Regularity in the Obstacle Problem for Parabolic Non-divergence Operators of Hörmander type

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In this paper we continue the study initiated in [FGN] concerning the obstacle problem for a class of parabolic non-divergence operators structured on a set of vector fields X = X₁,...,Xq in Rⁿ with C¹-coefficients satisfying Hörmander's finite rank condition, i.e., the rank of LieX₁,...,Xq equals n at every point in Rⁿ. In [FGN] we proved, under appropriate assumptions on the operator and the obstacle, the existence and uniqueness of strong solutions to a general obstacle problem. The main result of this paper is that we establish further regularity, in the interior as well as at the initial state, of strong solutions. Compared to [FGN] we in this paper assume, in addition, that there exists a homogeneous Lie group G = (Rⁿ, ∘, δλ) such that X₁,...,Xq are left translation invariant on G and such that X₁,...,Xq are δλ-homogeneous of degree one.