I will first give a brief survey of the weighted Radon (or X-ray) transform in dimension 2, with some selected breakthroughs. In particular, I will focus on the so-called local injectivity problem and sketch how one can obtain stability estimates for inverse local Radon transforms. The ideas are best illustrated in the case of constant weights, i.e. for the standard Radon transform. If time allows I might go into more details on how to treat some interesting special cases of non-constant weights.
This talk is based on the preprint arXiv:1405.3922, which is joint work with Jan Boman, Stockholm University.