An inversion formula for the X-ray normal operator over closed hyperbolic surfaces

TL;DR

This paper develops an explicit inversion formula for Guillarmou's normal operator on closed hyperbolic surfaces, enabling reconstruction of invariant distributions and advancing integral geometry in negatively curved spaces.

Contribution

The paper introduces a new explicit inversion formula for Guillarmou's normal operator on closed hyperbolic surfaces of constant negative curvature.

Findings

Constructed an inversion formula for the normal operator.

Extended the inversion to the attenuated normal operator on the Poincaré disk.

Enabled explicit construction of invariant distributions with prescribed pushforward.

Abstract

We construct an explicit inversion formula for Guillarmou's normal operator on closed surfaces of constant negative curvature. This normal operator can be defined as a weak limit for an "attenuated normal operator", and we prove this inversion formula by first constructing an additional inversion formula for this attenuated normal operator on both the Poincar\'e disk and closed surfaces of constant negative curvature. A consequence of the inversion formula is the explicit construction of invariant distributions with prescribed pushforward over closed hyperbolic manifolds.