On the representation as exterior differentials of closed forms with L1-coefficients

Let $N\ge 2$. If $g\in L_c^1({\mathbf{R}}^N)$ has zero integral, then the equation $\mathrm{div}X=g$ need not have a solution $X\in W_{\mathrm{loc}}^{1,1}({\mathbf{R}}^N;{\mathbf{R}}^N)$ 6] or even $X\in L_{\mathrm{loc}}^{N/(N?1)}$ $({\mathbf{R}}^N;{\mathbf{R}}^N)$ 2]. Using these results, we prove that, whenever $N\ge 3$ and $2\le ?\le N?1$, there exists some ?-form $f\in L_c^1({\mathbf{R}}^N;{\mathrm{\Lambda }}^{?})$ such that $\mathrm{d}f=0$ and the equation $\mathrm{d}\lambda =f$ has no solution $\lambda \in W_{\mathrm{loc}}^{1,1}({\mathbf{R}}^N;{\mathrm{\Lambda }}^{??1})$. This provides a negative answer to a question raised by Baldi, Franchi, and Pansu 1].