# Revista Matemática Iberoamericana

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**Volume 30, Issue 1, 2014, pp. 79–108**

**DOI: 10.4171/RMI/770**

Published online: 2014-03-23

Maximal and quadratic Gaussian Hardy spaces

Pierre Portal^{[1]}(1) Université Lille 1, Villeneuve d'Ascq, France

Building on the author’s recent work with Jan Maas and Jan van Neerven, this paper establishes the equivalence of two norms (one using a maximal function, the other a square function) used to define a Hardy space on $\mathbb{R}^{n}$ with the Gaussian measure, that is adapted to the Ornstein–Uhlenbeck semigroup. In contrast to the atomic Gaussian Hardy space introduced earlier by Mauceri and Meda, the $h^{1}(\mathbb{R}^{n};d\gamma)$ space studied here is such that the Riesz transforms are bounded from $h^{1}(\mathbb{R}^{n};d\gamma)$ to $L^{1}(\mathbb{R}^{n};d\gamma)$. This gives a Gaussian analogue of the seminal work of Fefferman and Stein in the case of the Lebesgue measure and the usual Laplacian.

*Keywords: *Hardy spaces, Gaussian measure, Ornstein–Uhlenbeck operator, square functions, molecular decompositions

Portal Pierre: Maximal and quadratic Gaussian Hardy spaces. *Rev. Mat. Iberoamericana* 30 (2014), 79-108. doi: 10.4171/RMI/770