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

We prove optimal improvements of the Hardy inequality on the hyperbolic space. Here, optimal means that the resulting operator is critical in the sense of Devyver, Fraas, and Pinchover (2014), namely the associated inequality cannot be further improved. Such inequalities arise from more general, optimal ones valid for the operator $P_{lambda}=-Delta-lambda$ where $0 leq lambda leq lambda_1$ and $lambda_1$ is the bottom of the L^2 spectrum of the laplacian, a problem that had been studied in Berchio, Ganguly, and Grillo (2017) only for $lambda=lambda_1$. A different, critical and new inequality on the hyperbolic space, locally of Hardy type is also shown. Such results have in fact greater generality since they are proved on general Cartan-Hadamard manifolds under curvature assumptions, possibly depending on the point. Existence/nonexistence of extremals for the related Hardy-Poincaré 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 $P_{lambda}$.