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 $u\in C_c^{\mathrm{\infty }}({\mathbb{H}}^4)$ with ${\int }_{{\mathbb{H}}^4}(P_{2}u)ud{v}_g\le 1.$ 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.