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

## TL;DR

This paper establishes injectivity of ray and broken ray transforms on spherically symmetric manifolds with low regularity metrics, introducing generalized Abel transforms to handle the challenges posed by the low regularity, with applications in geophysics.

## Contribution

The paper introduces and analyzes generalized Abel transforms to address low regularity in ray transforms on spherically symmetric manifolds, extending previous injectivity results.

## Key findings

- Injectivity of the X-ray transform on $L^2$ functions under low regularity conditions.
- Injectivity results for broken ray transforms with and without periodicity.
- Development of generalized Abel transforms suitable for low regularity settings.

## 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.