Levi foliations in pseudoconvex boundaries and vector fields that commute approximately with partial derivative

Boas and Straube proved a general sufficient condition for global regularity of the (&PARTIAL;) over bar -Neumann problem in terms of families of vector fields that commute approximately with (&PARTIAL;) over bar. In this paper, we study the existence of these vector fields on a compact subset of the boundary whose interior is foliated by complex manifolds. This question turns out to be closely related to properties of interest from the point of view of foliation theory.