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

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