A restriction estimate using polynomial partitioning

Abstract: If $S$ is a smooth compact surface in $\mathbb{R}^3$ with strictly positive second fundamental form, and $E_S$ is the corresponding extension operator, then we prove that for all $p > 3.25$, $| E_S f|{L^p(\mathbb{R}^3)} \le C(p,S) | f |{L^\infty(S)}$. The proof uses polynomial partitioning arguments from incidence geometry.