Optimal Sobolev Inequalities in the Hyperbolic Space

We find the optimal function norm on the left-hand side of the mth order Sobolev-type inequality parallel to u parallel to Y ((Hn)) <= C parallel to del(m)(g)u parallel to X-(Hn) in the n-dimensional hyperbolic space H-n, 1 <= 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 >= 3, seem to provide new, improved inequalities in these limiting cases.