The Almost Optimal Multilinear Restriction Estimate for Hypersurfaces with Curvature: The Case of n-1 Hypersurfaces in ℝn

In this paper, we establish the optimal multilinear restriction estimate for $n-1$ hypersurfaces with some curvature, where $n$ is the dimension of the underlying space. The result is sharp up to the end point and the role of curvature is made precise in terms of the shape operator.

A Trilinear Restriction Estimate for the Hyperbolic Paraboloid with Sharp Dependence on Transversality

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.