Compactness of the Complex Green Operator on CR-Manifolds of Hypersurface Type

Details Compactness of the Complex Green Operator on CR-Manifolds of Hypersurface Type Compactness of the Complex Green Operator on CR-Manifolds of Hypersurface Type

2008-10-14

arxiv.org/abs/0810.2553v2

Mathematics

Complex Variables

33 pages, v2. I fixed the statement of the main theorem and improved the exposition of the paper

Scientific paper

The purpose of this article is to study compactness of the complex Green operator on CR manifolds of hypersurface type. We introduce (CR-P_q), a potential theoretic condition on $(0,q)$-forms that generalizes Catlin's property (P_q) to CR manifolds of arbitrary codimension. We prove that if an embedded CR-manifold of hypersurface type satisfies (CR-P_q) and (CR-P_{n-1-q}) and is of real dimension at least five, then the complex Green operator is a compact operator on the Sobolev spaces $H^s_{0,q}(M)$, if $1\leq q \leq n-2$ and $s\geq 0$. We use CR-plurisubharmonic functions to build a microlocal norm that controls the totally real direction of the tangent bundle.

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