Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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An approximation theorem for $\overline \partial$-closed forms of type $(n, n-1)$
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by Barnet M. Weinstock
Proc. Amer. Math. Soc. 26 (1970), 625-628
DOI: https://doi.org/10.1090/S0002-9939-1970-0265638-4

Abstract:

Let $D$ be a bounded open set in ${C^n}$ with smooth boundary. Then every closed form of type $(n,n - 1)$ which is ${C^\infty }$ on $\bar D$ can be approximated uniformly on $\bar D$ by $(n,n - 1)$ forms which are closed in a neighborhood of $\bar D$. If ${C^n} - D$ is connected these forms can be chosen to be closed in ${C^n}$. This is applied to prove that a continuous function on the connected boundary of a bounded domain in ${C^n}$ admits a holomorphic extension to the interior if and only if it is a weak solution of the tangential Cauchy-Riemann equations.
References
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Bibliographic Information
  • © Copyright 1970 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 26 (1970), 625-628
  • MSC: Primary 32.70
  • DOI: https://doi.org/10.1090/S0002-9939-1970-0265638-4
  • MathSciNet review: 0265638