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

International audienceThis is the third of a sequence of four papers [21], [22], [23], [24] dedicated to the construction and the control of a parametrix to the homogeneous wave equation g φ = 0, where 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 R of g is a major step of the proof of the bounded L 2 curvature conjecture proposed in [12], and solved by S. Klainerman, I. Rodnianski and the author in [17]. 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