Non-divergence evolution operators modeled on H\"ormander's vector fields with Dini continuous coefficients

In this paper we construct a fundamental solution for operators of the form H = aᵢj (x, t) Xᵢ Xⱼ - d/dt (having adopted Einstein's convention on repeated indexes) and we show that the latter satisfies suitable Gaussian estimates. Here the Xᵢ are H\"ormander's vector fields generating a Carnot group and A = (aᵢj) is a symmetric and uniformly positive-definite matrix with bounded double Dini continuous entries. As a consequence of this procedure we also prove an existence result for the related Cauchy problem, under a Dini-type condition on the source.