Global Lᵖ estimate for some kind of Kolmogorov-Fokker-Planck Equations in nondivergence form

In this paper, we mainly investigate a class of Kolmogorov-Fokker-Planck operator with 4 different scalings in nondivergence form. And we assume the coefficients aⁱʲ are only measurable in t and satisfy the vanishing mean oscillation in space variables. We establish a global priori estimates of ∇ₓᵘ, u and u in Lᵖ space which extend the work of Dong and Yastrzhembskiy {ref49} where they focus on the 3 different scalings KFP operator. Moreover we establish a kind of Poincare inequality for homogeneous equations.