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

Subelliptic Spin_C Dirac operators, II Basic Estimates