frequencies. We first integrate the quadratic terms using a normal form transformation[^4]. We are then left with several cubic terms which oscillate along different phases. We account for the elliptic phases (which never vanish) using another iteration of the normal form transformation. This produces quartic terms which can be easily estimated. We account for the remaining non-elliptic phase by exploiting its more “hyperbolic” nature: it is somewhat local in space and “commutes[^5]” at first order with the position operator $f\mapsto xf$.

As mentioned before, this is similar to the approach used in [@GerMasSha2] for the water-wave problem, which has a similar structure. However, we introduce some different ideas, such as more involved space–frequency analysis, that we hope can find further applications.

First of all, we only work on the linear profile and derive all our estimates from information about it. This was already done in [@GNT] in $3D$ but encounters difficulties in $2D$ due to the fact that we need an endpoint dispersion estimate ($L^\infty$). This forces us to localize in space using the resolution of unity associated with the position operator in order to create a norm having the nice properties of $L^2$ but which is stronger than the 