Propagation of singularities for gravity-capillary water waves

We obtain two results of propagation for the gravity-capillary water wave system. The first result shows the propagation of oscillations and the spatial decay at infinity; the second result shows a microlocal smoothing effect under the nontrapping condition of the initial free surface. These results extend the works of Craig, Kappeler and Strauss (1995), Wunsch (1999) and Nakamura (2005) to quasilinear dispersive equations. These propagation results are stated for water waves with asymptotically flat free surfaces, of which we also obtain the existence. To prove these results, we generalize the paradifferential calculus of Bony (1979) to weighted Sobolev spaces and develop a semiclassical paradifferential calculus. We also introduce the quasihomogeneous wavefront sets which characterize, in a general manner, the oscillations and the spatial growth/decay of distributions.