We consider strongly degenerate parabolic operators of the form$\mathcal{L}:={\mathrm{\nabla }}_X\cdot (A(X,Y,t){\mathrm{\nabla }}_X)+X\cdot {\mathrm{\nabla }}_Y-{\partial }_t$ in unbounded domains$\mathrm{\Omega }=\{(X,Y,t)=(x,x_m,y,y_m,t)\in {\mathbb{R}}^{m-1}\times \mathbb{R}\times {\mathbb{R}}^{m-1}\times \mathbb{R}\times \mathbb{R}|x_m>\psi (x,y,t)\}.$ We assume that $A=A(X,Y,t)$ is bounded, measurable and uniformly elliptic (as a matrix in ${\mathbb{R}}^m$) and concerning 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 $\mathcal{L}$. We prove, assuming in addition that ψ is independent of the variable $y_m$, 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 $A_{\infty }$-weight with respect to the surface measure.

我们考虑以下形式的强退化抛物线算子$\mathcal{升}：={\mathrm{\nabla }}_X\cdot (\mathrm{一种}(X,是,吨){\mathrm{\nabla }}_X)+X\cdot {\mathrm{\nabla }}_{是}-{\partial }_{吨}$ 在无界域中$\mathrm{\Omega }=\{(X,是,吨)=(X,X_{米},是,{是}_{米},吨)\in {\mathbb{电阻}}^{米-1}\times \mathbb{电阻}\times {\mathbb{电阻}}^{米-1}\times \mathbb{电阻}\times \mathbb{电阻}|X_{米}>\psi (X,是,吨)\}.$ 我们假设 $\mathrm{一种}=\mathrm{一种}(X,是,吨)$ 是有界的、可测量的和均匀椭圆的（作为矩阵 ${\mathbb{电阻}}^{米}$) 并且关于ψ和 Ω，我们假设 Ω 是我们所说的（无界）Lipschitz 域：ψ满足适用于膨胀结构和算子下面的（非欧几里得）李群的统一 Lipschitz 条件$\mathcal{升}$. 我们证明，另外假设ψ与变量无关${是}_{米}$，ψ满足根据 Carleson 测度制定的附加正则条件，以及A上的附加条件，相关抛物线测度相对于表面测度是绝对连续的，并且相关联的 Radon-Nikodym 导数定义了${\mathrm{一种}}_{\infty }$- 相对于表面尺寸的重量。