we get $\hat{\rho}(t,k_{12},0)=\hat{A}(t,k_{12},0)+\hat{\rho}(t,k_{12},0)\ast\hat{\mathcal{L}}(t,k_{12},0),$ and

$$\begin{aligned}
&e^{2\pi\lambda'_{0}\eta_{k1}}e^{2\pi\lambda'_{0}\eta_{k2}}\hat{\rho}(t,k_{12},0)=e^{2\pi\lambda'_{0}\eta_{k1}}e^{2\pi\lambda'_{0}\eta_{k2}}
\hat{A}(t,k_{12},0)+e^{2\pi\lambda'_{0}\eta_{k1}}e^{2\pi\lambda'_{0}\eta_{k2}}\hat{\rho}(t,k_{12},0)\ast\hat{\mathcal{L}}(t,k_{12},0).\end{aligned}$$

Then $$\begin{aligned}
&\|\Phi(t,k_{12},0)\|_{L^{2}(dt)}
=\|\widetilde{\Phi}(\omega,k_{12},0)\|_{L^{2}}\leq\|\widetilde{\mathcal{A}}(\omega,k_{12},0)\|_{L^{2}}+\|\widetilde{\Phi}(\omega,k_{12},0)\|_{L^{2}}
\|\widetilde{\mathcal{K}}^{0}(\omega,k)\|_{L^{\infty}}\notag\\
&\leq\|e^{2\pi\lambda'_{0}\nu_{k_{12}}}\hat{A}(t,k_{12},0)\|_{L^{2}}+\|e^{2\pi\lambda'_{0}\nu_{k_{12}}}\hat{\rho}(t,k_{12},0)\|_{L^{2}}
\|\widetilde{\mathcal{K}}^{0}(\omega,k_{12},0)\|_{L^{\infty}},\notag\\\end{aligned}$$ where $\nu_{k_{12}}=|\eta_{k1}|+|\eta_{k2}|.$

Next we have to estimate $\|\widetilde{\mathcal{K}}^{0}(\omega,k_{12},0)\|_{L^{\infty}}.$

Indeed,

$$\|\widetilde{\mathcal{K}}^{0}(\omega,k_{12},0)\|_{L^{\infty}}
\leq\sup_{\omega}\frac{q}{m}\bigg[\bigg|\frac{1}{\omega}\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{3}}\int_{\mathbb{T}}e^{2\pi it\omega}e^{2\pi\lambda'_{0}\nu_{k_{12}}}
e^{-2\pi i\eta_{k2}v'_{2}}e^{-2\pi i\eta_{k1}v'_{1}}(v_{3}\partial_{x_{3}})\cdot(\widehat{W}_{1}\partial_{v'_{1}}f^{0}$$ $$-\widehat{W}_{2}\partial_{v'_{2}}f^{0})
dx_{3}dvdt\bigg|+\frac{q}{m}\bigg|\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{3}}\int_{\mathbb{T}}e^{2\pi it\omega}
e^{2\pi\lambda_{0}\nu_{k_{12}}}\cdot(\widehat{W}_{1}\partial_{v'_{1}}f^{0}+\widehat{W}_{2}\partial_{v'_{2}}f^{0})\cdot e^{-2\pi i\eta_{k2}v'_{2}}$$ $$e^{-2\pi i\eta_{k1}v'_{1}}
dx_{3}dvdt\bigg|-\frac{q}{m}(\omega,k)\frac{1}{\omega}\int_{\mathbb{R}^{+}}\bigg|\int_{\mathbb{R}^{3}}\int_{\mathbb{T}}e^{2\pi it\omega}
e^{2\pi\lambda_{0}\nu_{k}}e^{-2\pi i\eta_{k2}v'_{2}}e^{-2\pi i\eta_{k1}v'_{1}}\cdot(v_{1}\widehat{W}_{1}+v_{2}\widehat{W}_{2})k_{3}$$ $$\begin{aligned}
&\partial_{v'_{3}}f^{0}
dv\bigg|dt+\frac{q}{m}(\omega,k)\frac{1}{\omega}\int_{\mathbb{R}^{+}}\bigg|\int_{\mathbb{R}^{3}}e^{2\pi it\omega}e^{-2\pi i\eta_{k2}v'_{2}}e^{2\pi \lambda_{0}\nu_{k}}e^{-2\pi i\eta_{k1}v'_{1}}
\cdot(k_{1}\widehat{W}_{2}-k_{2}\widehat{W}_{1})\notag\\
&(v_{2}\partial_{v'_{1}}f^{0}
-v_{1}\partial_{v'_{2}}f^{0})dv\bigg|dt\bigg]=I+II+III+IV.\notag\\\end{aligned}$$ In fact, we only need to estimate one term of (1.10) because of similar processes of other terms. Without loss of generality, we give an estimate for $ I.$ In the same way, we only estimate one term of $I,$ here we still denote $ I. $ $$I=\sup_{\omega}\frac{q}{m}\bigg|\frac{1}{\omega}\int_{\mathbb{R}^{+}}\int_{\mathbb{R}}\int_{\mathbb{T}}\widehat{W}_{2}(k_{12},0)e^{2\pi it\omega}(2\pi i\eta_{k1})
e^{2\pi\lambda_{0}|\eta_{k1}|}e^{2\pi\lambda_{0}|\eta_{k2}|}\cdot\partial_{x_{3}}\widehat{v_{3}f^{0}}(\eta_{k1},\eta_{k2},v_{3})
dx_{3}dv_{3}dt\bigg|$$ $$\leq\frac{q}{mv_{Te}}e^{-c_{0}v_{Te}},$$ where in the last inequality we use the facts that if $k_{3}=0,$ then $v_{3}\gg v_{Te},$ and the assumption (i) and (iv). Then there exists some constant $0<\kappa<1$ such that $\|\widetilde{\mathcal{K}}^{0}(\omega,k)\|_{L^{\infty}}\leq\kappa.$

First through $Lemma $ 1.1, we can study the asymptotic behavior of the electric field and the magnetic field $E(t,x),B(t,x).$

Under the assumptions of Theorem 1.1, and $E(t,x),B(t,x)$ satisfy the Maxwell equations of (1.1), then for any for any $\lambda_{0}''<\lambda_{0}'<\lambda_{0},$ we have $$\begin{aligned}
&|\widehat{E}(t,k)|\leq e^{-2\pi\lambda'_{0}|\eta_{k1}|}e^{-2\pi\lambda'_{0}|\eta_{k2}|}e^{-2\pi\lambda'_{0}|k_{3}|t},
|\widehat{B}(t,k)|\leq te^{-2\pi\lambda''_{0}|\eta_{k1}|}e^{-2\pi\lambda''_{0}|\eta_{k2}|}e^{-2\pi\lambda'_{0}|k_{3}|t}.
 \end{aligned}$$

$Proof.$ Since $\partial_{t}\widehat{B}(t,k)=k\times\widehat{E}(t,k),\widehat{E}(t,k)=\widehat{W}(k)\hat{\rho}(t,k)=(\hat{W}_{1}(k),
\widehat{W}_{2}(k),0)\hat{\rho}(t,k),$ then $\partial_{t}\widehat{B}(t,k)=(-k_{3}\widehat{W}_{2}(k),k_{3}\widehat{W}_{1}(k),k_{1}\widehat{W}_{2}(k)-k_{2}\widehat{W}_{1}(k))\hat{\rho}(t,k).$ By Lemma 1.1, $|\partial_{t}\widehat{B}(t,k)|\leq C(k,\Omega,W)$ $e^{-2\pi\lambda'_{0}|\eta_{k1}|}$ $e^{-2\pi\lambda'_{0}|\eta_{k2}|}$ $e^{-2\pi\lambda'_{0}|k_{3}|t}.$ Hence, from $\widehat{B}(0,k)=0,$ 