Maximal and quadratic Gaussian Hardy spaces

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.