Executive Summary : | This research proposal consists of two main problems: 1. pth-Poincare-Hardy inequality, when $p \neq 2.$ 2. Rellich (or higher order) inequalities on Cartan Hadamard manifolds and the Heisenberg group. The first problem when $p =2$ is very well studied in recent years but for $p \neq 2$ very less is known. We would like to pursue this problem in this project and our main focus is to find the sharp constant of the resulting inequality. In the second problem, we would like to study Poincare-Rellich inequalities on Cartan Hadamard manifolds and the Heisenberg group. For symmetric manifolds, we exploit the spherical nature of the metric but in general manifolds, we do not have such a structure on the metric. It would be a very interesting problem to look into it. |