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

Part of: Partial differential equations Abstract harmonic analysis Real functions Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  02 April 2018

Óscar Ciaurri ,
Luz Roncal  and
Sundaram Thangavelu
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.
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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.

Keywords

Hardy inequalityDunkl harmonic oscillatorfractional order operatorLaguerre expansionsheat semigroup

MSC classification

Primary: 26A33: Fractional derivatives and integrals
Secondary: 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 35A08: Fundamental solutions 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 43A90: Spherical functions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 2 , May 2018 , pp. 513 - 544
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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