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

Quốc Anh Ngô, Van Hoang Nguyen

• 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.