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

$$C(n,m,p) = \left\{\begin{array}{ll} \left( p p^{\prime}/(n-1)^{2} \right)^{m/2}&\text{if}~m~\text{is even},\\ (p/(n-1))\left( p p^{\prime}/(n-1)^{2}\right)^{(m-1)/2} &\text{if}~m~\text{is odd}, \end{array}\right. $$

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.