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Sobolev estimates for the complex Green operator on CR submanifolds of hypersurface type

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An Erratum to this article was published on 17 December 2014

Abstract

Let \(M\) be a pseudoconvex, oriented, bounded and closed CR submanifold of \(\mathbb {C}^{n}\) of hypersurface type. Our main result says that when a certain \(1\)-form on \(M\) is exact on the null space of the Levi form, then the complex Green operator on \(M\) satisfies Sobolev estimates. This happens in particular when \(M\) admits a set of plurisubharmonic defining functions or when \(M\) is strictly pseudoconvex except for the points on a simply connected complex submanifold.

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Notes

  1. In [7], the authors only needed to assume that the defining function is plurisubharmonic at points of the boundary. Our proof does use plurisubharmonicity in some (arbitrarily small) neighborhood of \(M\). This may be an artifact of the proof. On the other hand, in terms of actually verifying the assumption, not much is lost. The role of \(\alpha _{M}\) is not made explicit in [7].

  2. Often referred to as the Lewy extension theorem, but see [25, 38].

  3. Via the continuity principle (see e.g. [37], Theorem 5.8 in section 5.4). Indeed, if there were a (small) disc with boundary in \(\pi (M)\) and non-empty intersection with the pseudoconcave side of \(\pi (M)\), translating it along the normal to \(\pi (M)\) at \(\pi (\gamma (t_{0}))\) would produce a one parameter family of discs that contradicts the continuity principle on the pseudoconvex side of \(\pi (M)\).

  4. \(\pi (M)\) may be Levi flat near \(\pi (\gamma (t))\), so that both sides are pseudoconvex. However, the pseudoconvex side of \(M\) is defined globally (it is given by \(iJT\)). The local projections \(\pi \) near a point in \(M\) then transfer this direction/side “downstairs”.

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Acknowledgments

The authors are grateful for very useful correspondence from Joseph Kohn and Andreea Nicoara concerning estimates for \(\overline{\partial }_{M}\) and \(\overline{\partial }_{M}^{*}\) in [29, 32], and from Alex Tumanov concerning the construction of the ‘strip’ manifold \(\widehat{M}\) via analytic discs. They also thank C. Denson Hill for a discussion on finite dimensionality vs. triviality of cohomology groups that led them to reference [11]. Finally, they thank the referee for very helpful comments on the exposition.

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA

    Emil J. Straube

  2. Department of Mathematics and Statistics, University of Michigan-Dearborn, Dearborn, MI, 48128, USA

    Yunus E. Zeytuncu

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  1. Emil J. Straube
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Correspondence to Yunus E. Zeytuncu.

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Research supported in part by NSF grant DMS 0758534.

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Straube, E.J., Zeytuncu, Y.E. Sobolev estimates for the complex Green operator on CR submanifolds of hypersurface type. Invent. math. 201, 1073–1095 (2015). https://doi.org/10.1007/s00222-014-0564-6

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