Sharp Constant for Poincaré-Type Inequalities in the Hyperbolic Space

Abstract

In this note, we establish a Poincaré-type inequality on the hyperbolic space ℍn, namely∥u∥p≤C(n,m,p)∥∇gmu∥p for any u∈ Wm,p(ℍn). We prove that the sharp constant C(n,m,p) for the above inequality is C(n,m,p)={(pp′/(n−1)2)m/2ifmis even,(p/(n−1))(pp′/(n−1)2)(m−1)/2ifmis odd, with p′ = p/(p − 1) and this sharp constant is never achieved in Wm,p(ℍn). Our proofs rely on the symmetrization method extended to hyperbolic spaces.