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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

An approximation theorem for $ \overline \partial $-closed forms of type $ (n,\,n-1)$


Author: Barnet M. Weinstock
Journal: Proc. Amer. Math. Soc. 26 (1970), 625-628
MSC: Primary 32.70
MathSciNet review: 0265638
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1970-0265638-4
PII: S 0002-9939(1970)0265638-4
Keywords: Complex differential form, $ {C^\infty }$ topology, theory of distributions, boundary values of analytic functions, tangential Cauchy-Riemann equations
Article copyright: © Copyright 1970 American Mathematical Society