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 $p>3.25$, $\Vert E_{S}f{\Vert }_{L^p({\mathbb{R}}^3)}\le C(p,S)\Vert f{\Vert }_{L^{\infty }(S)}$. The proof uses polynomial partitioning arguments from incidence geometry.

42 pages. Minor revisions. Accepted for publication in JAMS