Sharp Constant for Poincaré-Type Inequalities in the Hyperbolic Space

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.

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Funding

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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Correspondence to Quốc Anh Ngô.

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A Note Added

After announcing our work on arXiv, see [13], it has come to our attention that the sharpness of C(n, 1,p) can be realized by a different argument by considering the upper half space model for \(\mathbb {H}^{n}\), see [5].

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Ngô, Q.A., Nguyen, V.H. Sharp Constant for Poincaré-Type Inequalities in the Hyperbolic Space. Acta Math Vietnam 44, 781–795 (2019). https://doi.org/10.1007/s40306-018-0269-9

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Keywords

  • Poincaré inequality
  • Sharp constant
  • Symmetrization method
  • Hyperbolic space

Mathematics Subject Classification (2010)

  • 26D10
  • 46E35
  • 31C12