Sharp Adams–Moser–Trudinger type inequalities in the hyperbolic space

• Quốc Anh Ngô ^[1] ; Van Hoang Nguyen ^[2]
  1. [1] National University

     National University

     Estados Unidos

     
  2. [2] Paul Sabatier University

     Paul Sabatier University

     Arrondissement de Toulouse, Francia

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 36, Nº 5, 2020, págs. 1409-1467
• Idioma: inglés
• DOI: 10.4171/rmi/1171
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• Resumen

  • The purpose of this paper is to establish some Adams–Moser–Trudinger inequalities, which are the borderline cases of the Sobolev embedding, in the hyperbolic space Hn. First, we prove a sharp Adams’ inequality of order two with the exact growth condition in Hn. Then we use it to derive a sharp Adams-type inequality and an Adachi–Tanakat-ype inequality. We also prove a sharp Adams-type inequality with Navier boundary condition on any bounded domain of Hn, which generalizes the result of Tarsi to the setting of hyperbolic spaces. Finally, we establish a Lions-type lemma and an improved Adams-type inequality in the spirit of Lions in Hn. Our proofs rely on the symmetrization method extended to hyperbolic spaces.