Restriction estimates for hyperbolic paraboloids in higher dimensions via bilinear estimates

Alex Barron ^[1]
1. [1] Department of Mathematics, University of Illinois
Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 38, Nº 5, 2022, págs. 1453-1471
Idioma: inglés
DOI: 10.4171/RMI/1310
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Resumen
- 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