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Subelliptic $$\mathrm{Spin}_{\mathbb C}$$ Dirac operators. II: Basic estimates. (English) Zbl 1154.32017
This is the second paper of a series on modified $$\overline\partial$$-Neumann operators $$\eth$$. It contains the analytical setup and the main subelliptic estimates that are needed for the results in part I [Ann. Math. (2) 166, No. 1, 183–214 (2007; Zbl 1154.32016)].
Let $$X$$ be a compact $$\text{Spin}_{\mathbb C}$$ manifold with a compatible almost complex structure near its boundary $$Y$$ that is integrable to infinite order, and such that $$Y$$ is strictly pseudoconvex or strictly pseudoconcave. Let $$E\to X$$ be a complex vector bundle satisfieing a similar integrability condition near $$Y$$.
The author recalls the extended Heisenberg calculus that he has developed together with Melrose, and describes various Calderon and Szegő projectors in this setup. He then derives subelliptic estimates for the modified $$\overline\partial$$-Neumann operators $$\eth_{{\mathcal R}_\pm}$$ and $$\eth_{\text{Id}-{\mathcal R}_\mp}$$ introduced in part I and shows that they give rise to Fredholm operators. The estimates imply that each of these operators has a complete basis of smooth eigenforms.

##### MSC:
 32W05 $$\overline\partial$$ and $$\overline\partial$$-Neumann operators 32W25 Pseudodifferential operators in several complex variables 58J32 Boundary value problems on manifolds
Zbl 1154.32016
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