# Lifting in Besov Spaces

## TL;DR

This paper explores the lifting problem in Besov spaces, examining conditions under which a unit modulus function can be expressed as an exponential of a real-valued function within the same Besov space, extending prior Sobolev space results.

## Contribution

It extends the lifting problem analysis from Sobolev to Besov spaces, revealing new properties of Besov spaces, especially for the case when q>p.

## Key findings

- Established conditions for the existence of a lifting function in Besov spaces.
- Proved new properties of Besov spaces, including a non restriction property for q>p.
- Extended classical Sobolev space results to the broader Besov space setting.

## Abstract

Let $\Omega$ be a smooth bounded domain in $\mathbb R^n$ and u be a measurable function on $\Omega$ such that $|u(x)|=1$ almost everywhere in $\Omega$. Assume that u belongs to the $B^s_{p,q}(\Omega)$ Besov space. We investigate whether there exists a real-valued function $\varphi \in B^s_{p,q}$ such that $u=e^{i\varphi}$. This extends the corresponding study in Sobolev spaces due to Bourgain, Brezis and the first author. The analysis of this lifting problem leads us to prove some interesting new properties of Besov spaces, in particular a non restriction property when $q>p$.