In this paper, we study the Hardy-Rellich type inequalities and uncertainty principle on the geodesic sphere. Firstly, we derive $L^p$-Hardy inequalities via divergence theorem, which are in turn used to establish the $L^p$-Rellich inequalities. We also establish Heisenberg uncertainty principle on the sphere via the Hardy-Rellich type inequalities. The best constants appearing in the inequalities are shown to be sharp.
"$L^p$-Hardy-Rellich and uncertainty principle inequalities on the sphere." Adv. Oper. Theory 3 (4) 745 - 762, Autumn 2018. https://doi.org/10.15352/aot.1712-1282