Parametrix for wave equations on a rough background III: space-time regularity of the phase
Jeremie Szeftel
TL;DR
The paper advances the control of a null-phase parametrix for the wave equation on rough spacetimes by establishing time-space regularity of the eikonal phase on a background with $L^2$-controlled curvature. It develops a robust geometric-analytic toolkit, including maximal and $u$-foliations, null structure and Bianchi identities, commutator formulas, and a comprehensive geometric Littlewood-Paley framework on $P_{t,u}$, $\mathcal{H}_u$, and $\Sigma_t$. The main contributions are (i) proving absence of conjugate points and intersection of null geodesics up to time $t=1$, (ii) deriving precise bounds for $n$, $k$, and derivatives of the phase $u$ with respect to the angular parameter $\omega$, and (iii) establishing invariant LP/Besov estimates and harmonic-coordinate control to propagate initial regularity from $\Sigma_0$ through spacetime. Collectively, these results underpin a parametrix construction for $\square_{\bf g}$ with error terms controlled in $L^2$, contributing to the bounded $L^2$ curvature conjecture and offering sharp $L^2$ bounds for Fourier integral operators on rough phases.
Abstract
This is the third 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.