  We consider the Kolmogorov equation, where the right-hand side is given by a
non-local integro-differential operator comparable to the fractional Laplacian
in velocity with possibly time, space and velocity dependent density. We prove
that this equation admits kinetic maximal $L^p_\mu$-regularity under suitable
assumptions on the density and on $p$ and $\mu$. We apply this result to prove
short-time existence of strong $L^p_\mu$-solutions to quasilinear fractional
kinetic partial differential equations.
