Weighted norm inequalities of higher-order Riesz transforms associated with Laguerre expansions
The Anh Bui
TL;DR
This work resolves the weighted $L^p$ boundedness of higher-order Riesz transforms associated with Laguerre expansions for the full parameter range $\nu\in(-1,\infty)^n$. By deriving sharp heat-kernel and derivative estimates, the authors overcome obstacles that arise in higher dimensions and when Calderón–Zygmund techniques fail. The core contribution is a precise description of weighted boundedness: $\delta_\nu^k\mathcal{L}_\nu^{-|k|/2}$ is bounded on $L^p_w(\mathbb{R}^n_+)$ for $p$ in $\big(\frac{1}{1-\gamma_\nu}, \frac{1}{\gamma_{\nu+\sigma(k)}}\big)$ with weights $w\in A_{p(1-\gamma_\nu)}\cap RH_{(\frac{1}{p\gamma_{\nu+\sigma(k)}})'}$, where $\gamma_\nu$ encodes $\nu$-dependence and $\sigma(k)$ encodes the parity of $k$. The analysis hinges on refined kernel bounds for $e^{-t\mathcal{L}_\nu}$ and its derivatives, and a BD-type weighted boundedness criterion applied to localized operator decompositions. The results extend and complete previous partial analyses, providing a full description of weighted higher-order Laguerre-Riesz transforms in any dimension and for all admissible $~\nu$. This has implications for harmonic analysis on Laguerre-type spaces and related spectral multiplier theory.
Abstract
Let $ν=(ν_1,\ldots,ν_n)\in (-1,\vc)^n$, $n\ge 1$, and let $\mathcal{L}_ν$ 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 $L_ν$ 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 $δ_ν^k\mathcal L^{-|k|/2}_ν, k\in \mathbb N^n$, where $δ_ν^k=δ_{ν_n}^{k_n}\ldots δ_{ν_1}^{k_1}$. This completes the description of the boundedness of the higher-order Riesz transforms with the full range $ν\in (-1,\vc)^n$.