Sharp subcritical and critical $L^{p}$ Hardy inequalities on the sphere

TL;DR

This paper establishes sharp Hardy inequalities for Sobolev functions on the sphere in both subcritical and critical cases, with a focus on optimal constants and comprehensive proof techniques.

Contribution

It introduces new sharp Hardy inequalities on the sphere for Sobolev spaces, including the critical case, with detailed optimality analysis.

Findings

Proved sharp Hardy inequalities on the sphere

Established inequalities in both subcritical and critical cases

Provided optimal constants and proof methods

Abstract

We prove sharp inequalities of Hardy type for functions in the Sobolev space $W^{1,p}$ on the unit sphere $\mathbb{S}^{n-1}$ in $\mathbb{R}^{n}$. We achieve this in both the subcritical and critical cases. The method we use to show optimality takes into account all the constants involved in our inequalities.