Optimal Sobolev inequalities in the hyperbolic space

We find the optimal function norm on the left-hand side of the [Formula: see text]th order Sobolev type inequality [Formula: see text] in the [Formula: see text]-dimensional hyperbolic space [Formula: see text], [Formula: see text]. The optimal function norm in the inequality among all rearrangement-invariant function norms is completely characterized. A variety of concrete examples of optimal function norms is provided. The examples include delicate limiting cases and, especially when [Formula: see text], seem to provide new, improved inequalities in these limiting cases.