Optimal critical exponent $L^p$ inequalities of Hardy type on the sphere via Xiao's method

Abstract

First, we correct the proof presented in [Abimbola Abolarinwa, Kamilu Rauf, Songting Yin, Sharp $L^p$ Hardy type and uncertainty principle inequalities on the sphere, Journal of Mathematical Inequalities, 13, 4 (2019), 1011 - 1022] and obtain a correct sharp version of an $L^p$ Hardy inequality on the sphere ${\mathbb{S}}^n$ for all $2\le p<n$. Secondly, we prove sharp critical exponent $L^n$ inequalities on the sphere ${\mathbb{S}}^n$ in ${\mathbb{R}}^{n+1}$, $n\ge 2$. The singularity in this problem is the geodesic distance from an arbitrary point on the sphere.