Sharp higher-order Sobolev inequalities in the hyperbolic space $${\mathbb{H}^n}$$

In this paper, we obtain the sharp k-th order Sobolev inequalities in the hyperbolic space $${\mathbb{H}^n}$$ for all k = 1, 2, 3, . . . . This gives an answer to an open question raised by Aubin in [Aubin, Princeton University Press, Princeton (1982), pp. 176–177] for $${W^{k,2}(\mathbb{H}^n)}$$ with k > 1. In addition, we prove that the associated Sobolev constants are optimal.