Second order Sobolev type inequalities in the hyperbolic spaces

We establish several Poincar'e–Sobolev type inequalities for the Lapalce–Beltrami operator ${\mathrm{\Delta }}_g$ in the hyperbolic space ${\mathbb{H}}^n$ with $n\ge 5$. These inequalities could be seen as the improved second order Poincar'e inequality with remainder terms involving with the sharp Rellich inequality or sharp Sobolev inequality in ${\mathbb{H}}^n$. The novelty of these inequalities is that it combines both the sharp Poincar'e inequality and the sharp Rellich inequality or the sharp Sobolev inequality for ${\mathrm{\Delta }}_g$ in ${\mathbb{H}}^n$. As a consequence, we obtain the Poincar'e–Sobolev inequality for the second order GJMS operator $P_2$ in ${\mathbb{H}}^n$. In dimension $4$, we obtain an improvement of the sharp Adams inequality and an Adams inequality with exact growth for radial functions in ${\mathbb{H}}^4$.