Second order Sobolev type inequalities in the hyperbolic spaces

Abstract

We establish several Poincaré--Sobolev type inequalities for the Lapalce--Beltrami operator ${\Delta }_g$ in the hyperbolic space ${\mathbb{H}}^n$ with $n\ge 5$. These inequalities could be seen as the improved second order Poincaré 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 Poincaré inequality and the sharp Rellich inequality or the sharp Sobolev inequality for ${\Delta }_g$ in ${\mathbb{H}}^n$. As a consequence, we obtain the Poincaré--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 ${\mathbb{H}}^4$.

arXiv comment26 pages, comments are welcome