On a class of hypoelliptic operators with unbounded coefficients in ${matbb R}^N$

We consider a class of non-trivial perturbations ${mathscr A}$ of the degenerate Ornstein-Uhlenbeck operator in ${mathbb R}^N$. In fact we perturb both the diffusion and the drift part of the operator (say $Q$ and $B$) allowing the diffusion part to be unbounded in ${mathbb R}^N$. Assuming that the kernel of the matrix $Q(x)$ is invariant with respect to $xin {mathbb R}^N$ and the Kalman rank condition is satisfied at any $xin{mathbb R}^N$ by the same $m<N$, and developing a revised version of Bernsteins method we prove that we can associate a semigroup ${T(t)}$ of bounded operators (in the space of bounded and continuous functions) with the operator ${mathscr A}$. Moreover, we provide several uniform estimates for the spatial derivatives of the semigroup ${T(t)}$ both in isotropic and anisotropic spaces of (Holder-) continuous functions. Finally, we prove Schauder estimates for some elliptic and parabolic problems associated with the operator ${mathscr A}$.