The Bergman kernel on forms: General theory
HTML articles powered by AMS MathViewer
- by Andrew Raich PDF
- Proc. Amer. Math. Soc. 146 (2018), 4683-4692 Request permission
Abstract:
The goal of this paper is to explore the Bergman projection on forms. In particular, we show that some of most basic facts used to construct the Bergman kernel on functions, such as pointwise evaluation in $L^2_{0,q}(\Omega )\cap \ker \bar \partial _q$, fail for $(p,q)$-forms, $q \geq 1$, $p\geq 0$. We do, however, provide a careful construction of the Bergman kernel and explicitly compute the Bergman kernel on $(0,n-1)$-forms. For the ball in $\mathbb {C}^2$, we also show that the size of the Bergman kernel on $(0,1)$-forms is not governed by the control metric, in stark contrast to the Bergman kernel on functions.References
- Steven R. Bell, The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1228442
- 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
- Alain Berlinet and Christine Thomas-Agnan, Reproducing kernel Hilbert spaces in probability and statistics, Kluwer Academic Publishers, Boston, MA, 2004. With a preface by Persi Diaconis. MR 2239907, DOI 10.1007/978-1-4419-9096-9
- David Catlin, Necessary conditions for subellipticity of the $\bar \partial$-Neumann problem, Ann. of Math. (2) 117 (1983), no. 1, 147–171. MR 683805, DOI 10.2307/2006974
- David Catlin, Subelliptic estimates for the $\overline \partial$-Neumann problem on pseudoconvex domains, Ann. of Math. (2) 126 (1987), no. 1, 131–191. MR 898054, DOI 10.2307/1971347
- Philippe Charpentier and Yves Dupain, Estimates for the Bergman and Szegö projections for pseudoconvex domains of finite type with locally diagonalizable Levi form, Publ. Mat. 50 (2006), no. 2, 413–446. MR 2273668, DOI 10.5565/PUBLMAT_{5}0206_{0}8
- 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
- John P. D’Angelo, A note on the Bergman kernel, Duke Math. J. 45 (1978), no. 2, 259–265. MR 473231
- John P. D’Angelo, An explicit computation of the Bergman kernel function, J. Geom. Anal. 4 (1994), no. 1, 23–34. MR 1274136, DOI 10.1007/BF02921591
- Charles Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1–65. MR 350069, DOI 10.1007/BF01406845
- G. B. Folland and J. J. Kohn. The Neumann problem for the Cauchy-Riemann Complex, volume 75 of Ann. of Math. Stud. Princeton University Press, Princeton, New Jersey, 1972.
- A.-K. Herbig and J. D. McNeal, Regularity of the Bergman projection on forms and plurisubharmonicity conditions, Math. Ann. 336 (2006), no. 2, 335–359. MR 2244376, DOI 10.1007/s00208-006-0005-y
- Lars Hörmander, The analysis of linear partial differential operators. I, 2nd ed., Springer Study Edition, Springer-Verlag, Berlin, 1990. Distribution theory and Fourier analysis. MR 1065136, DOI 10.1007/978-3-642-61497-2
- Phillip S. Harrington and Andrew S. Raich, Closed range for $\overline \partial$ and $\overline \partial _b$ on bounded hypersurfaces in Stein manifolds, Ann. Inst. Fourier (Grenoble) 65 (2015), no. 4, 1711–1754 (English, with English and French summaries). MR 3449195
- J. J. Kohn and L. Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math. 18 (1965), 443–492. MR 181815, DOI 10.1002/cpa.3160180305
- T.V. Khanh and A. Raich, Local regularity of the Bergman projection on a class of pseudoconvex domains of finite type, submitted. arXiv:1406.6532.
- Steven G. Krantz, Function theory of several complex variables, AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1992 edition. MR 1846625, DOI 10.1090/chel/340
- Jeffery D. McNeal, Boundary behavior of the Bergman kernel function in $\textbf {C}^2$, Duke Math. J. 58 (1989), no. 2, 499–512. MR 1016431, DOI 10.1215/S0012-7094-89-05822-5
- Jeffery D. McNeal, Estimates on the Bergman kernels of convex domains, Adv. Math. 109 (1994), no. 1, 108–139. MR 1302759, DOI 10.1006/aima.1994.1082
- J. D. McNeal and E. M. Stein, Mapping properties of the Bergman projection on convex domains of finite type, Duke Math. J. 73 (1994), no. 1, 177–199. MR 1257282, DOI 10.1215/S0012-7094-94-07307-9
- A. Nagel, J.-P. Rosay, E. M. Stein, and S. Wainger, Estimates for the Bergman and Szegő kernels in $\textbf {C}^2$, Ann. of Math. (2) 129 (1989), no. 1, 113–149. MR 979602, DOI 10.2307/1971487
- Alexander Nagel and Elias M. Stein, The $\overline {\partial }_b$-complex on decoupled boundaries in $\Bbb C^n$, Ann. of Math. (2) 164 (2006), no. 2, 649–713. MR 2247970, DOI 10.4007/annals.2006.164.649
- Alexander Nagel, Elias M. Stein, and Stephen Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), no. 1-2, 103–147. MR 793239, DOI 10.1007/BF02392539
- D. H. Phong and E. M. Stein, Estimates for the Bergman and Szegö projections on strongly pseudo-convex domains, Duke Math. J. 44 (1977), no. 3, 695–704. MR 450623
Additional Information
- Andrew Raich
- Affiliation: Department of Mathematical Sciences, SCEN 309, University of Arkansas, Fayetteville, Arkansas 72701
- MR Author ID: 634382
- ORCID: 0000-0002-3331-9697
- Email: araich@uark.edu
- Received by editor(s): June 2, 2017
- Received by editor(s) in revised form: July 20, 2017
- Published electronically: August 14, 2018
- Additional Notes: The author was partially supported by NSF grant DMS-1405100.
- Communicated by: Harold P. Boas
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 4683-4692
- MSC (2010): Primary 32A25, 32A55, 32W05
- DOI: https://doi.org/10.1090/proc/13921
- MathSciNet review: 3856137