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

TL;DR

This paper establishes sharp Adams-Moser-Trudinger inequalities in hyperbolic space, extending classical results with new sharp inequalities, boundary conditions, and symmetrization techniques specific to hyperbolic geometry.

Contribution

It introduces new sharp Adams inequalities and their variants in hyperbolic space, including boundary conditions and Lions-type lemmas, expanding the theoretical framework of functional inequalities.

Findings

Proved a sharp Adams inequality of order two in hyperbolic space.

Derived new Adams-type and Adachi-Tanaka-type inequalities.

Extended symmetrization methods to hyperbolic spaces.

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-Tanaka-type 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.