Kinetic maximal $L^p$-regularity with temporal weights and application to quasilinear kinetic diffusion equations

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.