Limiting case Hardy inequalities on the sphere

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.