function $f=f(x,v),$ $\|\int_{\mathbb{R}^{3}}fdv\|_{\mathcal{F}^{\lambda|\tau|+\mu}}\leq\|f\|_{\mathcal{Z}^{\lambda,\mu;1}_{\tau}}.$

$Proof.$ Here we only give the proof of (v). By the invariance under the action of free transport, it is sufficient to do the proof for $t=0.$ Applying the Fourier transform formula, we have $$\nabla^{m}_{v}(v\hat{f})(k,v)=\int_{\mathbb{R}}\partial_{\eta}\tilde{f}(k,\eta)(2i\pi\eta)^{m}e^{2i\pi\eta\cdot v}d\eta,$$ and therefore $$\sum_{m\in\mathbb{N}_{0}}\frac{\lambda^{m}}{m!}\int_{\mathbb{R}}\bigg|\int_{\mathbb{R}}
\partial_{\eta}\tilde{f}(k,\eta)(2i\pi\eta)^{m}e^{2i\pi\eta\cdot v}d\eta\bigg|dv
\leq\sum_{m\in\mathbb{N}_{0}}\frac{\lambda^{m}}{m!}\sup_{\eta\in\mathbb{R}}|(2\pi\eta)^{m}\partial_{\eta}\tilde{f}(k,\eta)|
\int_{\mathbb{R}}\bigg|\int_{\mathbb{R}}e^{2i\pi\eta\cdot v}d\eta\bigg|dv$$ $$\leq C(\lambda,\bar{\lambda})\sum_{m\in\mathbb{N}_{0}}\frac{\bar{\lambda}^{m}}{m!}\sup_{\eta\in\mathbb{R}}|(2\pi\eta)^{m}\tilde{f}(k,\eta)|\leq
C(\lambda,\bar{\lambda})\sum_{m\in\mathbb{N}_{0}}
\frac{\bar{\lambda}^{m}}{m!}\int_{\mathbb{R}}|\nabla^{m}_{v}\hat{f}(k,v)|dv,$$ where the last second inequality uses the property (iv), then we have $$e^{2\pi\mu|k|}\sum_{m\in\mathbb{N}_{0}}\frac{\lambda^{m}}{m!}\int_{\mathbb{R}}\bigg|\int_{\mathbb{R}}
\partial_{\eta}\tilde{f}(k,\eta)(2i\pi\eta)^{m}e^{2i\pi\eta\cdot v}d\eta\bigg|dv
\leq e^{2\pi\mu|k|}\sum_{m\in\mathbb{N}_{0}}\frac{\bar{\lambda}^{m}}{m!}\int_{\mathbb{R}}|\nabla^{m+1}_{v}\hat{f}(k,v)|dv.$$ This establishes (v).

For any $X\in\{\mathcal{C},\mathcal{F},\mathcal{Z}\}$ and any $t,\tau\in\mathbb{R},$ $$\|f\circ S^{0}_{\tau}\|_{X^{\lambda,\mu}_{\tau}}=\|f\|_{X^{\lambda,\mu}_{t+\tau}}.$$

Let $\lambda,\mu\geq0,t\in\mathbb{R},$ and consider two functions $F,G:\mathbb{T}\times\mathbb{R}\rightarrow\mathbb{T}\times\mathbb{R}.$ Then there is $ \varepsilon\in(0,\frac{1}{2})$ such that if $F,G$ satisfy $$\begin{aligned}
\|\nabla(F-Id)\|_{\mathcal{Z}^{\lambda',\mu'}_{\tau}}\leq\varepsilon,\end{aligned}$$ where $\lambda'=\lambda +2\|F-G\|_{\mathcal{Z}^{\lambda ,\mu}_{\tau}},\quad\mu'=\mu+2(1+|\tau|)\|F-G\|_{\mathcal{Z}^{\lambda ,\mu}_{\tau}} ,$ then $F$ is invertible and $$\begin{aligned}
&\|F^{-1}\circ G-Id\|_{\mathcal{Z}^{\lambda ,\mu}_{\tau}}\leq2\|F-G\|_{\mathcal{Z}^{\lambda ,\mu}_{\tau}}.\end{aligned}$$

For any $\lambda,\mu\geq0$ and any $p\in[1,\infty],\tau\in\mathbb{R},\sigma\in\mathbb{R},a\in\mathbb{R}\setminus\{0\}$ and $b\in\mathbb{R},$ we have $$\|f(x+bv'+X(x,v'),av'+V(x,v'))\|_{\mathcal{Z}^{\lambda,\mu;p}_{\tau}}\leq|a|^{-\frac{3}{p}}\|f\|_{\mathcal{Z}^{\alpha,\beta;p}_{\sigma}},$$ where $\alpha=\lambda|a|+\|V\|_{\mathcal{Z}^{\lambda,\mu}_{\tau}},\quad \beta=\mu+\lambda|b+\tau-a\sigma|+\|X-\sigma V\|_{\mathcal{Z}^{\lambda,\mu}_{\tau}}.$

Let $G=G(x,v)$ and $R=R(x,v)$ be valued in $\mathbb{R},$ and $\beta(x)=\int_{\mathbb{R}}(G\cdot R)(x',v')dv'.$ Then for any $\lambda,\mu,t\geq0$ and any $b>-1,$ we have $$\|\beta\|_{\mathcal{F}^{\lambda t+\mu}}\leq3\|G\|_{\mathcal{Z}_{\tau-\frac{bt}{1+b}}^{\lambda(1+b),\mu;1}}
\|R\|_{\mathcal{Z}_{\tau-\frac{bt}{1+b}}^{\lambda(1+b),\mu}}.$$

Linear cyclotron damping revisited
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In this section, we recast the linear damping in the hybrid analytic norms.

For any $\eta,v\in\mathbb{R}^{3},k\in\mathbb{N}_{0}^{3},$ we assume that the following conditions hold in equations (0.9).

-   $ W(x)$ is an odd function on $x_{3},$ $|\widehat{W}(k)|\leq\frac{1}{1+|k|^{\gamma}},\gamma>1,$ where $ W(x)=(W_{1}(x),W_{2}(x),0);$

-   $\|\nabla_{v}f^{0}\|_{\mathcal{C}^{\lambda_{0};1}}\leq C^{0};\|f_{0}\|_{\mathcal{Z}^{\lambda_{0},\mu_{0};1}_{0}}\leq\delta_{0}$ for 