Global $L^p$ estimate for some kind of Kolmogorov-Fokker-Planck Equations in nondivergence form

TL;DR

This paper establishes global $L^p$ estimates for a class of Kolmogorov-Fokker-Planck equations with nondivergence form and variable coefficients, extending previous work to multiple scalings and proving a Poincare inequality.

Contribution

It introduces new global $L^p$ estimates for Kolmogorov-Fokker-Planck equations with four scalings and measurable coefficients, expanding prior results to more general settings.

Findings

Established global $L^p$ estimates for derivatives of solutions.

Extended previous work to four different scalings.

Proved a Poincare inequality for homogeneous equations.

Abstract

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 $a^{ij}$ are only measurable in $t$ and satisfy the vanishing mean oscillation in space variables. We establish a global priori estimates of $\nabla_x^u$, $( -\Delta_y )^{1/3} u$ and $( -\Delta_z )^{1/5} u$ in $L^p$ space which extend the work of Dong and Yastrzhembskiy \cite{ref49} where they focus on the 3 different scalings KFP operator. Moreover we establish a kind of Poincare inequality for homogeneous equations.