Abstract. We construct the fundamental solutions Γ and γ for the nondivergence form operators ∑ i, j ai, j(x, t) XiXj − ∂t and ∑ i, j ai, j(x) XiXj, where the Xi’s are Hörmander vector fields generating a stratified group G and (ai,j)i,j is a positive-definite matrix with Hölder continuous entries. We also provide Gaussian estimates of Γ and its derivatives and some results for the relevant Cauchy problem. Suitable long-time estimates of Γ allow us to construct γ using both t-saturation and approximation arguments. 1. Introduction an
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