Existence and non-existence of extremizers for certain $k$-plane transform inequalities

TL;DR

This paper establishes sharp inequalities for the $k$-plane transform on spheres and hyperbolic spaces, proving extremizers do not exist on hyperbolic spaces, extending prior Euclidean results.

Contribution

It provides the first sharp forms of $k$-plane transform inequalities on curved spaces and demonstrates the non-existence of extremizers on hyperbolic space.

Findings

Sharp inequalities for $k$-plane transform on $ ext{S}^d$ and $ ext{H}^d$

Non-existence of extremizers on hyperbolic space

Extension of Euclidean results to curved geometries

Abstract

We provide sharp forms of $k$-plane transform inequalities on the $d$-dimensional sphere $\mathbb{S}^d$ and the $d$-dimensional hyperbolic space $\mathbb{H}^d$. In particular, we prove that extremizers do not exist for $\mathbb{H}^d$. This work is a natural extension of some results for the $k$-plane transform on $\mathbb{R}^d$.