Second order Sobolev type inequalities in the hyperbolic spaces

TL;DR

This paper develops improved second order Sobolev inequalities in hyperbolic spaces, combining sharp Poincaré, Rellich, and Sobolev inequalities, with applications to GJMS operators and Adams inequalities.

Contribution

It introduces novel combined inequalities in hyperbolic spaces that unify and strengthen existing second order inequalities, including new results in dimension four.

Findings

Established improved second order inequalities in hyperbolic spaces.

Derived Poincaré--Sobolev inequalities for the GJMS operator.

Provided enhancements of Adams inequalities in dimension four.

Abstract

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 $\mathbb H^4$.