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

  • Óscar Ciaurri (a1), Luz Roncal (a2) and Sundaram Thangavelu (a3)

Abstract

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 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.

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

  • Óscar Ciaurri (a1), Luz Roncal (a2) and Sundaram Thangavelu (a3)

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