Optimal Sobolev inequalities in the hyperbolic space

We find the optimal function norm on the left-hand side of the $m$th order Sobolev type inequality $\mathrm{\parallel }u{\mathrm{\parallel }}_{Y({\mathbb{H}}^n)}\le C\mathrm{\parallel }{\mathrm{\nabla }}_g^{m}u{\mathrm{\parallel }}_{X({\mathbb{H}}^n)}$ in the $n$-dimensional hyperbolic space ${\mathbb{H}}^n$, $1\le m<n$. 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 $m\ge 3$, seem to provide new, improved inequalities in these limiting cases.