The Fourier restriction conjecture in three dimensions for positively curved surfaces: bounded-data and diagonal forms

Principal manuscript, Theorem 1.1: for every compact smooth positively curved $\Sigma\subset\mathbb R^3$, possibly with smooth boundary, and every $3<p<\infty$, $\|E_\Sigma f\|_{L^p(\mathbb R^3)}\le C_{\Sigma,p}\|f\|_{L^p(\Sigma)}$; the proof in fact bounds $L^3$ data into $L^p$. A corollary is free Schrodinger local smoothing in two space dimensions for $p>3$, $s>2-6/p$. Companion: for the sphere, $L^\infty(S^2)\to L^p(\mathbb R^3)$ for every $p>3$, and through Bourgain factorization the strict mixed-norm range $L^a\to L^b$ with $b>3$, $b>2a'$. Not shown: the scaling-line endpoint $p=2q'$ that the full conjecture includes, surfaces with vanishing or mixed-sign curvature, and higher dimensions.

Principal manuscript, Theorem 1.1: for every compact smooth positively curved $\Sigma\subset\mathbb R^3$, possibly with smooth boundary, and every $3<p<\infty$, $\|E_\Sigma f\|_{L^p(\mathbb R^3)}\le C_{\Sigma,p}\|f\|_{L^p(\Sigma)}$; the proof in fact bounds $L^3$ data into $L^p$. A corollary is free Schrodinger local smoothing in two space dimensions for $p>3$, $s>2-6/p$. Companion: for the sphere, $L^\infty(S^2)\to L^p(\mathbb R^3)$ for every $p>3$, and through Bourgain factorization the strict mixed-norm range $L^a\to L^b$ with $b>3$, $b>2a'$. Not shown: the scaling-line endpoint $p=2q'$ that the full conjecture includes, surfaces with vanishing or mixed-sign curvature, and higher dimensions.