0$ and $f\in H^{N_0}$ we define $$\|f\|_{Z_J}:=\sup_{k\in\mathbb{Z}}(2^{k/10}+2^{10k})\Big[\|P_kf\|_{L^2}+\sum_{j\in\mathbb{Z}_+}2^{\min(j,2J-j)}\|\varphi_j(x)\cdot P_kf(x)\|_{L^2}\Big],$$ and notice that $$\|f\|_{Z_J}\leq\|f\|_{Z},\qquad \|f\|_{Z_J}\lesssim_J\|f\|_{H^{N_0}}.$$ It suffices to prove that for any $t,t'\in [0,T]$ with $0\leq t'-t\leq 1$, and any $J\in\mathbb{Z}_+$ we have $$\label{cu1}
 \|e^{it'\Lambda}U(t')-e^{it\Lambda}U(t)\|_{Z_J}\leq C\big(T,\sup_{t\in[0,T]}\|U(t)\|_{H^{N_0}},\|U_0\|_Z\big)\cdot |t'-t|(1+\sup_{s\in[t,t']}\|e^{is\Lambda}U(s)\|_{Z_J}).$$ Indeed, assuming , it follows easily that $$\begin{split}
&\sup_{t\in[0,T]}\|e^{it\Lambda}U(t)\|_{Z_J}\leq C\big(T,\sup_{t\in[0,T]}\|U(t)\|_{H^{N_0}},\|U_0\|_Z\big),\\
&\|e^{it'\Lambda}U(t')-e^{it\Lambda}U(t)\|_{Z_J}\leq C\big(T,\sup_{t\in[0,T]}\|U(t)\|_{H^{N_0}},\|U_0\|_Z\big)\cdot |t'-t|, \qquad t,t'\in[0,T],
\end{split}$$ uniformly in $J$, and the desired conclusions follow by letting $J\to\infty$.

To prove we define $V_\pm(t)=e^{\pm it\Lambda}U(t)$, as in section \[Bootstrap\], and use the formula , $$\label{cu1.5}
\begin{split}
&\widehat{V_+}(\xi,t')-\widehat{V_+}(\xi,t)\\
&=\sum_{(\mu,\nu)\in\{(+,+),(+,-),(-,-)\}}\int_{t}^{t'}\int_{\mathbb{R}^2}e^{is[\Lambda(\xi)-\mu\Lambda(\xi-\eta)-\nu\Lambda(\eta)]}m_{\mu\nu}(\xi,\eta)\widehat{V_\mu}(\xi-\eta,s)\widehat{V_\nu}(\eta,s)\,d\eta ds.
\end{split}$$ Letting $$\widehat{A_{\mu\nu}}(\xi,s):=\int_{\mathbb{R}^2}e^{is[\Lambda(\xi)-\mu\Lambda(\xi-\eta)-\nu\Lambda(\eta)]}m_{\mu\nu}(\xi,\eta)\widehat{V_\mu}(\xi-\eta,s)\widehat{V_\nu}(\eta,s)\,d\eta,$$ it suffices to prove that for any $s\in[0,T]$ and $(\mu,\nu)\in\{(+,+),(+,-),(-,-)\}$ $$\|A_{\mu\nu}(s)\|_{Z_J}\leq C\big(s,\|V_+(s)\|_{H^{N_0}},\|V_+(0)\|_Z\big)\cdot(1+\|V_+(s)\|_{Z_J}),\qquad \text{ uniformly in }J.$$ In view of the definition, it suffices to prove that for any $k\in\mathbb{Z}$ and $s\in[0,T]$ $$\label{cu2}
\begin{split}
\|P_k A_{\mu\nu}(s)\|_{L^2}&+\sum_{j\geq 0}2^{\min(j,2J-j)}\|\varphi_j\cdot P_k A_{\mu\nu}(s)\|_{L^2}\\
&\leq C\big(s,\|V_+(s)\|_{H^{N_0}},\|V_+(0)\|_Z\big)\cdot(1+\|V_+(s)\|_{Z_J})(2^{k/10}+2^{10 k})^{-1}.
\end{split}$$ We prove this bound in several steps: first we estimate the sum over small values of $j$, see . Then we estimate the contribution of the very low frequencies of the inputs $V_\mu,V_\nu$, see . Finally, we pass to the physical space to estimate the remaining contributions using localization, see and .

For simplicity of notation, in the rest of the proof we let $\widetilde{C}$ denote constants that may depend only on $s,\|V_+(s)\|_{H^{N_0}},\|V_+(0)\|_Z$, 