Dispersion for the wave equation inside strictly convex domains II: the general case

We consider the wave equation on a strictly convex domain of dimension at least two with smooth non empty boundary and with Dirichlet boundary conditions. We construct a sharp local in time parametrix and then proceed to obtain dispersion estimates: our fixed time decay rate for the Green function exhibits a $t^{1/4}$ loss with respect to the boundary less case. Moreover, we precisely describe where and when these losses occur and relate them to swallowtail type singularities in the wave front set, proving that the resulting decay is optimal. Moreover, we derive better than expected Strichartz estimates, balancing lossy long time estimates at a given incidence with short time ones with no loss: in three dimensions, it heuristically means that, on average the decay loss is only $t^{1/6}$.