Lifting in Besov Spaces

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 $q>p$.