On the fine properties of parabolic measures associated to strongly degenerate parabolic operators of Kolmogorov type

Abstract

We consider strongly degenerate parabolic operators of the form \[ \mathcal{L}:=\nabla_X\cdot(A(X,Y,t)\nabla_X)+X\cdot\nabla_Y-\partial_t \] in unbounded domains \[ \Omega=\{(X,Y,t)=(x,x_{m},y,y_{m},t)\in\mathbb R^{m-1}\times\mathbb R\times\mathbb R^{m-1}\times\mathbb R\times\mathbb R\mid x_m>\psi(x,y,t)\}. \] We assume that $A=A(X,Y,t)$ is bounded, measurable and uniformly elliptic (as a matrix in $\mathbb R^{m}$) and concerning $\psi$ and $\Omega$ we assume that $\Omega$ is what we call an (unbounded) Lipschitz domain: $\psi$ satisfies a uniform Lipschitz condition adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator $\mathcal{L}$. We prove, assuming in addition that $\psi$ is independent of the variable $y_m$, that $\psi$ satisfies an additional regularity condition formulated in terms of a Carleson measure, and additional conditions on $A$, that 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 the surface measure.