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 $L^2$ type energy bounds to prove a family of Strichartz estimates. Keywords:oscillatory integrals, water waves, surface tension, Strichartz estimatesCategories:76B07, 76B15, 76B45