Range characterizations and Singular Value Decomposition of the geodesic X-ray transform on disks of constant curvature

For a one-parameter family of simple metrics of constant curvature (4 for 2 .-1; 1/) on the unit disk M, we first make explicit the Pestov-Uhlmann range characterization of the geodesic X-ray transform, by constructing a basis of functions making up its range and co-kernel. Such a range characterization also translates into moment conditions à la Helgason-Ludwig or Gel'fand-Graev. We then derive an explicit Singular Value Decomposition for the geodesic X-ray transform. Computations dictate a specific choice of weighted L2-L2 setting which is equivalent to the L2.M; dVol / ! L2.@CSM; d 2/ one for any 2 .-1; 1/.