Weighted norm inequalities of some singular integrals associated with Laguerre expansions

Abstract

Let $\nu =({\nu }_1,\dots ,{\nu }_n)\in (-1,\infty {)}^n$, $n\ge 1$, 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 investigate the weighted estimates of singular integrals in the Laguerre setting including the maximal function, the Riesz transform and the square functions associated to the Laguerre operator ${\mathcal{L}}_{\nu }$. In the special case of the Riesz transform, the paper completes the description of the Riesz transform for the full range of $\nu \in (-1,\infty {)}^n$ which significant improves the result in [J. Funct. Anal. 244 (2007), 399--443] for ${\nu }_i\ge -1/2,{\nu }_i\notin (-1/2,1/2)$ for $i=1,\dots ,n$.

31 pages