Parametrix for wave equations on a rough background I: regularity of the phase at initial time

TL;DR

This paper develops a parametrix construction for wave equations on rough spacetime backgrounds, focusing on the regularity of the phase at initial time, which is crucial for understanding Einstein's vacuum equations.

Contribution

It introduces a method to control the parametrix and its error with only $L^2$ curvature bounds, advancing the analysis of wave equations on rough metrics.

Findings

Constructed a parametrix for the wave equation on rough backgrounds.

Established $L^2$ bounds for Fourier integral operators with rough phases.

Progressed towards the bounded $L^2$ curvature conjecture.

Abstract

This is the first of a sequence of four papers \cite{param1}, \cite{param2}, \cite{param3}, \cite{param4} dedicated to the construction and the control of a parametrix to the homogeneous wave equation $\square_{\bf g} \phi=0$, where ${\bf g}$ is a rough metric satisfying the Einstein vacuum equations. Controlling such a parametrix as well as its error term when one only assumes $L^2$ bounds on the curvature tensor ${\bf R}$ of ${\bf g}$ is a major step of the proof of the bounded $L^2$ curvature conjecture proposed in \cite{Kl:2000}, and solved jointly with S. Klainerman and I. Rodnianski in \cite{boundedl2}. On a more general level, this sequence of papers deals with the control of the eikonal equation on a rough background, and with the derivation of $L^2$ bounds for Fourier integral operators on manifolds with rough phases and symbols, and as such is also of independent interest.