Weighted norm inequalities of higher-order Riesz transforms associated with Laguerre expansions

Abstract

Let $\nu=(\nu_1,\ldots,\nu_n)\in (-1,\vc)^n$, $n\ge 1$, and let $\mathcal{L}_\nu$ be a self-adjoint extension of the differential operator \[ L_\nu := \sum_{i=1}^n \left[-\frac{\partial^2}{\partial x_i^2} + x_i^2 + \frac{1}{x_i^2}(\nu_i^2 - \frac{1}{4})\right] \] on $C_c^\infty(\mathbb{R}_+^n)$ as the natural domain. The $j$-th partial derivative associated with $L_{\nu}$ is given by \[ \delta_{\nu_j} = \frac{\partial}{\partial x_j} + x_j-\frac{1}{x_j}\Big(\nu_j + \f{1}{2}\Big), \ \ \ \ j=1,\ldots, n. \] In this paper, we investigate the weighted estimates of the higher-order Riesz transforms $\delta_\nu^k\mathcal L^{-|k|/2}_\nu, k\in \mathbb N^n$, where $\delta_\nu^k=\delta_{\nu_n}^{k_n}\ldots \delta_{\nu_1}^{k_1}$. This completes the description of the boundedness of the higher-order Riesz transforms with the full range $\nu \in (-1,\vc)^n$.