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.

中文翻译：

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分数次Ornstein-Uhlenbeck半群的平滑特性和零可控性

我们研究作用于分数次椭圆的Ornstein-Uhlenbeck算子 ${\mathrm{大号}}^2（{\mathbb{[R}}^{ñ}）$满足卡尔曼等级条件。我们证明了这些算子生成的半群具有Gevrey正则化效应。从该平滑特性中衍生出两种副产物。一方面，我们从摆在整个空间上的相关抛物方程的厚控制子集证明了任何正时的零可控性。另一方面，通过使用插值理论，我们得到了全局${\mathrm{大号}}^2$ 这些算子的亚椭圆估计。