Fundamental solutions for non-divergence form operators on stratified groups

We construct the fundamental solutions $\Gamma$ and $\gamma$for the non-divergence form operators $\,{\textstyle\sum_{i,\,j}\,} a_{i,\,j}(x,t)\,X_iX_j\,-\,\partial_t\,$ and ${\,\textstyle\sum_{i,\,j}}\,a_{i,\,j}(x)\,X_iX_j$, where the $X_i$'s are Hörmander vector fields generating a stratified group $\mathbb{G}$ and $(a_{i,j})_{i,j}$ is a positive-definite matrix with Hölder continuous entries. We also provide Gaussian estimates of $\Gamma$ and its derivatives and some results for the relevant Cauchy problem. Suitable long-time estimates of $\Gamma$ allow us to construct $\gamma$ using both $t$-saturation and approximation arguments.