$L^p$-Heisenberg-Pauli-Weyl Uncertainty Inequalities on Laguerre Hypergroup

TL;DR

This paper establishes $L^p$-Heisenberg-Pauli-Weyl uncertainty inequalities on the Laguerre hypergroup, extending previous results and providing a comprehensive $L^p$ framework for uncertainty principles in this setting.

Contribution

It introduces new $L^p$-uncertainty inequalities on the Laguerre hypergroup, improving earlier bounds and extending Euclidean and Lie group results without using heat kernel methods.

Findings

Derived $L^p$-HPW inequalities for different $p$ ranges.

Improved $L^2$-HPW inequality valid for all positive $a, b$.

Extended Euclidean and Lie group uncertainty results to Laguerre hypergroup.

Abstract

In this article, we establish the $L^p$-Heisenberg-Pauli-Weyl uncertainty inequalities on the Laguerre hypergroup $\mathbb{K}$, the natural setting for radial analysis on the Heisenberg group. For $1 \leq p < 2$, under the condition $b > Q(1/p - 1/2)$, and for $2 \leq p < \infty$, with $0 < a < Q/p$ and $0 < b < 4$, we derive $L^p$-HPW uncertainty inequalities and as a consequence, we obtain a refined $L^2$-HPW inequality on $\mathbb{K}$, valid for all $a, b > 0$, improving upon the earlier result of Atef (2013) which required $a, b \geq 1$. Our proofs rely on the Fourier-Laguerre transform, dilation and rescaling invariance, and Hausdorff-Young and Plancherel inequalities, thus avoiding heat kernel methods. These results extend Xiao's Euclidean $L^p$-HPW uncertainty inequalities (2022) and parallel recent developments on nilpotent Lie groups, thereby providing a complete $L^p$-framework for uncertainty inequalities on the Laguerre hypergroup.