Adams inequality with exact growth in the hyperbolic space H4 and Lions lemma

In this paper, we prove Adams inequality with exact growth condition in the four-dimensional hyperbolic space H-4, integral(H4) e(32 pi 2u2 )- 1/(1 + vertical bar u vertical bar)(2) dv (g) <= C parallel to u parallel to(L2)((2)(H4)), for all u is an element of C-c(infinity)(H-4) with integral(H4) (P(2)u)udv(g) <= 1. We will also establish an Adachi-Tanalcatype inequality in this setting. Another aspect of this paper is the Lions lemma in the hyperbolic space. We prove Lions lemma for the Moser functional and for a few cases of the Adams functional on the whole hyperbolic space.