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
DOI:
https://doi.org/10.1090/S0002-9939-1970-0265638-4
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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Additional Information
Keywords:
Complex differential form,
<!– MATH ${C^\infty }$ –> <IMG WIDTH="38" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${C^\infty }$"> topology,
theory of distributions,
boundary values of analytic functions,
tangential Cauchy-Riemann equations
Article copyright:
© Copyright 1970
American Mathematical Society