  Based on work of R. Lazarsfeld and M. Popa, we use the derivative complex
associated to the bundle of the holomorphic p-forms to provide inequalities for
all the Hodge numbers of a special class of irregular compact Kaehler
manifolds. For 3-folds and 4-folds we give an asymptotic bound for all the
Hodge numbers in terms of the irregularity. As a byproduct, via the BGG
correspondence, we also bound the regularity of the exterior cohomology modules
of bundles of holomorphic p-forms.
