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Transactions of the American Mathematical Society

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Estimates for the complex Green operator: Symmetry, percolation, and interpolation

Authors: Séverine Biard and Emil J. Straube
Journal: Trans. Amer. Math. Soc. 371 (2019), 2003-2020
MSC (2010): Primary 32W10, 32V20
Published electronically: October 17, 2018
MathSciNet review: 3894042
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Abstract: Let $ M$ be a pseudoconvex, oriented, bounded, and closed CR-submanifold of $ \mathbb{C}^{n}$ of hypersurface type. We show that Sobolev estimates for the complex Green operator hold simultaneously for forms of symmetric bidegrees; that is, they hold for $ (p,q)$-forms if and only if they hold for $ (m-p,m-1-q)$-forms. Here $ m$ equals the CR-dimension of $ M$ plus one. Symmetries of this type are known to hold for compactness estimates. We further show that with the usual microlocalization, compactness estimates for the positive part percolate up the complex; i.e., if they hold for $ (p,q)$-forms, they also hold for $ (p,q+1)$-forms. Similarly, compactness estimates for the negative part percolate down the complex. As a result, if the complex Green operator is compact on $ (p,q_{1})$-forms and on $ (p,q_{2})$-forms ( $ q_{1}\leq q_{2}$), then it is compact on $ (p,q)$-forms for $ q_{1}\leq q\leq q_{2}$. It is interesting to contrast this behavior of the complex Green operator with that of the $ \overline {\partial }$-Neumann operator on a pseudoconvex domain.

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Additional Information

Séverine Biard
Affiliation: Department of Mathematics, School of Engineering and Natural Sciences, University of Iceland, IS-107 Reykjavík, Iceland

Emil J. Straube
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843

Keywords: Complex Green operator, $\overline{\partial}_{M}$, CR-submanifolds of hypersurface type, compactness estimates, Sobolev estimates, form level symmetry and percolation
Received by editor(s): April 14, 2017
Received by editor(s) in revised form: June 12, 2017, and August 14, 2017
Published electronically: October 17, 2018
Additional Notes: This research was supported in part by Qatar National Research Fund Grant NPRP 7-511-1-98
Article copyright: © Copyright 2018 American Mathematical Society