Endpoint estimates for first-order Riesz transforms associated to the Ornstein-Uhlenbeck operator

In the setting of Euclidean space with the Gaussian measure g, we consider all first-order Riesz transforms associated to the infinitesimal generator of the Ornstein-Uhlenbeck semigroup. These operators are known to be bounded on L^p(g), for 1<p<\infty. We determine which of them are bounded from H^1(g) to L^1(g) and from L^\infty(g)$ to \BMO(g). Here H^1(g) and BMO(g) are the spaces introduced in this setting by the first two authors. Surprisingly, we find that the results depend on the dimension of the ambient space.