal coordinates at a fixed point p. Then u has the coordinate expression u = u I dx I . We have Combining these computations with Lemma 3.1, we see that Proof of Lemma 3.3. First examine −tδu ⊥ . By using the expansion of δ in coordinates, at a point p ∈ ∂M , where {e j } are a basis for T ∂M at p. Since N is a geodesic vector field, ∇ N N ♭ = 0, and thus u ⊥ (e j , X 1 , .., ∇ ej X l , .., X k−1 ) u ⊥ (∇ ′ ej e j + II(e j , e j ), X 1 , .., X k−1 ) where II : T ∂M ⊕ T ∂M → N ∂M is the second fundamental form of ∂M relative to its embedding in M . Since all of e j and X k in the above sum belong to T ∂M and u ⊥ needs an element of N ∂M in its argument to survive, we have that u ⊥ (e j , X 1 , .., ∇ ′ ej X l + II(e j , X l ), .., X k−1 )) = u ⊥ (e j , X 1 , .., II(e j , X l ), .., X k−1 )).

Then
Using this together with (10.1) gives Now consider tδu . Again, using the expansion of δ in coordinates, Since N is a geodesic vector field, ∇ N N = 0. Moreover, i N u = 0 by the definition of u . Therefore i N ∇ N u = 0. Then for the remaining sums, we let {X 1 , ..X k−1 } be sections of T ∂M and compute u || (e j , X 1 , .., ∇ ej X l , .., X k−1 ) .
Proof of Lemma 3.4. As in the previous proof, we'll begin by examining the u ⊥ part. Using the expansion of d in terms of sections, if {X 1 , . . . , X k } are local sections of T ∂M , (N, X 1 , ..,X j , .., X k ) + ∇ N u ⊥ (X 1 , .., X k ) Here we have identified X j with its extension to a neighbourhood of ∂M in the interior of M by parallel transport along normal geodesics, so that (N |X j ) vanishes identically. Then the last term becomes ∇ N u ⊥ (X 1 , .., X k ) = N (u ⊥ (X 1 , .., X k )) =0 − k l=1 u ⊥ (X 1 , .., ∇ N X l =0 , ..X k ) = 0.
Putting this together with the calculation for the normal part gives To prove Lemma 3.5, we will first require the following intermediate lemma.
Lemma 10.1. For all smooth functions f , and u ∈ Ω k (M ) such that tu = 0, we have that Proof. We first split ∇ ∇f into its normal and tangential parts: ∇ ∇f = −∂ ν f ∇ N + ∇ ∇f || and where ∇f || is defined by ∇f = −(∂ ν f )N + ∇f || . We first compute the tangential derivative We now observe that ti N ∇ N u || = 0 and t∇ N u ⊥ = 0 so that we have Plugging all this into the equality above we have  N ] i N u. For the derivative in the normal direction we write Therefore tδBu =δ ′ tBu + iht 2∇ N i N ∇ grad(ϕc) u + ∇ N ∆ϕ c i N u + ih(S − (n − 1)κ) 2t∇ grad(ϕc) i N u − 2∂ ν ϕ c t∇ N i N u + ∆ϕ c ti N u Now by Lemma 10.1, Substituting this into the previous equation finishes the proof.
This shows that ∇ ∂1 dx I = 0 for all I, and therefore any k-form u = u I dx I satisfies ∇ ∂1 (u I dx I ) = ∂ 1 u I dx I .
Thus ∇ ∂1 u = 0 if and only if each u I only depends on x ′ . In general, if u is a k-form on R × M 0 we have the unique decomposition u = dx 1 ∧ u ′ + u ′′ where u ′ (x 1 , · ) is a (k − 1)-form in M 0 and u ′′ (x 1 , · ) is a k-form in M 0 , depending smoothly on the parameter x 1 . If ∇ ∂1 u = 0, then u = dx 1 ∧ u ′ + u ′′ where u ′ and u ′′ are differential forms in M 0 . Suppose now that u = dx 1 ∧ u ′ + u ′′ where u ′ and u ′′ are forms in M 0 . Denote by d x ′ and δ x ′ the exterior derivative and codifferential in x ′ . Clearly The identity δ = − n j=1 i ej ∇ ej , where e j is an orthonormal frame in T (R × M 0 ) with e 1 = ∂ 1 , together with the fact that ∇ ∂1 u ′′ = 0, implies that Finally, computing in Riemannian normal coordinates at p gives that It follows directly from these facts that