The optimal trilinear restriction estimate for a class of hypersurfaces with curvature

TL;DR

This paper improves trilinear restriction estimates for certain hypersurfaces with curvature, extending previous results by establishing $L^p$ bounds for a broader range of exponents, with a focus on double-conic surfaces.

Contribution

It demonstrates that curvature enhances the range of exponents for $L^p$ estimates in trilinear restriction problems, identifying a universal threshold for these estimates.

Findings

Established $L^p$ estimates for $p > \frac{2(n+4)}{3(n+2)}$ for double-conic surfaces.

Curvature improves the range of exponents compared to transversality-only results.

Identified the exponent $\frac{2(n+4)}{3(n+2)}$ as the universal threshold for the estimates.

Abstract

Bennett, Carbery and Tao established nearly optimal $L^1$ trilinear restriction estimates in $\mathbb{R}^{n+1}$ under transversality assumptions only. In this paper we show that the curvature improves the range of exponents, by establishing $L^p$ estimates, for any $p > \frac{2(n+4)}{3(n+2)}$ in the case of double-conic surfaces. The exponent $\frac{2(n+4)}{3(n+2)}$ is shown to be the universal threshold for the trilinear estimate.