Revista Matemática Iberoamericana

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Published online first: 2020-02-14
DOI: 10.4171/rmi/1171

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

Quốc Anh Ngô[1] and Van Hoang Nguyen[2]

(1) Duy Tân University, Dá Nang, Vietnam and Vietnam National University, Hanoi, Vietnam
(2) Université Paul Sabatier, Toulouse, France and Vietnam Academy of Science and Technology, Hanoi, Vietnam

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

Keywords: Hyperbolic space, sharp Moser–Trudinger inequality, sharp Adams inequality, Lions lemma, exact growth condition

Ngô Quốc Anh, Nguyen Van Hoang: Sharp Adams–Moser–Trudinger type inequalities in the hyperbolic space. Rev. Mat. Iberoam. Electronically published on February 14, 2020. doi: 10.4171/rmi/1171 (to appear in print)