Sharp constant for Poincar\'e-type inequalities in the hyperbolic space

TL;DR

This paper determines the exact sharp constant for Poincaré-type inequalities in hyperbolic space, revealing the precise bounds and their dependence on parameters, and shows these bounds are not attained within the function space.

Contribution

It provides the explicit form of the sharp constant for Poincaré inequalities in hyperbolic space and extends symmetrization techniques to this setting.

Findings

Sharp constant formula for even and odd m

The constant depends explicitly on p, n, and m

The sharp constant is not attained in the space

Abstract

In this note, we establish a Poincar\'e-type inequality on the hyperbolic space $\mathbb H^n$, namely \[ \|u\|_{p} \leqslant C(n,m,p) \|\nabla^m_g u\|_{p} \] for any $u \in W^{m,p}(\mathbb H^n)$. We prove that the sharp constant $C(n,m,p)$ for the above inequality is \[ C(n,m,p) = \begin{cases} \left( p p'/(n-1)^2 \right)^{m/2}&\mbox{if $m$ is even},\\ (p/(n-1))\left( p p'/(n-1)^2\right)^{(m-1)/2} &\mbox{if $m$ is odd}, \end{cases} \] with $p' = p/(p-1)$ and this sharp constant is never achieved in $W^{m,p}(\mathbb H^n)$. Our proofs rely on the symmetrization method extended to hyperbolic spaces.