# Maximal and quadratic Gaussian Hardy spaces

### Pierre Portal

Université Lille 1, Villeneuve d'Ascq, France

## Abstract

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.

## Cite this article

Pierre Portal, Maximal and quadratic Gaussian Hardy spaces. Rev. Mat. Iberoam. 30 (2014), no. 1, pp. 79–108

DOI 10.4171/RMI/770