Multi-scale bilinear restriction estimates for general phases

TL;DR

This paper establishes new bilinear restriction estimates for general phases across multiple scales, with explicit phase dependence, impacting PDE analysis and restriction problems for surfaces with degenerate curvature.

Contribution

It introduces a comprehensive framework for bilinear restriction estimates at various scales for general phases, including explicit phase dependence and applications to PDEs.

Findings

New bilinear restriction estimates for elliptic phases and wave/Klein-Gordon interactions.

Refined Strichartz inequality for Klein-Gordon equation.

Extension of estimates to adapted function spaces using a transference principle.

Abstract

We prove (adjoint) bilinear restriction estimates for general phases at different scales in the full non-endpoint mixed norm range, and give bounds with a sharp and explicit dependence on the phases. These estimates have applications to high-low frequency interactions for solutions to partial differential equations, as well as to the linear restriction problem for surfaces with degenerate curvature. As a consequence, we obtain new bilinear restriction estimates for elliptic phases and wave/Klein-Gordon interactions in the full bilinear range, and give a refined Strichartz inequality for the Klein-Gordon equation. In addition, we extend these bilinear estimates to hold in adapted function spaces by using a transference type principle which holds for vector valued waves.