Inversion formulas and range characterizations for the attenuated geodesic ray transform

TL;DR

This paper develops new range characterizations and inversion formulas for the attenuated geodesic X-ray transform on simple surfaces, advancing the mathematical understanding of this integral transform.

Contribution

It introduces two novel range characterizations for the transform and derives new inversion formulas for sums of functions and one-forms, enhancing previous theoretical frameworks.

Findings

Range characterized over sums of functions and one-forms

Implicit conditions separate sub-ranges

New inversion formulas derived for sums

Abstract

We present two range characterizations for the attenuated geodesic X-ray transform defined on pairs of functions and one-forms on simple surfaces. Such characterizations are based on first isolating the range over sums of functions and one-forms, then separating each sub-range in two ways, first by implicit conditions, second by deriving new inversion formulas for sums of functions and one-forms.