Regularity in the obstacle problem for parabolic non-divergence operators of Hörmander type

In this paper we continue the study initiated in [15] 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 C∞-coefficients satisfying Hörmander's finite rank condition, i.e., the rank of Lie [X1, ..., Xq] equals n at every point in Rn. In [15] 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 [15] we in this paper assume, in addition, that there exists a homogeneous Lie group G = (Rn, {ring operator}, δλ) such that X1, ..., Xq are left translation invariant on G and such that X1, ..., Xq are δλ-homogeneous of degree one.