Smoothness of extremizers for certain inequalities of the Radon transform

TL;DR

This paper proves that extremizers for certain Radon transform inequalities are infinitely differentiable when specific exponents are integers, extending previous results and using manifold-based methods.

Contribution

It establishes the smoothness of extremizers for Radon transform inequalities in the case where exponents are integers, adapting methods to the manifold setting.

Findings

Extremizers are infinitely differentiable when exponents are integers.

All derivatives of extremizers are in L^p and show decay in weighted L^p spaces.

The method adapts Christ and Xue's approach to the manifold context.

Abstract

The Radon transform is a bounded operator from $L^p$ of Euclidean space to $L^q$ of the manifold of all affine hyperplanes in $\mathbb{R}^n$ for certain exponents depending dimension. Extremizers have been determined for certain values of $q$ and $p$, but most remain open. We show that extremizers are infinitely differentiable whenever the exponents in the associated Euler-Lagrange equation, $q-1$ and $\frac1{p-1}$, are integers. The proof adapts the method of Christ and Xue, to the case where the underlying space is a manifold. The proof is carried out in the setting of the $k$-plane transform, which takes functions on $\mathbb{R}^n$ to functions on the manifold of all affine $k$-planes in $\mathbb{R}^n$ by integrating the function over the $k$-dimensional plane. We show that when $q-1$ and $\frac1{p-1}$ are intergers, all nonnegative critical points of the functional \[ \|T_{n,k}f\|_{L^q(M)}/\|f\|_{L^p(\mathbb{R}^n)}\] are infinitely differentiable, all derivatives are in $L^p$ and exhibit some additional decay measured in a weighted $L^p$-space.