Parametrix for wave equations on a rough background IV: control of the error term
Jeremie Szeftel
TL;DR
The paper tackles the problem of bounding the error term in the plane-wave parametrix for the homogeneous wave equation on a rough Einstein vacuum spacetime, using only $L^2$ bounds on curvature. It develops a robust, invariant approach combining dyadic frequency and angular decompositions, geometric Littlewood–Paley projections, and careful energy estimates to control the Fourier integral operator with phase $u(t,x,\omega)$ and symbol $b^{-1}(t,x,\omega)\mathrm{tr}\chi(t,x,\omega)$. The main achievement is the $L^2$ bound ${\|Ef\|}_{L^2(\mathcal M)} \lesssim {\varepsilon}{\|\lambda f\|}_{L^2(\mathbb{R}^3)}$, ensuring the error term is controlled under minimal regularity assumptions and contributing to the bounded $L^2$ curvature conjecture. These results are of independent interest for Fourier integral operator theory on manifolds with rough phases and have implications for the nonlinear stability analysis in general relativity.
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 $\square_{\bf g} φ=0$, 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.