Parametrix for wave equations on a rough background I: regularity of the phase at initial time
Jeremie Szeftel
TL;DR
This work develops a geometric parametrix framework for the wave equation on rough Einstein vacuum backgrounds by constructing a phase function $u(0,x,\omega)$ with controlled asymptotics and regularity under minimal $L^2$ curvature hypotheses. A parabolic-type constraint $\mathrm{tr}\theta - k_{NN}=1-a$ on the initial slice drives a robust regularization of the lapse $a$, enabling precise control of normal and tangential derivatives and yielding a foliation geometry with quantified bounds. The authors build a detailed analytic toolkit, including geometric Littlewood–Paley theory on leaves $P_u$, commutator/product/parabolic estimates, and a Nash–Moser scheme to realize the foliation with high regularity in $x$ and $\omega$. The resulting estimates supply the essential $L^2$ control of the parametrix and its error, contributing to the bounded $L^2$ curvature conjecture and offering insights into Fourier integral operators with rough phases on manifolds.
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 $\square_{\bf g} φ=0$, 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 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.