Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The Bergman projection on Hartogs domains in $ {\bf C}\sp 2$


Authors: Harold P. Boas and Emil J. Straube
Journal: Trans. Amer. Math. Soc. 331 (1992), 529-540
MSC: Primary 32H10; Secondary 32F15
DOI: https://doi.org/10.1090/S0002-9947-1992-1062188-1
MathSciNet review: 1062188
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Estimates in $ {L^2}$ Sobolev norms are proved for the Bergman projection in certain smooth bounded Hartogs domains in $ {{\mathbf{C}}^2}$. In particular, (1) if the domain is pseudoconvex and "nonwormlike" (the normal vector does not wind on a critical set in the boundary), then the Bergman projection is regular; and (2) Barrett's counterexample domains with irregular Bergman projection nevertheless admit a priori estimates.


References [Enhancements On Off] (What's this?)

  • [1] David E. Barrett, Irregularity of the Bergman projection on a smooth bounded domain in $ {{\mathbf{C}}^2}$, Ann. of Math. (2) 119 (1984), 431-436. MR 740899 (85e:32030)
  • [2] -, Regularity of the Bergman projection and local geometry of domains, Duke Math. J. 53 (1986), 333-343. MR 850539 (87j:32073)
  • [3] -, Behavior of the Bergman projection on the Diederich-Fornass worm, Acta Math, (to appear). MR 1149863 (93c:32033)
  • [4] David E. Barrett and John Erik Fornaess, Uniform approximation of holomorphic functions on bounded Hartogs domains in $ {{\mathbf{C}}^2}$, Math. Z. 191 (1986), 61-72. MR 812603 (87e:32022)
  • [5] Eric Bedford and John Erik Fornaess, Domains with pseudoconvex neighborhood systems, Invent. Math. 47 (1978), 1-27. MR 0499316 (58:17215)
  • [6] Mechthild Behrens, Plurisubharmonische definierende Funktionen pseudokonvexer Gebiete, Schriftenreihe des Math. Inst. Univ. Münster, Ser. 2, Heft 31, Univ. Münster, Münster, 1984. MR 741466 (85d:32036)
  • [7] Steven R. Bell, Biholomorphic mappings and the $ \bar \partial $-problem, Ann. of Math. (2) 114 (1981), 103-113. MR 625347 (82j:32039)
  • [8] Harold P. Boas, Small sets of infinite type are benign for the $ \bar \partial $-Neumann problem, Proc. Amer. Math. Soc. 103 (1988), 569-578. MR 943086 (89g:32026)
  • [9] Harold P. Boas, So-Chin Chen, and Emil J. Straube, Exact regularity of the Bergman and Szegö projections on domains with partially transverse symmetries, Manuscripta Math. 62 (1988), 467-475. MR 971689 (90a:32032)
  • [10] Harold P. Boas and Emil J. Straube, Complete Hartogs domains in $ {{\mathbf{C}}^2}$ have regular Bergman and Szegö projections, Math. Z. 201 (1989), 441-454. MR 999739 (90h:32052)
  • [11] -, Equivalence of regularity for the Bergman projection and the $ \bar \partial $-Neumann operator, Manuscripta Math. 67 (1990), 25-33. MR 1037994 (90k:32057)
  • [12] -, Sobolev estimates for the $ \bar \partial $-Neumann operator on domains in $ {{\mathbf{C}}^n}$ admitting a defining function that is plurisubharmonic on the boundary, Math. Z. 206 (1991), 81-88. MR 1086815 (92b:32027)
  • [13] Aline Bonami and Philippe Charpentier, Une estimation Sobolev $ 1/2$ pour le projecteur de Bergman, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), 173-176. MR 955546 (90b:32048)
  • [14] -, Boundary values for the canonical solution to $ \bar \partial $-equation and $ {W^{1/2}}$ estimates, preprint.
  • [15] Klas Diederich and John Erik Fornaess, A strange bounded smooth domain of holomorphy, Bull. Amer. Math. Soc. 82 (1976), 74-76. MR 0397019 (53:879)
  • [16] -, Pseudoconvex domains: an example with nontrivial Nebenhülle, Math. Ann. 225 (1977), 275-292. MR 0430315 (55:3320)
  • [17] G. B. Folland and J. J. Kohn, The Neumann problem for the Cauchy-Riemann complex, Ann. of Math. Stud., No. 75, Princeton Univ. Press, Princeton, N.J., 1972. MR 0461588 (57:1573)
  • [18] John Erik Fornaess, Plurisubharmonic defining functions, Pacific J. Math. 80 (1979), 381-388. MR 539424 (80h:32035)
  • [19] John Erik Fornaess and Berit Stensønes, Lectures on counterexamples in several complex variables, Princeton Univ. Press, Princeton, N. J., 1987. MR 895821 (88f:32001)
  • [20] Christer Kiselman, A study of the Bergman projection in certain Hartogs domains, paper presented at the 1989 AMS Summer Research Institute on Several Complex Variables, Santa Cruz.
  • [21] I. Ramadanov, Sur une propriété de la fonction de Bergman, C. R. Acad. Bulgare Sci. 20 (1967), no. 20, 759-762. MR 0226042 (37:1632)
  • [22] Stephen Scheinberg, Uniform approximation by functions analytic on a Riemann surface, Ann. of Math. (2) 108 (1978), 257-298. MR 0499183 (58:17111)
  • [23] Nessim Sibony, Un exemple de domaine pseudoconvexe régulier où l'équation $ \bar \partial u= f$ n'admet pas de solution bornée pour $ f$ bornée, Invent. Math. 62 (1980), 235-242. MR 595587 (82c:32020)
  • [24] Emil J. Straube, Exact regularity of Bergman, Szegö, and Sobolev space projections in non pseudoconvex domains, Math. Z. 192 (1986), 117-128. MR 835396 (87k:32045)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32H10, 32F15

Retrieve articles in all journals with MSC: 32H10, 32F15


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1062188-1
Keywords: Bergman projection, $ \overline \partial $-Neumann operator, Hartogs domain, worm domain, a priori estimate
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society