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

Published online by Cambridge University Press:  02 April 2018

Óscar Ciaurri
Affiliation:
Departamento de Matemáticas y Computación, Universidad de La Rioja, 26006 Logroño, Spain (oscar.ciaurri@unirioja.es)
Luz Roncal*
Affiliation:
Departamento de Matemáticas y Computación, Universidad de La Rioja, 26006 Logroño, Spain and BCAM – Basque Center for Applied Mathematics, 48009 Bilbao, Spain (lroncal@bcamath.org)
Sundaram Thangavelu
Affiliation:
Department of Mathematics, Indian Institute of Science, 560 012 Bangalore, India (veluma@math.iisc.ernet.in)
*
*Corresponding author.

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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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