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

TL;DR

This work extends the theory of the weighted $k$-plane (totally-geodesic Radon) transform from Euclidean spaces to simply connected spaces of constant curvature $\mathbb{R}^n$, $\mathbb{H}^n$, and $\mathbb{S}^n$. By introducing radial volume-growth weights tied to the curvature, the authors establish existence criteria and sharp weighted $L^p$-boundedness (including endpoint Lorentz bounds) for radial data, and they derive explicit transform formulae on $\mathbb{H}^n$ and $\mathbb{S}^n$ for general functions. The analysis combines duality, Lorentz-space interpolation, and hypergeometric-function techniques to characterize end-points and obtain precise necessary and sufficient conditions for boundedness, both in the radial-restricted setting and in full generality via formulae. The results significantly deepen the understanding of $k$-plane transforms in non-Euclidean geometries and provide tools for sharp weighted estimates and potential inversion-type results in constant-curvature manifolds. The paper also identifies natural end-points for existence and boundedness, clarifying the limitations and optimality of prior Euclidean results in hyperbolic and spherical contexts.

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.

Figures (7)

• Figure 1: Parameterizing the totally-geodesic submanifolds on spaces of constant curvature
• Figure 2: Collection of points at a fixed distance from the origin in a space of consatnt curvature.
• Figure 3: Region in $\mathbb{R}^2$ for choices of $\gamma_1$ and $\gamma_2$.
• Figure 4: Parameterizing points on a line using angles at origin.
• Figure 5: A line in a plane obtained from a direction of $\mathbb{S}^{n - 1}$.

Theorems & Definitions (154)

• Theorem 1.1: Quinto
• Theorem 1.2: Oberlin and Stein OberlinStein
• Theorem 1.3: Drury Drury
• Theorem 1.4: Kumar and Ray
• Theorem 1.5: Kumar and Ray
• Theorem 2.1
• Definition 2.2: Totally-Geodesic Submanifold
• Remark 2.3
• Theorem 2.4
• Theorem 2.5: LeeRM