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ISSN 1088-6850(e) ISSN 0002-9947(p)

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

Author(s): Harold P. Boas; Emil J. Straube
Journal: Trans. Amer. Math. Soc. 331 (1992), 529-540.
MSC: Primary 32H10; Secondary 32F15
MathSciNet review: 1062188
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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:

[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 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 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)

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