Boundary Behavior of Subelliptic Parabolic Equations on Time-Dependent Domains

In this paper we study the boundary behavior of solutions of a divergence-form subelliptic heat equation in a time-varying domain Ωin R^{n+1}, structured on a set of vector fields X = (X_1, ... X_m) with smooth coefficients satisfying Hörmander's finite rank condition. Assuming that Ωis an X-NTA domain, we first prove a Dahlberg type estimate comparing the X-caloric measure of Ωand the Green function of the subelliptic heat operator. We then prove a backward Harnack inequality, the doubling property for the X-caloric measure of Ω, the Hölder continuity at the boundary for quotients of solutions of H, and a Fatou theorem.

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