any $(\mu,\nu)\in\{(+,+),(+,-),(-,-)\}$ $$\label{lo1}
\sup_{t\in[0,T]}\big\|F_{\mu\nu}(t)\big\|_{H^{N_0}}+\sup_{t\in[0,T]}\big\|F_{\mu\nu}(t)\big\|_{Z}\lesssim\varepsilon_0^2,$$ where $$\widehat{F_{\mu\nu}}(\xi,t):=e^{it\Lambda(\xi)}\int_{\mathbb{R}^2}\frac{m_{\mu\nu}(\xi,\eta)}{\Phi_{\mu\nu}(\xi,\eta)}\widehat{U_\mu}(\xi-\eta,t)\widehat{U_\nu}(\eta,t)\,d\eta.$$

[**[Step 1.]{}**]{} We prove first certain $L^2$ estimates. For any $k\in\mathbb{Z}$ and $t\in[0,T]$ we have $$\label{lo2}
\begin{split}
\widehat{P_kF_{\mu\nu}}(\xi,t)&=\varphi_k(\xi)e^{it\Lambda(\xi)}\int_{\mathbb{R}^2}\frac{m_{\mu\nu}(\xi,\eta)}{\Phi_{\mu\nu}(\xi,\eta)}\widehat{U_\mu}(\xi-\eta,t)\widehat{U_\nu}(\eta,t)\,d\eta\\
&=\sum_{(k_1,k_2)\in\mathcal{X}_k^1\cup\mathcal{X}_k^2}\varphi_k(\xi)e^{it\Lambda(\xi)}\int_{\mathbb{R}^2}\frac{m_{\mu\nu}(\xi,\eta)}{\Phi_{\mu\nu}(\xi,\eta)}\varphi_{k_1}(\xi-\eta)\widehat{U_\mu}(\xi-\eta,t)\varphi_{k_2}(\eta)\widehat{U_\nu}(\eta,t)\,d\eta\\
&=\sum_{(k_1,k_2)\in\mathcal{X}_k^1\cup\mathcal{X}_k^2}e^{it\Lambda(\xi)}\int_{\mathbb{R}^2}P_{k,k_1,k_2}^{\mu\nu}(\xi,\eta)\widehat{P_{k_1}U_\mu}(\xi-\eta,t)\widehat{P_{k_2}U_\nu}(\eta,t)\,d\eta
\end{split}$$ where, as before, $$\begin{split}
&\mathcal{X}^1_k=\{(k_1,k_2)\in\mathbb{Z}\times\mathbb{Z}:|\max(k_1,k_2)-k|\leq 4\},\\
&\mathcal{X}^2_k=\{(k_1,k_2)\in\mathbb{Z}\times\mathbb{Z}:\max(k_1,k_2)\geq k+4\text{ and }|k_1-k_2|\leq 4\},
\end{split}$$ and $$\label{km2}
P_{k,k_1,k_2}^{\mu\nu}(\xi,\eta):=\varphi_k(\xi)\frac{m_{\mu\nu}(\xi,\eta)}{\Phi_{\mu\nu}(\xi,\eta)}\varphi_{[k_1-1,k_1+1]}(\xi-\eta)\varphi_{[k_2-1,k_2+1]}(\eta).$$

We use the estimates and : for any $k\leq 0$ and $t\in[0,T]$ it follows that $$\label{lo15}
\begin{split}
\|P_kF_{\mu\nu}(t)\|_{L^2}&\lesssim\sum_{(k_1,k_2)\in\mathcal{X}_k^2}(1+2^{3k_1})2^{k_1}\|P_{k_1}U_\pm(t)\|_{L^2}
\min(2^k\|P_{k_2}U_\pm(t)\|_{L^2},\|P_{k_2}U_\pm(t)\|_{L^\infty})\\
&+\sum_{(k_1,k_2)\in\mathcal{X}_k^1,\,k_1\leq k_2}2^k\|P_{k_1}U_\pm(t)\|_{L^2}\|P_{k_2}U_\pm(t)\|_{L^\infty}\\
&\lesssim \varepsilon_0^2\min((1+t)^{-1},2^k).
\end{split}$$ Similarly, for any $k\geq 0$ and $t\in[0,T]$, $$\label{lo16}
\begin{split}
\|P_kF_{\mu\nu}(t)\|_{L^2}&\lesssim\sum_{(k_1,k_2)\in\mathcal{X}_k^2}2^{2k}(1+2^{3k_1})2^{k_1}\|P_{k_1}U_\pm(t)\|_{L^\infty}\|P_{k_2}U_\pm(t)\|_{L^2}\\
&+\sum_{(k_1,k_2)\in\mathcal{X}_k^1,\,-20k_2\leq k_1\leq k_2}2^{2k}(1+2^{3k_1})2^k\|P_{k_1}U_\pm(t)\|_{L^\infty}\|P_{k_2}U_\pm(t)\|_{L^2}\\
&+\sum_{(k_1,k_2)\in\mathcal{X}_k^1,\,k_1\leq -20k_2}2^{2k}(1+2^{3k_1})2^k\|P_{k_1}U_\pm(t)\|_{L^2}\|P_{k_2}U_\pm(t)\|_{L^\infty}\\
&\lesssim \varepsilon_0^2\min(2^{-(N-3)k},(1+t)^{-1}2^{-15k}).
\end{split}$$ The bound on the first term in follows easily from the last two inequalities.

[**[Step 2.]{}**]{} We prove now the bound on the second term in , i.e. for any $k\in\mathbb{Z}$ and $t\in[0,T]$ $$\sum_{j\in\mathbb{Z}_+}2^j\|\varphi_j\cdot P_kF_{\mu\nu}(t)\|_{L^2}\lesssim\varepsilon_0^2(2^{k/10}+2^{10k})^{-1}.$$ Using and , it suffices to prove that $$\sum_{j\geq J_{k,t}}2^j\|\varphi_j\cdot P_kF_{\mu\nu}(t)\|_{L^2}\lesssim\varepsilon_0^22^{-10k_+},$$ where, for some sufficiently large constant $D$, $$\label{lo16.5}
2^{J_{k,t}}=D(1+t+2^{-41k/40})(1+2^{2k}).$$ Using and the formulas $\widehat{U_\mu}(\xi,t)=e^{-\mu it\Lambda(\xi)}\widehat{V_\mu}(\xi,t)$ we write $$\label{lu3.5}
\begin{split}
&P_k F_{\mu\nu}(t)=\sum_{(k_1,k_2)\in \mathcal{X}_k}F_{k,k_1,k_2}^{\mu\nu}(t),\\
&\widehat{F_{k,k_1,k_2}^{\mu\nu}}(\xi,t):=\int_{\mathbb{R}^2}e^{it[\Lambda(\xi)-\mu\Lambda(\xi-\eta)-\nu\Lambda(\eta)]}P_{k,k_1,k_2}^{\mu\nu}(\xi,\eta)\widehat{P_{k_1}V_\mu}(\xi-\eta,t)\widehat{P_{k_2}V_\nu}(\eta,t)\,d\eta,
\end{split}$$ where, as before, $\mathcal{X}_k=\mathcal{X}_k^1\cup\mathcal{X}_k^2$. Using this decomposition, it suffices to prove that $$\label{lo21}
\sum_{j\geq J_{k,t}}\sum_{(k_1,k_2)\in \mathcal{X}_k}2^j\|\varphi_j\cdot F_{k,k_1,k_2}^{\mu\nu}(t)\|_{L^2}\lesssim\varepsilon_0^22^{-10k_+},$$ for any $k\in\mathbb{Z}$ and $t\in[0,T]$. This is similar to the proof of Proposition \[LocExLemma\] (iii) in the previous section, and follows from the bounds , , and below.

[**[Step 