Smoothness of solutions of a convolution equation of restricted-type on the sphere

TL;DR

This paper proves that solutions to a specific convolution equation on the sphere are infinitely smooth, extending previous results to higher dimensions and more general exponents, with implications for Fourier restriction inequalities.

Contribution

It establishes the smoothness of solutions to a convolution equation on the sphere for a broad class of parameters, generalizing prior work to higher dimensions and even exponents.

Findings

Solutions are in C^(^{d-1}) for the convolution equation.

All critical points of the associated Fourier restriction inequality are smooth.

Extends previous results to arbitrary dimensions and general even exponents.

Abstract

Let $\mathbb{S}^{d-1}$ denote the unit sphere in Euclidean space $\mathbb{R}^d$, $d\geq 2$, equipped with surface measure $\sigma_{d-1}$. An instance of our main result concerns the regularity of solutions of the convolution equation \[ a\cdot(f\sigma_{d-1})^{\ast {(q-1)}}\big\vert_{\mathbb{S}^{d-1}}=f,\text{ a.e. on }\mathbb{S}^{d-1}, \] where $a\in C^\infty(\mathbb{S}^{d-1})$, $q\geq 2(d+1)/(d-1)$ is an integer, and the only a priori assumption is $f\in L^2(\mathbb{S}^{d-1})$. We prove that any such solution belongs to the class $C^\infty(\mathbb{S}^{d-1})$. In particular, we show that all critical points associated to the sharp form of the corresponding adjoint Fourier restriction inequality on $\mathbb{S}^{d-1}$ are $C^\infty$-smooth. This extends previous work of Christ & Shao to arbitrary dimensions and general even exponents, and plays a key role in a companion paper.