Higher-order Riesz operators for the Ornstein-Uhlenbeck semigroup

We prove that the second-order Riesz transforms associated to the Ornstein- Uhlenbeck semigroup are weak type (1,1) with respect to the Gaussian measur e in finite dimension. We also show that they are given by a principal valu e integral plus a constant multiple of the identity. For the Riesz transfor ms of order three or higher, we present a counterexample showing that the w eak type (1,1) estimate fails.