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2D Inverse with a Foliation Condition

The inverse problem considers reconstructing functions from some data that are easier to measure, which is typically the integral of that function along geodesics. Our paper proves that if the domain have foliation structure, then this reconstruction process is possible.


Introduction
The inverse problem on the geodesic ray transformation is a generalization of the Radon transformation, and it can be formulated as follows: On a Riemannian manifold (X, g), the information we have are integrals like I ̺ f (γ) := γ ̺f (z)dz, where γ are geodesic segment in a neighborhood O p of p ∈ X, and ̺ is a density function on T * X, the cotangent bundle of X. The goal we want to achieve is to recover the function f , hence we focus on the injectivity of the ray transform. In this paper, we work on local ray transform. This problem is resolved by Uhlmann and Vasy [6] in dimension ≥ 3 under convexity assumption. And using the same framework, Paternain et al. [3] extended this result to the case with matrix weights. And resolution of the real analytic case traces back to Boman and Quinto [2]. Pestov and Uhlmann [4] considered the two dimensional compact simple case.
For the general case, Boman proved [1] the transformation is not locally injective in the 2-dimensional case with density. On the other hand, the injectivity holds under certain restrictions on the function or geometric structure. Here the additional condition we impose is that there exists a convex foliation of the manifold X as in [6] and the function is adapted to it in the sense we define below. Roughly speaking, this means that the direction that the function changes is conormal to the layer structure of the manifold. In the real world application, this can be interpreted as the situation where the data is sensitive to depth but not the position along layers.
From a microlocal perspective, the major difference between the 2dimensional case and the higher dimensional case is that, without the extra dimension to allow for certain orthogonality, ellipticity on the entire cotangent bundle fails. However, the ellipticity still holds if we restrict the directions on the fibre part of the cotangent bundle. So we identify the directions on which our operator behaves well, and modify the symbol on other directions to obtain complete ellipticity.
In the next section, we introduce notations, the definition of the geodesic ray transform and state the main result. In Section 3 we recall some basic facts about Sobolev spaces and scattering calculus. We describe and prove important properties of the conjugated normal operator of the geodesic ray transform in Section 4. Finally in Section 5 we prove the main theorem.

Notations and results
2.1. General notations. (X, g) is a Riemannian manifold. Most of our discussions are valid for general dimension, but our major results concerns only the 2 dimensional case. It is convenient to consider a larger region containing X. So suppose X is embedded as a strictly convex domain in a Riemannian manifold (X, g) (we have used the same notation to indicate the smooth extension of the metric). Here convexity means when a geodesic is tangent to ∂X, it is tangent and curving away from X. In terms of Hamiltonian dynamics, let G be the dual metric function on T * X and let ρ be its boundary defining function. If at some β ∈ T * pX we have (H G ρ)(β) = 0, then (H 2 G ρ)(β) < 0.
We emphasize that introducingX is an important technique in this framework. Take a neighborhood U of p ∈ ∂X inX. In the 2dimensional case, the local coordinate on T * X (actually we only need T * X part) we use is (z, ζ) = (x, y, ξ, η). We consider local geodesic transform near p ∈ ∂X in a neighborhood O p of it (more properties of this neighborhood are specified below).
Denote the boundary defining function ofX, the closure of X, by ρ(z). ρ(z) vanishes on ∂X, ρ(z) > 0 on X, and satisfy the nondegeneracy condition dρ = 0 when ρ = 0. We introduce another boundary defining functionx satisfying dx(p) = −dρ(p),x(p) = 0, whose level sets are strictly convex from the sublevel sets. This allows us to introduce another artificial boundary to enforce our discussion to be local. In terms of a new parameter c, the region O p is Ω c := {z ∈ X :x(z) ≥ −c, ρ(z) ≥ 0} for c ≥ 0. We can choosex such thatΩ c is compact. Our proof for the local result is valid for all small c.
Shrinking the region if necessary, we can assume the neighborhood we are working on is entirely in a local coordinate patch. We take where | · | means the Euclidean norm in this coordinate patch, and this term is introduced to enforce the region characterized byx to be compact. If we define Ω c using the inequality involving only ρ(z), the region might be non-compact (even when c is small, it might be a long thin strip near the boundary). So we use a modification of −ρ making the level sets less convex to enforce its intersection with ∂X happen in a compact region. Furthurmore, the class of 'adapted' function is determined by the foliation given byx, which makesx even more important in two dimensional case compared with higher dimensional cases.
2.2. The foliation condition. We now turn to the foliation condition we need in the two dimensional case. Suppose there is a foliation of X by level sets ofx. That is, a family of hypersurfaces {Σ t =x −1 (−t), 0 ≤ t ≤ T }. In addition, we assume each Σ t is convex in the sense that any geodesic tangent to it curves away from {x ≤ −t}. Next we define the adapted function class. Definition 1. With notations above, Fx(X) is defined to be the function space consists of functions which are constant on each Σ t , and we say such a function is 'adapted to the foliationx'. In addition, 2.3. The geodesic ray transform. By 'a geodesic', we mean a geodesic determined by the metric g. Define Ω c := {p ∈ X :x(p) ≥ −c} for c > 0. We replacex by x =x + c, so that x itself becomes the defining function of the artificial boundary. In an open set O ⊂X, for a geodesic segment γ ⊂ O, we call it O-local geodesic if its endpoints are on ∂X, and all geodesic segments we consider below are assumed to be O p −local. Next we introduce strictly positive density functions, which is needed in the discussion of geodesic ray transform. We use T X to denote the tangent bundle of X, and notice that each point on it (i.e., a point with a tangent vector living at that point) determines a geodesic. Before defining the function class, we define G X := {(s, z)|s ∈ T X, z ∈ X lies on the geodesic determined by s}. G X is a submanifold of T X × X.
The upper bound condition, which is assumed to ensure integability, can be weakened. As we mentioned in the introduction, the geodesic ray transform weighted by ̺ of a function f is defined by: where ̺ is an O p −strictly positive density function, s ∈ T X, γ is the geodesic determined by s. So our geodesic ray transform is a function on T X. Condition 1. above implies that for s 1 , s 2 projecting to the same pont on X and has parallel nonzero fibre part, they determine the same function ̺(s 1 , ·) = ̺(s 2 , ·) of z.
Our injectivity result for the geodesic ray transform implies the injectivity for the unweighted case. To reduce to the unweighted case, we take ̺ ≡ 1. The purpose of adding this notion of density function is to make our theorem more general, and ̺ here should be considered as 'known' and our injectivity claim is for f only. For density functions without condition 1 of Definition 2, i.e., for density functions may also depend on the choice of the starting point of the geodesic, we have more information since our given information in the injectivity problem is the vanishing of these geodesic ray transforms. In that case, we have many integrals for a single geodesic. This means our formulation is the case where we need the 'least information'.
2.4. The main result. We use exponentially weighted Sobolev spaces: where the additional subscript constant F indicates the exponential conjugation. For exponentially weighted Sobolev spaces on other manifolds, we use the same notation with O p replaced by that manifold. Furthermore, SX| Op is the restriction of the sphere bundle to O p . With all these preparations, the main theorem is: Theorem. For p ∈ ∂X, with density ̺ as above, we can choose O p = {x > −c} ∩X, so that the local geodesic transform is injective on In the corollary below, X, Σ t are defined as above, and in addition we assume thatX is compact.
Corollary. If K T := X\∪ t∈[0,T ) Σ t has measure zero, the global geodesic X-ray transform is injective on F 0 x (X) := Fx(X) ∩ L 2 (X). If K T has empty interior, the global geodesic transform is injective on F s x (X) := Fx(X) ∩ H s (X) for s > n 2 . Remark. We added 'global' because the function are not restricted to O p anymore. The geodesic X-ray transform appears in this paper is weighted.
Proof. Assuming the theorem holds, we prove the corollary. For nonzero f ∈ L 2 (X) and K T has measure zero case, suppf has non-zero measure by the definition of L 2 (X). Consider τ := inf suppf (−x). If τ ≥ T , then suppf ⊂ K T , which has measure zero, contradiction. So τ < T and by definition f ≡ 0 on Σ t with t < τ . By the definition of τ , closedness of suppf and compactness ofX, we know there exists q ∈ Σ τ ∩ suppf . However, consider the manifold given by {x < −τ }, to which we can apply our theorem. Since we have local injectivity near q, we conclude that q has a neighborhood disjoint with suppf , contradiction.
If f ∈ H s (X), s > n 2 , f = 0, then f is continuous by the Sobolev embedding theorem and consequently suppf has non-empty interior since .Then apply local result to a fixed point in suppf gives the contradiction.

Sobolev spaces and the scattering calculus
In this section, we recall some basic facts of pesudodifferential operators, their symbols, and the process of quantization, and also some basic facts about Sobolev spaces.
3.1. Sobolev spaces. We state some inclusion relationship between weighted Sobolev spaces. SupposeM is a compact manifold with boundary whose interior is M. Let V b (M ) be the collection of all smooth vector fields tangent to ∂M. Suppose x is a global boundary defining function, we set V sc (M ) = xV b (M ).
Then the L 2 -integrability with respect to the scattering density x −(n+1) dxdy gives L 2 sc (M ). Here the density comes from the identification through x = r 1 −1 , and the ordinary volume form in the polar coordinate is r 1 n−1 dr 1 dy, where y denotes the spherecal variables and r 1 denotes the radial variable. The corresponding polynomially weighted Sobolev space H s,r sc (M ) consists of functions u such that With these definitions, we know See Section 2.3 of [6] for more details.

3.2.
Scattering calculus on the Euclidean space. a(z, ζ) ∈ C ∞ (R n z × R n ζ ) is said to be a scattering symbol of order (m, l) if and only if: , with |z| being the Euclidean norm, and similarly for ζ . The space consists of such symbols is denoted by S m,l (R n , R n ), or S m,l for short. Then the space of pesudodifferential operators Ψ m,l sc (R n ) is defined as 'left quantizations' of such symbols. Explicitly, they are operators of the form: And right quantization could be defined in the same way by replacing a(z, ζ) by a(z ′ , ζ), and the space of operators we obtain is unchanged. Both the space of symbols and that of pseudodifferential operators increase with respect to m, l. This family of spaces Ψ * , * sc (R n ) forms a filtered * −algebra under the composition and taking adjoints relative to the Euclidean metric, i.e.
. The next important notion is the principal symbol. For A ∈ Ψ m,l sc , its principal symbol is the equivalence class of a in S m,l /S m−1,l−1 where a is the symbol whose left quantization is A. This equivalence class captures the behaviour and properties of A modulo lower order operators. We say A ∈ Ψ m,l sc (R n ) is elliptic if its principal symbol is invertible in the sense that there exists b ∈ S −m,−l such that ab−1 ∈ S −1,−1 . Whether b exists or not does not depend on the choice of representative of a in that class. When A is elliptic, the standard parametrix construction gives , which means the error term operator has a Schwartz function on R 2n as its Schwartz kernel. Hence R := AB − Id mapping H s,r to H s−m,r−l is compact for any s, r, m, l, with H s,r defined in the Sobolev space section. Compactness and parametrix construction together gives Fredholm property of elliptic operator A. That is, it has closed range, finite dimensional kernel and cokernel, with following estimate for any N: whereF can be taken as a finite rank operator in Ψ −∞,−∞ sc (R n ). In order to facilitate the generalization to general manifolds with boundary, we compactify the R n in both base and phase factors. Concretely, we compactify R n to a closed ballR n by adding the 'sphere at infinity' S n−1 . Using r 1 to denote the radial variable, we first identify R n \{0} with (0, +∞) r 1 × S n−1 θ through polar coordinates (r 1 , θ) → r 1 θ.
. And now glueing a sphere to x = 0, or extending the range of x to [0, ∞) is equivalent to attaching a sphere at infinity in the original coordinates. So formally R n is obtained by taking disjoint union of R n and [0, +∞) x ×S n−1 modulo the identification given above. Now x = r 1 −1 is a boundary defining function near ∂R n . By modifying it in the 'large x small r 1 ' part, this gives us a global boundary defining function ρ. Decay properties can be rephrased as regularity on this compatified space: Schwartz functions on R n are exactly restrictions to R n of C ∞ functions onR n .
As we have mentioned, we can campactify both factors of R n z × R n ζ to define the scattering symbols ofR n z ×R n ζ . We denote the defining function ofR n z ('the position factor') by ρ ∂ , and that ofR n ζ ('the momentum factor') by ρ ∞ . We also define Ψ m,l sc (R n ) := Ψ m,l sc (R n ). And the ellipticity of A ∈ Ψ m,l sc (R n ) is equivalent to the non-vanishing property of ρ l ∂ ρ m ∞ a, where a is a left symbol (whose left quantization is that operator) of A. In particular, by the compactness of the boundary sphere, there exists C > 0 such that |a| ≥ Cρ −l ∂ ρ −m ∞ , which is convenient to use in practice.
3.3. Generalization to manifolds. LetM be a manifold with boudary and denote its interior by M. Then the scattering pseudodifferential operators Ψ m,l sc (M) is obtained by locally identifying the manifold with R n . And on such charts U × U, the Schwartz kernel of the operator has the same property as the case ofR n , and we also allow additional globally Schwartz terms in the Schwartz kernel. And all those algebraic properties of Ψ m,l sc (R n ) generalize to the manifold case. In addition, the weighted Sobolev spaces H s,r sc (M ) are also obtained by locally identi-fyingM withR n . A clarification on how we define L 2 sc (M) might be useful. After locally identifyM withR n , we use the scattering density r 1 n−1 dr 1 dy = x −n−1 dxdy to 3.4. Preparation for decomposing operators. Our approach is to construct an elliptic pseudodifferential operator A = A 0 + A 1 to obtain an elliptic type estimate, then shrink the region to obtain invertibility. The two summands are specified below. The first lemma points out that operators whose left symbols are supported away from (or outside a conic neighborhood of) the '(1,0) direction' annihilate functions adapted to the foliation. And the second lemma tells us that when an operator is elliptic near '(1,0) direction', we can add another operator satisfying conditions in Lemma 1 to it and make the sum elliptic in every direction. Recall that in our setting, (X, g) is a Riemannian manifold with boundary, and O p is a neighborhood of p ∈ ∂X.
Proof. As we mentioned before, we are working inside a coordinate patch, hence although the condition on a 1 is formulated using a specific coordinate system, it is well defined because we can multiply a localizer to make a 1 supported in the coordinate patch, say T * Op X. We use f (x ′ ) to denote f (x ′ , y ′ ) since it only depends on the first variable by the definition of Fx(X).
Proof. In the proof below, we only consider the region R 1 := {(x, y, ζ)||ζ| > 1} because we can extend the resulting symbol into R c 1 , keeping smoothness.
We consider a smooth χ satisfying: (1) suppχ ⊂ (−∞, C]; (2) C 2 ≤ |χ(t) + t| ≤ 2C for t ≤ C; (3) Its derivatives of any order is bounded. This can be achieved by taking χ(t) = −t + Cχ 1 (t) where χ 1 is a nondecreasing smooth function with uniformly bounded derivatives of any order, being 1 2 on (−∞, C 2 ], and is t C on [C, ∞). Consequently χ(t) = 0 on [C, ∞). Then the first two properties follows by definition, and notice that for ζ = (ξ, 0), b 0 ≥ C. The last condition follows by the observation that χ 1 has trivial derivatives on (−∞, C 2 ], and it coincide with t C for t ≥ C. So the boundedness essentially concerns [ C 2 , C], which is compact, and the result follows by the smoothness of χ 1 . Then we set b 1 (z, ζ) = χ(b 0 ). The desired properties of b 1 is included in the construction of χ. The fact that b 1 is again a symbol of at most the order that of b 0 follows from the boundedness of derivatives of χ and the symbolic property of |ζ|a 0 . And since we are in the region where |ζ| ≥ 1, so a i = |ζ| −1 b i , i = 0, 1 are well defined.

Ellpiticity of the normal operator
In this section we prove the ellipticity of the exponentially conjugated microlocalized normal operator. The exponential conjugation is needed because although the Schwartz kernel of A in the previous section behaves well when X = x ′ −x x 2 , Y = y ′ −y x are bounded, it is not so when (X, Y ) → ∞. This conjugation gives additional exponential decay to resolve this issue. 4.1. The fibre infinity behaviour. Using the notation (z, ζ) = (x, y, ξ, η) ∈ T * X, ζ = 0, we define where γ x,y,λ,ω (t) is the geodesic starting at (x, y) with initial tangent vector (λ, ω), ω = ±1. By the compactness ofΩ c discussed after 1 and |(λ, ±1)| ≥ 1, there exits a uniform bound T g of the escape time ofŌ p . Thus |t| ≤ T g below. v is a function defined on the space of O p −local geodesic segments, whose prototype is the geodesic ray transform y, λ, ω), γ x,y,λ,ω (t))dt, in which f (γ(t)) can be replaced by higher order tensors, coupling witḣ γ(t) in all of its slots in more general situations. I ̺ is the original X-ray transform operator and L is its adjoint if we ignore χ and assume good decay conditions on integrands. So their composition is the model of the normal operator. We define (3) y, λ, ω), γ x,y,λ,ω (t))dt|dν|, with |dν| = |dλdω| a smooth density.
Proof. For the derivation of the decay property of A F 's Schwartz kernel and consequently the membership A F ∈ Ψ −1,0 sc , we refer readers to Section 3.5 of [6] or Section 5 of [5]. We focus on ellipticity here.
We have included the β−term in the definition of Γ (2) . Then we make the change of variablest By the support condition of χ, the integrand is none-zero whenλ is in a compact interval. However, the bound ont is |t| ≤ Tg x , which is not uniformly bounded, we amend this by treating it in two regions separately. Using these new variables, we rewrite our phase as φ = ξ(λt+αt 2 +xt 3 Γ (1) (x, y, xλ, ω, xt))+η(ωt+xt 2 Γ (2) (x, y, xλ, ω, xt)).
We first consider the critical points of the phase at x = 0, where the phase becomes ξ(λt + αt 2 ) +tηω.
When |t| ≥ ǫ t , the derivative with respect toλ vanishes only when ξ = 0. ξ = 0 implies η = 0, hence ηω can not vanish. So the region |t| ≥ ǫ t gives rapid decay contribution. The case x > 0 can be dealt with the same method, but with more complicated computation. Notice that, α, Γ (i) take λ = xλ, t = xt as variables, and produces an extra x factor when we take partial derivatives with respect toλ,t. Concretely, the derivative with respect toλ is: Recall that |t| ≤ T g and we can choose T g to be small by shringking O p , so this can not vanish when |ξ| ≥ C|η|,t ≥ ǫ t . Next we consider the region |t| < ǫ t , whose closure is compact, and consequently we can apply the stationary phase lemma. The same as before, we consider the condition that the derivative with respect toλ andt vanish. First consider the x = 0, in which case the expression can be significantly simplified: We exclude ξ = 0 case since that implies η = 0, but ξ, η can't vanish at the same time. This is the major difference between 2 dimensional case case and the higher dimensional case. Then we have the condition for critical points:t = 0, ξλ + ηω = 0.
Consequently when |ξ| ≥ C|η| the variable on the exponent is uniformly bounded, therefore we have which proves ellipticity of the boundary principal symbol for |ξ| ≥ C|η|. Now we amend the compact support issue. Let χ be a Gaussian as above, which generates an elliptic operator, then we pick a sequence χ n ∈ C ∞ c (R) converges toχ in the Schwartz function space S(R). Then we can obtain the convergence ofχ n toχ in Schwartz function space. This gives us the convergence of X−Forier transform in S(R). And the Y −Fourier transform step is also continuous with respect to the topology of S(R). In particular, we obtain the convergence of |ζ|a n,F (z, ζ), the symbol obtained from χ n , in the C 0 topology, which is enough to derive an elliptic type estimate for χ n with large enough n.

The proof of the main theorem
Fix c 0 small and apply results in previous sections to Ω c 0 , estimates above are uniform with respect to c ∈ (0, c 0 ]. We let c vary and take f ∈ Fx(X) such that on the region Ω c we have x ≤ f and f = 0 when x = 0. Construct B c as follows: for Ω c , denote A F in Section 4 constructed for Ω c by A F,c . By Proposition 1, we can apply Lemma 2 with A F,c being A 0 in Lemma 2, there exists A 1,c such that B 1 := A F,c + A 1,c is elliptic. And A 1,c f = 0 for f ∈ Fx(X) by Lemma 1, hence estimate of B c is equivalent estimate for A F,c on Fx(X). By the ellipticity of B c we have its parametrix G c such that G c B c = Id + E 0c , E 0c ∈ Ψ −∞,−∞ sc (X). Consider the map Ψ c (x, y) = (x+c, y), and let This conjugation is introduced to make this family of operators to be defined on a fixed regionM 0 := {x ≥ 0}. We have an estimate of the error term in terms of f . To be more precise, we consider the Schwartz kernel K Ec of E c , which satisfies |x −N x ′−N K Ec | ≤ C N on Ω c . Then we insert a truncation factor φ c compactly supported in Ω c , and being identically 1 on smaller compact set K c , such that The n+1 power terms are introduced for the scattering density. Then we apply Schur's lemma on the integral operator bound (together with the aforementioned N − th power estimate) to conclude that ||φ c E c φ c || L 2 sc (M 0 ) ≤ C ′′ N f (t) 2N . In particular, we can take c 0 so that this norm < 1 when c ∈ (0, c 0 ]. Since those conjugations are invertible,this guarantees that φ c G c B c φ c = Id + φ c E 0c φ c is invertible. So for the functions supported on K c , B c is injective. And K c can be arbitrary compact subset of Ω c for arguments up to now. Support conditions are encoded by subscripts below. . Finally we consider the boundedness of operators involved. We consider the decomposition A = L • I ̺ , and show that L is bounded. In order to prove this, we decompose L into L = M 2 • Π • M 1 , with M 2 , Π, M 1 being Consider the boundedness of M 1 when s ∈ N first. The general case follows from interpolation. Consider derivatives of x s χ( λ x )u(x, y, λ, ω) up to order s. Each order of differentiation on χ gives an x −1 factor, which is cancelled by x s and the remaining part belongs to L 2 by smoothness of χ and u ∈ H s . M 2 is bounded by the definition of the space on the right hand side. The operator Π is a pushforward map, integrating over |λ| ≤ C|x| (notice the support condition after we apply M 1 ), hence bounded.
On the other hand, I ̺ itself is a bounded operator. This comes from the decomposition I ̺ =Π • Φ * , where Φ is the geodesic coordinate representation Φ(z, ν, t) = γ z,ν (t) andΠ is integrating against t, which is bounded as a pushforward map. Because the initial vector always has length 1 on the tangent component, the travel time is uniformly bounded. Φ is one component of Γ and the later is a diffeomorphism when we shrink the region. So Φ has surjective differential, hence the pull back is bounded. Consequently I ̺ is bounded.
The boundedness of L gives us an estimate , where we require f to have supported in K c , and used the fact R λ × {±1} ω parametrizes S 1 apart from two poles, and this completes the proof.