Linear dispersive decay estimates for the 3+1 dimensional water wave equation with surface tension

We consider the linearization of the three-dimensional water waves equation with surface tension about a flat interface. Using oscillatory integral methods, we prove that solutions of this equation demonstrate dispersive decay at the somewhat surprising rate of t^-5/6. This rate is due to competition between surface tension and gravitation at O(1) wave numbers and is connected to the fact that, in the presence of surface tension, there is a so-called "slowest wave". Additionally, we combine our dispersive estimates with L² type energy bounds to prove a family of Strichartz estimates.