by $$\begin{aligned}
\label{Equiv}
\|f\|_{\alpha,p}\asymp\|f\|_{p}+\|\Delta^{\frac{\alpha-[\alpha]}{2}}\nabla^{[\alpha]} f\|_p,\end{aligned}$$ where $[\alpha]$ is the integer part of real number $\alpha$, and we have used the convention $\Delta^0:=\mI$. Notice that for $\alpha\in(0,1]$ and $p>1$, $$\begin{aligned}
\label{LI0}
\|f(\cdot+x)-f(\cdot)\|_p\preceq \|\Delta^{\frac{\alpha}{2}}f\|_p|x|^\alpha,\end{aligned}$$ and in particular, $$\begin{aligned}
\label{LI}
\|f(\cdot+x)-f(\cdot)\|_p\preceq \|f\|_{\alpha,p}(|x|^\alpha\wedge 1).\end{aligned}$$ Moreover, we also have the following interpolation inequality: for any $0\leq\alpha<\beta<\infty$, $$\begin{aligned}
\label{Inter}
\|f\|_{\alpha,p}\leq C(p,d,\alpha,\beta)\|f\|^{\frac{\beta-\alpha}{\beta}}_p\|f\|^{\frac{\alpha}{\beta}}_{\beta,p},\end{aligned}$$ and the following Sobolev embedding results hold: for any $\alpha\in(0,1)$, if $p\alpha>d$, then $$\begin{aligned}
\label{Em0}
\|f\|_\infty+\sup_{x\not=y}\frac{|f(x)-f(y)|}{|x-y|^\gamma}\leq C(p,d,\alpha,\gamma)\|f\|_{\alpha,p},\ \gamma\in(0,\alpha-\tfrac{d}{p});\end{aligned}$$ if $p\alpha<d$, then $$\begin{aligned}
\label{Em1}
\|f\|_{q}\leq C(p,d,\alpha,q)\|f\|_{\alpha,p},\ \ q\in[p,\tfrac{pd}{d-p\alpha}].\end{aligned}$$ All the above facts are standard and can be found in [@Be-Lo] and [@St].

To treat the kinetic Fokker-Planck operator, we introduce the following anisotropic Bessel potential spaces. Let $C^\infty_c(\mR^{2d})$ be the space of all smooth functions on $\mR^{2d}$ with compact supports. For $\alpha,\beta\geq 0$, we define the Bessel potential space $H^{\alpha,\beta}_p:=H^{\alpha,\beta}_p(\mR^{2d})$ as the completion of $C^\infty_c(\mR^{2d})$ with respect to norm: $$\|f\|_{\alpha,\beta; p}:=\|(\mI-\Delta_x)^{\frac{\alpha}{2}}f\|_p+\|(\mI-\Delta_\v)^{\frac{\beta}{2}}f\|_p.$$ Notice that by the Mihlin multiplier theorem (cf. [@Be-Lo]), $$\|f\|_{\alpha,\beta;p}\asymp \|f\|_p+\|\Delta_x^{\frac{\alpha}{2}}f\|_p+\|\Delta_\v^{\frac{\beta}{2}}f\|_p
\asymp \|((\mI-\Delta_x)^{\frac{\alpha}{2}}+(\mI-\Delta_\v)^{\frac{\beta}{2}})f\|_p.$$ In the following, we simply write $$H^{\infty,\infty}_p:=H^{\infty,\infty}_p(\mR^{2d}):=\cap_{\alpha,\beta\geq 0}H^{\alpha,\beta}_p(\mR^{2d}).$$ (i) For any $\alpha,\beta\geq 0$, $\theta\in[0,1]$ and $p>1$, there is a constant $C=C(\alpha,\beta,\theta,p,d)>0$ such that $$\begin{aligned}
&\|(\mI-\Delta_x)^{\frac{\theta\alpha}{2}}(\mI-\Delta_\v)^{\frac{(1-\theta)\beta}{2}}f\|_p\leq C\|f\|_{\alpha,\beta;p},\label{Em8}\\
&\quad \|\Delta_x^{\frac{\theta\alpha}{2}}\Delta_\v^{\frac{(1-\theta)\alpha}{2}}f\|_p\leq C\|(\Delta_x+\Delta_\v)^{\frac{\alpha}{2}}f\|_p.\label{Em88}\end{aligned}$$ In particular, for any $\alpha\geq 0$, $\beta\geq 1$ and $p>1$, we have $$\begin{aligned}
