Sharp Strichartz estimates for the wave equation on a rough background

TL;DR

This paper establishes sharp Strichartz estimates for wave equations on rough Lorentzian backgrounds, crucial for low regularity well-posedness and the proof of the bounded $L^2$ curvature conjecture.

Contribution

It provides the final step in proving sharp Strichartz estimates for wave equations on rough metrics, advancing the low regularity analysis of quasilinear wave equations.

Findings

Sharp Strichartz estimates obtained for rough Lorentzian metrics.

Estimates hold under minimal regularity assumptions.

Supports the proof of the bounded $L^2$ curvature conjecture.

Abstract

In this paper, we obtain sharp Strichartz estimates for solutions of the wave equation $\square_\gg\phi=0$ where $\gg$ is a rough Lorentzian metric on a 4 dimensional space-time $\MM$. This is the last step of the proof of the bounded $L^2$ curvature conjecture proposed in [3], and solved by S. Klainerman, I. Rodnianski and the author in [8], which also relies on the sequence of papers [16][17][18][19]. Obtaining such estimates is at the core of the low regularity well-posedness theory for quasilinear wave equations. The difficulty is intimately connected to the regularity of the Eikonal equation $\gg^{\a\b}\pr_\a u\pr_\b u=0$ for a rough metric $\gg$. In order to be consistent with the final goal of proving the bounded $L^2$ curvature conjecture, we prove Strichartz estimates for all admissible Strichartz pairs under minimal regularity assumptions on the solutions of the Eikonal equation.