∂x

Abstract. The main result of our work ([2]) is the C1,αloc regularity for subelliptic p-harmonic functions in the case of the Grušin vector fields. To this goal we prove a Calderón-Zygmund inequality and an estimate for strong solutions of a linear subelliptic equation in nondivergence form with L ∞ coefficients. Let Ω be a bounded domain in Rn and X = (X1, X2, · · · , Xq) be a system of C ∞ vector fields defined in Ω, q ≤ n. We suppose that the system X satisfies the following Hörmander or bracket generating condition of step m in Ω (see [5]): the vector fields Xi together with their commutators of length at most m span Rn at every point of Ω. In particular we will discuss about the following Grušin vector field