Regularity equivalence of the Szegö projection and the complex Green operator
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- by Phillip S. Harrington, Marco M. Peloso and Andrew S. Raich PDF
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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
- MR Author ID: 799501
- ORCID: 0000-0002-1398-0162
- 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
- MR Author ID: 634382
- ORCID: 0000-0002-3331-9697
- Email: araich@uark.edu
- 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
- © Copyright 2014 American Mathematical Society
- 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
- MathSciNet review: 3272760