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In this paper we develop a potential theory for strongly degenerate parabolicoperators of the form \[\mathcal{L}:=\nabla_X\cdot(A(X,Y,t)\nabla_X)+X\cdot\nabla_{Y}-\partial_t, \] inunbounded domains of the form \[ \Omega=\{(X,Y,t)=(x,x_{m},y,y_{m},t)\in\mathbbR^{m-1}\times\mathbb R\times\mathbb R^{m-1}\times\mathbb R\times\mathbb R\midx_m>\psi(x,y,y_m,t)\}, \] where $\psi$ is assumed to satisfy a uniformLipschitz condition adapted to the dilation structure and the (non-Euclidean)Lie group underlying the operator $\mathcal{L}$. Concerning $A=A(X,Y,t)$ weassume that $A$ is bounded, measurable, symmetric and uniformly elliptic (as amatrix in $\mathbb R^{m}$). Beyond the solvability of the Dirichlet problem andother fundamental properties our results include scale and translationinvariant boundary comparison principles, boundary Harnack inequalities anddoubling properties of associated parabolic measures. All of our estimates aretranslation- and scale-invariant with constants only depending on the constantsdefining the boundedness and ellipticity of $A$ and the Lipschitz constant of$\psi$. Our results represent a version, for operators of Kolmogorov type withbounded, measurable coefficients, of the by now classical results of Fabes andSafonov, any several others, concerning boundary estimates for uniformlyparabolic equations in (time-dependent) Lipschitz type domains.

Potential theory for a class of strongly degenerate parabolic operators of Kolmogorov type with rough coefficients