Description of the smoothing effects of semigroups generated by fractional Ornstein–Uhlenbeck operators and subelliptic estimates

Abstract

We study semigroups generated by general fractional Ornstein–Uhlenbeck operators acting on \(L^2({\mathbb {R}}^n)\). We characterize geometrically the partial Gevrey-type smoothing properties of these semigroups, and we sharply describe the blow-up of the associated seminorms for short times, generalizing the hypoelliptic and the quadratic cases. As a byproduct of this study, we establish partial subelliptic estimates enjoyed by fractional Ornstein–Uhlenbeck operators on the whole space by using interpolation theory.