# Inequalities involving Aharonov-Bohm magnetic potentials in dimensions 2 and 3

## TL;DR

This paper explores nonlinear inequalities related to Schrödinger operators with Aharonov-Bohm magnetic potentials across various geometries, revealing new magnetic Hardy inequalities and emphasizing the role of symmetry and optimality in magnetic fields.

## Contribution

It introduces novel results on inequalities involving magnetic potentials in 2D and 3D, focusing on symmetry, rigidity, and optimality, with new applications to Hardy inequalities.

## Key findings

- New magnetic Hardy inequalities in dimensions 2 and 3
- Results on symmetry and optimality in magnetic field contexts
- Development of methods for nonlinear interpolation inequalities

## Abstract

This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schr{\"o}dinger operators involving Aharonov-Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sphere or a two-dimensional torus, and also the Euclidean spaces of dimensions two and three. Most of the results are new and we put the emphasis on the methods, as very little is known on symmetry, rigidity and optimality in presence of a magnetic field. The most spectacular applications are new magnetic Hardy inequalities in dimensions 2 and 3.