The Proof of restriction conjecture In $\mathbb{R}^{3}$

Hoyoung Song

Published 2023-04-03Version 1

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 $p > 3$, $\left\|E_S f\right\|_{L^p\left(\mathbb{R}^3\right)} \leq C(p, S)\|f\|_{L^{\infty}(S)}.$ The proof of restriction conjecture in $\mathbb{R}^{3}$ implies that Kakeya set conjecture is true when n=3.