  We assume that the manifold with boundary, X, has a Spin_C-structure with
spinor bundle S. Along the boundary, this structure agrees with the structure
defined by an infinite order integrable almost complex structure and the metric
is Kahler. The induced CR-structure on bX is integrable and either strictly
pseudoconvex or strictly pseudoconcave. We assume that E->X is a complex vector
bundle, which has an infinite order integrable complex structure along bX,
compatible with that defined along bX. In this paper use boundary layer methods
to prove subelliptic estimates for the twisted Spin_C- Dirac operator acting on
sections on S\otimes E. We use boundary conditions that are modifications of
the classical dbar-Neumann condition. These results are proved by using the
extended Heisenberg calculus.
