Some sharp Hardy inequalities on spherically symmetric domains

TL;DR

This paper establishes sharp Hardy inequalities on spherically symmetric domains, including the sphere and half-space, using symmetrization and weighted one-dimensional inequalities, advancing understanding of singularity behavior in these geometries.

Contribution

It introduces new sharp Hardy inequalities for spherical and half-space domains with singularities, utilizing symmetrization and weighted inequalities, which were not previously established.

Findings

Proved a sharp Hardy inequality on the unit sphere with a point singularity.

Derived a Hardy inequality for functions on the half-space vanishing on a hyperplane.

Developed a general form of a weighted one-dimensional Hardy inequality.

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 $\{x_{n+1}=0\}$, with singularity along the $x_{n+1}$-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.