Smoothness of extremizers for certain inequalities of the Radon transform

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.