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 $2\leq p<n$. Secondly, we prove sharp critical exponent $L^{ n}$ inequalities on the sphere $\mathbb{ S}^{ n}$ in $\mathbb{ R}^{ n+1}$, $n\geq 2$. The singularity in this problem is the geodesic distance from an arbitrary point on the sphere.
Categories: math.FA math.AP