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

TL;DR

This paper investigates the smoothing properties of semigroups generated by fractional Ornstein-Uhlenbeck operators, providing geometric characterizations and subelliptic estimates that generalize hypoelliptic cases.

Contribution

It offers a geometric characterization of Gevrey-type smoothing and establishes subelliptic estimates for fractional Ornstein-Uhlenbeck operators on the entire space.

Findings

Characterization of partial Gevrey smoothing properties.

Description of seminorm blow-up for short times.

Establishment of partial subelliptic estimates.

Abstract

We study semigroups generated by general fractional Ornstein-Uhlenbeck operators acting on $L2(\mathbb R^n)$. 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.