The X-Ray Transform for a Generic Family of Curves and Weights

Abstract

Abstract. We study the weighted integral transform on a compact manifold with boundary over a smooth family of curves Γ. We prove generic injectivity and a stability estimate under the condition that the conormal bundle of Γ covers T * M .

A Support Theorem for the Geodesic Ray Transform on Functions

“…This estimate also shows that the conormals to γ 0 are not in the analytic wavefront set of f ; see [14,Defn 6.1]. We also mention a recent work of Frigyik-Stefanov-Uhlmann in [5] where similar ideas based on the complex stationary phase method are used.…”

The Geodesic Ray Transform on Riemannian Surfaces with Conjugate Points

“…The analysis can be easily generalized to more general geodesic-like curves as in [10] or to the even more general case of "regular exponential maps" [40] as in [37]. For the simplicity of the exposition, we consider the geodesic case only.…”

“…The latter allows the use of the clean intersection calculus of Duistermaat and Guillemin [9] to show that X * X is a pseudodifferential operator (ΨDO), elliptic when κ = 0. In the geodesic case under consideration, a microlocal study of Xf has been done in [31,32,10,35,37,39], and in some of those works, f can even be a tensor field.…”

“…We do not study the uniqueness question here but we will just mention that if (M, g) is real analytic, in many cases, even ones with conjugate points, one can have injectivity based on analytic microlocal arguments. An example of that is a small tubular neighborhood of γ 0 as above; then the geodesics normal or close to normal to γ 0 carry enough information to prove injectivity based on analyticity arguments as in [10,35,5,21], for example. Stability however, will be always lost for even weights, for example, if there are conjugate points.…”

“…Then X * X is a ΨDO of order −1. It is elliptic at (x, ξ) if and only if κ(x, θ) = 0 for at least one θ with θ, ξ = 0, see, e.g., [10,35]. Then one can recover singularities conormal to all geodesics over which we integrate by elliptic regularity.…”