Calderón–Zygmund Operators and Endpoint Spaces for Hermite Expansions

Let \(\mathcal {L}=-\Delta +|x|^2\) $\mathcal{L}=-\mathrm{\Delta }+{|x|}^2$ be the Hermite operator on \({\mathbb {R}}^n\) ${\mathbb{R}}^n$ , and T be a Calderon–Zygmund type operator that is modelled on certain singular integrals related to \(\mathcal {L}\) $\mathcal{L}$ . 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 \({\mathcal {L}}\) $\mathcal{L}$ . We then apply our results to study the boundedness of the Riesz transforms and pseudo-multipliers associated to \({\mathcal {L}}\) $\mathcal{L}$ .