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

In this paper we analyze operators H = a^{ij}(x,t) X_i X_j - d/dt (having adopted Einstein's convention on repeated indexes), where the X_i's are H\"ormander vector fields generating a Carnot group and A = [a_{ij}] is a symmetric and uniformly positive-definite matrix whose entries satisfy double Dini continuity, a strictly weaker condition than H\"older continuity. For these operators, we build a fundamental solution and show a two-sided Gaussian estimate for the latter, as well as upper Gaussian estimates for its derivatives up to weight 2. As a consequence of the whole procedure, we prove an existence result for the related Cauchy problem, under a Dini-type condition on the source.