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The Dolbeault complex in infinite dimensions II


Author: László Lempert
Journal: J. Amer. Math. Soc. 12 (1999), 775-793
MSC (1991): Primary 32F20, 46G20
DOI: https://doi.org/10.1090/S0894-0347-99-00296-9
Published electronically: April 13, 1999
Part I: J. Amer. Math. Soc. (1998), 485-520
MathSciNet review: 1665984
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Abstract | References | Similar Articles | Additional Information

Abstract: We study the equation $\overline {\partial }u=f$ on a ball $B(R)\subset l^{1}$, and prove that it is solvable if $f$ is a Lipschitz continuous, closed $(0,1)$-form.


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

  • G. Coeuré, Les équations de Cauchy–Riemann sur un espace de Hilbert, manuscript.
  • Robert Deville, Gilles Godefroy, and Václav Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1211634
  • Seán Dineen, Complex analysis in locally convex spaces, North-Holland Mathematics Studies, vol. 57, North-Holland Publishing Co., Amsterdam-New York, 1981. Notas de Matemática [Mathematical Notes], 83. MR 640093
  • Hans Grauert and Ingo Lieb, Das Ramirezsche Integral und die Lösung der Gleichung $\bar \partial f=\alpha $ im Bereich der beschränkten Formen, Rice Univ. Stud. 56 (1970), no. 2, 29–50 (1971) (German). MR 273057
  • G. M. Henkin, Integral representation of functions in strongly pseudoconvex regions, and applications to the $\overline \partial $-problem, Mat. Sb. (N.S.) 82 (124) (1970), 300–308 (Russian). MR 0265625
  • 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
  • J. Kurzweil, On approximations in real Banach spaces, Studia Math. 14 (1954), 214–231.
  • L. Lempert, The Dolbeault complex in infinite dimensions I, J. Amer. Math. Soc. 11 (1998), 485–520.
  • Pierre Mazet, Analytic sets in locally convex spaces, North-Holland Mathematics Studies, vol. 89, North-Holland Publishing Co., Amsterdam, 1984. Notas de Matemática [Mathematical Notes], 93. MR 756238
  • Pierre Raboin, Le problème du $\bar \partial $ sur un espace de Hilbert, Bull. Soc. Math. France 107 (1979), no. 3, 225–240 (French, with English summary). MR 544520
  • Raymond A. Ryan, Holomorphic mappings on $l_1$, Trans. Amer. Math. Soc. 302 (1987), no. 2, 797–811. MR 891648, DOI https://doi.org/10.1090/S0002-9947-1987-0891648-7

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Additional Information

László Lempert
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907–1395
MR Author ID: 112435
Email: lempert@math.purdue.edu

Keywords: $\overline {\partial }$ equation, Banach spaces
Received by editor(s): September 22, 1998
Published electronically: April 13, 1999
Additional Notes: This research was partially supported by an NSF grant.
Article copyright: © Copyright 1999 American Mathematical Society