Uniqueness of extremizers for an endpoint inequality of the $k$-plane transform

TL;DR

This paper characterizes all extremizers for a specific endpoint inequality related to the $k$-plane transform, a key operator in integral geometry, providing a complete understanding of optimal functions at this critical case.

Contribution

It uniquely identifies all extremizers for the endpoint inequality of the $k$-plane transform, a significant step in understanding its extremal behavior.

Findings

All extremizers are explicitly characterized.

The extremizers are unique up to symmetries.

The result applies to the general $k$-plane transform at the endpoint case.

Abstract

The $k$-plane transform is a bounded operator from $\lp$ to $L^q$ of the Grassmann manifold of all affine $k$-planes in $\R^n$ for certain exponents depending on $k$ and $n$. In the endpoint case $q=n+1$, we identify all extremizers of the associated inequality for the general $k$-plane transform.