Anisotropic Maximal $L^p$-regularity Estimates for a Hypoelliptic Operator

TL;DR

This paper establishes anisotropic maximal $L^p$-regularity estimates for a hypoelliptic Ornstein-Uhlenbeck operator, advancing understanding of regularity in Vlasov-Fokker-Planck equations.

Contribution

It proves anisotropic maximal regularity estimates for a specific hypoelliptic operator and introduces new weak (1,1) estimates for related operators.

Findings

Existence of solutions with anisotropic maximal regularity

Weak (1,1) estimate for the operator $L$

Pointwise estimates of the fundamental solution

Abstract

We consider the maximal regularity of a specific Vlasov-Fokker-Planck equation $\mathcal{A}u=f$ in the Euclidean space. The operator $\mathcal{A}=\Delta_{y}u-y\cdot \nabla_x{u}$ is an example of the Ornstein-Uhlenbeck operators. We prove the existence of a solution that satisfies the anisotropic maximal regularity estimates. To prove this we also show a similar estimates and a weak (1, 1) estimate for $L=\partial_t-\mathcal{A}$, which is of independent interest. These results rely on the pointwise estimates of the fundamental solution of $L$.