Generalised Hardy type and Rellich type inequalities on the Heisenberg group

This paper is primarily devoted to a class of interpolation inequalities of Hardy and Rellich types on the Heisenberg group ${\mathbb{H}}^n$. Consequently, several weighted Hardy type, Heisenberg-Pauli-Weyl uncertainty principle and Hardy-Rellich type inqualities are established on ${\mathbb{H}}^n$. Moreover, new weighted Sobolev type embeddings are derived. Finally, an integral inequality for vector fields in a domain of the Heisenberg group is obtained, leading to several specific weighted Hardy type inequalities by making careful choices of vector fields.