  We introduce the concept of kinetic maximal $L^p$-regularity with temporal
weights and prove that this property is satisfied for the (fractional)
Kolmogorov equation. We show that solutions are continuous with values in the
trace space and prove, in particular, that the trace space can be characterized
in terms of anisotropic Besov spaces. We further extend the property of kinetic
maximal $L^p_\mu$-regularity to the Kolmogorov equation with variable
coefficients. Finally, we show how kinetic maximal $L^p_\mu$-regularity can be
used to obtain local existence of solutions to a class of quasilinear kinetic
equations and illustrate our result with a quasilinear kinetic diffusion
equation.
