  Given a finite collection of $C^1$ complex vector fields on a $C^2$ manifold
$M$ such that they and their complex conjugates span the complexified tangent
space at every point, the classical Newlander-Nirenberg theorem gives
conditions on the vector fields so that there is a complex structure on $M$
with respect to which the vector fields are $T^{0,1}$. In this paper, we give
intrinsic, diffeomorphic invariant, necessary and sufficient conditions on the
vector fields so that they have a desired level of regularity with respect to
this complex structure (i.e., smooth, real analytic, or have Zygmund regularity
of some finite order). By addressing this in a quantitative way we obtain a
holomorphic analog of the quantitative theory of sub-Riemannian geometry
initiated by Nagel, Stein, and Wainger. We call this sub-Hermitian geometry.
Moreover, we proceed more generally and obtain similar results for manifolds
which have an associated formally integrable elliptic structure. This allows us
to introduce a setting which generalizes both the real and complex theories.
