Fundamental solutions for non-divergence form operators on stratified groups

Abstract:

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.