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

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 $\mathbb H^n$. 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.