pointwise estimate to our knowledge in the general case and we shall prove that, by means of concentration [@bourgjams], that a large number of norms are bounded. Several difficulties appear: we mention them now.

First we need to identify the norms we would like to control: we realize, after some tries, that the right candidates are $\| Iu
\|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (\mathbb{R})}$, with\
$\left( 2(p-1)-,2(p-1)+ \right)$ defined in Theorem \[Thm:Scat\].

Then we have to control the errors generated by this multiplier $I$, not only in order to establish upper bounds (see for example (\[Eqn:proc2\]) ) but also lower bounds (see (\[Eqn:IncreaseRate2\]) ). These bounds should not depend on time $T$, if not it would kill the scattering: this can be done by letting the estimates only depend on the parameter $N$, taking advantage of the local smoothing properties of the solutions. If we neglect, in a first approximation, the variation of the mollified energy in order to estimate the norm $\| I u
\|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}}$, we expect to find a tower-type exponential bound of this norm, by analogy with the bound of the $L_{t}^{q} L_{x}^{r}$ norm of the solutions of (\[Eqn:NlkgWdat\]) with data $(u_{0},u_{1}) \in H^{1} \times L^{2}$, say formally (by 