TAILIEUCHUNG - Đề tài " Subelliptic Spin C Dirac operators, II Basic estimates "

We assume that the manifold with boundary, X, has a SpinC -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 K¨hler. In this case the SpinC -Dirac operator . agrees with a ¯ ¯ ∂ + ∂ ∗ along the boundary. 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. | Annals of Mathematics Subelliptic Spin C Dirac operators II Basic estimates By Charles L. Epstein Annals of Mathematics 166 2007 723-777 Subelliptic Spin C Dirac operators II Basic estimates By Charles L. Epstein This paper dedicated to Peter D. Lax on the occasion of his Abel Prize. Abstract We assume that the manifold with boundary X has a Spine-structure with spinor bundle . Along the boundary this structure agrees with the structure defined by an infinite order integrable almost complex structure and the metric is Kahler. In this case the Spine-Dirac operator 3 agrees with d d along the boundary. 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 we use boundary layer methods to prove subelliptic estimates for the twisted Spine-Dirac operator acting on sections on E. We use boundary conditions that are modifications of the classical Ỡ-Neumann condition. These results are proved by using the extended Heisenberg calculus. Introduction Let X be an even dimensional manifold with a Spine-structure see 11 . A compatible choice of metric g defines a Spine-Dirac operator 3 which acts on sections of the bundle of complex spinors . This bundle splits as a direct sum e o. The metric on TX induces a metric on the bundle of spinors. We let ơ ơ g denote the pointwise inner product. This in turn defines an inner product on the space of sections of by setting ơ ơ x J ơ ơ g dVg . Research partially supported by NSF grants DMS99-70487 and DMS02-03795 and the Francis J. Carey term chair. 724 CHARLES L. EPSTEIN If X has an almost complex structure then this structure defines a Spine-structure see 4 . If the complex structure is integrable then the bundle of complex spinors is canonically identified with q oA q. We use the notation 1 LJJ Ae A 2q q 0 AO mi. q If the

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