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Compactness of the complex Green operator on CR-manifolds of hypersurface type

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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 of real dimension at least five satisfies (CR-P q ) and (CR-P n-1-q), then the complex Green operator is a compact operator on the Sobolev spaces \({H^s_{0,q}(M)}\) and \({H^s_{0,n-1-q}(M)}\) , if 1 ≤  q ≤  n−2 and s ≥  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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Raich, A. Compactness of the complex Green operator on CR-manifolds of hypersurface type. Math. Ann. 348, 81–117 (2010). https://doi.org/10.1007/s00208-009-0470-1

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