Revista de la
Unión Matemática Argentina
Endpoint estimates for higher-order Gaussian Riesz transforms
Fabio Berra, Estefanía Dalmasso, and Roberto Scotto

Volume 69, no. 1 (2026), pp. 25–43    

Published online (final version): November 19, 2025

https://doi.org/10.33044/revuma.4878

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Abstract

We 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 any dimension $n$. We also prove that the classical Gaussian Riesz transforms of higher order are bounded from an appropriate subspace of $\mathcal{H}^1(\mathbb{R}^n, \gamma)$ into $L^1(\mathbb{R}^n, \gamma)$, extending T. Bruno (2019) to the first-order case.

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