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This article surveys recent results aiming at obtaining refined mappingestimates for the X-ray transform on a Riemannian manifold with boundary, whichleverage the condition that the boundary be strictly geodesically convex. Thesequestions are motivated by classical inverse problems questions (e.g. rangecharacterization, stability estimates, mapping properties on Hilbert scales),and more recently by uncertainty quantification and operator learningquestions.

Non-standard Sobolev scales and the mapping properties of the X-ray transform on manifolds with strictly convex boundary