A Parametrix for Step-Two Hypoelliptic Diffusion Equations

In this paper I construct a parametrix for the hypoelliptic diffusion equations $(\partial /\partial t - L)u = 0$, where $L = \sum \nolimits _{a = 1}^n {g_a^2}$ and where the ${g_a}$ are vector fields which satisfy the property that they, together with all of the commutators $[{g_{a,}}{g_b}]$ for $a < b$, are at each point linearly independent and span the tangent space.