Hardy-Type Inequalities for Fractional Powers of the Dunkl–Hermite Operator

Abstract

AbstractWe prove Hardy-type inequalities for a fractional Dunkl–Hermite operator, which incidentally gives Hardy inequalities for the fractional harmonic oscillator as well. The idea is to useh-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 Franket al. [‘Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators,J. Amer. Math. Soc.21(2008), 925–950’] in the Euclidean setting, to obtain 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.