Appendix A. Estimates of Nonlinear Terms
In this appendix, we list the estimates of nonlinear terms in the space of analytic functions. Lemma A.1–A.3 will be used to prove the local well-posedness.
First, we estimate nonlinear terms of the form \(f\cdot \nabla g\).
Lemma A.1
For \(f, g, h\in {\mathcal {D}}(e^{\tau A}: H^{r+\frac{1}{2}})\), where \(r>2\) and \(\tau \geqq 0\), one has
$$\begin{aligned} \begin{aligned} \Big |\Big \langle A^r e^{\tau A} (f\cdot \nabla g), A^r e^{\tau A} h \Big \rangle \Big | \le C_r\Big [&(\Vert A^r e^{\tau A} f\Vert + |{\hat{f}}_0| ) \Vert A^{r+\frac{1}{2}} e^{\tau A} g\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} h\Vert \\&+ \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \Vert A^{r} e^{\tau A} g\Vert \Vert A^{r} e^{\tau A} h\Vert \Big ]. \end{aligned} \end{aligned}$$
Proof
First, notice that \(\Big |\Big \langle A^r e^{\tau A} (f\cdot \nabla g), A^r e^{\tau A} h \Big \rangle \Big | = \Big |\Big \langle f\cdot \nabla g, A^r e^{\tau A} H \Big \rangle \Big | \), where \(H = A^r e^{\tau A} h\). We use Fourier representation of f, g and H, in which we can write
$$\begin{aligned}&f({\varvec{x}}) = \sum \limits _{{\varvec{j}}\in {\mathbb {Z}}^3} {\hat{f}}_{{\varvec{j}}} e^{2\pi i{\varvec{j}}\cdot {\varvec{x}}}, \quad g({\varvec{x}}) = \sum \limits _{{\varvec{k}}\in {\mathbb {Z}}^3} {\hat{g}}_{{\varvec{k}}} e^{2\pi i{\varvec{k}}\cdot {\varvec{x}}}, \\&A^r e^{\tau A} H({\varvec{x}}) = \sum \limits _{{\varvec{l}}\in {\mathbb {Z}}^3} |{\varvec{l}}|^r e^{\tau |{\varvec{l}}|}{\hat{H}}_{{\varvec{l}}} e^{2\pi i{\varvec{l}}\cdot {\varvec{x}}}. \end{aligned}$$
Therefore,
$$\begin{aligned} \Big |\Big \langle f\cdot \nabla g, A^r e^{\tau A} H \Big \rangle \Big | \le \sum \limits _{{\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0} |{\hat{f}}_{{\varvec{j}}}||{\varvec{k}}||{\hat{g}}_{{\varvec{k}}}||{\varvec{l}}|^r e^{\tau |{\varvec{l}}|} |{\hat{H}}_{{\varvec{l}}}|. \end{aligned}$$
From \(|{\varvec{l}}| = |{\varvec{j}}+{\varvec{k}}| \le |{\varvec{j}}|+|{\varvec{k}}|\) we have the following inequalities:
$$\begin{aligned} |{\varvec{l}}|^r \le (|{\varvec{j}}|+|{\varvec{k}}|)^r \le C_r(|{\varvec{j}}|^r + |{\varvec{k}}|^r), \;\;\; e^{\tau |{\varvec{l}}|} \le e^{\tau |{\varvec{j}}|} e^{\tau |{\varvec{k}}|}. \end{aligned}$$
Applying these inequalities, we have
$$\begin{aligned} \Big |\Big \langle f\cdot \nabla g, A^r e^{\tau A} H \Big \rangle \Big | \le \sum \limits _{{\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0} C_r|{\hat{f}}_{{\varvec{j}}}||{\varvec{k}}||{\hat{g}}_{{\varvec{k}}}|(|{\varvec{j}}|^r+|{\varvec{k}}|^r)e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{k}}|}|{\varvec{l}}|^r e^{\tau |{\varvec{l}}|}|{\hat{h}}_{{\varvec{l}}}|. \end{aligned}$$
Since \(|{\varvec{k}}|, |{\varvec{j}}|, |{\varvec{l}}|\) are all nonnegative, we have \(|{\varvec{k}}|^{\frac{1}{2}} \le (|{\varvec{j}}|+|{\varvec{l}}|)^{\frac{1}{2}} \le |{\varvec{j}}|^{\frac{1}{2}} + |{\varvec{l}}|^{\frac{1}{2}}\), therefore,
$$\begin{aligned}&\Big |\Big \langle f\cdot \nabla g, A^r e^{\tau A} H \Big \rangle \Big | \\&\quad \le \sum \limits _{{\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0} C_r|{\hat{f}}_{{\varvec{j}}}||{\varvec{k}}|^{\frac{1}{2}}(|{\varvec{j}}|^{\frac{1}{2}} + |{\varvec{l}}|^{\frac{1}{2}})|{\hat{g}}_{{\varvec{k}}}|(|{\varvec{j}}|^r+|{\varvec{k}}|^r)e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{k}}|}|{\varvec{l}}|^r e^{\tau |{\varvec{l}}|}|{\hat{h}}_{{\varvec{l}}}| \nonumber \\&\quad \le \sum \limits _{{\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0} C_r \Big (|{\varvec{k}}|^{\frac{1}{2}}|{\varvec{j}}|^{r+\frac{1}{2}} |{\varvec{l}}|^{r} + |{\varvec{k}}|^{r+\frac{1}{2}}|{\varvec{j}}|^{\frac{1}{2}} |{\varvec{l}}|^{r} + |{\varvec{k}}|^{\frac{1}{2}}|{\varvec{j}}|^{r} |{\varvec{l}}|^{r+\frac{1}{2}} + |{\varvec{k}}|^{r+\frac{1}{2}}|{\varvec{l}}|^{r+\frac{1}{2}} \Big ) \nonumber \\&\qquad \times e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{k}}|} e^{\tau |{\varvec{l}}|} |{\hat{f}}_{{\varvec{j}}}| |{\hat{g}}_{{\varvec{k}}}| |{\hat{h}}_{{\varvec{l}}}| =: A_1 + A_2 + A_3 + A_4. \end{aligned}$$
Thanks to the Cauchy–Schwarz inequality, since \(r>2\), we have
$$\begin{aligned} A_1= & {} \sum \limits _{{\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0} C_r |{\varvec{k}}|^{\frac{1}{2}}|{\varvec{j}}|^{r+\frac{1}{2}} |{\varvec{l}}|^{r} e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{k}}|} e^{\tau |{\varvec{l}}|} |{\hat{f}}_{{\varvec{j}}}| |{\hat{g}}_{{\varvec{k}}}| |{\hat{h}}_{{\varvec{l}}}| \\= & {} C_r \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ {\varvec{k}}\ne 0 \end{array} } |{\varvec{k}}|^{\frac{1}{2}} |{\hat{g}}_{{\varvec{k}}}| e^{\tau |{\varvec{k}}|} \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ {\varvec{j}}\ne 0, -{\varvec{k}} \end{array} } |{\varvec{j}}|^{r+\frac{1}{2}} e^{\tau |{\varvec{j}}|}|{\hat{f}}_{{\varvec{j}}}| |{\varvec{j}}+{\varvec{k}}|^{r}e^{\tau |{\varvec{j}}+{\varvec{k}}|}|{\hat{h}}_{-{\varvec{j}}-{\varvec{k}}}| \\\le & {} C_r \Big ( \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ {\varvec{k}}\ne 0 \end{array} } |{\varvec{k}}|^{1-2r}\Big )^{\frac{1}{2}} \Big ( \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ {\varvec{k}}\ne 0 \end{array} } |{\varvec{k}}|^{2r} e^{2\tau |{\varvec{k}}|} |{\hat{g}}_{{\varvec{k}}}|^2\Big )^{\frac{1}{2}} \\&\times \sup \limits _{{\varvec{k}}\in {\mathbb {Z}}^3}\Big ( \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ {\varvec{j}}\ne 0, -{\varvec{k}} \end{array} } |{\varvec{j}}|^{2r+1}e^{2\tau |{\varvec{j}}|} |{\hat{f}}_{{\varvec{j}}}|^2\Big )^{\frac{1}{2}} \Big ( \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ {\varvec{j}}\ne 0, -{\varvec{k}} \end{array} } |{\varvec{j}}+{\varvec{k}}|^{2r}e^{2\tau |{\varvec{j}}+{\varvec{k}}|} |{\hat{h}}_{-{\varvec{j}}-{\varvec{k}}}|^2\Big )^{\frac{1}{2}} \\\le & {} C_r \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \Vert A^{r} e^{\tau A} g\Vert \Vert A^{r} e^{\tau A} h\Vert , \end{aligned}$$
Similarly, we have
$$\begin{aligned} A_2= & {} \sum \limits _{{\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0} C_r |{\varvec{k}}|^{r+\frac{1}{2}}|{\varvec{j}}|^{\frac{1}{2}} |{\varvec{l}}|^{r} e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{k}}|} e^{\tau |{\varvec{l}}|} |{\hat{f}}_{{\varvec{j}}}| |{\hat{g}}_{{\varvec{k}}}| |{\hat{h}}_{{\varvec{l}}}| \\\le & {} C_r \Vert A^{r} e^{\tau A} f\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} g\Vert \Vert A^{r} e^{\tau A} h\Vert \text { and } \\ A_3= & {} \sum \limits _{{\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0} C_r |{\varvec{k}}|^{\frac{1}{2}}|{\varvec{j}}|^{r} |{\varvec{l}}|^{r+\frac{1}{2}} e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{k}}|} e^{\tau |{\varvec{l}}|} |{\hat{f}}_{{\varvec{j}}}| |{\hat{g}}_{{\varvec{k}}}| |{\hat{h}}_{{\varvec{l}}}| \\\le & {} C_r \Vert A^{r} e^{\tau A} f\Vert \Vert A^{r} e^{\tau A} g\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} h\Vert . \end{aligned}$$
For \(A_4\), thanks to the Cauchy–Schwarz inequality, since \(r>2\), we have
$$\begin{aligned} A_4= & {} \sum \limits _{{\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0} C_r |{\varvec{k}}|^{r+\frac{1}{2}}|{\varvec{l}}|^{r+\frac{1}{2}} e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{k}}|} e^{\tau |{\varvec{l}}|} |{\hat{f}}_{{\varvec{j}}}| |{\hat{g}}_{{\varvec{k}}}| |{\hat{h}}_{{\varvec{l}}}| \\= & {} C_r \sum \limits _{{\varvec{j}}\in {\mathbb {Z}}^3 } e^{\tau |{\varvec{j}}|}|{\hat{f}}_{{\varvec{j}}}| \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ {\varvec{k}}\ne 0, -{\varvec{j}} \end{array} } |{\varvec{k}}|^{r+\frac{1}{2}} |{\hat{g}}_{{\varvec{k}}}| e^{\tau |{\varvec{k}}|} |{\varvec{j}}+{\varvec{k}}|^{r+\frac{1}{2}}e^{\tau |{\varvec{j}}+{\varvec{k}}|}|{\hat{h}}_{-{\varvec{j}}-{\varvec{k}}}| \\\le & {} C_r \Big \{ |{\hat{f}}_0| + \Big ( \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ {\varvec{j}}\ne 0 \end{array} } |{\varvec{j}}|^{-2r}\Big )^{\frac{1}{2}} \Big ( \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ {\varvec{j}}\ne 0 \end{array} } |{\varvec{j}}|^{2r} e^{2\tau |{\varvec{j}}|} |{\hat{f}}_{{\varvec{j}}}|^2\Big )^{\frac{1}{2}} \Big \} \\&\times \sup \limits _{{\varvec{j}}\in {\mathbb {Z}}^3} \Big ( \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ {\varvec{k}}\ne 0, -{\varvec{j}} \end{array} } |{\varvec{k}}|^{2r+1}e^{2\tau |{\varvec{k}}|} |{\hat{g}}_{{\varvec{k}}}|^2\Big )^{\frac{1}{2}} \Big ( \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ {\varvec{k}}\ne 0, -{\varvec{j}} \end{array} } |{\varvec{j}}+{\varvec{k}}|^{2r+1}e^{2\tau |{\varvec{j}}+{\varvec{k}}|} |{\hat{h}}_{-{\varvec{j}}-{\varvec{k}}}|^2\Big )^{\frac{1}{2}} \\\le & {} C_r (\Vert A^{r} e^{\tau A} f\Vert + |{\hat{f}}_0|) \Vert A^{r+\frac{1}{2}} e^{\tau A} g\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} h\Vert . \end{aligned}$$
Combine the estimates for \(A_1\) to \(A_4\), and since \(\Vert A^{r} e^{\tau A} g\Vert \le \Vert A^{r+\frac{1}{2}} e^{\tau A} g\Vert \), \(\Vert A^{r} e^{\tau A} h\Vert \le \Vert A^{r+\frac{1}{2}} e^{\tau A} h\Vert \), we achieve the desired inequality. \(\square \)
Similarly, we estimate \((\nabla \cdot f)g\) in the following:
Lemma A.2
For \(f, g, h\in {\mathcal {D}}(e^{\tau A}: H^{r+\frac{1}{2}})\), where \(r>2\) and \(\tau \geqq 0\), one has
$$\begin{aligned} \begin{aligned} \Big |\Big \langle A^r e^{\tau A} \big ((\nabla \cdot f)g\big ), A^r e^{\tau A} h \Big \rangle \Big | \le&C_r\Big [ (\Vert A^r e^{\tau A} g\Vert + |{\hat{g}}_0|) \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} h\Vert \\&+ \Vert A^{r+\frac{1}{2}} e^{\tau A} g\Vert \Vert A^{r} e^{\tau A} f\Vert \Vert A^{r} e^{\tau A} h\Vert \Big ]. \end{aligned} \end{aligned}$$
The proof of Lemma A.2 is almost the same as Lemma A.1, so we omit it.
Finally, we provide an estimate for \((\int _0^z \nabla \cdot f({\varvec{x}}',s)ds)\partial _z g\) in the following:
Lemma A.3
For \(f, g, h\in {\mathcal {D}}(e^{\tau A}: H^{r+\frac{1}{2}})\), where \(r>2\), \(\tau \geqq 0\), and \({\overline{f}}=0\), one has
$$\begin{aligned} \begin{aligned}&\Big |\Big \langle A^r e^{\tau A} \big ((\int _0^z \nabla \cdot f({\varvec{x}}',s)ds)\partial _z g\big ), A^r e^{\tau A} h \Big \rangle \Big | \\&\quad \le C_r\Big ( \Vert A^r e^{\tau A} f\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} g\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} h\Vert \\&\qquad + \Vert A^r e^{\tau A} g\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} h\Vert \\&\quad + \Vert A^r e^{\tau A} h\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} g\Vert \Big ). \end{aligned} \end{aligned}$$
Proof
First, \(\Big |\Big \langle A^r e^{\tau A} \big ((\int _0^z \nabla \cdot f({\varvec{x}}',s)ds)\partial _z g\big ), A^r e^{\tau A} h \Big \rangle \Big | = \Big |\Big \langle (\int _0^z \nabla \cdot f({\varvec{x}}',s)ds)\partial _z g, A^r e^{\tau A} H \Big \rangle \Big | \). Since \({\overline{f}}=0\), one has \( f({\varvec{x}}) = \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3 \ne 0 \end{array}} {\hat{f}}_{{\varvec{j}}} e^{2\pi ( i{\varvec{j}}' \cdot {\varvec{x}}' + ij_3 z)} \) where \({\varvec{j}}'=(j_1,j_2)\). Then we have
$$\begin{aligned} \int _0^z \nabla \cdot f({\varvec{x}}',s) ds = \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3 \ne 0, {\varvec{j}}' \ne 0 \end{array}} \frac{1}{j_3}{\varvec{j}}'\cdot {\hat{f}}_{{\varvec{j}}} e^{2\pi ( i{\varvec{j}}' \cdot {\varvec{x}}' + ij_3 z)} - \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3 \ne 0, {\varvec{j}}' \ne 0 \end{array}} \frac{1}{j_3}{\varvec{j}}'\cdot {\hat{f}}_{{\varvec{j}}} e^{2\pi i{\varvec{j}}'\cdot {\varvec{x}}'}. \end{aligned}$$
Therefore,
$$\begin{aligned}&\Big |\Big \langle (\int _0^z \nabla \cdot f(s)ds)\partial _z g, A^r e^{\tau A} H \Big \rangle \Big |\\&\le \Big |\Big \langle (\sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3 \ne 0, {\varvec{j}}' \ne 0 \end{array}} \frac{1}{j_3}{\varvec{j}}'\cdot {\hat{f}}_{{\varvec{j}}} e^{2\pi ( i{\varvec{j}}' \cdot {\varvec{x}}' + ij_3 z)})\partial _z g, A^r e^{\tau A} H \Big \rangle \Big | \nonumber \\&\qquad + \Big |\Big \langle (\sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3 \ne 0, {\varvec{j}}' \ne 0 \end{array}} \frac{1}{j_3}{\varvec{j}}'\cdot {\hat{f}}_{{\varvec{j}}} e^{i{\varvec{j}}'\cdot {\varvec{x}}'})\partial _z g, A^r e^{\tau A} H \Big \rangle \Big | =: I_1 + I_2 . \end{aligned}$$
Let us estimate \(I_2\) first. For \({\varvec{l}} = ({\varvec{l}}', l_3) = (-{\varvec{j}}'-{\varvec{k}}', -k_3)\), by using the inequalities
$$\begin{aligned} |{\varvec{j}}'|^{\frac{1}{2}} \le C(|{\varvec{k}}|^{\frac{1}{2}} + |{\varvec{l}}|^{\frac{1}{2}}), \;\;\; |{\varvec{k}}|^{\frac{1}{2}} \le C(|{\varvec{j}}'|^{\frac{1}{2}} + |{\varvec{l}}|^{\frac{1}{2}}), \;\;\; |{\varvec{l}}|^r \le C_r (|{\varvec{j}}'|^r+|{\varvec{k}}|^r), \end{aligned}$$
one has
$$\begin{aligned} I_2\le & {} \sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}} C_r\frac{1}{|j_3|}|{\varvec{j}}'||k_3||{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}|(|{\varvec{j}}'|^r+|{\varvec{k}}|^r)e^{\tau |{\varvec{j}}'|}e^{\tau |{\varvec{k}}|}|{\varvec{l}}|^r e^{\tau |{\varvec{l}}|}|{\hat{h}}_{{\varvec{l}}}|\\\le & {} \sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}} C_r\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}|(|{\varvec{j}}'|^{r+1}|{\varvec{k}}|+|{\varvec{j}}'||{\varvec{k}}|^{r+1})e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{k}}|}|{\varvec{l}}|^r e^{\tau |{\varvec{l}}|}|{\hat{h}}_{{\varvec{l}}}| \\\le & {} \sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}} C_r \frac{1}{|j_3|}\Big (|{\varvec{k}}|^{\frac{3}{2}}|{\varvec{j}}'|^{r+\frac{1}{2}} |{\varvec{l}}|^{r} + |{\varvec{k}}||{\varvec{j}}'|^{r+\frac{1}{2}} |{\varvec{l}}|^{r+\frac{1}{2}} + |{\varvec{j}}'|^{\frac{3}{2}}|{\varvec{k}}|^{r+\frac{1}{2}} |{\varvec{l}}|^{r}\\&+ |{\varvec{j}}'||{\varvec{k}}|^{r+\frac{1}{2}}|{\varvec{l}}|^{r+\frac{1}{2}} \Big ) e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{k}}|} e^{\tau |{\varvec{l}}|} |{\hat{f}}_{{\varvec{j}}}| |{\hat{g}}_{{\varvec{k}}}| |{\hat{h}}_{{\varvec{l}}}| =: B_1 + B_2 + B_3 + B_4. \end{aligned}$$
When \(k_3 \ne 0\) and \(r>2\), we know that \(|{\varvec{k}}|^{1-r} \le |({\varvec{k}}',\pm 1)|^{1-r}\) and \(\sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} |({\varvec{k}}',\pm 1)|^{2-2r} \le C_r\) is finite. Thanks to the Cauchy–Schwarz inequality, we have
$$\begin{aligned} B_1= & {} \sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}} C_r \frac{1}{|j_3|} |{\varvec{k}}|^{\frac{3}{2}}|{\varvec{j}}'|^{r+\frac{1}{2}} |{\varvec{l}}|^{r} e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{k}}|} e^{\tau |{\varvec{l}}|} |{\hat{f}}_{{\varvec{j}}}| |{\hat{g}}_{{\varvec{k}}}| |{\hat{h}}_{{\varvec{l}}}| \\= & {} C_r \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ k_3\ne 0 \end{array} } |{\varvec{k}}|^{\frac{3}{2}} |{\hat{g}}_{{\varvec{k}}}| e^{\tau |{\varvec{k}}|} \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}' \ne 0 \end{array} }\\&\frac{1}{|j_3|} |{\varvec{j}}'|^{r+\frac{1}{2}} e^{\tau |{\varvec{j}}|}|{\hat{f}}_{{\varvec{j}}}| |({\varvec{j}}'+{\varvec{k}}',k_3)|^{r}e^{\tau |({\varvec{j}}'+{\varvec{k}}',k_3)|}|{\hat{h}}_{-({\varvec{j}}'+{\varvec{k}}',k_3)}| \\\le & {} C_r \sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} |({\varvec{k}}',\pm 1)|^{1-r} \sum \limits _{k_3\ne 0} |{\varvec{k}}|^{r+\frac{1}{2}} |{\hat{g}}_{{\varvec{k}}}| e^{\tau |{\varvec{k}}|} \Big ( \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ {\varvec{j}}\ne 0 \end{array} } |{\varvec{j}}|^{2r+1} e^{2\tau |{\varvec{j}}|} |{\hat{f}}_{{\varvec{j}}}|^2\Big )^{\frac{1}{2}} \\&\times \Big ( \sum \limits _{j_3\ne 0} \frac{1}{|j_3|^2} \sum \limits _{{\varvec{j}}'\in {\mathbb {Z}}^2 } |({\varvec{j}}'+{\varvec{k}}',k_3)|^{2r}e^{2\tau |({\varvec{j}}'+{\varvec{k}}',k_3)|}|{\hat{h}}_{-({\varvec{j}}'+{\varvec{k}}',k_3)}|^2\Big )^{\frac{1}{2}} \\\le & {} C_r \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} |({\varvec{k}}',\pm 1)|^{1-r} \Big (\sum \limits _{k_3\ne 0} |{\varvec{k}}|^{2r+1} |{\hat{g}}_{{\varvec{k}}}|^2 e^{2\tau |{\varvec{k}}|} \Big )^{\frac{1}{2}} \\&\times \Big ( \sum \limits _{k_3\ne 0} \sum \limits _{{\varvec{j}}'\in {\mathbb {Z}}^2 } |({\varvec{j}}'+{\varvec{k}}',k_3)|^{2r}e^{2\tau |({\varvec{j}}'+{\varvec{k}}',k_3)|}|{\hat{h}}_{-({\varvec{j}}'+{\varvec{k}}',k_3)}|^2\Big )^{\frac{1}{2}} \\\le & {} C_r \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \Vert A^{r} e^{\tau A} h\Vert \Big (\sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} |({\varvec{k}}',\pm 1)|^{2-2r}\Big )^{\frac{1}{2}}\\&\Big ( \sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} \sum \limits _{k_3\ne 0} |{\varvec{k}}|^{2r+1} |{\hat{g}}_{{\varvec{k}}}|^2 e^{2\tau |{\varvec{k}}|} \Big )^{\frac{1}{2}} \\\le & {} C_r \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} g\Vert \Vert A^{r} e^{\tau A} h\Vert . \end{aligned}$$
The estimate for \(B_2\) is similar as \(B_1\), and we can get \(B_2 \le C_r \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \Vert A^{r} e^{\tau A} g\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} h\Vert \). For \(B_3\), thanks to the Cauchy–Schwarz inequality, since \(r>2\), we have
$$\begin{aligned} B_3= & {} \sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}} C_r \frac{1}{|j_3|} |{\varvec{j}}'|^{\frac{3}{2}}|{\varvec{k}}|^{r+\frac{1}{2}} |{\varvec{l}}|^{r} e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{k}}|} e^{\tau |{\varvec{l}}|} |{\hat{f}}_{{\varvec{j}}}| |{\hat{g}}_{{\varvec{k}}}| |{\hat{h}}_{{\varvec{l}}}| \\= & {} C_r \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}'\ne 0 \end{array} } \frac{1}{|j_3|}|{\varvec{j}}'|^{\frac{3}{2}} |{\hat{f}}_{{\varvec{j}}}| e^{\tau |{\varvec{j}}|}\\&\sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ k_3 \ne 0 \end{array} } |{\varvec{k}}|^{r+\frac{1}{2}} e^{\tau |{\varvec{k}}|}|{\hat{g}}_{{\varvec{k}}}| |({\varvec{j}}'+{\varvec{k}}',k_3)|^{r}e^{\tau |({\varvec{j}}'+{\varvec{k}}',k_3)|}|{\hat{h}}_{-({\varvec{j}}'+{\varvec{k}}',k_3)}| \\\le & {} C_r \Big (\sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}'\ne 0 \end{array} } \frac{1}{|j_3|^2}|{\varvec{j}}'|^{2-2r} \Big )^{\frac{1}{2}} \Big (\sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}'\ne 0 \end{array} } |{\varvec{j}}|^{2r+1} |{\hat{f}}_{{\varvec{j}}}|^2 e^{2\tau |{\varvec{j}}|} \Big )^{\frac{1}{2}}\\&\Big ( \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ k_3\ne 0 \end{array} } |{\varvec{k}}|^{2r+1} e^{2\tau |{\varvec{k}}|} |{\hat{g}}_{{\varvec{k}}}|^2\Big )^{\frac{1}{2}} \\&\times \sup \limits _{{\varvec{j}}\in {\mathbb {Z}}^3} \Big ( \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ k_3\ne 0 \end{array}} |({\varvec{j}}'+{\varvec{k}}',k_3)|^{2r}e^{2\tau |({\varvec{j}}'+{\varvec{k}}',k_3)|}|{\hat{h}}_{-({\varvec{j}}'+{\varvec{k}}',k_3)}|^2\Big )^{\frac{1}{2}} \\\le & {} C_r \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} g\Vert \Vert A^{r} e^{\tau A} h\Vert . \end{aligned}$$
The estimate for \(B_4\) is similar as \(B_3\), and we can get \(B_4 \le C_r \Vert A^{r} e^{\tau A} f\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} g\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} h\Vert \). The estimates of \(B_1\) to \(B_4\) indicate that \(I_2\) satisfies the desired inequality.
Now let us estimate on \(I_1\). For \( {\varvec{j}}+{\varvec{k}}+ {\varvec{l}}=0\), by using the inequalities
$$\begin{aligned} |{\varvec{j}}|^{\frac{1}{2}} \le C(|{\varvec{k}}|^{\frac{1}{2}} + |{\varvec{l}}|^{\frac{1}{2}}), \;\;\; |{\varvec{k}}|^{\frac{1}{2}} \le C(|{\varvec{j}}|^{\frac{1}{2}} + |{\varvec{l}}|^{\frac{1}{2}}), \;\;\; |{\varvec{l}}|^r \le C_r (|{\varvec{j}}|^r+|{\varvec{k}}|^r), \end{aligned}$$
we have
$$\begin{aligned} I_1\le & {} \sum \limits _{\begin{array}{c} {\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0\\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}} C_r\frac{1}{|j_3|}|{\varvec{j}}'||k_3||{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}|(|{\varvec{j}}|^r+|{\varvec{k}}|^r)e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{k}}|}|{\varvec{l}}|^r e^{\tau |{\varvec{l}}|}|{\hat{h}}_{{\varvec{l}}}|\\\le & {} \sum \limits _{\begin{array}{c} {\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0\\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}} C_r\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}|(|{\varvec{j}}|^{r+1}|{\varvec{k}}|+|{\varvec{j}}||{\varvec{k}}|^{r+1})e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{k}}|}|{\varvec{l}}|^r e^{\tau |{\varvec{l}}|}|{\hat{h}}_{{\varvec{l}}}| \\\le & {} \sum \limits _{\begin{array}{c} {\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0\\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}} C_r \frac{1}{|j_3|}\Big (|{\varvec{k}}|^{\frac{3}{2}}|{\varvec{j}}|^{r+\frac{1}{2}} |{\varvec{l}}|^{r} + |{\varvec{k}}||{\varvec{j}}|^{r+\frac{1}{2}} |{\varvec{l}}|^{r+\frac{1}{2}} + |{\varvec{j}}|^{\frac{3}{2}}|{\varvec{k}}|^{r+\frac{1}{2}} |{\varvec{l}}|^{r} \\&+ |{\varvec{j}}||{\varvec{k}}|^{r+\frac{1}{2}}|{\varvec{l}}|^{r+\frac{1}{2}} \Big ) e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{k}}|} e^{\tau |{\varvec{l}}|} |{\hat{f}}_{{\varvec{j}}}| |{\hat{g}}_{{\varvec{k}}}| |{\hat{h}}_{{\varvec{l}}}| =: {\widetilde{B}}_1 + {\widetilde{B}}_2 + {\widetilde{B}}_3 + {\widetilde{B}}_4. \end{aligned}$$
Thanks to the Cauchy–Schwarz inequality, since \(r>2\), we have
$$\begin{aligned} {\widetilde{B}}_1= & {} \sum \limits _{\begin{array}{c} {\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0\\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}} C_r \frac{1}{|j_3|} |{\varvec{k}}|^{\frac{3}{2}}|{\varvec{j}}|^{r+\frac{1}{2}} |{\varvec{l}}|^{r} e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{k}}|} e^{\tau |{\varvec{l}}|} |{\hat{f}}_{{\varvec{j}}}| |{\hat{g}}_{{\varvec{k}}}| |{\hat{h}}_{{\varvec{l}}}| \\= & {} C_r \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ k_3\ne 0 \end{array} } |{\varvec{k}}|^{\frac{3}{2}} |{\hat{g}}_{{\varvec{k}}}| e^{\tau |{\varvec{k}}|} \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}' \ne 0 \end{array} } \frac{1}{|j_3|} |{\varvec{j}}|^{r+\frac{1}{2}} e^{\tau |{\varvec{j}}|}|{\hat{f}}_{{\varvec{j}}}| |{\varvec{j}}+{\varvec{k}}|^{r}e^{\tau |{\varvec{j}}+{\varvec{k}}|}|{\hat{h}}_{-{\varvec{j}}-{\varvec{k}}}| \\\le & {} C_r \sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} |({\varvec{k}}',\pm 1)|^{1-r} \sum \limits _{k_3\ne 0} |{\varvec{k}}|^{r+\frac{1}{2}} |{\hat{g}}_{{\varvec{k}}}| e^{\tau |{\varvec{k}}|} \Big ( \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ {\varvec{j}}\ne 0 \end{array} } |{\varvec{j}}|^{2r+1} e^{2\tau |{\varvec{j}}|} |{\hat{f}}_{{\varvec{j}}}|^2\Big )^{\frac{1}{2}} \\&\times \Big ( \sum \limits _{j_3\ne 0} \frac{1}{|j_3|^2} \sum \limits _{{\varvec{j}}'\in {\mathbb {Z}}^2 } |({\varvec{j}}'+{\varvec{k}}',j_3+k_3)|^{2r}e^{2\tau |({\varvec{j}}'+{\varvec{k}}',j_3+k_3)|}|{\hat{h}}_{-({\varvec{j}}'+{\varvec{k}}',j_3+k_3)}|^2\Big )^{\frac{1}{2}} \\\le & {} C_r \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} |({\varvec{k}}',\pm 1)|^{1-r} \Big (\sum \limits _{k_3\ne 0} |{\varvec{k}}|^{2r+1} |{\hat{g}}_{{\varvec{k}}}|^2 e^{2\tau |{\varvec{k}}|} \Big )^{\frac{1}{2}} \\&\times \Big ( \sum \limits _{j_3\ne 0} \frac{1}{|j_3|^2} \sum \limits _{k_3\ne 0} \sum \limits _{{\varvec{j}}'\in {\mathbb {Z}}^2 } |({\varvec{j}}'+{\varvec{k}}',j_3+k_3)|^{2r}e^{2\tau |({\varvec{j}}'+{\varvec{k}}',j_3+k_3)|}|{\hat{h}}_{-({\varvec{j}}'+{\varvec{k}}',j_3+k_3)}|^2\Big )^{\frac{1}{2}} \\\le & {} C_r \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \Vert A^{r} e^{\tau A} h\Vert \Big (\sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} |({\varvec{k}}',\pm 1)|^{2-2r}\Big )^{\frac{1}{2}} \Big (\sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} \sum \limits _{k_3\ne 0} |{\varvec{k}}|^{2r+1} |{\hat{g}}_{{\varvec{k}}}|^2 e^{2\tau |{\varvec{k}}|} \Big )^{\frac{1}{2}} \\\le & {} C_r \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} g\Vert \Vert A^{r} e^{\tau A} h\Vert , \end{aligned}$$
where in the second inequality, we use Fubini theorem to exchange the order of \(\sum \limits _{j_3\ne 0}\) and \(\sum \limits _{k_3\ne 0}\). The estimate for \({\widetilde{B}}_2\) is similar to \({\widetilde{B}}_1\), and we can get \({\widetilde{B}}_2 \le C_r \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \Vert A^{r} e^{\tau A} g\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} h\Vert \). For \({\widetilde{B}}_3\), thanks to the Cauchy–Schwarz inequality, since \(r>2\), we have
$$\begin{aligned} {\widetilde{B}}_3= & {} \sum \limits _{\begin{array}{c} {\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0\\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}} C_r \frac{1}{|j_3|} |{\varvec{j}}|^{\frac{3}{2}}|{\varvec{k}}|^{r+\frac{1}{2}} |{\varvec{l}}|^{r} e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{k}}|} e^{\tau |{\varvec{l}}|} |{\hat{f}}_{{\varvec{j}}}| |{\hat{g}}_{{\varvec{k}}}| |{\hat{h}}_{{\varvec{l}}}| \\= & {} C_r \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}' \ne 0 \end{array} } \frac{1}{|j_3|}|{\varvec{j}}|^{\frac{3}{2}} e^{\tau |{\varvec{j}}|}|{\hat{f}}_{{\varvec{j}}}| \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ k_3\ne 0 \end{array} } |{\varvec{k}}|^{r+\frac{1}{2}} |{\hat{g}}_{{\varvec{k}}}| e^{\tau |{\varvec{k}}|} |{\varvec{j}}+{\varvec{k}}|^{r}e^{\tau |{\varvec{j}}+{\varvec{k}}|}|{\hat{h}}_{-{\varvec{j}}-{\varvec{k}}}| \\\le & {} C_r \Big (\sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}'\ne 0 \end{array} } \frac{1}{|j_3|^2}|{\varvec{j}}'|^{2-2r} \Big )^{\frac{1}{2}}\\&\Big (\sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}'\ne 0 \end{array} } |{\varvec{j}}|^{2r+1} |{\hat{f}}_{{\varvec{j}}}|^2 e^{2\tau |{\varvec{j}}|} \Big )^{\frac{1}{2}} \Big ( \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ k_3\ne 0 \end{array} } |{\varvec{k}}|^{2r+1} e^{2\tau |{\varvec{k}}|} |{\hat{g}}_{{\varvec{k}}}|^2\Big )^{\frac{1}{2}} \\&\times \sup \limits _{{\varvec{j}}\in {\mathbb {Z}}^3} \Big ( \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ k_3\ne 0 \end{array}} |{\varvec{j}}+{\varvec{k}}|^{2r}e^{2\tau |{\varvec{j}}+{\varvec{k}}|}|{\hat{h}}_{-{\varvec{j}}-{\varvec{k}}}|^2\Big )^{\frac{1}{2}}\\&\le C_r \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} g\Vert \Vert A^{r} e^{\tau A} h\Vert , \end{aligned}$$
where in the first inequality we use \(|{\varvec{j}}|^{2-2r}\le |{\varvec{j}}'|^{2-2r}\) due to \(r>2\). The estimate for \({\widetilde{B}}_4\) is similar as \({\widetilde{B}}_3\), and we can get \({\widetilde{B}}_4 \le C_r \Vert A^{r} e^{\tau A} f\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} g\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} h\Vert \). The estimates of \({\widetilde{B}}_1\) to \({\widetilde{B}}_4\) indicate that \(I_1\) satisfies the desired inequality. \(\square \)
Lemma A.4–A.7 play an essential role in the proof of Theorem 6.1. First, let us state the following:
Lemma A.4
For \(f, g, h\in {\mathcal {D}}(e^{\tau A}: H^{r+\frac{1}{2}})\), where \(r>\frac{5}{2}\) and \(\tau \geqq 0\), one has
$$\begin{aligned}&\Big |\Big \langle A^r e^{\tau A} (f\cdot \nabla g), A^r e^{\tau A} h \Big \rangle - \Big \langle f\cdot \nabla A^r e^{\tau A} g, A^r e^{\tau A} h \Big \rangle \Big | \\&\quad \le C_r \Vert A^r f\Vert \Vert A^{r} g\Vert \Vert A^{r} h\Vert + C_r \tau \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} g\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} h\Vert . \end{aligned}$$
Lemma A.5
For \(f, g, h\in {\mathcal {D}}(e^{\tau A}: H^{r+\frac{1}{2}})\), where \(r>\frac{5}{2}\) and \(\tau \geqq 0\), one has
$$\begin{aligned}&\Big |\Big \langle A^r e^{\tau A} \big ( (\nabla \cdot f)g\big ), A^r e^{\tau A} h \Big \rangle - \Big \langle (\nabla \cdot A^r e^{\tau A} f)g, A^r e^{\tau A} h \Big \rangle \Big | \\&\quad \le C_r \Vert A^r f\Vert \Vert A^{r} g\Vert \Vert A^{r} h\Vert + C_r \tau \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} g\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} h\Vert . \end{aligned}$$
The proof of Lemma A.4 is similarly to that of Lemma 8 in [49] since it involves nonlinear term similar to that appearing in the Euler equations. The proof of Lemma A.5 is similarly to that of Lemma A.4. Therefore, they are omitted.
The next two lemmas provide the estimates for nonlinear terms which are specific to the structure of the PEs.
Lemma A.6
For \(f\in {\mathcal {D}}(e^{\tau A}: H^{r+\frac{3}{2}})\) and \(g, h\in {\mathcal {D}}(e^{\tau A}: H^{r+\frac{1}{2}})\), where \(r>\frac{5}{2}\), \(\tau \geqq 0\), and \({\overline{f}} = 0\), one has
$$\begin{aligned}&\Big |\Big \langle A^r e^{\tau A} \Big ( (\int _0^z \nabla \cdot f({\varvec{x}}',s)ds) \partial _z g \Big ), A^r e^{\tau A} h \Big \rangle \\&\quad - \Big \langle (\int _0^z \nabla \cdot f({\varvec{x}}',s)ds)A^r e^{\tau A} \partial _z g , A^r e^{\tau A} h \Big \rangle \Big | \\&\quad \le C_r \Vert A^{r+1} f\Vert \Vert A^{r} g\Vert \Vert A^{r} h\Vert + C_r \tau \Vert A^{r+\frac{3}{2}} e^{\tau A} f\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} g\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} h\Vert . \end{aligned}$$
Lemma A.7
For \(g\in {\mathcal {D}}(e^{\tau A}: H^{r+\frac{3}{2}})\) and \(f, h\in {\mathcal {D}}(e^{\tau A}: H^{r+\frac{1}{2}})\), where \(r>\frac{5}{2}\), \(\tau \geqq 0\), and \({\overline{f}} = 0\), one has
$$\begin{aligned}&\Big |\Big \langle A^r e^{\tau A} \Big ( (\int _0^z \nabla \cdot f({\varvec{x}}',s)ds) \partial _z g \Big ), A^r e^{\tau A} h \Big \rangle \\&\quad - \Big \langle \partial _z g A^r e^{\tau A} (\int _0^z \nabla \cdot f({\varvec{x}}',s)ds) , A^r e^{\tau A} h \Big \rangle \Big | \\&\quad \le C_r \Vert A^{r+1} g\Vert \Vert A^{r} f\Vert \Vert A^{r} h\Vert + C_r \tau \Vert A^{r+\frac{3}{2}} e^{\tau A} g\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} h\Vert . \end{aligned}$$
Since the proof of Lemma A.6 is similar to that of Lemma A.7, we first focus below on the proof of Lemma A.7, and later we sketch the proof of Lemma A.6 with emphasis on the main differences.
Proof
(proof of Lemma A.7) First, denote by \(H = A^r e^{\tau A} h\), and let
$$\begin{aligned} I:= & {} \Big |\Big \langle A^r e^{\tau A} \Big ( (\int _0^z \nabla \cdot f({\varvec{x}}',s)ds) \partial _z g \Big ), A^r e^{\tau A} h \Big \rangle \\&- \Big \langle \partial _z g A^r e^{\tau A} (\int _0^z \nabla \cdot f({\varvec{x}}',s)ds) , A^r e^{\tau A} h \Big \rangle \Big | \\= & {} \Big | \Big \langle (\int _0^z \nabla \cdot f({\varvec{x}}',s)ds) \partial _z g, A^r e^{\tau A} H \Big \rangle \\&- \Big \langle \partial _z g A^r e^{\tau A} (\int _0^z \nabla \cdot f({\varvec{x}}',s)ds) , H \Big \rangle \Big |. \end{aligned}$$
Similarly to the proof of Lemma A.3, using Fourier representation of f, since \({\overline{f}}=0\), we have
$$\begin{aligned} \int _0^z \nabla \cdot f({\varvec{x}}',s) ds = \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3 \ne 0 \end{array}} \frac{1}{j_3}{\varvec{j}}'\cdot {\hat{f}}_{{\varvec{j}}} e^{2\pi ( i{\varvec{j}}' \cdot {\varvec{x}}' + ij_3 z)} - \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3 \ne 0 \end{array}} \frac{1}{j_3}{\varvec{j}}'\cdot {\hat{f}}_{{\varvec{j}}} e^{2\pi i{\varvec{j}}'\cdot {\varvec{x}}'}, \end{aligned}$$
where \({\varvec{j}}'=(j_1,j_2)\). Using Fourier representation of g and H, we have
$$\begin{aligned} I\le & {} C \sum \limits _{\begin{array}{c} {\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0\\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}} \frac{1}{|j_3|} |{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{H}}_{{\varvec{l}}}| |{\varvec{j}}'||{\varvec{k}}| \Big | |{\varvec{l}}|^r e^{\tau |{\varvec{l}}|} - |{\varvec{j}}|^r e^{\tau |{\varvec{j}}|} \Big | \\&+ C \sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{H}}_{{\varvec{l}}}| |{\varvec{j}}'||{\varvec{k}}| \Big | |{\varvec{l}}|^r e^{\tau |{\varvec{l}}|} - |({\varvec{j}}',0)|^r e^{\tau |({\varvec{j}}',0)|} \Big |{:=} I_1 {+} I_2. \end{aligned}$$
We estimate \(I_2\) first. By virtue of the following observation [49]: For \(r\geqq 1\) and \(\tau \geqq 0\), and for all positive \(\xi ,\eta \in {\mathbb {R}}\), we have
$$\begin{aligned}&|\xi ^r e^{\tau \xi } - \eta ^r e^{\tau \eta }| \le C_r|\xi - \eta |\nonumber \\&\Big ( |\xi - \eta |^{r-1} + \eta ^{r-1} + \tau (|\xi - \eta |^{r} + \eta ^r)e^{\tau |\xi -\eta |}e^{\tau \eta } \Big ); \end{aligned}$$
(A.1)
with \(\xi = |{\varvec{l}}|\), \(\eta = |({\varvec{j}}',0)| = |{\varvec{j}}'|\), and \(|\xi - \eta | = \Big ||{\varvec{l}}|- |({\varvec{j}}',0)| \Big |\le \Big |-{\varvec{l}}-({\varvec{j}}',0)\Big | = |{\varvec{k}}|\), inequality (A.1) implies
$$\begin{aligned}&I_2 \le C_r\sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{H}}_{{\varvec{l}}}| |{\varvec{j}}'||{\varvec{k}}|^2\nonumber \\&\Big ( |{\varvec{k}}|^{r-1} + |{\varvec{j}}'|^{r-1} + \tau (|{\varvec{k}}|^{r} + |{\varvec{j}}'|^{r})e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|} \Big ). \end{aligned}$$
(A.2)
By the definition of H, and since \(e^x \le 1+xe^x\) for any \(x\geqq 0\), we have
$$\begin{aligned} |{\hat{H}}_{{\varvec{l}}}| = |{\varvec{l}}|^r e^{\tau |{\varvec{l}}|} |{\hat{h}}_{{\varvec{l}}}| \le |{\varvec{l}}|^r(1+\tau |{\varvec{l}}|e^{\tau |{\varvec{l}}|}) |{\hat{h}}_{{\varvec{l}}}| \le |{\varvec{l}}|^r |{\hat{h}}_{{\varvec{l}}}| + \tau (|{\varvec{j}}'|+|{\varvec{k}}|) |{\hat{H}}_{{\varvec{l}}}|. \end{aligned}$$
Therefore, one obtains that
$$\begin{aligned}&|{\hat{H}}_{{\varvec{l}}}| \Big ( |{\varvec{k}}|^{r-1} + |{\varvec{j}}'|^{r-1} + \tau (|{\varvec{k}}|^{r} + |{\varvec{j}}'|^{r})e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|} \Big ) \\&\quad \le \Big (|{\varvec{l}}|^r |{\hat{h}}_{{\varvec{l}}}| + \tau (|{\varvec{j}}'|+|{\varvec{k}}|) |{\hat{H}}_{{\varvec{l}}}|\Big ) \Big ( |{\varvec{k}}|^{r-1} + |{\varvec{j}}'|^{r-1} \Big )\\&+ |{\hat{H}}_{{\varvec{l}}}| \Big (\tau (|{\varvec{k}}|^{r} + |{\varvec{j}}'|^{r})e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|} \Big ) \\&\quad \le |{\hat{h}}_{{\varvec{l}}}| |{\varvec{l}}|^r(|{\varvec{k}}|^{r-1} + |{\varvec{j}}'|^{r-1}) + \tau C_r |{\hat{H}}_{{\varvec{l}}}|(|{\varvec{k}}|^{r} + |{\varvec{j}}'|^{r})e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|}. \end{aligned}$$
Based on this, one has
$$\begin{aligned} I_2\le & {} C_r\sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{h}}_{{\varvec{l}}}| |{\varvec{j}}'||{\varvec{k}}|^2 |{\varvec{l}}|^r(|{\varvec{k}}|^{r-1} + |{\varvec{j}}'|^{r-1}) \\&+ \tau C_r\sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{H}}_{{\varvec{l}}}| |{\varvec{j}}'||{\varvec{k}}|^2 (|{\varvec{k}}|^{r} + |{\varvec{j}}'|^{r})e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|}:= I_{21} + I_{22}. \end{aligned}$$
Here
$$\begin{aligned} I_{21}= & {} C_r\Big (\sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{h}}_{{\varvec{l}}}| |{\varvec{j}}'||{\varvec{k}}|^{r+1} |{\varvec{l}}|^r + \frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{h}}_{{\varvec{l}}}| |{\varvec{j}}'|^{r}|{\varvec{k}}|^{2} |{\varvec{l}}|^r\Big ) \\:= & {} I_{211}+ I_{212}. \end{aligned}$$
Thanks to the Cauchy–Schwarz inequality, since \(r> \frac{5}{2}\), we have
$$\begin{aligned} I_{211}= & {} C_r \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}'\ne 0 \end{array} } \frac{1}{|j_3|}|{\varvec{j}}'| |{\hat{f}}_{{\varvec{j}}}| \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ k_3\ne 0 \end{array} } |{\varvec{k}}|^{r+1} |({\varvec{j}}'+{\varvec{k}}',k_3)|^r |{\hat{g}}_{{\varvec{k}}}||{\hat{h}}_{-({\varvec{j}}'+{\varvec{k}}',k_3)}| \\\le & {} C_r \Big (\sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}'\ne 0 \end{array} } \frac{1}{|j_3|^2}|{\varvec{j}}'|^{2-2r} \Big )^{\frac{1}{2}} \Big (\sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}'\ne 0 \end{array} } |{\varvec{j}}|^{2r} |{\hat{f}}_{{\varvec{j}}}|^2 \Big )^{\frac{1}{2}} \Big ( \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ k_3\ne 0 \end{array} } |{\varvec{k}}|^{2r+2} |{\hat{g}}_{{\varvec{k}}}|^2\Big )^{\frac{1}{2}} \\&\times \sup \limits _{{\varvec{j}}\in {\mathbb {Z}}^3} \Big ( \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ k_3\ne 0 \end{array}} |({\varvec{j}}'+{\varvec{k}}',k_3)|^{2r}|{\hat{h}}_{-({\varvec{j}}'+{\varvec{k}}',k_3)}|^2\Big )^{\frac{1}{2}}\\&\le C_r \Vert A^r f\Vert \Vert A^{r+1} g\Vert \Vert A^r h \Vert , \text { and } \end{aligned}$$
$$\begin{aligned} I_{212}= & {} C_r \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ k_3\ne 0 \end{array} } |{\varvec{k}}|^{2} |{\hat{g}}_{{\varvec{k}}}| \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}' \ne 0 \end{array} } \frac{1}{|j_3|} |{\varvec{j}}'|^{r} |{\hat{f}}_{{\varvec{j}}}| |({\varvec{j}}'+{\varvec{k}}',k_3)|^{r}|{\hat{h}}_{-({\varvec{j}}'+{\varvec{k}}',k_3)}| \\\le & {} C_r \sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} |({\varvec{k}}',\pm 1)|^{1-r} \sum \limits _{k_3\ne 0} |{\varvec{k}}|^{r+1} |{\hat{g}}_{{\varvec{k}}}| \Big ( \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ {\varvec{j}} \ne 0 \end{array} } |{\varvec{j}}|^{2r} |{\hat{f}}_{{\varvec{j}}}|^2\Big )^{\frac{1}{2}} \\&\times \Big ( \sum \limits _{j_3\ne 0} \frac{1}{|j_3|^2} \sum \limits _{{\varvec{j}}'\in {\mathbb {Z}}^2 } |({\varvec{j}}'+{\varvec{k}}',k_3)|^{2r}|{\hat{h}}_{-({\varvec{j}}'+{\varvec{k}}',k_3)}|^2\Big )^{\frac{1}{2}} \nonumber \\\le & {} C_r \Vert A^{r} f\Vert \sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} |({\varvec{k}}',\pm 1)|^{1-r} \Big (\sum \limits _{k_3\ne 0} |{\varvec{k}}|^{2r+2} |{\hat{g}}_{{\varvec{k}}}|^2 \Big )^{\frac{1}{2}} \\&\times \Big ( \sum \limits _{k_3\ne 0} \sum \limits _{{\varvec{j}}'\in {\mathbb {Z}}^2 } |({\varvec{j}}'+{\varvec{k}}',k_3)|^{2r}|{\hat{h}}_{-({\varvec{j}}'+{\varvec{k}}',k_3)}|^2\Big )^{\frac{1}{2}} \\\le & {} C_r \Vert A^r f\Vert \Vert A^r h \Vert \Big (\sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} |({\varvec{k}}',\pm 1)|^{2-2r}\Big )^{\frac{1}{2}} \Big (\sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} \sum \limits _{k_3\ne 0} |{\varvec{k}}|^{2r+2} |{\hat{g}}_{{\varvec{k}}}|^2 \Big )^{\frac{1}{2}}\\\le & {} C_r \Vert A^r f\Vert \Vert A^{r+1} g\Vert \Vert A^r h \Vert . \end{aligned}$$
Next, for \(I_{22}\), we have
$$\begin{aligned} I_{22}= & {} \tau C_r\sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{H}}_{{\varvec{l}}}| |{\varvec{j}}'||{\varvec{k}}|^{r+2} e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|} \\&+ \tau C_r\sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{H}}_{{\varvec{l}}}| |{\varvec{j}}'|^{r+1} |{\varvec{k}}|^{2} e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|} \\:= & {} I_{221} + I_{222}. \end{aligned}$$
Noticing that \(|{\varvec{k}}|^{\frac{1}{2}}\le C(|{\varvec{j}}'|^{\frac{1}{2}} + |{\varvec{l}}|^{\frac{1}{2}})\) and \(|{\varvec{j}}'|^{\frac{1}{2}} \le C(|{\varvec{k}}|^{\frac{1}{2}} + |{\varvec{l}}|^{\frac{1}{2}})\), thanks to the Cauchy–Schwarz inequality, since \(r> \frac{5}{2}\), we have
$$\begin{aligned} I_{221}= & {} \tau C_r\sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{H}}_{{\varvec{l}}}| |{\varvec{j}}'||{\varvec{k}}|^{r+2} e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|} \\\le & {} \tau C_r\sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{h}}_{{\varvec{l}}}| |{\varvec{j}}'| |{\varvec{l}}|^r |{\varvec{k}}|^{r+\frac{3}{2}}(|{\varvec{j}}'|^{\frac{1}{2}}+ |{\varvec{l}}|^{\frac{1}{2}}) e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{l}}|}\\\le & {} \tau C_r\sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{h}}_{{\varvec{l}}}| |{\varvec{j}}'|^{\frac{3}{2}} |{\varvec{l}}|^{r+\frac{1}{2}} |{\varvec{k}}|^{r+\frac{3}{2}} e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{l}}|}\\\le & {} \tau C_r \Big (\sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}'\ne 0 \end{array} } \frac{1}{|j_3|^2}|{\varvec{j}}'|^{2-2r} \Big )^{\frac{1}{2}} \Big (\sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}'\ne 0 \end{array} } |{\varvec{j}}|^{2r+1} e^{2\tau |{\varvec{j}}|} |{\hat{f}}_{{\varvec{j}}}|^2 \Big )^{\frac{1}{2}}\\&\Big ( \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ k_3\ne 0 \end{array} } |{\varvec{k}}|^{2r+3} | e^{2\tau |{\varvec{k}}|} {\hat{g}}_{{\varvec{k}}}|^2\Big )^{\frac{1}{2}} \\&\times \sup \limits _{{\varvec{j}}\in {\mathbb {Z}}^3} \Big ( \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ k_3\ne 0 \end{array}} |({\varvec{j}}'+{\varvec{k}}',k_3)|^{2r+1} e^{2\tau |({\varvec{j}}'+{\varvec{k}}',k_3)|} |{\hat{h}}_{-({\varvec{j}}'+{\varvec{k}}',k_3)}|^2\Big )^{\frac{1}{2}} \nonumber \\\le & {} \tau C_r \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \Vert A^{r+\frac{3}{2}} e^{\tau A} g\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} h\Vert , \text { and } \end{aligned}$$
$$\begin{aligned} I_{222}= & {} \tau C_r\sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{H}}_{{\varvec{l}}}| |{\varvec{j}}'|^{r+1}|{\varvec{k}}|^{2} e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|} \\\le & {} \tau C_r\sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{h}}_{{\varvec{l}}}| |{\varvec{j}}'|^{r+\frac{1}{2}}|{\varvec{k}}|^{2}|{{\varvec{l}}}|^r(|{\varvec{k}}|^{\frac{1}{2}}+ |{\varvec{l}}|^{\frac{1}{2}}) e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{l}}|} \\\le & {} \tau C_r\sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{h}}_{{\varvec{l}}}| |{\varvec{j}}'|^{r+\frac{1}{2}}|{\varvec{k}}|^{\frac{5}{2}}|{{\varvec{l}}}|^{r+\frac{1}{2}} e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{l}}|} \\\le & {} \tau C_r \sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} |({\varvec{k}}',\pm 1)|^{1-r} \sum \limits _{k_3\ne 0} |{\varvec{k}}|^{r+\frac{3}{2}} e^{\tau |{\varvec{k}}|} |{\hat{g}}_{{\varvec{k}}}| \Big ( \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ {\varvec{j}} \ne 0 \end{array} } |{\varvec{j}}|^{2r+1} e^{2\tau |{\varvec{j}}|} |{\hat{f}}_{{\varvec{j}}}|^2\Big )^{\frac{1}{2}} \\&\times \Big ( \sum \limits _{j_3\ne 0} \frac{1}{|j_3|^2} \sum \limits _{{\varvec{j}}'\in {\mathbb {Z}}^2 } |({\varvec{j}}'+{\varvec{k}}',k_3)|^{2r+1} e^{2\tau |({\varvec{j}}'+{\varvec{k}}',k_3)|}|{\hat{h}}_{-({\varvec{j}}'+{\varvec{k}}',k_3)}|^2\Big )^{\frac{1}{2}} \\\le & {} \tau C_r \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} |({\varvec{k}}',\pm 1)|^{1-r} \Big (\sum \limits _{k_3\ne 0} |{\varvec{k}}|^{2r+3} e^{2\tau |{\varvec{k}}|} |{\hat{g}}_{{\varvec{k}}}|^2 \Big )^{\frac{1}{2}} \\&\times \Big ( \sum \limits _{k_3\ne 0} \sum \limits _{{\varvec{j}}'\in {\mathbb {Z}}^2 } |({\varvec{j}}'+{\varvec{k}}',k_3)|^{2r+1} e^{2\tau |({\varvec{j}}'+{\varvec{k}}',k_3)|}|{\hat{h}}_{-({\varvec{j}}'+{\varvec{k}}',k_3)}|^2\Big )^{\frac{1}{2}} \\\le & {} \tau C_r \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} h\Vert \Big (\sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} |({\varvec{k}}',\pm 1)|^{2-2r}\Big )^{\frac{1}{2}}\\&\Big (\sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} \sum \limits _{k_3\ne 0} |{\varvec{k}}|^{2r+3} e^{2\tau |{\varvec{k}}|} |{\hat{g}}_{{\varvec{k}}}|^2 \Big )^{\frac{1}{2}} \\\le & {} \tau C_r \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \Vert A^{r+\frac{3}{2}} e^{\tau A} g\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} h\Vert . \end{aligned}$$
Therefore, \(I_2\) satisfies the desired estimates.
To estimate \(I_1\), we use (A.1) with \(\xi = |{\varvec{l}}|\), \(\eta = |{\varvec{j}}|\), and with \(|\xi - \eta | = \Big ||{\varvec{l}}|- |{\varvec{j}}| \Big |\le |-{\varvec{l}}-{\varvec{j}}| = |{\varvec{k}}|\), to obtain
$$\begin{aligned} I_1 \le C_r \sum \limits _{\begin{array}{c} {\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0\\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}} \frac{1}{|j_3|} |{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{H}}_{{\varvec{l}}}| |{\varvec{j}}'||{\varvec{k}}|^2 \Big ( |{\varvec{k}}|^{r-1} + |{\varvec{j}}|^{r-1} + \tau (|{\varvec{k}}|^{r} + |{\varvec{j}}|^{r})e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|} \Big ). \end{aligned}$$
(A.3)
Thanks to (A.3), one obtains that
$$\begin{aligned} I_1\le & {} C_r \sum \limits _{\begin{array}{c} {\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0\\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{h}}_{{\varvec{l}}}| |{\varvec{j}}'||{\varvec{k}}|^2 |{\varvec{l}}|^r(|{\varvec{k}}|^{r-1} + |{\varvec{j}}|^{r-1}) \\&+ \tau C_r \sum \limits _{\begin{array}{c} {\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0\\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{H}}_{{\varvec{l}}}| |{\varvec{j}}'||{\varvec{k}}|^2 (|{\varvec{k}}|^{r} + |{\varvec{j}}|^{r})e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|}:= I_{11} + I_{12}. \end{aligned}$$
Here
$$\begin{aligned} I_{11}\le & {} C_r\Big (\sum \limits _{\begin{array}{c} {\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0\\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{h}}_{{\varvec{l}}}| |{\varvec{j}}||{\varvec{k}}|^{r+1} |{\varvec{l}}|^r + \frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{h}}_{{\varvec{l}}}| |{\varvec{j}}|^{r}|{\varvec{k}}|^{2} |{\varvec{l}}|^r\Big ) \\:= & {} I_{111}+ I_{112}. \end{aligned}$$
Thanks to the Cauchy–Schwarz inequality, since \(r> \frac{5}{2}\), we have
$$\begin{aligned} I_{111}= & {} C_r \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}'\ne 0 \end{array} } \frac{1}{|j_3|}|{\varvec{j}}| |{\hat{f}}_{{\varvec{j}}}| \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ k_3\ne 0 \end{array} } |{\varvec{k}}|^{r+1} |{\varvec{j}}+{\varvec{k}}|^r |{\hat{g}}_{{\varvec{k}}}||{\hat{h}}_{-({\varvec{j}}+{\varvec{k}})}|\\\le & {} C_r \Big (\sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}'\ne 0 \end{array} } |{\varvec{j}}|^{2-2r} \Big )^{\frac{1}{2}} \Big (\sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}'\ne 0 \end{array} } |{\varvec{j}}|^{2r} |{\hat{f}}_{{\varvec{j}}}|^2 \Big )^{\frac{1}{2}} \Big ( \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ k_3\ne 0 \end{array} } |{\varvec{k}}|^{2r+2} |{\hat{g}}_{{\varvec{k}}}|^2\Big )^{\frac{1}{2}} \\&\times \sup \limits _{{\varvec{j}}\in {\mathbb {Z}}^3} \Big ( \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ k_3\ne 0 \end{array}} |{\varvec{j}}+{\varvec{k}}|^{2r}|{\hat{h}}_{-({\varvec{j}}+{\varvec{k}})}|^2\Big )^{\frac{1}{2}} \le C_r \Vert A^r f\Vert \Vert A^{r+1} g\Vert \Vert A^r h \Vert , \text { and } \end{aligned}$$
$$\begin{aligned} I_{112}= & {} C_r \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ k_3\ne 0 \end{array} } |{\varvec{k}}|^{2} |{\hat{g}}_{{\varvec{k}}}| \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}' \ne 0 \end{array} } \frac{1}{|j_3|} |{\varvec{j}}|^{r} |{\hat{f}}_{{\varvec{j}}}| |{\varvec{j}}+{\varvec{k}}|^{r}|{\hat{h}}_{-({\varvec{j}}+{\varvec{k}})}| \\\le & {} C_r \Big (\sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ {\varvec{k}} \ne 0 \end{array} } |{\varvec{k}}|^{2-2r} \Big )^{\frac{1}{2}} \Big (\sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ {\varvec{k}} \ne 0 \end{array} } |{\varvec{k}}|^{2r+2} |{\hat{g}}_{{\varvec{k}}}|^2 \Big )^{\frac{1}{2}} \Big ( \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ {\varvec{j}} \ne 0 \end{array} } |{\varvec{j}}|^{2r} |{\hat{f}}_{{\varvec{j}}}|^2\Big )^{\frac{1}{2}} \\&\times \sup \limits _{{\varvec{k}}\in {\mathbb {Z}}^3} \Big ( \sum \limits _{{\varvec{j}}\in {\mathbb {Z}}^3} |{\varvec{j}}+{\varvec{k}}|^{2r}|{\hat{h}}_{-({\varvec{j}}+{\varvec{k}})}|^2\Big )^{\frac{1}{2}} \le C_r \Vert A^r f\Vert \Vert A^{r+1} g\Vert \Vert A^r h \Vert . \end{aligned}$$
Next, for \(I_{12}\), we have
$$\begin{aligned} I_{12}\le & {} \tau C_r\sum \limits _{\begin{array}{c} {\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0\\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{H}}_{{\varvec{l}}}| |{\varvec{j}}||{\varvec{k}}|^{r+2} e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|} \\&+ \tau C_r\sum \limits _{\begin{array}{c} {\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0\\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{H}}_{{\varvec{l}}}| |{\varvec{j}}|^{r+1} |{\varvec{k}}|^{2} e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|} := I_{121} + I_{122}. \end{aligned}$$
Since \(|{\varvec{k}}|^{\frac{1}{2}}\le C(|{\varvec{j}}|^{\frac{1}{2}} + |{\varvec{l}}|^{\frac{1}{2}})\) and \(|{\varvec{j}}|^{\frac{1}{2}} \le C(|{\varvec{k}}|^{\frac{1}{2}} + |{\varvec{l}}|^{\frac{1}{2}})\), thanks to the Cauchy–Schwarz inequality, since \(r> \frac{5}{2}\), we have
$$\begin{aligned} I_{121}= & {} \tau C_r\sum \limits _{\begin{array}{c} {\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0\\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{H}}_{{\varvec{l}}}| |{\varvec{j}}||{\varvec{k}}|^{r+2} e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|} \\\le & {} \tau C_r\sum \limits _{\begin{array}{c} {\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0\\ j_3, k_3, {\varvec{j}}', {\varvec{l}} \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{h}}_{{\varvec{l}}}| |{\varvec{j}}| |{\varvec{l}}|^r |{\varvec{k}}|^{r+\frac{3}{2}}(|{\varvec{j}}|^{\frac{1}{2}}+ |{\varvec{l}}|^{\frac{1}{2}}) e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{l}}|}\\\le & {} \tau C_r\sum \limits _{\begin{array}{c} {\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0\\ j_3, k_3, {\varvec{j}}' , {\varvec{l}}\ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{h}}_{{\varvec{l}}}| |{\varvec{j}}|^{\frac{3}{2}} |{\varvec{l}}|^{r+\frac{1}{2}} |{\varvec{k}}|^{r+\frac{3}{2}} e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{l}}|}\\\le & {} \tau C_r \Big (\sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}'\ne 0 \end{array} } |{\varvec{j}}|^{2-2r} \Big )^{\frac{1}{2}} \Big (\sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}'\ne 0 \end{array} } |{\varvec{j}}|^{2r+1} e^{2\tau |{\varvec{j}}|} |{\hat{f}}_{{\varvec{j}}}|^2 \Big )^{\frac{1}{2}} \Big ( \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ k_3\ne 0 \end{array} } |{\varvec{k}}|^{2r+3} | e^{2\tau |{\varvec{k}}|} {\hat{g}}_{{\varvec{k}}}|^2\Big )^{\frac{1}{2}} \\&\times \sup \limits _{{\varvec{j}}\in {\mathbb {Z}}^3} \Big ( \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ k_3\ne 0 \end{array}} |{\varvec{j}}+{\varvec{k}}|^{2r+1} e^{2\tau |{\varvec{j}}+{\varvec{k}}|} |{\hat{h}}_{-({\varvec{j}}+{\varvec{k}})}|^2\Big )^{\frac{1}{2}} \\\le & {} \tau C_r \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \Vert A^{r+\frac{3}{2}} e^{\tau A} g\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} h\Vert , \text { and } \\ I_{122}= & {} \tau C_r\sum \limits _{\begin{array}{c} {\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0\\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{H}}_{{\varvec{l}}}| |{\varvec{j}}|^{r+1}|{\varvec{k}}|^{2} e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|} \\\le & {} \tau C_r\sum \limits _{\begin{array}{c} {\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0\\ j_3, k_3, {\varvec{j}}', {\varvec{l}} \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{h}}_{{\varvec{l}}}| |{\varvec{j}}|^{r+\frac{1}{2}}|{\varvec{k}}|^{2}|{{\varvec{l}}}|^r(|{\varvec{k}}|^{\frac{1}{2}}+ |{\varvec{l}}|^{\frac{1}{2}}) e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{l}}|}\\\le & {} \tau C_r\sum \limits _{\begin{array}{c} {\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0\\ j_3, k_3, {\varvec{j}}', {\varvec{l}} \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{h}}_{{\varvec{l}}}| |{\varvec{j}}|^{r+\frac{1}{2}}|{\varvec{k}}|^{\frac{5}{2}}|{{\varvec{l}}}|^{r+\frac{1}{2}} e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|}e^{\tau |{\varvec{l}}|}\\\le & {} \tau C_r \Big (\sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ {\varvec{k}} \ne 0 \end{array} } |{\varvec{k}}|^{2-2r}\Big )^{\frac{1}{2}} \Big (\sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ {\varvec{k}} \ne 0 \end{array} } |{\varvec{k}}|^{2r+3} e^{2\tau |{\varvec{k}}|} |{\hat{g}}_{{\varvec{k}}}|^2\Big )^{\frac{1}{2}} \Big ( \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ {\varvec{j}} \ne 0 \end{array} } |{\varvec{j}}|^{2r+1} e^{2\tau |{\varvec{j}}|} |{\hat{f}}_{{\varvec{j}}}|^2\Big )^{\frac{1}{2}} \\&\times \sup \limits _{{\varvec{k}}\in {\mathbb {Z}}^3}\Big (\sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ {\varvec{j}} \ne 0 \end{array} } |{\varvec{j}}+{\varvec{k}}|^{2r+1} e^{2\tau |{\varvec{j}}+{\varvec{k}}|}|{\hat{h}}_{-({\varvec{j}}+{\varvec{k}})}|^2\Big )^{\frac{1}{2}} \\\le & {} \tau C_r \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \Vert A^{r+\frac{3}{2}} e^{\tau A} g\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} h\Vert . \end{aligned}$$
Therefore, \(I_1\) satisfies the desired estimates. The proof is completed. \(\square \)
Finally, we sketch the proof of Lemma A.6.
Proof
(proof of Lemma A.6) Similar to the proof of Lemma A.7, we have
$$\begin{aligned} I:= & {} \Big |\Big \langle A^r e^{\tau A} \Big ( (\int _0^z \nabla \cdot f({\varvec{x}}',s)ds) \partial _z g \Big ), A^r e^{\tau A} h \Big \rangle \\&- \Big \langle (\int _0^z \nabla \cdot f({\varvec{x}}',s)ds)A^r e^{\tau A} \partial _z g , A^r e^{\tau A} h \Big \rangle \Big | \\= & {} \Big | \Big \langle (\int _0^z \nabla \cdot f({\varvec{x}}',s)ds) \partial _z g, A^r e^{\tau A} H \Big \rangle - \Big \langle (\int _0^z \nabla \cdot f({\varvec{x}}',s)ds)A^r e^{\tau A} \partial _z g , H \Big \rangle \Big |. \\\le & {} C \sum \limits _{\begin{array}{c} {\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0\\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}} \frac{1}{|j_3|} |{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{H}}_{{\varvec{l}}}| |{\varvec{j}}'||{\varvec{k}}| \Big | |{\varvec{l}}|^r e^{\tau |{\varvec{l}}|} - |{\varvec{k}}|^r e^{\tau |{\varvec{k}}|} \Big | \\&+ C \sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{H}}_{{\varvec{l}}}| |{\varvec{j}}'||{\varvec{k}}| \Big | |{\varvec{l}}|^r e^{\tau |{\varvec{l}}|} - |{\varvec{k}}|^r e^{\tau |{\varvec{k}}|} \Big |:= I_1 + I_2. \end{aligned}$$
For \(I_1\), since \({\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0\), we use (A.1) with \(\xi = |{\varvec{l}}|\), \(\eta = |{\varvec{k}}|\) and \(|\xi - \eta | = \Big ||{\varvec{l}}|- |{\varvec{k}}| \Big |\le \Big |-{\varvec{l}}-{\varvec{k}}\Big | = |{\varvec{j}}|\), to conclude
$$\begin{aligned}&I_1 \le C_r \sum \limits _{\begin{array}{c} {\varvec{j}}+{\varvec{k}}+{\varvec{l}}=0\\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}} \frac{1}{|j_3|} |{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{H}}_{{\varvec{l}}}| |{\varvec{j}}'||{\varvec{j}}||{\varvec{k}}|\nonumber \\&\Big ( |{\varvec{k}}|^{r-1} + |{\varvec{j}}|^{r-1} + \tau (|{\varvec{k}}|^{r} + |{\varvec{j}}|^{r})e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|} \Big ). \end{aligned}$$
(A.4)
For \(I_2\), since \(({\varvec{j}}',0)+{\varvec{k}}+{\varvec{l}}=0\), we use (A.1) with \(\xi = |{\varvec{l}}|\), \(\eta = |{\varvec{k}}|\) and \(|\xi - \eta | = \Big ||{\varvec{l}}|- |{\varvec{k}}| \Big |\le \Big |-{\varvec{l}}-{\varvec{k}}\Big | = |{\varvec{j}}'|\), to obtain
$$\begin{aligned}&I_2 \le C_r\sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}}\frac{1}{|j_3|}|{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}||{\hat{H}}_{{\varvec{l}}}| |{\varvec{j}}'|^2|{\varvec{k}}|\nonumber \\&\Big ( |{\varvec{k}}|^{r-1} + |{\varvec{j}}'|^{r-1} + \tau (|{\varvec{k}}|^{r} + |{\varvec{j}}'|^{r})e^{\tau |{\varvec{k}}|}e^{\tau |{\varvec{j}}|} \Big ). \end{aligned}$$
(A.5)
Observe that the difference between the sums in the right-hand sides of (A.4) and (A.3) is manifested in the factors \(|{\varvec{j}}'||{\varvec{j}}||{\varvec{k}}|\) and \(|{\varvec{j}}'||{\varvec{k}}|^2\), and between (A.5) and (A.2) is manifested in the factors \(|{\varvec{j}}'|^2|{\varvec{k}}|\) and \(|{\varvec{j}}'||{\varvec{k}}|^2\). Therefore, one can follow the estimates of \(I_1\) in (A.3) and \(I_2\) in (A.2), and obtain that \(I_1\) in (A.4) and \(I_2\) in (A.5) satisfy the desired bound in Lemma A.6. \(\square \)
