A sharp stability estimate for the geodesic ray transform

TL;DR

This paper establishes a precise stability estimate for the geodesic X-ray transform of low-order tensor fields on simple Riemannian manifolds, highlighting the transform's microlocal structure and range properties.

Contribution

It provides a sharp $L^2\to H^{1/2}$ stability estimate for the geodesic X-ray transform, introducing a family of equivalent norms tailored to the transform's microlocal range.

Findings

Proves a sharp stability estimate in $L^2\to H^{1/2}$ norm.

Shows the stability holds for a family of non-topologically equivalent norms.

Highlights the microlocal structure of the transform's range.

Abstract

We prove a sharp $L^2\to H^{1/2}$ stability estimate for the geodesic X-ray transform of tensor fields of order $0$, $1$ and $2$ on a simple Riemannian manifold with a suitable chosen $H^{1/2}$ norm. We show that such an estimate holds for a family of such $H^{1/2}$ norms, not topologically equivalent, but equivalent on the range of the transform. The reason for this is that the geodesic X-ray transform has a microlocally structured range.