using (\[Eqn:BoundInitMolNrj\]) )

$$\begin{array}{l}
\| I u \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}} \lesssim \left( ( N^{1-s} A)^{\alpha} \right)^{ ( N^{1-s} A)^{\alpha}}
\end{array}$$

for some $\alpha >0$. Then combining this bound with the polynomial decay ( $\frac{1}{N^{\frac{5-p}{2}-}} $) of the almost conservation law we expect to find after iteration of the local theory (see Proposition \[prop:LocalBd\]) an error of the form

$$\begin{array}{l}
\frac{ \left( ( N^{1-s} A)^{\alpha} \right)^{ ( N^{1-s} A)^{\alpha}} } {N^{\frac{5-p}{2}-}}
\end{array}$$

and it seems at first sight that there do not exist $s_{0}:=s_{0}(p) < 1$ and $N:=N(A,p,s)$ such that we can make the error smaller than the upper bound $N^{2(1-s)} A^{p+1}$ (see Proposition \[Prop:EstInitMolNrj\]) of the mollified energy at initial time for $s \geq s_{0}$, since the denominator grows polynomially whereas the numerator grows exponentially. Notice, however, that

$$\begin{array}{l}
\lim_{s \rightarrow 1^{-}} \frac{ \left( ( N^{1-s} A)^{\alpha} \right)^{ ( N^{1-s} A)^{\alpha}} } { N^{2(1-s)} A^{p+1} N^{\frac{5-p}{2}-}}  =
\frac{ (A^{\alpha})^{A^{\alpha}}}{A ^{p+1} N^{\frac{5-p}{2}-}}
\end{array}$$

and therefore, choosing $N:= C  \left( (A^{\alpha})^{A^{\alpha}} \right)^{\frac{2}{5-p}+}$ (with $C:=C(p)>>1$), we see that there exists $s_{1}:=s_{1}(A,p) <1$ such that the error is small compare to $N^{2(1-s)} A^{p+1}$ for $ s_{1} \leq s < 1$: this is why the range of $s$ for which there is scattering is expected not 