Context: The Poincaré-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 $u\in W^{1,p}\Omega^k(U)$ with $p>1$, such that $du=0$ in the sense of distributions (cf. arXiv:1301.4978).
Now, I am only interested in the case $U\subset \mathbb{R}^d$, star-shaped and $p=1$. I'm an analysis student, so I'm quite lost.
Q1: Is there a Poincare-lemma in the case $p=1$? I.e. Does a potential even exist?
My thoughts: It is generally very difficult to find information on differential forms from an analytic point of view. I've looked into many papers, but I haven't come across a single mention of the case $p=1$. It may be obvious for experts in this field, but I'm at a loss. My understanding is, that the techniques employed to show the Poincare-lemma in the weak setting require a weak Hodge decomposition, which I have encountered in the reference "Scott, C.: L p theory of differential forms on manifolds", but only for $p>1$.
Q2: Is it natural to expect a similar 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} for $u\in\Omega^k(\mathbb{R}^d)$ would be possible to generalise via density.
Q3: 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 Q2 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 $\|Dv\|\leq C\|dv\|$ here (probably not true in general). Also, from a PDE point of view, it does not seem natural to expect any $L^1$ bounds. One can show $L^p$ estimates for $p>1$ by constructing a different potential, which is CZ-singular. But then, singular integral theory suggests, that one should require $L\log L$ bounds on $u$, so as to get $L^1$ bounds on the gradient of $v$. Still, since the de-Rham setting is quite special, one could possibly obtain stronger estimates.
