[Paper Review] Hardy inequalities for fractional $(k,a)$-generalized harmonic oscillator

Wentao Teng|arXiv (Cornell University)|Aug 3, 2020

Mathematical Analysis and Transform Methods18 references4 citations

TL;DR

This paper introduces $a$-deformed Laguerre operators and holomorphic semigroups to derive a spherical harmonic expansion of the $(k,a)$-generalized Laguerre semigroup, which reduces to a Bochner-type identity at $z = \pi i/2$. It establishes a fractional Hardy inequality for the $a$-deformed Dunkl harmonic operator $\triangle_{k,a}$, recovering known results for $a=2$ (Dunkl-Hermite) and $a=1$ (Kobayashi-Mano).

ABSTRACT

In this paper, we will define $a$-deformed Laguerre operators $L_{a,α}$ and $a$-deformed Laguerre holomorphic semigroups on $L^2\left(\left(0,\infty ight),dμ_{a,α} ight)$. 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 $ riangle_{k,a}:=\left|x ight|^{2-a} riangle_k-\left|x ight|^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 $\mathcal H_k^m\left(\mathbb{R}^N ight)$ for $λ_{k,a,m}:=\frac{2m+2\left\langle k ight angle+N-2}a\geq -1/2$ defined via decomposition of unitary representation.

Motivation & Objective

• To define $a$-deformed Laguerre operators $L_{a,\alpha}$ and their associated holomorphic semigroups on $L^2((0,\infty), d\mu_{a,\alpha})$.
• To establish a spherical harmonic expansion of the $(k,a)$-generalized Laguerre semigroup via decomposition of unitary representations.
• To derive a Bochner-type identity as the boundary value $z = \pi i/2$ of the semigroup expansion.
• To prove a fractional Hardy inequality for the $a$-deformed Dunkl harmonic operator $\triangle_{k,a}$ using the semigroup expansion.
• To recover known results for $a=2$ (Dunkl-Hermite) and $a=1$ (Kobayashi-Mano) as special cases of the generalized inequality.

Proposed method

• Define $a$-deformed Laguerre operators $L_{a,\alpha}$ as infinitesimal generators of $a$-deformed Laguerre holomorphic semigroups $I_{a,\alpha;z} = e^{-\frac{z}{a}L_{a,\alpha}}$ on $L^2((0,\infty), d\mu_{a,\alpha})$.
• Use the decomposition of unitary representations of $S\widetilde{L}(2,\mathbb{R})$ to express the radial part $\Omega_{k,a}^{(m)}(\gamma_z)$ of the $(k,a)$-generalized Laguerre semigroup on each $k$-spherical component $\mathcal{H}_k^m(\mathbb{R}^N)$.
• Establish the identity $\Omega_{k,a}^{(m)}(\gamma_z)f(s) = s^m I_{a,\lambda_{k,a,m};z}((\cdot)^{-m}f)(s)$, linking the semigroup to the $a$-deformed Laguerre semigroup.
• Derive the spherical harmonic expansion of $\mathcal{I}_{k,a}(z)f(x) = \sum_{m,j} Y_{m,j}(x') r^m I_{a,\lambda_{k,a,m};z}((\cdot)^{-m}f)(r)$ for $f \in L^2(\mathbb{R}^N, \vartheta_{k,a}(x)dx)$.
• Prove that the boundary value $z = \pi i/2$ yields the $a$-deformed Hankel transform $H_{a,\alpha}(f)$, recovering known transforms as special cases.
• Apply the semigroup expansion to derive a fractional Hardy inequality for $\triangle_{k,a}$, valid when $\lambda_{k,a,m} \geq -1/2$.

Experimental results

Research questions

• RQ1 How can the $(k,a)$-generalized Laguerre semigroup be decomposed via spherical harmonics and $a$-deformed Laguerre operators?
• RQ2 What is the precise form of the radial part $\Omega_{k,a}^{(m)}(\gamma_z)$ in terms of $a$-deformed Laguerre semigroups?
• RQ3 How does the boundary value $z = \pi i/2$ of the semigroup relate to the $a$-deformed Hankel transform?
• RQ4 Can a fractional Hardy inequality be established for the $a$-deformed Dunkl harmonic operator $\triangle_{k,a}$ using this semigroup structure?
• RQ5 How do the results specialize to known cases such as $a=2$ (Dunkl-Hermite) and $a=1$ (Kobayashi-Mano)?

Key findings

• The spherical harmonic expansion of the $(k,a)$-generalized Laguerre semigroup is given by $\mathcal{I}_{k,a}(z)f(x) = \sum_{m,j} Y_{m,j}(x') r^m I_{a,\lambda_{k,a,m};z}((\cdot)^{-m}f)(r)$, valid for $f \in L^2(\mathbb{R}^N, \vartheta_{k,a}(x)dx)$ under the condition $4\langle k\rangle + 2N + a - 4 \geq 0$.
• The $a$-deformed Laguerre semigroup $I_{a,\alpha;z}$ reduces to the $a$-deformed Hankel transform $H_{a,\alpha}$ at $z = \pi i/2$, with $e^{(\alpha+1)\pi i/2} I_{a,\alpha;\pi i/2}(f) = H_{a,\alpha}(f)$.
• The radial part $\Omega_{k,a}^{(m)}(\gamma_z)$ of the semigroup is explicitly expressed as $\Omega_{k,a}^{(m)}(\gamma_z)f(s) = s^m I_{a,\lambda_{k,a,m};z}((\cdot)^{-m}f)(s)$, with $\lambda_{k,a,m} = \frac{2m + 2\langle k\rangle + N - 2}{a}$.
• The fractional Hardy inequality for $\triangle_{k,a}$ is proven under the condition $\lambda_{k,a,m} \geq -1/2$, generalizing known results for $a=2$ (Dunkl-Hermite) and $a=1$ (Kobayashi-Mano).
• The infinitesimal generator of $\Omega_{k,a}^{(m)}(\gamma_z)$ is $-\frac{1}{a} L_{a,\lambda_{k,a,m}}$, confirming the connection between the $a$-deformed Laguerre operator and the radial part of the semigroup.
• For radial functions $f = f_0(|\cdot|)$ with $f_0 \in \mathcal{S}(\mathbb{R}_+)$, the operator $\triangle_{k,a}$ acts as $\triangle_{k,a}f(x) = -L_{a,\lambda_a}(f_0)(r)$, recovering the known formula for the Dunkl Laplacian on radial functions.