  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.
