Resumen de Restriction estimates for hyperbolic paraboloids in higher dimensions via bilinear estimates

Alex Barron

• Let \mathbb{H}H be a (d-1)(d−1)-dimensional hyperbolic paraboloid in \mathbb{R}^dRd and let EfEf be the Fourier extension operator associated to \mathbb{H}H, with ff supported in B^{d-1}(0,2)Bd−1(0,2). We prove that \lVert Ef \rVert_{L^p (B(0,R))} \leq C_{\varepsilon}R^{\varepsilon}\lVert f \rVert_{L^p}∥Ef∥Lp(B(0,R))≤CεRε∥f∥Lp for all p \geq 2(d+2)/dp≥2(d+2)/d whenever d/2\geq m + 1d/2≥m+1, where mm is the minimum between the number of positive and negative principal curvatures of \mathbb{H}H. Bilinear restriction estimates for \mathbb{H}H proved by S. Lee and Vargas play an important role in our argument