Calder\'on-Zygmund operators and endpoint spaces for Hermite expansions

TL;DR

This paper investigates the boundedness of Calderón-Zygmund operators related to the Hermite operator on various endpoint function spaces, providing necessary and sufficient conditions and applications to Riesz transforms and pseudo-multipliers.

Contribution

It establishes new criteria for the boundedness of Hermite-related Calderón-Zygmund operators on Hardy and Lipschitz spaces, extending classical results to this setting.

Findings

Boundedness criteria for operators on Hardy and Lipschitz spaces

Application to Riesz transforms associated with the Hermite operator

Analysis of pseudo-multipliers in the Hermite setting

Abstract

Let $L=-\Delta +|x|^2$ be the Hermite operator on $\mathbb{R}^n$, and $T$ be a Calder\'on-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 $L$.