Improved higher order poincaré inequalities  on the hyperbolic space  via Hardy-type remainder terms

Elvise Berchio; Debdip  Ganguly

doi:10.3934/cpaa.2016020

Article Contents

2016, Volume 15, Issue 5: 1871-1892. Doi: 10.3934/cpaa.2016020

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Improved higher order poincaré inequalities on the hyperbolic space via Hardy-type remainder terms

Elvise Berchio^1, and
Debdip Ganguly^2,

1.
Dipartimento SEMEQ, Università del Piemonte Orientale, via E. Perrone 18, Novara, 28100
2.
Department of Mathematics, Technion-Israel Institute of Technology, Haifa, 32000

Received: November 2015

Revised: April 2016

Published: September 2016

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Abstract

The paper deals about Hardy-type inequalities associated with the following higher order Poincaré inequality: $$ \left( \frac{N-1}{2} \right)^{2(k -l)} :=\displaystyle \inf_{ u \in C^{\infty}_{0}(\mathbb{H}^{N} ) \setminus \{0\}} \qquad \frac{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{k} u|^2 \ dv_{\mathbb{H}^{N}}}{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{l} u|^2 \ dv_{\mathbb{H}^{N}} }, $$ where $0 \leq l < k$ are integers and $\mathbb{H}^{N}$ denotes the hyperbolic space. More precisely, we improve the Poincaré inequality associated with the above ratio by showing the existence of $k$ Hardy-type remainder terms. Furthermore, when $k = 2$ and $l = 1$ the existence of further remainder terms are provided and the sharpness of some constants is also discussed. As an application, we derive improved Rellich type inequalities on upper half space of the Euclidean space with non-standard remainder terms.

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Mathematics Subject Classification: Primary: 26D10, 46E35; Secondary: 31C12.

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