An endline bilinear restriction estimate for paraboloids

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}.