JournalsrmiVol. 36, No. 5pp. 1409–1467

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

  • Quốc Anh Ngô

    Duy Tân University, Dá Nang, Vietnam and Vietnam National University, Hanoi, Vietnam
  • Van Hoang Nguyen

    Université Paul Sabatier, Toulouse, France and Vietnam Academy of Science and Technology, Hanoi, Vietnam
Sharp Adams–Moser–Trudinger type inequalities in the hyperbolic space cover

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Abstract

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\mathbb H^n. First, we prove a sharp Adams’ inequality of order two with the exact growth condition in Hn\mathbb H^n. 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\mathbb H^n, 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\mathbb H^n. Our proofs rely on the symmetrization method extended to hyperbolic spaces.

Cite this article

Quốc Anh Ngô, Van Hoang Nguyen, Sharp Adams–Moser–Trudinger type inequalities in the hyperbolic space. Rev. Mat. Iberoam. 36 (2020), no. 5, pp. 1409–1467

DOI 10.4171/RMI/1171