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Weighted Bergman spaces and the $ \partial-$equation


Author: Bo-Yong Chen
Journal: Trans. Amer. Math. Soc. 366 (2014), 4127-4150
MSC (2010): Primary 32A25, 32A36, 32A40, 32W05
DOI: https://doi.org/10.1090/S0002-9947-2014-06113-8
Published electronically: March 26, 2014
MathSciNet review: 3206454
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Abstract: We give a Hörmander type $ L^2-$estimate for the $ \bar {\partial }-$equation with respect to the measure $ \delta _\Omega ^{-\alpha }dV$, $ \alpha <1$, on any bounded pseudoconvex domain with $ C^2-$boundary. Several applications to the function theory of weighted Bergman spaces $ A^2_\alpha (\Omega )$ are given, including a corona type theorem, a Gleason type theorem, together with a density theorem. We investigate in particular the boundary behavior of functions in $ A^2_\alpha (\Omega )$ by proving an analogue of the Levi problem for $ A^2_\alpha (\Omega )$ and giving an optimal Gehring type estimate for functions in $ A^2_\alpha (\Omega )$. A vanishing theorem for $ A^2_1(\Omega )$ is established for arbitrary bounded domains. Relations between the weighted Bergman kernel and the Szegő kernel are also discussed.


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  • [1] Aldo Andreotti and Edoardo Vesentini, Carleman estimates for the Laplace-Beltrami equation on complex manifolds, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 81-130. MR 0175148 (30 #5333)
  • [2] Frank Beatrous Jr., $ L^p$-estimates for extensions of holomorphic functions, Michigan Math. J. 32 (1985), no. 3, 361-380. MR 803838 (87b:32023), https://doi.org/10.1307/mmj/1029003244
  • [3] Steven R. Bell and Harold P. Boas, Regularity of the Bergman projection and duality of holomorphic function spaces, Math. Ann. 267 (1984), no. 4, 473-478. MR 742893 (86a:32047), https://doi.org/10.1007/BF01455965
  • [4] Bo Berndtsson, The extension theorem of Ohsawa-Takegoshi and the theorem of Donnelly-Fefferman, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 4, 1083-1094 (English, with English and French summaries). MR 1415958 (97k:32019)
  • [5] Bo Berndtsson, Weighted estimates for the $ \overline \partial $-equation, Complex analysis and geometry (Columbus, OH, 1999) Ohio State Univ. Math. Res. Inst. Publ., vol. 9, de Gruyter, Berlin, 2001, pp. 43-57. MR 1912730 (2003f:32049)
  • [6] Bo Berndtsson and Philippe Charpentier, A Sobolev mapping property of the Bergman kernel, Math. Z. 235 (2000), no. 1, 1-10. MR 1785069 (2002a:32039), https://doi.org/10.1007/s002090000099
  • [7] Zbigniew Błocki, The Bergman metric and the pluricomplex Green function, Trans. Amer. Math. Soc. 357 (2005), no. 7, 2613-2625 (electronic). MR 2139520 (2006d:32013), https://doi.org/10.1090/S0002-9947-05-03738-4
  • [8] Harold P. Boas, The Szegő projection: Sobolev estimates in regular domains, Trans. Amer. Math. Soc. 300 (1987), no. 1, 109-132. MR 871667 (88d:32030), https://doi.org/10.2307/2000590
  • [9] Harold P. Boas and Emil J. Straube, Global regularity of the $ \overline \partial $-Neumann problem: a survey of the $ L^2$-Sobolev theory, Several complex variables (Berkeley, CA, 1995-1996) Math. Sci. Res. Inst. Publ., vol. 37, Cambridge Univ. Press, Cambridge, 1999, pp. 79-111. MR 1748601 (2002m:32056)
  • [10] David Catlin, Boundary behavior of holomorphic functions on pseudoconvex domains, J. Differential Geom. 15 (1980), no. 4, 605-625 (1981). MR 628348 (83b:32013)
  • [11] B.-Y. Chen, A simple proof of the Ohsawa-Takegoshi extension theorem, arXiv:1105.2430v1.
  • [12] Bo-Yong Chen and Siqi Fu, Comparison of the Bergman and Szegő kernels, Adv. Math. 228 (2011), no. 4, 2366-2384. MR 2836124 (2012h:32004), https://doi.org/10.1016/j.aim.2011.07.013
  • [13] Jean-Pierre Demailly, Estimations $ L^{2}$ pour l'opérateur $ \bar \partial $ d'un fibré vectoriel holomorphe semi-positif au-dessus d'une variété kählérienne complète, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 3, 457-511 (French). MR 690650 (85d:32057)
  • [14] Harold Donnelly and Charles Fefferman, $ L^{2}$-cohomology and index theorem for the Bergman metric, Ann. of Math. (2) 118 (1983), no. 3, 593-618. MR 727705 (85f:32029), https://doi.org/10.2307/2006983
  • [15] Klas Diederich and John Erik Fornaess, Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions, Invent. Math. 39 (1977), no. 2, 129-141. MR 0437806 (55 #10728)
  • [16] Klas Diederich and John Erik Fornaess, Pseudoconvex domains: an example with nontrivial Nebenhülle, Math. Ann. 225 (1977), no. 3, 275-292. MR 0430315 (55 #3320)
  • [17] Klas Diederich and Takeo Ohsawa, An estimate for the Bergman distance on pseudoconvex domains, Ann. of Math. (2) 141 (1995), no. 1, 181-190. MR 1314035 (95j:32039), https://doi.org/10.2307/2118631
  • [18] Miroslav Engliš, Toeplitz operators and weighted Bergman kernels, J. Funct. Anal. 255 (2008), no. 6, 1419-1457. MR 2565714 (2010k:47061), https://doi.org/10.1016/j.jfa.2008.06.026
  • [19] John B. Garnett, Bounded analytic functions, 1st ed., Graduate Texts in Mathematics, vol. 236, Springer, New York, 2007. MR 2261424 (2007e:30049)
  • [20] F. W. Gehring, On the radial order of subharmonic functions, J. Math. Soc. Japan 9 (1957), 77-79. MR 0086139 (19,131e)
  • [21] R. E. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Mathematics, vol. 699, Springer, Berlin, 1979. MR 521983 (81a:53002)
  • [22] Monique Hakim and Nessim Sibony, Spectre de $ A(\bar \Omega )$ pour des domaines bornés faiblement pseudoconvexes réguliers, J. Funct. Anal. 37 (1980), no. 2, 127-135 (French, with English summary). MR 578928 (81g:46072), https://doi.org/10.1016/0022-1236(80)90037-3
  • [23] Lars Hörmander, $ L^{2}$ estimates and existence theorems for the $ \bar \partial $ operator, Acta Math. 113 (1965), 89-152. MR 0179443 (31 #3691)
  • [24] Lars Hörmander, Generators for some rings of analytic functions, Bull. Amer. Math. Soc. 73 (1967), 943-949. MR 0226387 (37 #1977)
  • [25] Lars Hörmander, $ L^{p}$ estimates for (pluri-) subharmonic functions, Math. Scand. 20 (1967), 65-78. MR 0234002 (38 #2323)
  • [26] Lars Hörmander, An introduction to complex analysis in several variables, 3rd ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990. MR 1045639 (91a:32001)
  • [27] Marek Jarnicki and Peter Pflug, Extension of holomorphic functions, de Gruyter Expositions in Mathematics, vol. 34, Walter de Gruyter & Co., Berlin, 2000. MR 1797263 (2001k:32017)
  • [28] Masahiko Kanai, Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds, J. Math. Soc. Japan 37 (1985), no. 3, 391-413. MR 792983 (87d:53082), https://doi.org/10.2969/jmsj/03730391
  • [29] Maciej Klimek, Pluripotential theory, London Mathematical Society Monographs. New Series, vol. 6, The Clarendon Press Oxford University Press, New York, 1991. Oxford Science Publications. MR 1150978 (93h:32021)
  • [30] J. J. Kohn, Global regularity for $ \bar \partial $ on weakly pseudo-convex manifolds, Trans. Amer. Math. Soc. 181 (1973), 273-292. MR 0344703 (49 #9442)
  • [31] Steven G. Krantz, Function theory of several complex variables, AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1992 edition. MR 1846625 (2002e:32001)
  • [32] Ewa Ligocka, The Sobolev spaces of harmonic functions, Studia Math. 84 (1986), no. 1, 79-87. MR 871847 (88b:46057)
  • [33] Jeffery D. McNeal, On large values of $ L^2$ holomorphic functions, Math. Res. Lett. 3 (1996), no. 2, 247-259. MR 1386844 (97e:32004)
  • [34] Joachim Michel and Mei-Chi Shaw, The $ \overline \partial $-Neumann operator on Lipschitz pseudoconvex domains with plurisubharmonic defining functions, Duke Math. J. 108 (2001), no. 3, 421-447. MR 1838658 (2002f:32067), https://doi.org/10.1215/S0012-7094-01-10832-6
  • [35] Alexander Nagel, Elias M. Stein, and Stephen Wainger, Boundary behavior of functions holomorphic in domains of finite type, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), no. 11, 6596-6599. MR 634936 (82k:32027)
  • [36] Takeo Ohsawa and Kenshō Takegoshi, On the extension of $ L^2$ holomorphic functions, Math. Z. 195 (1987), no. 2, 197-204. MR 892051 (88g:32029), https://doi.org/10.1007/BF01166457
  • [37] Takeo Ohsawa, On the extension of $ L^2$ holomorphic functions. V. Effects of generalization, Nagoya Math. J. 161 (2001), 1-21. MR 1820210 (2001m:32011)
  • [38] Peter Pflug, Quadratintegrable holomorphe Funktionen und die Serre-Vermutung, Math. Ann. 216 (1975), no. 3, 285-288 (German). MR 0382717 (52 #3599)
  • [39] Rolf Richberg, Stetige streng pseudokonvexe Funktionen, Math. Ann. 175 (1968), 257-286 (German). MR 0222334 (36 #5386)
  • [40] Yum-Tong Siu, The Fujita conjecture and the extension theorem of Ohsawa-Takegoshi, Geometric complex analysis (Hayama, 1995) World Sci. Publ., River Edge, NJ, 1996, pp. 577-592. MR 1453639 (98f:32033)
  • [41] E. M. Stein, Boundary behavior of holomorphic functions of several complex variables, Princeton University Press, Princeton, N.J., 1972. Mathematical Notes, No. 11. MR 0473215 (57 #12890)
  • [42] M. Tsuji, Potential theory in modern function theory, Maruzen Co. Ltd., Tokyo, 1959. MR 0114894 (22 #5712)
  • [43] Kehe Zhu, Spaces of holomorphic functions in the unit ball, Graduate Texts in Mathematics, vol. 226, Springer-Verlag, New York, 2005. MR 2115155 (2006d:46035)

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Additional Information

Bo-Yong Chen
Affiliation: Department of Mathematics, Tongji University, Shanghai 200092, People’s Republic of China
Email: boychen@tongji.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-2014-06113-8
Keywords: $\partial-$equation, $L^2-$estimate, Bergman space, weighted Bergman kernel, Szeg\H{o} kernel
Received by editor(s): July 29, 2012
Published electronically: March 26, 2014
Additional Notes: This work was supported by the Key Program of NSFC No. 11031008
Dedicated: Dedicated to Professor Jinhao Zhang on the occasion of his seventieth birthday
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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