HARDY-TYPE INEQUALITIES FOR FRACTIONAL POWERS OF THE DUNKL-HERMITE OPERATOR

We 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 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 Frank et al. ['Hardy-Lieb-Thirring inequalities for fractional Schrodinger 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.