Abstract
A sufficient condition for the infinite dimensionality of the Bergman space of a pseudoconvex domain is given. This condition holds on any pseudoconvex domain that has at least one smooth boundary point of finite type in the sense of D’Angelo.
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Adachi, M., Brinkschulte, J.: A global estimate for the Diederich-Fornæss index of weakly pseudoconvex domains. Nagoya Math. J. 220, 67–80 (2015)
Ahn, T., Gaussier, H., Kim, K.-T.: Positivity and Completeness of Invariant Metrics. J. Geom. Anal. 26(2), 1173–1185 (2016)
Carleson, L.: Selected Problems on Exceptional Sets. Van Nostrand Mathematical Studies, No. 13. D. Van Nostrand Co., Inc., Princeton (1967)
Chen, B.-Y.: An essay on Bergman completeness. Ark. Mat. 51(2), 269–291 (2013)
Chen, B.-Y., Zhang, J.-H.: The Bergman metric on a Stein manifold with a bounded plurisubharmonic function. Trans. Am. Math. Soc. 354(8), 2997–3009 (electronic) (2002)
Cho, S.: A lower bound on the Kobayashi metric near a point of finite type in \({ C}^n\). J. Geom. Anal. 2(4), 317–325 (1992)
D’Angelo, J.P.: Real hypersurfaces, orders of contact, and applications. Ann. Math. (2) 115(3), 615–637 (1982)
Demailly, J.-P.: Analytic methods in algebraic geometry. Surveys of Modern Mathematics, 1. International Press/Higher Education Press, Somerville, MA/Beijing (2012)
Diederich, K., Fornæss, J.E.: Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions. Invent. Math. 39(2), 129–141 (1977)
Dinew, Ż.: The Ohsawa-Takegoshi extension theorem on some unbounded sets. Nagoya Math. J. 188, 19–30 (2007)
Globevnik, J.: On Fatou-Bieberbach domains. Math. Z. 229(1), 91–106 (1998)
Harvey, F.R., Lawson Jr., H.B.: Geometric plurisubharmonicity and convexity: an introduction. Adv. Math. 230(4–6), 2428–2456 (2012)
Harvey, F.R., Lawson Jr., H.B.: \(p\)-convexity, \(p\)-plurisubharmonicity and the Levi problem. Indiana Univ. Math. J. 62(1), 149–169 (2013)
Harz, T., Shcherbina, N., Tomassini, G.: On defining functions for unbounded pseudoconvex domains. arXiv:1405.2250, ver. 3 (2014)
Herbort, G.: Über das Randverhalten der Bergmanschen Kernfunktion und Metrik für eine spezielle Klasse schwach pseudokonvexer Gebiete des \({ C}^{n}\). Math. Z. 184(2), 193–202 (1983)
Hörmander, L.: \(L^{2}\) estimates and existence theorems for the \(\bar{\partial }\) operator. Acta Math. 113, 89–152 (1965)
Jucha, P.: A remark on the dimension of the Bergman space of some Hartogs domains. J. Geom. Anal. 22(1), 23–37 (2012)
Kobayashi, S.: Geometry of bounded domains. Trans. Am. Math. Soc. 92, 267–290 (1959)
Krantz, S.G., Peloso, M.M., Stoppato, C.: Bergman kernel and projection on the unbounded Diederich–Fornæss worm domain. Annali della Scuola Normale Superiore. doi:10.2422/2036-2145.201503_012 (2014)
Ligocka, E.: On the Forelli-Rudin construction and weighted Bergman projections. Studia Math. 94(3), 257–272 (1989)
Pflug, P., Zwonek, W.: Bergman completeness of unbounded Hartogs domains. Nagoya Math. J. 180, 121–133 (2005)
Riemenschneider, O.: Über den Flächeninhalt analytischer Mengen und die Erzeugung \(k\)-pseudokonvexer Gebiete. Invent. Math. 2, 307–331 (1967)
Słodkowski, Z., Tomassini, G.: Minimal kernels of weakly complete spaces. J. Funct. Anal. 210(1), 125–147 (2004)
Wiegerinck, J.J.O.O.: Domains with finite-dimensional Bergman space. Math. Z. 187(4), 559–562 (1984)
Acknowledgments
Special thanks to the Institute of Mathematics, particularly Professor Ines Kath, at the University of Greifswald for their hospitality and support of the first author during the academic year 2015–2016. The second author was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean government (MSIP) (No. 2011-0030044).
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Gallagher, AK., Harz, T. & Herbort, G. On the Dimension of the Bergman Space for Some Unbounded Domains. J Geom Anal 27, 1435–1444 (2017). https://doi.org/10.1007/s12220-016-9725-8
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DOI: https://doi.org/10.1007/s12220-016-9725-8