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 formL:=∇X⋅(A(X,Y,t)∇X)+X⋅∇Y−∂t in unbounded domainsΩ={(X,Y,t)=(x,xm,y,ym,t)∈Rm−1×R×Rm−1×R×R|xm>ψ(x,y,t)}. We assume that A=A(X,Y,t) is bounded, measurable and uniformly elliptic (as a matrix in Rm) and concerning ψ and Ω we assume that Ω is what we call an (unbounded) Lipschitz domain: ψ satisfies a uniform Lipschitz condition adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator L. We prove, assuming in addition that ψ is independent of the variable ym, that ψ 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∞-weight with respect to the surface measure.