The Poincare-lemma is a central statement in differential geometry. It shows that a k-form is closed iff it is exact. A special case is as follows:
Let $\omega\in\Omega^k(U)$ with $\omega=\sum_{I\in\mathcal{I}^d_k}\omega_Idx^I$ be a smooth k-form with $U\subset\mathbb{R}^d$ bounded and star-shaped (wlog with respect to $0$). $\mathcal{I}^d_k$ denotes the corresponding index set and $\omega_I$ the $I$-th coefficient of $\omega$. Then we define its potential:
\begin{equation} P^{k-1}(\omega)(x):=\sum_{I\in\mathcal{I}^d_k}\sum_{a=1}^k(-1)^a\int_0^1t^{k-1}\omega_I(tx)x^{i_a}dt dx^{\hat{I}_a}, \end{equation}
where $\hat{I}_a$ means we omit the $a$-th part of the multi-index $I$. If $d\omega=0$ i.e. if it's closed, then $dP^{k-1}(\omega)=\omega$, that is $\omega$ is exact.
It has been shown, that exactness and closed-ness are equivalent for weakly closed forms as well, that is functions $u\in W^{1,p}\Omega^k(U)$, such that $du=0$ in the sense of distributions (cf. arXiv:1301.4978). I am only interested in the case $U\subset \mathbb{R}^d$ and $p=1$.
Q1: Is it natural to expect the same potential formula in the weak context, that is for $P^{k-1}(u)$?
My thoughts: One could probably use a density argument here. It is true, that $\Omega^k$ is dense in $W^{1,p}\Omega^k$, at least for $p>1$, so one should also expect for closed k-forms to be dense in the space of weakly closed k-forms. The case $p=1$ eludes me. A suitable Poincare-type inequality would also be quite helpful here. For example, something like \begin{equation} \|u-dP(u)\|\leq C \|du\| \end{equation}
Q2: Let $v$ be such that $dv=u$. Can we show, that if $u\in W^{1,1}$, then $v\in W^{2,1}$? In other words, does the potential of $u$ admit $L^1$-gradient bounds with respect to $u$?
My thoughts: If Q1 can be answered in the affirmative, then this would be rather easy. But one could also work with a different choice of a potential or prove it straight out of $dv=u$, though I don't quite see how or if one could get $\|Dv\|\leq C\|dv\|$. Does not seem very clear to me.