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 Rn+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.
Some sharp Hardy inequalities on spherically symmetric domains
CHIACCHIO, FRANCESCO;RICCIARDI, TONIA
2009
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 Rn+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.File in questo prodotto:
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