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

Abstract

Let $\nu \in [-1/2,\infty {)}^n$, $n\ge 1$, and let ${\mathcal{L}}_{\nu }$ be a self-adjoint extension of the differential operator \[ L_ν:= \sum_{i=1}^n \left[-\frac{\partial^2}{\partial x_i^2} + x_i^2 + \frac{1}{x_i^2}(ν_i^2 - \frac{1}{4})\right] \] on $C_c^{\infty }({\mathbb{R}}_{+}^n)$ 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.

48 pages; some typos were corrected