The sharp Poincar\'e--Sobolev type inequalities in the hyperbolic spaces $\mathbb H^n$

TL;DR

This paper establishes sharp $L^p$ Poincaré--Sobolev inequalities in hyperbolic spaces, linking classical inequalities with optimal constants using rearrangement techniques and Euclidean comparisons.

Contribution

It introduces a new $L^p$ framework for Poincaré--Sobolev inequalities in hyperbolic spaces, achieving sharp constants and extending classical inequalities.

Findings

Derived sharp $L^p$ Poincaré--Sobolev inequalities in $ ext{H}^n$

Extended inequalities to Poincaré--Gagliardo--Nirenberg and Poincaré--Morrey--Sobolev forms

Generalized Sobolev inequalities in hyperbolic spaces beyond previous results

Abstract

In this note, we establish a $L^p-$version of the Poincar\'e--Sobolev inequalities in the hyperbolic spaces $\mathbb H^n$. The interest of this result is that it relates both the Poincar\'e (or Hardy) inequality and the Sobolev inequality with the sharp constant in $\mathbb H^n$. Our approach is based on the comparison of the $L^p-$norm of gradient of the symmetric decreasing rearrangement of a function in both the hyperbolic space and the Euclidean space, and the sharp Sobolev inequalities in Euclidean spaces. This approach also gives the proof of the Poincar\'e--Gagliardo--Nirenberg and Poincar\'e--Morrey--Sobolev inequalities in the hyperbolic spaces $\mathbb H^n$. Finally, we discuss several other Sobolev inequalities in the hyperbolic spaces $\mathbb H^n$ which generalize the inequalities due to Mugelli and Talenti in $\mathbb H^2$.