On 3D water waves system above a flat bottom

As a starting point of studying the long time behavior of the $3D$ water waves system in the flat bottom setting, in this paper, we try to improve the understanding of the Dirichlet-Neumann operator in this setting. As an application, we study the $3D$ gravity waves system and derive a new $L^2-L^{\mathrm{\infty }}$ type energy estimate, which has a good structure in the $L^{\mathrm{\infty }}-$type space. In our second paper, base on the results we obtained in this paper, we prove the global regularity of the $3D$ gravity waves system for suitably small initial data.