\end{split}$$ $$\begin{split}
&e^{-is\Lambda(\xi)}H_{+-}(\xi,s)\\
&=\sum_{(\mu,\nu)\in\{(+,+),(+,-),(-,-)\}}\int_{\mathbb{R}^2\times\mathbb{R}^2}\frac{m_{+-}(\xi,\eta)}{\Phi_{+-}(\xi,\eta)}\big[\widehat{U_-}(\eta,s)m_{\mu\nu}(\xi-\eta,\chi)\widehat{U_\mu}(\xi-\eta-\chi,s)\widehat{U_\nu}(\chi,s)\big]\,d\eta d\chi\\
&+\sum_{(\mu,\nu)\in\{(-,-),(-,+),(+,+)\}}\int_{\mathbb{R}^2\times\mathbb{R}^2}\frac{m_{+-}(\xi,\eta)}{\Phi_{+-}(\xi,\eta)}\big[\widehat{U_+}(\xi-\eta,s)m'_{\mu\nu}(\eta,\chi)\widehat{U_\mu}(\eta-\chi,s)\widehat{U_\nu}(\chi,s)\big]\,d\eta d\chi,
\end{split}$$ and $$\begin{split}
&e^{-is\Lambda(\xi)}H_{--}(\xi,s)\\
&=\sum_{(\mu,\nu)\in\{(-,-),(-,+),(+,+)\}}\int_{\mathbb{R}^2\times\mathbb{R}^2}\frac{m_{--}(\xi,\eta)}{\Phi_{--}(\xi,\eta)}\big[\widehat{U_-}(\eta,s)m'_{\mu\nu}(\xi-\eta,\chi)\widehat{U_\mu}(\xi-\eta-\chi,s)\widehat{U_\nu}(\chi,s)\big]\,d\eta d\chi\\
&+\sum_{(\mu,\nu)\in\{(-,-),(-,+),(+,+)\}}\int_{\mathbb{R}^2\times\mathbb{R}^2}\frac{m_{--}(\xi,\eta)}{\Phi_{--}(\xi,\eta)}\big[\widehat{U_-}(\xi-\eta,s)m'_{\mu\nu}(\eta,\chi)\widehat{U_\mu}(\eta-\chi,s)\widehat{U_\nu}(\chi,s)\big]\,d\eta d\chi.
\end{split}$$

After reorganizing the terms, it follows that $$\begin{split}
H_{++}(\xi,s)&+H_{+-}(\xi,s)+H_{--}(\xi,s)\\
&=e^{is\Lambda(\xi)}\int_{\mathbb{R}^2\times\mathbb{R}^2}m_{+++}(\xi,\eta,\chi)\widehat{U_+}(\xi-\eta,s)\widehat{U_+}(\eta-\chi,s)\widehat{U_+}(\chi,s)\,d\eta d\chi\\
&+e^{is\Lambda(\xi)}\int_{\mathbb{R}^2\times\mathbb{R}^2}m_{++-}(\xi,\eta,\chi)\widehat{U_+}(\xi-\eta,s)\widehat{U_+}(\eta-\chi,s)\widehat{U_-}(\chi,s)\,d\eta d\chi\\
&+e^{is\Lambda(\xi)}\int_{\mathbb{R}^2\times\mathbb{R}^2}m_{+--}(\xi,\eta,\chi)\widehat{U_+}(\xi-\eta,s)\widehat{U_-}(\eta-\chi,s)\widehat{U_-}(\chi,s)\,d\eta d\chi\\
&+e^{is\Lambda(\xi)}\int_{\mathbb{R}^2\times\mathbb{R}^2}m_{---}(\xi,\eta,\chi)\widehat{U_-}(\xi-\eta,s)\widehat{U_-}(\eta-\chi,s)\widehat{U_-}(\chi,s)\,d\eta d\chi,
\end{split}$$ where $$\label{mult1}
m_{+++}(\xi,\eta,\chi):=\frac{m_{++}(\xi,\chi)m_{++}(\xi-\chi,\eta-\chi)}{\Phi_{++}(\xi,\chi)}+\frac{m_{++}(\xi,\eta)m_{++}(\eta,\chi)}{\Phi_{++}(\xi,\eta)}+\frac{m_{+-}(\xi,\eta)m'_{++}(\eta,\chi)}{\Phi_{+-}(\xi,\eta)},$$ $$\label{mult2}
\begin{split}
&m_{++-}(\xi,\eta,\chi):=\frac{m_{++}(\xi,\xi-\eta)m_{+-}(\eta,\chi)}{\Phi_{++}(\xi,\xi-\eta)}+\frac{m_{++}(\xi,\eta)m_{+-}(\eta,\chi)}{\Phi_{++}(\xi,\eta)}+\frac{m_{+-}(\xi,\chi)m_{++}(\xi-\chi,\eta-\chi)}{\Phi_{+-}(\xi,\chi)}\\
&+\frac{m_{+-}(\xi,\eta)m'_{-+}(\eta,\eta-\chi)}{\Phi_{+-}(\xi,\eta)}+\frac{m_{--}(\xi,\chi)m'_{++}(\xi-\chi,\xi-\eta)}{\Phi_{--}(\xi,\chi)}+\frac{m_{--}(\xi,\xi-\chi)m'_{++}(\xi-\chi,\xi-\eta)}{\Phi_{--}(\xi,\xi-\chi)},
\end{split}$$ $$\label{mult3}
\begin{split}
&m_{+--}(\xi,\eta,\chi):=\frac{m_{++}(\xi,\xi-\eta)m_{--}(\eta,\chi)}{\Phi_{++}(\xi,\xi-\eta)}+\frac{m_{++}(\xi,\eta)m_{--}(\eta,\chi)}{\Phi_{++}(\xi,\eta)}+\frac{m_{+-}(\xi,\chi)m_{+-}(\xi-\chi,\eta-\chi)}{\Phi_{+-}(\xi,\chi)}\\
&+\frac{m_{+-}(\xi,\eta)m'_{--}(\eta,\eta-\chi)}{\Phi_{+-}(\xi,\eta)}+\frac{m_{--}(\xi,\chi)m'_{-+}(\xi-\chi,\xi-\eta)}{\Phi_{--}(\xi,\chi)}+\frac{m_{--}(\xi,\xi-\chi)m'_{-+}(\xi-\chi,\xi-\eta)}{\Phi_{--}(\xi,\xi-\chi)},
\end{split}$$ and $$\label{mult4}
m_{---}(\xi,\eta,\chi):=\frac{m_{+-}(\xi,\chi)m_{--}(\xi-\chi,\eta-\chi)}{\Phi_{+-}(\xi,\chi)}+\frac{m_{--}(\xi,\chi)m'_{--}(\xi-\chi,\eta-\chi)}{\Phi_{--}(\xi,\chi)}+\frac{m_{--}(\xi,\eta)m'_{--}(\eta,\chi)}{\Phi_{--}(\xi,\eta)}.$$

To summarize, we proved the following:

\[formul\] For $t\in[0,T]$ let $$\begin{split}
&U_+(t)=U(t),\qquad U_-(t)=\overline{U(t)},\qquad V_+(t)=e^{it\Lambda}U_+(t),\qquad V_-(t)=e^{-it\Lambda}U_-(t),\\
&\widehat{W}(\xi,t)=e^{it\Lambda(\xi)}\widehat{U}(\xi,t)+ie^{it\Lambda(\xi)}\sum_{(\mu,\nu)\in\{(+,+),(+,-),(-,-)\}}\int_{\mathbb{R}^2}\frac{m_{\mu\nu}(\xi,\eta)}{\Phi_{\mu\nu}(\xi,\eta)}\widehat{U_\mu}(\xi-\eta,t)\widehat{U_\nu}(\eta,t)\,d\eta,
\end{split}$$ where $m_{++},m_{+-},m_{--}$ are defined in – and $$\Phi_{\mu\nu}(\xi,\eta)=\Lambda(\xi)-\mu\Lambda(\xi-\eta)-\nu\Lambda(\eta),\qquad (\mu,\nu)\in\{(+,+),(+,-),(-,-)\}.$$ Then $$\label{formul1}
\frac{d}{ds}[\widehat{V_+}(\xi,s)]=\sum_{(\mu,\nu)\in\{(+,+),(+,-),(-,-)\}}e^{is\Lambda(\xi)}\int_{\mathbb{R}^2}m_{\mu\nu}(\xi,\chi)\widehat{U_\mu}(\xi-\chi,s)\widehat{U_\nu}(\chi,s)\,d\chi,$$ and $$\label{formul2}
\begin{split}
\widehat{W}(\xi,t)-\widehat{W}(\xi,0)&=i\int_0^t\int_{\mathbb{R}^2\times\mathbb{R}^2}e^{is\Lambda(\xi)}m_{+++}(\xi,\eta,\chi)\widehat{U_+}(\xi-\eta,s)\widehat{U_+}(\eta-\chi,s)\widehat{U_+}(\chi,s)\,d\eta d\chi ds\\
&+i\int_0^t\int_{\mathbb{R}^2\times\mathbb{R}^2}e^{is\Lambda(\xi)}m_{++-}(\xi,\eta,\chi)\widehat{U_+}(\xi-\eta,s)\widehat{U_+}(\eta-\chi,s)\widehat{U_-}(\chi,s)\,d\eta d\chi ds\\
&+i\int_0^t\int_{\mathbb{R}^2\times\mathbb{R}^2}e^{is\Lambda(\xi)}m_{+--}(\xi,\eta,\chi)\widehat{U_+}(\xi-\eta,s)\widehat{U_-}(\eta-\chi,s)\widehat{U_-}(\chi,s)\,d\eta d\chi ds\\
&+i\int_0^t\int_{\mathbb{R}^2\times\mathbb{R}^2}e^{is\Lambda(\xi)}m_{---}(\xi,\eta,\chi)\widehat{U_-}(\xi-\eta,s)\widehat{U_-}(\eta-\chi,s)\widehat{U_-}(\chi,s)\,d\eta d\chi ds,
\end{split}$$ where the symbols $m_{+++},m_{++-},m_{+--},m_{---}$ are defined in –.

Estimates {#estim}
---------

In this subsection we prove the bound , as a consequence of Lemma \[ble1\] and Lemma \[ble2\] below. Using the dispersive estimate $$\label{disper}
\|P_le^{-it\Lambda}f\|_{L^\infty}\lesssim (1+t)^{-1}(1+2^{2l})\|f\|_{L^1},\qquad t\in\mathbb{R},$$ and the definition of the $Z$-norm, it follows that for any $l\in\mathbb{Z}$ and $t\in [0,T]$ $$\|P_lU_{\pm}(t)\|_{L^\infty}\lesssim (1+t)^{-1}(1+2^{2l})\|P_lV_{\pm}\|_{L^1}\lesssim (1+t)^{-1}\frac{1+2^{2l}}{2^{l/10}+2^{10l}}\varepsilon_0.$$ Moreover $$\|\widehat{P_lU_{\pm}}(t)\|_{L^\infty}=\|\widehat{P_lV_{\pm}}(t)\|_{L^\infty}\lesssim \|P_lV_{\pm}\|_{L^1}\lesssim (2^{l/10}+2^{10l})^{-1}\varepsilon_0.$$ Therefore, using the assumption $\|U\|_{X_T^N}\leq\varepsilon_0$ and the definitions –, for any $t\in[0,T]$ $$\label{lo7}
\begin{split}
&\|P_lU_{\pm}(t)\|_{L^2}+\|P_lV_{\pm}(t)\|_{L^2}\lesssim\varepsilon_0 \min(2^{-N_0l},(1+t)^{\delta}2^{-Nl}),\qquad l\geq 0,\\
&\|P_lU_{\pm}(t)\|_{L^2}+\|P_lV_{\pm}(t)\|_{L^2}\lesssim\varepsilon_0 2^{9l/10},\qquad l\leq 0,\\
&(2^{l/10}+2^{8l})\|P_lU_{\pm}(t)\|_{L^\infty}\lesssim \varepsilon_0\min(2^{2l},(1+t)^{-1}),\qquad l\in\mathbb{Z},\\
&\|\widehat{P_lU_{\pm}}(t)\|_{L^\infty}+\|\widehat{P_lV_{\pm}}(t)\|_{L^\infty}\lesssim \varepsilon_0(2^{l/10}+2^{10l})^{-1},\qquad l\in\mathbb{Z}.
\end{split}$$

\[ble1\] We have $$\sup_{t\in[0,T]}\|e^{it\Lambda}U(t)-W(t)\|_{Y^{N_0}}\lesssim \varepsilon_0^2.$$

Using the definitions, see Lemma \[formul\], it suffices to prove that for 