Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group

TL;DR

This paper introduces magnetic fields in the Heisenberg group, analyzing their effects on spectral properties and Hardy inequalities, including new inequalities and spectral uplift results for magnetic sub-Laplacians.

Contribution

It defines magnetic fields in the Heisenberg group and establishes spectral and Hardy inequality improvements, including a Hardy inequality for the Folland--Stein operator.

Findings

Uniform magnetic fields raise the spectrum's bottom.

Magnetic fields vanishing at infinity improve Hardy inequalities.

Established a Hardy inequality for the Folland--Stein operator.

Abstract

In this paper we introduce a notion of magnetic field in the Heisenberg group and we study its influence on spectral properties of the corresponding magnetic (sub-elliptic) Laplacian. We show that uniform magnetic fields uplift the bottom of the spectrum. For magnetic fields vanishing at infinity, including Aharonov--Bohm potentials, we derive magnetic improvements to a variety of Hardy-type inequalities for the Heisenberg sub-Laplacian. In particular, we establish a sub-Riemannian analogue of Laptev and Weidl sub-criticality result for magnetic Laplacians in the plane. Instrumental for our argument is the validity of a Hardy-type inequality for the Folland--Stein operator, that we prove in this paper and has an interest on its own.