On the improvement of the Hardy inequality due to singular magnetic fields

TL;DR

This paper enhances the classical Hardy inequality by incorporating effects of singular magnetic fields, specifically the Aharonov-Bohm field and a diverging magnetic field in 3D, revealing new sharp inequalities.

Contribution

It provides new sharp Hardy inequalities that account for singular magnetic fields, extending classical results to magnetic contexts in various dimensions.

Findings

Established a sharp Hardy inequality for the Aharonov-Bohm field in all dimensions.

Derived a non-trivial magnetic Hardy inequality in 3D with a magnetic field diverging along a plane.

Quantified the influence of magnetic flux and field divergence on Hardy inequalities.

Abstract

We establish magnetic improvements upon the classical Hardy inequality for two specific choices of singular magnetic fields. First, we consider the Aharonov-Bohm field in all dimensions and establish a sharp Hardy-type inequality that takes into account both the dimensional as well as the magnetic flux contributions. Second, in the three-dimensional Euclidean space, we derive a non-trivial magnetic Hardy inequality for a magnetic field that vanishes at infinity and diverges along a plane.