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 L := del(X) . (A(X, Y, t)del(X)) + X . del(Y) - partial derivative(t) in unbounded domains Omega = {(X, Y, t) = (x, x(m), y, y(m), t) is an element of Rm-1 x R x Rm-1 x R x R vertical bar x(m) > psi(x, y, t)}. We assume that A = A(X, Y, t) is bounded, measurable and uniformly elliptic (as a matrix in 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 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(infinity)-weight with respect to the surface measure. (C) 2021 The Author(s). Published by Elsevier Inc.