Parametrix for wave equations on a rough background II: construction and control at initial time

TL;DR

This paper constructs and controls a parametrix for the wave equation on a rough Einstein vacuum background, advancing the proof of the bounded $L^2$ curvature conjecture and studying Fourier integral operators on rough manifolds.

Contribution

It develops methods to control a parametrix and its error for wave equations on rough metrics, crucial for the bounded $L^2$ curvature conjecture.

Findings

Successful construction of a parametrix on rough backgrounds

Establishment of $L^2$ bounds for Fourier integral operators

Progress towards the bounded $L^2$ curvature conjecture

Abstract

This is the second 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 by S. Klainerman, I. Rodnianski and the author 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.