  Let $d\geq 2$ be an integer, $1\leq l\leq d-1$ and $\varphi$ be a
differential $l$-form on ${\mathbb R}^d$ with $\dot{W}^{1,d}$ coefficients. It
was proved by Bourgain and Brezis (\cite[Theorem 5]{MR2293957}) that there
exists a differential $l$-form $\psi$ on ${\mathbb R}^d$ with coefficients in
$L^{\infty}\cap \dot{W}^{1,d}$ such that $d\varphi=d\psi$. Bourgain and Brezis
also asked whether this result can be extended to differential forms with
coefficients in the fractional Sobolev space $\dot{W}^{s,p}$ with $sp=d$. We
give a positive answer to this question, in the more general context of
Triebel-Lizorkin spaces, provided that $d-\kappa\leq l\leq d-1$, where $\kappa$
is the largest positive integer such that $\kappa<\min(p,d)$. The proof relies
on an approximation result for functions in $\dot{W}^{s,p}$ by functions in
$\dot{W}^{s,p}\cap L^{\infty}$, even though $\dot{W}^{s,p}$ does not embed into
$L^{\infty}$ in this critical case.
