Calderón-Zygmund operators and endpoint spaces for Hermite expansions

Abstract

Let $L=-\Delta +|x{|}^2$ be the Hermite operator on ${\mathbb{R}}^n$, and $T$ be a Calderón-Zygmund type operator that is modelled on certain singular integrals related to $L$. We establish necessary and sufficient conditions for $T$ to be bounded on various function spaces including the Hardy spaces and the Lipschitz spaces associated to $L$. We then apply our results to study the boundedness of the Riesz transforms and pseudo-multipliers associated to $L$.