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arXiv · 1611.08413

Improved L$^p$-Poincar\'e inequalities on the hyperbolic space

Elvise Berchio·Lorenzo D'Ambrosio·Debdip Ganguly·Gabriele Grillo
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Abstract

We investigate the possibility of improving the $p$-Poincar\'e inequality $\|\nabla_{\mathbb{H}^N} u\|_p \ge \Lambda_p \|u\|_p$ on the hyperbolic space, where $p>2$ and $\Lambda_p:=[(N-1)/p]^{p}$ is the best constant for which such inequality holds. We prove several different, and independent, improved inequalities, one of which is a Poincar\'e-Hardy inequality, namely an improvement of the best $p$-Poincar\'e inequality in terms of the Hardy weight $r^{-p}$, $r$ being geodesic distance from a given pole. Certain Hardy-Maz'ya-type inequalities in the Euclidean half-space are also obtained.

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Elvise Berchio, Lorenzo D'Ambrosio, Debdip Ganguly, Gabriele Grillo. 2016-11-25. Improved L$^p$-Poincar\'e inequalities on the hyperbolic space. https://doi.org/10.1016/j.na.2017.03.016

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