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

Abstract

In the setting of Euclidean space with the Gaussian measure $\gamma$, 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 $L^p(\gamma )$, for $1<p<\infty$. We determine which of them are bounded from $H^1(\gamma )$ to $L^1(\gamma )$ and from $L^{\infty }(v)$ to BMO($\gamma$). Here $H^1(\gamma )$ and BMO($\gamma$) 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.