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

TL;DR

This paper corrects and refines $L^{p}$ Hardy inequalities on the sphere, establishing sharp inequalities for all relevant $p$ and the critical case $p=n$, using Xiao's method to address singularities related to geodesic distance.

Contribution

It provides corrected proofs and sharp $L^{p}$ Hardy inequalities on the sphere, including the critical case, extending previous results with a new methodological approach.

Findings

Corrected proof of $L^{p}$ Hardy inequality on the sphere

Established sharp inequalities for the critical case $p=n$

Extended inequalities to all $2 \\leq p < n$

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.