Wave packet decompositions and sharp bilinear estimates for rough Hamiltonian flows

The goal of this paper is to prove bilinear Lᵖ estimates for rough dispersive evolutions satisfying non-degeneracy and transversality assumptions. The estimates generalize the sharp Fourier extension estimates for the cone and the paraboloid. To this end, we require a wave packet decomposition with localization properties in space-time and space-time frequencies. Secondly, we construct a refined wave packet parametrix for dispersive equations with C1,1-coefficients by using the FBI transform. As a consequence, we obtain bilinear estimates for solutions to dispersive equations with C1,1 coefficients provided that the solutions interact transversely.