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 $x\in {\mathbb R}^N$ and the Kalman rank condition is satisfied at any $x\in{\mathbb R}^N$ by the same $m<N$, and developing a revised version of Bernstein's 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 (H\"older-) continuous functions. Finally, we prove Schauder estimates for some elliptic and parabolic problems associated with the operator ${\mathscr A}$.