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Sharp Constant for Poincaré-Type Inequalities in the Hyperbolic Space

  • Quốc Anh Ngô
  • Van Hoang Nguyen
Article

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.

Keywords

Poincaré inequality Sharp constant Symmetrization method Hyperbolic space 

Mathematics Subject Classification (2010)

26D10 46E35 31C12 

Notes

Funding Information

The second author receives support from the CIMI postdoctoral research fellowship. The research of the first author is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2016.02. He also receives support from the VNU University of Science under project number TN.16.01.

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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of Mathematics, College of ScienceVietnam National UniversityHanoiVietnam
  2. 2.Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouse cédex 09France

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