A note on maximal Fourier Restriction for spheres in all dimensions

Abstract: We prove a maximal Fourier restriction theorem for the sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^{d}$ for any dimension $d\geq 3$ in a restricted range of exponents given by the Stein-Tomas theorem. The proof consists of a simple observation. When $d=3$ the range corresponds exactly to the full Stein-Tomas one, but is otherwise a proper subset when $d>3$. We also present an application regarding the Lebesgue points of functions in $\mathcal{F}(L^p)$ when $p$ is sufficiently close to 1.