Sharp Constant for Poincaré-Type Inequalities in the Hyperbolic Space

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