Kinetic maximal $L^{p}$-regularity for nonlocal Kolmogorov equation and application

We study the linear and nonlinear variable coefficients Kolmogorov equations. The equations include the abstract operator $A=A\left(x\right) $ in a Fourier type Banach space $E$ and convolution terms. Here, the kinetic maximal $L^{p}$-regularity for the linear equatıon is derived in terms of $ E$-valued Sobolev spaces. Moreover, we show that the solution $u$ is also regular in time and space variables when $u$ is assumed to have a certain amount of regularity in velocity. Finally, the kinetic maximal $L^{p}$ -regularity for the linear equation can be used to obtain local existence and uniqueness of solutions to a quasilinear nonlocal Kolmogorov type kinetic equation.