Some sharp Hardy inequalities on spherically symmetric domains

Abstract

We prove some sharp Hardy inequalities for domains with a spherical symmetry. In particular, we prove an inequality for domains of the unit n-dimensional sphere with a point singularity, and an inequality for functions defined on the half-space R n+1 + vanishing on the hyperplane {xn+1 = 0}, with singularity along the xn+1-axis. The proofs rely on a one-dimensional Hardy inequality involving a weight function related to the volume element on the sphere, as well as on symmetrization arguments. The one-dimensional inequality is derived in a general form.