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.
Categories: math.AP math.PR