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

Let $\nu =({\nu }_1,\dots ,{\nu }_n)\in (-1,\text{vc}{)}^n$, $n\ge 1$, and let ${\mathcal{L}}_{\nu }$ be a self-adjoint extension of the differential operator $$L_{\nu }:=\sum _{i=1}^n[-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_i^2}+x_i^2+\frac{1}{x_i^2}({\nu }_i^2-\frac{1}{4})]$$ on $C_c^{\mathrm{\infty }}({\mathbb{R}}_{+}^n)$ as the natural domain. The $j$-th partial derivative associated with $L_{\nu }$ is given by $${\delta }_{{\nu }_j}=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}+x_j-\frac{1}{x_j}({\nu }_j+\text{f}12),j=1,\dots ,n\mathrm{.}$$ In this paper, we investigate the weighted estimates of the higher-order Riesz transforms ${\delta }_{\nu }^k{\mathcal{L}}_{\nu }^{-\mathrm{\mid }k\mathrm{\mid }\mathrm{/}2},k\in {\mathbb{N}}^n$, where ${\delta }_{\nu }^k={\delta }_{{\nu }_n}^{k_n}\dots {\delta }_{{\nu }_1}^{k_1}$. This completes the description of the boundedness of the higher-order Riesz transforms with the full range $\nu \in (-1,\text{vc}{)}^n$.