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
- Proc. Amer. Math. Soc. 143 (2015), 353-367
- DOI: https://doi.org/10.1090/S0002-9939-2014-12393-8
- Published electronically: September 18, 2014
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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.
References
- Heungju Ahn, Luca Baracco, and Giuseppe Zampieri, Non-subelliptic estimates for the tangential Cauchy-Riemann system, Manuscripta Math. 121 (2006), no. 4, 461–479. MR 2283474, DOI 10.1007/s00229-006-0049-z
- Luca Baracco, Erratum to: The range of the tangential Cauchy-Riemann system to a CR embedded manifold [MR2981820], Invent. Math. 190 (2012), no. 2, 511–512. MR 2981821, DOI 10.1007/s00222-012-0425-0
- Luca Baracco, The range of the tangential Cauchy-Riemann system to a CR embedded manifold, Invent. Math. 190 (2012), no. 2, 505–510. MR 2981820, DOI 10.1007/s00222-012-0387-2
- Albert Boggess, CR manifolds and the tangential Cauchy-Riemann complex, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1991. MR 1211412
- Harold P. Boas and Mei-Chi Shaw, Sobolev estimates for the Lewy operator on weakly pseudoconvex boundaries, Math. Ann. 274 (1986), no. 2, 221–231. MR 838466, DOI 10.1007/BF01457071
- Harold P. Boas and Emil J. Straube, Equivalence of regularity for the Bergman projection and the $\overline \partial$-Neumann operator, Manuscripta Math. 67 (1990), no. 1, 25–33. MR 1037994, DOI 10.1007/BF02568420
- So-Chin Chen and Mei-Chi Shaw, Partial differential equations in several complex variables, AMS/IP Studies in Advanced Mathematics, vol. 19, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2001. MR 1800297, DOI 10.1090/amsip/019
- G. B. Folland and J. J. Kohn, The Neumann problem for the Cauchy-Riemann complex, Annals of Mathematics Studies, No. 75, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. MR 0461588
- Lars Hörmander, $L^{2}$ estimates and existence theorems for the $\bar \partial$ operator, Acta Math. 113 (1965), 89–152. MR 179443, DOI 10.1007/BF02391775
- P. Harrington and A. Raich, Closed range for $\bar \partial$ and $\bar \partial _b$ on bounded hypersurfaces in Stein manifolds, submitted. arXiv:1106.0629.
- Phillip S. Harrington and Andrew Raich, Regularity results for $\overline \partial _b$ on CR-manifolds of hypersurface type, Comm. Partial Differential Equations 36 (2011), no. 1, 134–161. MR 2763350, DOI 10.1080/03605302.2010.498855
- J. J. Kohn, Global regularity for $\bar \partial$ on weakly pseudo-convex manifolds, Trans. Amer. Math. Soc. 181 (1973), 273–292. MR 344703, DOI 10.1090/S0002-9947-1973-0344703-4
- J. J. Kohn, The range of the tangential Cauchy-Riemann operator, Duke Math. J. 53 (1986), no. 2, 525–545. MR 850548, DOI 10.1215/S0012-7094-86-05330-5
- Andreea C. Nicoara, Global regularity for $\overline \partial _b$ on weakly pseudoconvex CR manifolds, Adv. Math. 199 (2006), no. 2, 356–447. MR 2189215, DOI 10.1016/j.aim.2004.12.006
- Andrew Raich, Compactness of the complex Green operator on CR-manifolds of hypersurface type, Math. Ann. 348 (2010), no. 1, 81–117. MR 2657435, DOI 10.1007/s00208-009-0470-1
- Andrew S. Raich and Emil J. Straube, Compactness of the complex Green operator, Math. Res. Lett. 15 (2008), no. 4, 761–778. MR 2424911, DOI 10.4310/MRL.2008.v15.n4.a13
- Mei-Chi Shaw, $L^2$-estimates and existence theorems for the tangential Cauchy-Riemann complex, Invent. Math. 82 (1985), no. 1, 133–150. MR 808113, DOI 10.1007/BF01394783
- Emil J. Straube, Lectures on the $\scr L^2$-Sobolev theory of the $\overline {\partial }$-Neumann problem, ESI Lectures in Mathematics and Physics, European Mathematical Society (EMS), Zürich, 2010. MR 2603659, DOI 10.4171/076
- Giuseppe Zampieri, Complex analysis and CR geometry, University Lecture Series, vol. 43, American Mathematical Society, Providence, RI, 2008. MR 2400390, DOI 10.1090/ulect/043
Bibliographic 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