We address the local existence of solutions of the 2D and 3D water wave problems. For the space dimension three, we consider the irrotational datum u (0) and prove that the local in time existence holds for initial velocities belonging to H (2.5+delta) , where delta > 0 is arbitrary. For the space dimension two, the data does not need to be irrotational. We prove the local in time existence when u (0) belongs to H (2+delta) and to H (1.5+delta) , where delta > 0 is arbitrary.