{\lesssim_{A}}1 $ and $\| I u \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (\mathbb{R})} {\lesssim_{A}}1$, by time reversal symmetry. Indeed, notice first that it is enough by time reversal symmetry to establish these bounds restricted to $[t_{l}, \infty)$; then it is enough to apply $\mathcal{P}_{C,Q}(0)$ to $u$

-   $\mathcal{P}_{C,Q}(\sim A^{p+1 + \alpha})$ is true: it is enough to apply Proposition \[prop:EstSmallNrj\]

-   we want to prove that $\mathcal{P}_{C,Q}(l+1) \Longrightarrow \mathcal{P}_{C,Q}(l)$. Let $T> t_{l}$. We shall first prove that the bounds (\[Eqn:ControlIwStrich\]) and (\[Eqn:ControlEIw\]) hold if we are restricted to $[t_{l},T]$ [^8]. Then it is enough to let $T$ go to infinity in order to fully prove (\[Eqn:ControlIwStrich\]) and (\[Eqn:ControlEIw\]).\
    We introduce

    $$F_{T}:= \left\{
    T^{'} \in [t_{l}, \,T]:
    \begin{array}{ll}
    \sup_{t \in [t_{l}, \, T^{'}] } E(Iw_{l}(t))  \leq  \\
    (C_{E} N^{2(1-s)} A^{p+1} - l(1 - \epsilon_{0}) (N^{s-1} \eta_{1} A^{-1})^{\alpha})\left( 1+ \frac{\beta}{2} \right) \\
    \| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([t_{l},T^{'}])} \leq X_{C,Q}
    \end{array}
    \right\}
    $$

    We want to show that $F_{T}=[t_{l}, \, T]$ . Indeed

    -   $F_{T} \neq \emptyset $ since $ t_{l} \in F_{T}$, by (\[Eqn:DefCi\]).

    -   $F_{T}$ is closed by continuity

    -   $F_{T}$ is an open set of $[t_{l},T]$. Let $\widetilde{T'} \in F_{T}$. By continuity there exists $ \delta > 0$ such that for all $T^{'} 