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

  1. Ciaurri, Ó. 1
  2. Roncal, L. 12
  3. Thangavelu, S. 3
  1. 1 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

  2. 2 Basque Center for Applied Mathematics
    info

    Basque Center for Applied Mathematics

    Bilbao, España

    ROR 03b21sh32

  3. 3 Department of Mathematics, Indian Institute of Science, 560 012 Bangalore, India
Revista:
Proceedings of the Edinburgh Mathematical Society

ISSN: 0013-0915

Año de publicación: 2018

Páginas: 1-32

Tipo: Artículo

DOI: 10.1017/S0013091517000311 SCOPUS: 2-s2.0-85044777540 GOOGLE SCHOLAR

Otras publicaciones en: Proceedings of the Edinburgh Mathematical Society

Objetivos de desarrollo sostenible

Resumen

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. Copyright © Edinburgh Mathematical Society 2018