Functional relations, sharp mapping properties and regularization of the X-ray transform on disks of constant curvature

TL;DR

This paper establishes new functional relations for the geodesic X-ray transform on constant curvature disks, leading to sharp mapping properties and insights into regularized inversion methods.

Contribution

It introduces novel functional relations involving elliptic differential operators with boundary degeneracy, advancing understanding of the X-ray transform on curved disks.

Findings

Derived new functional relations for the X-ray transform.

Established sharp mapping properties for the transform and normal operator.

Discussed potential for rigorous regularized inversion methods.

Abstract

On simple geodesic disks of constant curvature, we derive new functional relations for the geodesic X-ray transform, involving a certain class of elliptic differential operators whose ellipticity degenerates normally at the boundary. We then use these relations to derive sharp mapping properties for the X-ray transform and its corresponding normal operator. Finally, we discuss the possibility of theoretically rigorous regularized inversions for the X-ray transform when defined on such manifolds.