# New characterizations of magnetic Sobolev spaces

## TL;DR

This paper introduces two novel characterizations of magnetic Sobolev spaces using nonlocal functionals, extending classical results and analyzing convergence properties.

## Contribution

It provides new nonlocal characterizations of magnetic Sobolev spaces for Lipschitz magnetic fields, linking to BBM formulas and classical Sobolev space work.

## Key findings

- New characterizations of magnetic Sobolev spaces established
- Convergence properties in almost everywhere and L^1 sense analyzed
- Connections made to BBM formula and classical Sobolev spaces

## Abstract

We establish two new characterizations of magnetic Sobolev spaces for Lipschitz magnetic fields in terms of nonlocal functionals. The first one is related to the BBM formula, due to Bourgain, Brezis, and Mironescu. The second one is related to the work of the first author on the classical Sobolev spaces. We also study the convergence almost everywhere and the convergence in $L^1$ appearing naturally in these contexts.