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Sharp Constant for Poincaré-Type Inequalities in the Hyperbolic Space
Quốc Anh Ngô
,
Van Hoang Nguyen
Abstract
In this note, we establish a Poincaré-type inequality on the hyperbolic space $\mathbb H^n$, namely $$\|u\|_{p} \leq C(n,m,p) \|{\nabla^{m}_{g}} u\|_{p}$$ 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 $p^\prime = p/(p − 1)$ 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.