Calderón–Zygmund Operators and Endpoint Spaces for Hermite Expansions

Let L=-Δ+|x|2 be the Hermite operator on Rn, and T be a Calderon–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.