Proposition \[Prop:SepLocMol\]) and that there exists a constant $L(l+1)$ such that

$$\begin{array}{ll}
\| I w_{l+1} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}(J)} &  \leq L(l+1) \\
& \leq X_{C,Q}
\end{array}
\label{Eqn:BoundIw}$$

Then there exist $ C_{2}:= C_{2}(p)>> 1 $ and $\bar{C}:=\bar{C}(p) >> 1 $ such that if

$$\begin{array}{ll}
k & \leq \frac{1}{C_{2}^{C_{2}L(l+1) + 1}}
\end{array}
\label{Eqn:Boundk}$$

and

$$\begin{array}{ll}
\| I v_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([T_{l},b])} \leq k
\end{array}
\label{Eqn:BoundIv}$$

then

$$\begin{array}{l}
\| I  w_{l}  \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} ([T_{l},b])} \lesssim L(l+1)
\end{array}$$

provided that $(N,s)$ satisfies (\[Eqn:Cond2\]) and

$$\begin{array}{l}
\frac{\bar{C}^{\bar{C} X_{C,Q}} (N^{1-s} A^{\frac{p+1}{2}})^{p} X_{C,Q} } {N^{ \min{ \left( \left( s_{c} -\frac{1}{2} \right)-, (1-s_{c})-
\right) }}} << \eta_{0}
\end{array}
\label{Eqn:Cond4}$$

(with $\eta_{0}$ defined in Proposition \[prop:LocalBd\])

Here, again, $\beta$ is defined in (\[Eqn:DefBeta\]).

\[prop:PerturbArg\]

[**“ Initial mollified energy small implies control of $\| Iw \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (\mathbb{R})}$ and control of $\sup_{t \in \mathbb{R}} E(Iw(t))$ ”**]{} Assume that $w$ satisfies (\[Eqn:NlkgWdat\]) and that there exists $(i, t_{i}) \in
\mathbb{N} \times \mathbb{R}$ such that

$$\begin{array}{ll}
E(Iw (t_{i})) & \leq C_{E} N^{2(1-s)} A^{p+1} - i (1- \epsilon_{0}) (N^{s-1} \eta_{1} A^{-1})^{\alpha} \\
& \leq \frac{1}{10^{8}}
\end{array}
\label{Eqn:BoundInitSmall}$$

then

$$\begin{array}{l}
\| I w \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (\mathbb{R})}  \lesssim 1
\end{array}$$

and

$$\begin{array}{l}
\sup_{t \in \mathbb{R}} E(Iw(t)) \lesssim  \left( C_{E} N^{2(1-s)} A^{p+1} - i (1- \epsilon_{0}) 