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Sharp Constant for Poincaré-Type Inequalities in the Hyperbolic Space

$icon-email$ Quốc Anh Ngô , Van Hoang Nguyen

Abstract

In this note, we establish a Poincaré-type inequality on the hyperbolic space $H^{n}$ , namely $‖ u ‖_{p} \leq C (n, m, p) ‖ \nabla_{g}^{m} u ‖_{p}$ for any $u \in W^{m, p} (H^{n})$ . We prove that the sharp constant $C (n, m, p)$ for the above inequality is $C (n, m, p) = {\begin{cases} {(p p^{'} / (n - 1)^{2})}^{m / 2} & if m is even, \\ (p / (n - 1)) {(p p^{'} / (n - 1)^{2})}^{(m - 1) / 2} & if m is odd, \end{cases}$ 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.

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Sharp Constant for Poincaré-Type Inequalities in the Hyperbolic Space

Abstract