of notation, we write $$\mL^p(T):=\mL^p(0,T),\ \ \mH^{\alpha,\beta}_p(T):=\mH^{\alpha,\beta}_p(0,T).$$

We assume that $a:[0,T]\times\mR^{2d}\to\mR^d\otimes\mR^d$ is symmetric and satisfies that for some $K\geq 1$ and $\delta\in(0,1)$, $$\begin{aligned}
\tag{\bf H$^{\delta,p}_K$}
\left\{
\begin{aligned}
&K^{-1}\cdot\mI\leq a_t(z)\leq K\cdot\mI,\,\,(t,z)\in[0,T]\times\mR^{2d}\\
&\omega_a(\delta):=\sup_{|z-z'|\leq\delta}\sup_{t\in[0,T]}\|a_t(z)-a_t(z')\|\leq\tfrac{1}{2(C_p+1)}
\end{aligned}
\right\},\end{aligned}$$ where $C_p$ is the same as in . Here and in the remainder of this paper, $\|\cdot\|$ denotes the Hilbert-Schmidt norm. For $\lambda\geq 0$, consider the following backward kinetic Fokker-Planck equation $$\begin{aligned}
\label{PDE}
\p_t u+\sL^{a,b}_t u-\lambda u+f=0, \ \ u_T=0,\end{aligned}$$ where $f_t(x,\v):[0,T]\times\mR^{2d}\to\mR$ is a Borel function. We first introduce the following notion of solutions to the above equation. \[Def31\] Let $p\in(1,\infty)$ and $f\in\mL^p(T)$. A Borel function $u\in \mH^{0,2}_p(T)$ is called a solution of if for any $\varphi\in C^\infty_c(\mR^{2d})$ and all $t\in[0,T]$, $$\begin{aligned}
\label{Sol}
\begin{split}
\<u_t,\varphi\>&=\int^T_t\<\tr(a_s\cdot\nabla^2_\v u_s),\varphi\>\dif s-\int^T_t\<\v\cdot\nabla_x\varphi, u_s\>\dif s\\
&\quad+\int^T_t\<b_s\cdot\nabla_\v u_s,\varphi\>\dif s-\lambda\int^T_t\<u_s,\varphi\>\dif s+\int^T_t\<f_s,\varphi\>\dif s,
\end{split}\end{aligned}$$ where $\<u_t,\varphi\>:=\int_{\mR^{2d}}u_t(z)\varphi(z)\dif z$. The main aim of this section is to show that \[Th32\] Let $\alpha\in[0,\frac{2}{3})$, $\beta\in(1,2)$ and $p>\frac{2}{(2-3\alpha)\wedge(2-\beta)}$ be not equal to $\frac{d(\alpha+\beta)}{\alpha(\beta-1)}$. Suppose that $a$ satisfies [**(H$^{\delta,p}_K$)**]{}, and for some $q\in[p\vee \frac{d(\alpha+\beta)}{\alpha(\beta-1)},\infty]$, $$\kappa_0:=\|b\|_{L^p([0,T]; L^q(\mR^{2d}))}<\infty.$$

(i) For any $f\in \mL^p(T)$, there exists a unique solution $u=u^\lambda$ to in the sense of Definition \[Def31\] with $$\begin{aligned}
    \label{Pri0}
    \|u^\lambda\|_{\mH^{2/3,2}_p(T)}\leq C\|f\|_{\mL^p(T)}, \ \end{aligned}$$ and for all $t\in[0,T]$, $$\begin{aligned}
    \label{Pri00}
    \|u^\lambda_t\|_{\alpha,\beta;p}\leq C((T-t)\wedge\lambda^{-1})^{1-\frac{1}{p}-\frac{(3\alpha)\vee\beta}{2}}\|f\|_{\mL^p(t,T)},\end{aligned}$$ where the constant 