End-point estimates of the totally-geodesic Radon transform on simply connected spaces of constant curvature: A Unified Approach

TL;DR

This paper provides a unified geometric approach to establish end-point estimates for the totally-geodesic Radon transform on spaces of constant curvature, simplifying and unifying previous disparate results.

Contribution

It introduces a unified proof method for end-point estimates of the $k$-plane transform on constant curvature spaces, highlighting geometric similarities.

Findings

Unified proof for end-point estimates across different geometries

Derived a general formula for the $k$-plane transform of radial functions

Established inequalities for special functions as applications

Abstract

In this article, we give a unified proof of the end-point estimates of the totally-geodesic $k$-plane transform of radial functions on spaces of constant curvature. The problem of getting end-point estimates is not new and some results are available in literature. However, these results were obtained independently without much focus on the similarities between underlying geometries. We give a unified proof for the end-point estimates on spaces of constant curvature by making use of geometric ideas common to the spaces of constant curvature, and obtaining a unified formula for the $k$-plane transform of radial functions. We also give some inequalities for certain special functions as an application to one of our lemmata.