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

Abstract

Abstract We obtain sharp inequalities of Hardy type for functions in the Sobolev space W 1 , p $W^{1,p}$ on the unit sphere S n − 1 $\mathbb{S}^{n-1}$ in R n $\mathbb{R}^{n}$ . 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. We apply our results to obtain lower bounds for the the first eigenvalue of the p-Laplacian on the sphere.

Keywords

• Sobolev Space on manifolds
• L p $L^{p}$ Hardy inequalities
• Density agrument
• Unit sphere