Bourgain-Brezis spaces obtained by real interpolation

TL;DR

This paper extends a known divergence equality property from Bourgain-Brezis spaces to their real interpolation spaces, using a general interpolation method for linear equations.

Contribution

It demonstrates that the divergence equality property is preserved under real interpolation of Bourgain-Brezis spaces, broadening the class of spaces where this property holds.

Findings

The divergence equality ($ ext{div}(L^{ ext{infty}}igcap X)= ext{div} X$) holds for interpolation spaces.

The proof introduces a general method for interpolating solutions of linear equations.

The result applies to a wide class of function spaces satisfying the divergence property.

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.