Endpoint results for the Riesz transform of the Ornstein-Uhlenbeck operator

TL;DR

This paper introduces a new Hardy space adapted to the Gauss measure and proves the boundedness of the Riesz transform of the Ornstein-Uhlenbeck operator from this space to L^1, along with a simplified proof of its weak-type (1,1).

Contribution

It develops a novel atomic Hardy space for the Gauss measure and establishes boundedness results for the Riesz transform in this setting.

Findings

Boundedness of Riesz transform from $X^1(\gamma)$ to $L^1(\gamma)$

New atomic Hardy space $X^1(\gamma)$ for Gauss measure

Simplified proof of weak-type (1,1) for the Riesz transform

Abstract

In this paper we introduce a new atomic Hardy space $X^1(\gamma)$ adapted to the Gauss measure $\gamma$, and prove the boundedness of the first order Riesz transform associated with the Ornstein-Uhlenbeck operator from $X^1(\gamma)$ to $L^1(\gamma)$. We also provide a new, short and almost self-contained proof of its weak-type $(1,1)$.