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

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.

Keywords:

• Hardy inequality
• singular magnetic field
• Aharonov-Bohm potential

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

The research of D.K. was partially supported by the GACR grant No. 18-08835S. The research of L.V. was partially supported by ERCEA Advanced Grant 669689-HADE and the MINECO projects PGC 2018-095422-B-100 and SEV-2017-0718.