Restriction estimates for hyperbolic paraboloids in higher dimensions via bilinear estimates (Q2104839)

Let \[ \mathbb{H}=\{\xi=(\xi_1,\ldots,\xi_d)\in\mathbb{R}^d:\ \xi_d=\xi_1^2+\xi_2^2+\cdots+\xi^2_{d-m-1}-\xi^2_{d-m}-\cdots-\xi^2_{d-1}\} \] be a \((d-1)\)-dimensional hyperbolic paraboloid in \(\mathbb{R}^d\), and let \(Ef\) be the Fourier extension operator associated with \(\mathbb{H}\) for any function \(f\) supporting in \(B^{d-1}(0,2)\), that is, \[ Ef(x,t)=\int_{\mathbb{R}^{d-1}}f(\xi)e^{2\pi i (x\cdot \xi+t(\xi_1^2++\xi_2^2+\cdots+\xi^2_{d-m-1}-\xi^2_{d-m}-\cdots-\xi^2_{d-1}))}\,d\xi. \] Here \(m\) is the minimum between the number of positive and negative principal curvatures in \(\mathbb{H}\). Applying Lee and Vargas's bilinear restriction estimates [\textit{S. Lee}, Trans. Am. Math. Soc. 358, No. 8, 3511--3533 (2006; Zbl 1092.42003); \textit{A. Vargas}, Math. Z. 249, No. 1, 97--111 (2005; Zbl 1071.42009)] for \(\mathbb{H}\), together with an elementary orthogonality estimate in place of \(k\)-linear restriction estimates by \textit{J. Bennett} et al. [Acta Math. 196, No. 2, 261--302 (2006; Zbl 1203.42019)], and the \(\ell^2\) decoupling theorem of \textit{J. Bourgain} and \textit{C. Demeter} [J. Anal. Math. 133, 279--311 (2017; Zbl 1384.42016)], it is proved in this paper that, when \(d/2\ge m+1\), then, for any \(p\ge 2(d+2)/d\), \(R\ge 1\) and \(\epsilon >0\), one has the restriction estimate \[ \|Ef\|_{L^p(B(0,R))}\le C_\epsilon R^\epsilon \|f\|_{L^p}. \]