Second order Sobolev type inequalities in the hyperbolic spaces

Abstract

In this paper, we establish several Poincaré–Sobolev type inequalities concerning to the Laplace–Beltrami operator Δg in the hyperbolic space Hn with n≥5. These inequalities could be seen as the improved second order Poincaré inequality with remainder terms involving with the sharp Rellich inequality or the sharp Sobolev inequality in Hn. The novelty of these inequalities is that it combines both the sharp Poincaré inequality and the sharp Rellich inequality or the sharp Sobolev inequality for Δg in Hn. As a consequence, we obtain the Poincaré–Sobolev inequality for the second order GJMS operator P2 in Hn. In dimension 4, we obtain an improvement of the sharp Adams inequality and an Adams inequality with exact growth for radial functions.