Sharp Poincar\'e-Hardy and Poincar\'e-Rellich inequalities on the hyperbolic space

By Elvise Berchio, D. Ganguly, G. Grillo. 2015

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We study Hardy-type inequalities associated to the quadratic form of the shifted Laplacian $-\Delta_{\mathbb H^N}-(N-1)^2/4$ on the hyperbolic space ${\mathbb H}^N$, $(N-1)^2/4$ being, as it is well-known, the bottom of the $L^2$-spectrum of $-\Delta_{\mathbb H^N}$. We find the optimal constant in the resulting Poincar\'e-Hardy inequality, which includes a further remainder term which makes it sharp also locally. A related inequality under suitable curvature assumption on more general manifolds is also shown. Similarly, we prove Rellich-type inequalities associated with the shifted Laplacian, in which at least one of the constant involved is again sharp.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/17c26a1f4f943b30726750ee0d46f15d4ff2f37c

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In Applied Mathematics

1. Guozhen Lu
2. Qiaohua Yang

Citations in the corpus

• Paneitz operators on hyperbolic spaces and high order Hardy-Sobolev-Maz'ya inequalities on half spaces (2019, methodology): “We refer to [11, 12] for Poincaré-Hardy inequalities on Hn (n ≥ 3).”