Hardy inequalities for fractional $(k,a)$-generalized harmonic oscillator

Abstract

In this paper, we will 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 $z=\frac{\pi i}{2}$, of the $(k,a)$-generalized Laguerre semigroup introduced by S. Ben Saïd, T. Kobayashi and B. Ørsted. And then we prove a Hardy inequality for fractional powers of the $a$-deformed Dunkl harmonic oscillator ${△}_{k,a}:={\left|x\right|}^{2-a}{△}_k-{\left|x\right|}^a$ using this expansion. When $a=2$, the fractional Hardy inequality reduces to that of Dunkl--Hermite operators given by Ó. Ciaurri, L. Roncal and S. 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 ${\lambda }_{k,a,m}:=\frac{2m+2⟨k⟩+N-2}{a}\ge -1/2$ defined via decomposition of unitary representation.

16 pages; accepted for publication in "Journal of Lie Theory"; fixed some typos