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

Abstract

Let $ν=(ν_1,\ldots,ν_n)\in (-1,\vc)^n$, $n\ge 1$, 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. The $j$-th partial derivative associated with $L_{\nu }$ is given by \[ δ_{ν_j} = \frac{\partial}{\partial x_j} + x_j-\frac{1}{x_j}\Big(ν_j + \f{1}{2}\Big), \ \ \ \ j=1,\ldots, n. \] In this paper, we investigate the weighted estimates of the higher-order Riesz transforms ${\delta }_{\nu }^k{\mathcal{L}}_{\nu }^{-|k|/2},k\in {\mathbb{N}}^n$, 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 $ν\in (-1,\vc)^n$.

27 pages. arXiv admin note: text overlap with arXiv:2411.19404