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

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