Sharp constant for Poincaré-type inequalities in the hyperbolic space

Abstract

In this note, we establish a Poincaré-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 $p^{\prime }=p/(p-1)$ 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.

14 pages