is a circle; in the $\hat{z}$ direction, the damping still occurs. Therefore, there is no immediate tendency toward trapping. This is a crucial point for cyclotron damping. And in the latter case, the motion of plasma particles moves along spiral trajectories.

Before proving $Theorem$ 1.1, we give a key lemma.

Under the assumptions of Theorem 0.1, let $\Phi(t,k)=e^{2\pi\lambda'_{0}|k_{3}|t}$ $\cdot e^{2\pi\lambda'_{0}|\eta_{k1}|}e^{2\pi\lambda'_{0}|\eta_{k2}|}\hat{\rho}(t,k),$ $A(t,k)=\int_{\mathbb{T}^{3}}\int_{\mathbb{R}^{3}}
         e^{-2\pi ix\cdot k}f_{0}(x'(0,x,v),v'(0,x,v))dv'dx,$ where $\lambda_{0}>\lambda'_{0}>0,\lambda_{0}$ is defined in Theorem 0.1, we have $$\begin{aligned}
\|\Phi\|_{L^{\infty}(dt)}\leq\bigg(1+\frac{C(k,W,\Omega)}{(\lambda_{0}-\lambda'_{0})^{\frac{3}{2}}}\bigg)\|e^{\lambda_{0}\nu_{k}}A\|_{L^{\infty}(dt)},\end{aligned}$$ where $\eta_{k1}=\frac{1}{\Omega}(-k_{2}\cos\Omega t+k_{2}-k_{1}\sin\Omega t),\eta_{k2}=\frac{1}{\Omega}(-k_{2}\sin\Omega t-k_{1}+k_{1}\cos\Omega t), \nu_{k}=|\eta_{k1}|+|\eta_{k2}|+|k_{3}|t.$

$Proof.$ First, we consider the case that $\omega\neq0,k_{3}\neq0,$ $$\begin{aligned}
&\tilde{\rho}(\omega,k)=\int_{\mathbb{R}^{+}}\int_{\mathbb{T}^{3}}\int_{\mathbb{R}^{3}}e^{2\pi it\omega}e^{-2\pi ix\cdot k}f(t,x,v)dxdvdt\notag\\
&=\int_{\mathbb{R}^{+}}\int_{\mathbb{T}^{3}}\int_{\mathbb{R}^{3}}e^{2\pi it\omega}e^{-2\pi ix\cdot k}f_{0}(x'(0,x,v),v'(0,x,v))dvdxdt-\frac{q}{m}\int_{\mathbb{R}^{+}}\int_{\mathbb{T}^{3}}\int^{t}_{0}\int_{\mathbb{R}^{3}}\notag\\
&\cdot e^{2\pi it\omega}e^{-2\pi ix\cdot k}[(E+v'(\tau,x,v)\times B)\cdot\nabla'_{v}f^{0}](\tau,x'(\tau,x,v),v'(\tau,x,v))dvdxd\tau dt.\notag\\\end{aligned}$$ Recall $E(t,x)=W(x)\ast\rho(t,x),$ $\partial_{t}B(t,x)=\nabla_{x}\times E(t,x),$ then taking the Fourier transform in the variables $t,x,$ $$\widetilde{E}(\omega,k)=\widehat{W}(k)\tilde{\rho}(\omega,k),\quad\omega\widetilde{B}(\omega,k)=k\times\widetilde{E}(\omega,k).$$ Furthermore, we get $$\begin{aligned}
&v\times\widetilde{B}(\omega,k)=\frac{1}{\omega}[v\times(k\times\widetilde{E}(\omega,k))]\notag\\
&=\frac{1}{\omega}\bigg(v_{2}(k_{1}\widehat{W}_{2}-k_{2}\widehat{W}_{1})-v_{3}k_{3}\widehat{W}_{1},-v_{1}(k_{1}
\widehat{W}_{2}-k_{2}\widehat{W}_{1})+v_{3}k_{3}\widehat{W}_{2},
v_{1}k_{3}\widehat{W}_{1}+v_{2}k_{3}\widehat{W}_{2}\bigg)\tilde{\rho}(\omega,k).\notag\\\end{aligned}$$ Combining (1.3)-(1.4) and (2.3)-(2.4), and note that $dv\rightarrow dv'$ preserves the measure, we can change between $dv$ and $dv'$ whenever we need, but in order to simply the notations, we don’t differentiate the notations $dv$ and $dv'$ in this paper, so we have

$$\tilde{\rho}(\omega,k)=\int_{\mathbb{R}^{+}}\int_{\mathbb{T}^{3}}\int_{\mathbb{R}^{3}}e^{2\pi it\omega}e^{-2\pi ix\cdot k}f_{0}(x'(0,x,v),v'(0,x,v))dvdxdt$$ $$+\frac{q}{m}\tilde{\rho}(\omega,k)\frac{1}{\omega}\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{3}}e^{2\pi it\omega}e^{-2\pi ik_{3}v_{3}t}
e^{-2\pi i\eta_{k2}v'_{2}}e^{-2\pi i\eta_{k1}v'_{1}}(k_{3}v_{3})(\widehat{W}_{1}\partial_{v'_{1}}f^{0}-\widehat{W}_{2}\partial_{v'_{2}}f^{0})dvdt$$ $$+\frac{q}{m}\tilde{\rho}(\omega,k)\frac{1}{\omega}\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{3}}e^{2\pi it\omega}e^{-2\pi ik_{3}v_{3}t}
e^{-2\pi i\eta_{k2}v'_{2}}e^{-2\pi i\eta_{k1}v'_{1}}(k_{1}\widehat{W}_{2}-k_{2}\widehat{W}_{1})$$ $$\cdot(v_{2}\partial_{v'_{1}}f^{0}-v_{1}\partial_{v'_{2}}f^{0})dvdt-\frac{q}{m}\tilde{\rho}(\omega,k)\frac{1}{\omega}\int_{\mathbb{R}^{+}}
\int_{\mathbb{R}^{3}}e^{2\pi it\omega}e^{-2\pi ik_{3}v_{3}t}
e^{-2\pi i\eta_{k2}v'_{2}}e^{-2\pi i\eta_{k1}v'_{1}}$$ $$\begin{aligned}
&\cdot(v_{1}\widehat{W}_{1}+v_{2}\widehat{W}_{2})k_{3}\partial_{v'_{3}}f^{0}
dvdt-\frac{q}{m}\tilde{\rho}(\omega,k)\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{3}}e^{2\pi it\omega}e^{-2\pi ik_{3}v_{3}t}
e^{-2\pi 