Hardy-type inequalities for fractional powers of the Dunkl--Hermite operator

TL;DR

This paper establishes Hardy-type inequalities for fractional Dunkl--Hermite operators by reducing the problem to Laguerre operators using h-harmonic expansions and adapting non-local ground state techniques.

Contribution

It introduces a novel approach to derive Hardy inequalities for fractional Dunkl--Hermite operators via spectral reduction and extends the method to an abstract setting.

Findings

Hardy inequalities are proven for fractional Dunkl--Hermite operators.

The technique is adaptable to abstract settings with spectral theorems.

The approach also yields Hardy inequalities for fractional harmonic oscillators.

Abstract

We prove Hardy-type inequalities for a fractional Dunkl--Hermite operator which incidentally give Hardy inequalities for the fractional harmonic oscillator as well. The idea is to use $h$-harmonic expansions to reduce the problem in the Dunkl--Hermite context to the Laguerre setting. Then, we push forward a technique based on a non-local ground representation, initially developed by R. L. Frank, E. H. Lieb and R. Seiringer in the Euclidean setting, to get a Hardy inequality for the fractional-type Laguerre operator. The above-mentioned method is shown to be adaptable to an abstract setting, whenever there is a "good" spectral theorem and an integral representation for the fractional operators involved.