Non-divergence equations structured on Hörmander vector fields: heat kernels and Harnack inequalities.

In this work we deal with linear second order partial differential operators of the following type:% \[ H=\partial_{t}-L=\partial_{t}-\sum_{i,j=1}^{q}a_{ij}\left( t,x\right) X_{i}X_{j}-\sum_{k=1}^{q}a_{k}\left( t,x\right) X_{k}-a_{0}\left( t,x\right) \] where $X_{1},X_{2},\ldots,X_{q}$ is a system of real H\"{o}rmander's vector fields in some bounded domain $\Omega$ of $\mathbb{R}^{n}$, $A=\left\{ a_{ij}\left( t,x\right) \right\} _{i,j=1}^{q}$ is a real symmetric uniformly positive definite matrix in $\left( T_{1},T_{2}\right) \times\Omega$. The coefficients $a_{ij},a_{k},a_{0}$ are H\"{o}lder continuous on $\left( T_{1},T_{2}\right) \times\Omega$ with respect to the parabolic CC-metric $d_{P}\left( \left( t,x\right) ,\left( s,y\right) \right) =\sqrt{d\left( x,y\right) ^{2}+\left\vert t-s\right\vert }$(where $d\ $is the Carnot-Carath\'{e}odory distance induced by the vector fields $X_{i}$'s). We prove the existence of a fundamental solution $h\left( t,x;s,y\right) $ for $H$, satisfying natural properties and sharp Gaussian bounds of the kind:% \begin{gather*} \frac{e^{-cd\left( x,y\right) ^{2}/\left( t-s\right) }}{c\left\vert B\left( x,\sqrt{t-s}\right) \right\vert }\leq h\left( t,x;s,y\right) \leq c\frac{e^{-d\left( x,y\right) ^{2}/c\left( t-s\right) }}{\left\vert B\left( x,\sqrt{t-s}\right) \right\vert }\\ \left\vert X_{i}h\left( t,x;s,y\right) \right\vert \leq\frac{c}{\sqrt{t-s}% }\frac{e^{-d\left( x,y\right) ^{2}/c\left( t-s\right) }}{\left\vert B\left( x,\sqrt{t-s}\right) \right\vert }\\ \left\vert X_{i}X_{j}h\left( t,x;s,y\right) \right\vert +\left\vert \partial_{t}h\left( t,x;s,y\right) \right\vert \leq\frac{c}{t-s}% \frac{e^{-d\left( x,y\right) ^{2}/c\left( t-s\right) }}{\left\vert B\left( x,\sqrt{t-s}\right) \right\vert }% \end{gather*} where $\left\vert B\left( x,r\right) \right\vert $ denotes the Lebesgue measure of the $d$-ball $B\left( x,r\right) $. We then use these properties of $h$ as a starting point to prove a \textit{scaling invariant }Harnack inequality for positive solutions to $Hu=0$, when $a_{0}\equiv0$. All the constants in our estimates and inequalities will depend on the coefficients $a_{ij},a_{k},a_{0}$ only through their H\"{o}lder norms and the ellipticity constant of the matrix $A$.