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 Hörmander'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:

$$\lambda^{-1}\left\vert \xi\right\vert ^{2}\leq\sum_{i,j=1}^{q}a_{ij}\left( t,x\right) \xi_{i}\xi_{j}\leq\lambda\left\vert \xi\right\vert ^{2}$$ $$\forall\xi\in\mathbb{R}^{q},x\in\Omega,t\in\left( T_{1},T_{2}\right)$$

for a suitable constant $\lambda>0$ a for some real numbers $T_{1}<T_{2}.$ The coefficients $a_{ij},a_{k},a_{0}$ are Hölder 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-Carathéodory 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:

$\frac{e^{-cd\left( x,y\right) ^{2}/\left( t-s\right) }}{c\left\ve... ...^{2}/c\left( t-s\right) }}{\left\vert B\left( x,\sqrt{t-s}\right) \right\vert }$
$\left\vert X_{i}h\left( t,x;s,y\right) \right\vert \leq\frac{c}{\... ...^{2}/c\left( t-s\right) }}{\left\vert B\left( x,\sqrt{t-s}\right) \right\vert }$
$\left\vert X_{i}X_{j}h\left( t,x;s,y\right) \right\vert +\left\ve... ...^{2}/c\left( t-s\right) }}{\left\vert B\left( x,\sqrt{t-s}\right) \right\vert }$

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 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 Hölder norms and the number $\lambda$.