论文标题
hardy不平等$（k，a）$广义谐波振荡器
Hardy inequalities for fractional $(k,a)$-generalized harmonic oscillator
论文作者
论文摘要
在本文中，我们将定义$ a $ a-deformed laguerre运营商$ l_ {a，α} $和$ a $ deformed laguerre holomorphic semigroups in $ l^2 \ left（\ left（\ left（0，\ infty \ right），dμ___{a，a，α} \ right（0，\ infty \ right）。然后，我们给出一个球形谐波扩展，当取边界值$ z = \ frac {πi} 2 $时，它会减少到bochner型的身份，这是$（k，a）$广义的Laguerre Semigroup，由S. S. BenSaïd，T。Kobayashi和B.。然后，我们证明了$ a $ a $ a的dunkl谐波振荡器$ \ triangle_ {k，a}：= \ left | x \ firt | x \ right |^{2-a} \ triangle_k- \ weft | x \ weft | x \ prirt | x \ right |^a $使用此扩展。当$ a = 2 $时，hardy不平等的分数将减少到邓尔（Dunkl-Hermite）运算符的dunkl-Hermite运算符。 Ciaurri，L。Roncal和S. Thangavelu。操作员$ l_ {a，α} $还给出了$（k，a）$ - $ k $ -spherical component $ \ mathcal h_k h_k^m \ weft（\ mathbb {r}^n \ right）上的$（k，a）$ - $ k $ -spherical component $ \ m ythcal h_k h_k^m \ mathcal $ \ mathcal h_k^m \ $λ_{k，a，m}：= \ frac {2m+2 \ left \ langle k \ right \ rangle+n -2} a \ geq -1/2 $通过单位表示的分解而定义。
In this paper, we will define $a$-deformed Laguerre operators $L_{a,α}$ and $a$-deformed Laguerre holomorphic semigroups on $L^2\left(\left(0,\infty\right),dμ_{a,α}\right)$. Then we give a spherical harmonic expansion, which reduces to the Bochner-type identity when taking the boundary value $z=\frac{πi}2$, 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 $\triangle_{k,a}:=\left|x\right|^{2-a}\triangle_k-\left|x\right|^a$ using this expansion. When $a=2$, the fractional Hardy inequality reduces to that of Dunkl--Hermite operators given by Ó. Ciaurri, L. Roncal and S. Thangavelu. The operators $L_{a,α}$ 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 $λ_{k,a,m}:=\frac{2m+2\left\langle k\right\rangle+N-2}a\geq -1/2$ defined via decomposition of unitary representation.
