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

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.