Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group

  1. Roncal, L. 1
  2. Thangavelu, S. 2
  1. 1 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

  2. 2 Department of Mathematics, Indian Institute of Science, Bangalore, India
Revista:
Advances in Mathematics

ISSN: 0001-8708

Año de publicación: 2016

Volumen: 302

Páginas: 106-158

Tipo: Artículo

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DOI: 10.1016/J.AIM.2016.07.010 SCOPUS: 2-s2.0-84979299124 WoS: WOS:000384510600006 GOOGLE SCHOLAR

Otras publicaciones en: Advances in Mathematics

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Resumen

We prove Hardy inequalities for the conformally invariant fractional powers of the sublaplacian on the Heisenberg group Hn. We prove two versions of such inequalities depending on whether the weights involved are non-homogeneous or homogeneous. In the first case, the constant arising in the Hardy inequality turns out to be optimal. In order to get our results, we will use ground state representations. The key ingredients to obtain the latter are some explicit integral representations for the fractional powers of the sublaplacian and a generalized result by M. Cowling and U. Haagerup. The approach to prove the integral representations is via the language of semigroups. As a consequence of the Hardy inequalities we also obtain versions of Heisenberg uncertainty inequality for the fractional sublaplacian. © 2016 Elsevier Inc.