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.