Second order Sobolev type inequalities in the hyperbolic spaces

We establish several Poincar\'e--Sobolev type inequalities for the Lapalce--Beltrami operator $\Delta_g$ in the hyperbolic space $\mathbb H^n$ with $n\geq 5$. 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 $\Delta_g$ 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 $\mathbb H^4$.