Endpoint estimates for higher order Gaussian Riesz transforms

We will show that, contrary to the behavior of the higher order Riesz transforms studied so far on the atomic Hardy space ${\mathcal{H}}^1({\mathbb{R}}^n,\gamma )$, associated with the Ornstein-Uhlenbeck operator with respect to the $n$-dimensional Gaussian measure $\gamma$, the new Gaussian Riesz transforms are bounded from ${\mathcal{H}}^1({\mathbb{R}}^n,\gamma )$ to $L^1({\mathbb{R}}^n,\gamma )$, for any order and dimension $n$. We will also prove that the classical Gaussian Riesz transforms of higher order are bounded from an adequate subspace of ${\mathcal{H}}^1({\mathbb{R}}^n,\gamma )$ into $L^1({\mathbb{R}}^n,\gamma )$, extending Bruno’s result (J. Fourier Anal. Appl. 25, 4 (2019), 1609–1631) for the first order case.