Restriction estimates for hyperbolic paraboloids in higher dimensions via bilinear estimates

Abstract

Let $\mathbb{H}$ be a $(d-1)$-dimensional hyperbolic paraboloid in ${\mathbb{R}}^d$ and let $Ef$ be the Fourier extension operator associated to $\mathbb{H}$, with $f$ supported in $B^{d-1}(0,2)$. We prove that $\Vert Ef{\Vert }_{L^p(B(0,R))}\le C_{\epsilon }{R}^{\epsilon }\Vert f{\Vert }_{L^p}$ for all $p\ge 2(d+2)/d$ whenever $d/2\ge m+1$, where $m$ is the minimum between the number of positive and negative principal curvatures of $\mathbb{H}$. Bilinear restriction estimates for $\mathbb{H}$ proved by S. Lee and Vargas play an important role in our argument.