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We prove that the geodesic X-ray transform is injective on scalar functionsand (solenoidally) on one-forms on simple Riemannian manifolds $(M,g)$ with $g\in C^{1,1}$. In addition to a proof, we produce a redefinition of simplicitythat is compatible with rough geometry. This $C^{1,1}$-regularity is optimal onthe H\"older scale. The bulk of the article is devoted to setting up a calculusof differential and curvature operators on the unit sphere bundle atop thisnon-smooth structure.

Pestov identities and X-ray tomography on manifolds of low regularity