End-point estimates of the totally-geodesic Radon transform on spaces of constant curvature: a unified approach

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 improve the known results about the end-point estimates and provide a unified approach to prove them on spaces of constant curvature by making use of geometric ideas common to these spaces. In this process we also obtain a unified formula for the k-plane transform of radial functions. Lastly, we give some inequalities for certain special functions as an application to one of our lemmata.