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Regularity equivalence of the Szegö projection and the complex Green operator


Authors: Phillip S. Harrington, Marco M. Peloso and Andrew S. Raich
Journal: Proc. Amer. Math. Soc. 143 (2015), 353-367
MSC (2010): Primary 32W10, 32W05, 35N15, 32V20, 32Q28
DOI: https://doi.org/10.1090/S0002-9939-2014-12393-8
Published electronically: September 18, 2014
MathSciNet review: 3272760
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Abstract: In this paper we prove that on a CR manifold of hypersurface type that satisfies the weak $ Y(q)$ condition, the complex Green operator $ G_q$ is exactly (globally) regular if and only if the Szegö projections $ S_{q-1}, S_q$ and a third orthogonal projection $ S'_{q+1}$ are exactly (globally) regular. The projection $ S'_{q+1}$ is closely related to the Szegö projection $ S_{q+1}$ and actually coincides with it if the space of harmonic $ (0,q+1)$-forms is trivial.

This result extends the important and by now classical result by H. Boas and E. Straube on the equivalence of the regularity of the $ \bar {\partial }$-Neumann operator and the Bergman projections on a smoothly bounded pseudoconvex domain.

We also prove an extension of this result to the case of bounded smooth domains satisfying the weak $ Z(q)$ condition on a Stein manifold.


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Additional Information

Phillip S. Harrington
Affiliation: Department of Mathematical Sciences, SCEN 301, University of Arkansas, Fayetteville, Arkansas 72701
Email: psharrin@uark.edu

Marco M. Peloso
Affiliation: Departimento di Matematica, Università degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy
Email: marco.peloso@unimi.it

Andrew S. Raich
Affiliation: Department of Mathematical Sciences, SCEN 301, University of Arkansas, Fayetteville, Arkansas 72701
Email: araich@uark.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12393-8
Keywords: $\partial_b$, close range, Kohn's weighted theory, CR manifold, hypersurface type, tangential Cauchy-Riemann operator, $Y(q)$, weak $Y(q)$, $Z(q)$, weak $Z(q)$, complex Green operator, $\partial$-Neumann operator, Stein manifolds
Received by editor(s): May 1, 2013
Published electronically: September 18, 2014
Additional Notes: The first author was partially supported by NSF grant DMS-1002332
This paper was written while the second author was visiting the University of Arkansas. He wishes to thank this institution for its hospitality and for providing a very pleasant working environment.
The third author was partially supported by NSF grant DMS-0855822
Communicated by: Franc Forstneric
Article copyright: © Copyright 2014 American Mathematical Society