slowly in time; this growth appears to be necessary in $2$ dimensions, due to the non-integrable factor $(1+t)^{-1}$ in the dispersive bound below, but is not necessary in $3$ dimensions (see [@Guo]).

We also control an intermediate energy norm $H^{N_0}$, for some $N_0$ chosen smaller than $N$, uniformly in time. This intermediate norm is mostly for convenience and can be removed.

Finally, we control the key $Z$ norm described in , which captures the dispersive nature of our flow. Variants of this norm are of course possible, see for example the similar norms used in [@GerMas; @GerMasSha; @GerMasSha2]. We make the specific choice described in in order to achieve two basic inequalities, $$\label{keychoice}
\begin{split}
&\|R_1f\|_{Z}+\|R_2f\|_{Z}\lesssim \|f\|_{Z},\\
&\sup_{k\in\mathbb{Z}}(2^{k/10}+2^{10k})\|P_k f\|_{L^1}\lesssim \|f\|_{Z}. 
\end{split}$$ The first inequality in is a consequence of the proof of Lemma \[tech1\] and is necessary because our nonlinearity contains several Riesz transforms, see . The second inequality in is an easy consequence of the definition and explains, in particular, the choice of the $l^1$ sum in $j$. It can be combined with the basic dispersive estimate $$\label{keydisperse}
 \|P_le^{-it\Lambda}f\|_{L^\infty}\lesssim (1+t)^{-1}(1+2^{2l})\|f\|_{L^1},\qquad t\in\mathbb{R},$$ to show that $$\label{keydisperse2}
\|P_ke^{-it\Lambda}f\|_{L^\infty}\lesssim (1+t)^{-1}(2^{k/10}+2^{8k})^{-1}\|f\|_{Z},\qquad t\in\mathbb{R},k\in\mathbb{Z}.$$

We can now state our main theorem in terms of the function 