Optimal multilinear restriction estimates for a class of surfaces with curvature

TL;DR

This paper demonstrates that curvature assumptions on hypersurfaces enhance the range of exponents in multilinear restriction estimates across various levels of multilinearity, extending previous near-optimal results.

Contribution

It establishes almost sharp multilinear restriction estimates for hypersurfaces with curvature for all levels of multilinearity between 4 and n, improving upon prior transversality-based bounds.

Findings

Curvature assumptions improve restriction estimate ranges.

Established almost sharp estimates for hypersurfaces with curvature.

Extends multilinear restriction estimates to all levels k between 4 and n.

Abstract

Bennett, Carbery and Tao considered the $k$-linear restriction estimate in $\mathbb{R}^{n+1}$ and established the near optimal $L^\frac2{k-1}$ estimate under transversality assumptions only. We have shown that the trilinear restriction estimate improves its range of exponents under some curvature assumptions. In this paper we establish almost sharp multilinear estimates for a class of hypersurfaces with curvature for $4 \leq k \leq n$. Together with previous results in the literature, this shows that curvature improves the range of exponents in the multilinear restriction estimate at all levels of lower multilinearity, that is when $k \leq n$.