also integration by parts and the formulas $-|\nabla|=\partial_1R_1+\partial_2R_2$ and $\partial_jR_m=\partial_jR_m$, $$\begin{split}
&|II_P|\lesssim\sum_{j,m=1}^2\Big|\int_{\mathbb{R}^2}PR_mY\cdot P\partial_m(R_jY\cdot R_jY)\,dx\Big|\\
&\lesssim \sum_{j,m=1}^2\Big[\Big|\int_{\mathbb{R}^2}PR_mY\cdot \partial_m[P(R_jY\cdot R_jY)-bR_jY\cdot PR_jY]\,dx\Big|
+\Big|\int_{\mathbb{R}^2}PR_mY\cdot \partial_m(R_jY\cdot PR_jY)\,dx\Big|\Big]\\
&\lesssim \|U\|_{H^\sigma}^2\|U\|_{Z'}+\sum_{j,m=1}^2\Big|\int_{\mathbb{R}^2}PR_mY\cdot \partial_jPR_mY\cdot R_jY\,dx\Big|\\
&\lesssim \|U\|_{H^\sigma}^2\|U\|_{Z'},
\end{split}$$ and similarly, $$\begin{split}
|VI_P|&\lesssim\sum_{j,m=1}^2\Big|\int_{\mathbb{R}^2}|\nabla|\Lambda^{-1}X\cdot R_jPY\cdot \partial_jP\big(R_mY\cdot R_mY\big)\,dx\Big|\\
&\lesssim \sum_{j,m=1}^2\Big|\int_{\mathbb{R}^2}|\nabla|\Lambda^{-1}X\cdot R_jPY\cdot \partial_j(PR_mY\cdot R_mY\big)\,dx\Big|\\
&+\sum_{j,m=1}^2\Big|\int_{\mathbb{R}^2}|\nabla|\Lambda^{-1}X\cdot R_jPY\cdot \partial_j\big[P\big(R_mY\cdot R_mY\big)-bPR_mY\cdot R_mY\big]\,dx\Big|\\
&\lesssim \sum_{j,m=1}^2\Big|\int_{\mathbb{R}^2}\big(|\nabla|\Lambda^{-1}X\cdot R_mY\big)\cdot R_jPY\cdot \partial_mPR_jY\,dx\Big|+\|U\|_{H^\sigma}^2\|U\|_{Z'}\\
&\lesssim \|U\|_{H^\sigma}^2\|U\|_{Z'},
\end{split}$$ where $b=1$ if $P=\mathrm{Id}$ and $b=2$ if $P=D^\alpha$, $|\alpha|\geq 1$. Therefore $$\label{ale2}
\sum_{P=D^\alpha,\,|\alpha|\leq\sigma}\big[|II_P|+|III_P|+|V_P|+|VI_P|\big]\lesssim \|U\|_{H^\sigma}^2\|U\|_{Z'}.$$

In addition, using and the bound , $$\begin{split}
|I_P+IV_P|&=2\Big|\sum_{j=1}^2\int_{\mathbb{R}^2}R_jPX\cdot \big[P\Lambda (|\nabla|\Lambda^{-1}X\cdot R_jY)-\Lambda(|\nabla|\Lambda^{-1}X\cdot R_jPY)\big]\,dx\Big|\\
&\lesssim \sum_{j=1}^2\Big|\int_{\mathbb{R}^2}R_jPX\cdot \Lambda \big[P|\nabla|\Lambda^{-1}X\cdot R_jY\big]\,dx\Big|+\|U\|_{H^\sigma}^2\|U\|_{Z'}\\
&\lesssim \sum_{j=1}^2\Big|\int_{\mathbb{R}^2}R_jPX\cdot P|\nabla| X\cdot R_jY\,dx\Big|+\|U\|_{H^\sigma}^2\|U\|_{Z'}.
\end{split}$$ Since $|\nabla|=-\partial_1R_1-\partial_2R_2$ and $\partial_jR_m=\partial_mR_j$ we can further estimate $$\label{ale3}
\begin{split}
|I_P+IV_P|&\lesssim \sum_{j,m=1}^2\Big|\int_{\mathbb{R}^2}R_jPX\cdot P\partial_mR_mX\cdot R_jY\,dx\Big|+\|U\|_{H^\sigma}^2\|U\|_{Z'}\\
&\lesssim \sum_{j,m=1}^2\Big|\int_{\mathbb{R}^2}\partial_jR_mPX\cdot PR_mX\cdot R_jY\,dx\Big|+\|U\|_{H^\sigma}^2\|U\|_{Z'}\\
&\lesssim \|U\|_{H^\sigma}^2\|U\|_{Z'}.
\end{split}$$

Notice also that, for any $m,j\in\{1,2\}$ $$\begin{split}
&\Big|\int_{\mathbb{R}^2}\partial_m|\nabla|\Lambda^{-1}X\cdot R_jPY\cdot \partial_mR_jPY\,dx\Big|\\
&\lesssim \Big|\int_{\mathbb{R}^2}\partial_m|\nabla|\Lambda^{-1}X\cdot P_{(-\infty,0]}R_jPY\cdot \partial_mR_jPY\,dx\Big|+\Big|\int_{\mathbb{R}^2}\partial_m|\nabla|\Lambda^{-1}X\cdot P_{[1,\infty)}R_jPY\cdot \partial_mR_jPY\,dx\Big|\\
&\lesssim \|\nabla U\|^2_{H^\sigma}\big(\|P_{(-\infty,0]}R_jPY\|_{L^\infty}+\|\partial_m|\nabla|\Lambda^{-1}X\|_{L^\infty}\big)\\
&\lesssim \|\nabla U\|^2_{H^\sigma}\|U\|_{H^3}.
\end{split}$$ Since $\|U\|_{H^3}$ is small, it follows that $$\sum_{P=D^\alpha,\,|\alpha|\leq\sigma}VII_P\geq 0.$$ The desired inequality follows for $0\leq s\leq t\leq T^\varepsilon$, using also and . A simple continuity argument, using this inequality with $\sigma=3$ and the bound $\|U(t)\|_{Z'}\lesssim \|U(t)\|_{H^3}$, shows that the solution $U^\varepsilon$ can be extended up to time $1$, which completes the proof of the lemma.

We estimate now differences of smooth solutions.

\[enerdif\] (i) 