We consider Riesz transforms of any order associated to an Ornstein–Uhlenbeck operator with covariance given by a real, symmetric and positive definite matrix, and with drift given by a real matrix whose eigenvalues have negative real parts. In this general Gaussian context, we prove that a Riesz transform is of weak type (1, 1) with respect to the invariant measure if and only if its order is at most 2.

中文翻译：

---

一般 Ornstein-Uhlenbeck 半群的 Riesz 变换

我们考虑与 Ornstein-Uhlenbeck 算子相关联的任意阶 Riesz 变换，协方差由实数、对称和正定矩阵给出，漂移由实矩阵给出，其特征值具有负实部。在这个一般的高斯上下文中，我们证明 Riesz 变换是关于不变测度的弱类型 (1, 1) 当且仅当其阶数至多为 2。