  Let $\mathbb{H}$ be a $(d-1)$-dimensonal 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 $\|Ef\|_{L^p
(B(0,R))} \leq C_{\epsilon}R^{\epsilon}\|f\|_{L^p}$ for all $p \geq
\frac{2(d+2)}{d}$ whenever $ \frac{d}{2} \geq 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.
