The complex Green operator on CR-submanifolds of $\mathbb{C}^{n}$ of hypersurface type: Compactness

Abstract

Abstract. We establish compactness estimates for ∂ b on a compact pseudoconvex CR-submanifold of C n of hypersurface type that satisfies property(P). When the submanifold is orientable, these estimates were proved by A. Raich in [32] using microlocal methods. Our proof deduces the estimates from (a slight extension, when q > 1, of) those known on hypersurfaces via the fact that locally, CR-submanifolds of hypersurface type are CR-equivalent to a hypersurface. The relationship between two potential theoretic conditions is also clarified.

Closed Range for $\bar{\partial }$ and $\bar{\partial }_b$ on Bounded Hypersurfaces in Stein Manifolds

“…Unfortunately, while the results of Theorem 1.1 do not depend on the metric, our condition appears to depend on the metric (see Proposition 6.3). See [Str11] for discussion of analogous difficulties surrounding the apparent metric dependence of Property (P q ). This is another benefit of working in generic Stein manifolds, since our example in Section 6 requires a non-Euclidean metric.…”

Global Regularity for the∂-b-Equation onCRManifolds of Arbitrary Codimension

“…where ( , ) = ( , ) + ( * , * ). For this estimate and further results on the compactness of the complex Green operator see, e.g., [16][17][18][19]. Applying (32) for Λ 1 , we obtain…”

Compactness of the complex Green operator on non-pseudoconvex CR manifolds

“…In the case that M is a smooth, embedded, orientable CR manifold of hypersurface type, Raich showed that G q is compact if M satisfies (CR-P q ) and (CR-P n−1−q ) where (CR-P q ) is a CR version of P q . Straube later relaxed the orientability condition [21] and also showed the equivalence of P q and (CR-P q ) if M is orientable and pseudoconvex. Khanh, Pinton, and Zampieri later relaxed the embeddedness requirement [15].…”