Abel transforms with low regularity with applications to x-ray tomography on spherically symmetric manifolds

Abstract

We study ray transforms on spherically symmetric manifolds with a piecewise C 1,1 metric. Assuming the Herglotz condition, the X-ray transform is injective on the space of L 2 functions on such manifolds. We also prove injectivity results for broken ray transforms (with and without periodicity) on such manifolds with a C 1,1 metric. To make these problems tractable in low regularity, we introduce and study a class of generalized Abel transforms and study their properties. This low regularity setting is relevant for geophysical applications.