we have $|\widehat{B}(t,k)|\leq te^{-2\pi\lambda'_{0}|\eta_{k1}|}e^{-2\pi\lambda'_{0}|\eta_{k2}|}e^{-2\pi\lambda'_{0}|k_{3}|t}.$ Then $|\widehat{B}(t,k)|\leq e^{-2\pi\lambda''_{0}|\eta_{k1}|}e^{-2\pi\lambda''_{0}|\eta_{k2}|}e^{-2\pi\lambda''_{0}|k_{3}|t}.$ $$$$ $Proof$ $of$ $Theorem$ $ 0.1.$ From (1.1), we have $$\begin{aligned}
&f(t,x,v)=f_{0}(x'(0,x,v),v'(0.x,v))-\frac{q}{m}\int^{t}_{0}[(E+v'\times B)\cdot\nabla'_{v}f^{0}](\tau.x'(\tau,x,v),v'(\tau,x,v))d\tau.\end{aligned}$$

Taking the Fourier-Laplace transform in variables $x,v,t,$ we find $$\begin{aligned}
&\tilde{f}(\omega,k,\eta)=\int_{\mathbb{R}^{+}}\int_{\mathbb{T}^{3}}\int_{\mathbb{R}^{3}}e^{2\pi it\cdot\omega}e^{-2\pi ix\cdot k}e^{-2\pi iv\cdot\eta}f_{0}(x'(0,x,v),v'(0,x,v))dvdxdt\notag\\
&-\frac{q}{m}\int_{\mathbb{R}^{+}}\int_{\mathbb{T}^{3}}\int_{\mathbb{R}^{3}}e^{2\pi it\cdot\omega}e^{-2\pi ix\cdot k}e^{-2\pi iv\cdot\eta}\int^{t}_{0}[B\cdot\nabla'_{v}\times(f^{0}v')](\tau,x'(\tau,x,v),v'(\tau,x,v))d\tau dvdxdt\notag\\
&-\frac{q}{m}\int_{\mathbb{R}^{+}}\int_{\mathbb{T}^{3}}\int_{\mathbb{R}^{3}}e^{2\pi it\cdot\omega}e^{-2\pi ix\cdot k}e^{-2\pi iv\cdot\eta}\int^{t}_{0}[E\cdot\nabla'_{v}f^{0}](\tau,x'(\tau,x,v),v'(\tau,x,v))d\tau dvdxdt\notag\\
&=I+II+III.\end{aligned}$$ $$\begin{aligned}
&I=\int_{\mathbb{R}^{+}}\int_{\mathbb{T}^{3}}\int_{\mathbb{R}^{3}}e^{2\pi it\cdot\omega}e^{-2\pi iv\cdot\eta}\cdot f_{0}(x,v')\exp(-2\pi i[x+(\frac{1}{\Omega}(v'_{2}-v_{2}),
-\frac{1}{\Omega}(v'_{1}-v_{1}),v_{3}t)]\cdot k)dvdxdt\notag\\
&=\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{3}}e^{2\pi it\cdot\omega}e^{-2\pi iv_{3}(\eta_{3}+k_{3}t)}e^{-2\pi iv'_{2}(\eta_{2}+\eta_{k2})}
e^{-2\pi iv'_{1}(\eta_{1}+\eta_{k1})}\hat{f}_{0}(k,v')dvdt.\notag\\
$$

As the above process, we have

$$II=-\frac{q}{m}\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{3}}e^{2\pi it\cdot\omega}e^{-2\pi i(k_{3}t+\eta_{3})v_{3}}e^{-2\pi iv'_{2}(\eta_{2}+\eta_{k2})}
e^{-2\pi iv'_{1}(\eta_{1}+\eta_{k1})}$$ $$\cdot\int^{t}_{0} (\hat{B}(\tau,k)\cdot (v_{3}\partial_{v'_{2}}f^{0}-v'_{2}\partial_{v_{3}}f^{0},v'_{}\partial_{v_{3}}f^{0}-v_{3}\partial_{v'_{1}}f^{0},
v'_{2}\partial_{v'_{1}}f^{0}-v'_{1}\partial_{v'_{2}}f^{0})d\tau dvdt$$ $$\begin{aligned}
&=-\frac{q}{m}\int_{\mathbb{R}^{+}}e^{2\pi i\tau\cdot\omega}\hat{B}(\tau,k)d\tau\cdot\int_{\mathbb{R}^{+}}e^{2\pi it\cdot\omega}\int_{\mathbb{R}^{3}}
e^{-2\pi i(k_{3}t+\eta_{3})v_{3}}e^{-2\pi iv'_{2}(\eta_{2}+\eta_{k2})}
e^{-2\pi iv'_{1}(\eta_{1}+\eta_{k1})}\notag\\
&\cdot(v_{3}\partial_{v'_{2}}f^{0}-v'_{2}\partial_{v_{3}}f^{0},v'_{}\partial_{v_{3}}f^{0}-v_{3}\partial_{v'_{1}}f^{0},
v'_{2}\partial_{v'_{1}}f^{0}-v'_{1}\partial_{v'_{2}}f^{0}) dvdt\notag\\
&=-\frac{q}{m}\tilde{\rho}(\omega,k)\frac{1}{\omega}\int_{\mathbb{R}^{+}}e^{2\pi it\cdot\omega}\int_{\mathbb{R}^{3}}
e^{-2\pi i(k_{z}t+\eta_{3})v_{3}}e^{-2\pi iv'_{2}(\eta_{2}+\eta_{k2})}
e^{-2\pi iv'_{1}(\eta_{1}+\eta_{k1})}(k\times\hat{W}(k))\notag\\
&\cdot (v_{3}\partial_{v'_{2}}f^{0}-v'_{2}\partial_{v_{3}}f^{0},v'_{}\partial_{v_{3}}f^{0}-v_{3}\partial_{v'_{1}}f^{0},
v'_{2}\partial_{v'_{1}}f^{0}-v'_{1}\partial_{v'_{2}}f^{0}) dvdt.\notag\\\end{aligned}$$ Since $\widehat{W}(k)=(\widehat{W}_{1}(k),\widehat{W}_{2}(k),0),$ then $ k\times\widehat{W}(k)=(-k_{3}\widehat{W}_{2},k_{3}\widehat{W}_{1},k_{1}\widehat{W}_{2}-k_{2}\widehat{W}_{1}),$ and $$\begin{aligned}
&II=\frac{q}{m}\tilde{\rho}(\omega,k)\frac{1}{\omega}\int_{\mathbb{R}^{+}}e^{2\pi it\cdot\omega}\int_{\mathbb{R}}e^{-2\pi i(k_{3}t+\eta_{3})v_{3}}
e^{-2\pi iv'_{2}(\eta_{2}+\eta_{k2})}
e^{-2\pi iv'_{1}(\eta_{1}+\eta_{k1})}
\cdot (k_{3}\widehat{W}_{2}(-v_{3}\partial_{v'_{2}}f^{0}\notag\\
&+v'_{2}\partial_{v_{3}}f^{0})+k_{3}\widehat{W}_{1}(v'_{1}\partial_{v_{3}}f^{0}-v_{3}\partial_{v'_{1}}f^{0})+
(k_{1}\widehat{W}_{2}-k_{2}\widehat{W}_{1})
(v'_{2}\partial_{v'_{1}}f^{0}-v'_{1}\partial_{v'_{2}}f^{0}))dvdt.\notag\\
$$

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

Therefore, from(1.17)-(1.21), we have $$\begin{aligned}
&\omega\tilde{f}(\omega,k,v',\eta_{3})=\omega (I+II+III).\end{aligned}$$

Furthermore, $$\begin{aligned}
&\partial_{t}\hat{f}(t,k,\eta)=\partial_{t}\int_{\mathbb{R}^{3}}e^{-2\pi i(k_{3}t+\eta_{3})v_{3}}e^{-2\pi iv'_{2}(\eta_{2}+\eta_{k2})}
e^{-2\pi iv'_{1}(\eta_{1}+\eta_{k1})}\hat{f}_{0}(k,v')dv\notag\\
&-\frac{q}{m}\hat{\rho}(t,k)\ast_{t} \int_{\mathbb{R}^{3}}e^{-2\pi i(k_{3}t+\eta_{3})v_{3}}e^{-2\pi iv'_{2}(\eta_{2}+\eta_{k2})}
e^{-2\pi iv'_{1}(\eta_{1}+\eta_{k1})}\cdot (k_{3}\widehat{W}_{2}(-v_{3}\partial_{v'_{2}}f^{0}\notag\\
&+v'_{2}\partial_{v_{3}}f^{0})+k_{3}\widehat{W}_{1}(v'_{1}\partial_{v_{3}}f^{0}-v_{3}\partial_{v'_{1}}f^{0})+
(k_{1}\widehat{W}_{2}-k_{2}\widehat{W}_{1})
(v'_{2}\partial_{v'_{1}}f^{0}-v'_{1}\partial_{v'_{2}}f^{0}))dvdt\notag\\
&-\frac{q}{m}\hat{\rho}(t,k)\ast_{t}\partial_{t}\int_{\mathbb{R}^{3}}
e^{-2\pi i(k_{3}t+\eta_{3})v_{3}}e^{-2\pi iv'_{2}(\eta_{2}+\eta_{k2})}
e^{-2\pi iv'_{1}(\eta_{1}+\eta_{k1})}(\widehat{W}_{1}\partial_{v'_{1}}f^{0}+\widehat{W}_{2}\partial_{v'_{2}}f^{0})dvdt\notag\\
&=IV+V+VI.\end{aligned}$$

Now we estimate $IV,V,VI,$ respectively. $$\begin{aligned}
&IV=\partial_{t}\int_{\mathbb{R}^{3}}e^{-2\pi i(k_{3}t+\eta_{3})v_{3}}e^{-2\pi iv'_{2}(\eta_{2}+\eta_{k2})}
e^{-2\pi iv'_{1}(\eta_{1}+\eta_{k1})}\hat{f}_{0}(k,v')dv=\partial_{t}\hat{f}_{0}(k,\eta_{1}+\eta_{k1},\eta_{2}+\eta_{k2},k_{3}t+\eta_{3})\notag\\
&\leq C(\Omega,k)e^{-2\pi\lambda_{0}|\eta_{1}+\eta_{k1}|}e^{-2\pi\lambda_{0}|\eta_{2}+\eta_{k2}|}e^{-2\pi\lambda_{0}|k_{3}t+\eta_{3}|},
$$ here we use the assumption that $|\hat{f}_{0}(k,\eta)|\leq C_{0} e^{-2\pi\lambda_{0}|\eta_{1}|}e^{-2\pi\lambda_{0}|\eta_{2}|}e^{-2\pi\lambda_{0}|\eta_{3}|}.$

$$V=-\frac{q}{m}\hat{\rho}(t,k)\ast_{t} \int_{\mathbb{R}^{3}}e^{-2\pi i(k_{3}t+\eta_{3})v_{3}}
e^{-2\pi iv'_{2}(\eta_{2}+\eta_{k2})}
e^{-2\pi iv'_{1}(\eta_{1}+\eta_{k1})}\cdot (k_{3}\widehat{W}_{2}(-v_{3}\partial_{v'_{2}}f^{0}$$ $$+v'_{2}\partial_{v_{3}}f^{0})+k_{3}\widehat{W}_{1}(v'_{1}\partial_{v_{3}}f^{0}-v_{3}\partial_{v'_{1}}f^{0})+
(k_{1}\widehat{W}_{2}-k_{2}\hat{W}_{1})
(v'_{2}\partial_{v'_{1}}f^{0}-v'_{1}\partial_{v'_{2}}f^{0}))dv$$ $$=-\frac{q}{m}\hat{\rho}(t,k)\ast_{t}[k_{3}\widehat{W}_{2}(\widehat{v'_{2}\partial_{v_{3}}f^{0}}
-\widehat{v_{3}\partial_{v'_{2}}f^{0}})(\eta_{1}+\eta_{k1},\eta_{2}+\eta_{k2},k_{3}t+\eta_{3})$$ 