  We regard the Cauchy problem for a particular Whitham-Boussinesq system
modelling surface waves of an inviscid incompressible fluid layer. We are
interested in well-posedness at a very low level of regularity. We derive
dispersive and Strichartz estimates, and implement them together with a fixed
point argument to solve the problem locally. Hamiltonian conservation
guarantees global well-posedness for small initial data in the one dimensional
settings.
