3.]{}**]{} Since $$\sup_{\xi,\eta\in\mathbb{R}^2}|P_{k,k_1,k_2}^{\mu\nu}(\xi,\eta)|\lesssim (1+2^{\min(k_1,k_2)})2^{\max(k_1,k_2)},$$ it follows from that $$\begin{split}
\|F_{k,k_1,k_2}^{\mu\nu}(s)\|_{L^2}&\lesssim (2^{\max(k_1,k_2)}+2^{k_1+k_2})\min\big(\|\widehat{P_{k_1}V_\mu}(s)\|_{L^1}\|\widehat{P_{k_2}V_\nu}(s)\|_{L^2},\|\widehat{P_{k_1}V_\mu}(s)\|_{L^2}\|\widehat{P_{k_2}V_\nu}(s)\|_{L^1}\big)\\
&\lesssim\varepsilon_0^2(2^{\max(k_1,k_2)}+2^{k_1+k_2})2^{(19/10)\min(k_1,k_2)}(1+2^{\max(k_1,k_2)})^{-N_0}.
\end{split}$$ Therefore $$\label{lu4}
\sum_{j\geq J_{k,t}}\sum_{(k_1,k_2)\in \mathcal{X}_k,\,\min(k_1,k_2)\leq-2j/3}2^j\|F_{k,k_1,k_2}^{\mu\nu}(s)\|_{L^2}\lesssim \varepsilon_0^22^{-10 k_+}.$$

[**[Step 4.]{}**]{} It remains to bound the contribution $$\sum_{j\geq J_{k,t}}\sum_{(k_1,k_2)\in \mathcal{X}_k,\,\min(k_1,k_2)\geq-2j/3}2^{j}\|\varphi_j\cdot F_{k,k_1,k_2}^{\mu\nu}(t)\|_{L^2}.$$ For this we write, using , $$\begin{split}
&F_{k,k_1,k_2}^{\mu\nu}(x,t)=\int_{\mathbb{R}^2\times\mathbb{R}^2}P_{k_1}V_\mu(y,t)P_{k_2}V_\nu(z,t)L_{k,k_1,k_2}^{\mu\nu}(x,y,z,t)\,dydz,\\
&L_{k,k_1,k_2}^{\mu\nu}(x,y,z,t):=c\int_{\mathbb{R}^2\times\mathbb{R}^2}e^{it[\Lambda(\xi)-\mu\Lambda(\xi-\eta)-\nu\Lambda(\eta)]}e^{i(x\cdot\xi-y\cdot(\xi-\eta)-z\cdot\eta)}P_{k,k_1,k_2}^{\mu\nu}(\xi,\eta)\,d\xi d\eta.
\end{split}$$ Since $$\sup_{\xi\in\mathbb{R}^2}|D^\alpha \Lambda(\xi)|\lesssim_{|\alpha|}1$$ it follows by integration by parts that $$\label{lu5}
\begin{split}
&|L_{k,k_1,k_2}^{\mu\nu}(x,y,z,t)|\lesssim (1+2^{4k_1}+2^{4k_2})(|x-y|+|x-z|)^{-20}\\
\text{if }&2^j\geq D(1+t),\,\,\min(k,k_1,k_2)\geq -40j/41,\,\,|x-y|+|x-z|\geq 2^{j-4},
\end{split}$$ provided that the constant $D$ is fixed sufficiently large. Letting $$\begin{split}
&G_{j,k,k_1,k_2}^{\mu\nu}(x,t):=\int_{\mathbb{R}^2\times\mathbb{R}^2}V^\mu_{j,k_1}(y,t)V^\nu_{j,k_2}(z,t)L_{k,k_1,k_2}^{\mu\nu}(x,y,z,t)\,dydz,\\
&V^\mu_{j,l}(x,t)=\varphi_{[j-4,j+4]}(x)\cdot P_{l}V_\mu(x,t),
\end{split}$$ it follows from that $$\|\varphi_j\cdot [F_{k,k_1,k_2}^{\mu\nu}(t)-G_{j,k,k_1,k_2}^{\mu\nu}(t)]\|_{L^2}\lesssim 2^{-10j}(1+2^{4k_1}+2^{4k_2})\|P_{k_1}V_+(t)\|_{L^2}\|P_{k_2}V_+(t)\|_{L^2}.$$ Since $\|P_{l}V_+(t)\|_{L^2}\lesssim 2^{-N_0l_+}\|V_+(t)\|_{H^{N_0}}$, $l\in\mathbb{Z}$, it follows that $$\label{lu7}
\sum_{j\geq J_{k,t}}\sum_{(k_1,k_2)\in \mathcal{X}_k,\,\min(k_1,k_2)\geq-2j/3}2^j\|\varphi_j\cdot [F_{k,k_1,k_2}^{\mu\nu}(t)-G_{j,k,k_1,k_2}^{\mu\nu}(t)]\|_{L^2}\leq\varepsilon_0^22^{-10 k_+}.$$

[**[Step 5.]{}**]{} Finally we rewrite $G_{j,k,k_1,k_2}^{\mu\nu}(t)$ in the frequency space, $$\widehat{G_{j,k,k_1,k_2}^{\mu\nu}}(\xi,t)=\int_{\mathbb{R}^2}e^{it[\Lambda(\xi)-\mu\Lambda(\xi-\eta)-\nu\Lambda(\eta)]}P^{\mu\nu}_{k,k_1,k_2}(\xi,\eta)\widehat{V^\mu_{j,k_1}}(\xi-\eta,t)\widehat{V_{j,k_2}^\nu}(\eta,t)\,d\eta,$$ and estimate $$\|\widehat{G_{j,k,k_1,k_2}^{\mu\nu}}(t)\|_{L^2}\lesssim \|P^{\mu\nu}_{k,k_1,k_2}\|_{L^\infty}2^{\min(k_1,k_2)}\|\widehat{V^\mu_{j,k_1}}(t)\|_{L^2}\|\widehat{V_{j,k_2}^\nu}(t)\|_{L^2}.$$ Therefore $$\label{lu12}
\begin{split}
\sum_{j\geq J_{k,t}}&\sum_{(k_1,k_2)\in \mathcal{X}_k,\,\min(k_1,k_2)\geq-2j/3}2^{j}\|\varphi_j\cdot G_{j,k,k_1,k_2}^{\mu\nu}(t)\|_{L^2}\\
&\lesssim\sum_{(k_1,k_2)\in \mathcal{X}_k,\,k_1\leq k_2}\sum_{j\geq 10}2^{j}2^{k_1+k_2}(1+2^{\min(k_1,k_2)})\|V^\pm_{j,k_1}(t)\|_{L^2}\|V_{j,k_2}^\pm(t)\|_{L^2}.
\end{split}$$ We estimate the sum in the right-hand side of in two different ways, depending on the relative sizes of $k,k_1,k_2$. If $k_1=\min(k_1,k_2)\notin[100-10k,k-10]$ then we estimate $\|V^\pm_{j,k_1}(t)\|_{L^2}\lesssim \varepsilon_02^{-N_0\max(k_1,0)}$, and the corresponding sum in the right-hand side of is bounded by $$C\sum_{(k_1,k_2)\in \mathcal{X}_k,\,k_1\leq k_2,\,k_1\notin[100-10k,k-10]}\varepsilon_02^{k_1+k_2}2^{(1-N_0)\max(k_1,0)}(2^{k_2/10}+2^{10k_2})^{-1}\|V_+(t)\|_{Z}\lesssim \varepsilon_0^22^{-10 k_+}.$$ On the other hand, if $k_1=\min(k_1,k_2)\in[100-10k,k-10]$ (in particular $k\geq 10$, $|k_2-k|\leq 4$) then we use $\|V^\pm_{j,k_2}(t)\|_{L^2}\lesssim \varepsilon_0 2^{-N_0k}$, and the corresponding sum in the right-hand side of is bounded by $$C\sum_{(k_1,k_2)\in 