new term $(B[f^{n}]\circ\Omega^{n}_{s,t}(x,v))\cdot$ $
            (\nabla'_{v}h^{n+1}\times V^{0}_{s,t}(x,v)\circ\Omega^{n}_{s,t}(x,v))$ that have the information of the stage $n+1.$ Of course, this is due to the reason that we regard the perturbation of the magnetic field as a negligible term. To get a self-consistent estimate, we have to deal with this term and have little choice but to come back the equation of $h^{n+1}$. This leads to different kinds of resonances (in term of different norms), for example, $v\times B[f^{n}]\cdot\nabla_{v}h^{n+1}$ may generate resonance in $\mathcal{Z}^{\lambda,\mu;1}_{\tau}$ norm on $h^{n+1},$ except in $\mathcal{F}^{\lambda\tau+\mu}$ norm on $\rho[h^{n+1}],$ because both $B[f^{n}]$ and $h^{n+1}$ contain the space variable. But there are no these problems in \[23\], and Landau damping in \[23\] only has the resonances in $\mathcal{F}^{\lambda\tau+\mu}$ norm on $\rho[h^{n+1}].$

$\gamma>1$ of Theorems 0.1 and 0.3 can be extended to $\gamma\geq1,$ the difference between $\gamma>1$ and $\gamma=1$ is the proof of the growth integral in section 7. The proof of $\gamma=1$ is similar to section 7 in \[23\], here we omit this case.

Indeed, we don’t need to assume that $ W(x)=(W_{1}(x),W_{2}(x),0)$ and $B_{0}=(0,0,B_{0}),$ only need to assume that the electric field $E(t,x)$ is perpendicular to the background magnetic field $B_{0}$.

From the physics viewpoint, the condition of the damping 