Determining an unbounded potential from Cauchy data in admissible geometries

In a previous article of Dos Santos Ferreira, Kenig, Salo and Uhlmann, anisotropic inverse problems were considered in certain admissible geometries, that is, on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. In particular, it was proved that a bounded smooth potential in a Schr\"odinger equation was uniquely determined by the Dirichlet-to-Neumann map in dimensions n \geq 3. In this article we extend this result to the case of unbounded potentials, namely those in Ln/2. In the process, we derive Lp Carleman estimates with limiting Carleman weights similar to the Euclidean estimates of Jerison-Kenig and Kenig-Ruiz-Sogge.


Introduction
In this paper we consider the problem of proving L p estimates for limiting Carleman weights on Riemannian manifolds, and the related inverse problem of recovering an L n/2 potential from the Dirichlet-to-Neumann map (DN map) related to the Schrödinger equation. The main motivation comes from the inverse conductivity problem posed by Date: March 30, 2011. 1 Calderón [1]. This problem asks to determine the conductivity function of a medium from electrical measurements made on its boundary.
In mathematical terms, if Ω ⋐ R n is the medium of interest having a positive conductivity coefficient γ, in the Calderón problem one considers the conductivity equation ∇ · γ∇u = 0 in Ω and defines the DN map by This operator maps the voltage at the boundary to the current given by γ times the normal derivative, which encodes the electrical measurements at the boundary. The inverse problem of Calderón asks to determine γ from the knowledge of Λ γ . This problem has been extensively studied and we refer to [30] for a recent survey. The anisotropic Calderón problem considers the case where the conductivity γ is a symmetric positive definite matrix instead of a scalar function. This corresponds to situations where the electrical properties of the medium depend on direction. The problem is open in general in dimensions n ≥ 3, see [4] for known results and more details. Following [18] the problem may be recast as the determination of the metric g on a compact Riemannian manifold (M, g) with boundary from the corresponding DN map. In [4] progress was made on the anisotropic Calderón problem in the following class of conformal smooth manifolds.
Definition. A compact Riemannian manifold (M, g), with dimension n ≥ 3 and with boundary ∂M, is called admissible if M ⋐ R × M 0 for some (n−1)-dimensional simple manifold (M 0 , g 0 ), and if g = c(e⊕g 0 ) where e is the Euclidean metric on R and c is a smooth positive function on M.
Here, a compact manifold (M 0 , g 0 ) with boundary is simple if for any p ∈ M 0 the exponential map exp p with its maximal domain of definition is a diffeomorphism onto M 0 , and if ∂M 0 is strictly convex (that is, the second fundamental form of ∂M 0 ֒→ M 0 is positive definite).
In [4] it was proved that a Riemannian metric in a conformal class of admissible geometries is uniquely determined by the DN map. This was obtained as a corollary of a result for the Schrödinger equation in a fixed admissible manifold, stating that a bounded smooth potential q is determined by the corresponding DN map. In [4] all coefficients were assumed infinitely differentiable. In this paper we relax this requirement and show that a complex potential q ∈ L n/2 (M) is determined by the DN map.
To state the main result, assume that (M, g) is a compact Riemannian manifold with smooth boundary ∂M, and let ∆ g be the Laplace-Beltrami operator. Given a complex function q ∈ L n/2 (M), where n ≥ 3 is the dimension of the manifold M, we consider the Dirichlet problem (−∆ g + q)u = 0 in M, u| ∂M = f.
We assume throughout that 0 is not a Dirichlet eigenvalue for this problem, and then standard arguments (see Appendix A) show that there is a well-defined DN map Λ g,q : The following uniqueness theorem is the main result for the inverse problem. (The assumption q ∈ L n/2 may be considered optimal in the context of the standard wellposedness theory for the Dirichlet problem with L p potentials, and also for the strong unique continuation principle to hold [10].) Theorem 1.1. Let (M, g) be admissible and let q 1 , q 2 be complex functions in L n/2 (M). If Λ g,q 1 = Λ g,q 2 , then q 1 = q 2 .
In the case where M is a bounded domain in R n and g is the Euclidean metric, this result is due to Lavine and Nachman [17] following the earlier result of Jerison and Kenig for q j ∈ L n/2+ε (M) for some ε > 0 (see Chanillo [2] for an account and also for a related result with q j in a Fefferman-Phong class with small norm). As mentioned above, if q is a smooth function on an admissible manifold M this result was proved in [4] by using L 2 Carleman estimates. In fact, smoothness of q is not essential, and by inspecting the proof of [4] the uniqueness result can be extended to bounded continuous q (with the complex geometrical optics construction in the proof going through for q ∈ L n (M)). However, the proof for q ∈ L n/2 requires to replace the L 2 Carleman estimates in [4] with corresponding L p Carleman estimates.
The other main result in this paper is a L p Carleman estimate for limiting Carleman weights on Riemannian manifolds. The concept of limiting Carleman weights was introduced in [14] as part of a general procedure for producing special complex geometrical optics solutions to elliptic equations, with applications to inverse problems. We refer to [4] for a precise definition and more careful analysis of limiting Carleman weights, also on Riemannian manifolds. For present purposes it is sufficient to mention that the existence of a limiting Carleman weight on (M, g) in dimensions n ≥ 3 is locally equivalent with the manifold being admissible, and that typical limiting Carleman weights in R n , n ≥ 3, include the linear weight ϕ(x) = x 1 and logarithmic weight ϕ(x) = log |x|.
The last two weights are featured in the literature of Carleman estimates and unique continuation, in particular in the scale invariant L p Carleman estimates of Kenig-Ruiz-Sogge [12] for the linear weight and of Jerison-Kenig [10] for the logarithmic weight. We prove an analogue of these estimates on more general Riemannian manifolds. Note that the existence of a limiting Carleman weight requires at least locally a product structure on the manifold, and therefore the following result is stated for the linear weight on a product manifold. The result, in the case when the manifold (M 0 , g 0 ) below is the standard n−1 dimensional torus, is due to Shen [23]. Theorem 1.2. Let (M 0 , g 0 ) be an (n − 1)-dimensional compact manifold without boundary, and equip R × M 0 with the metric g = e ⊕ g 0 where e is the Euclidean metric. The Euclidean coordinate is denoted by x 1 . For any compact interval I ⊆ R there exists a constant C I > 0 such that if |τ | ≥ 4 and then we have when u ∈ C ∞ 0 (I × M 0 ). The proof of the L 2 Carleman estimates for limiting Carleman weights in [4] is based on integration by parts and cannot be used in the L p setting. However, in [13] another proof of the L 2 Carleman estimate is given; this proof is based on Fourier analysis and gives an explicit inverse for the conjugated Laplacian. We will derive the L p bounds from this explicit inverse operator. This follows the proof of the L p Carleman estimate of Jerison-Kenig [10] using Jerison's approach [9], [26, Section 5.1] based on the spectral cluster estimates of Sogge [26]. Finally, if one allows strongly pseudoconvex Carleman weights then much stronger estimates are available (see for instance [15,16]), however for the applications to inverse problems it seems necessary to restrict to limiting Carleman weights. Remark 1.3. The above theorems are in the setting of (conformal) product manifolds. However, the results also apply to warped products. If f : R → R is a smooth function and (M 0 , g 0 ) is an (n−1)-dimensional manifold, the warped product R × e 2f M 0 is the manifold M = R × M 0 endowed with the metric .
We choose coordinates y 1 = η(x 1 ), y ′ = x ′ for a suitable smooth strictly increasing function η. In fact, if then η ′ (t) −2 = e 2f (t) and the metric in y coordinates becomes a conformal multiple of a product metric, .
Warped products have a natural limiting Carleman weight ϕ(y) = y 1 , and Theorem 1.1 remains true in conformal multiples of warped products whenever (M 0 , g 0 ) is a simple manifold.
The paper is organized as follows. Section 1 is the introduction. In Section 2 we prove the L p Carleman estimate complemented with the usual L 2 Carleman estimates. Section 3 presents the construction of complex geometrical optics solutions for Schrödinger equations with L n/2 potentials in admissible geometries. The proof of Theorem 1.1 is contained in Section 4, modulo a uniqueness result for an analogue of the attenuated geodesic ray transform acting on unbounded functions. This last result has a different character than the rest of the proof, and it is therefore established separately in Section 5. There are two appendices concerning the wellposedness of the Dirichlet problem and the normal operator for the attenuated ray transform.
Acknowledgements. The last named author would like to thank Adrian Nachman for explaining his unpublished argument with Richard Lavine [17] which proves a uniqueness result for L n/2 potentials in Euclidean space. C.K. is supported partly by NSF grant DMS-0968472, and M.S. is supported in part by the Academy of Finland. D. DSF. would like to thank the University of Chicago for its hospitality.

L p Carleman estimates
The aim of this section is to prove Theorem 1.2, which is an analogue of the L p Carleman estimates obtained in the Euclidean case by Jerison and Kenig [10] (for logarithmic weights) or by Kenig, Ruiz and Sogge [12] (for linear weights). In fact, we prove a more general result which implies Theorem 1.
The case when (M 0 , g 0 ) is the standard n − 1 dimensional torus is due to Shen [23].
where C 0 is independent of τ (but may depend on I).

Remark 2.2.
In the Euclidean case, L p Carleman estimates with linear weights can be obtained from L p Carleman estimates with pseudoconvex Carleman weights by scaling. Indeed, suppose that the following Carleman estimate , holds for all ε ≤ ε 0 and all u ∈ C ∞ 0 (K), then applying this estimate to u µ = u(µ ·) with µ ≥ 1 and u ∈ C ∞ 0 (K), one gets e .
Choosing µ = √ τ , and using the fact that e x 2 1 /2ε ≃ C ε on K, one gets the Carleman estimate , for all u ∈ C ∞ 0 (K). However, in the anisotropic case, one has to find another way.
To prepare for the proof of Proposition 2.1 consider the Laplace-Beltrami operator on N, x 1 + ∆ g 0 and the corresponding conjugated operator (by the limiting Carleman weight x 1 ) We denote by λ 0 = 0 < λ 1 ≤ λ 2 ≤ . . . the sequence of eigenvalues of −∆ g 0 on M 0 and (ψ j ) j≥0 the corresponding sequence of eigenfunctions forming an orthonormal basis of L 2 (M 0 ), We denote by π j : the corresponding Fourier coefficients of a function u on M 0 . We define the spectral clusters as Note that these are projection operators, χ 2 k = χ k , and they constitute a decomposition of the identity We end this paragraph by recalling the spectral cluster estimates of Sogge [24,26] that we will need: The first estimate is given in [26, Corollary 5.1.2] and the second one is a consequence of the first one by duality.
Proof of Proposition 2.1. Recall that our main goal is to prove with D x 1 = −i∂ x 1 . The inverse operator in (2.6) is actually easy to write down, as was done in [13]. The same procedure appears in [9] and [26,Section 5.1]. Writing f = ∞ j=0 π j f and similarly for u, the equation formally becomes f for x 1 on the real line and for j ≥ 0. The symbol of the operator on the left is ξ 2 1 + 2iτ ξ 1 − τ 2 + λ j , and this is always nonzero provided that τ 2 = λ j for all j. Thus, if an inverse operator may be obtained as The operatorG τ is the same as G τ in [13,Section 4], except that in the present setting {ψ j } is a basis of L 2 (M 0 ) on a compact manifold (M 0 , g 0 ) without boundary instead of being a basis of Dirichlet eigenfunctions on a compact manifold with boundary. Let by the corresponding Sobolev spaces. The proof of [13, Proposition 4.1] goes through forG τ without changes and shows that for any fixed δ > 1/2, if |τ | ≥ 1 and τ 2 / ∈ Spec(−∆ g 0 ) then the equation with χ ∈ C ∞ 0 (R) which equals 1 on I. It is then clear that all the statements in the proposition except for the L p estimate follow from the results forG τ explained above.
It is sufficient to prove the L p estimate in the case where τ ≥ 4 and τ 2 / ∈ Spec(−∆ g 0 ). We first record a lemma.
Proof. This follows by writing 1 and by noting that for α > 0 and similarly for α < 0.
Furthermore we have and this concludes the proof of the lemma.
From the decomposition (2.3), the spectral cluster estimate (2.4), and the fact that spectral clusters are projections (χ 2 k = χ k ), we get the following string of estimates By Minkowski's inequality, we have using once again the spectral cluster estimate (2.4), we finally get Using Lemma 2.3, we estimate when k ≤ τ < k + 1 e −(τ −k−1)|t| when τ ≥ k + 1 with k > 0. (Note that when k = 0, a majorant is e −(τ /2)|t| for τ ≥ 4). This allows us to estimate the series By an obvious change of variables we have From the estimates (2.8) and (2.9), we obtain and we conclude using the Hardy-Littlewood-Sobolev inequality . This completes the proof of Proposition2.1.

Complex geometrical optics
In this section we will construct the complex geometrical optics solutions that will be used to recover an L n/2 potential. Throughout the section, let (M, g) be a compact manifold with smooth boundary, and let (M, g) ⋐ (T, g) ⋐ (T , g) where T = R × M 0 ,T = R ×M 0 , and g = e ⊕ g 0 , and (M 0 , g 0 ) ⋐ (M 0 , g 0 ) are two (n − 1)-dimensional simple manifolds. We also assume that n ≥ 3.
First we state a consequence of Proposition 2.1 for the manifold M (this follows easily by embedding (M 0 , g 0 ) in some compact manifold without boundary and using suitable restrictions and extensions by zero).
This operator satisfies where C 0 is independent of τ .
Let us first construct the required complex geometrical optics solutions for the case where no potential is present. This is analogous to [4, Proposition 5.1] for q = 0.
It is enough to take r 0 = G τ f and to note that The L 2 and H 1 estimates follow from Proposition 3.1. The L 2n n−2 estimate follows from the H 1 estimate and Sobolev embedding, or alternatively from the L 2n n+2 → L 2n n−2 estimate for G τ .
We next consider the case with a potential q ∈ L n/2 (M), and try to find a solution to (−∆ g + q)u = 0 in M of the form u = u 0 + e −τ x 1 r 1 .
Since q is only in L n/2 (M), here we need to use the L 2n n+2 → L 2n n−2 estimates for G τ . We follow the argument of Lavine and Nachman [17]. It will be convenient to symmetrize the situation slightly. Later on, the L n functions m j in the next lemma are chosen to be essentially |q| 1/2 . Lemma 3.3. Let m 1 , m 2 ∈ L n (M) be two fixed functions. Then for |τ | ≥ τ 0 outside a countable set, Further, Proof. The Hölder inequality and Proposition 3.1 imply that (One can take for instance m ♯ j = m j χ {|m j |≤µ} for large enough µ.) It follows from the L 2 estimates for G τ and (3.2) that The last expression is bounded by ε f L 2 if |τ | is sufficiently large. This proves (3.3).
We now finish the construction of complex geometrical optics solutions. Proof. As explained above, we let u 0 be the harmonic function given in Proposition 3.2, and look for a solution of the form u = u 0 +e −τ x 1 r 1 . We Then |q| 1/2 , m ∈ L n (M) with L n norms equal to q 1/2 L n/2 . We obtain a solution u provided that (3.1) holds. Trying r 1 in the form r 1 = G τ |q| 1/2 v, we see that v should satisfy By Lemma 3.3, for |τ | sufficiently large one has mG τ |q| 1/2 Now u is of the form given in the statement of the proposition, provided thatr = r 0 + r 1 .
Finally, to prove that u ∈ H 1 (M), it is enough to consider a compact manifold (M , g) which is slightly larger than (M, g) and extend q by zero outside M, and to perform the above construction of solutions in M. One obtains a solution u ∈ L

Recovering the potential
We are now ready to give the proof of the main uniqueness result.
Proof of Theorem 1.1. Assume, as before, that (M, g) ⋐ (T, g) ⋐ (T , g) where T = R × M 0 ,T = R ×M 0 , and (M 0 , g 0 ) ⋐ (M 0 , g 0 ) are two (n − 1)-dimensional simple manifolds where n ≥ 3. Also assume that g = e ⊕ g 0 . From the assumption Λ g,q 1 = Λ g,q 2 , writing q = q 1 − q 2 , we know from Lemma A.1 that Here λ is a fixed real number, b ∈ C ∞ (S n−2 ) is a fixed function, and x = (x 1 , r, θ) are coordinates inT where (r, θ) are polar normal coordinates in (M 0 , g 0 ) with center at a fixed point ω ∈M 0 \ M 0 . Also, the remainder terms satisfy r j L as τ → ∞. Inserting the solutions in (4.1) and noting that dV g = |g| 1/2 dx 1 dr dθ, we obtain that where a 1 , a 2 are smooth functions in M independent of τ . We show that the right hand side converges to 0 as τ → ∞. To do this, fix ε > 0 and write q = q ♯ + q ♭ where q ♯ ∈ L ∞ (M), q ♯ L n/2 ≤ q L n/2 , and q ♭ L n/2 ≤ ε. The right hand side of (4.2) is bounded by M q(a 1 r 2 + a 2 r 1 + r 1 r 2 ) dV n−2 ). Using the bounds for r j , if τ is sufficiently large then the last quantity is ε. This shows that the right hand side of (4.2) goes to 0 as τ → ∞.
Extend q by zero into T and interpret the left hand side of (4.2) as an integral over T . Taking the limit as τ → ∞, we obtain that This statement is true for all choices of ω ∈M 0 \M 0 , for all real numbers λ, and for all functions b ∈ C ∞ (S n−2 ). Since q ∈ L 1 (M), Fubini's theorem shows that q( · , r, θ) is in L 1 (R) for a.e. (r, θ). Consequently If |λ| is sufficiently small, it follows from Lemma 5.1 below that the vanishing of the integrals (4.3) for all choices ω and b implies that f λ = 0. Since q( · , r, θ) is a compactly supported function in L 1 (R) for a.e. (r, θ), the Paley-Wiener theorem shows that q( · , r, θ) = 0 for such (r, θ). Consequently q 1 = q 2 .

Attenuated ray transform
It remains to show the following lemma which was used in the proof of Theorem 1.1. where (r, θ) are polar normal coordinates in (M 0 , g 0 ) centered at some ω ∈ ∂M 0 , and τ (ω, θ) is the time when the geodesic r → (r, θ) exits M 0 . If |λ| is sufficiently small, and if these integrals vanish for all ω ∈ ∂M 0 and all b ∈ C ∞ (S n−2 ), then f = 0.
The last result is related to the vanishing of the attenuated geodesic ray transform of the function f on M 0 . For the following facts see [3], [20], [22]. To define the ray transform, we consider the unit sphere bundle This manifold has boundary ∂(SM 0 ) = {(x, ξ) ∈ SM 0 ; x ∈ ∂M 0 } which is the union of the sets of inward and outward pointing vectors, Here ν is the outer unit normal vector to ∂M 0 . Note that ∂ + (SM 0 ) is a manifold whose boundary consists of all the tangential directions {(x, ξ) ∈ ∂(SM 0 ) ; ξ, ν = 0}. Thus the space C ∞ 0 ((∂ + (SM 0 )) int ) contains all smooth functions on ∂ + (SM 0 ) vanishing near tangential directions.
The geodesic ray transform, with constant attenuation −λ, acts on functions on M 0 by In The previous argument together with Proposition 5.2 proves Lemma 5.1 for smooth f . However, this requires well defined restrictions of f to all geodesics and it is not obvious how to do this when f ∈ L 1 . We circumvent this problem by using duality and the ellipticity of the normal operator T * λ T λ .
We will need a few facts. Below we write where µ(x, ξ) = − ξ, ν(x) and dN is the natural Riemannian volume form on a manifold N.
Proof. By the Santaló formula This proves Lemma 5.4.
Lemma 5.5. T * λ T λ is a self-adjoint elliptic pseudodifferential operator of order −1 in M int 0 . Proof. This is contained in [6,Proposition 2], but for completeness we also include a proof in Appendix B.
The bilinear form related to this problem is where · , · is the complex-linear inner product of 1-forms and dV is the volume form on (M, g). By the Sobolev embedding H 1 (M) ⊆ L 2n n−2 (M) and by Hölder's inequality, B is a bounded bilinear form on H 1 0 (M). Writing q = q ♯ + q ♭ where q ♯ ∈ L ∞ (M) and q ♭ L n/2 (M ) is small, we obtain from Poincaré's inequality that . This shows that B + C is coercive, and by the Lax-Milgram lemma, compact Sobolev embedding and the Fredholm theorem, the equation (A.1) has a unique solution u ∈ H 1 0 (M) for any F ∈ H −1 (M) if one is outside a countable set of eigenvalues.
We can now consider the Dirichlet problem We assume that 0 is not a Dirichlet eigenvalue, and it follows from the above discussion that for any f ∈ H 1/2 (∂M) there is a unique solution u ∈ H 1 (M). The DN map is formally defined as the map Λ g,q : More precisely, if f ∈ H 1/2 (∂M) we define Λ g,q f weakly as the function in H −1/2 (∂M) which satisfies As a consequence, we have the basic integral identity used in the uniqueness proof.
Lemma A.1. If q 1 , q 2 ∈ L n/2 (M) and Λ g,q 1 = Λ g,q 2 , then Proof. Follows from the computation and the definition of the DN maps.

Appendix B. Normal operator
The setting is the compact simple Riemannian manifold (M 0 , g 0 ) of dimension n − 1. Let T λ be the attenuated ray transform as in Section 5. We will prove Lemma 5.5. Write We compute the normal operator τ (x,−ξ)+τ (x,ξ) 0 e −λt f (γ(t, ψ(x, ξ))) dt dS x (ξ) and using changes of variables we get for the last integral expression Changing variables y = exp x (sξ) shows that with ϕ ± = 2τ (x, ∓ grad y d g 0 (x, y)) ± d g 0 (x, y). The functions ϕ ± are smooth away from the diagonal x = y, and their k-th order derivatives behave as d g 0 (x, y) −k . Note that det(exp −1 ) ′ stands for the Jacobian determinant of x (y). The kernel of the normal operator is symmetric and the singular support of this kernel is the diagonal in M 0 × M 0 .
We will now prove that the operator P λ with kernel K λ is actually a pseudodifferential operator. The first observation in that direction is that in coordinates d 2 g 0 (x, y) = a jk (x, y)(x j − y j )(x k − y k ) (B.1) with a jk (x, x) = g jk 0 (x). Indeed the square of the distance vanishes at second order and its Hessian at x = y is twice the metric. This can be seen from the well known formula ∇ 2 ϕ(y)(θ, θ) = ∂ 2 ∂t 2 ϕ(exp y tθ) t=0 and the fact that if |θ| g 0 = 1 then d 2 g 0 (exp y tθ, y) = t 2 . To prove that P λ is a pseudodifferential operator in Ψ −1 (M int ) we need to show that for any couple of cutoff functions (ψ, χ) supported in charts of M int , the operator with kernel K λ (x, y) = ψ(x)K λ (x, y) det g 0 (y)χ(y) expressed in coordinates 1 , is a pseudodifferential operator on R n−1 with symbol in S −1 . Because of its form and of (B.1), the kernel satisfies and has compact support in R n−1 × R n−1 .
Such operators are pseudodifferential operators and this can easily be seen in the following way: the symbol associated with such an operator isp For cutoff functions ψ and χ whose supports don't intersect, the previous symbol is a Schwartz function because the kernel is a smooth compactly supported function. So we are only interested in those symbols corresponding to kernelsK λ (x, x − y) which are supported close to R n−1 × {0}. In that case, we use a dyadic partition of unity with χ a function supported in an annulus, to decompose the symbol as a sum of terms of the form 2 µ(n−1) e i2 µ y·ξ χ(y)K λ (x, x − 2 µ y) dy.
Note that because of the compact support of the kernel, these terms vanish when µ is large, so we are mainly concerned with the terms where µ is less than some positive integer, say N. Because of the behaviour (B.2), the rescaled kernelK λ (x, x − 2 µ y) is uniformly bounded by 2 −µ(n−2) as well as all its derivatives. Applying the non-stationary phase when |ξ| ≥ 1 and 2 µ ξ is large we get Repeating this argument for the derivatives of this function, we get thatp λ is a classical symbol of order −1.