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

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}$^N)$isthebottomofthe$L^2$spectrumof$-\Delta_{$\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 $P_{\lambda }\mathrm{.}$