Restriction estimates for hyperbolic paraboloids in higher dimensions via bilinear estimates

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