JournalsrmiVol. 30, No. 1pp. 79–108

Maximal and quadratic Gaussian Hardy spaces

  • Pierre Portal

    Université Lille 1, Villeneuve d'Ascq, France
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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 Rn\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 h1(Rn;dγ)h^{1}(\mathbb{R}^{n};d\gamma) space studied here is such that the Riesz transforms are bounded from h1(Rn;dγ)h^{1}(\mathbb{R}^{n};d\gamma) to L1(Rn;dγ)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