Affiliation:
1. Department of Mathematics, National University of Colombia, Bogotá, Colombia
Abstract
Abstract
The problem of restriction in harmonic analysis revolves around the operator $\mathcal{R}:f\mapsto \hat{f}|_{S}$, where $S$ is a manifold in $\mathbb{R}^n$. A fundamental inequality of the multilinear theory of the restriction operator is $\big \lVert{\prod _{k=1}^{n}\mathcal{R}^{*} f_{k}}\big \rVert _{L^{2/(n-1)}(\mathbb{R}^{n})}\le C\prod _{k=1}^{n}\lVert{f_k}\rVert _{L^{2}(S_k)}$, where $\mathcal{R}^{*}$ is the dual operator and the hypersurfaces $S_k$ satisfy a condition of transversality. This estimate was proven by Bennett, Carbery, and Tao, up to logarithmic losses, but without optimal dependence on transversality in the constant $C$. For subsets of the paraboloid in $\mathbb{R}^{3}$, Ramos got the sharp dependence on transversality. In this paper we extend this result to subsets of the hyperbolic paraboloid.
Funder
Ministry of Economy and Competitiveness
Publisher
Oxford University Press (OUP)