\mathcal{X}_k,\,k_1\leq k_2}\sum_{j\geq 10}2^{\min(j,2J-j)}2^{k_1+k_2}\|V^\pm_{j,k_1}(s)\|_{L^2}\|V_{j,k_2}^\pm(s)\|_{L^2}.
\end{split}$$ We estimate the sum in the right-hand side of in two different ways, depending on the relative sizes of $k,k_1,k_2$. If[^8] $k_1=\min(k_1,k_2)\notin[100-10k,k-10]$ then we estimate $\|V^\pm_{j,k_1}(s)\|_{L^2}\leq \widetilde{C}2^{-N_0\max(k_1,0)}$, and the corresponding sum in the right-hand side of is bounded by $$\sum_{(k_1,k_2)\in \mathcal{X}_k,\,k_1\leq k_2,\,k_1\notin[100-10k,k-10]}\widetilde{C}2^{k_1+k_2}2^{-N_0\max(k_1,0)}(2^{k_2/10}+2^{10k_2})^{-1}\|V_+(s)\|_{Z_J}\leq \widetilde{C}\|V_+(s)\|_{Z_J}2^{-10 k_+}.$$ On the other hand, if $k_1=\min(k_1,k_2)\in[100-10k,k-10]$ (in particular $k\geq 10$, $|k_2-k|\leq 4$) then we use $\|V^\pm_{j,k_2}(s)\|_{L^2}\leq \widetilde{C}2^{-N_0k}$, and the corresponding sum in the right-hand side of is bounded by $$\sum_{(k_1,k_2)\in \mathcal{X}_k,\,k_1\leq k_2,\,k_1\in[100-10k,k-10]}\widetilde{C}2^{k_1+k_2}2^{-N_0k}(2^{k_1/10}+2^{10k_1})^{-1}\|V_+(s)\|_{Z_J}\leq \widetilde{C}\|V_+(s)\|_{Z_J}2^{-10 k}.$$ Therefore $$\label{cu15}
\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}\\
\leq\widetilde{C}\|V_+(s)\|_{Z_J}2^{-10 k_+},
\end{split}$$ which completes the proof.

Proof of Proposition \[Norm\] {#Bootstrap}
=============================

Recall first some of the notation from section \[section2\]: for $f\in H^1(\mathbb{R}^2)$ we define $|\nabla|f$, $R_jf$, $\Lambda f$, and $P_kf$ by multiplication with the Fourier multipliers $$\label{not1}
\xi\to|\xi|,\qquad \xi\to i\xi_j/|\xi|,\qquad\xi\to\Lambda(\xi)=\sqrt{a|\xi|^2+b},\qquad \xi\to\varphi_k(\xi)=\varphi(|\xi|/2^k)-\varphi(|\xi|/2^{k-1})$$ respectively, where $a,b\in(0,\infty)$, $j\in\{1,2\}$, $k\in\mathbb{Z}$, and $\varphi:\mathbb{R}\to[0,1]$ is an even smooth function supported in $[-8/5,8/5]$ and equal to $1$ in $[-5/4,5/4]$. For any set $I\in\mathbb{R}$ let $$\varphi_I=\sum_{m\in I\cap\mathbb{Z}}\varphi_m.$$ For any $k\in\mathbb{Z}$ let $$k_+:=\max(k,0).$$ Recall also the definition of our main spaces of “smooth” solutions $X^N_T(\mathbb{R}^2\times[0,T])$, $Y^{N_0}(\mathbb{R}^2)$, and $Z(\mathbb{R}^2)$ (see and ), where $T\geq 1$, $N=30$ and $N_0=20$.

The rest of the section is concerned with the proof 