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

TL;DR

This paper establishes sharp Poincaré-Hardy and Rellich inequalities on hyperbolic space, identifying optimal constants and remainder terms, with extensions to more general manifolds under curvature conditions.

Contribution

It derives the first sharp Poincaré-Hardy and Rellich inequalities on hyperbolic space, including optimal constants and local sharpness, and extends results to manifolds with curvature assumptions.

Findings

Optimal constants in Poincaré-Hardy inequalities are identified.

Sharp Rellich-type inequalities are established.

Extensions to manifolds with curvature conditions are demonstrated.

Abstract

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.