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

In this paper we analyze operators $H = \partial_t-\sum_{i,j} a_{ij}(x,t) X_i X_j$, where the $X_i$'s are Hörmander 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 Hölder 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, we prove an existence result for the related Cauchy problem, under a Dini-type condition on the source.