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.