Anisotropic bilinear $L^2$ estimates related to the 3d wave equation

TL;DR

This paper reviews and extends $L^2$ bilinear estimates for the 3D wave equation, emphasizing anisotropic Fourier restrictions and their effects on solution concentration behaviors.

Contribution

It introduces new anisotropic bilinear $L^2$ estimates and refinements that detect Fourier space concentration and nonconcentration phenomena.

Findings

Extended $L^2$ bilinear estimates for 3D wave solutions.

Established how anisotropic restrictions affect these estimates.

Developed refinements to identify Fourier space concentration patterns.

Abstract

We first review the $L^2$ bilinear generalizations of the $L^4$ estimate of Strichartz for solutions of the homogeneous 3D wave equation, and give a short proof based solely on an estimate for the volume of intersection of two thickened spheres. We then go on to prove a number of new results, the main theme being how additional, anisotropic Fourier restrictions influence the estimates. Moreover, we prove some refinements which are able to simultaneously detect both concentrations and nonconcentrations in Fourier space.