An optimal improvement for the Hardy inequality on the hyperbolic space and related manifolds

TL;DR

This paper establishes optimal Hardy inequalities on hyperbolic space and related manifolds, providing critical operators, new inequalities, and applications such as improved Rellich inequalities and uncertainty principles.

Contribution

It introduces the first optimal Hardy inequalities for the operator P_λ on hyperbolic space, extending to general Cartan-Hadamard manifolds with curvature conditions.

Findings

Proved optimal Hardy inequalities on hyperbolic space.

Derived a new critical Hardy-type inequality on hyperbolic space.

Applied results to improve Rellich inequalities and uncertainty principles.

Abstract

We prove \emph{optimal} improvements of the Hardy inequality on the hyperbolic space. Here, optimal means that the resulting operator is critical in the sense of [J.Funct.Anal. 266 (2014), pp. 4422-89], namely the associated inequality cannot be further improved. Such inequalities arise from more general, \emph{optimal} ones valid for the operator $ P_{\lambda}:= -\Delta_{\mathbb{H}^N} - \lambda$ where $0 \leq \lambda \leq \lambda_{1}(\mathbb{H}^N)$ and $\lambda_{1}(\mathbb{H}^N)$ is the bottom of the $L^2$ spectrum of $-\Delta_{\mathbb{H}^N} $, a problem that had been studied in [J.Funct.Anal. 272 (2017), pp. 1661-1703 ] only for the operator $P_{\lambda_{1}(\mathbb{H}^N)}$. A different, critical and new inequality on $\mathbb{H}^N$, locally of Hardy type, is also shown. Such results have in fact greater generality since there are shown on general Cartan-Hadamard manifolds under curvature assumptions, possibly depending on the point. Existence/nonexistence of extremals for the related Hardy-Poincar\'e inequalities are also proved using concentration-compactness technique and a Liouville comparison theorem. As applications of our inequalities we obtain an improved Rellich inequality and we derive a quantitative version of Heisenberg-Pauli-Weyl uncertainty principle for the operator $P_\lambda.$