Limiting case Hardy inequalities on the sphere

TL;DR

This paper establishes sharp Hardy inequalities on the sphere $\

Contribution

It provides the first sharp limiting case Hardy inequalities on the sphere and shows the optimal constants are unattainable by nonzero functions in $H^1(\

Findings

Optimal constants are unattainable by any nonzero $H^1$ function.

The inequalities are sharp and relate to geodesic distance singularities.

The results extend Hardy inequalities to spherical geometry.

Abstract

We give sharp limiting case Hardy inequalities on the sphere $\mathbb{S}^{2}$ and show that their optimal constants are unattainable by any $f\in H^{1}\left(\mathbb{S}^{2}\right)\setminus\{0\}$. The singularity of the problem is related to the geodesic distance from a point on the sphere.