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

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\mid x_m>\psi(x,y,t)\}. assume that $A=A(X,Y,t)$ is bounded, measurable and uniformly elliptic (as a matrix $\mathbb R^{m}$) concerning $\psi$ $\Omega$ we what call an (unbounded) Lipschitz domain: satisfies uniform condition adapted to dilation structure (non-Euclidean) Lie group underlying operator $\mathcal{L}$. prove, assuming addition independent variable $y_m$, additional regularity formulated terms Carleson measure, conditions on $A$, associated measure absolutely continuous with respect surface Radon-Nikodym derivative defines $A_\infty$-weight measure.