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

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 Poincaré-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.

arXiv commentFinal version. To appear in JFA