We prove that the geodesic X-ray transform is injective on \$L^2(M)\$ when the Riemannian metric is simple but the metric tensor is only finitely differentiable. The number of derivatives needed depends explicitly on dimension, and in dimension 2 we assume \$g\in C^{10}\$. Our proof is based on microlocal analysis of the normal operator: we establish 10 ellipticity and a smoothing property in a suitable sense and then use a recent injectivity result on Lipschitz functions [1]. The proof of this injectivity result in the smooth setting [2] uses the parametrix construction for elliptic pseudodifferential operators which is a standard result in PDE theory [3]; when the metric tensor is \$C^k\$, the Schwartz kernel is not smooth but \$C^{k-2}\$ off the diagonal, which makes standard smooth microlocal analysis inapplicable as the corresponding symbol only satisfies the pseudodifferential operators symbol estimates up to a finite degree.
As an application we use the injectivity of the geodesic X-ray transform to prove that even for metrics at low regularity - the scattering relation on a compact two-dimensional simple manifolds determines the Dirichlet-to-Neumann map - a major component of the proof of boundary rigidity for simple metrics proved in [4] for smooth geometry.
References:
[1] Joonas Ilmavirta and Antti Kykkänen, Pestov identities and X-ray tomography on manifolds of low regularity, Inverse Problems and Imaging, December 2021.
[2] Paternain, Gabriel P. and Salo, Mikko and Uhlmann, Gunther, Geometric inverse problems—with emphasis on two dimensions, volume 4 of Cambridge Studies in Advanced Mathematics, Cambridge University Press (2023).
[3] Taylor Michael E, Partial differential equations. III, volume 117 of Applied Mathematical Sciences, Springer-Verlag, New York (1997).
[4] Leonid Pestov and Gunther Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid. Ann. of Math. (2) , 161(2):1093–1110 (2005).