norms) $$\|f\|_{\mathcal{C}^{\lambda,\mu}}=\sum_{m,n\in\mathbb{N}_{0}^{3}}\frac{\lambda^{n}}{n!}\frac{\mu^{m}}{m!}
            \|\nabla^{m}_{x'_{\tau}}\nabla_{v'_{\tau}}^{n}f\|_{L^{\infty}(\mathbb{T}_{x'}^{3}\times\mathbb{R}^{3}_{v'})},\quad
            \|f\|_{\mathcal{F}^{\lambda,\mu}}=\sum_{k\in\mathbb{Z}^{3}}\int_{\mathbb{R}^{3}}|\tilde{f}(k,\eta)|
           e^{2\pi\lambda|\eta|}e^{2\pi\mu|k|}d\eta,$$ $$\|f\|_{\mathcal{Z}^{\lambda,\mu}}=\sum_{l\in\mathbb{Z}^{3}}\sum_{n\in\mathbb{N}_{0}^{3}}
            \frac{\lambda^{n}}{n!}e^{2\pi\mu|l|}\|\widehat{\nabla_{v'_{\tau}}^{n}f(l,v)}\|_{L^{\infty}(\mathbb{R}_{v'}^{3})}.$$

(Time-shift pure and hybrid analytic norms) For any $\lambda,\mu\geq0,p\in[1,\infty],$ we define $$\|f\|_{\mathcal{C}^{\lambda,\mu}_{t,\tau}}=\|f\circ S^{0}_{t,\tau}(x',v')\|_{\mathcal{C}^{\lambda,\mu}}=\sum_{m,n\in\mathbb{N}_{0}^{3}}\frac{\lambda^{n}}{n!}\frac{\mu^{m}}{m!}
            \|\nabla^{m}_{x'_{\tau}}[\nabla_{v'_{\tau}}+\bigg(\frac{\sin\Omega(t-\tau)}{\Omega},\frac{\sin\Omega(t-\tau)}{\Omega},t-\tau\bigg)
            \cdot\nabla_{x'_{\tau}}]^{n}f\|_{L^{\infty}(\mathbb{T}_{x}^{3}\times\mathbb{R}^{3}_{v})},$$ $$\|f\|_{\mathcal{F}^{\lambda,\mu}_{t,\tau}}=\|f\circ S^{0}_{t,\tau}(x',v')\|_{\mathcal{F}^{\lambda,\mu}}=\sum_{k\in\mathbb{Z}^{3}}\int_{\mathbb{R}^{3}}|\tilde{f}(k,\eta)|
           e^{2\pi\lambda|\eta+k\cdot(\frac{\sin\Omega(t-\tau)}{\Omega},\frac{\sin\Omega(t-\tau)}{\Omega},(t-\tau))|}e^{2\pi\mu|k|}d\eta,$$ $$\|f\|_{\mathcal{Z}^{\lambda,\mu}_{t,\tau}}=\|f\circ S^{0}_{t,\tau}(x,v)\|_{\mathcal{Z}^{\lambda,\mu}}=\sum_{l\in\mathbb{Z}^{3}}\sum_{n\in\mathbb{N}_{0}^{3}}
            \frac{\lambda^{n}}{n!}e^{2\pi\mu|l|}\|[\nabla_{v'_{\tau}}+2i\pi\bigg(\frac{\sin\Omega(t-\tau)}{\Omega},\frac{\sin\Omega(t-\tau)}{\Omega},t-\tau\bigg)\cdot l]^{n}\hat{f}(l,v)\|
            _{L^{\infty}(\mathbb{R}_{v}^{3})},$$ $$\|f\|_{\mathcal{Z}^{\lambda,\mu;p}_{t,\tau}}=\sum_{l\in\mathbb{Z}^{3}}\sum_{n\in\mathbb{N}_{0}^{3}}
            \frac{\lambda^{n}}{n!}e^{2\pi\mu|l|}\|[\nabla_{v'_{\tau}}+2i\pi\bigg(\frac{\sin\Omega(t-\tau)}{\Omega},\frac{\sin\Omega(t-\tau)}{\Omega},t-\tau\bigg)\cdot l]^{n}\hat{f}(l,v)\|_{L^{p}(\mathbb{R}_{v'}^{3})},$$ $$\|f\|_{\mathcal{Y}^{\lambda,\mu}_{t,\tau}}=\|f\|_{\mathcal{F}^{\lambda,\mu;\infty}_{t,\tau}}
            =\sup_{k\in\mathbb{Z}^{3},\eta\in\mathbb{R}^{3}}e^{2\pi\mu|k|}
            e^{2\pi\lambda|\eta+k\cdot(\frac{\sin\Omega(t-\tau)}{\Omega},\frac{\sin\Omega(t-\tau)}{\Omega},(t-\tau))|}
            |\hat{f}(k,\eta)|.$$

From the above definitions, we can state some simple and important propositions, and the related proofs can be found in \[23\], so we remove the proofs. From (2.6), we know that the damping occurs only in the $\hat{z}$ direction. Therefore, we will mainly focus on the third component of the “phase space" $(x,v)$ when referring to the damping mechanics of the wave propagation.

For any $\tau\in\mathbb{R},\lambda,\mu\geq 0,$

-   if $f$ is a function only of $x,$ that is, $t=\tau$ in the second equality of (2.5), then $\|f\|_{\mathcal{C}^{\lambda,\mu}_{\tau}}=\|f\|_{\mathcal{C}^{\lambda|\tau|+\mu}},\|f\|_{\mathcal{F}^{\lambda,\mu}_{\tau}}
               =\|f\|_{\mathcal{Z}^{\lambda,\mu}_{\tau}}=\|f\|_{\mathcal{F}^{\lambda|\tau|+\mu}};$

-   if $f$ is a function only of $v,$ then $\|f\|_{\mathcal{C}^{\lambda,\mu;p}_{\tau}}=\|f\|_{\mathcal{Z}^{\lambda,\mu;p}_{\tau}}=\|f\|_{\mathcal{C}^{\lambda,;p}};$

-   for any $\lambda>0,$ then $\|f\circ(Id+G)\|_{\mathcal{F}^{\lambda}}\leq\|f\|_{\mathcal{F}^{\lambda+\nu}},\nu=\|G\|_{\dot{\mathcal{F}}^{\lambda}};$

-   for any $\bar{\lambda}>\lambda,p\in[1,\infty],$ $\|\nabla f\|_{\mathcal{C}^{\lambda;p}}\leq
                \frac{1}{\lambda e\log(\frac{\bar{\lambda}}{\lambda} )}
                \| f\|_{\mathcal{C}^{\bar{\lambda};p}},$ $\|\nabla f\|_{\mathcal{F}^{\lambda;p}}\leq\frac{1}{2\pi e(\bar{\lambda}-\lambda )}
                \| f\|_{\mathcal{F}^{\bar{\lambda};p}},$

-   for any $\bar{\lambda}>\lambda>0,\mu>0,$ then $\|vf\|_{\mathcal{Z}^{\lambda,\mu;1}_{\tau}}\leq\|f\|_{\mathcal{Z}^{\bar{\lambda},\mu;1}_{\tau}};$

-   for any $\bar{\lambda}>\lambda,\bar{\mu}>\mu,$ $\|\nabla_{v}f\|_{\mathcal{Z}_{\tau}^{\lambda,\mu;p}}
              \leq C(d)\bigg(\frac{1}{\lambda\log(\frac{\bar{\lambda}}{\lambda})}\|f\|_{\mathcal{Z}_{\tau}^{\bar{\lambda},\bar{\mu};p}}+\frac{\tau}{\bar{\mu}-\mu}
     \|f\|_{\dot{\mathcal{Z}}_{\tau}^{\bar{\lambda},\bar{\mu};p}}\bigg);$

-   for any $\bar{\lambda}>\lambda,$ $\|(\nabla_{v}+\tau\nabla_{x})f\|_{\mathcal{Z}_{\tau}^{\lambda,\mu;p}}\leq \frac{1}{C(d)\lambda\log(\frac{\bar{\lambda}}{\lambda})}
     \|f\|_{\mathcal{Z}_{\tau}^{\bar{\lambda},\mu;p}};$

-   for any $\bar{\lambda}\geq\lambda\geq0,\bar{\mu}\geq\mu\geq0,$ then $\|f\|_{\mathcal{Z}_{\tau}^{\lambda,\mu}}
      \leq_{\mathcal{Z}_{\tau}^{\bar{\lambda},\bar{\mu}}}.$ Moreover, for any $\tau,\bar{\tau}\in\mathbb{R},$ $p\in[1,\infty],$ we have $\|f\|_{\mathcal{Z}_{\tau}^{\lambda,\mu;p}}\leq\|f\|_{\mathcal{Z}_{\bar{\tau}}^{\lambda,\mu+\lambda|\tau-\bar{\tau}|;p}};$

-   $\|f\|_{\mathcal{Y}^{\lambda,\mu}_{\tau}}\leq\|f\|_{\mathcal{Z}^{\lambda,\mu;1}_{\tau}};$

-   for any 