Our goal in this volume is to apply a normal forms method

to estimate the Sobolev norms of the solutions of the water

waves equation. We construct a paradifferential change of

unknown, without derivatives losses, which eliminates the

part of the quadratic terms that bring non zero contributions

in a Sobolev energy inequality. Our approach is purely

Eulerian: we work on the Craig-Sulem-Zakharov formulation

of the water waves equation.

In addition to these Sobolev estimates, we also prove L ²-estimates

for the (...)^Alpha_xZ^ß -derivatives of the solutions of the water

waves equation, where Z is the Klainerman vector field

t(...)_t + 2 x(...)_x. These estimates are used in the paper [ 6 ]. In

that reference, we prove a global existence result for the

water waves equation with smooth, small, and decaying at

infinity Cauchy data, and we obtain an asymptotic description

in physical coordinates of the solution, which shows

that modified scattering holds. The proof of this global in

time existence result relies on the simultaneous bootstrap

of some Hölder and Sobolev a priori estimates for the action

of iterated Klainerman vector fields on the solutions of

the water waves equation. The present volume contains the

proof of the Sobolev part of that bootstrap.