Sharp Poincaré-Hardy and Poincaré-Rellich inequalities on the hyperbolic space

We study Hardy-type inequalities associated to the quadratic form of the shifted Laplacian −ΔHN−(N−1)2/4 on the hyperbolic space HN, (N−1)2/4 being, as it is well-known, the bottom of the L2-spectrum of −ΔHN. We find the optimal constant in a resulting Poincaré-Hardy inequality, which includes a further remainder term which makes it sharp also locally: the resulting operator is in fact critical in the sense of [17]. A related improved Hardy inequality on more general manifolds, under suitable curvature assumption and allowing for the curvature to be possibly unbounded below, is also shown. It involves an explicit, curvature dependent and typically unbounded potential, and is again optimal in a suitable sense. Furthermore, with a different approach, we prove Rellich-type inequalities associated with the shifted Laplacian, which are again sharp in suitable senses.