Bourgain-Brezis spaces obtained by real interpolation

Abstract

In 2002, Bourgain and Brezis proved that for the space $X=W^{1,d}$ (on $\mathbb{T}^{d}$, with $d\geq2$) we have the equality of images \begin{equation} \operatorname{div} (L^{\infty}\cap X)=\operatorname{div} X, \tag{$\ast$} \end{equation} i.e., given a vector field $v\in X$ there exists a vector field $u\in L^{\infty }\cap X$ such that $\operatorname{div} u=\operatorname{div] v $. In this paper we show that if $X$ is a function space satisfying ($\ast$) then, any real interpolation space $X_{\theta,q}=(L^{\infty},X)_{\theta,q}$ (where $\theta\in (0,1)$ and $q\in [1,\infty)$) also satisfies ($\ast$). The proof is based on a general method that allows us to interpolate solutions of linear equations.