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

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 ${\square }_{\mathbf{g}}\varphi =0$, where $\mathbf{g}$ 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 $\mathbf{R}$ of $\mathbf{g}$ 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.

References in corpus (1)

• Parametrix for wave equations on a rough background I: regularity of the phase at initial time

Cited by in corpus (2)

• Parametrix for wave equations on a rough background I: regularity of the phase at initial time
• Overview of the proof of the Bounded $L^2$ Curvature Conjecture