The $A_\infty$-property of the Kolmogorov measure

Abstract

We consider the Kolmogorov–Fokker–Planck operator

$$\mathcal{K}:=\sum _{i=1}^m{\partial }_{x_i{x}_i}+\sum _{i=1}^m{x}_i{\partial }_{y_i}-{\partial }_t$$

in unbounded domains of the form

$$\Omega =\left\{\left(x,x_m,y,y_m,t\right)\in {ℝ}^{N+1}\mid x_m>\psi \left(x,y,t\right)\right\}.$$

Concerning $\psi$ and $\Omega$, we assume that $\Omega$ is what we call an (unbounded) admissible ${Lip}_K$-domain: $\psi$ satisfies a uniform Lipschitz condition, adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator $\mathcal{K}$, as well as an additional regularity condition formulated in terms of a Carleson measure. We prove that in admissible ${Lip}_K$-domains the associated parabolic measure is absolutely continuous with respect to a surface measure and that the associated Radon–Nikodym derivative defines an $A_{\infty }$ weight with respect to this surface measure. Our result is sharp.