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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 $sp=d$. 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