A restriction estimate in ${\mathbb{R}}^3$ using brooms

Abstract

If $f$ is a function supported on the truncated paraboloid in ${\mathbb{R}}^3$ and $E$ is the corresponding extension operator, then we prove that for all $p>3+3/13$, $\Vert Ef{\Vert }_{L^p({\mathbb{R}}^3)}\le C\Vert f{\Vert }_{L^{\infty }}$. The proof combines Wolff's two ends argument with polynomial partitioning techniques. We also observe some geometric structures in wave packets.

53 pages, revised version incorporating referees' suggestions