Carleman estimates for geodesic X-ray transforms

TL;DR

This paper develops Carleman estimates for the geodesic X-ray transform on negatively curved manifolds, establishing invertibility results and advancing the understanding of geometric inverse problems using frequency localization techniques.

Contribution

It introduces a novel approach using Carleman estimates with logarithmic and linear weights to analyze the geodesic X-ray transform on various negatively curved manifolds.

Findings

Invertibility of the geodesic X-ray transform on negatively curved simple manifolds.

Carleman estimates with logarithmic weights for negatively curved manifolds.

Localization of the Pestov energy identity in frequency domain.

Abstract

In this article we introduce an approach for studying the geodesic X-ray transform and related geometric inverse problems by using Carleman estimates. The main result states that on compact negatively curved manifolds (resp. nonpositively curved simple or Anosov manifolds), the geodesic vector field satisfies a Carleman estimate with logarithmic weights (resp. linear weights) on the frequency side. As a particular consequence, on negatively curved simple manifolds the geodesic X-ray transform with attenuation given by a general connection and Higgs field is invertible modulo natural obstructions. The proof is based on showing that the Pestov energy identity for the geodesic vector field completely localizes in frequency. Our approach works in all dimensions $\geq 2$, on negatively curved manifolds with or without boundary, and for tensor fields of any order.