  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
$L^2-L^\infty$ type energy estimate, which has a good structure in the
$L^\infty-$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.
