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Nebenhülle and the Gleason problem


Author: Linus Carlsson
Journal: Proc. Amer. Math. Soc. 138 (2010), 267-273
MSC (2000): Primary 32A65, 32W05, 46J20
DOI: https://doi.org/10.1090/S0002-9939-09-10064-3
Published electronically: August 24, 2009
MathSciNet review: 2550192
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Abstract: This article concerns the Gleason property as a local phenomenon. We prove that there always exists an open set where the domain $ D\Subset \mathbb{C}^{2}$ has the Gleason $ \mathcal{B}$ property whenever the boundary of the Nebenhülle of $ D$ coincides with a $ C^{2}$ smooth part of the boundary $ bD$; here $ \mathcal{B}$ is either one of the Banach algebras, $ H^{\infty}$ or $ A$. As an easy consequence of this, we see that if the extremal boundary points are $ C^{2}$-smooth, then $ D$ has the Gleason $ \mathcal{B}$ property close to those points. Also a $ \overline{\partial} $-problem for locally supported forms is solved.


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

  • [Bea80] Frank Beatrous, Jr.
    Hölder estimates for the $ \bar \partial $ equation with a support condition.
    Pacific J. Math., 90(2):249-257, 1980. MR 600628 (82b:32029)
  • [BF91] U. Backlund and A. Fällström.
    A pseudoconvex domain with nonschlicht $ H\sp \infty$-envelope.
    In Geometrical and algebraical aspects in several complex variables (Cetraro, 1989), volume 8 of Sem. Conf., pages 13-18. EditEl, Rende, 1991. MR 1222202 (94f:32034)
  • [BF95] Ulf Backlund and Anders Fällström.
    The Gleason problem for $ A(\Omega)$.
    New Zealand J. Math., 24(1):17-22, 1995. MR 1348049 (96f:46099)
  • [BR80] F. Beatrous, Jr., and R. Michael Range.
    On holomorphic approximation in weakly pseudoconvex domains.
    Pacific J. Math., 89(2):249-255, 1980. MR 599118 (82h:32022)
  • [Car08a] Linus Carlsson.
    Analytic properties in the spectrum of certain Banach algebras.
    Math. Z., 261(1):189-200, 2009. MR 2452644
  • [Car08b] Linus Carlsson.
    Trivial generators for nontrivial fibres.
    Math. Bohem., 133(2):121-131, 2008. MR 2428308
  • [CCF07] Linus Carlsson, Urban Cegrell, and Anders Fällström.
    Spectrum of certain Banach algebras and $ \overline\partial$-problems.
    Ann. Polon. Math., 90(1):51-58, 2007. MR 2283112 (2008d:32009)
  • [Ceg93] Urban Cegrell.
    On the spectrum of $ A(\Omega)$ and $ H\sp \infty(\Omega)$.
    Ann. Polon. Math., 58(2):193-199, 1993. MR 1239023 (94f:32031)
  • [DF77] Klas Diederich and John Erik Fornaess.
    Pseudoconvex domains: an example with nontrivial Nebenhülle.
    Math. Ann., 225(3):275-292, 1977. MR 0430315 (55:3320)
  • [FØ83] J. E. Fornæss and N. Øvrelid.
    Finitely generated ideals in $ A(\Omega )$.
    Ann. Inst. Fourier (Grenoble), 33(2):v, 77-85, 1983. MR 699487 (84h:32019)
  • [Gle64] Andrew M. Gleason.
    Finitely generated ideals in Banach algebras.
    J. Math. Mech., 13:125-132, 1964. MR 0159241 (28:2458)
  • [Hen71] G. M. Henkin.
    The approximation of functions in pseudo-convex domains and a theorem of Z. L. Leĭbenzon.
    Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 19:37-42, 1971. MR 0287027 (44:4234)
  • [Jak84] Piotr Jakóbczak.
    On Fornæss' imbedding theorem.
    Univ. Iagel. Acta Math., 24:273-294, 1984. MR 815913 (87f:32038)
  • [Koh77] J. J. Kohn.
    Methods of partial differential equations in complex analysis.
    In Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 1, Williams Coll., Williamstown, Mass., 1975), pages 215-237. Amer. Math. Soc., Providence, RI, 1977. MR 0477156 (57:16699)
  • [LW02] O. Lemmers and J. Wiegerinck.
    Solving the Gleason problem on linearly convex domains.
    Math. Z., 240(4):823-834, 2002. MR 1922731 (2003h:32008)
  • [OA87] Joaquın Ma and Ortega Aramburu.
    On Gleason's decomposition for $ A\sp \infty(\overline D)$.
    Math. Z., 194(4):565-571, 1987. MR 881710 (88f:32053)
  • [Øvr71] Nils Øvrelid.
    Integral representation formulas and $ L\sp{p}$-estimates for the $ \bar \partial $-equation.
    Math. Scand., 29:137-160, 1971. MR 0324073 (48:2425)
  • [Ran86] R. Michael Range.
    Holomorphic functions and integral representations in several complex variables, volume 108 of Graduate Texts in Mathematics.
    Springer-Verlag, New York, 1986. MR 847923 (87i:32001)

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Additional Information

Linus Carlsson
Affiliation: Department of Mathematics and Mathematical Statistics, Umeå University, S-901 87 Umeå, Sweden
Email: linus.carlsson@math.umu.se

DOI: https://doi.org/10.1090/S0002-9939-09-10064-3
Keywords: Holomorphic functions, Banach algebras, Nebenh\"{u}lle, $\overline {\partial }$-problems
Received by editor(s): December 9, 2008
Received by editor(s) in revised form: May 26, 2009
Published electronically: August 24, 2009
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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