
Journal of Lie Theory 32 (2022), No. 4, 10071023 Copyright Heldermann Verlag 2022 Hardy Inequalities for Fractional (k,a)Generalized Harmonic Oscillators Wentao Teng School of Science, Kwansei Gakuin University, Sanda, Hyogo, Japan wentaoteng6@sina.com [Abstractpdf] We define $a$deformed Laguerre operators $L_{a,\alpha}$ and $a$deformed Laguerre holomorphic semigroups on $L^2\left(\left(0,\infty\right),d\mu_{a,\alpha}\right)$. Then we give a spherical harmonic expansion, which reduces to the Bochnertype identity when taking the boundary value $z=\pi i/2$, of the $(k,a)$generalized Laguerre semigroup introduced by Ben Sa\"id, Kobayashi and \O rsted. We prove a Hardy inequality for fractional powers of the $a$deformed Dunkl harmonic oscillator $\smash{\triangle_{k,a}:=\leftx\right^{2a}\triangle_k\leftx\right^a}$ using this expansion. When $a=2$, the fractional Hardy inequality reduces to that of DunklHermite operators given by Ciaurri, Roncal and Thangavelu. The operators $L_{a,\alpha}$ also give a tangible characterization of the radial part of the $(k,a)$generalized Laguerre semigroup on each $k$spherical component $\mathcal H_k^m\left(\mathbb{R}^N\right)$ for $$ \smash{\lambda_{k,a,m}:= \frac{2m+2\left\langle k\right\rangle+N2}{a}\geq \frac12} $$ defined via a decomposition of the unitary representation. Keywords: Spherical harmonic expansion of (k,a)generalized Laguerre semigroup, adeformed Laguerre operators, fractional Hardy inequality, (k,a)generalized harmonic oscillator. MSC: 22E46, 26A33, 17B22, 47D03, 33C55, 43A32, 33C45. [ Fulltextpdf (181 KB)] for subscribers only. 