Hardy inequalities for magnetic $p$-Laplacians

TL;DR

This paper proves improved Hardy inequalities for magnetic $p$-Laplacians, showing that magnetic fields, especially Aharonov-Bohm fields, increase the sharp constant, and discusses related open problems.

Contribution

It introduces new Hardy inequalities for magnetic $p$-Laplacians and demonstrates the strict increase of the sharp constant with Aharonov-Bohm magnetic fields.

Findings

Improved Hardy inequalities for magnetic $p$-Laplacians.

Sharp constant increases with Aharonov-Bohm magnetic fields.

Open problems related to magnetic Hardy inequalities.

Abstract

We establish improved Hardy inequalities for the magnetic $p$-Laplacian due to adding nontrivial magnetic fields. We also prove that for Aharonov-Bohm magnetic fields the sharp constant in the Hardy inequality becomes strictly larger than in the case of a magnetic-free $p$-Laplacian. We also post some remarks with open problems.