  For $p\in(1,\infty)$, let $u(t,x,v)$ and $f(t,x,v)$ be in $L^p(\mathbb{R}
\times \mathbb{R}^d \times \mathbb{R}^d)$ and satisfy the following nonlocal
kinetic Fokker-Plank equation on $\mathbb{R}^{1+2d}$ in the weak sense: $$
\partial_t u+v\cdot\nabla_x u=\Delta^{{\alpha}/{2}}_v u+f, $$ where
$\alpha\in(0,2)$ and $\Delta^{{\alpha}/{2}}_v$ is the usual fractional
Laplacian applied to $v$-variable. We show that there is a constant
$C=C(p,\alpha,d)>0$ such that for any $f(t, x, v)\in L^p(\mathbb{R} \times
\mathbb{R}^d \times \mathbb{R}^d)=L^p(\mathbb{R}^{1+2d})$, $$
\|\Delta_x^{{\alpha}/{(2(1+\alpha))}}u\|_p+\|\Delta_v^{{\alpha}/{2}}u\|_p\leq
C\|f\|_p, $$ where $\|\cdot\|_p$ is the usual $L^p$-norm in
$L^p(\mathbb{R}^{1+2d}; d z)$. In fact, in this paper the above inequality is
established for a large class of time-dependent non-local kinetic Fokker-Plank
equations on $\mathbb{R}^{1+2d}$, with $U_t v$ and
$\mathscr{L}^{\nu_t}_{\sigma_t}$ in place of $v\cdot \nabla_x$ and
$\Delta^{\alpha/2}_v$. See Theorem 3.3 for details.
