Bourgain-Brezis spaces obtained by real interpolation
Eduard Curc\u{a}

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 (on , with ) we have the equality of images \begin{equation} \operatorname{div} (L^{\infty}\cap X)=\operatorname{div} X, \tag{} \end{equation} i.e., given a vector field there exists a vector field such that \operatorname{div} u=\operatorname{div] v . In this paper we show that if is a function space satisfying () then, any real interpolation space (where and ) also satisfies (). The proof is based on a general method that allows us to interpolate solutions of linear equations.
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