A restriction estimate in R3 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 $p>3+3∕13$, $\Vert Ef{\Vert }_{L^p({\mathbb{R}}^3)}\le C\Vert f{\Vert }_{L^{\mathrm{\infty }}}$. The proof combines Wolff’s two ends argument with polynomial partitioning techniques. We also observe some geometric structures in the wave packets.