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

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.