Let $X$ be a compact complex manifold with (regular) holomorphic foliation given by a holomorphic subbundle $\mathcal F$ of the tangent bundle. The foliation induces a filtration on differential forms. The $\bar\partial$-operator on differential forms respects this filtration and thus the associated elliptic complex is also filtered.

Can one construct a parametrix $P$ for $\bar \partial$ that also respects the filtration? More precisely we want that if $\alpha$ is annullated by $\Lambda^k\mathcal F$ then the same is true for $P\alpha$.

If necessary we can put all sorts of simplyfing assumptions on the foliation, e.g. assume to work with a Lie-foliation.