Range description for a spherical mean transform on spaces of constant   curvatures
  We describe the range of a restricted spherical mean transform, which sends a function supported inside a closed ball in a hyperbolic space to its mean values on the geodesics spheres centered at the boundary of the ball. The description resembles that of the same transform on the Euclidean spaces obtained by Mark Agranovsky, David Finch, and Peter Kuchment [Inverse Problems and Imaging, 3(3):373--382, 2009] and Mark Agranovsky and Linh V. Nguyen [J. Anal. Math., 112:351--367, 2010].   We also derive a similar characterization for the corresponding transform on the two dimensional spherical space. 
Linh V. Nguyen
math.DG math.AP math.MG