Maximal and quadratic Gaussian Hardy spaces

TL;DR

This paper proves the equivalence of maximal and square function norms defining Gaussian Hardy spaces on ^n, establishing boundedness of Riesz transforms and extending classical harmonic analysis results to the Gaussian setting.

Contribution

It introduces a Gaussian Hardy space with norm equivalence and Riesz transform boundedness, extending Fefferman-Stein theory to Gaussian measures.

Findings

Equivalence of maximal and square function norms for Gaussian Hardy spaces

Boundedness of Riesz transforms on the Gaussian Hardy space

Extension of classical harmonic analysis results to Gaussian measure setting

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 $\R^{n}$ 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 $h^{1}(\R^{n};d\gamma)$ space studied here is such that the Riesz transforms are bounded from $h^{1}(\R^{n};d\gamma)$ to $L^{1}(\R^{n};d\gamma)$. This gives a gaussian analogue of the seminal work of Fefferman and Stein in the case of the Lebesgue measure and the usual Laplacian.