Appendix
This appendix provides the complete proof of Lemma 4.1.
Proof
Proof of Lemma 4.1
$$\begin{aligned} \begin{aligned} -\int _{\Omega }\Delta _q(f\cdot \nabla g)\cdot \Delta _q h\,\hbox {d}x\hbox {d}y&=-\int _{\Omega }\Delta _q(f^1\partial _1 g)\cdot \Delta _q h\,\hbox {d}x\hbox {d}y\\&\quad -\int _{\Omega }\Delta _q(f^2\partial _2 g)\cdot \Delta _q h\,\hbox {d}x\hbox {d}y\\&\triangleq P+Q. \end{aligned} \end{aligned}$$
For P, by Bony’s decomposition, we can divide it into the following three terms,
$$\begin{aligned} P&=-\int _{\Omega }\Delta _q(f^1 \partial _1 g)\cdot \Delta _q h\,\hbox {d}x\hbox {d}y\nonumber \\&=-\sum _{|k-q|\le 2}\int _{\Omega }\Delta _q(S_{k-1}f^1 \Delta _k\partial _1 g)\cdot \Delta _q h\,\hbox {d}x\hbox {d}y\nonumber \\&\quad -\sum _{|k-q|\le 2}\int _{\Omega }\Delta _q(\Delta _kf^1 S_{k-1}\partial _1 g)\cdot \Delta _q h\,\hbox {d}x\hbox {d}y\nonumber \\&\quad -\sum _{k\ge q-1}\sum _{|k-l|\le 1}\int _{\Omega }\Delta _q(\Delta _kf^1 \Delta _l\partial _1 g)\cdot \Delta _q h\,\hbox {d}x\hbox {d}y\nonumber \\ \triangleq&P_1+P_2+P_3. \end{aligned}$$
For \(P_1\), using decomposition (2.3), we can rewrite it as
$$\begin{aligned} \begin{aligned} P_1&=-\sum _{|k-q | \le 2} \int _{\Omega } \Delta _{q}\left( S_{k-1} {\widetilde{f}}^{1} \partial _{1} \Delta _{k} {\widetilde{g}}\right) \cdot \Delta _{q} {\bar{h}}\,\hbox {d}x \hbox {d}y\\&\quad -\sum _{|k-q | \le 2} \int _{\Omega } \Delta _{q}\left( S_{k-1} {\widetilde{f}}^{1} \partial _{1} \Delta _{k} {\widetilde{g}}\right) \cdot \Delta _{q} {\widetilde{h}}\,\hbox {d}x \hbox {d}y\\&\quad -\sum _{|k-q | \le 2} \int _{\Omega } \Delta _{q}\left( S_{k-1} {\bar{f}}^{1} \partial _{1} \Delta _{k} {\widetilde{g}}\right) \cdot \Delta _{q} {\bar{h}}\,\hbox {d}x \hbox {d}y\\&\quad -\sum _{|k-q | \le 2} \int _{\Omega } \Delta _{q}\left( S_{k-1} {\bar{f}}^{1} \partial _{1} \Delta _{k} {\widetilde{g}}\right) \cdot \Delta _{q} {\widetilde{h}}\,\hbox {d}x \hbox {d}y\\&\triangleq P_{11}+P_{12}+P_{13}+P_{14}. \end{aligned} \end{aligned}$$
For \(P_{11}\), by anisotropic Hölder inequality and Poincaré inequality (2.5),
$$\begin{aligned} \begin{aligned} P_{11}&=-\sum _{| k-q | \le 2} \int _{\Omega }\Delta _q\left( S_{k-1} {\tilde{f}}^{1} \partial _{1} \Delta _{k} {\widetilde{g}}\right) \cdot \Delta _{q} {\bar{h}}\,\hbox {d}x \hbox {d}y \\&\le C \sum _{| k-q | \le 2}\left\| S_{k-1} {\widetilde{f}}^{1}\right\| _{L^\infty _yL_{x}^{2}}\left\| \partial _{1} \Delta _k{\widetilde{g}}\right\| _{L^{2}} \left\| \Delta _{q} {\bar{h}}\right\| _{L_{y}^{2}} \\&\le C\sum _{| k-q | \le 2}\left\| S_{k-1} {\widetilde{f}}^{1}\right\| _{L^2}^{\frac{1}{2}}\left\| S_{k-1}\partial _2 {\widetilde{f}}^{1}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _{1} \Delta _k{\widetilde{g}}\right\| _{L^{2}} \left\| \Delta _{q} {\bar{h}}\right\| _{L_{y}^{2}} \\&\le C\sum _{| k-q | \le 2}\left\| \partial _1S_{k-1} {\widetilde{f}}^{1}\right\| _{L^2}^{\frac{1}{2}}\left\| S_{k-1}\partial _1\partial _2 {\widetilde{f}}^{1}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _{1} \Delta _k{\widetilde{g}}\right\| _{L^{2}} \left\| \Delta _{q} {\bar{h}}\right\| _{L_{y}^{2}} \\&\le C2^{-2qs}b_q\left\| \partial _1{f}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _1\partial _2 {f}\right\| _{L^2}^{\frac{1}{2}} \left\| \partial _{1} {g}\right\| _{H^s} \left\| h\right\| _{H^s}. \end{aligned} \end{aligned}$$
For \(P_{12}\), along the same method,
$$\begin{aligned} P_{12}&=-\sum _{| k-q | \le 2} \int _{\Omega }\Delta _q\left( S_{k-1} {\tilde{f}}^{1} \partial _{1} \Delta _{k} {\widetilde{g}}\right) \cdot \Delta _{q} {\widetilde{h}}\,\hbox {d}x \hbox {d}y \\&\le C \sum _{| k-q | \le 2}\left\| S_{k-1} {\widetilde{f}}^{1}\right\| _{L^\infty _yL_{x}^{2}}\left\| \partial _{1} \Delta _k{\widetilde{g}}\right\| _{L^{2}} \left\| \Delta _{q} {\widetilde{h}}\right\| _{L^\infty _xL_{y}^{2}} \\&\le C\sum _{| k-q | \le 2}\left\| S_{k-1} {\widetilde{f}}^{1}\right\| _{L^2}^{\frac{1}{2}}\left\| S_{k-1}\partial _2 {\widetilde{f}}^{1}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _{1} \Delta _k{\widetilde{g}}\right\| _{L^{2}} \left\| \Delta _{q} {\widetilde{h}}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \Delta _{q}\partial _1 {\widetilde{h}}\right\| _{L^{2}}^{\frac{1}{2}} \\&\le C\sum _{| k-q | \le 2}\left\| S_{k-1} {\widetilde{f}}^{1}\right\| _{L^2}^{\frac{1}{2}}\left\| S_{k-1}\partial _1\partial _2 {\widetilde{f}}^{1}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _{1} \Delta _k{\widetilde{g}}\right\| _{L^{2}} \left\| \Delta _{q} {\widetilde{h}}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \Delta _{q}\partial _1 {\widetilde{h}}\right\| _{L^{2}}^{\frac{1}{2}} \\&\le C2^{-2qs}b_q\left\| {f}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _1\partial _2 {f}\right\| _{L^2}^{\frac{1}{2}} \left\| \partial _{1} {g}\right\| _{H^s} \left\| h\right\| _{H^s}^{\frac{1}{2}}\left\| \partial _1 h\right\| _{H^s}^{\frac{1}{2}}. \end{aligned}$$
According to the duality property of the operator \(\Delta _q\),
$$\begin{aligned} \begin{aligned} P_{13}&=-\sum _{|k-q | \le 2} \int _{\Omega } \Delta _{q}\left( S_{k-1} {\bar{f}}^{1} \partial _{1} \Delta _{k} {\widetilde{g}}\right) \cdot \Delta _{q} {\bar{h}}\,\hbox {d}x \hbox {d}y\\&=-\sum _{|k-q | \le 2} \int _{\Omega } \left( S_{k-1} {\bar{f}}^{1} \partial _{1} \Delta _{k} {\widetilde{g}}\right) \cdot \Delta _{q}^2 {\bar{h}}\,\hbox {d}x \hbox {d}y\\&=-\sum _{|k-q | \le 2} \int _{{{\mathbb {R}}}}S_{k-1} {\bar{f}}^{1}\Delta _{q}^2 {\bar{h}}\cdot \bigg (\int _{{\mathbb {T}}} \partial _{1} \Delta _{k} {\widetilde{g}}(x,y)~\hbox {d}x\bigg )\,\hbox {d}y\\&=0. \end{aligned} \end{aligned}$$
Also by anisotropic Hölder inequality, interpolation (2.6) and Poincaré inequality (2.5),
$$\begin{aligned} \begin{aligned} P_{14}&=-\sum _{| k-q | \le 2} \int _{\Omega }\Delta _q\left( S_{k-1} {\bar{f}}^{1} \partial _{1} \Delta _{k} {\widetilde{g}}\right) \cdot \Delta _{q} {\widetilde{h}}\,\hbox {d}x \hbox {d}y \\&\le C \sum _{| k-q | \le 2}\left\| S_{k-1} {\bar{f}}^{1}\right\| _{L^\infty _y}\left\| \partial _{1} \Delta _k{\widetilde{g}}\right\| _{L^{2}} \left\| \Delta _{q} {\widetilde{h}}\right\| _{L^{2}} \\&\le C\sum _{| k-q | \le 2}\left\| S_{k-1} {\bar{f}}^{1}\right\| _{L^2}^{\frac{1}{2}}\left\| S_{k-1}\partial _2 {\bar{f}}^{1}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _{1} \Delta _k{\widetilde{g}}\right\| _{L^{2}} \left\| \partial _1\Delta _{q} {\widetilde{h}}\right\| _{L_2} \\&\le C2^{-2qs}b_q\left\| {f}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _2 {f}\right\| _{L^2}^{\frac{1}{2}} \left\| \partial _{1} {g}\right\| _{H^s} \left\| \partial _1h\right\| _{H^s}. \end{aligned} \end{aligned}$$
Combining the estimates of \(P_{11}-P_{14}\), we obtain the estimate for \(P_1\) that,
$$\begin{aligned} \begin{aligned} P_{1}&\le C2^{-2qs}b_q(\left\| \partial _1{f}\right\| _{L^2}\left\| \partial _1\partial _2 {f}\right\| _{L^2} \left\| h\right\| _{H^s}^2 +\left\| {f}\right\| _{L^2}^2\left\| \partial _1\partial _2 {f}\right\| _{L^2}^2 \left\| h\right\| _{H^s}^2 )\\&\quad +C\varepsilon _02^{-2qs}b_q(\left\| \partial _{1} {g}\right\| _{H^s}^2+ \left\| \partial _1 h\right\| _{H^s}^2)\\&\quad +2^{-2qs}b_q\left\| {f}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _2 {f}\right\| _{L^2}^{\frac{1}{2}} (\left\| \partial _{1} {g}\right\| _{H^s}^2+ \left\| \partial _1h\right\| _{H^s}^2). \end{aligned} \end{aligned}$$
Then, we estimate \(P_2\); also, by decomposition (2.3), we can write \(P_2\) into the following three terms,
$$\begin{aligned} P_2&=-\sum _{|k-q|\le 2}\int _{\Omega }\Delta _q(\Delta _kf^1 S_{k-1}\partial _1 g)\cdot \Delta _q h\,\hbox {d}x\hbox {d}y\\&=-\sum _{|k-q | \le 2} \int _{\Omega } \Delta _{q}\left( \Delta _{k}{\widetilde{f}}^{1} \partial _{1} S_{k-1} {g}\right) \cdot \Delta _{q} {h}\,\hbox {d}x \hbox {d}y\\&\quad -\sum _{|k-q | \le 2} \int _{\Omega } \Delta _{q}\left( \Delta _{k}{\bar{f}}^{1} \partial _{1} S_{k-1} {\widetilde{g}}\right) \cdot \Delta _{q} {\bar{h}}\,\hbox {d}x \hbox {d}y\\&\quad -\sum _{|k-q | \le 2} \int _{\Omega } \Delta _{q}\left( \Delta _{k}{\bar{f}}^{1} \partial _{1} S_{k-1} {\widetilde{g}}\right) \cdot \Delta _{q} {\widetilde{h}}\,\hbox {d}x \hbox {d}y\\&\triangleq P_{21}+P_{22}+P_{23}. \end{aligned}$$
We write that owing to anisotropic Hölder inequality, interpolation (2.6) and Poincaré inequality (2.5),
$$\begin{aligned} P_{21}&\le C \sum _{| k-q | \le 2}\left\| \Delta _k {\widetilde{f}}^{1}\right\| _{L^\infty _xL_{y}^{2}}\left\| \partial _{1} S_{k-1}{g}\right\| _{L^{\infty }_yL^2_x} \left\| \Delta _{q} {h}\right\| _{L^{2}} \\&\le C\sum _{| k-q | \le 2}\left\| \Delta _k {\widetilde{f}}^{1}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _1\Delta _k {\widetilde{f}}^{1}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _{1} S_{k-1}{g}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \partial _{1}\partial _2 S_{k-1}{g}\right\| _{L^2}^{\frac{1}{2}} \left\| \Delta _{q} {h}\right\| _{L^{2}} \\&\le C\sum _{| k-q | \le 2} \left\| \partial _{1} {g}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \partial _{1}\partial _2 {g}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _1\Delta _k {\widetilde{f}}^{1}\right\| _{L^2} \left\| \Delta _{q} {h}\right\| _{L^{2}}\\&\le C2^{-2qs}b_q\left\| \partial _1{g}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _1\partial _2 {g}\right\| _{L^2}^{\frac{1}{2}} \left\| \partial _{1} {f}\right\| _{H^s} \left\| h\right\| _{H^s}. \end{aligned}$$
Similar as \(P_{13}\), it is easy to see that \(P_{22}=0\). To bound \(P_{23}\), we write that
$$\begin{aligned} P_{23}&=\sum _{| k-q | \le 2} \int _{\Omega }\Delta _q\left( \Delta _{k}{\bar{f}}^{1} S_{k-1}\partial _{1} {\widetilde{g}}\right) \cdot \Delta _{q} {\widetilde{h}}~\hbox {d}x \hbox {d}y \\&\le C \sum _{| k-q | \le 2}\left\| \Delta _k {\bar{f}}^{1}\right\| _{L^2_y}\left\| \partial _{1} S_{k-1}{\widetilde{g}}\right\| _{L^{\infty }_yL^{2}_x} \left\| \Delta _{q} {\widetilde{h}}\right\| _{L^{2}} \\&\le C\sum _{| k-q | \le 2}\left\| \Delta _k {\bar{f}}^{1}\right\| _{L^2_y}\left\| \partial _{1} S_{k-1}{\widetilde{g}}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _{1}\partial _2 S_{k-1}{\widetilde{g}}\right\| _{L^{2}}^{\frac{1}{2}} \left\| \partial _1\Delta _{q} {\widetilde{h}}\right\| _{L^2} \\&\le C2^{-2qs}b_q\left\| {\partial _1g}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _1\partial _2 {g}\right\| _{L^2}^{\frac{1}{2}} \left\| {f}\right\| _{H^s} \left\| \partial _1h\right\| _{H^s}. \end{aligned}$$
Making use of Young’s inequality, we deduce the estimate of \(P_2\) that
$$\begin{aligned} \begin{aligned} P_{2}&\le C2^{-2qs}b_q\left\| \partial _1{g}\right\| _{L^2}\left\| \partial _1\partial _2 {g}\right\| _{L^2}( \left\| {f}\right\| _{H^s}^2+ \left\| h\right\| _{H^s}^2)\\&\quad + C\varepsilon _02^{-2qs}b_q( \left\| \partial _1{f}\right\| _{H^s}^2+ \left\| \partial _1h\right\| _{H^s}^2). \end{aligned} \end{aligned}$$
For \(P_3\), first we use decomposition (2.3),
$$\begin{aligned} P_3&=-\sum _{k\ge q-1}\sum _{|k-l|\le 1}\int _{\Omega }\Delta _q(\Delta _kf^1 \Delta _l\partial _1 g)\cdot \Delta _q h\,\hbox {d}x \hbox {d}y\\&=-\sum _{k\ge q-1}\sum _{|k-l|\le 1} \int _{\Omega } \Delta _{q}\left( \Delta _{k}{\widetilde{f}}^{1} \partial _{1} \Delta _l {g}\right) \cdot \Delta _{q} {h}\,\hbox {d}x \hbox {d}y\\&\quad -\sum _{k\ge q-1}\sum _{|k-l|\le 1} \int _{\Omega } \Delta _{q}\left( \Delta _{k}{\bar{f}}^{1} \partial _{1} \Delta _l {\widetilde{g}}\right) \cdot \Delta _{q} {\bar{h}}\,\hbox {d}x \hbox {d}y\\&\quad -\sum _{k\ge q-1}\sum _{|k-l|\le 1} \int _{\Omega } \Delta _{q}\left( \Delta _{k}{\bar{f}}^{1} \partial _{1} \Delta _l {\widetilde{g}}\right) \cdot \Delta _{q} {\widetilde{h}}\,\hbox {d}x \hbox {d} y\\&\triangleq P_{31}+P_{32}+P_{33}. \end{aligned}$$
The term \(P_{31}\) can be handled in the same way as \(P_{21}\) and utilizing Bernstein inequality,
$$\begin{aligned} P_{31}&\le C \sum _{k\ge q-1}\sum _{|k-l|\le 1}\left\| \Delta _k {\widetilde{f}}^{1}\right\| _{L^\infty _xL_{y}^{2}}\left\| \partial _{1} \Delta _l{g}\right\| _{L^2} \left\| \Delta _{q} {h}\right\| _{L^\infty _yL^{2}_x} \\&\le C\sum _{k\ge q-1}\sum _{|k-l|\le 1}\left\| \Delta _k {\widetilde{f}}^{1}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _1\Delta _k {\widetilde{f}}^{1}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _{1} \Delta _l{g}\right\| _{L^{2}} \left\| \Delta _{q} {h}\right\| _{L^{2}}^{\frac{1}{2}} \left\| \partial _2\Delta _{q} {h}\right\| _{L^{2}}^{\frac{1}{2}}\\&\le C\sum _{k\ge q-1}2^{\frac{q}{2}}2^{-\frac{k}{2}}\left\| \partial _1\Delta _k {\widetilde{f}}^{1}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _1\nabla \Delta _k {\widetilde{f}}^{1}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _{1} \Delta _k{g}\right\| _{L^{2}} \left\| \Delta _{q} {h}\right\| _{L^{2}} \\&\le C2^{-2qs}b_q\left\| \partial _1{f}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _1\nabla {f}^1\right\| _{L^2}^{\frac{1}{2}} \left\| \partial _{1} {g}\right\| _{H^s} \left\| h\right\| _{H^s}\\&\le C2^{-2qs}b_q\left\| \partial _1{f}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _1\partial _2{f}\right\| _{L^2}^{\frac{1}{2}} \left\| \partial _{1} {g}\right\| _{H^s} \left\| h\right\| _{H^s}, \end{aligned}$$
where we have used the relation \(\nabla f^1=(\partial _1f^1,\partial _2f^1)=(-\partial _2f^2,\partial _2f^1)\) in the last step.
Similar as \(P_{13}\), it is easy to check that \(P_{32}=0\). To bound \(P_{33}\), we write that
$$\begin{aligned} \begin{aligned} P_{33}&=-\sum _{k\ge q-1}\sum _{|k-l|\le 1} \int _{\Omega } \Delta _{q}\left( \Delta _{k}{\bar{f}}^{1} \partial _{1} \Delta _l {\widetilde{g}}\right) \cdot \Delta _{q} {\widetilde{h}}\,\hbox {d}x \hbox {d}y\\&\le C \sum _{k\ge q-1}\sum _{|k-l|\le 1}\left\| \Delta _k {\bar{f}}^{1}\right\| _{L_{y}^{2}}\left\| \partial _{1} \Delta _l{\widetilde{g}}\right\| _{L^2} \left\| \Delta _{q} {\widetilde{h}}\right\| _{L^\infty _yL^{2}_x} \\&\le C \sum _{k\ge q-1}\sum _{|k-l|\le 1}\left\| \Delta _k {\bar{f}}^{1}\right\| _{L_{y}^{2}}^{\frac{1}{2}} 2^{-\frac{k}{2}}\left\| \nabla \Delta _k {\bar{f}}^{1}\right\| _{L_{y}^{2}}^{\frac{1}{2}} \left\| \partial _{1} \Delta _l{\widetilde{g}}\right\| _{L^2} 2^{\frac{q}{2}}\left\| \Delta _{q} {\widetilde{h}}\right\| _{L^2_yL^{2}_x} \\&\le C \sum _{k\ge q-1}\sum _{|k-l|\le 1}2^{\frac{q}{2}-\frac{k}{2}}\left\| \Delta _k {\bar{f}}^{1}\right\| _{L_{y}^{2}}^{\frac{1}{2}} \left\| \partial _1\Delta _k {\bar{f}}^{1}\right\| _{L_{y}^{2}}^{\frac{1}{2}} \left\| \partial _{1} \Delta _l{\widetilde{g}}\right\| _{L^2} \left\| \Delta _{q} {\widetilde{h}}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _1\Delta _{q} {\widetilde{h}}\right\| _{L^2}^{\frac{1}{2}} \\&\quad +C \sum _{k\ge q-1}\sum _{|k-l|\le 1}2^{\frac{q}{2}-\frac{k}{2}}\left\| \Delta _k {\bar{f}}^{1}\right\| _{L_{y}^{2}}^{\frac{1}{2}} \left\| \partial _2\Delta _k {\bar{f}}^{1}\right\| _{L_{y}^{2}}^{\frac{1}{2}} \left\| \partial _{1} \Delta _l{\widetilde{g}}\right\| _{L^2} \left\| \partial _1\Delta _{q} {\widetilde{h}}\right\| _{L^2} \\&\le C2^{-2qs}b_q(\left\| {f}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _1 {f}^1\right\| _{L^2}^{\frac{1}{2}} \left\| \partial _{1} {g}\right\| _{H^s} \left\| h\right\| _{H^s}^{\frac{1}{2}}\left\| \partial _1h\right\| _{H^s}^{\frac{1}{2}}\\&\quad +\left\| {f}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _2 {f}^1\right\| _{L^2}^{\frac{1}{2}} \left\| \partial _{1} {g}\right\| _{H^s} \left\| \partial _1h\right\| _{H^s}). \end{aligned} \end{aligned}$$
Combining with Young’s inequality, we obtain the estimate for \(P_3\) that
$$\begin{aligned} \begin{aligned} P_{3}&\le C2^{-2qs}b_q(\left\| {f}\right\| _{L^2}^{2}\left\| \partial _1 {f}^1\right\| _{L^2}^{2} \left\| h\right\| _{H^s}^{2}+\left\| \partial _1{f}\right\| _{L^2}\left\| \partial _1\partial _2 {f}^1\right\| _{L^2} \left\| h\right\| _{H^s}^2)\\&\quad +C2^{-2qs}b_q\left\| {f}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _2 {f}^1\right\| _{L^2}^{\frac{1}{2}} (\left\| \partial _{1} {g}\right\| _{H^s}^2+ \left\| \partial _1h\right\| _{H^s}^2) \\&\quad +C\varepsilon _0 2^{-2qs}b_q( \left\| \partial _{1} {g}\right\| _{H^s}^2+\left\| \partial _1h\right\| _{H^s}^2). \end{aligned} \end{aligned}$$
Next, we estimate Q; first, we divide it into three parts,
$$\begin{aligned} \begin{aligned} -\int _{{{\mathbb {R}}}^2}\Delta _q(u^2 \partial _2 f)\cdot \Delta _q f\,\hbox {d}x\hbox {d}y =Q_1+Q_2+Q_3, \end{aligned} \end{aligned}$$
(8.1)
with
$$\begin{aligned} Q_1= & {} -\sum _{|k-q|\le 2}\int _{\Omega }\Delta _q(S_{k-1}f^2 \Delta _k\partial _2 g)\cdot \Delta _q h\,\hbox {d}x\hbox {d}y, \\ Q_2= & {} -\sum _{|k-q|\le 2}\int _{\Omega }\Delta _q(\Delta _kf^2 S_{k-1}\partial _2 g)\cdot \Delta _q h\,\hbox {d}x\hbox {d}y \end{aligned}$$
and
$$\begin{aligned} Q_3=-\sum _{k\ge q-1}\sum _{|k-l|\le 1}\int _{\Omega }\Delta _q(\Delta _kf^2 \Delta _l\partial _2 g)\cdot \Delta _q h\,\hbox {d}x\hbox {d}y. \end{aligned}$$
Because f satisfies the divergence-free condition and according to property (2.7), we can rewrite \(Q_1\) as
$$\begin{aligned} Q_1&=-\sum _{|k-q|\le 2}\int _{\Omega }\Delta _q(S_{k-1}{\widetilde{f}}^2 \Delta _k\partial _2 g)\cdot \Delta _q h\,\hbox {d}x\hbox {d}y\\&=-\sum _{|k-q|\le 2}\int _{\Omega }[\Delta _q, S_{k-1}{\widetilde{f}}^2\partial _2]\Delta _k g\cdot \Delta _q h\,\hbox {d}x\hbox {d}y\\&\quad -\sum _{|k-q|\le 2}\int _{\Omega }({ S_{k-1}{\widetilde{f}}^2-S_q{\widetilde{f}}^2})\partial _2\Delta _q \Delta _kg\cdot \Delta _q h\,\hbox {d}x\hbox {d}y\\&\quad -\int _{\Omega }{S}_{q}{\widetilde{f}}^2\partial _2\Delta _q g\cdot \Delta _q h\,\hbox {d}x\hbox {d}y\\&\triangleq Q_{11}+Q_{12}+Q_{13}. \end{aligned}$$
For \(Q_{11}\), we decompose it into the following four terms,
$$\begin{aligned} Q_{11}&=-\sum _{|k-q|\le 2}\int _{\Omega }[\Delta _q, S_{k-1}{\widetilde{f}}^2\partial _2]\Delta _k {\bar{g}} \cdot \Delta _q {\bar{h}}\,\hbox {d}x\hbox {d}y\\&\quad -\sum _{|k-q|\le 2}\int _{\Omega }[\Delta _q, S_{k-1}{\widetilde{f}}^2\partial _2]\Delta _k {\bar{g}} \cdot \Delta _q {\widetilde{h}}\,\hbox {d}x\hbox {d}y\\&\quad -\sum _{|k-q|\le 2}\int _{\Omega }[\Delta _q, S_{k-1}{\widetilde{f}}^2\partial _2]\Delta _k {\widetilde{g}} \cdot \Delta _q {\bar{h}}\,\hbox {d}x\hbox {d}y\\&\quad -\sum _{|k-q|\le 2}\int _{\Omega }[\Delta _q, S_{k-1}{\widetilde{f}}^2\partial _2]\Delta _k {\widetilde{g}} \cdot \Delta _q {\widetilde{h}}\,\hbox {d}x\hbox {d}y,\\&\triangleq Q_{111}+Q_{112}+Q_{113}+Q_{114}, \end{aligned}$$
where \([X,Y]\triangleq XY-YX\) defining the standard commutator.
According to the definition of decomposition (2.3), it is not difficult to check that \(Q_{111}=0\). For \(Q_{112}\), according to the definition of \(\Delta _q\),
$$\begin{aligned}{}[\Delta _q, S_{k-1}{{\widetilde{f}}^2\partial _2}]\Delta _k {\bar{g}}&=\int _{\Omega }h_q({\mathbf {x}}-{\mathbf {x}}')(S_{k-1}{\widetilde{f}}^2({\mathbf {x}}')\partial _2\Delta _k {\bar{g}}({\mathbf {x}}'))\,\hbox {d}{\mathbf {x}}'\\&\quad -S_{k-1}{\widetilde{f}}^2({\mathbf {x}})\int _{\Omega }h_q({\mathbf {x}}-{\mathbf {x}}')\partial _2\Delta _k {\bar{g}}({\mathbf {x}}')\,\hbox {d}{\mathbf {x}}'\\&=\int _{\Omega }h_q({\mathbf {x}}-{\mathbf {x}}')(S_{k-1}{\widetilde{f}}^2({\mathbf {x}}')-S_{k-1}{\widetilde{f}}^2({\mathbf {x}}))\partial _2\Delta _k {\bar{g}}({\mathbf {x}}')\,\hbox {d}{\mathbf {x}}'\\&=\int _{\Omega }h_q({\mathbf {x}}-{\mathbf {x}}')\int _0^1({\mathbf {x}}'-{\mathbf {x}})\cdot \nabla S_{k-1}{\widetilde{f}}^2(s\mathbf {x'}\\&\quad +(1-s){\mathbf {x}})~\hbox {d}s\partial _2\Delta _k {\bar{g}}({\mathbf {x}}')\,\hbox {d}{\mathbf {x}}'\\&=\int _{\Omega }\int _0^1h_q({\mathbf {z}}){\mathbf {z}}\cdot \nabla S_{k-1}{\widetilde{f}}^2({\mathbf {x}}-s{\mathbf {z}})\partial _2\Delta _k {\bar{g}}({\mathbf {x}}-{\mathbf {z}})\,\hbox {d}s \hbox {d}{\mathbf {z}}, \end{aligned}$$
where \(h_q\) is defined as in (2.2).
Making use of the anisotropic Hölder inequality, interpolation (2.6), Poincaré inequality (2.5) and Bernstein inequality,
$$\begin{aligned} \begin{aligned} Q_{112}&=-\sum _{|k-q|\le 2}\int _{\Omega }[\Delta _q, S_{k-1}{\widetilde{f}}^2\partial _2]\Delta _k {\bar{g}} \cdot \Delta _q {\widetilde{h}}\,\hbox {d}x\hbox {d}y \\&\le C \sum _{| k-q | \le 2}2^{-q}\left\| S_{k-1}\nabla {\widetilde{f}}^{2}\right\| _{L^\infty _yL^2_x}\left\| \partial _{2} \Delta _k{\bar{g}}\right\| _{L^{2}_y} \left\| \Delta _{q} {\widetilde{h}}\right\| _{L^{2}} \\&\le C\sum _{| k-q | \le 2}2^{k-q}\left\| S_{k-1}\nabla {\widetilde{f}}^{2}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _2S_{k-1}\nabla {\widetilde{f}}^{2}\right\| _{L^2}^{\frac{1}{2}}\left\| \Delta _k{g}\right\| _{L^{2}} \left\| \partial _1\Delta _{q} {\widetilde{h}}\right\| _{L^{2}} \\&\le C2^{-2qs}b_q\left\| {\partial _1f}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _1\partial _2 {f}\right\| _{L^2}^{\frac{1}{2}} \left\| {g}\right\| _{H^s} \left\| \partial _1h\right\| _{H^s}, \end{aligned} \end{aligned}$$
where we have used the relation \(\nabla {\widetilde{f}}^2=(\partial _1{\widetilde{f}}^1,\partial _2{\widetilde{f}}^2)=(\partial _1{\widetilde{f}}^1,-\partial _1{\widetilde{f}}^1)\) in the last step.
The similar conclusion can also be drawn for \(Q_{113}\) and \(Q_{114}\) that
$$\begin{aligned} \begin{aligned} Q_{113}&\le C2^{-2qs}b_q\left\| {\partial _1f}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _1\partial _2 {f}\right\| _{L^2}^{\frac{1}{2}} \left\| \partial _1{g}\right\| _{H^s} \left\| h\right\| _{H^s},\\ Q_{114}&\le C2^{-2qs}b_q\left\| {\partial _1f}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _1\partial _2 {f}\right\| _{L^2}^{\frac{1}{2}} \left\| \partial _1{g}\right\| _{H^s} \left\| h\right\| _{H^s}. \end{aligned} \end{aligned}$$
Next, we deal with \(Q_{12}\), also making use decomposition (2.3),
$$\begin{aligned} Q_{12}&=-\sum _{|k-q|\le 2}\int _{\Omega }({ S_{k-1}{\widetilde{f}}^2-S_q{\widetilde{f}}^2})\partial _2\Delta _q \Delta _kg\cdot \Delta _q h\,\hbox {d}x\hbox {d}y\\&=-\sum _{|k-q|\le 2}\int _{\Omega }({ S_{k-1}{\widetilde{f}}^2-S_q{\widetilde{f}}^2})\partial _2\Delta _q \Delta _k{\bar{g}}\cdot \Delta _q {\bar{h}}\,\hbox {d}x\hbox {d}y\\&\quad -\sum _{|k-q|\le 2}\int _{\Omega }({ S_{k-1}{\widetilde{f}}^2-S_q{\widetilde{f}}^2})\partial _2\Delta _q \Delta _k{\bar{g}}\cdot \Delta _q {\widetilde{h}}\,\hbox {d}x\hbox {d}y\\&\quad -\sum _{|k-q|\le 2}\int _{\Omega }({ S_{k-1}{\widetilde{f}}^2-S_q{\widetilde{f}}^2})\partial _2\Delta _q \Delta _k{\widetilde{g}}\cdot \Delta _q {\bar{h}}\,\hbox {d}x\hbox {d}y\\&\quad -\sum _{|k-q|\le 2}\int _{\Omega }({ S_{k-1}{\widetilde{f}}^2-S_q{\widetilde{f}}^2})\partial _2\Delta _q \Delta _k{\widetilde{g}}\cdot \Delta _q {\widetilde{h}}\,\hbox {d}x\hbox {d}y\\&\triangleq Q_{121}+Q_{122}+Q_{123}+Q_{124}. \end{aligned}$$
The same reasoning as \(Q_{111}\) implies that \(Q_{121}=0\). For \(Q_{122}\), by the anisotropic Hölder inequality, Bernstein inequality, interpolation (2.6) and Poincaré inequality (2.5),
$$\begin{aligned} \begin{aligned} Q_{122}&\le C \sum _{| k-q | \le 2}2^{-q}\left\| \nabla \Delta _k {\widetilde{f}}^{2}\right\| _{L^\infty _yL^2_x}\left\| \partial _{2} \Delta _k{\bar{g}}\right\| _{L^{2}_y} \left\| \Delta _{q} {\widetilde{h}}\right\| _{L^{2}} \\&\le C\sum _{| k-q | \le 2}2^{k-q}\left\| \nabla \Delta _k {\widetilde{f}}^{2}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _2\nabla \Delta _k {\widetilde{f}}^{2}\right\| _{L^2}^{\frac{1}{2}}\left\| \Delta _k{g}\right\| _{L^{2}} \left\| \partial _1\Delta _{q} {\widetilde{h}}\right\| _{L^{2}} \\&\le C2^{-2qs}b_q\left\| {\nabla f^2}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _2\nabla {f}^2\right\| _{L^2}^{\frac{1}{2}} \left\| {g}\right\| _{H^s} \left\| \partial _1h\right\| _{H^s}\\&\le C2^{-2qs}b_q\left\| {\partial _1f}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _1\partial _2 {f}\right\| _{L^2}^{\frac{1}{2}} \left\| {g}\right\| _{H^s} \left\| \partial _1h\right\| _{H^s}, \end{aligned} \end{aligned}$$
where we have used the relations \(\partial _2f^2=-\partial _1f^1\) and \(\nabla f^2=(\partial _1 f^2, \partial _2 f^2)=(\partial _1 f^2, -\partial _1f^1)\) in the last step.
Similar arguments apply to the terms \(Q_{123}\) and \(Q_{124}\); we can see that
$$\begin{aligned} \begin{aligned} Q_{123}&\le C2^{-2qs}b_q\left\| {\partial _1f}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _1\partial _2 {f}\right\| _{L^2}^{\frac{1}{2}} \left\| \partial _1{g}\right\| _{H^s} \left\| h\right\| _{H^s},\\ Q_{124}&\le C2^{-2qs}b_q\left\| {\partial _1f}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _1\partial _2 {f}\right\| _{L^2}^{\frac{1}{2}} \left\| \partial _1{g}\right\| _{H^s} \left\| h\right\| _{H^s}. \end{aligned} \end{aligned}$$
Gathering the estimates for \(Q_{11}\) and \(Q_{12}\), we get the bound for \(Q_{1}\) that
$$\begin{aligned} \begin{aligned} Q_{1}&\le C2^{-2qs}b_q\left\| {\partial _1f}\right\| _{L^2}\left\| \partial _1\partial _2 {f}\right\| _{L^2} (\left\| g\right\| _{H^s}^2+\left\| h\right\| _{H^s}^2) \\&\quad +C\varepsilon _02^{-2qs}b_q (\left\| \partial _1{g}\right\| _{H^s}^2+\left\| \partial _1{h}\right\| _{H^s}^2)\\&\quad -\int _{\Omega }{S}_{q}{\widetilde{f}}^2\partial _2\Delta _q g\cdot \Delta _q h\,\hbox {d}x\hbox {d}y. \end{aligned} \end{aligned}$$
Then, we estimate \(Q_2\). We first decompose \(Q_2\) as follows:
$$\begin{aligned} Q_2&=-\sum _{|k-q|\le 2}\int _{\Omega }\Delta _q(\Delta _kf^2 S_{k-1}\partial _2 g)\cdot \Delta _q h\,\hbox {d}x\hbox {d}y\\&=-\sum _{|k-q|\le 2}\int _{\Omega }\Delta _q(\Delta _k{\widetilde{f}}^2 S_{k-1}\partial _2 g)\cdot \Delta _q h\,\hbox {d}x\hbox {d}y\\&=-\sum _{|k-q|\le 2}\int _{\Omega }\Delta _q(\Delta _k{\widetilde{f}}^2 S_{k-1}\partial _2 {\bar{g}})\cdot \Delta _q {\bar{h}}\,\hbox {d}x\hbox {d}y\\&\quad -\sum _{|k-q|\le 2}\int _{\Omega }\Delta _q(\Delta _k{\widetilde{f}}^2 S_{k-1}\partial _2 {\bar{g}})\cdot \Delta _q {\widetilde{h}}\,\hbox {d}x\hbox {d}y\\&\quad -\sum _{|k-q|\le 2}\int _{\Omega }\Delta _q(\Delta _k{\widetilde{f}}^2 S_{k-1}\partial _2 {\widetilde{g}})\cdot \Delta _q {\bar{h}}\,\hbox {d}x\hbox {d}y\\&\quad -\sum _{|k-q|\le 2}\int _{\Omega }\Delta _q(\Delta _k{\widetilde{f}}^2 S_{k-1}\partial _2 {\widetilde{g}})\cdot \Delta _q {\widetilde{h}}\,\hbox {d}x\hbox {d}y\\&\triangleq Q_{21}+Q_{22}+Q_{23}+Q_{24}. \end{aligned}$$
Similar as \(Q_{111}\), we can check at once that \(Q_{21}=0\). Owing to anisotropic Hölder inequality, Bernstein inequality, interpolation (2.6) and Poincaré inequality (2.5), we write that
$$\begin{aligned} Q_{22}&=-\sum _{|k-q|\le 2}\int _{\Omega }\Delta _q(\Delta _k{\widetilde{f}}^2 S_{k-1}\partial _2 {\bar{g}})\cdot \Delta _q {\widetilde{h}}\,\hbox {d}x\hbox {d}y \\&\le C \sum _{| k-q | \le 2}\left\| \Delta _k {\widetilde{f}}^{2}\right\| _{L^{\infty }_yL^{2}_x}\left\| \partial _{2} S_{k-1}{\bar{g}}\right\| _{L^2_y} \left\| \Delta _{q} {\widetilde{h}}\right\| _{L^{2}} \\&\le C \sum _{| k-q | \le 2}\left\| \Delta _k {\widetilde{f}}^{2}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _2\Delta _k {\widetilde{f}}^{2}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _{2} S_{k-1}{\bar{g}}\right\| _{L^2_y} \left\| \Delta _{q} {\widetilde{h}}\right\| _{L^{2}} \\&\le C \sum _{| k-q | \le 2}\left\| \Delta _k {\widetilde{f}}^{2}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _1\Delta _k {\widetilde{f}}^{1}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _{2} S_{k-1}{\bar{g}}\right\| _{L^2_y} \left\| \partial _1\Delta _{q} {\widetilde{h}}\right\| _{L^{2}} \\&\le C2^{-2qs}b_q\left\| {\partial _2g}\right\| _{L^2} \left\| \partial _1{f}\right\| _{H^s} \left\| \partial _1h\right\| _{H^s}. \end{aligned}$$
Along the same way,
$$\begin{aligned} \begin{aligned} Q_{23}&=-\sum _{|k-q|\le 2}\int _{\Omega }\Delta _q(\Delta _k{\widetilde{f}}^2 S_{k-1}\partial _2 {\widetilde{g}})\cdot \Delta _q {\bar{h}}\,\hbox {d}x\hbox {d}y\\&\le C \sum _{| k-q | \le 2}\left\| \Delta _k {\widetilde{f}}^{2}\right\| _{L^{\infty }_yL^{2}_x}\left\| \partial _{2} S_{k-1}{\widetilde{g}}\right\| _{L^2} \left\| \Delta _{q} {\bar{h}}\right\| _{L^{2}_y} \\&\le C \sum _{| k-q | \le 2}\left\| \Delta _k {\widetilde{f}}^{2}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _2\Delta _k {\widetilde{f}}^{2}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _1\partial _{2} S_{k-1}{\widetilde{g}}\right\| _{L^2} \left\| \Delta _{q} {\bar{h}}\right\| _{L^{2}_y} \\&\le C \sum _{| k-q | \le 2}\left\| \partial _1\Delta _k {\widetilde{f}}^{2}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _1\Delta _k {\widetilde{f}}^{1}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _1\partial _{2} S_{k-1}{\widetilde{g}}\right\| _{L^2} \left\| \Delta _{q} {\bar{h}}\right\| _{L^{2}_y} \\&\le C2^{-2qs}b_q\left\| {\partial _1\partial _2g}\right\| _{L^2} \left\| \partial _1{f}\right\| _{H^s} \left\| h\right\| _{H^s}. \end{aligned} \end{aligned}$$
In the same manner, we can see that
$$\begin{aligned} \begin{aligned} Q_{24} \le C2^{-2qs}b_q\left\| {\partial _1\partial _2g}\right\| _{L^2} \left\| \partial _1{f}\right\| _{H^s} \left\| h\right\| _{H^s}. \end{aligned} \end{aligned}$$
Thus, we conclude that
$$\begin{aligned} \begin{aligned} Q_{2}&\le C2^{-2qs}b_q\left\| {\partial _1\partial _2g}\right\| _{L^2}^2\left\| h\right\| _{H^s}^2 + C\varepsilon _02^{-2qs}b_q\left\| \partial _1{f}\right\| _{H^s}^2\\&\quad +C2^{-2qs}b_q\left\| {\partial _2g}\right\| _{L^2} (\left\| \partial _1{f}\right\| _{H^s}^2+ \left\| \partial _1h\right\| _{H^s}^2). \end{aligned} \end{aligned}$$
Finally, we deal with \(Q_3\). Similar as \(Q_2\), we first divide it into the following four parts,
$$\begin{aligned} Q_3&=-\sum _{k\ge q-1}\sum _{|k-l|\le 1}\int _{\Omega }\Delta _q(\Delta _kf^2 \Delta _l\partial _2 g)\cdot \Delta _q h\,\hbox {d}x\hbox {d}y\\&=-\sum _{k\ge q-1}\sum _{|k-l|\le 1}\int _{\Omega }\Delta _q(\Delta _k{\widetilde{f}}^2 \Delta _l\partial _2 g)\cdot \Delta _q h\,\hbox {d}x\hbox {d}y\\&=-\sum _{k\ge q-1}\sum _{|k-l|\le 1}\int _{\Omega }\Delta _q(\Delta _k{\widetilde{f}}^2 \Delta _l\partial _2 {\bar{g}})\cdot \Delta _q {\bar{h}}\,\hbox {d}x\hbox {d}y\\&\quad -\sum _{k\ge q-1}\sum _{|k-l|\le 1}\int _{\Omega }\Delta _q(\Delta _k{\widetilde{f}}^2 \Delta _l\partial _2 {\bar{g}})\cdot \Delta _q {\widetilde{h}}\,\hbox {d}x\hbox {d}y\\&\quad -\sum _{k\ge q-1}\sum _{|k-l|\le 1}\int _{\Omega }\Delta _q(\Delta _k{\widetilde{f}}^2 \Delta _l\partial _2 {\widetilde{g}})\cdot \Delta _q {\bar{h}}\,\hbox {d}x\hbox {d}y\\&\quad -\sum _{k\ge q-1}\sum _{|k-l|\le 1}\int _{\Omega }\Delta _q(\Delta _k{\widetilde{f}}^2 \Delta _l\partial _2 {\widetilde{g}})\cdot \Delta _q {\widetilde{h}}\,\hbox {d}x\hbox {d}y\\&\triangleq Q_{31}+Q_{32}+Q_{33}+Q_{34}. \end{aligned}$$
A trivial verification shows that \(Q_{31}=0\). For \(Q_{32}\), we can handle it by
$$\begin{aligned} Q_{32}&=-\sum _{k\ge q-1}\sum _{|k-l|\le 1}\int _{\Omega }\Delta _q(\Delta _k{\widetilde{f}}^2 \Delta _l\partial _2 {\bar{g}})\cdot \Delta _q {\widetilde{h}}\,\hbox {d}x\hbox {d}y\\&\le C \sum _{k\ge q-1}\sum _{|k-l|\le 1}\left\| \Delta _k {\widetilde{f}}^{2}\right\| _{L^2}\left\| \partial _{2} \Delta _l{\bar{g}}\right\| _{L^2_y} \left\| \Delta _{q} {\widetilde{h}}\right\| _{L^\infty _yL^{2}_x} \\&\le C\sum _{k\ge q-1}\sum _{|k-l|\le 1}\left\| \Delta _k {\widetilde{f}}^{2}\right\| _{L^2}^{\frac{1}{2}}2^{-\frac{k}{2}}\left\| \nabla \Delta _k {\widetilde{f}}^{2}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _{2} \Delta _l{\bar{g}}\right\| _{L^{2}_y} 2^{\frac{q}{2}}\left\| \Delta _{q} {\widetilde{h}}\right\| _{L^{2}}\\&\le C\sum _{k\ge q-1}2^{\frac{q}{2}}2^{-\frac{k}{2}}\left\| \partial _1\Delta _k {\widetilde{f}}^{2}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _1\Delta _k {\widetilde{f}}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _{2} {g}\right\| _{L^{2}} \left\| \partial _1\Delta _{q} {\widetilde{h}}\right\| _{L^{2}} \\&\le C2^{-2qs}b_q\left\| \partial _2{g}\right\| _{L^2} \left\| \partial _{1} {f}\right\| _{H^s} \left\| \partial _1h\right\| _{H^s}, \end{aligned}$$
where we have used the relation \(\nabla {\widetilde{f}}^2=(\partial _1{\widetilde{f}}^2,\partial _2{\widetilde{f}}^2)=(\partial _1{\widetilde{f}}^2,-\partial _1{\widetilde{f}}^1)\).
\(Q_{33}\) can be bounded similarly,
$$\begin{aligned} Q_{33}= & {} -\sum _{k\ge q-1}\sum _{|k-l|\le 1}\int _{\Omega }\Delta _q(\Delta _k{\widetilde{f}}^2 \Delta _l\partial _2 {\widetilde{g}})\cdot \Delta _q {\bar{h}}\,\hbox {d}x\hbox {d}y\\\le & {} C \sum _{k\ge q-1}\sum _{|k-l|\le 1}\left\| \Delta _k {\widetilde{f}}^{2}\right\| _{L^2}\left\| \partial _{2} \Delta _l{\widetilde{g}}\right\| _{L^2} \left\| \Delta _{q} {\bar{h}}\right\| _{L^\infty _y} \\\le & {} C\sum _{k\ge q-1}\sum _{|k-l|\le 1}\left\| \Delta _k {\widetilde{f}}^{2}\right\| _{L^2}^{\frac{1}{2}}2^{-\frac{k}{2}}\left\| \nabla \Delta _k {\widetilde{f}}^{2}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _{2} \Delta _l{\widetilde{g}}\right\| _{L^{2}} 2^{\frac{q}{2}}\left\| \Delta _{q} {\bar{h}}\right\| _{L^{2}_y}\\\le & {} C\sum _{k\ge q-1}2^{\frac{q}{2}}2^{-\frac{k}{2}}\left\| \partial _1\Delta _k {\widetilde{f}}\right\| _{L^2}\left\| \partial _1\partial _{2} {\widetilde{g}}\right\| _{L^{2}} \left\| \Delta _{q} {\bar{h}}\right\| _{L^{2}_y} \\\le & {} C2^{-2qs}b_q\left\| \partial _1\partial _2{g}\right\| _{L^2} \left\| \partial _{1} {f}\right\| _{H^s} \left\| h\right\| _{H^s}. \end{aligned}$$
The same estimate remains valid for \(Q_{34}\) that
$$\begin{aligned} \begin{aligned} Q_{34}\le C2^{-2qs}b_q\left\| \partial _1\partial _2{g}\right\| _{L^2} \left\| \partial _{1} {f}\right\| _{H^s} \left\| h\right\| _{H^s}. \end{aligned} \end{aligned}$$
Using Young’s inequality, we can get the estimate for \(Q_3\) that
$$\begin{aligned} \begin{aligned} Q_{3}&\le C2^{-2qs}b_q\left\| \partial _1\partial _2{g}\right\| _{L^2}^2\left\| h\right\| _{H^s}^2+C\varepsilon _02^{-2qs}b_q\left\| \partial _{1} {f}\right\| _{H^s}^2\\&\quad +C2^{-2qs}b_q\left\| \partial _2{g}\right\| _{L^2} (\left\| \partial _{1} {f}\right\| _{H^s}^2+ \left\| \partial _1h\right\| _{H^s}^2). \end{aligned} \end{aligned}$$
Taking the estimates for \(Q_{1}-Q_{3}\), \(P_{1}-P_{3}\) into account, finally we can obtain
$$\begin{aligned} \begin{aligned}&-\int _{\Omega }\Delta _q(f\cdot \nabla g)\cdot \Delta _q h\,\hbox {d}x\hbox {d}y\\&\quad \le C2^{-2qs}b_q(\left\| \partial _1{f}\right\| _{L^2}\left\| \partial _1\partial _2 {f}\right\| _{L^2}+\left\| {f}\right\| _{L^2}^2\left\| \partial _1 {f}\right\| _{L^2}^2+\left\| {f}\right\| _{L^2}^2\left\| \partial _1\partial _2 {f}\right\| _{L^2}^2\\&\qquad +\left\| \partial _1{g}\right\| _{L^2}\left\| \partial _1\partial _2 {g}\right\| _{L^2}+\left\| {\partial _1\partial _2g}\right\| _{L^2}^2)\times (\left\| {f}\right\| _{H^s}^2+ \left\| g\right\| _{H^s}^2+\left\| h\right\| _{H^s}^2 ) \\&\qquad +2^{-2qs}b_q(\left\| {f}\right\| _{L^2}^{\frac{1}{2}}\left\| \partial _2 {f}\right\| _{L^2}^{\frac{1}{2}}+\left\| {\partial _2g}\right\| _{L^2})\times (\left\| \partial _1{f}\right\| _{H^s}^2+\left\| \partial _{1} {g}\right\| _{H^s}^2+ \left\| \partial _1h\right\| _{H^s}^2)\\&\qquad +C\varepsilon _02^{-2qs}b_q(\left\| \partial _1{f}\right\| _{H^s}^2+\left\| \partial _{1} {g}\right\| _{H^s}^2+ \left\| \partial _1 h\right\| _{H^s}^2)-\int _{\Omega }{S}_{q}{\widetilde{f}}^2\partial _2\Delta _q g\cdot \Delta _q h\,\hbox {d}x\hbox {d}y, \end{aligned} \end{aligned}$$
which completes the proof of this lemma. \(\square \)
