Let $M$ be an $n$-dimensional manifold. Then the space of currents $\mathcal D^k(M)$ of degree $k$ on $M$ is the space of continuous linear functionals on the space of test $n-k$-forms. Two typical examples of currents are $$ \mathcal{F}(\omega) = \int_\Gamma \omega $$ where $\Gamma$ is a $n-k$-chain, and $$ \mathcal{G}(\omega) = \int_M \eta \wedge \omega $$ where $\eta$ is a $k$-form.
One can extend the action of Lie derivative from forms to currenst simply by $$ L_X \mathcal{H}(\omega) = \mathcal{H}(L_X \omega) $$ So the algebra of differential operators on $M$ acts on the space of $k$-currents.
Is it true that the two examples of currents above generate the space of currents as a module over the ring of differential operators, i.e. any current can be obtained from a current like $\mathcal{F}$ or $\mathcal{G}$ by successive applications of Lie derivatives? If no, can the statement be repaired by adding more closure operations/generators? Or is it a completely wrong attitude, and a general current is a significantly more "singular" object?