Abstract
AbstractWe obtain sharp inequalities of Hardy type for functions in the Sobolev space $W^{1,p}$
W
1
,
p
on the unit sphere $\mathbb{S}^{n-1}$
S
n
−
1
in $\mathbb{R}^{n}$
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.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics,Analysis
Reference17 articles.
1. Abdelhakim, A.A.: Limiting case Hardy inequalities on the sphere. Math. Inequal. Appl. 21(4), 1079–1090 (2018)
2. Abolarinwa, A., Rauf, K., Yin, S.: Sharp $L^{p}$ Hardy type and uncertainty principle inequalities on the sphere. J. Math. Inequal. 13(4), 1011–1022 (2019)
3. Adriano, L., Xia, C.: Hardy type inequalities on complete Riemannian manifolds. Monatshefte Math. 163, 115–129 (2011)
4. D’Ambrosioa, L., Dipierrob, S.: Hardy inequalities on Riemannian manifolds and applications. Ann. Inst. Henri Poincaré C, Anal. Non Linéaire 31(3), 449–475 (2014)
5. Eichhorn, J.: Global Analysis on Open Manifolds. Nova Publ. (Nova Science Publishers), New York (2007)