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

TL;DR

This paper reviews recent advances in understanding the X-ray transform on convex-boundary Riemannian manifolds, focusing on refined mapping estimates relevant to inverse problems and operator learning.

Contribution

It surveys new results on mapping properties of the X-ray transform using non-standard Sobolev scales on manifolds with convex boundary.

Findings

Refined mapping estimates for the X-ray transform

Applications to inverse problems and stability analysis

Connections to uncertainty quantification and operator learning

Abstract

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