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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 aparametrix 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 ofthe 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 theeikonal equation on a rough background, and with the derivation of $L^2$ boundsfor Fourier integral operators on manifolds with rough phases and symbols, andas such is also of independent interest.

Parametrix for wave equations on a rough background III: space-time regularity of the phase