Non-Divergence Equations Structured on Hörmander Vector Fields: Heat Kernels and Harnack Inequalities

In this work the authors deal with linear second order partial differential operators of the following type $$H=\partial_{t}-L=\partial_{t}-\sum_{i,j=1}^{q}a_{ij}(t,x) X_{i}X_{j}-\sum_{k=1}^{q}a_{k}(t,x)X_{k}-a_{0}(t,x)$$ where $X_{1},X_{2},\ldots,X_{q}$ is a system of real Hormander's vector fields in some bounded domain $\Omega\subseteq\mathbb{R}^{n}$, $A=\left\{ a_{ij}\left( t,x\right) \right\} _{i,j=1}^{q}$ is a real symmetric uniformly positive definite matrix such that $$\lambda^{-1}\vert\xi\vert^{2}\leq\sum_{i,j=1}^{q}a_{ij}(t,x) \xi_{i}\xi_{j}\leq\lambda\vert\xi\vert^{2}\forall\xi\in\mathbb{R}^{q}, x \in\Omega,t\in(T_{1},T_{2})$$ for a suitable constant $\lambda>0$ a for some real numbers $T_{1} < T_{2}$.