A Kirchoff-Sobolev parametrix for the wave equation and applications

We construct a first order, physical space, parametrix for solutions to covariant, tensorial, wave equations on a general Lorentzian manifold. The construction is entirely geometric; that is both the parametrix and the error terms generated by it have a purely geometric interpretation. In particular, when the background Lorentzian metric satisfies the Einstein vacuum equations, the error terms, generated at some point p of the space-time, depend, roughly, only on the flux of curvature passing through the boundary of the past causal domain of p. The virtues of our specific geometric construction becomes apparent in applications to realistic problems. Though our main application is to General Relativity, which we discuss in [14], another simpler application shown here is to give a gauge invariant proof of the classical regularity result of Eardley - Moncrief [4, 5] for the Yang - Mills equations in ℝ¹⁺³.