similar to the constant in the right-hand side of .

[**[Step 2.]{}**]{} Notice first that $$\|\widehat{P_kA_{\mu\nu}}(s)\|_{L^\infty}\lesssim 2^{-(N_0-1)k_+}\|V_+(s)\|_{H^{N_0}}^2.$$ Therefore $\|P_kA_{\mu\nu}(s)\|_{L^2}\lesssim 2^k2^{-(N_0-1)k_+}\|V_+(s)\|_{H^{N_0}}^2$ and it follows that $$\label{cu2.5}
 \|P_k A_{\mu\nu}(s)\|_{L^2}+\sum_{j\leq\max(-41k/40,2k)+C(s)}2^j\|\varphi_j\cdot P_k A_{\mu\nu}(s)\|_{L^2}\leq \widetilde{C}(2^{k/10}+2^{10 k})^{-1}.$$

[**[Step 3.]{}**]{} To estimate the remaining sum $$\sum_{j\geq\max(-41k/40,2k)+C(s)}2^{\min(j,2J-j)}\|\varphi_j\cdot P_k A_{\mu\nu}(s)\|_{L^2},$$ we decompose $$\label{cu3.5}
\begin{split}
&P_k A_{\mu\nu}(s)=\sum_{(k_1,k_2)\in \mathcal{X}_k}A_{k,k_1,k_2}^{\mu\nu}(s),\\
&\widehat{A_{k,k_1,k_2}^{\mu\nu}}(\xi,s):=\int_{\mathbb{R}^2}e^{is[\Lambda(\xi)-\mu\Lambda(\xi-\eta)-\nu\Lambda(\eta)]}\varphi_k(\xi)m_{\mu\nu}(\xi,\eta)\widehat{P_{k_1}V_\mu}(\xi-\eta,s)\widehat{P_{k_2}V_\nu}(\eta,s)\,d\eta,
\end{split}$$ where $$\begin{split}
&\mathcal{X}^1_k:=\{(k_1,k_2)\in\mathbb{Z}\times\mathbb{Z}:|\max(k_1,k_2)-k|\leq 4\},\\
&\mathcal{X}^2_k:=\{(k_1,k_2)\in\mathbb{Z}\times\mathbb{Z}:\max(k_1,k_2)\geq k+4\text{ and }|k_1-k_2|\leq 4\},\\
&\mathcal{X}_k:=\mathcal{X}_k^1\cup\mathcal{X}_k^2.
\end{split}$$ Since $$|m_{\mu\nu}(\xi,\eta)|\lesssim |\xi|+|\eta|+|\xi-\eta|,$$ see –, it follows from that $$\label{cu7.5}
\begin{split}
\|\widehat{P_l V_{\pm}}(s)\|_{L^\infty}&\lesssim \|\widehat{P_l V_{\pm}}(0)\|_{L^\infty}+(1+s)\Big[2^l\|P_{[l-4,l+4]}V_+(s)\|_{L^2}\|P_{(-\infty,l+4]}V_+(s)\|_{L^2}\\
&+\sum_{(k_1,k_2)\in\mathcal{X}_l^2}(2^{k_1}+2^{k_2}) \|P_{k_1}V_+(s)\|_{L^2}\|P_{k_2}V_+(s)\|_{L^2}\Big]\\
&\leq \widetilde{C}(2^{l/10}+2^{10l})^{-1}.
\end{split}$$ Therefore $$\begin{split}
\|A_{k,k_1,k_2}^{\mu\nu}(s)\|_{L^2}&\lesssim 2^{\max(k,k_1,k_2)}\min\big(\|\widehat{P_{k_1}V_\mu}(s)\|_{L^1}\|\widehat{P_{k_2}V_\nu}(s)\|_{L^2},\|\widehat{P_{k_1}V_\mu}(s)\|_{L^2}\|\widehat{P_{k_2}V_\nu}(s)\|_{L^1}\big)\\
&\leq \widetilde{C}2^{\max(k_1,k_2)}(1+2^{\max(k_1,k_2)})^{-N_0}2^{(19/10)\min(k_1,k_2)}.
\end{split}$$ Therefore $$\label{cu4}
\sum_{j\geq \max(-41k/40,2k)}\sum_{(k_1,k_2)\in \mathcal{X}_k,\,\min(k_1,k_2)\leq-2j/3}2^j\|A_{k,k_1,k_2}^{\mu\nu}(s)\|_{L^2}\leq \widetilde{C}2^{-10 k_+}.$$

[**[Step 4.]{}**]{} It remains to bound the contribution $$\sum_{j\geq \max(-41k/40,2k)+C(s)}\sum_{(k_1,k_2)\in \mathcal{X}_k,\,\min(k_1,k_2)\geq-2j/3}2^{\min(j,2J-j)}\|\varphi_j\cdot A_{k,k_1,k_2}^{\mu\nu}(s)\|_{L^2}.$$ For this we write, using , $$\begin{split}
&A_{k,k_1,k_2}^{\mu\nu}(x,s)=\int_{\mathbb{R}^2\times\mathbb{R}^2}P_{k_1}V_\mu(y,s)P_{k_2}V_\nu(z,s)K_{k,k_1,k_2}^{\mu\nu}(x,y,z)\,dydz,\\
&K_{k,k_1,k_2}^{\mu\nu}(x,y,z):=c\int_{\mathbb{R}^2\times\mathbb{R}^2}e^{is[\Lambda(\xi)-\mu\Lambda(\xi-\eta)-\nu\Lambda(\eta)]}e^{i(x\cdot\xi-y\cdot(\xi-\eta)-z\cdot\eta)}m_{k,k_1,k_2}^{\mu\nu}(\xi,\eta)\,d\xi d\eta,\\
&m_{k,k_1,k_2}^{\mu\nu}(\xi,\eta):=m_{\mu\nu}(\xi,\eta)\varphi_k(\xi)\varphi_{[k_1-1,k_1+1]}(\xi-\eta)\varphi_{[k_2-1,k_2+1]}(\eta).
\end{split}$$ Since $$\sup_{\xi\in\mathbb{R}^2}|D^\alpha \Lambda(\xi)|\lesssim_{|\alpha|}1,\qquad |\alpha|\geq 1,$$ it follows by integration by parts that $$\label{cu5}
\begin{split}
&|K_{k,k_1,k_2}^{\mu\nu}(x,y,z)|\lesssim (1+2^{4k_1}+2^{4k_2})(|x-y|+|x-z|)^{-20}\\
\text{if }&j\geq C(s),\,\,\min(k,k_1,k_2)\geq -40j/41,\,\,|x-y|+|x-z|\geq 2^{j-4},
\end{split}$$ provided that the constant $C(s)$ is fixed sufficiently large. Letting $$\begin{split}
&B_{j,k,k_1,k_2}^{\mu\nu}(x,s):=\int_{\mathbb{R}^2\times\mathbb{R}^2}V^\mu_{j,k_1}(y,s)V^\nu_{j,k_2}(z,s)K_{k,k_1,k_2}^{\mu\nu}(x,y,z)\,dydz,\\
&V^\mu_{j,l}(x,s):=\varphi_{[j-4,j+4]}(x)\cdot P_{l}V_\mu(x,s),
\end{split}$$ it follows from that $$\|\varphi_j\cdot [A_{k,k_1,k_2}^{\mu\nu}(s)-B_{j,k,k_1,k_2}^{\mu\nu}(s)]\|_{L^2}\lesssim 2^{-10j}(1+2^{4k_1}+2^{4k_2})\|P_{k_1}V_+(s)\|_{L^2}\|P_{k_2}V_+(s)\|_{L^2}.$$ Since $\|P_{l}V_+(s)\|_{L^2}\lesssim 2^{-N_0l_+}\|V_+(s)\|_{H^{N_0}}$, $l\in\mathbb{Z}$, it follows that $$\label{cu7}
\sum_{j\geq \max(-41k/40,2k)+C(s)}\sum_{(k_1,k_2)\in \mathcal{X}_k,\,\min(k_1,k_2)\geq-2j/3}2^j\|\varphi_j\cdot [A_{k,k_1,k_2}^{\mu\nu}(s)-B_{j,k,k_1,k_2}^{\mu\nu}(s)]\|_{L^2}\leq \widetilde{C}2^{-10 k_+}.$$

[**[Step 5.]{}**]{} Finally we rewrite $B_{j,k,k_1,k_2}^{\mu\nu}(s)$ in the frequency space, $$\widehat{B_{j,k,k_1,k_2}^{\mu\nu}}(\xi,s)=\int_{\mathbb{R}^2}e^{is[\Lambda(\xi)-\mu\Lambda(\xi-\eta)-\nu\Lambda(\eta)]}m^{\mu\nu}_{k,k_1,k_2}(\xi,\eta)\widehat{V^\mu_{j,k_1}}(\xi-\eta,s)\widehat{V_{j,k_2}^\nu}(\eta,s)\,d\eta,$$ and estimate $$\|\widehat{B_{j,k,k_1,k_2}^{\mu\nu}}(s)\|_{L^2}\lesssim 2^{\max(k_1,k_2)}2^{\min(k_1,k_2)}\|\widehat{V^\mu_{j,k_1}}(s)\|_{L^2}\|\widehat{V_{j,k_2}^\nu}(s)\|_{L^2}.$$ Therefore $$\label{cu12}
\begin{split}
\sum_{j\geq \max(-41k/40,2k)+C(s)}&\sum_{(k_1,k_2)\in \mathcal{X}_k,\,\min(k_1,k_2)\geq-2j/3}2^{\min(j,2J-j)}\|\varphi_j\cdot B_{j,k,k_1,k_2}^{\mu\nu}(s)\|_{L^2}\\
&\lesssim\sum_{(k_1,k_2)\in 