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

We consider second order linear partial differential operators $A$ on $R^N$ which are not assumed to be uniformly elliptic and whose coefficients in the second order part may grow quadratically, while the drift part has essentially linear growth. Instead of uniform ellipticity, we require a much weaker hypothesis of uniform hypoellipticity, which in an equivalent formulation connects the behaviour of the diffusion and the drift coefficients, by requiring that a Kalman-type condition is satisfied for them. By refining Bernstein's method we prove the existence of a semigroup {$T(t)$} of bounded linear operators (in the space of bounded and continuous functions) associated to the operator $A$. We also show uniform estimates for the spatial derivatives of the semigroup {$T(t)$} in (an)isotropic spaces of (Hölder-) continuous functions. As a consequence, we obtain Hölder estimates for the solutions of some elliptic and parabolic problems associated to the operator $A$.