An endline bilinear restriction estimate for paraboloids

TL;DR

This paper establishes a new bilinear restriction estimate for elliptic paraboloids at the endline, including the endpoint case, advancing the understanding of Fourier restriction phenomena in harmonic analysis.

Contribution

It proves the first bilinear restriction estimate at the endline for paraboloids, including the endpoint, using novel induction-on-scale and wave-table techniques.

Findings

Proves bilinear restriction estimate at the endline for paraboloids.

Includes the endpoint case $q=r=(n+3)/(n+1)$.

Extends previous results to the full range, confirming a conjecture.

Abstract

We prove an $L^2\times L^2\to L^q_tL^r_x$ bilinear adjoint Fourier restriction estimate for $n$-dimensional elliptic paraboloids, with $n\ge 2$ and $1\le q \le \infty$, $1\le r\le 2$ being on the endline $\frac{1}{q}=\frac{n+1}{2}\bigl(1-\frac{1}{r}\bigr)$ except for the critical index. This includes the endpoint case when $q=r=\frac{n+3}{n+1}$, a question left unsettled in Tao \cite{TaoGFA}. Apart from the critical index, it improves the sharp non-endline result of Lee-Vargas \cite{LeeVargas} to the full range, confirming a conjecture in the spirit of Foschi and Klainerman \cite{FoKl} on the elliptic paraboloid. Our proof is accomplished by uniting the \emph{profound} induction-on-scale tactics based on the wave-table theory and the method of descent both stemming from \cite{TaoMZ}.