Marco Bramanti 
Non-Divergence Equations Structured on Hoermander 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 Hoermander’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 $T_{1} < T_{2}$.