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 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 \[\lambda^{-1}\vert\xi\vert^{2}\leq\sum_{i,j=1}^{q}a_{ij}(t,x) \xi_{i}\xi_{j}\leq\lambda\vert\xi\vert^{2}\forall\xi\in\mathbb{R}^{q}, x \in\Omega,t\in(T_{1},T_{2})\] for a suitable constant \(\lambda>0\) a for some real numbers \(T_{1} < T_{2}\).