Bilinear estimates associated to the Schrödinger equation with a nonelliptic principal part

TL;DR

The paper investigates bilinear estimates in Fourier restriction spaces for the 2D Schrödinger equation with a nonelliptic principal part $a(\xi)=\xi_1^2-\xi_2^2$. It shows a sharp dichotomy: estimates hold for $s\ge 0$ but fail for $s<0$, with the proof combining trilinear form analysis, Strichartz-type dispersion bounds, and Knapp-type constructions near the hyperbola where $a(\xi)=0$. The noncompact zero set of $a(\xi)$ governs the breakdown, highlighting geometric obstructions in the hyperbolic setting and aligning with, yet distinct from, paraboloid-based results of Tao-Vargas. These results clarify when tempered distributions must actually be functions for the bilinear estimates to be valid.

Abstract

We discuss bilinear estimates of tempered distributions in the Fourier restriction spaces for the two-dimensional Sch\"odinger equation whose principal part is the d'Alembertian. We prove that the bilinear estimates hold if and only if the tempered distributions are functions.

Theorems & Definitions (6)

• Theorem 1: CDKSKPV1
• Theorem 2
• proof : Proof of (i) of Theorem \ref{['theorem:main']}
• Lemma 3
• proof : Proof of Lemma \ref{['theorem:ste']}
• proof : Proof of (ii) of Theorem \ref{['theorem:main']}