Sharp subcritical and critical $L^p$ Hardy inequalities on the sphere

Abstract

We prove sharp inequalities of Hardy type for functions in the Sobolev space $W^{1,p}$ on the unit sphere ${\mathbb{S}}^{n-1}$ in ${\mathbb{R}}^n$. We achieve this in both the subcritical and critical cases. The method we use to show optimality takes into account all the constants involved in our inequalities.