Some remarks on Fourier restriction estimates

We provide $L^p\to L^q$ refinements on some Fourier restriction estimates obtained using polynomial partitioning. Let $S\subset {\mathbb{R}}^3$ be a compact $C^{\mathrm{\infty }}$ surface with strictly positive second fundamental form. We derive sharp $L^p(S)\to L^q({\mathbb{R}}^3)$ estimates for the associated Fourier extension operator for $q>3.25$ and $q\ge 2p^{\mathrm{\prime }}$ from an estimate of Guth that was used to obtain $L^{\mathrm{\infty }}(S)\to L^q({\mathbb{R}}^3)$ bounds for $q>3.25$. We present a slightly weaker result when $S$ is the hyperbolic paraboloid in ${\mathbb{R}}^3$ based on the work of Cho and Lee. Finally, we give some refinements for the truncated paraboloid in higher dimensions.