Parametrix for wave equations on a rough background IV: control of the error term

TL;DR

This paper finalizes the construction and error control of a parametrix for the wave equation on a rough Einstein vacuum background, crucial for proving the bounded $L^2$ curvature conjecture.

Contribution

It introduces methods to control the parametrix and its error term under minimal regularity assumptions, advancing the analysis of wave equations on rough geometries.

Findings

Successfully controls the error term of the parametrix

Establishes $L^2$ bounds for Fourier integral operators on rough manifolds

Contributes to the proof of the bounded $L^2$ curvature conjecture

Abstract

This is the last 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.