Subelliptic Spin_C Dirac operators, II Basic Estimates

TL;DR

This paper establishes subelliptic estimates for twisted Spin_C-Dirac operators on manifolds with boundary, using boundary layer methods and the extended Heisenberg calculus, under specific geometric conditions.

Contribution

It introduces boundary conditions based on modifications of the dbar-Neumann condition and applies advanced calculus techniques to prove new subelliptic estimates.

Findings

Proved subelliptic estimates for twisted Spin_C-Dirac operators.

Utilized boundary layer methods and extended Heisenberg calculus.

Handled manifolds with boundary having compatible CR-structures.

Abstract

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.