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 Sa\"id, T. Kobayashi and B. \O rsted. And then we prove a Hardy inequality for fractional powers of the $a$-deformed Dunkl harmonic oscillator $\triangle_{k,a}:=\left|x\right|^{2-a}\triangle_k-\left|x\right|^a$ using this expansion. When $a=2$, the fractional Hardy inequality reduces to that of Dunkl--Hermite operators given by \'O. 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 $\lambda_{k,a,m}:=\frac{2m+2\left\langle k\right\rangle+N-2}a\geq -1/2$ defined via decomposition of unitary representation.