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 $d\ge 2$) 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.

arXiv comment15 pages, 1 figure