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Journal of Lie Theory 32 (2022), No. 4, 1007--1023
Copyright Heldermann Verlag 2022

Hardy Inequalities for Fractional (k,a)-Generalized Harmonic Oscillators

Wentao Teng
School of Science, Kwansei Gakuin University, Sanda, Hyogo, Japan


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 Bochner-type 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 $\smash{\triangle_{k,a}:=\left|x\right|^{2-a}\triangle_k-\left|x\right|^a}$ using this expansion. When $a=2$, the fractional Hardy inequality reduces to that of Dunkl-Hermite 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 $$ \smash{\lambda_{k,a,m}:= \frac{2m+2\left\langle k\right\rangle+N-2}{a}\geq -\frac12} $$ defined via a decomposition of the unitary representation.

Keywords: Spherical harmonic expansion of (k,a)-generalized Laguerre semigroup, a-deformed Laguerre operators, fractional Hardy inequality, (k,a)-generalized harmonic oscillator.

MSC: 22E46, 26A33, 17B22, 47D03, 33C55, 43A32, 33C45.

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