Adams' inequality with exact growth in the hyperbolic space $\mathbb{H}^4$ and Lions lemma

TL;DR

This paper establishes Adams' inequality with exact growth in four-dimensional hyperbolic space, extends Lions lemma to hyperbolic settings, and introduces related inequalities, advancing the understanding of functional inequalities in non-Euclidean geometries.

Contribution

It proves Adams' inequality with precise growth conditions in hyperbolic space and extends Lions lemma to this setting, including new inequalities and functional analysis tools.

Findings

Proved Adams' inequality with exact growth in $ ext{H}^4$.

Extended Lions lemma to hyperbolic space for specific functionals.

Established an Adachi-Tanaka type inequality in hyperbolic geometry.

Abstract

In this article we prove Adams inequality with exact growth condition in the four dimensional hyperbolic space $\mathbb{H}^4,$ \begin{align} \int_{\mathbb{H}^4} \frac{e^{32 \pi^2 u^2} - 1}{(1 + |u|)^2} \ dv_g \leq C ||u||^2_{L^2({\mathbb{H}^4})}. \end{align} for all $u \in C^{\infty}_c(\mathbb{H}^4) $ with $\int_{\mathbb{H}^4} (P_2 u) u \ dv_g \leq 1.$ We will also establish an Adachi-Tanaka type inequality in this settings. Another aspect of this article is the P.L.Lions lemma in the hyperbolic space. We prove P.L.Lions lemma for the Moser functional and for a few cases of the Adams functional on the whole hyperbolic space.