Riesz transforms, Hardy spaces and Campanato spaces associated with Laguerre expansions

Let $\nu \in [-1\mathrm{/}2,\mathrm{\infty }{)}^n$, $n\ge 1$, and let ${\mathcal{L}}_{\nu }$ be a self-adjoint extension of the differential operator $$L_{\nu }:=\sum _{i=1}^n[-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_i^2}+x_i^2+\frac{1}{x_i^2}({\nu }_i^2-\frac{1}{4})]$$ on $C_c^{\mathrm{\infty }}({\mathbb{R}}_{+}^n)$ as the natural domain. In this paper, we first prove that the Riesz transform associated with ${\mathcal{L}}_{\nu }$ is a Calderón-Zygmund operator, answering the open problem in [JFA, 244 (2007), 399-443]. In addition, we develop the theory of Hardy spaces and Campanato spaces associated with ${\mathcal{L}}_{\nu }$. As applications, we prove that the Riesz transform related to ${\mathcal{L}}_{\nu }$ is bounded on these Hardy spaces and Campanato spaces, completing the description of the boundedness of the Riesz transform in the Laguerre expansion setting.