$L^p$-maximal hypoelliptic regularity of nonlocal kinetic Fokker-Planck operators

For $p\in (1,\mathrm{\infty })$, 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: $${\mathrm{\partial }}_{t}u+v\cdot {\mathrm{\nabla }}_{x}u={\mathrm{\Delta }}_v^{\alpha \mathrm{/}2}u+f,$$ where $\alpha \in (0,2)$ and ${\mathrm{\Delta }}_v^{\alpha \mathrm{/}2}$ 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$|\cdot|_p$istheusual$L^p$-normin$L^p($\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 ${\mathcal{L}}_{{\sigma }_t}^{{\nu }_t}$ in place of $v\cdot {\mathrm{\nabla }}_x$ and ${\mathrm{\Delta }}_v^{\alpha \mathrm{/}2}$. See Theorem 3.3 for details.