Endpoint results for the Riesz transform of the Ornstein-Uhlenbeck operator

In this paper we introduce a new atomic Hardy space $X^1(\gamma )$ adapted to the Gauss measure $\gamma$, and prove the boundedness of the first order Riesz transform associated with the Ornstein-Uhlenbeck operator from $X^1(\gamma )$ to $L^1(\gamma )$. We also provide a new, short and almost self-contained proof of its weak-type $(1,1)$.