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arXiv · 2006.06214

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

Ahmed A. Abdelhakim
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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 $2\leq p<n$. 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.

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Ahmed A. Abdelhakim. 2020-06-11. Optimal critical exponent $L^{p}$ inequalities of Hardy type on the sphere via Xiao's method. https://arxiv.org/abs/2006.06214

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