Generalised Hardy type and Rellich type inequalities on the Heisenberg group

TL;DR

This paper develops a broad class of interpolation inequalities of Hardy and Rellich types on the Heisenberg group, establishing new inequalities, embeddings, and integral inequalities relevant to analysis on this non-commutative structure.

Contribution

It introduces generalized Hardy and Rellich inequalities on the Heisenberg group, including weighted versions, uncertainty principles, and Sobolev embeddings, advancing the mathematical understanding of analysis in this setting.

Findings

Established new weighted Hardy type inequalities on the Heisenberg group.

Derived Heisenberg-Pauli-Weyl uncertainty principles and Rellich inequalities.

Proved novel weighted Sobolev embeddings and integral inequalities for vector fields.

Abstract

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.