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

W. Teng

Journal of Lie theory, Tome 32 (2022) no. 4, pp. 1007-1023

Résumé

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.

Détail

Classification : 22E46, 26A33, 17B22, 47D03, 33C55, 43A32, 33C45
Mots-clés : Spherical harmonic expansion of (k,a)-generalized Laguerre semigroup, a-deformed Laguerre operators, fractional Hardy inequality, (k,a)-generalized harmonic oscillator