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

Let ${\mathbb{S}}^{d-1}$ denote the unit sphere in Euclidean space ${\mathbb{R}}^d$, $d\ge 2$, equipped with surface measure ${\sigma }_{d-1}$. An instance of our main result concerns the regularity of solutions of the convolution equation $$a\cdot (f{\sigma }_{d-1}{)}^{\ast (q-1)}{\mid }_{{\mathbb{S}}^{d-1}}=f,\text{ a.e. on }{\mathbb{S}}^{d-1},$$ where $a\in C^{\mathrm{\infty }}({\mathbb{S}}^{d-1})$, $q\ge 2(d+1)\mathrm{/}(d-1)$ 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 $C^{\mathrm{\infty }}({\mathbb{S}}^{d-1})$. 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 $C^{\mathrm{\infty }}$-smooth. This extends previous work of Christ & Shao to arbitrary dimensions and general even exponents, and plays a key role in a companion paper.