The Fourier Extension Conjecture in three dimensions

Abstract: The Fourier extension conjecture in $n\geq 2$ dimensions is, \begin{equation*} \left\Vert \widehat{fd\sigma {n-1}}\right\Vert _{L^{p}\left( \mathbb{R}% ^{n}\right) }\leq C{p}\left\Vert f\right\Vert _{L^{p}\left( \sigma _{n-1}\right) },\ \text{for }f\in L^{p}\left( \sigma _{n-1}\right) \text{ and }p>\frac{2n}{n-1}, \end{equation*} where $\sigma _{n-1}$ is surface measure on the sphere $\mathbb{S}^{n-1}$. We give a proof of this conjecture in dimension $n=3$ that uses trilinear estimates for Fourier transforms of smooth Alpert wavelets, corresponding local linear Fourier estimates for smooth Alpert wavelets with geometric decay, and the deterministic estimates from the author's paper on probabilistic Fourier extension.