Certain Weighted $L^p$-improving estimates for the totally-geodesic $k$-plane transform on simply connected spaces of constant curvature

Abstract

In this article we study the $L^p$-improving mapping properties of the totally-geodesic $k$-plane transform on simply connected spaces of constant curvature, namely, $\mathbb{R}^n$, $\mathbb{H}^n$ and $\mathbb{S}^n$. We begin our study by answering the question of the existence of the totally-geodesic $k$-plane transform on weighted $L^p$ spaces {with radial weights arising from the volume growth on these spaces}. These weights arise naturally from the geometry of these spaces. We then derive necessary and sufficient conditions for the $k$-plane transform of radial functions to be bounded on weighted Lebesgue spaces, with radial power weights. {Following an idea of Kurusa,} we also {derive} formulae for the totally-geodesic $k$-plane transform of general functions on the hyperbolic space and the sphere. Using this formula, and an elementary technique of Minkowski inequality, we prove weighted $L^p$-$L^p$ boundedness of the $k$-plane transform of general functions as well. Along with this, we also study the end-point behaviour of the transform, where the ``end-point" naturally arises due to either the existence conditions or the necessary conditions for boundedness.