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

Abstract

In this paper, we establish the first $L^p$-Heisenberg--Pauli--Weyl uncertainty inequalities on the Laguerre hypergroup for the full range $1\le p\le2$. These results extend Xiao's Euclidean $L^p$ theory to the setting of the Laguerre hypergroup, which is the fundamental manifold of the radial function space for the Heisenberg group. The analysis is carried out through the Fourier--Laguerre transform and exploits the mixed discrete--continuous spectral structure of the Laguerre hypergroup, requiring estimates adapted to its Plancherel measure and dilation structure. As a consequence, in the endpoint case $p=2$, we obtain a refined $L^2$-Heisenberg--Pauli--Weyl uncertainty inequality valid for all positive exponents $a,b>0$, thereby improving the earlier result of Atef (2013), where the assumptions $a,b\ge1$ arose from the heat kernel methods. Our proofs rely on the Fourier--Laguerre transform, dilation and scaling invariance, the Hausdorff--Young inequality and the Plancherel identity, completely avoiding heat kernel techniques. These results provide a unified Fourier-analytic framework for Heisenberg--Pauli--Weyl uncertainty inequalities on the Laguerre hypergroup and further strengthen the connections between Euclidean, Heisenberg and hypergroup harmonic analysis.