# Class of flat currents stable under $\overline{\partial}$ operator

Given $U\subset\mathbb{C}^n$, open domain, a locally flat current on $U$ is a $k-$current $T$ such that for every $f\in\mathcal{D}(U)$ (smooth functions with compact support in $U$) there exist a compactly supported $k-$vectorfield $\xi$ of class $L^1(U, dV)$ and a compactly supported $(k+1)-$vectorfield $\eta$ also of class $L^1(U, dV)$ such that
$$T(f\omega)=\int_{U}\langle \xi,f\omega\rangle dV+(-1)^{k+1}\int_{U}\langle\eta,d(f\omega)\rangle dV$$
for every $\omega\in\mathcal{D}^k(U)$. In other words, $T=\xi\llcorner dV + b(\eta\llcorner dV)$, where $b$ is the boundary operator between currents.

The flat currents are obviously stable under $b$: if $T$ is locally flat, then $bT$ is too.

I would like to know if there is a subclass of flat currents which is stable under the $\overline{\partial}$ operator or, which is the same, stable under $b$ and under taking $(p,q)-$components.

The class of currents associated with $\mathcal{C}^\infty$ vectorfields obviously works, but I would be interested in larger classes.

The "obvious" idea $\xi+\overline{\partial}\eta$ does not work as, for example, $$\overline{\partial}\left(z^{-1}\frac{\partial}{\partial z}\wedge\frac{\partial}{\partial \bar{z}}\right)=\delta_0\frac{\partial}{\partial z}$$
and this isn't flat.

Remark An idea only a little less obvious works, but it is still not satisfactory for me:
denote by $W_{p,q}(U)$ the set of $(p,q)-$vectorfields $\xi$ which are in $L^1(U, dV)$ and such that $\overline{\partial}\mathrm{iv}\xi$ belongs to $L^1(U, dV)$ as well (here $\overline{\partial}\mathrm{iv}$ is the $(p,q-1)-$ component of the divergence);
given $\xi\in W_{p,q}(U)$, the current
$T_\xi(\omega)=\int_{U}\langle\xi,\omega\rangle dV$
is such that $\overline{\partial} T$ is associated to $\overline{\partial}\mathrm{iv}\xi$, which is obviously in $L^1(U, dV)$, but then for every current we know that $\overline{\partial}\overline{\partial}T=0$;
this means that the current $S=\overline{\partial}T$ is in $L^1(U, dV)$ and also $\overline{\partial}S$ is (because the latter is $0$).

Why is this not satisfactory? Well, this works for the $\overline{\partial}$, but not for a decomposition in $(p,q)-$components (if we define $W_k(U)$ in the same way, it does not hold that $W_k=\bigoplus W_{p,q}$); moreover, I would like to see if there exists a larger class, inside flat currents, stable under $\overline{\partial}$.

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