The water-waves system, with both gravity and surface tension taken into account, forms a nonlinear dispersive system. On small time scales, dispersion usually implies a class of inequalities known as Strichartz estimates. We will show how to prove these estimates for the gravity-capillarity water-waves system, using a paradifferential reduction and constructing a parametrix for high-frequency scales. Then we will show how those estimates can be used to prove local existence for low-regularity initial data, where the initial velocity field is not Lipschitz. This is a joint work with Quang Huy Nguyen.