Smoothing Properties of Fractional Ornstein-Uhlenbeck Semigroups and Null-Controllability

We study fractional hypoelliptic Ornstein-Uhlenbeck operators acting on $L^2({\mathbb{R}}^n)$ satisfying the Kalman rank condition. We prove that the semigroups generated by these operators enjoy Gevrey regularizing effects. Two byproducts are derived from this smoothing property. On the one hand, we prove the null-controllability in any positive time from thick control subsets of the associated parabolic equations posed on the whole space. On the other hand, by using interpolation theory, we get global $L^2$ subelliptic estimates for the these operators.