Weighted dispersion and Strichartz estimates for the Liouville equation in 1D

We consider in dimension 1 the Liouville equation associated to a nonnegative metric g∈𝒞²_b( $\mathbb {R}$ ). We prove global in time dispersion and Strichartz estimates with a weight dependent on g. This weighted norm appears because of a phenomenon of concentration of trajectories when g is close to 0.